H
ILLIN I
S
UNIVERSITY OF ILLINOIS AT URBANACHAMPAIGN
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University of Illinois at
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UNIVERSITY OF" ILLINOIS tBUtL.LLETIN
Vol. XXI March 17, 1924 No. sP
[Entered as secondclasa matter De e 11, 1912 at the pot oce at Urbana, Illnois, under
the Act of Augst 24, 1912. Aceptance for m1 at thpe l rate of otae provided
Sfor in section 1108. Act of October 8, 1917, an July 31, 1918.
A>N INVESTIGATION
OF THE MAXIMUM TEMPERATURES
AND PRESSURES ATTAINABLE
IN THE COMBUSTION OF GASEOUS
AND LIQUID FUELS
BY
G. A. GOODENOUGH
AND
G. T. FELBECK
f.jLASw
BULLETIN No. 139
ENGINEERING EXPERIMENT STATI
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T IE Engineering Experiment Station was established by act of
the Board of Trustees of the University of Illinois on Deeeim
ber 8, 1903. It is the purpose of the Station to conduct inves
tigations and make studies of importance to the engineering,
manufacturing, railway, mining, and other industrial interests of the
State.
The management of the Engineering Experiment Station is vested
in an Executive Staff composed of the Director and his Assistant, the
Heads of the several Departments in the College of Engineering, and
the Professor of Industrial Chemistry. This Staff is responsible for
the establishment of general policies governing the work of the Station,
including the approval of material for publication. All members of
the teaching staff of the College are encouraged to engage in scientific
research, either directly or in cooperation with the Research Corps
composed of fulltime research assistants, research graduate assistants,
and special investigators.
To render the results of its scientific investigations available to
the public, the Engineering Experiment Station publishes and distrib
utes a series of bulletins. Occasionally it publishes circulars of timely
interest, presenting information of importance, compiled from various
sources which may not readily be accessible to the clientele of the
Station.
The volume and number at the top of the front cover page are
merely arbitrary numbers and refer to the general publications of the
University. Either above the title or below the seal is given the
number of the Engineering Experiment Station bulletin or circular
which should be used in referring to these publications.
For copies of bulletins or circulars or for other information
address
TH ENGINBEEING EXPERIMNT STATION,
UNIvFrSTY OP ILJUNOIS,
URBAgA, ILuoiR.
UNIVERSITY OF ILLINOIS
ENGINEERING EXPERIMENT STATION
BULLETIN No. 139
MARCH, 1924
AN INVESTIGATION OF THE
MAXIMUM TEMPERATURES AND PRESSURES
ATTAINABLE IN THE
COMBUSTION OF GASEOUS AND LIQUID FUELS
BY
G. A. GOODENOUGH
PROFESSOR OF THERMODYNAMICS, UNIVERSITY OF ILLINOIS
AND
GEORGE T. FELBECK
RESEARCH ASSISTANT, ENGINEERING EXPERIMENT STATION
ENGINEERING EXPERIMENT STATION
PUBLISHED BT THE UNIVERSITY OF ILLINOIS, URBANA
CONTENTS
PAGE
1. OBJECT AND SCOPE OF THE INVESTIGATION . . . . . 11
1. Statement of the Problem . . . . . . . 11
2. Data Required . . . . . . . . . . 13
3. Plan and Scope of the Discussion . . . . . 13
[I. THE LAWS OF GAS MIXTURES . . . . . . . .. 13
4. Universal Gas Constant . . . . . . .. 13
5. Gas Mixtures; Partial Pressures . . . . 14
6. Specific Heat of Gases . . . . . . .. 15
7. Energy of a Gas Mixture . . . . . .. 16
8. Thermal Potential of a Gas Mixture . . . . 16
9. Heat of Combustion . . . . . . . . . 17
10. Higher and Lower Heat of Combustion . . . 19
11. Entropy of a Gas Mixture . . . . . . . 19
12. Thermodynamic Potentials . . . . . . . 20
III. THERMODYNAMICS OF GAS REACTIONS . . . . . . . 20
13. Chemical Equilibrium . . . . . . . . 20
14. Energy Changes in Chemical Reactions . . . 21
15. Introduction of Thermodynamic Potentials . . 22
16. Conditions of Equilibrium . .. . ... . 24
17. Reaction CO + 02 = CO2 . . . . . . . 25
18. Mixture of CO, H2 and Air . . . . .. . 32
19. Mixture with a Hydrocarbon Constituent . . . 35
20. The General Case . . . . . . . . .. 38
21. Reactions with Increasing Temperature . . . 40
22. The WaterGas Equilibrium . . . . .. 41
23. Constants in the Equilibrium Equation . . . 42
IV. CALCULATION OF MAXIMUM TEMPERATURES . . . . . 44
24. The Energy Equation . . . . . . .. 44
25. Other Expressions for the Energy Equation . .47
26. System of Two Equations; Combustion of CO .50
3
CONTENTS (CONTINUED)
PAGE
27. System of Three Equations; General Case . 54
28. Equilibrium and Maximum Temperature . . 56
V. EXPLOSION EXPERIMENTS OF DAVID AND OF BONE AND
HAWARD . .. . . . . . . . . 59
29. Experimental Data Available . . . . . . 59
30. Experimental Methods . . . . . . .. 60
31. Experimental Results . . . . . . . . 60
32. Chemical Analysis of Cambridge Coal Gas 60
33. Calculation of Temperatures . . . . .. 61
34. Comparison of Experimental and Calculated Tem
peratures . . . . . . . . . . . 62
35. Experiments of Bone and Haward . . . .. 64
VI. EFFECT OF VARYING CONDITIONS IN THE COMBUSTION OF
GASEOUS FUELS . . . . .. . . .68
36. Problems Investigated . . . . . . . . 68
37. Effect of Initial Pressure . . . . . . . 68
38. Effect of Initial Temperature . . . . . . 70
39. Effect of Excess Air and Heat Loss . . . . 70
40. Comparison of Various GasAir Mixtures . . . 73
VII. LIQUID FUELS . . . . . . . . . . . . . 76
41. Method of Tizard and Pye . . . . . . . 76
42. Recalculation of Tizard and Pye's Values . . . 77
43. Mixtures of Gasoline and Kerosene Vapors and Air 81
44. Water Injection. . . . . . . . . . 82
VIII. WELDING FLAMES . .. . . . . . . . . . .85
45. Dissociation of Hydrogen into Atoms . . . . 85
46. Possibilities of Error in Flame Temperature Cal
culations . . . . . . . . . . . 85
47. Oxyhydrogen Flame Temperatures . . . 86
48. Chemical Reactions Occurring in Oxyacetylene
Flame . . . . . . . . . . . . 88
49. Oxyacetylene Flame Temperatures . . . 90
CONTENTS (CONTINUED)
PAGE
APPENDIX I. SPECIFIC HEATS OF GASES . . . . . . . .93
1. Methods of Measurement . . . . . . . . . . 93
2. Results of Experiments . . . .. . . . . . 93
3. Diatomic Gases . . . . . .. . . . . . 95
4. Carbon Dioxide . . . . . .. . . . . .97
5. Water Vapor . ........ ....99
6. Comparison with Pye's Values of Specific Heat .. .100
7. Specific Heats of Various Hydrocarbons . . .. . 101
Methane (CH) .......... .. 101
Ethylene (C2H,) . . . . . .. . . . . 102
Ethane (CH6) ....... ... ..102
Acetylene (C2H2) . . . . .. . . . . . 102
Benzene Vapor (CH6) . . . .. . . . . 103
Gasoline and Kerosene Vapor . . . . . . 103
8. Specific Heat of Amorphous Carbon . . . . . 103
APPENDIX II. HEATS OF COMBUSTION . . . . . . . 107
1. Hydrogen (H,) . . . . . . . . . . . . 107
2. Carbon Monoxide (CO) . . . . . ... . . 112
3. Methane (CH,) . . . . . . . . .. . . 114
4. Acetylene (CH2) . . . . . . . .. . . 114
5. Ethylene (CH,) . . . . . . . .. . . . 115
6. Ethane (C2H,) ......... . ..115
7. Benzene Vapor (CH,) . . . . .. . . . . 116
8. Gasoline . . . . . . . . . . . . . . 116
9. Kerosene . . . . . . . . . . . . . . 117
10. Amorphous Carbon . . . . .... . . . . 117
APPENDIX III. CHEMICAL EQUILIBRIUM . . . . . .. . 120
1. Statement of the Problem . . . . . . . . . . 120
2. Experimental Methods . . . . . .. . . . 120
Streaming Method . . . . .. . . . . 120
SemiPermeable Membrane Method . .. . . 121
Maximum Explosion Pressure Method . . . . 121
Method of the Heated Catalyst . . . . .. . 121
Iridium Dust Method . . . . . . . . . 121
Measurement of Equilibrium in the Bunsen Flame . 122
6 CONTENTS (CONTINUED)
PAGE
2. Experimental Methods (Continued)
Direct Determination of Equilibrium . . . . . 122
Equilibrium from Density Measurement . . . 122
3. The Reaction CO + 0 = CO, . .. . .. . 123
4. The Reaction H2 02 = H2O . . . . . . . 125
5. The WaterGas Reaction H2 + CO, = 1,20 + CO . .128
6. The Reaction C + 2H2 = CH . . . . . . .. 130
7. The Reaction C CO = 2 CO . . . . . .. . 131
8. The Reaction CH, + 2 O, = CO2 + 2 H2O . . 132
9. The Reaction N2 + 02 = 2 NO . . . . . .. . 135
10. The Reaction H2 = 2 H . . . . . . . . .. 135
APPENDIX IV . . . . . . . . . . . 138
Tables 24 to 36 . . . . . . . . 138153
References . . . . . . . . . . . . 155158
LIST OF FIGURES
NO. PAGE
1. Driving Force and Work of Chemical Reaction . . . . . ... .24
2. Temperature Curve for Chemical Reaction . . .. . . . . . 24
3. Combustion of Carbon Monoxide at Constant Volume . . . . .. .51
4. Combustion of Carbon Monoxide at Constant Pressure . .... .53
5. Graphical Solution; General Case . . . . . . . . . . .. . 56
6. Approach of Actual Combustion to Equilibrium . . ... . . . 57
7. Experimental and Calculated Explosion Temperature Curves for Cam
bridge Coal Gas and Air Mixtures . . . .. . . . . . 62
8. Effect of Initial Pressure . . . . . . . . . . . . .. 69
9. Effect of Initial Temperature .. . . . . . . . . . . . 69
10. Calculated Explosion Curves for Mixtures of Cambridge Coal Gas and Air 71
11. Calculated Explosion Curves for Mixtures of Hydrogen and Air . . 71
12. Calculated Explosion Curves for Mixtures of Carbon Monoxide and Air 75
13. Calculated Explosion Curves for Various GasAir Mixtures . . . . 75
14. Calculated Explosion Temperature Curves for Mixtures of Liquid Fuel
Vapors and Air when Exploded Adiabatically in an Engine having
a Compression Ratio of 5 .. . . . . . . . . . . . 81
15. Effect of Water Injection . . . . . ... . ...... . 84
16. Calculated Maximum Temperature Curves for Oxyhydrogen Flames . 89
17. Calculated Maximum Temperature Curves for Oxyacetylene Flames . 89
18. Specific Heat of Diatomic Gases 02, N2, CO . . . . . . . .. 96
19. Specific Heat of Carbon Dioxide . . . . . . . . . ... .98
20. Specific Heat of Water Vapor . . . . . .. . . . . . . 99
21. Specific Heat of Amorphous Carbon . . . . . . . . . .. 104
22. Agreement between Calculated and Experimental Values for the Equili
brium Constant for the Reaction CO + 02 = CO, . . . . .. .125
23. Agreement between Calculated and Experimental Values for the Equili
brium Constant for the Reaction H. + A 02 = H20 . . . . .. .126
24. Agreement between Calculated and Experimental Values for the Equili
brium Constant for the WaterGas Reaction . ...... . 128
25. Agreement between Calculated and Experimental Values for the Equili
brium Constant for the Reaction C + 211 = CH. .... .. . 132
8 LIST OF FIGURES (CONTINUED)
NO. PAGE
26. Agreement between Calculated and Experimental Values for the Equili
briumConstant for the Reaction C + CO2 = 2CO . . . . . . 134
27. Agreement between Calculated and Experimental Values for the Equili
brium Constant for the Reaction H2 = 2H . . . . . . .. 136
LIST OF TABLES
NO. PAGE
1. Comparison of Products of Combustion Obtained from Cambridge Coal
Gas Analysis with the Experimental Values of David .. . . . 61
2. Experimental and Calculated Results for Explosions of Mixtures of Cam
bridge Coal Gas and Air . . . . . . . . .. . . . 66
3. Experimental and Calculated Results for High Pressure Explosions . . 67
4. Effect of Initial Pressure .. . . . . . . .. . . . 68
5. Effect of Initial Temperature . . . .... . . . . . . . 70
6. Calculated Explosion Data for Three GasAir Mixtures .. . . . 72
7. Gas Analyses . . . . . . . . . . . . . . . .. . . 73
8. Calculated Explosion Data for Various GasAir Mixtures .. . . . 74
9. Calculated Results for BenzeneAir Mixtures compared with the Results
of Tizard and Pye .. . . . . . . . . . . . . 79
10. Calculated Explosion Data for Mixtures of Gasoline Vapor and Air and
Kerosene Vapor and Air . . . . . .. . . . . . . 82
11. Effect of Water Injection on GasolineAir Combustion . . . . . . 83
12. Oxyhydrogen Flame Temperatures . . . .. .... . . . .87
13. Oxyacetylene Flame Temperatures . . . . . . . . . . .92
14. Mean Specific Heat per mol between 100 deg. C. and t deg. C. . . . 100
15. Specific Heat Equations . . . . . . . . . . . . ... .106
16. Heats of Combustion (Low) in Mean B. t. u .. . . . . ... .119
17. Dissociation of CO, at One Atmosphere Pressure . . . . . . . 124
18. Dissociation of H.O at One Atmosphere Pressure . . . . . . .127
19. Experimental Data on the Equilibrium Constant for the WaterGas
Reaction ................. . 129
20. Experimental Data on the Equilibrium Constant for the Reaction
C+ 2H2,=CH. ......... .... . . . . .131
21. Experimental Data on the Equilibrium Constant for the Reaction
C CO, = 2C0 .... ........... .133
22. Experimental Data on the Equilibrium Constant for the Reaction
H,=2H . ................ . 135
23. Equilibrium Equations . . . . . . . . . . .. . . . 137
24. Thermal Energy Equations . . . . . . . . . .. . . . 138
25. Thermal Energy of Carbon Dioxide in B. t. u. per lb. mol . . . . 139
26. Thermal Energy of Water Vapor in B. t. u. per lb. mol . . . . . 140
27. Thermal Energy of the Diatomic Gases CO, O,, and N, in B. t. u. per
lb. mol. .. ................ . . . 141
9
10 LIST OF TP B (CONTINUED)
NO. PAGE
28. Thermal Energy of Hydrogen in B. t. u. per lb. mol . .. ... . 142
29. Thermal Energy of Hydrocarbon Gases in B. t. u. per lb. mol . . 143
30. Values of Log,, Kp for the Reaction CO + I O0 = CO, . .. 144145
31. Values of Logo, Kp for the Reaction H2 + A 02 = HO . . . . 146147
32. Values of Kp for the Reaction H2 + CO, = H0O + CO . . . . .148149
33. Equations for Heat of Combustion . . . . . . . . .. . 150
34. Heat of Combustion of Carbon Monoxide at Constant Volume in B. t. u.
per lb. mol . . . . .. . . . . . . .. . . . 151
35. Lower Heat of Combustion of Hydrogen at Constant Volume in B. t. u.
per lb. mol .. .. ... ...... . ... 152
36. Lower Heats of Combustion of Hydrocarbon Gases at Constant Volume
in B. t.u. per lb. mol ............. . . 153
AN INVESTIGATION OF THE MAXIMUM TEMPERATURES
AND PRESSURES ATTAINABLE IN THE COMBUSTION
OF GASEOUS AND LIQUID FUELS
I. OBJECT AND SCOPE OF THE INVESTIGATION
1. Statement of the Problem.Certain chemical reactions which
are associated with high temperatures possess great technical im
portance. Among these, especially interesting to the engineer, are the
combustion reactions such as occur in boiler practice, in the internal
combustion engine, in the gas producer, and in the oxyhydrogen and
the oxyacetylene flame.
Reactions at high temperatures present certain phenomena which
have not received the attention they deserve from engineers. The
reaction energy, that is, the available work of the reaction, is not
generally the exact equivalent of the heat of reaction; however, this is
not always recognized. Again, at high temperatures a reaction can
not proceed to completion but must halt when the constituents attain
a state of equilibrium. The maximum temperature that can be at
tained is thus definitely limited by the law of chemical equilibrium.
A concrete example will serve to illustrate the effect of equilibrium
conditions on the temperatures attained. Clerk* has discussed the
great discrepancy between the observed and the calculated pressures
in the explosion of coal gas with various proportions of air to gas.
The figures are as follows:
Ratio of air to gas .... 14 13 12 11 9 7 6
Observed pressure ...... 40 51.5 60 61 78 87 90
Calculated pressure .... 89.5 96 103 112 134 168 192
The pressures are in lb. per sq. in. above atmospheric pressure.
The calculated pressures are nearly double the observed pressures,
and this means that calculated temperatures are likewise nearly double
the temperatures actually attained.
The explanation of the large discrepancy lies in the crudeness of
the method used in calculating the temperatures. Three assumptions
were made, not one of which was justified:
(1) It was assumed that the combustion process was adiabatic;
that is, that all the heat of combustion was used in raising the tempera
ture of the products of combustion. The recent experiments 9f David
* D. Clerk, "The Gas, Petrol, and Oil Engine," Vol. 1, p. 136.
11
ILLINOIS ENGINEERING EXPERIMENT STATION
show that during explosion an appreciable amount of heat is dissipated
by conduction and radiation, and is, therefore, not available for raising
the temperature of the mixture.
(2) The specific heat of the products mixture was taken as con
stant. It is now known that the specific heats of all gaseous con
stituents increase with the temperature. The effect of such variation
in the specific heat is a considerable reduction in the calculated
temperature.
(3) Complete combustion at the end of the explosion period was
assumed. This assumption is also untenable. According to the law
of chemical equilibrium the mixture at the point of maximum pressure
must contain a considerable amount of uncombined carbon monoxide
and a smaller amount of uncombined hydrogen. Consequently, at this
point the whole of the heat of combustion has not been developed.
Clerk explained the discrepancy between actual and calculated
temperatures as due largely to dissociation of the combustion prod
ucts.* The following quotation gives a clear statement of the situa
tion:
"It is quite evident, then, that at the highest temperatures produced by
combustion, the product cannot exist in a state of complete combination. It will
be mixed to a certain extent with the free constituents which cannot combine
further until the temperature falls; as the temperature falls, combustion will con
tinue till all the free gases are combined. The subject, from its nature, is a
difficult one in experiment, and accordingly different observers do not agree upon
temperatures and percentages of dissociation, but all are agreed that dissociation
places a rigid barrier in the way of combustion at high temperatures, and prevents
the attainment of temperatures, by combustion, which are otherwise quite
possible."'
At the present time fairly consistent and reliable data on the
dissociation of the main products of combustion are available, and
apparently there is a possibility of taking account of dissociation
phenomena in the calculation of temperatures and pressures. In
fact, Tizard and Pye,t using the figures of Nernst on the dissociation
of CO, and HO2, have succeeded in calculating the maximum tempera
tures attained in the combustion of benzene and similar fuels. Though
the method was laborious, and a number of approximations were used,
the results obtained are surprisingly good.
The equilibrium conditions for a chemical reaction have been
determined from the laws of thermodynamics, and for the principal
dissociation reactions, as CO, = CO + ½O and HO = H2 02,
* D. Clerk, "The Gas, Petrol, and Oil Engine," Vol. 1, p. 123.
t The Automobile Engineer, I, Feb., 1921, p. 55, II, March, 1921, p. 98, III, April,
1921, p. 134.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 13
these conditions are furnished by equilibrium equations which can be
established with a fair degree of accuracy. These equilibrium equa
tions, along with an equation connecting the heat of combustion with
various energy changes, furnish a system from which the tempera
ture attained at equilibrium may be calculated in a straightforward
manner.
The problem proposed for investigation is, therefore, the follow
ing: first, to investigate the conditions of equilibrium and establish
the necessary equilibrium equations; and, second, to incorporate these
equilibrium equations into a formulation by means of which the maxi
mum temperature resulting from the combustion of a fuel under
predetermined conditions may be calculated.
2. Data Required.The constants in the equilibrium equations
involve certain thermal data: namely, the heat of combustion and the
specific heats of the various constituents. The accuracy of the equili
brium equations depends, therefore, in the first instance, on the accur
acy of these thermal data. The available experimental evidence on
the specific heats of various constituents has been examined critically,
and the discussion is given in Appendix I. In the same way the ex
perimental evidence on heats of combustion is reviewed in Appendix
II. One constant in the equilibrium equations is independent of
thermal magnitudes, and its determination requires a chemical analysis
of a gas mixture in the equilibrium state. In Appendix III, the
chemical data for various reactions are assembled, and the unknown
constants are determined for such reactions.
3. Plan and Scope of the Discussion.A brief resume of the
important laws of gas mixtures is desirable and is given in Chapter
II. Then in Chapter III follows the general theory of chemical
equilibrium, and the derivation of the equilibrium equations for
various reactions. In Chapter IV the energy equation is developed,
and the method of solving the system of equations for the maximum
temperature is shown. With this section the theoretical discussion is
completed. The remaining sections are devoted to comparisons and
applications of the theory.
II. THE LAWS OF GAS MIXTURES
4. Universal Gas Constant.Calculations relating to gas mix
tures are simplified by the introduction of a unit of weight called the
mol. Let m denote the molecular weight of a gas; then 1 mol = m
pounds or m grams, according as the pound or the gram is taken as
the unit. Thus, for oxygen, 1 mol = 32 lb., for ammonia, 1 mol = 17
lb., etc. At a temperature of 32 deg. F. and at normal atmospheric
pressure the weight of 1 cubic foot of oxygen is 0.08922 lb.; hence the
ILLINOIS ENGINEERING EXPERIMENT STATION
volume of 1 mol of oxygen is 32  0.08922 = 358.7 cu. ft. From the
relation between the volumes and molecular weights of gases, it is
evident that 358.7 cu. ft. is the volume of 1 mol of any gas at 32 deg.
F. and atmospheric pressure. If the standard temperature is taken
as 62 deg. F. instead of 32 deg. F. the volume of 1 mol is 380.6 cu. ft.
Let both members of the usual gas equation
pv = BT
be multiplied by the molecular weight m. The resulting equation is
p mv = mBT
Taking now the standard condition, T = 32 + 459.6 = 491.6, p 
2116.3 lb. per sq. ft. (atmospheric pressure), the product my is the
volume of 1 mol or 358.7 cu. ft.; hence
2116.3 X 358.7
mB = 1544.
491.6
Denoting the constant 1544 by R, the equation of a gas is
pv= RT... .... ............. (1)
provided v denotes the volume of 1 mol. The constant R is the same
for all gases.
In the following sections the thermal unit (B. t. u.) rather than
the mechanical unit (ft. lb.) will generally be used. In the preceding
equation the product pv is in foot pounds, hence the constant R has
the dimension ft. lb. per degree. To change from ft. lb. to B. t. u.
we divide by the mechanical equivalent J = 777.64. The value of R
thus obtained lies between 1.985 and 1.986. The value 1.985 is usually
taken.
5. Gas Mixtures; Partial Pressures.The composition of a gas
mixture as determined by chemical analysis is expressed in terms of
volumes. That is, the total volume V of the mixture, having a pres
sure p and temperature T, is divided into partial volumes V2, V2, etc.,
that would be occupied by the individual constituents at the same
pressure and temperature. Thus the composition
CO, = 0.30
HO = 0.20
N, = 0.50
1.00
signifies that if the CO2 is separated from the mixture its volume will
be 30 per cent of the volume of the mixture at the same pressure and
temperature.
Another interpretation of a volume composition is useful. Be
cause of the invariable relation between the volume of a gas and its
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 15
weight expressed in mols, the numbers in the volume composition are
proportional to the number of mols in the several constituents. Thus
1 mol of the mixture just given contains 0.3 mol of CO,, 0.2 mol of
HO0, and 0.5 mol of N,.
When the separated constituents having the volumes V1, V2, etc.,
and a common pressure p and temperature T are allowed to mix, the
volume V, of the first constituent increases to the total volume V of
the mixture, and
V V, + V, + V, + ......
The pressure of this constituent must change from the original com
mon pressure p to a smaller value pl, given by the relation
pV, pV
Likewise for the other constituents
pV2 = p2V
pV, = p8V
A combination of these equations gives the important relation
P 2 + P + P + ** ** * * = P
That is, the pressure of the mixture is the sum of the partial pressures
of the constituents (Dalton's law). Also
P : P2: p3 .... V:7: 3 V7: ......
or the partial pressures are proportional to the partial volumes given
by the composition. For example, let the pressure of the mixture
just quoted be 14.7 lb. per sq. in.; then the partial pressure of the CO2
constituent is 14.7 X 0.3 = 4.41 lb., that of the N2 constituent 14.7 X
0.5 = 7.35 lb.
6. Specific Heat of Gases.In the problems that are attacked in
the following sections the mol rather than the pound will be used
as the unit; hence the specific heat per mol is required. Let c denote
the specific heat referred to the pound, y the specific heat referred to
the mol, and m the molecular weight. Then
y = me yp = mcp, y = mcv
The difference c,  c, between the specific heats at constant pressure
and constant volume is the product AB, where A is the reciprocal of
J, the mechanical equivalent of heat; hence taking R in thermal units,
y,  , = R = 1.985 .............. (2)
If for a gas the specific heat at constant pressure is given by an ex
pression such as
yp = a + bT + cT2,
ILLINOIS ENGINEERING EXPERIMENT STATION
the specific heat at constant volume is given by the expression
where y a' + bT + cT2,
a' = a 1.985.
The equations for the specific heats of various gases are examined
in Appendix I, and a collection of the equations finally chosen will
be found on page 106.
7. Energy of a Gas Mixture.The intrinsic energy of one mol
of a single component is given by the expression
u= f ,,dT ............. ...... . . (3)
Thus the energy of one mol of H2 is
u= (4.015+0.6667 X 103 T) dT
=4.015 T+0.3333 X 103T2 +u
in which uo denotes the energy of the gas when T = 0.
The energy of a gas mixture is the sum of the energies of the in
dividual constituents. If nL, n2 n3.... denote the number of mols
of the first, second, third, etc., constituents respectively, the energy
of the mixture is
U=nl fy,dT+ n2 f dT+nl3dT + ............ (4)
8. Thermal Potential of a Gas Mixture.The thermal potential
i of one mol is defined by the equation
i= u + Apv .................. (5)
in which u denotes the energy and v the volume of one mol. For
Apv may be substituted ART, or simply RT if R is taken as 1.985.
Then
i = u + RT
di = du + RdT = yvdT + RdT
= (yv + R) dT = ypdT
Hence for one mol
i= fpdT
and for a mixture
I*=ni 7'dT + n2 7dT + ns ,dT .............. (6)
* The usual convention of small and capital letters is adopted. Thus u denotes the
energy of 1 mol, U the energy of M mols, whence U = Mu; likewise I = Mi.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 17
9. Heat of Combustion.In the experimental determination of
the heat of combustion the temperature of the products is brought
back to the temperature of the original fuel mixture and the heat
generated is absorbed by cold water. The usual energy equation may
be applied to the process; thus
V2
Q2=U2UiA pd ........ .........(7)
JVi
In this equation Q2, denotes heat absorbed; therefore, since heat is re
jected in the process, 1Q2 is intrinsically negative. Hence, denoting by
H the heat of combustion,
H= 1Q2= U1U2AfpdV
V1
Two cases come. under consideration: (a) If the volume is kept con
stant, the heat of combustion He is given by the equation
H,= U,  ................. (8)
(b) If the pressure is kept constant, the heat of combustion is given
by the equation
Hp = U,  U2  (ApV2  ApV1).
Since by definition I = U + ApV, this equation becomes
Hp= I,I2 ................ (9)
For constant volume, the heat of combustion is the difference between
the energy of the fuel mixture and the energy of the products mixture,
both at the same temperature. Similarly, the difference between the
thermal potentials of the two mixtures, at the same temperature, gives
the heat of combustion at constant pressure.
A general expression for the heat of combustion at a given ab.
solute temperature T is useful. This may be derived as follows: As
suming that the products of combustion with oxygen are CO2 and H20,
the reaction equation may be written
1 mol fuel + n, mols 02 = n' mols C2 + n" mols H,O.
For example, in the combustion of H2, n2 = J, n' = 0, n" = 1, in
the combustion of CH,, n2 = 2, n' = 1, n" = 2. Let the expressions
for the four specific heats involved be:
For the fuel y,= + T +S T2
For the oxygen yp a + 2 T 2 T2
For the CO2 yp=a' + /'T+ 'T2
For the H20 y, = a" + P"T + 8"T2
18 ILLINOIS ENGINEERING EXPERIMENT STATION
The thermal potential of the original mixture of fuel and oxygen is
I = f (al + BIT + 8,T2) dT + nf(a2 + 62T + 2T2)dT
Likewise the thermal potential of the products mixture is
I = n' (a' + 'T + 'T2)dT+ ±n"f (a" + "T+ "T2) dT
Performing the integrations and substituting the results in equation
(9), the result is
H,=T[ai +n2a2n'a'n"a" + T21 + nS'Tn'3'n" "
+1T3 iol+n2i02ni2onio (10)
+T3 [,+n2 n'6'n"6"]+ Lio±+fl0 n'2'f i"0 (10)
The first three terms in brackets are denoted by a', a", a", respectively,
and the last term by Ho. Therefore,
H, = Ho + T(a' + ½a"T + aj'"T') ........... (11)
To get an expression for H,, we take the specific heats at constant
volume; namely,
y,= (ol R) + 31T + 61T2, etc.
and by integration obtain expressions for U1 and U2. The second and
third terms in the second member of equation (10) will remain the
same; the first term will be
T[(a R) n2(a2 R) n'(a'R)n"(a"R)
= T[a +, nan'a'n"a"] RT[1 + n n'n"]
and the last term will be
U01 + n2U02 n'uo  n"u"o
Now uo is the energy of a constituent and io is the corresponding
thermal potential of the constituent when T = 0. Since
i = u + Apv = u + RT,
i,= uo when T = 0. Consequently the last terms are identical, and
H,,=Ho + T (7'+ "T T+ or"'T2)RT [ + n'n"] 
The expression 1 + n2  n'  n" gives the decrease in the molecular
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 19
volume incurred in the reaction, and this may be denoted by n. Thus,
in the reaction CO + 02, = CO2, n = 1 + j  1 = J; in the reaction
CH, 20 =CO 2 H,O, n 1+212=0; and in the re
action C2H1O +30 2C + 33HO, n=1+323=1.
The expression for H, is, therefore,
Hv =Ho  T (u' + i a"T + J r"'T2) nRT .........(12)
and the difference between the two heats of combustion is
H,H = nRT ...............(13)
10. Higher and Lower Heat of Combustion.In the experi
mental determination of the heat of combustion, the temperature of
the products is usually reduced below the saturation temperature cor
responding to the partial pressure of the H20 constituent. Part of
the water vapor is condensed and the latent heat released by such con
densation is added to Hp, giving the socalled higher heat of combus
tion. A correction must be made to reduce the experimental higher
value to the true lower value, which is the only one that is useful in the
present discussion. The method used is given in Appendix II. Values
of the heat of combustion for various reactions are given in Table
16. page 119.
11. Entropy of a Gas Mixture.The expression for the entropy
of unit weight (one pound) of a gas at pressure p and temperature T
is deduced from the energy equation
dq = cdT + Apdv
From the characteristic equation pv = BT,
Apdv = ABdT  Avdp
Combining these equations, and taking AB = c,  Cv, the result is
dq = cCdT  Avdp
dq
Then from the defining equation ds = 
T
dT Av
ds p  dp
TT
v B
or, since
T p
dT dp
ds = c   .
P
ILLINOIS ENGINEERING EXPERIMENT STATION
This expression is applicable when the pound is the unit of weight;
if the mol is taken as the unit, the expression becomes
dT dp
ds = ................ (14)
T p
with the understanding that R = 1.985.
The entropy of 1 mol of a gas is therefore given by the expres
sion
s=JpdT Rlogep + so............. . (15)
In a mixture of gases each gas occupies the total volume V, and this
being the case, the entropy of the mixture is the sum of the entropies
of the individual constituents. If nl, n2, n,, etc., denote the number
of mols of the several constituents and p,, P2, Ps, etc., the correspond
ing partial pressures, the total entropy is
f dT dT d
S=lY T l+2 7p + 37,y +....
 R(nl logpl + n2 logep2 + n3 loge p ... )
+ n1o01 + n2802 + n303 + ........... .. (16)
12. Thermodynamic Potentials.Two important functions that
will be required in the following chapter are the thermodynamic po
tentials at constant volume and at constant pressure, respectively.
These are denoted by F, and FP, and they are defined by the follow
ing equations:
F, = U  TS ................... ......(17)
Fp = U  TS + ApV = I  TS ........(18)
By a combination of the preceding equations a general expression for
F, or F, of a mixture of gases may be readily derived; but such a
procedure is unnecessary, as it is more convenient to operate with the
individual terms, as U, I, and S, and ultimately combine the results.
III. THERMODYNAMICS OF GAS REACTIONS.
13. Chemical Equilibrium.In a mixture of gases in which
chemical action is possible, for example, a mixture of CO, 02, H2, N2,
O20, reactions may proceed with an accompanying change of tempera
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 21
ture; but at some definite state of the mixture, a state determined by
the temperature and the partial pressures of the constituents of the
mixture, chemical action ceases, and the mixture is said to be in
equilibrium.
The determination of the conditions under which a gas mixture
attains the equilibrium state is a fundamental problem in the in
vestigation of temperatures and pressures due to combustion. The
equations giving these conditions are deduced by various methods, all
of which, however, are essentially identical with the analysis of Prof.
J. Willard Gibbs, who first developed the general theory of chemical
equilibrium. Haber first makes use of the entropy principle and after
wards deduces the fundamental equation with the aid of Van't Hoff's
"equilibrium box." Nernst also uses the device of Van't Hoff.
Planck makes use of the method of thermodynamic potentials, which
is the original method of Gibbs.
In this section as much of the theory is presented as is necessary
for solution of the problem proposed. The method of thermodynamic
potentials is used because it is peculiarly applicable when several
simultaneous equilibrium equations are to be established.
14. Energy Changes in Chemical Reactions.When a system is
subjected to any change, either physical or chemical, the intrinsic
energy of the system will in general increase or decrease, the system
will absorb heat from, or give heat to, the surroundings, and external
work will be done by, or upon, the system. Let  AU denote the de
crease of energy,  AQ the heat given up to the surroundings, and
AW the work done by the system. Then the energy  AU liberated
is partly expended in doing the work AW and the remainder is the
heat  AQ rejected to external systems. This statement expressed
symbolically gives the energy equation
AU = AWAQ ............. (19)
In a chemical reaction the decrease of energy  AU when 1 mol
of the substance is combined is the heat of reaction Hv at constant
volume. If AQ = 0, the total energy given up is available for the
performance of work; but in general, AQ is not equal to zero and the
work attainable is not equal to the heat of reaction.
The work AW is in every case the product of two factors, one of
which has the quality of a force. Thus the work done by an expand
ing gas is AW = pAV, in which the pressure p is the intensity factor
or force; in a cell a quantity of electricity Ae is generated and the
work is AW = EAe, and E is the electromotive force. Similarly, the
work obtainable in a chemical reaction may be taken as proportional
to the amount Ax of a constituent transformed; and if the expression
for work is written
AW = XAx
ILLINOIS ENGINEERING EXPERIMENT STATION
the factor X may be regarded as a force. It is sometimes called the
driving force of the reaction. If there are several fuel constituents
in the gas mixture the work is given by the expression
AW = X1Axz + j±A Xz+ xAx3
and if there is a change in volume during the combustion the work
ApAV done against the external pressure must be added. Hence the
energy equation takes the form
 AU = XAx,  X2Ax2 +X,3Ax + ...... + ApAV AQ (20)
15. Introduction of Thermodynamic Potentials.The thermo
dynamic potential Fv is defined by the equation
F, = U TS
which upon differentiation gives
dF, = dU  TdS  SdT
The general expression for entropy provided the process is rever
sible, is
dS dQ
or
TdS = dQ
Substituting dQ for TdS in the preceding equation, the result is
dF, = dU dQ SdT
The energy equation (19) in differential form is
 dU = dW  dQ
and a combination of the two equations gives
dF ,=  dW  SdT............... (21)
In the case of an irreversible process TdS>dQ, and equation (21)
must be replaced by the inequality
dF,< dW  SdT ............. (22)
For an isothermal process dT= 0, and equation (21) becomes
dW =  dF,
Hence for a change of the system from state 1 to state 2,
1W 2 =Fv F 2 ................ (23)
For an isothermal change that is irreversible the inequality (22) gives
1W2 < FvlF2 ................. (24)
Hence the maximum work obtainable from an isothermal change of
state is equal to the decrease of the potential function F,.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 23
Since
U = F,7 + TS
U1U2 =F1 F2 + T (S,S2)..........(25)
when T is constant. The total energy released is therefore made up
of two parts: Fv,  Fv2, which is equal to the work obtainable and is
therefore the available part; and T (S,  82), which is the unavailable
part. The first of these is the socalled "free energy," the second the
"bound energy."
In the case of a chemical reaction,
dW = Xxdx, + X2dx2 + Xdx3 +........+ ApdV
If the temperature is kept constant dW = dFv; hence for T = const.,
 dF, = X dx, + X2dx2 + X3,dx +........+ ApdV.. (26)
The coefficients X,, X2, etc., are the partial derivatives of F, with re
spect to the variables xa, x2, etc. Thus
OF, OF, OF,,(
Xx O , X2  , X3= '............. (27)
aOx, ax' I Ox,'
The second potential function F, is defined by the equation
Fp = U  TS +ApV
The result of differentiation is
dFp = dU  TdS  SdT + ApdV + AVdp
= dF, + ApdV + AVdp ................ (28)
Combination of equations (26) and (28) gives the equation
 dFp = Xdx, + Xdx2 + Xdzs  ......+ AVdp. (29)
which is valid for a reversible isothermal process.
The expressions (27) for X,, X2, etc. were obtained by taking the
total volume V constant. If the pressure p is taken as constant, equa
tion (29) gives
aF, OF, Fp,
X,= X2 = X,= ,............. (30)
Ox, Ox2 Ox3
Since F, = I  TS, the decrease of the thermal potential I for a
change of state is
IlI2 =Fp. FP2+T(S1 S2) ..........:... .(31)
an equation analogous to equation (25). In the case of an isothermal
chemical combination U1  U, = H, and I,  I2 = H; hence equa
tions (25) and (31) may be written, respectively,
F,1F 2 =H,T(S,S2) ............... . (32)
F1l F, = H T(S1S2) ................. (33)
ILLINOIS ENGINEERING EXPERIMENT STATION
Equation (32) shows that the work obtainable from a reaction proceed
ing at constant temperature and volume is less than the heat of com
bustion by the amount T (S,  S). For a reaction at constant pres
sure and temperature, equation (33) shows that the work obtainable
is less than the heat of combustion H, by the bound energy
T(S,  S,).
16. Conditions of Equilibrium.It is a wellestablished fact that
in a chemical reaction the driving force changes continuously as the
reaction proceeds, and that at some definite composition of the gas
mixture, depending on the temperature, the driving force vanishes
and the reaction halts. The mixture has then attained a state of
equilibrium.
A
T
7;
' r
7 dT
dx
0 a5 /O x
FIG. 1. DRIVING FORCE AND WORK OF FIG. 2. TEMPERATURE CURVE FOR
CHEMICAL REACTION CHEMICAL REACTION
Consider, for example, the reaction CO + 10, = CO, and let x
denote the fraction of CO transformed to CO,. Values of x are given
by the distances along the line AB, Fig. 1; at A, x = 0, at B, where
the reaction is complete, x = 1. The magnitude of the driving force X
is given by the ordinate of the curve m. The driving force is greatest
when x = 0, and steadily decreases as x increases until for some value
of x, as xo, it becomes zero. For x greater than xo the driving force
changes sign, which means that the reaction tends to proceed in the
opposite direction; that is, for x>xo, CO, dissociates into CO and 02.
Curve n is the integral curve of curve m; that is, the ordinate of
curve n represents the work
W = f Xdx ................(34)
corresponding to the progress of the reaction from the beginning to
the point indicated by the value of x. The maximum ordinate of curve
n is OP corresponding to the value x = xo.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 25
The condition of equilibrium may now be established. The mix
ture of the three gases CO, 02 and CO, is in a state of equilibrium
when the driving force is zero, that is, when x x= o. The condition
is, therefore,
X = 0 ................... (35)
If there are several fuel constituents, the corresponding conditions are
X1= 0, X2 X 0, X,= 0, ..........(36)
If the reaction proceeds at constant temperature and the whole volume
is kept constant, these conditions are equivalent to the conditions
F O, aF, OF ,
 0, =0, 0, .............(37)
19x ' ax, 9xa
or if the pressure instead of the volume is constant, they are equivalent
to the conditions
OF, aF, OF,
= 0, 0, 0,. ..........(38)
In order to get these conditions in a useful form it is necessary
to express the composition of the mixture in terms of the variable x,
or in terms of the variables x., x2, x, .... if there are several com
bustible constituents. An expression for F, or FP is then obtained,
and by differentiation expressions for the driving forces X,, X2, etc.,
are found. Equating these to zero gives the required conditions of
equilibrium. In the following paragraphs the equilibrium conditions
are worked out in detail for several important cases.
17. Case 1. The Reaction CO + 0, = CO2.It is assumed that
CO is burned in air and that an excess of air may be used; thus for 1
mol of CO let e mols of 02 and f mols of N, be supplied. The initial
composition is
CO ==1 mol
O, = e mols
N, = f mols
Total ... 1+ e f = n mols
The composition of the mixture after the fraction x of CO has been
converted to C02, is
CO =1 x mol
CO= x mol
02= e  jx mols
N,== f mols
Total ...1 + e + f  ix = m = m  ix
Let P denote the total pressure of this mixture; then the partial pres
sures of the constituents are, respectively,
26 ILLINOIS ENGINEERING EXPERIMENT STATION
pco= P X ) Pco =
mia r miix ,om
ezx _ 1
Me  iX f (39)
S= P x, P = P f
mI x m1i2x
If the mixture is kept at constant volume during the reaction the total
pressure P changes, but the partial pressures may be expressed in
terms of Pi the initial pressure. For the initial mixture,
PV = mRT
for the final mixture,
PV = (m  x) . RT
Hence with T and V constant,
P P1
 = ..............(40)
m,  I mi
and the expressions for the partial pressures are
P1 P
. .....(41)
P P1
po = (ex), pN = 1
° = mi 2 
A certain function of the partial pressures will appear repeatedly
in the following developments. For this reaction it is
pco
for the reaction CH4 + 202 = C0 + 2 H20 it is
Pco, P 2H .
2* 1,
The function is a fraction in the numerator of which appear the par
tial pressures of the products and in the denominator the partial pres
sures of the factors. Each partial pressure is given an exponent equal
to the coefficient applied to the constituent in the reaction equation.
Let this function be denoted by K(p) ; then for the CO reaction under
discussion
p x
K(p) pco2   x
pco 72 1x ex2
xP eP x/
mlx mP i X
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 27
or Imx
K(p)= x \ x 1 .....m....... (42)
IX elx c 'x **p
If the volume is kept constant as well as the temperature,
K (p)=   .... ...... 43)
In deriving the equation of equilibrium either the pressure or the
volume may be taken as constant; the result is the same. The algebraic
work is perhaps somewhat simpler when constant volume is assumed,
and the potential Fv is used.
Taking the defining equation
F = U TS,
the expression for the driving force X is
OF, OU OS
X . T.ýx ...........(44)
Ox Ox ox
OU
The derivative  is easily obtained. The energy of the final mixture
ax
is
U=(1x))Uco + uco0 + (ex)uo2 +fuN2
whence
OU
Uco + ou Uco
The second member of this equation is the heat of combustion at con
stant volume, hence
OU
H , .....................(45)
OS
To evaluate the remaining term T we must obtain an expression
ox
for the total entropy S of the mixture, differentiate with respect to x,
and multiply the result by T. To get the entropy S the entropy per
mol of each constituent is multiplied by the weight of the constituent
in mols, and the products are added.
The expression for the entropy of one mol of a gas is (Section 11)
s=f y R logp + so
I dT
For convenience, let h = fp  + So s; then
s = h  Rlogp
28 ILLINOIS ENGINEERING EXPERIMENT STATION
The expression for S will have two groups of terms; thosa contributed
by the h functions and those contributed by the terms of the form
Rlogp
The first group gives the sum
(1x)h.o + xho, + (e x)ho + fh,,
Since for any constituent h is a function of the temperature only, the
x derivative of this sum is simply
(hco + ho, hco X
The expression for the specific heat of a gas is
Ap = a + fT + T2
The function h is, therefore,
h =fyT +so=alogT+ T+ T2 + so ..... ..... (46)
Consequently, the expression for A in the present reaction is
X=hco +ho hoo
=logeT(aco + ½ao aco ) + T(,oo + 3, /300)
+ iT2 (8 + 80 2 co2) + (Soco + 2so0 So00)
The following symbols may be used for the terms in the parentheses:
o'= co + ~o aco "=* 03co + o0, I0co
0" =6o00 + ks2  k o+s=soco ,
Then
X=o'logeT+ "T + "'T2+k ...........(47)
The formation of the expressions for A, a',......k for any reac
tion is obvious. Thus for the reaction CH, + 202 = CO, + 2H20,
X=hcH + 2ho hoo 2h.o
0T'= atc + 20ao  aco 22aH 0
k=So0H4 + 2so0 Sc02 2SOH20
The coefficients are those in the reaction equation; terms involving the
factors are given the positive sign, those involving the products, the
negative sign.
The group of terms involving the partial pressures must now
receive attention. Each term has the form m R logep, in which m de
notes the number of mols of the constituent and p the partial pressure.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OP FUELS 29
1x
Thus for the CO constituent, m = 1  x, and p = P , or as the
1x
volume is considered constant, p = P1  The sum of the terms in
ni,
this group is
R [(1 x)log( x) + x logex + (e ex) loge(e  x)
= R (mi½x)loge 1 (1x)loge(1x) +x logex
Pmi lg
+ (e x)loge(e Ix) +f logof
The x derivative of a term such as (e  x) loge (e  x) is
 logc (e  x)  i
hence the x derivative of the preceding sum is
' [ log log,(lx )l+logex+1 i oge (e x)i
m1
Lx im 1
= RRlog 1 ex
x1@ 7n I p
= RRlogeK(p)
since 1 =K(p) (See page 27.)
1nc e x 4p/
Collecting the results, we have the following:
 A + R  R logeK (p)
Ox
49S
T  =  T + IRT  RTlogK (p)
ox
@H,
dU OS
x x ox
X KV + T a H, + RT  T  RTlogeK (p)
30 ILLINOIS ENGINEERING EXPERIMENT STATION
or, since H, + ½RT = H, in the case of the CO reaction,
X=II AT RTlogeK(p) ....... .......(48)
The heat of combustion H, is a function of the temperature (page 17),
that is
Hp = H, + VT + ½"Tj + i"Ta
Also
AT = a'TlogeT + u"T' + lu"'T3 + kT
whence
H,  XT = Ho  dTlogeT  ja"T2  do"T3  (k  ') T
Introducing this expression in equation (48), the result is
X = H,  "TlogeT  ½a"T2  d"T3 
(k a') T  RTlogeK(p)
The condition of equilibrium is X= 0; hence denoting by Kp the
value of the pressure function K(p) when equilibrium exists, the
equilibrium equation is
BTlogeKp = Ho  u'TlogeT  ao"T2  b.,d"T  (k  a') T,
or dividing both members by T,
H,
RlogeKp =  'log,T  Jo"T  a"T  (k a') (49)
Denoting by primes the partial pressures in the equilibrium state,
and by Xo, the value of x at equilibrium, we have
P'co, XO_ x [miýo 1
K P'co l= eX0 ............ (50)
Kp is called the equilibrium constant. For any temperature T, the
value of K, may be calculated from equation (49) ; then from equation
(50) the value of xo is found. The elimination of K, between equa
tions (49) and (50) gives the relation between T and x, that must
exist when the mixture is in the equilibrium state.
It is evident that K, is a function of the temperature T. The rate
of change of Kp with the temperature is obtained by differentiating
equation (49) ; thus
SalogK, Ho a  ,'
OT TP T
=  (Ho  a'T + ja"T + a"'T3) = (51)
12" 12
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 31
This result is entirely general and is important, though it has no
bearing on the present investigation.
A convenient expression for the driving force X is obtained by
combining equations (48) and (49), namely,
X=RT logeK,logeK(p)
or
X=RT loge, P,  loge P .............. (52)
p co p' o PcoP, "2
While the equilibrium equation (49) was developed for a par
ticular reaction, CO +  02 CO2, it is perfectly general. The same
method applied to any other reaction will give an equation of the same
form. The only question that demands attention is the expression of
the equilibrium constant K, (1) in terms of the partial pressures, (2)
in terms of Xo.
For the reaction H2 + 0 = HO, all the relations just de
veloped for the CO reaction are immediately applicable merely by sub
stituting H, for CO and HO for COz. Thus equation (50) gives the
relation between K, and xo equally for both reactions.
As another example we consider the reaction CH,  202 = CO2
 2HO. The composition of the mixture corresponding to the trans
formation of the fraction x is
CH4... ..1x mol
CO .......... mol
H2O........2x mols
Oz..... e2x mols
N ...........f mols
Total..... 1 + e + f = m
The partial pressures are, respectively,
1 x x 2x e2x f
poll P = ; p°o =P; PHo=P ; po =P P =P 
Smi mi mi
The pressure function is
pNop 2 H20
K (p)= .................... (53)
PCH'P 20
Hence the equilibrium constant is given by the expression
p'co,(pl ýo)2 xo(2xo)2
K(P)2 xo(2x)2 ......(54)
*p= pH,(po,)2 (1xo) (e2xo)2
32 ILLINOIS ENGINEERING EXPERIMENT STATION
It will be observed that the pressure P does not appear in the expres
sion for K, given by (54); hence the same expression applies for con
stant pressure and for constant volume. Such is the case in all reac
tions in which there is no change in molecular volume.
If oxygen alone is supplied without any excess the expressions
for K, are as follows:
In the case of the CO and HE reactions, e == , f = 0, m, ,
whence from equation (50)
K,= 3x ... ...... (55)
P1xo Nxo "P
In the case of CH1, e = 2, f = 0, and from equation (54)
K,=  o ....................(56)
18. Case 2. Mixture of CO, H,, and Air.Consider a mixture
of a mols of CO, b mols of H2, and an amount of air at least sufficient
for combustion. The progress of the CO reaction may be denoted by
x, that of the H2 reaction by y. Then the original mixture and the
mixture after combustion has progressed are given by the following
schedules:
Originl Mi Mixture during Combustion
Original Mixture
CO .......a (1a) mols
CO............... a mols H .......b (1y) mols
H,.......... ..... b mols CO ....... ax mols
O,................ e mols HO ...... by mols
N,.................f mols 02........e a (ax+ by) mols
__ N ........ f mols
Total ....... a + b + e + f = m Total...........m, (ax + by) = m
Taking P as the total pressure of the mixture, the partial pres
sures are, respectively,
P P P F 1b
poo=a(lx), pco = ax , o = e(ax +by)
P P P
p. = b(1y) , PH2o=M by, p =m f
From these partial pressures the two pressure functions are given by
the equations
pco, x m(ax+by) 1
p co m (ax by) .......(57)
poo x e(ax+by) p
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 33
PHO y m(ax+by) 1 ....... (58)
p s, l 1y eI(ax+by) p
A combination of (57) and (58) gives the relation
PH.o _ y(1x)
. .. . . (59)
PH2 pcoo x(1y)
which is the pressure function for the watergas reaction
H, + CO2 H20 + CO
In establishing the equations of equilibrium the method used in
case 1 is followed in all details. The expression for the energy of the
mixture is
U=a(1 x)uo, + b(l1y)u) + axuco + byuHo
+ [e(ax +by)] uo, + fu
The x and y partial derivatives are, respectively,
 a a Uco +Uo Uco =aHoco
 = b U 2 + O HO =bHv2
Vy + U2 20] 2
Similarly, the sum of the h functions furnishes the derivatives
 h= a(hco + ho hoo,)= aXoo
hh= b(h. + ho hHo) = bXHH
Taking the volume and the temperature as constant, the relation
P P
 =  holds, and the sum of the group of terms in the entropy equa
m my
tion that involve the pressures is
Ra(lx) log, E a(1) +b(1y)loge_ b(1y) +axloge, ax
+byloge by + ei(ax +by) logPle(ax+by)] +flog Pl1
m J 2 m1
ILLINOIS ENGINEERING EXPERIMENT STATION
After slight reduction the expression takes the form
R{[mi (ax+by)]loge ! + a(1x)loga(1x)
+ b(1 y)logeb(1 y) + axlogeax + bylogeby
+ [e(ax +by)]loge eI(ax +by)] +flogef
The x derivative of this expression is
aR Y loge~ logea(1 x))1 logeax
+1iloge[ei(ax+by)] }
SX mi 1
= taR aR logee  
=a Rlogxa e (ax + by) "P,
or, making use of equation (57), the expression for the x derivative
becomes
a[iRRlog.K(p)co]
poo
in which K (p) co denotes the pressure function P
Poo'Po,
Similarly, the y derivative is found to be
b [RRlogeK(p),]
Collecting the results, the two driving forces X and Y are given by
the expressions
X=a[HpooXcoTRTlogeK(p)oo]
I.. . (60)
Y=b[H,n2 2TRTlogeK(p)H,
The conditions of equilibrium are X = 0, Y = 0; inserting these
values in equation (60), the resulting equations are precisely those
that are obtained for the equilibrium of the two constituents when
taken separately. It should be noted, however, that the equilibrium
constant K, is a function of both xo and y, when the two constituents
are mixed.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 35
19. Case 3. Mixture Containing a Hydrocarbon Constitutent.
As a concrete example, let c mols of CH, be added to the mixture of
the preceding case, and let z denote the progress of the reaction
CH, + 20, = CO, + 2H20. Then the original mixture and the mix
ture after combustion has started are as follows:
Origirnal Mixture Mix.ure during Combuslion
CO.............. a mols CO.. .a (1x) mols
H ...b (ly) mols
H,..................b mols CH,. .c (1  z) mols
CH ,................ c mols CO,. .ax + c mols
O,.................e mols H1O .by + 2cz mols
. O...e i (ax + by)  2cz mols
N,... ...... ....... .f mols N .. .f mols
Total ....a+ b + c + e f = m Total... m = mi (ax + by)
If now the same process is followed as in case 2, the two equations
(60) will again be obtained; and in addition a third equation
Z=c[HpCH cH TRTTlogeK(p)c,]
Putting Z = 0, for equilibrium, and Kp for K(p),
H,
R log, K, 
T
The equilibrium equations for the CH, reaction are known approxi
mately and K, may be calculated for various assumed temperatures.
The equilibrium equations (see page 137) are:
For T<2900 deg.
348 330
4.571 log,, K, = + 14.9462 log,,T  3.606.103T
+ 0.001  10'T2  47
For T>2900 deg.
353 210
4.571 logo, K, = + 26.5651 logoT  5.346.103T
T
+ 0.101  106T2  84.7
From these equations the following results are calculated:
T = 1000 2000 3000 4000 5000
logoKp = 76.69 37.91 24.68 17.80 13.50
Kp= 576 837 4.824 6.317 3.213
1
That is, Kp is practically infinite, and  =0.
Kp
ILLINOIS ENGINEERING EXPERIMENT STATION
Now, from the composition, the expression for K, is
K co2, p2' o (ax + cz) (by + 2cz)2
KI =
PCH .p2o c(1z) (eax by)2
It follows that z must be practically equal to 1 at all temperatures.
In other words, at equilibrium there is no appreciable amount of CH,
present.
Tizard and Pye* have pointed out the impossibility of the forma
tion of methane at high temperature from the reaction
CO + 3H, = CH, + HO
by using the following approximate equilibrium equation due to
Nernst,
pco  PM 19000
logK,=log P P + 3.5logT+2.2
PCH 'P4 2 T
in which T = deg. C. (abs.). From this formula K, = about 1012 at
3000 deg. C. (abs.) and 108 at 2000 deg. C. (abs.). Methane, therefore,
will not be formed from this reaction at the high explosion tempera
tures. However, at T 1000 deg. C. (abs.) K, = about 50 so that
methane would be formed at low temperatures according to the above
reaction if free CO and H, were present. This fact, according to
Tizard and Pye, accounts for the presence of CH, in engine exhausts.
Tizard and Pye also show that the formation of formaldehyde
according to the reaction CO + H2 = H.CHO can only occur at low
temperatures. The equilibrium equation derived by the use of
Nernst's heat theorem is
pClo  pH 3700
log K = log  + 1.75logT+2.1
PH*OHO T
in which T = deg C. (abs.).
The formation of methane and formaldehyde in engine exhausts
will only be possible in case CO and H2 are present in appreciable
quantities. With sufficient oxygen in the mixture only very slight
traces of CO and H, will exist, these traces being due to nonhomogen
eous combustion. With insufficient oxygen in the original mixture,
appreciable quantities of free CO and H2 are found in the cooled
exhaust gas along with traces of methane and formaldehyde.
From the foregoing discussion it may be concluded that in the
equilibrium state the products of combustion of a mixture of H1, CO,
and CH, with sufficient air will be CO2, CO, H20, H2, 02, and N2 only.
* The Automobile Engineer, February, 1921. p. 59.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 37
(See Section 33, on possibility of formation of NO.) The equilibrium
state for such a mixture is specified by equation (60) alone.
The question naturally arises as to the constitution of the mix
ture of gases in the equilibrium state resulting from the combustion
of hydrocarbons other than methane with or without sufficient oxygen
present for complete combustion. In the case of insufficient oxygen
the excess hydrocarbon may be unaffected by the high temperature,
it may break down into simpler hydrocarbons, it may decompose into
its elements, or any one of a number of phenomena might occur. Ex
perimental evidence, however, points rather directly to one result,
namely, that the total hydrocarbon burns to CO and H, and that these
two products burning in part to CO, and HO consume any remaining
oxygen. This was found to be the case by W. A. Bone* and others in
explosions of CH,, C2H,, C2H4, C2H, with just sufficient oxygen
present to burn the carbon to CO. Only in the case of C2,H was any
carbon at all deposited. Methane and aldehyde vapors were found in
the products of combustion.
Further evidence is obtained from the analyses of exhaust gases
obtained from engines running with insufficient air supplied to the
mixture. The following table of analyses is given by Dr. Watsont
for dry exhaust gases from a petrol engine. Traces of aldehyde were
found in the condensed water obtained from the exhaust.
Ratio of Composition of Exhaust Gases
Air to Petrol
by Weight CO O0 CO H2 CH4 N2
9 6.7 0.0 12.0 4.3 1.4 75.6
10 8.2 0.0 9.6 3.5 1.2 77.5
11 9.6 0.0 7.3 2.6 0.9 79.6
12 11.1 0.0 5.0 1.8 0.6 81.5
13 12.5 0.0 2.6 0.9 0.3 82.7
14 13.5 0.4 0.4 0.1 0.0 85.6
From the evidence given in this table, from the experiments of
Bone, with the satisfactory explanation of the presence of methane
and aldehyde in the cooled products of combustion, and also from the
known general instability of hydrocarbons at high temperature, it
may be safely assumed that the mixture of gases in the equilibrium
state resulting from the combustion of any hydrocarbon will consist
* Jour. Chem. Soc., V. 71, pp. 2641.Phil. Trans. Roy. Soc., Ser. A. V. 215, (1915).
pp. 278318.
t Proc. Inst. of Auto. Engrs., 1909.
Also, A. W. Judge, "Automobile and Aircraft Engines," (1921), p. 23.
ILLINOIS ENGINEERING EXPERIMENT STATION
only of CO2, CO, H20, H2, 02, and N2. The only condition is that
sufficient oxygen be present to burn all the carbon to CO.
20. The General Case.The mixture contains CO, H,, one or
more hydrocarbons, as CH,, C2,H, and possibly CO2 and HO. When
equilibrium is attained the mixture will have the six constituents CO,
CO2, H2, H.,O O0, and N2. The relative proportions of these con
stituents will determine the two variables xo and yo; thus
CO, xz H20 Yo
CO 1xo ' H1 1yo
The first step in the application of the equilibrium equations is the
determination of the composition of the mixture in the equilibrium
state in terms of Xo and y,. For this purpose it is convenient to as
sume (a) that all the reactions are completed, and (b) that products
CO2 and H20 in this mixture are then partially dissociated. The
following example illustrates the procedure.
The 15 per cent mixture of coal gas and air used in David's ex
periments (see page 60) had the following composition:
Carbon monoxide ......................... CO..... 0.060
H ydrogen .............................. . H .....0.480
Methane .................. ............ CH4... .0.335
Ethylene ............................. CH,.... .0.035
Benzene ................................. C H . . ..0.010
Carbon dioxide ............................ CO ....0.020
Water vapor ............................. H,0.. .0.033
Oxygen ................................ 0 .....1.179
Nitrogen ...............................N. N2....4.515
Total ....... .6.667
Complete combustion of the fuel constituents gives the mixture
CO, = 0.545
HIO = 1.283
02 = 0.059
N, = 4.515
Total ..... 6.402
The equilibrium mixture is now obtained from this hypothetical mix
ture by the dissociation of the fraction 1  xz of CO, and the fraction
1  yo of H20. These dissociations produce
.  0.545 (1 x) + i 1.283 (1 yo) mols of 02,;
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 39
hence in the equilibrium mixture the oxygen present is
0.059 + 1(0.545 + 1.283)  0.2725xo  0.6415yo
The desired composition in the equilibrium state is, therefore,
CO ....0.545 (1 zo)
CO2... .0.545 xo
H2.....1.283 (1 yo)
H,O ... 1.283 y,
02..... 0.973  0.2725xo  0.6415yo = eo
N2 .....4.515
Total...... 7.316  0.2725xo  0.6415yo = mo
From this composition the following expressions for K, are obtained:
P
Sp'o 0.545xo 
o P coý I MO __ mp
Keco=pco. p, =( P½
SP0.545(1xo) P (eo
mo mo
X o Imo 1 . ...(61)
lxo e o VP
Kp(H)  Y__o o 1.... ...... ... (62)
1yo eo P/p
The formulas (61) and (62) are perfectly general. Provided
there is sufficient oxygen supplied for complete combustion, the mix
ture of products, assuming complete combustion, will contain n, mols
of CO2, n2 mols of H20, excess oxygen, and nitrogen. Then the com
position of the mixture at equilibrium must be
mols
CO .... n (1 Xo)
CO,.... .nXo
H2 ..... n (1  Yo)
H20 ... n2Yo
02 .....eo
N,.....f
Total....... n, + n, + eo + f mo
From this composition follow the expressions for Kp given by equations
(61) and (62).
ILLINOIS ENGINEERING EXPERIMENT STATION
A possible modification of the equilibrium composition should be
mentioned. If the temperature at equilibrium is sufficiently high the
endothermic reaction
N2 + 02 = 2 NO
may be present, and the mixture may contain NO along with the other
constituents. In this case the equilibrium equation for the reaction
in addition to equations (61) and (62) is required. It will be shown
subsequently that under most conditions the amount of NO formed
is so small as to be negligible.
21. Reactions with Increasing Temperature.In the discussion
of Section 16 the equilibrium conditions expressed by equations (37)
and (38) were based on the assumption of constant temperature; and
this same assumption is involved in the derivation of equations (48),
(49), and (50). In combustion reactions the temperature does not
remain constant but rises rapidly. It might be inferred, therefore,
that the relations that have been deduced are not applicable in such
reactions with varying temperature.
The assumption of constant temperature does not mean that the
temperature is necessarily the same throughout the whole course of
the reaction. Consider, for example, the reaction CO + ½02 = CO,
in a closed space. The increase of temperature T with the weight x
transformed is represented by some such curve as a, Fig. 2. For the
purpose of the analysis the actual continuous process may be regarded
as made up of a series of steps, as indicated by the broken line. First,
the combustion proceeds at constant temperature T while the amount
dx is transformed; secondly, the heat developed by the combustion is
used to raise the temperature of the mixture from T to TI + dT = T';
and so on. For each of the constant temperature transformations the
driving force X is given correctly by equation (48) when the proper
values of P, T, and x are introduced. The final constant temperature
transformation will be at the equilibrium temperature To, and for
this transformation X= 0, and K(p) K,. Equation (50) cor
rectly expresses K, as a function of xo provided P = P0, the pressure
of the mixture in the equilibrium state.
In the same way equations (61) and (62) are applicable in the
general case. The pressure P in these equations must be taken as Po,
the pressure in the equilibrium state; and the temperature T in equa
tion (49), which expresses K, as a function of T, must be the tempera
ture To of the mixture in equilibrium.
The equations deduced for the equilibrium condition are, there
fore, valid under all conditions. Whether equilibrium is attained at
constant temperature or varying temperature, at constant volume or
with a changing volume, such equations as (49), (61), and (62) will
be applicable in the equilibrium state.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 41
When the combustion is at constant pressure equations (61) and
(62) are immediately applicable, since P = const. When the combus
tion is at constant volume, the pressure is continuously changing and
it is convenient to replace the variable P by the variable T. The
change is really made through the characteristic equation of the gas
mixture. In the initial state there are m, mols at pressure P, and
temperature T,; in the equilibrium state mo mols at pressure Po and
temperature To. Hence with the volume constant
P,1V m,RT,, PV = rMRT,,
1 1 mIT
Po P1 moTF
Substituting this expression in (61) and (62),
K xo ý m, 1 2T
K1(co)  0 ...... (61a)
1X o eo To
Yo, ml 1 . *T **2 (62a)
KP(HI) ( ...... (62a)
K 1Yo eoo P ToP .
A more convenient form of equation (61a) is
logKp(co) + log To = log Xo log (1 xo) + ilog m,  logeo
+ logT1 logP .............. (61b)
In this equation the first member is a function of the equilibrium tem
perature To alone, and the second member is a function of xo and yo.
22. The WaterGas Equilibrium.The two equations (61) and
(62) may be combined into the single equation
Kp _ yo(1 xo)
K, co xo(1yo)
Introducing the partial pressures,
KpH_ P'H20 p'CO p _ P'CO P'o20
Kp co P' P'0 C P 'o, CO, P'H2
The ratio KpH2/Kpco is therefore the equilibrium constant of the
reaction
H, + CO, = H20 + CO
the watergas reaction. Hence
K (w.g.) ..............(63)
xo (1 Yo)
ILLINOIS ENGINEERING EXPERIMENT STATION
The composition of the mixture at equilibrium must be such that the
watergas equilibrium expressed by equation (63) is satisfied. This
equation (63) may replace either (61) or (62) in the specification
of the equilibrium state.
The value of the constant K, (w.g.) at a temperature T gives an
indication of the relative values of xo and y,. For example, at the
maximum temperature attained in the internal combustion engine
the value of this constant is about 7; hence
1 xo 7x0
1  o Yo
or the dissociation of CO2 is from 6 to 7 times the dissociation of H20
when x is 0.8 or greater.
23. Constants in the Equilibrium Equations.The general form
of the equilibrium equation is
R loge K, = uo _ 'logT  ½j"T  Jo"'T2  (k  ') . .. (49)
T
The constant H, is determined from the heat of combustion; the con
stants ', a", a"' from the specific heats of the components involved in
the reaction. These constants for the two principal equilibrium equa
tions are here given.
(a) The Reaction H,2  A2 = H20. The constant Ho = 102 820.
From the expressions for the specific heats of the three constituents,
we obtain:
For H2 yp = 6.00 + 0.6667 • 103T
For 02, ½y, = 3.465 + 0.06 106T2
Sum... 9.465 + 0.6667 10T  0.06 106T2
For H,20 ,= 8.33  0.276 103T + 0.423 106T2
Difference... 1.135 + 0.9427 103T 0.363 • 106T2
Hence
a' = 1.135, a" = 0.9427 . 103, a"'  0.363 106
2.3026 a' = 2.6135, o" = 0.4713  103, 4a"' =  0.0605 . 106
The constant k is determined by experiments on the chemical com
position of the mixture in the equilibrium state. An account of such
experiments is given in Appendix III. For the reaction under con
sideration, the experiments indicate the following:
k  ' =2.3, k=3.435
The first member of the equilibrium equation (49) may be transformed
as follows:
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 43
R logeKp = 2.3026 R logo K, = 2.3026 X 1.985 logo K,
= 4.571 logo K,
The final equilibrium equation is therefore
102 820
4.571 logo K, = _ 2.6135 log0o T  0.4713 _ 103T
T
S0.0605 . 106T2  2.3 .................... (64)
(b) The Reaction CO + O2 = CO,. The constant Ho is 120 930.
For temperatures below 2900 deg. F. (abs.) the following schedule of
specific heats applies:
For CO y, = 6.93 + 0.12 • 106'T
For O, jy, = 3.465 + 0.06 • 106T2
Sum... 10.395 + 0.18  106T2
For CO2 7.15 + 3.9 103T  0.60 • 106T2
Difference.. .3.245  3.9  103T 0.78 • 106T2
Hence
a = 3.245, " = 3.9 .103, '"= 0.78 10
2.3026 a = 7.4719, la" =  1.95 • 103, pa'" = 0.13 . 106
From Appendix III,
k  a =  0.6, k = 2.645.
With T<2900, the final equation is
120 930
4.571 logo K,= 120  7.4719 log0o T + 1.95  103T
T
 0.13 106T2 + 0.6 ................... ... (65)
For temperatures exceeding 2900 deg. the equilibrium equation
must be modified to conform with the different expression for y, of
CO,. The details of the transformation need not be repeated. The
final equation (T>2900) is
125 810
4.571 logo, T = 4.147 logo, T + 0.21  103T
 0.03  10TT2  37.107 .................. (66)
(c) The Watergas Reaction H + CO, H20 + CO. The
equilibrium equation for the watergas reaction is obtained by sub
tracting the equations for the H2 and CO reactions, r< spectively. Thus
for T<2900, equation (65) is subtracted from equation (64) giving
 18 110
4.571 logo K, (w.g.) = + 4.8584 log,, T
2.4213 3T + 0.1905 T 2.9 . . (67)
2.4213  103T + 0.1905  106T2 2.9 .............. (67)
ILLINOIS ENGINEERING EXPERIMENT STATION
Similarly, for T>2900, equation (66) is subtracted from equation (64).
In Table 30, page 144, the values of log,,K, for the CO reaction are
given for 10deg. intervals from T = 3000 deg. to T = 7000 deg. F.
(abs.). Similar values for the H2 reaction are given in Table 31, page
146. Subtraction of these values gives the logarithm of K, for the
watergas reaction in accordance with the equation
log 1oK,(w.g.) =log IoKpH, log 1oKpco.
Thus, for T = 4000,
logoK,(H2) = 2.86015, logoKpco = 2.10965
whence
logoK, (w.g.) = 2.86015  2.10965 = 0.75050, and Kp (w.g.) = 5.63.
Table 32, page 148, gives values of K, for the watergas reaction.
The expression for Kp in terms of Xo, y, will, in general, contain
some power of the pressure P. Thus in equations (50), (61), and (62)
the factor P1 appears. Only when there is no change in molecular
volume, as in the CH, reaction or the watergas reaction is K, in
dependent of the pressure. When K, involves the pressure, as in the
CO and H2 reactions, the pressure unit employed must be in some way
bound up in the equilibrium equation. It is evident that the constant
k is the only one that can be affected. Since most of the experiments
in equilibrium were made under a pressure of one atmosphere, it is
convenient to use the atmosphere as the unit of pressure. The con
stants 2.3, and 0.6 in equations (64) and (65) are based on this unit.
IV. CALCULATION OF MAXIMUM TEMPERATURES.
24. The Energy Equation.It is assumed in the first instance
that the maximum temperature is reached when the products mixture
attains the equilibrium state and the reactions halt. This assumption
is perhaps justified when the combustion is extremely rapid as in the
explosion of rich mixtures; it may not be justified in the case of the
relatively slow combustion of a weak mixture. The calculation of
the maximum temperature involves also the determination of the com
position of the products mixture in the equilibrium state, that is, the
determination of the unknown values of Xo and yo. There are, con
sequently, three unknown quantities, To, X0, and yo, and there must
be three independent equations from which to determine them. Two
of these are the equilibrium equations for the CO and H2 reactions,
respectively; the third is obtained by applying the energy equation
to the combustion process.
It is assumed that the combustion is at constant volume. Then
the energy equation applied to the process may be expressed as
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 45
follows: Heat absorbed = energy of products in equilibrium state 
energy of original mixture. The mixture of products in equilibrium
at the end of the process has the unknown higher temperature T,; the
original mixture has the temperature T,. Let subscripts e and m be
used to denote these two mixtures, and let a prime and double prime
be associated with the temperatures T, and To, respectively; then U"e
denotes the energy of the mixture of products at To and U'm the energy
of the original mixture at T1. Also let Qr denote the heat lost during
the process by radiation and conduction. Then the energy equation
may be written
 Qr = Ue'  Un'
The U's in the second member of this equation are determined uniquely
by the fixed initial and final states, hence Qr must be the same what
ever series of changes is traversed in passing from one state to the
other. We are at liberty, therefore, to replace the actual combustion
process by any hypothetical process that leads from the initial gas
mixture to the final mixture in equilibrium. The following assumed
changes lead directly to the desired result: (a) Let the mixture be
completely burned at the constant initial temperature T,; (b) let the
products CO, and HO dissociate at the same temperature T, until
the composition of the mixture is the same as that in the final state
of equilibrium; (c) finally, let the mixture at temperature T: be
heated without change of composition until the final temperature To
is attained. Constant volume is assumed in each of these changes.
The heat absorbed and rejected in the three changes may now be con
sidered. In the first change heat Q, is rejected; evidently Q, is
simply the heat of combustion of the mixture at temperature T,.
For the dissociations in the second change, heat Q2 must be supplied.
Denoting by n,, n, the number of mols of CO2 and H20, respectively,
and by H',co and H'vH2 the heat of combustion per mol of CO and H2,
respectively, at temperature T, we have
Q2 =nl(l Xo) H',co + n2(1 Yo) H',H
In the third change heat Q, is supplied to raise the temperature of the
products mixture from T, to T2; hence
Q = U  Uc'.
If the sum of Qs and Q, be subtracted from Q, the result is the net
heat rejected by the system during the three changes, and this is equal
to Qr, the heat rejected in the actual combustion process. Thus
Qr = Q1 (Q Q  Q3).
The heat Q,, the heat of combustion of the initial mixture at the tem
perature T2, may be denoted by H'm; and the heat rejected Q, by
ILLINOIS ENGINEERING EXPERIMENT STATION
fH'm, where f is a proper fraction. Then the energy equation takes
the form
(1f) H'm =nl(1Xo) H'.co + n2(1 o) H', + U", U'e(68)
This equation is of the first degree in xo and yo and may ultimately
be thrown into the form
Yo = b ax. ................ (69)
To show the application of the energy equation (68), the combus
tion of the Cambridge coal gas (see page 38) is taken. The following
schedule shows the calculation of Hm:
Constituent Mols H, per mol at 60' F. Product B.t.u.
CO 0.06 121 600 7 296
H, 0.48 103 000 49 440
CH, 0.335 345 930 115 887
CH, 0.035 575 400 20 139
CH, 0.010 1 360 000 13 600
H'm = 206 362
David's experiments showed a loss of heat equal to 8.9 per cent of the
heat of combustion; hence
(1 f)H',n = (1  0.089)  206 362 = 187 996
From the products composition at equilibrium (page 39),
n, = 0.545, n, = 1.283;
also for T, = 520, H'vco = 121 600, H'H2 = 103 000.
These values are substituted in equation (68) and the result is the
equation
U",  U', = 187 996  0.545(1  xo) 121 600
1.283(1yo) 103000
or
U",  U', = 66 272 xo + 132 150 y,  10 425
The initial temperature T, is known, hence an expression for the initial
energy U'e of the equilibrium mixture is readily obtained. The final
temperature T, is, of course, unknown. David's experiments indicated
a value of To in the vicinity of 4500 deg. F., hence we assume values
of To such as 4450, 4500, 4550 and work out the energy equation for
each temperature.
In Tables 24 to 29 are given the energies per mol of the various
constituents at different temperatures. From these tables the follow
ing values are obtained:
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 47
Energy per mol, B. t. u. Increase of Energy from
Gas 520 deg. to To
Ti=520 To =4450 To =4500 To =4550 5204450 5204500 5204550
COs 3186 44 719 45 324 45 929 41 533 42 138 42 743
H20 3282 37 927 38 606 39 294 34 645 35 324 36 012
H2 2178 24 468 24 818 25 169 22 290 22 640 22 991
CO,02,N2 2577 25 529 25 897 26 267 22 952 23 320 23 690
In the equilibrium composition the sum of the diatomic constituents
CO, 02, and N, is 6.033  0.8175xo  0.6415yo. The CO, is 0.545xo
mols, the HO0, 1.283yo mols, and the H,, 1.283(1yo) mols. For
each of the assumed temperatures the change of energy per mol given
in the preceding table is multiplied by the number of mols and the
products are added. Thus for To = 4450,
Ue"  U,' = 41 533 X 0.545xo + 34 645 X 1.283yo + 22 290
X 1.283 (1  yo) + 22 952 (6.033  0.8175xo  0.6415yo)
= 167 066 + 3872xo + 1128yo.
Introducing this expression for the first member of the preceding
equation, the result is the equation
62 400xo + 131 022yo = 177 491,
or
Yo = 1.3547  0.4763xo.
For T = 4500, the corresponding equation is
Yo = 1.3770  0.4768xo
and for T = 4550, it is
Yo = 1.3996  0.4772xo.
The preceding discussion has been based on the assumption that
the combustion proceeds at constant volume. A slight modification
fits the analysis to the case of combustion at constant pressure. All
that is necessary is to replace H, by H, and the energy U by the ther
mal potential I.
25. Other Expressions for the Energy Equation.A second
method of arriving at the energy equation is sometimes convenient.
The following series of changes in passing from the initial state to the
ILLINOIS ENGINEERING EXPERIMENT STATION
equilibrium state is assumed: (a) let the temperature of the initial
mixture be raised from T, to To at constant volume; (b) let the
mixture be completely burned at the temperature To; (c) let the pro
ducts CO, and H20 be dissociated until the equilibrium state char
acterized by the variables xo, yo is reached. In the first operation heat
Q' = U"m  U'm is absorbed; in the second process heat Q" is rejected,
and this heat is H"m, the heat of combustion of the mixture at tem
perature To; in the third process heat Q'" is absorbed, and the ex
pression for Q"' is
Q" =nl(1 xo)H" +co + n2(1yo)H",H
Denoting again by Qr the heat lost by radiation and conduction,
Qr = "  (Q' + Q'")
or
Q" Qr =Q' + Q'"
Inserting the proper expressions,
H",,Qr= U", U',+ nl(1xo)H"vco + n2 yo)H",V (70)
It will be observed that the expression (68) contains the heats of
combustion at the initial temperature T1 and the increase of energy
of the products of combustion; while expression (70) contains the
heats of combustion at the equilibrium temperature To and the in
crease of energy of the initial mixture.
A third expression for the energy equation is derived from equa
tion (68) by algebraic manipulation. The composition of the mix
ture at equilibrium is
mols
CO = n1 (1 Xo)
CO, = n1xo
H2 = 2n (1yo)
H20 = n2Xo
02 = el nlzo  in2yo
N, =f
Let the three diatomic gases having the same specific heat be
combined; thus,
CO + 02 N n + e + f  3 nXo  y
and let u'D, u"D denote the energy per mol of these gases at tempera
tures Ti and To respectively. Then the expression for the increase of
energy of the whole mixture is
U"e U'e = (u" u'D) (nl + e, +f inlxo  n 2yo)
+(u", u' H)n2(1y0) + (u"co 0 U'co )n1xo + (u"HO U'H o)n2y2
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 49
Inserting this expression in equation (68), the following equation is
obtained:
nixo H'co (u",u ) (u"oo U'co2
+n2YO H',~ + (u"Du'o)+(u".,u' ,)( u"IoU'o)
= (n, + el +f) (u", u',) + n2(u" u')  (1 f)H',
+ nlH'vco + n2H'v, .............. (71)
The terms in the brackets are simply the heats of combustion at the
final temperature To. The proof of this statement has the same basis
as the derivation of equation (10) (see page 18). For the temperature
T, the expression for the heat of combustion H, is
fT1 fTi
H', =Ho + f 'YvacordT f v(,,roduc.,)dT
or 0 0
H', = Ho + U'lfor,,  U"t(rodues
and likewise for the temperature To
H" = Ho + U "(atof) U"(products)
Combining these expressions,
H", = H'v + (U"  U') Vo,  (U"  U') (roduc.)
Now for the reaction CO + j 02 = CO, the factors CO and 02 are
diatomic gases having the same energy per mol; hence
U"(fato.) = U"co Uf + "o2 = 
and U"'rod,,a,)= U"co,. Therefore the expression in the first bracket
gives H",,o and likewise that in the second bracket gives H", .
The expression nH',co + 2H' H 2 (1 f)H'm is independent
of To, X0, yo, and in any given problem is a constant. Let it be denoted
by Qo. Equation (71) may now be written
n2yoH",s, =Qo + (nl + el +f) (u"Du',) + n2(u" H u's,)
nlXoH"co .................. (72)
In this form of the energy equation, it will be observed that the energy
of the products CO2 and H20 does not appear; but heats of combus
ILLINOIS ENGINEERING EXPERIMENT STATION
tion at both temperatures T1 and T2 are required. Necessarily the
three equations (68), (70), and (72) give the same expression when
reduced to the form yo = b  axo.
26. System of Two Equations; Combustion of CO.In the case
of the combustion of a single constituent, either CO or H2, there are
two unknown quantities to determine, namely To and Xo, and there
are two independent equations for the purpose, the equilibrium equa
tion and the energy equation. Four cases will be considered:
(a) Adiabatic combustion at constant volume with oxygen.
(b) Adiabatic combustion at constant volume with air.
(c) Adiabatic combustion at constant pressure of one at
mosphere with oxygen.
(d) Combustion at constant pressure of one atmosphere
with loss of heat due to conduction and radiation.
Case (a) The mixtures involved in the problem are
Initial State Equilibrium State
mols mols
CO 1x
CO 1 CO, xo
0, 0.5 O 0.5 0.5xo
mn = 1.5 mn = 1.5 0.5xo
The energy equation may be obtained from either equation (68) or
equation (70). If (68) is used, put f = 0, n, = 1, n,= 0; then the
equation becomes
XoH', =U"  U'e
= 1.5(1xo) (u"Du'D) +Xo(u"co u'c )
If (70) is used with Qr = 0, n2 = 0, the resulting equation is
xoH"v = U"m  U'm
=1.5 (u"  u'D)
Let the temperature and pressure of the mixture initially be T, = 520,
P1 = 1 atmosphere. Then taking values of H, and U from the ap
propriate tables the following results are obtained:
T = 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500
xo=0.061 0.096 0.132 0.171 0.212 0.258 0.305 0.357 0.411 0.468 0.530
These results plotted give the curve a, Fig. 3.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 51
The equilibrium equation for this case is equation (61b).
Taking m, = 1.5, eo = 0.5 (1 Xo), P1 = 1, T, = 520, the equation
becomes
log Kpco + I log To = log xo + log 3 +1.358
1 zX 1 Xz
With values of K, taken from Table 30, the following results are
found:
xo = 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95
To, 6850 6450 6100 5780 5460 5080 4620 4250
The curve b, Fig. 3, is determined from these results, and represents
equation (61b). The intersection P of the two curves gives the re
I6000
64000
42000
,1
77
7,
z
L
v
"0 0.2 04 06 08 /0
Va/ues of.r
FIG. 3. COMBUSTION OF CARBON MONOXIDE AT CONSTANT VOLUME
quired values of To and xo. Apparently To lies between 6100 and 6200.
To determine TO accurately, we assume tentatively three temperatures,
6100, 6150, and 6200, and calculate the two members of the preceding
equation for each temperature. The details of the computation are
as follows:
T = 6100 6150 6200
1.5 (u",  u',) = 54980 55688 56402
H", = 114570 114640 114710
x, = 0.4799 0.4858 0.4917
1 x 0.5201 0.5142 0.5083
Second member .... 1.70356 1.71629 1.72908
log Kp = 0.13562 0.17080 0.20547
j log T = 1.89267 1.89444 1.89620
First member ..... 1.75705 1.72364 1.69073
I
s~·i
6

ILLINOIS ENGINEERING EXPERIMENT STATION
The two sets of values are plotted against T, and the intersection of
the two lines gives T, = 6158, .o = 0.487.
Case (b) With the theoretical weight of air supplied the two
compositions become
Initial State Equilibrium State
inols mols
CO 1  x
CO 1 CO,2 x
O, 0.5 O, 0.5  0.5x
N2 1.89 N2 1.89
m = 3.39 mo = 3.39 0.5xo
The energy equation now becomes
xoH'" = 3.39 (u"  u'D)
that is, the values of x are those obtained in case (a) multiplied by
3.39/1.5. The resulting curve representing this energy equation is
curve a', Fig. 3.
m, 6.78
Since in this case m, = 3.39, the quotient l becomes 68, and
eo 1_ x
3
thus replaces  in equation (73). The result is a shifting of the
1  X
equilibrium curve to the position b', Fig. 3. The values of To and Xz
determined by the intersection P' are
To = 4950, xo = 0.795
Case (c) Adiabatic combustion at constant pressure of one at
mosphere.
The energy equation is in this case
xH"/= (i(",D i'),
in which iD = UD + 1.985T.
The equilibrium equation for constant pressure is simply
Xo 3  xo
log K, = log  + ilog 3 x  logP
1  xo 1 Xz
Taking P = 1 atmosphere, the last term becomes zero. The curve a,
Fig. 4, represents the simultaneous values of x and T from the energy
equation, the curve b, values of x and T from the equilibrium equa
tion. The intersection P gives
To = 5456, xo, = 0.511.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 53
/0000
S8000
 63000
4 000
S.'
I)
7
7/
_/
I
goeo'
0 0.2 0.4 06 08 10
Va/ues of a
FIG. 4. COMBUSTION OF CARBON MONOXIDE AT CONSTANT PRESSURE
Case (d) Combustion at constant pressure of one atmosphere,
but with a loss of heat equal to 15 000 B. t. u. per mol of CO, that is,
about oneeighth of H,.
For the energy equation in this case, we choose equation (68),
which becomes for constant pressure
xoH',15 000 =xo(i"co, i'o,) + 1 5(1 xo) (i",i'D)
whence
1.5(i"Di',) + 15 000
H', + 1.5(i",i',)  (i" ico
Various values of T are chosen and values of xo are calculated. The
simultaneous values when plotted give the curve c, Fig. 4. The
equilibrium curve is the same as in case (c); and the intersection P'
gives the results To = 5225, xo = 0.608.
These examples show clearly the marked effect of dissociation in
limiting the maximum temperatures attainable. If the curve a, Fig.
3, were continued it would meet the line x = 1 at the point T
S10 450; and curve a, Fig. 4, for the constant pressure case, would
meet it at the point T = 9020. These are the temperatures that would
be deduced from the energy equation alone for complete combustion.
The dissociation of the products reduces these apparent maximum
temperatures about 40 per cent.
~7 /2'
/
,
7
//
rL
7
i
t
/#
f
ILLINOIS ENGINEERING EXPERIMENT STATION
It will be observed also that in cases (a) and (c) the value of xo
is approximately 0.50; that is, the reaction halts when about onehalf
of the CO is consumed. As soon as the temperature falls, however, the
equilibrium is disturbed and more CO will be burned in the effort to
reestablish equilibrium. Ultimately, when the temperature has
dropped to 2000 deg. F. or 2500 deg. F., the equilibrium curve b
practically merges into the line x =1, which means that the combus
tion is complete. The point representing the coincident values of T
and x moves along the curve a until the point P is reached, and then
along the equilibrium curve b.
From the form of the equilibrium curve b, it is evident that a re
duction in temperature for any reason results in a larger value of o,,
that is, smaller dissociation. Thus, comparing cases (a) and (b) the
introduction of nitrogen results in a lower temperature throughout as
shown by the curve a'. As a consequence, 0o is increased to nearly
0.8. In case (d), the assumed loss of heat by conduction and radiation
reduces the temperature, and the result is an increase of xo.
27. System of Three Equations; General Case.For the deter
mination of the three unknowns xo, y,, To there are three independent
equations:
(a) The energy equation, having the form
Yo = b  axo
(b) The equation of the watergas equilibrium. Denoting
the equilibrium constant by c, this is
yo (1  Xo)
xao(1 o)
(c) The equilibrium equation for the CO reaction (or if
preferred, that of the H2 reaction). For a reaction at constant
volume equation (61b) is applicable:
log Kpco + I log T = log xo  log (1  xo) + j log m,  ½log eo
+ I logT,   logP,
If the reaction is at constant pressure, equation (61) is used in the
form
(c') log Kco = log xo  log (1  xo) + i log mn  log eo
 ilogP
The elimination of y, between equations (a) and (b) gives the
quadratic equation
(d) a(c1)x2o+ [a+bbc(bl)xob=0
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 55
For various assumed values of T the constants a, b, c are determined
and corresponding values of xo are obtained from the quadratic equa
tion (d). It will be observed that the first (or left hand) member
of the equilibrium equations (c) or (c') is a function of the tempera
ture T only; therefore, let it be denoted by L(T). The second (right
hand) member is a function of xo and y, and may be denoted by
R(xo,yo). For each assumed value of T simultaneous values of xo and
yo have been found from equation (c). The value of T substituted in
the first member of equation (c) gives a value of L(T); and the cor
responding values of x0 and y, substituted in the second member give
a value of R(xo,y0). If now these two functions are plotted against
T as abscissa, the intersection of the two curves gives the desired
values of T, Xo, y, that satisfy all three of the independent equations.
As an example we continue the calculation for the Cambridge
coal gas. The constants a and b for the assumed temperatures have
already been determined (see page 47). Values of the constant c, the
equilibrium constant for the watergas reaction, are found in Table 32.
Then the following results are obtained:
T = 4450 4500 4550
a = 0.47 630 0.47 680 0.47 720
b 1.35 470 1.37 700 1.39 960
c = 6.56 770 6.67 100 6.77 440
xo = 0.81 490 0.84 630 0.87 910
?o = 0.96 660 0.97 350 0.98 010
log T = 3.64 738 3.65 321 3.65 801
ilog T = 1.82 369 1.82 661 1.82 901
log Kp = 1.45 154 1.38 659 1.32 307
L(T) = 3.27 523 3.21 320 3.15 208
To determine R(xo,y0), we have eo = 0.973  0.2725 xo  0.6415 y,,
and m, = 6.667; P1i 1, log P1 = 0; T, = 520, i (log T, + log m,)
1.76 827.
The calculation proceeds as follows:
T = 4450 4500 4550
log Xo =1.91 110 1.92 752 1.94 404
i(log m, + log T,) = 1.76 827 1.76 827 1.76 827
Sum of + terms....... 1.67 937 1.69 579 1.71 231
eo = 0.13 080 0.11 790 0.10 470
1  x = 0.18 510 0.15 370 0.12 090
Jlog eo = 1.55 831 1.53 576 1.50 997
log (1xo) =1.26 741 1.18 667 1.08 243
Sum of terms...... 2.82 572 2.72 243 2.59 240
R (o,yo) = 2.85 365 2.97 336 3.11 991
ILLINOIS ENGINEERING EXPERIMENT STATION
The values of L(T) and R(xo,y,0) are plotted and give the curves
shown in Fig. 5. The intersection of the curves shows that T2 = 4557;
Te/mperAPHI re S n deGENERAL s.
FIG. 5. GRAPHICAL SOLUTION; GENERAL CASE
and for this temperature the values of xo, Yo from the xz curve and the
y, curve are approximately
xo = 0.884 yo = 0.981
By interpolation the more accurate value Xz = 0.8838 is found. As
a check these values of xz and y, may be substituted in equation (64);
thus
0.981 X 0.1162
c 6.789
0.8838 X 0.019
From Table 32 the value of c, that is, the value of K, for the water
gas reaction, for T = 4557 is 6.7899; therefore the system of values
satisfies the three equations.
28. Equilibrium and Maximum Temperature.In the preced
ing analysis the assumption has been made that the gas mixture has
attained equilibrium when the maximum temperature is reached; this
assumption requires examination.
Consider a combustion reaction proceeding at constant volume.
During an element of time dt the heat generated by the combustion
is Hvdx, the heat lost by conduction and radiation is dQ,, and the in
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 57
crease of thermal energy of the gas mixture is du. The energy equa
tion applied to the process is
Hdx dQ = du ................... (73)
The magnitude of the term dQ, depends upon the temperature of the
mixture and upon the time; that is,
dQ, = kf(T)dt
The exact form of the function f(T) is not essential at present. The
magnitude of the coefficient k depends on the character of the fuel and
the nature of the containing vessel.
Introducing the expression for dQr in equation (73), the result is
the equation
Hdxkf(T)dt = du,
or
dx du dx
dt dx dt
whence
dx _ kf(T) . ............(74)
dt du
HV
dx
The significance of this equation will be understood by reference
to Fig. 6, in which the curves a, b, c are reproduced from Fig. 4. The
FIG. 6. APPROACH OF ACTUAL COMBUSTION TO EQUILIBRIUM
curve c is merely a hypothetical locus obtained by assuming that for
any value of x the loss of heat is a fixed amount; thus, if 20 per cent
of CO is burned and 15 000 B. t. u. are lost, one point on the curve is
found.; if it is assumed that 40 per cent of CO is burned and during
ILLINOIS ENGINEERING EXPERIMENT STATION
the process the same amount, 15 000 B. t. u., is lost, a second point is
found; and so on. The curve a xepresents the relation between T and
x when the combustion is adiabatic. With a constantly increasing
loss of heat, aggregating Qr when maximum temperature is reached,
the actual curve m giving the relation between T and x will lie below
the curve a and will intersect the curve c at a point M. Now curve c
is the locus corresponding to a constant heat loss Qr and by hypothesis
Qr is the aggregate heat loss when maximum temperature is attained;
therefore, the intersection M must be the maximum point on the curve
dT
m. At this point M,  0, and since the energy u and potential i
dx
du di
are functions of T alone, we must have also at M,  0, 0.
dx dx
du
Putting  = 0 in equation (74), we have
dx
dx kf(T) ................(75)
dt Hv
which shows that the reaction is still progressing, and that equilibrium
has not been reached when the maximum temperature is attained.
After the point M is passed the temperature is decreasing and there
du dx
fore  is negative; hence the reaction velocity  continues to de
dx dt
crease. The equation shows, however, that this velocity cannot vanish
until du becomes infinite. In other words, equilibrium is not attained
dx
until the combustion is complete. The curve m approaches the equilib
rium curve b and both curves approach the line x = 1 as an asymp
tote.
In the case of the combustion of CO, the curve b is practically
coincident with x = 1 when T is reduced to about 2500 deg. F.; hence
at this same temperature, the mixture is practically in the equilibrium
state.
The assumption that equilibrium exists when maximum tempera
ture is attained requires that point M coincide with point P on the
equilibrium curve. The assumption is evidently untenable, because at
point P, as at all points on the equilibrium curve, the reaction velocity
is zero, while at point M it has the positive value given by equation
(75). The segment MP measures, in a way, the deviation of the state
of the gas mixture from the equilibrium state. As the curve m
approaches more closely to the equilibrium curve b the mixture ap
proaches the condition of equilibrium.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 59
The preceding calculation of maximum temperatures by the solu
tion of the energy equation and equilibrium equation gives the tem
perature at the point P, a temperature that must be somewhat greater
than the actual maximum temperature indicated by point M. The
calculated temperature must be regarded, therefore, as an upper
limit which will be attained in the actual combustion only when the
combustion is adiabatic. With rapid explosions of rich mixtures the
loss of heat is relatively small and the calculated temperature may
be a good approximation to the maximum temperature. If the mix
ture burns slowly, so that the loss of heat is large, the segment MP is
correspondingly large, and the calculated temperature must neces
sarily exceed the actual maximum temperature by a considerable
amount.
As will be shown by the experiments to be described in the follow
ing chapter, the time occupied by an explosion is only a fraction of a
second; consequently, the reaction velocity must be very high. It is a
question whether at such high velocities equilibrium can even be
approached. Again, in the actual explosion, there must be a variation
of temperature throughout the mass of burning gas, and therefore the
temperature deduced from the observed pressure must in a certain
sense be a mean temperature. It is not to be expected, therefore, that
the temperature calculated from an analysis of ideal equilibrium con
ditions will coincide accurately with observed experimental tempera
tures. The fairly good agreement between theory and experiment to
be described in the following chapter indicates that the mixture does
approach a condition of equilibrium.
V. EXPLOSION EXPERIMENTS OF DAVID AND OF BONE AND HAWARD
29. Experimental Data Available.A large amount of experi
mental work has been done along the line of gaseous explosions in
closed vessels. A complete bibliography is contained in Bulletin
No. 133 of the Engineering Experiment Station, University of Illinois,
entitled "A Study of Explosion of Gaseous Mixtures," by A. P.
KRATZ and C. Z. ROSECRANS. With one exception the experimental
results published to date do not lend themselves readily to the verifica
tion of a theoretical analysis because no attempt has been made to
measure the heat losses during the combustion period and the sub
sequent cooling period. The one exception is the recent work of
Dr. W. T. David* on "The Internal Energy of Inflammable Mixtures
of Coal Gas and Air after Explosion," in which explosion tempera
* Proc. Roy. Soc., V. 98A, p. 303, 1921.
ILLINOIS ENGINEERING EXPERIMENT STATION
tures and total heat losses are given for various points on the explo
sion curve. The results of Borne and Haward* on the explosions of
hydrogenair and carbonmonoxideair mixtures at high initial pres
sures are also useful in the verification of the theory.
30. Experimental Methods.David used a cylindrical explo
sion vessel of 0.788 cu. ft. capacity. Ignition was obtained by a spark
placed in the center of the explosion vessel. Pressures were measured
by a Hopkinson optical indicator. The spot of light was reflected
from the indicator to a photographic film mounted on a rotating drum.
Radiation measurements were made with a bolometer receiving the
heat radiated through a small window in the explosion vessel. Values
of the heat losses by conduction were obtained by measuring the rise
in temperature of a thin polished silver plate mounted on a backing
of linoleum and placed upon the inside surface of one of the end
covers of the explosion vessel. Various mixtures of Cambridge coal
gas and air were exploded from atmospheric pressure and room tem
perature.
Bone and Haward used a spherical explosion vessel 3 inches in
internal diameter equipped with a Petavel optical indicator. An ini
tial pressure of 50 atmospheres was used with room temperatures.
Various mixtures of hydrogen, carbon monoxide, and air were used
in the series of experiments which is of particular interest here.
31. Experimental Results.The experimental results of David
are given in Table 2. The temperatures given by David were cal
culated from the recorded pressures by means of the equation
PV = RT, making the proper correction for the change of molecular
volume upon combustion.
The experimental results of Bone and Haward are given in
Table 3.
32. Chemical Analysis of Cambridge Coal Gas used by David.
In the absence of any exact analysis of the Cambridge coal gas used
by David the following is taken as being a typical analysis:
Percent. by Vol.
Hydrogen H, ............................. 48.0
Methane CH, ............................. 33.5
Ethylene CH ............................. 3.5
Benzene CH, ............................. 1.0
Carbon Monoxide CO ....................... 6.0
Nitrogen N2 .............................. 5.5
Carbondioxide CO. ....................... 2.0
Water Vapor HRO ......................... 0.5
Total............................ 100.0
* Proc. Roy. Soc., V. 100A, p. 65. 1921,
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 61
Analyses of the products of combustion resulting from the ex
plosions of the various mixtures of gas and air are given by David,
and the lower heat of combustion is also stated. An analysis of Cam
bridge coal gas for a period previous to 1914 was obtained and a few
minor shifts were made so as to duplicate as nearly as possible the data
given by David. The preceding analysis is the result. A humidity
of 30 per cent at 60 deg. F. is assumed. The analyses of the products
of combustion given by David are contained in Table 1. The values
in parentheses are those resulting from the foregoing analysis.
TABLE 1
COMPARISON OF PRODUCTS OF COMBUSTION OBTAINED FROM CAMBRIDGE
COAL GAS ANALYSIS WITH THE EXPERIMENTAL VALUES OF DAVID
Percentage of Gas in Mixture ....................... . 15.0 12.4 9.7
H20 20.0 16.4
(20.0) (16.5)
CO2 8.5 7.0
(8.5) (7.0)
N2+UO 71.5 76.6
(71.5) (76.5)
The lower heat of combustion given by David is 145 000 calories
per cu. ft. (0 deg. C. and 760 mm.) or 206 394 B. t. u. per mol. The
lower heat of combustion at constant volume calculated from the
preceding analysis is 206 362 B. t. u. per mol. The products of com
bustion and the heat of combustion are the only factors resulting from
the gas analysis that enter into the computation of the explosion tem
perature; hence the calculated results will be the same even if the
assumed gas analysis does not duplicate precisely the actual analysis.
33. Calculation of Temperatures.The maximum temperature
for the 15 per cent mixture with 8.9 per cent loss of heat has been
given as an illustrative example in the preceding section. By the
same method the theoretical maximum temperatures for the other mix
tures are obtained, and also the temperatures subsequent to the ex
plosion when the mixture is cooling. The results are given in Table 2
and Fig. 7..
As suggested before, it is possible that at the highest temperatures
the reaction ½N2 + JO, = NO may require consideration. This be
ing an endothermic reaction, the result would be a decrease in the
maximum temperature. The equilibrium equation given by Haber is
38 700
R logeKI = +2.45
i i
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 7. EXPERIMENTAL AND CALCULATED EXPLOSION TEMPERATURE CURVES
FOR CAMBRIDGE COAL GAS AND AIR MIXTURES
in which
PNO
pso
K,=
Assuming that the values of xo and yo are but little changed by the
formation of NO the expression for K, is approximately
P.03z
K= (0.1044.515z) • 4. 515(1z)
in which s denotes the part of the N2 present that is used in the forma
tion of NO. This equation applies to the 15 per cent mixture. For
a temperature in the vicinity of 4500 deg. F. (abs.) the value of s is
found to be about 0.003. Therefore the heat absorbed in this reaction
is 38 700 X 9.03 X 0.003 = 1050 B. t. u. Taking this amount of heat
into consideration, and revising the original calculation, the maximum
temperature is found to be approximately 4530 instead of 4557. The
method of calculation is not at all exact, but the result is sufficiently
accurate to show that the effect of the NO reaction is negligible.
34. Comparison of Experimental and Calculated Temperatures.
A comparison of the maximum temperatures shows the following
differences:
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 63
Mixture 15 per cent 12.4 per cent 9.7 per cent
T calc. ............ 4557 4141 3327
T exp. ............ . 4390 3690 3000
Diff., per cent. .... 3.8 12.2 10.9
The difference of 3.8 per cent for the richer mixture is about what
might be expected. The theoretical temperature, 4557 deg. F., is based
on the assumption of an equilibrium that does not exist, and the
actual temperature must be slightly lower. In the case of the slower
burning 9.7 per cent mixture the gas at maximum temperature is still
further from equilibrium, and the difference between calculated and
experimental temperatures is greater, as it should be. The intermedi
ate 12.4 per cent mixture, however, shows a divergence that is difficult
to explain. With only 10 per cent heat loss compared with 18 per cent
for the weaker mixture, and with 0.08 second as the time of explosion
compared with 0.18 second for the weaker mixture, the difference
between calculated and experimental temperatures should certainly
be less for the 12.4 per cent mixture than for the 9.7 per cent mixture.
David,* in his discussion of these experiments, divides the heat of
combustion into three parts: (a) the increase of internal energy of
the mixture; (b) the available chemical energy remaining when
maximum temperature is attained; and (c) the heat loss by conduc
tion and radiation. The following results are deduced:
Internal Energy
81
78
72.5
Available
Chemical Energy
10
12
9.5
Here again the 12.4 per cent mixture is not in line with the other two.
There is no conceivable reason why this mixture should retain a
greater amount of available chemical energy than either the richer
or the weaker mixture. Apparently there must be some error in the
experimental results.
A comparison of the successive temperature differences for each
of the experiments is important:
* Proc. Roy. Soc., Ser. A, V. 98, p. 316.
Mixture
per cent
15
12.4
9.7
Heat of
Combustion
100
100
100
Heat
Loss
9
10
18
ILLINOIS ENGINEERING EXPERIMENT STATION
15 per cent Mixture
T calc................. 455;7 4092 3721 3442 3204 3017 2719 2505
T exp ................. 4390 4000 3640 3310 3080 2880 2570 2340
Diff., per cent.......... 3.8 2.3 2.25 4.0 4.03 4.76 5.8 7.06
12.4 per cent Mixture
T calc. ................. 4141 4064 3763 3483 3233 3051 2750 2534
T exp. ................. 3690 3650 3460 3240 3060 2900 2630 2390
Diff., per cent.......... 12.2 11.6 8.75 7.5 5.6 5.2 4.6 6.0
9.7 per cent Mixture
T calc ................. 3327 3237 3039 2877 2600 2441 ...........
T exp. ................ 3000 2970 2840 2740 2500 2360 ......
Diff., per cent.......... 10.9 5.0 7.0 5.0 4.0 3.4 ......
After the maximum temperature is passed the gas mixture should
be approaching nearer to the equilibrium state, and consequently the
difference between calculated and experimental temperatures should
be growing smaller. Such is the case in the two weaker mixtures, but
the 15 per cent mixture shows an opposite tendency. Several ex
planations may be advanced for the apparent conflict of theory and
experiment:
(a) The calculated values of T may be too large because of in
accuracy of the formulas for the specific heats used. However, a very
large increase in the specific heat of diatomic gases would be required
to bring the two sets of temperatures into reasonable agreement.
(b) The magnitude of the heat losses might have been under
estimated. A slight increase of the values given in column 6, Table
2, would reduce the calculated temperatures appreciably.
(c) Accurate measurements of pressure are difficult to attain
even with the best instrumental methods. It is possible that all the
measured pressures were slightly low, and consequently the tempera
tures deduced from them are likewise low.
After taking account of all the discrepancies herein noted, it must
be concluded that David's experiments, on the whole, furnish a thor
oughly satisfactory verification of the theoretical analysis.
35. Experiments of Bone and Haward.Various mixtures of
H2 and CO along with 02 and N, were exploded. Table 3 gives the
composition of the initial mixture and the maximum pressure attained.
The initial pressure in each case was 50 atmospheres. The proportions
of CO and H2 were varied, but the sum CO + H2 was kept approxi
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 65
mately constant. The volume of N, was approximately four times the
volume of O,.
The heat losses during the explosion were not determined; hence
a direct comparison between calculated and experimental tempera
tures is not possible. It is possible, however, to estimate the heat loss
that will bring the calculated temperature into coincidence with the
experimental temperature, assuming that maximum temperature and
equilibrium are attained together.
This calculation is performed as follows: Assuming heat losses of
0, 5, 10, and 15 per cent, the equilibrium state is calculated for each
case, that is, P, T, xo, and y( areobtained; a curve is plotted using the
calculated explosion pressures as ordinates and the assumed heat losses
as abscissae; from this curve the heat loss is obtained corresponding
to the experimental maximum pressure obtained from the explosion of
the given mixture; the required values of T, xo, and yo consistent with
the experimental explosion pressure are obtained from curves plotted
with heat loss as abscisse. The results of the calculation are the figures
given in column 3 of Table 3.
For experiments II, X, and XI, the calculated heat losses are
apparently too high, considering the short time interval occupied by
the explosion. It will be noted, however, that in these experiments
the H2 content of the mixture is high, and it was thought possible that
slight dissociation of HI into atomic hydrogen (see Section 45) would
account for the discrepancy. Calculation showed that the high maxi
mum pressure reduced the dissociation of H, practically to zero, so
that no error is introduced by neglecting this dissociation. The only
other explanation which presents itself is the possibility that the
Petavel indicator used was too slow to record accurately the maximum
pressures of the extremely rapid explosions. In the remaining ex
periments the figures for the heat loss are reasonable, and the varia
tions are no more than might be expected considering the nature of
the experiments.
It should be noted that the figures given for the heat loss are
based on the assumption that equilibrium exists at the point of
maximum pressure. For these rapid explosions it is probable that the
mixture is very near equilibrium at maximum pressure. However,
the calculated temperature, with a known heat loss, must slightly ex
ceed the actual temperature; consequently, the calculated values for
the heat loss are greater than the actual values.
The application of the system of calculation to the experiments
of Bone and Haward gives a satisfactory correlation. From the com
parisons made in this section it seems evident that the maximum tem
peratures resulting from the combustion of rich mixtures with high
reaction velocities may be calculated with a good degree of accuracy
from the equations deduced in the preceding chapters.
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 2
EXPERIMENTAL AND CALCULATED RESULTS FOR EXPLOSIONS OF
MIXTURES OF CAMBRIDGE COAL GAS AND AIR
Experimental Data by David
Time Heat Loss Expressed as a
after Mean Gas Percentage of the Heat of
Igni Temperature Combustion of the Coal Gas
tion
see. deg. C. deg. F Radiation Conduction Total
(abs.) (abs.) I
Calculated Values using Heat
Losses given in col. 6.
T deg. F.
(abs.)
15 per cent Mixture of Coal Gas and Air
0.884
0.965
0.990
0.998
1.000
1.000
1.000
1.000
0.981
0.994
0.998
1.000
1.000
1.000
1.000
1.000
12.4 per cent Mixture of Coal Gas and Air
0.08 2050 3690 5.5 4.5 10.0 4141 0.978 0.996
0.10 2030 3650 6.8 5.8 12.6 4064 0.983 0.997
0.15 1920 3460 12.5 9.7 22.2 3763 0.995 1.000
0.20 1800 3240 17.4 13.0 30.4 3483 1.000 1.000
0.25 1700 3060 21.7 15.6 37.3 3233 1.000 1.000
0.30 1610 2900 25.0 17.2 42.2 3051 1.000 1.000
0.40 1460 2630 30.2 19.8 50.0 2750 1.000 1.000
0.50 1330 2390 34.0 21.4 55.4 2534 1.000 1.000
9.7 per cent Mixture of Coal Gas and Air
1660 3000
1650 2970
1580 2840
1520 2740
1390 2500
1310 2360
11.0 7.0 18.0 3327
12.8 8.3 " 21.1 3237
16.5 11.2 27.7 3039
19.4 13.6 33.0 2877
24.4 17.4 41.8 2600
27.7 19.0 46.7 2441
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
0.18
0.20
0.25
0.30
0.40
0.50
I
I
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 67
c! c.
S00
4 0 00
1^4 g S3
co . . . Co C!o C! c ý 000
0 l'i','010100L1C CCo0 0
St0000000000.N'I0
S1 00000000000
+M C 12 00 O itl00000000GO
· (
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0 r
o; i "LJt'Ot't»t< iC >rom~~ a >Si;t
fr< I
Wz
o .X C
PI I <
S0 
CoC
o C)
S U.0
0" Sa
p5A I
00
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a r
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B
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E 5
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Co
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Ao
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0
I " "
'3
o
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Xli
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S~
c: 4
H^
0 " C C
000Co 0000000
Co 0 C DCC Co 0 10 C4
0004Co 0 0 C0
C) i Co C) C) )'C' ) tl  Co ' C '* C) C
00'oo'ooo'ra
0 0. Co 0 "I 0ý 0 C
01 0c NC C m C010 C 'A
olmmmmmm M=C
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0o Coý Coý 0 Coý Co 010 . o  C) a.
0000000000
000000000l 0
 0t ^ 0T Co1( CC COl Coa O CO
CO Co C oC)o Co C)' C) C)'
NCCNO000 NCOO"
0 0 '0 00 'Co^ C 04CoI o
CTSO>OiS0)0 ~N0 010o
dddooooooo *~d
l>COOO(N"t    
" ^Rxks^ I
ILLINOIS ENGINEERING EXPERIMENT STATION
VI. EFFECT OF VARYING CONDITIONS IN THE COMBUSTION
OF. GASEOUS FUELS
36. Problems Investigated.With a system of calculation avail
able, certain important problems connected with the combustion of
gaseous fuels may be attacked. In this chapter, therefore, attention is
given to the following topics:
(a) The effect of the initial pressure of the gas mixture.
(b) The effect of the initial temperature of the mixture.
(c) The effect of excess air; also the effect of the loss of heat
during combustion.
(d) A comparison of several wellknown fuel gases.
In all the cases considered, explosion at constant volume has been
assumed.
As has been stated, the calculated results are more accurate for
rich mixtures that burn rapidly and thus more nearly attain chemical
equilibrium at the point of maximum temperature. The absolute
values calculated for weaker and slower burning mixtures are un
doubtedly too high; still the results are of value in making a study of
the effects of various initial conditions on the explosion temperatures
and pressures. A more complete study of the gaseous fuels has been
made than of the vapors of liquid fuels, because the chemical and
physical constants for the gases are more accurately known; and the
gas fuels have a more definite chemical composition.
37. Effect of Initial Pressure.The 15 per cent mixture of Cam
bridge coal gas and air is used as an example. The initial temperature
is taken as 60 deg. F. and a heat loss of 10 per cent of the heat of com
bustion at 60 deg. F. is assumed. Initial pressures up to five at
mospheres are used. The calculated results are given in Table 4 and
Fig. 8.
TABLE 4
EFFECT OF INITIAL PRESSURE
Initial Press., atm.......
Max. Explosion Temp.,
deg. F. (abs.) ........
Max. Press., atm ......
Xo .....................
Y o .. .... . . . . . . . . . . . . . . .
0.25 050
4456 4494
2.076 4.182
0.844 0.868
0.973 0.978
1.00
1528
8.416
0.891
0.982
2.00
4558
16.921
0.911
0.986
3.00 4.00 5.00
4572 4 4583 4590
25.443 33.995 42.539
0.921 0.928 0.933
0.987 0.989 0.990
; "
I

MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 69
h~
wr~,c~iw~79 I7 /Vph~e aE(O ~.4')vA/
a/10ns6a10 PU/N / / .A'a' U _z y 1  ;
Sai/Co Lv ! a/nW/a0d ,nx' 90,710A7W N L!*
ILLINOIS ENGINEERING EXPERIMENT STATION
The linear relation between the initial and the maximum pres
sures has been verified experimentally by Bairstow and Alexander*
for mixtures of coal gas and air.
38. Effect of Initial Temperature.The same 15 per cent mix
ture of Cambridge coal gas and air is taken. T.he initial pressure is
one atmosphere and a heat loss of 10 per cent during the period of
attainment of maximum temnperature is assumed. Initial tempera
tures from 500 to 1300 deg. F. (abs.) are used. The results are
given in Table 5 and Fig. 9.
A linear relation is found to exist between the initial temperature
and the maximum temperature. The general effect of an increase in
initial temperature at a constant initial pressure is an increase of the
maximum temperature and a decrease in the maximum pressure with a
consequent increase of the dissociation of the products of combustion.
The falling off of the maximum pressure is due to the decrease in
the ratio of the maximum temperature to the initial temperature.
TABLE 5
EFFECT OF INITIAL TEMPERATURE
Initial Temp., deg. F. (abs.) ....... 520 700 900 1100 1300

Max. Temp.. deg. F. (abs.) ........ 4528 4591 4662 4732 4804
Max. Press., atm................. 8.41 6.35 5.03 4.18 3.60
X ........... . . ............. . 0.891 0.861 0.826 0.791 0.754
Yo .... ..... . ...... 0.982 0.977 0.971 0.964 0.957
39. Effect of Excess Air and Heat Loss.Curves of maximum
temperature and dissociation for various mixtures of Cambridge coal
gas and air, hydrogen and air, and carbon monoxide and air, with
various percentages of heat loss up to the time of attainment of maxi
mum temperature, are shown in Figs. 10, 11, and 12. The results are
given in Table 6. The initial temperature taken is 60 deg. F. and
initial pressure one atmosphere.
An increase either in the amount of air present or in the heat
loss tends to decrease the maximum temperature and makes the reac
tions go nearer to completion.
* Proc. Roy. Soc., V. 76A, p. 345.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 71
0
n / o 0 sa, /7/"1 ul 1 . 11. t 5u .
sgoy jag/V uq 1
,nws/Jadcual_. uo/so/dxy
,4
14
a;
0
w
0
IP
.,toxi yo san/lA 1 a
q i  %2O.9 9a^ v1
"..<, j/ ._f"ap u/
^/7/a/?^/0 tN 00 os 00
M f 0 00000 'f
00 N 0N
00 00 GO 00 00 0T
0 1 00 000
ON 0 0 i0 M '
ON G ON ®t
0 10 t 0  0 0
®~ tcl 'i 6 l
oc 0 CC 0
ND 000'^ O000
000000^
00000000;
00 (C ON 00 C
m R t(D ' 00
00000
00 ON 00
N00 m q 0
0 0 O Lo VO "
CD »o o =o "^*
Sc00 t000
C04s=0
oCl CVNCT3
lfN3 M^l0 M '
cWfforg
10 COD O 0 0a 0
NCO 0 0 0 00
000 000
c, ,0
ILLINOIS ENGINEERING EXPERIMENT STATION
The intermediate or equilibrium composition is then easily written
down as follows:
Intermediate Mixture
inols
CO, 6xo
CO 6(1xo)
H,O 3yo
H, 3(1 o)
0, (  3) 3x  1.5yo
N, 3.78n
m" = (m' + 5)  3x  1.5yo
The energy equation reduces to
yj 3H"H2 = [E + 3u"., + (2 + m')u"D IX [iH",coj
where
E= H'vc, + 6H',co + 3H',H 3u'H  (2 + m')u',
Single primes are used for the state at the end of compression and
double primes for the state at maximum explosion temperature.
The equilibrium equation becomes, with the proper values sub
stituted,
log K,,,o + log T = log xo + I log log(1.ro)
ilog[ (n3)3x0o1.5yo]
m,T, = mT, V 134.4m
* _. 134.4m,
PS P, V,
The third equation, namely, the quadratic used for the determina
tion of x, has the usual form.
The calculated results for the adiabatic constant volume explo
sions of benzene vapor and air mixture are given in Table 9 with
those of Tizard and Pye, and also in Fig. 14.
It can be seen from Fig. 14 that the agreement of the results
produced by the two methods of procedure is remarkably good. The
maximum variation is about 2.5 per cent, the values of Tizard and
Pye being the lower. The method of Tizard and Pye is undoubtedly
the more laborious and indirect. The many approximations used
greatly increase the probability of error in any wide application of
their method.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 81
Per Cen/, of/ 4r for Cozmp//ele Combus'o1n2
FIG. 14. CALCULATED EXPLOSION TEMPERATURE CURVES FOR MIXTURES OF LIQUID
FUEL VAPORS AND AIR WHEN EXPLODED ADIABATICALLY IN AN ENGINE
HAVING A COMPRESSION RATIO OF 5
43. Mixtures of Gasoline and Kerosene Vapors and Air.The
mixtures of gasoline vapor and air are assumed to be exploded adia
batically at constant volume in an engine having a compression ratio
of 5.
T, = 672 deg. F. (abs.), P, = 1 atmosphere.
The gasoline is assumed to have the molecular formula CH,,,.*
The calculated results are given in Table 10 and Fig. 14.
* Wilson and Barnard, Jour. S. A. E., V. 9, p. 313, 1921.
ILLINOIS ENGINEERING EXPERIMENT STATION
For kerosene vapor the same conditions are assumed as for the
gasoline vapor mixttres. The kerosene is assumed to have the molec
ular formula C12H,,.*
The calculated results are given in Table 10 and Fig. 14.
TABLE 10
CALCULATED EXPLOSION DATA FOR MIXTURES OF GASOLINE VAPOR AND AIR
AND KEROSENE VAPOR AND AIR
Mixtures Exploded in an Engine with a Compression Ratio of 5
Percentage
of Air Explosion
Required Temperature
for Complete deg. F.(abs.)
Combustion
20
Explosion Pressure Compression
o Pressure
Satm. lb. per Temp. lb
sq. in.(abs.) deg. F.(abs.) atm. lb. per
sq. in. (abas.)
Gasoline (C, H,)
70 4808 0.317 0.773 41.833 615 1096 8.154 120
80 5049 0.495 0.885 42.215 621 1108 8.244 121
90 5187 0.670 0.943 42.170 620 1121 8.340 123
100 5149 0.784 0.967 41.155 605 1131 8.415 124
125 4847 0.923 0.989 37.902 557 1152 8.572 126
150 4502 0.974 0.996 34.836 477 1166 8.676 128
Kerosene (C,, H,)
70 5222 0.306 0.784 45.691 672 1090 8.110 119
80 5394 0.473 0.886 45.394 667 1105 8.222 121
90 5425 0.620 0.934 44.476 654 1118 8.318 122
100 5359 0.726 0.958 43.179 635 1129 8.400 123
125 5058 0.883 0.983 39.764 584 1150 8.557 126
150 4718 0.952 0.993 36.635 538 1166 8.676 128
44. Water Injection.The advantage of water injection has
often been urged in connection with the operation of internal com
bustion engines. The effect of adding water to the fuel mixture may
readily be determined by an extension of the analytical method
developed in the preceding sections.
It is assumed that water in varying amounts is injected into
the fuel mixture at the end of the compression stroke. The fuel
chosen is gasoline, the theoretical amount of air is taken, and all
processes are assumed to be adiabatic.
The method used to determine the compression temperature and
pressure after the water has been injected is as follows:
The compression volume is determined from the perfect gas law,
the pressure, temperature, and the number of mols present being
* Wilson and Barnard, Jour. S. A. E., V. 9, p. 313, 1921.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS
known. The compression volume is equal to the total volume of
steam resulting from the water injected. The volume per pound of
steam is then known. From the steam tables the saturation tempera
ture and the energy required to transform one pound of water from
water at the temperature of injection to saturated steam are obtained.
The energy tables for water vapor (Appendix IV) are used for the
superheat range. The heat given up by the compressed gases is easily
computed by the use of the energy tables, and is equal to the heat
absorbed by the water in being transformed to superheated steam.
The compression temperature after water injection is therefore de
termined. The values are given in Table 11.
TABLE 11
EFFECT OF WATER INJECTION ON GASOLINEAIR COMBUSTION
Compression
Pressure
l'b. per sq.
atm. in. (abs.)
1131 8.415 123.7
1059 8.044 118.2
991 7.681 112.9
925 7.313 107.5
798 6.556 96.4
Temp.
deg. F.
(abs.)
5149
5059
4971
4884
4718
Explosion
Pressure
Values at
Explosion Temp.
lb. per sq.
atm. in. (abs.) o Y
41.155 605.0 0.784 0.967
41.163 605.1 0.809 0.970
41.164 605.1 0.830 0.974
41.155 605.0 0.850 0.976
41.128: 604.6 0.886 0.982
1
Let n' = the number of mols of water injected per mol of gaso
line; then the mixture compositions are as follows:
Initial Mixture
mols
CHI 1.00
02 12.50
N, 47.25
HO n'
mr = 60.75 + n'
The energy equation takes the forl
Intermediate Mixture
mols
CO, 8x,
CO 8(1 xo)
H,O (n' + 9)y,
H, (n' + 9) (1  yo)
ni
0, 8.5 + 4x o (n' + 9) y
N, 47.25
(72.75 + 3 n')  4x  I(n'  9) y,
2 )4o½n9y
o [(n'+9)H", = [E+(n'+9)u", +(63.75 + u",
X0 [8H" cooI
Temp.
deg. F.
(abs.)
Pounds of
Water In
jected per
Pound of
Gasoline
0.0
0.2
0.4
0.6
1.0
84 ILLINOIS ENGINEERING EXPERIMENT STATION
where
E H'rcC18 + 8H'rco + (n' + 9)H',H2
 (n' + 9)u'H (63.75+ ) u',
The equilibrium equation becomes
logKco + 'log T=lox + ogo + log( 1o)
½log[8.5 + 4xoI(n'+9)Uo
These two equations with the usual quadratic give the solutions
found in Table 11. The results are also represented by the curves of
Fig. 15.
n7,L
Pounds of WAfer /njecTOed
per Pound of Gaso/ine
FiG. 15. EFFECT OF WATER INJECTION
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 85
The outstanding result shown in Fig. 15 is the constancy of the
explosion pressure with variation in the amount of water injected.
With a water injection of one pound per pound of fuel vapor present
the compression pressure is lowered 22 per cent, but due to the addi
tional weight of charge and the more complete combustion the explo
sion pressure is unaffected. With injection of the water into the in
take manifold a reduction in the work of compression is probably
effected, but this saving is more than likely offset by a slowing down
of the combustion process. V. W. Brinkerhoff* found that for a class B
U. S. Army truck engine with a compression ratio of 3.71 no change
in power or efficiency was obtained when water up to 0.6 pound per
pound of gasoline was injected into the intake manifold. For amounts
above 0.6 pound there was a decided decrease in power.
VIII. WELDING FLAMES
BY GEORGE T. FELBECK
45. Dissociation of Hydrogen into Atoms.In oxyhydrogen and
oxyacetylene flames as ordinarily used in commercial practice the
partial pressures of the various gas constituents in the flame are very
low, the total pressure being one atmosphere. The low partial pres
sure of the hydrogen coupled with the high temperature causes an
appreciable dissociation of the molecular hydrogen into atomic hy
drogen. A quantitative determination of the extent of this dissocia
tion has been made by Dr. Irving Langmuir.t An equation for the
equilibrium constant for the reaction HI2 H + H has been derived
from the data given by.Langmuir. (See Appendix III.)
46. Possibilities of Error in Flame Temperature Calculations.
The conditions which tend to invalidate the calculated temperatures
for the oxyhydrogen and oxyacetylene flames may be briefly stated
as follows:
(a) The specific heat equations have been extended past the
range of experimental verification.
(b) The specific heat equation used for molecular hydrogen
may include some of the heat of dissociation of molecular hydro
gen into atomic hydrogen at the higher temperatures. This error
is probably small, since the specific heat equation is a linear rela
tion, and also, since the explosion method was used in determining
the specific heats, the dissociation was materially decreased by the
high pressure.
(c) There is a possibility of the dissociation of the molecular
oxygen into atoms.
* Report No. 45, Nat. Adv. Com. for Aeronautics.
t Jour. Am. Chem. Soc., V. 37, p. 417, 1915.
ILLINOIS ENGINEERING EXPERIMENT STATION
(d) The value used for the heat of dissociation of molecular
hydrogen into atomic hydrogen has a rather weak experimental
foundation.
(e) Experimental values for the equilibrium constant for
the reaction H2 = H + H have not been verified.
Because of the uncertainty of the data concerning the dissociation
of hydrogen into atoms, flame temperatures have been calculated both
with and without taking this dissociation into consideration.
47. Oxyhydrogen Flame Temperatures.The gases previous to
combustion are assumed to be at 60 deg. F.; the combustion takes place
at a pressure of one atmosphere; the combustion is assumed to be
adiabatic; the calculated temperatures are therefore the maximum
attainable.
Taking n mols of oxygen per mol of hydrogen, the mixtures given
are for the case where dissociation of hydrogen into atoms does not
occur.
Initial Mixture Intermediate Mixture
mols mols
H,O y
H, 1 H2 1y
O, n O, n y
m'= 1+ m" = m'iy
The equations for the determination of the maximum tempera
ture and extent of combustion become
i"H2 + ni02  i'2  i'o
y=
log K, 2 = log y + log (m' y) log(1y)  log (n y)
For the oxyhydrogen flame temperature with dissociation of
hydrogen into atoms, the procedure is as follows: Of the initial hydro
gen a portion equal to y burns to HzO. Of the remaining hydrogen
equal to (1 y) a portion z exists as molecular hydrogen.
Let the subscript a indicate the atomic state. The initial and
intermediate mixtures are:
Initial Mixture Intermediate Mixture
mols mols
H,O y
H, 1 H2 s(1y)
2O n H. 2(1z) (1y)
 O, n  y
m' 1 + n 2 nl
m" = 0.5 +n + (iz) (l y)
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 87
Following the usual procedure the equations established are:
yH"pH i", ni"o i'% , + ni'o (A
(1  )) = . .
(l) (1y)H",a
[4(1 z)2 (1 y)..
l H Z 0.5 + n + (3z) (1  .y)
SogK.lo +n( . n)(1~)y)] .
logKp, =log (C)1g7  ]......(C)
z(2 (y) (n y)l
The solutions of the three equations given above were obtained as
follows: Several probable values of y were assumed and equations (A)
and (B) solved by trial for values of a and T corresponding to each
value of y. Values of the righthand member of equation (C) were
obtained upon substitution of each set of values of y, s, and T. The
intersection of the curve of the righthand member of equation (C)
with the curve of the values of logKpH2 obtained from the tables
gives the solution.
The calculated temperatures for the oxyhydrogen flame are
given in Table 12 and Fig. 16. The third curve gives the percentage of
oxygen'present in the flame for the case with dissociation of hydrogen
into atoms. The dissociation of the hydrogen reduces the maximum
attainable temperature about 11 per cent. For welding purposes a
nonoxidizing flame is required. From the oxygen curve, Fig. 16, it
is seen that the oxygen content is inappreciable in the flames resulting
from mixtures having an oxygen to hydrogen ratio of 1 to 4 or less.
TABLE 12
OXYHYDROGEN FLAME TEMPERATURES
No Dissociation of
Hydrogen into Atoms
T
deg. F. (abs.) Y
4195
5552
6100
6169
6085
5844
0.300
0.493
0.638
0.712
0.790
0.869
With Dissociation of Hydrogen
into Atoms
T
deg. F. (abs.)
4096
5074
5648
5727
5713
5535
y
0.300
0.499
0.686
0.752
0.818
0.872
0.992
0.919
0.770
0.712
0.661
0.646
Vols. Oxygen
per vol.
Hydrogen
0.15
0.25
0.375
0.50
0.70
1.00
Percentage
Oxygen in
Flame
0.00
0.096
2.90
12.40
21.50
35.10
ILLINOIS ENGINEERING EXPERIMENT STATION
In commercial work this ratio of 1 to 4 for the oxygen to hydrogen in
the mixture is found to give a nonoxidizing flame, and is therefore
used. The temperature for the welding flame having this ratio is
5074 deg. F. (abs.), or say 4600 deg. F. This is slightly higher
than the usually accepted value of 4100 deg. F. which has been cal
culated by previous investigators. For cutting purposes where the
oxygen content can be increased, the maximum temperature may be
taken as 5000 deg. F.
48. Chemical Reactions Occurring in Oxyacetylene Flame.
The following statements are taken from a report of an experimental
investigation by W. A. Bone and J. C. Cain :
(1) When acetylene is exploded with less than its own volume of oxygen,
carbon monoxide and hydrogen are finally obtained in accordance with thp
equation C,H, + 0 = 2CO + H,.
(2) The excess of acetylene was for the greater part resolved into its
elements by the shock of the explosion wave. A small quantity (as much as 1
per cent in some cases) is, however, found in the products of combustion.
This may be due to acetylene which has escaped decomposition altogether,
or possibly to a recombination of H, and C in the rear of the explosion wave.
(3) No methane was formed.
(4) Possibly small amounts of CO, present.
(5) Carbon was deposited. In the cases of C2H, mixed with three
fourths of its own volume of oxygen a thick deposit of carbon was formed,
but when mixtures contained a larger proportion of oxygen much less carbon
was formed.
The primary phase in the flame may therefore be represented by
the reaction
C,H, + 0, = 2CO + H,
Any excess oxygen that may be present enters into a secondary phase,
namely, the formation of CO, and L2O.
At the high temperature attained in the flame there is possibility
of the dissociation of hydrogen into atoms, the dissociation of oxygen
into atoms, and the dissociation of carbon monoxide into carbon and
oxygen. The dissociation of hydrogen into atoms has been discussed.
There being no evidence available as to the dissociation of oxygen into
atoms, such dissociation is assumed not to occur. An equilibrium
equation for the reaction C + 20, = CO can be obtained by a com
bination of the equilibrium equations for the reactions CO + M0,
= CO2 and C + CO, = 2CO. From an equilibrium equation so estab
lished it has been determined that if carbon monoxide alone were
heated up to 8000 deg. F. (abs.) no dissociation whatever would
occur.
* Jour. Chem. Soc., V. 71, pp. 2641, 1897.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 89
I/Zas/d 0o9 /4/o0 Jo // ad // saj w u abnxP
%Sq % ^?L/f^7^/2900 deg. F. (abs.)
The following table gives a comparison of mean specific heats from
0 deg. to t deg. C. obtained from the three formulas:
t= 0 500 1000 1500 2000 2500 3000
Eq. (C) 8.922 10.249 11.253 11.935 12.382 12.727 13.019
Eq. (B) 8.750 10.239 11.260 11.931 12.370 12.696 13.030
Pier 8.790 10.215 11.240 11.940 12.390 12.665 12.840
It ap
equat
\t
pears that
ion (B).
equations (C) give results practically identical with
(3^ 
//I  
I A Swann
+ Regnau/t
o Ho/born ancd Aust1.o
1 00 Pier
/ • Bjerrum
_7 /11
P0
800 io00 10u euu0
Temm era/ure, /'C.
19. SPECIFIC HEAT OF CARBON DIOXIDE
"/"/ C04'L '
c Ou 3Wuu
i
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 99
In Fig. 19 the curve represents values derived from equations (C),
the points show the experimental results obtained by various ob
servers. Except for Swann's point at 0 deg., the agreement is satis
factory.
5. Water Vapor.For the high temperature range the experi
ments of Pier and Bjerrum are accepted. The values of y, (mean)
for the range 18 deg. to t deg. C. are plotted in Fig. 20. Again the
problem is to pass a satisfactory curve through these points and at
the same time make the curve represent fairly the available experi
mental data at low temperatures.
Holborn and Henning's equation when applied to a range 18 deg.
to t deg. C. gives the curve shown in the figure. This curve might be
accepted as conclusive if it were not for the experiments of Knoblauch
and Jakob. These had reference to superheated steam at various
pressures, and, as is well known, gave different curves for different
pressures. Since the water vapor in a gas mixture has usually a low
pressure, it is sufficient to take a single curve for a pressure of 1 or
2 lb. per sq. in. The Knoblauch and Jakob values reduced to mean
specific heat from 18 deg. to t deg. C. give the curve shown in the
figure. The two curves show a decidedly different trend; the HH
curve prolonged could be made to pass through the Bjerrum points,
but the KJ curve if prolonged would pass above them.
The curve chosen is a compromise between these two curves at
the lowtemperature end, and is made to give the best possible agree
/ 20 
/0
i ^ i tob/ch O__nd
Joob0 Bý/ //77
80 00 00 IZ00 /600 Z000 2400 00 9zX
Tepera1ture. ItC m1nus 1/8C
FIa. 20. SPECIF0IC hEATR OF WATER VAPOR
ILLINOIS ENGINEERING EXPERIMENT STATION
ment consistent with a second degree equation with the Bjerrum
points. The equation for the instantaneous specific heat per mol at
constant pressure deduced in this way is
y, = 8.33  0.276  103T + 0.423 106T2,
in which T denotes absolute temperature on the Fahrenheit scale.
6. Comparison with Pye's Values of Specific Heat.In Pye's
critical discussion of specific heats* is given a table of the specific
heats at constant volume. The values were obtained by plotting
energytemperature curves, taking as a basis the figures given by
Bjerrum and Swann. The table is here reproduced with the addition
of the values calculated from the preceding equations, which are given
in parentheses.
In the intermediate range, 1000 deg. to 2500 deg., the two sets
of values agree closely, the difference being less than one per cent ex
cept in one or two cases. For the low range 100 deg. to 500 deg. the
discrepancy is greater, and Pye's values are probably the more accu
rate. On the other hand, at 3000 deg. the values deduced from the
equations are probably nearer the truth than these given by Pye.
TABLE 14
MEAN SPECIFIC HEAT PER MOL BETWEEN 100 DEG. C. AND t DEG. C.
100 deg. C. up to
Gas
500 deg. 1000 deg. 1500 deg. 2000 deg. 2500 deg. 3000 deg.
Nitrogen............ 5.17 5.28 5.50 5.75 6.00 6.30
(5.08) (5.24) (5.46) (5.74) (6.09) (6.51)
Water Vapor......... 6.25 6.94 7.64 8.42 9.71 11.20
(6.53) (6.96) (7.61) (8.50) (9.61) (10.96)
Carbon Dioxide...... 8.25 9.55 10.07 10.50 10.87 10.95
(8.52) (9.50) (10.14) (10.56) (10.89) (11.17)
The probable accuracy of the values of y, deduced from the ex
plosion experiments is estimated by Pye.
For nitrogen the error may be within ± 1 per cent up to 2000 deg.
C. and + 2 per cent up to 3000 deg. C.
* Glazebrook's Dictionary of Applied Physics, V. 1, p. 418.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 101
For water vapor there may be an error of ± 5 per cent up to
3000 deg. C. Up to 2000 deg. C. it seems likely that the possible error
is not more than ± 2 per cent or ± 3 per cent.
For carbon dioxide, Bjerrum's values up to 2700 deg. C. may be
taken as correct to ± 3 per cent or ± 4 per cent.
The preceding equations, based largely on the experimental re
sults given by Bjerrum and Pier, may be regarded as having the
degree of accuracy indicated by the statement just quoted.
7. Specific Heats of Various Hydrocarbons.The experimental
evidence on the specific heats of the hydrocarbon fuel constituents,
such as CH,, CH4, C1,H, etc., is not sufficient for the deduction of
equations having any claim to extreme accuracy. However, the
amount of one of these constituents in a mixture of fuel and air is
usually small, so that a considerable error in the specific heat of a
single constituent may not introduce a perceptible error in the specific
heat of the whole mixture.
From such experimental data as are available the following con
clusions may be drawn:
(a) The specific heat may be taken as a firstdegree func
tion of the temperature.
(b) The rate of change of specific heat with temperature
for these hydrocarbon fuel constituents is very large compared
with the rate for nitrogen, CO2, and HO.
Methane (CH4)
Heuse's experiments at low temperatures give the following
values:
t deg. C. 80 55 30 5 15
*y 8.08 8.08 8.14 8.42 8.50
The linear equation
yp = 6.03 + 0.005 T (T = abs. temp. F.)
represents these values fairly well, and gives for the mean specific
heat between 18 deg. and 208 deg. C. the value 0.504, which agrees with
that determined by Regnault and Lussana.
The latest investigation, that of Dixon, Campbell, and Parker,
furnishes the equation
y, = 3.459 + 0.01056 T.
The two equations give about the same result at temperatures around
 30 deg. F.; but at higher temperatures the latter equation gives
larger values of y, because the rate of increase is more than double the
rate given by the first equation.
ILLINOIS ENGINEERING EXPERIMENT STATION
The choice between the two equations is based on the equilibrium
of the reaction C + 2H, = CH,. It is found that the equilibrium
equation obtained from the use of Dixon, Campbell, and Parker's
specific heat equation represents the experimental data much better
than the equilibrium equation deduced from the first equation for
specific heat. Hence for methane the expressions for specific heat are
7, = 1.473 + 0.01056 T
y, = 3.459 + 0.01056 T
Ethylene (C2H4)
Regnault and Lussana give the value 11.32 as the mean specific
heat y, between 10 deg. and 202 deg. C. Heuse has also determined
instantaneous values of yp at low temperatures. The equation
yp = 6.67  0.0068 T (T = abs. temp. F.)
gives 11.31 compared with 11.32 obtained by Regnault; and it re
presents quite accurately the experiments of Heuse, as shown by the
following comparison:
t deg. C. 18
7p (Heuse) ............. 10.2
7 (equation) ........... 10.2
Ethane (CzH6)
The experiments of Heuse at low temperatures are supplemented
by the experiments of Dixon, Campbell, and Parker at higher tem
peratures. The equation
y, = 7.10  0.0086 T (T = abs. temp. F.)
gives a fair compromise between the two sets of experiments.
Acetylene (C2H2)
Two values of the instantaneous specific heat are given by
Heuse, namely,
t deg. C. 71 18
Yp 9.13 10.43
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 103
A straight line through the two points giving these values has the
equation
yp = 6.19  0.0081 T (T = abs. temp. F.)
Benzene Vapor (C6H6)
For benzene vapor the equation
yp = 4.00  0.0318 T (T = abs. temp. F.)
represents accurately the available experiments, as shown by the
following comparison:
Investi
eat
Calculated
Wiedemann ....... 34115 23.337 (mean) 23.34
Wiedemann ....... 35180 25.913 " 25.91
Regnault ......... 116218 29.270 " 29.27
Thiabaut ......... 350 38.947 (inst.) 38.95
Gasoline and Kerosene Vapor
Wilson and Barnard give the equation
c, = 0.5 + 0.0006 t
for the instantaneous specific heat of gasoline and kerosene (liquid or
vapor) at constant pressure. Units are calories per gram and degrees
centigrade.
Wilson and Barnard represent gasoline by the formula CH,1 and
kerosene by C12H,,. Using these formulas and transferring to degrees
Fahrenheit absolute, the molecular specific heats are given by the
equations
Gasoline,
yp = 38.327  0.038 T
y, = 36.342 + 0.038 T
Kerosene,
y, = 57.154  0.05 667 T
y, = 55.169 + 0.05 667 T
8. Specific Heat of Amorphous Carbon.The reliable experi
mental data are as follows:
(a) Weber gives the following values for the mean specific heat
of one gram of wood charcoal:
0°24* C. 0.1653
099° C. 0.1933
0°  224° C. 0.2385
ILLINOIS ENGINEERING EXPERIMENT STATION
(b) Kunz gives the following values for the mean specific heat
of beechwood charcoal from 0 deg. to t deg. C.:
t deg. C. Specific Heat
435 0.243
561 0.290
728 0.328
925 0.358
1059 0.362
1197 0.378
1297 0.381
(c) Nernst has shown that the specific heat of carbon at ab
solute zero is zero.
From the preceding data an equation has been deduced by the
method of least squares for the instantaneous molecular specific heat of
amorphous carbon. With T in degrees centigrade absolute it is
y = 8.16 * 10'T  2.946  106 T2
(I
Yo
h
c1
Z
i
^,
<;
l
^Q
<
48
+
i .2 
40
'36 
/+
5
/0
. /Web
*e4   + Kunz
 !           
/0,
,0 _         /    
600 00 /000 /zOO
Temperoa/re i/ d/eg. C abs.
21. SPECIFIC 1EAT OF AMORPHOUS CARBON
L8v
·n ·r~n
4U /0/(
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 105
A comparison between the experimental values and the values
calculated from the above equation is given in the following table.
The data are shown graphically in Fig. 21.
SPECIFIC HEAT OF AMORPHOUS CARBON
No.
1
2
3
4
5
6
7
8
9
10
Temp. Range
deg. C. (abs.)
273 297
273 372
273 497
273 708
273 834
2731001
2731198
2731332
2731470
2731570
Mean Molecular Specific Heat
Calculated
2.086
2.322
2.693
3.247
3.520
3.872
4.198
4.376
4.522
4.605
Observed
1.984
2.322
2.862
2.916
3.480
3.936
4.296
4.344
4.536
4.572
Investigator
Weber
Weber
Weber
Kunz
Kunz
Kunz
Kunz
Kunz
Kunz
Kunz
The equation satisfied Nernst's value at absolute zero. The agree
ment between the values given by the equation and the experimental
values is satisfactory over the entire range of temperature.
The equation attains a maximum value at T = 1384 deg. C. (abs.)
or 2492 deg. F. (abs.).
Changing to Fahrenheit absolute, the equation becomes for the
instantaneous specific heat of amorphous carbon per mol,
y = 4.533  103 T  0.9092  106 T2
Attention may be drawn to the fact that while the specific heat
equations here developed are not entirely rational, they are service
able and will give correct results as long as their use is limited to the
range of temperatures defined by the experimental data. The un
certainty of extrapolated values is, of course, apparent.
The specific heat equations given in the foregoing discussion are
collected in Table 15.
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 15
SPECIFIC HEAT EQUATIONS
7p =Instantaneous Molecular Specific Heat at Constant Pressure
7= Instantaneous Molecular Specific Heat at Constant Volume
T =degree F. (abs.) y =5 ,+1 985
No.i Gas
Specific Heat Equation
I CO, 02, N2 7=6.93 +0.120010^T2
S.=4.945+0.1200.10T'2
2 H2 7p=6.00 +0.6667103T
7 = 4.015+0.6667103T
3 CO2 y =7.15 +3.90103T0.60.10T2
(T<2900) 7 =5.165+3.9010'T0.60106T3
4 CO, p =12.196+0.42103T
(T >2900) 7, =10.211+0.42103T
5 CO2 7 ,= 6.4587+5.066810 T1.2480106T2+0.1086 109T'
(Entire Range) Y, =4.5637+5.0668 103T1.2480106T2+0.108610'T'
6 H20 y,=8.33 0.27610 T+0.42310«T2
S= 6.3450.276103T +0.423.106T'
7 CH4 p =3.459+10.56103T
y =1.474+10.5610l T
8 CH112 ,=6.19 +8.1010aT
S,=4.205+8.10.103T
9 C2H4 Y =6.67 +6.8010BT
7,=4.685+6.80103T
10: C2H yp7.10 +8.6010aT
y7.=5.115+8.60103T
11 CeH yp=4.00 +31.8010sT
y =2.015+31.80.10l T
12 CsHis (Gasoline) jp =38.327+38.00103T
y, =36.342+38.00103T
13 C12 12 (Kerosene) , =57.154+56.67 103T
7 =55.169+56.6710ST
14 Amorphous = +4.533103T0.9092.10 T2
Carbon
NOTESee "References on Specific Heat," page 155.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 107
APPENDIX II
HEATS OF COMBUSTION
1. Hydrogen (H,).Leaving out of consideration investiga
tions previous to 1849, the following is a brief outline of the methods
used by the various investigators since that date for the determination
of the heat of combustion of hydrogen:
Andrews, in 1848, first used the bomb calorimeter. Hydrogen
and oxygen collected over water were introduced into the bomb in
the theoretical proportions for combustion. The gas mixture, under a
total pressure of one atmosphere, was ignited by an electric spark,
the heat generated being absorbed by water surrounding the bomb.
Andrews found as the average of four experiments the higher heat of
combustion of one standard liter (0 deg. C., 760 mm.) of dry hydrogen
at 20 deg. C. and constant volume to be 3036 calories (20degree
calorie).
Favre and Silbermann in 1852 burned hydrogen with oxygen in
a closed vessel at a constant pressure of 16 centimeters of water above
atmospheric. This burning was accomplished by leading two metal
tubes, one for hydrogen and one for oxygen, into a small metal
chamber. By regulating the flow of the gases a steady flame was
maintained. The heat of combustion was absorbed by water which
completely surrounded the combustion chamber. Since the flame was
completely enclosed, the water formed by the combustion could not
escape and so was condensed. At the beginning and end of each
experiment the combustion chamber was weighed, the increase in
weight being that due to the water formed. Correction was made for
the. noncondensed water vapor within the combustion chamber by
weighing the chamber filled with wet and then with dry gases, the
difference being the weight of noncondensed vapor. This weight of
vapor was multiplied by the latent heat of steam at 18 deg. C. and
the result was added to the experimental result. As the average of
six experiments the higher heat of combustion of 1 mol of hydrogen
at 18 deg. C. and constant pressure was found to be 68 924 calories
(20degree calorie).
J. Thomson in 1873, using the same method as Favre and Silber
mann with minor changes in the apparatus, found the higher heat of
combustion of one mol of hydrogen at 18 deg. C. and at constant
pressure to be 68 357 calories (20degree calorie). This is the average
of three experiments.
ILLINOIS ENGINEERING EXPERIMENT STATION
In 1877 Schiiller and Wartha used a Bunsen ice calorimeter
wherein the heat of combustion was determined by measuring the
amount of ice melted at 0 deg. C. The amount of ice melted was
measured by the contraction in volume of a mixture of ice and water.
The product of the volume of ice melted, the specific weight, and the la
tent heat of fusion of ice gave the heat absorbed by the ice. The closed
end of a test tube was projected into the mixture of ice and water.
Inside of the test tube was placed a previously weighed glass combus
tion pipette. This combustion pipette consisted simply of a small
glass bulb into which were led two glass tubes, one for hydrogen and
one for oxygen.
Burning occurred at constant pressure inside this bulb. Since
there was no outlet the water formed condensed and remained in the
bulb. At the end of an experiment the glass pipette was again weighed,
the increase in weight being that due to the water formed. The ex
periments lasted 3 to 4 hours so that the small amount of vapor left
uncondensed in the pipette introduced a negligible error. As an
average of five experiments the higher heat of combustion of 1 mol of
hydrogen at 0 deg. C. and at constant pressure was found to be
68 250 calories (mean calorie 0 deg. to 100 deg. C.).
In 1881 Than used the Bunsen ice calorimeter with a constant
volume combustion pipette. In order to get an appreciable volume
of gases in the pipette the calorimeter was quite large. Hydrogen
and oxygen were introduced into the pipette under a total pressure
of one atmosphere (barometer 760 mm.) and the mixture was exploded
by an electric spark. The gases before combustion were saturated with
water vapor so that all water vapor formed was condensed. The
average result obtained from five experiments for the higher heat of
combustion of 1 mol of hydrogen burned at 0 deg. C. and at con
stant volume was 67 644 calories (15degree calorie).
Berthelot in 1883 revived Andrew's bomb calorimeter and much
improved it. The bomb was first filled with dry hydrogen under a
pressure of one atmosphere. Wet oxygen was next introduced into
the bomb from a cylinder of compressed oxygen until the total gas
pressure in the bomb was about 1.7 atmospheres. The excess of oxygen
was used because the compressed oxygen contained a small percentage
of nitrogen. The mixture was exploded by an electric spark. Since
all the water vapor formed was condensed no correction was needed.
Berthelot gives as the higher heat of combustion of 1 mol of hydrogen
at constant volume and at 10 deg. C., 68 000 calories (10degree
calorie).
In 1903 Mixter, at Yale University, made further improvements
on the bomb calorimeter. Dry hydrogen was first introduced into the
bomb at atmospheric pressure (barometer 14.743 lb. per sq. in.). The
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 109
weight of hydrogen present was calculated from its known pressure,
temperature, and volume. Dry .oxygen was next introduced into the
bomb until the total pressure was in excess of 1.5 atmospheres. The
average of 14 experiments by Mixter gives for the higher heat of
combustion of 1 mol of hydrogen at 18 deg. C. and at constant
volume, omitting the correction for noncondensed water vapor, which
will be considered later, 66 835 calories (20degree calorie).
The latest work done on the heat of combustion of hydrogen is
that of G. Rumelin in 1907. He used a bomb calorimeter of more
elaborate design than those previously used. Dry hydrogen and
oxygen in the theoretical proportions for combustion were introduced
into the bomb under a total pressure of one atmosphere (barometer
14.32. lb. per sq. in.). As an average of six experiments, omitting the
correction for noncondensed vapor left in the bomb, the observed
result for the higher heat of combustion of 1 mol of hydrogen at 18
deg. C. and at constant volume was 66 940 calories (20degree calorie).
Certain corrections must be applied to the experimental results
before a proper comparison of the values can be made. In the first
place, a common heat unit must be taken. For this, the mean B. t. u.
is chosen. The experimental values are thus multiplied by 1.8 and
by one of the following correction factors, which have been deduced
from Callendar's equation for the specific heat of water. Secondly,
Temperature
Deg. C. Deg. F.
10 50
15 59
20 68
Correction
Factor
1.00 150
0.99 962
0.99 842
Logarithm of
Correction Factor
0.0 006 499
1.9 998 352
1.9 993 100
the experimental values must be corrected by the subtraction of the
heat obtained by the condensation of the water. The socalled higher
heats of combustion, which include the heat of condensation, are de
pendent on so many accidental circumstances that a comparison of
them is not possible. For a fixed temperature, on the other hand, the
lower heat of combustion is an invariant; hence comparison of such
values are valid.
The following notation is used:
H', = higher heat of combustion at constant pressure.
H, =.lower heat of combustion at constant pressure.
ILLINOIS ENGINEERING EXPERIMENT STATION
H', = higher heat of combustion at constant volume.
H, = lower heat of combustion at constant volume.
The difference H',  H, is the heat given up by the condensation of
the water when the experiment is conducted at constant pressure.
The method of calculation employed in reducing from the high
to the low heat of combustion may best be illustrated by two examples,
one for combustion at constant volume, the other for combustion at
constant pressure.
(a) Rumelin's Experiment
Take dry hydrogen and oxygen under a pressure of 14.32 lb. per
sq. in. and at a temperature of 64 deg. F. Consider the reaction equa
tion
H + 02, = HO2
The volume occupied by 1 mol or 18 lb. of H0O vapor after combustion
is the same as that occupied by 1 mols of mixture before combustion.
At 62 deg. F. and a pressure of 14.7 lb. the volume of 1 mol is 380.6
cu. ft. Hence the volume of 11 mols at 64 deg. and a pressure of 14.32
lb. is
524 14.7
380.6 X 1.5 X X = 588.3 cu. ft.
522 14.32
This is the volume of 18 lb. of water vapor; hence the volume per
pound is 588.3 18 = 32.68 cu. ft. Reference to the steam table
shows that at a temperature of 202 deg. F. one pound of saturated
steam has nearly this volume; hence in cooling the products of com
bustion the water vapor will start to condense at about 202 deg. and
will continue condensing until the final temperature 64 deg. is reached.
At 64 deg. the volume of 1 lb. of saturated steam is 1055 cu. ft.;
hence the quality of the steam in the final state is 32.68 1055
= 0.031. Since the water vapor remains at constant volume in con
densing, the heat given up is equal to the decrease in energy. The
energy of 1 lb. of saturated steam at 202 deg. F. is 1075.9 B. t. u.;
that of the mixture at 64 deg. F. having a quality 0.031 is 32.1
+ 0.031 X 998.4 = 63 B. t. u. The heat given up per pound is 1075.9
 63 = 1012.9 B. t. u. and per mol, 1012.9 X 18 = 18 232 B. t. u. If
the water vapor had remained in the gaseous state during the cooling
from 202 deg. F. to 64 deg. F. the decrease of energy would have been
662
yedT
524
in which
., = 6.345  0.276  103 T + 0.423  106T2
the specific heat per mol at constant volume. The result is 874
B. t. u. The difference 18 232  874 = 17 358 B. t. u. is the amount
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 111
by which the higher heat of combustion exceeds the lower, and is
therefore the amount to be subtracted from H', to get H,. The ob
served H', reduced to mean B. t. u. per mol is
H'v = 66 940 X 1.8 X 0.99 842 = 120 300 B. t. u. per mol.
Hence
H, = 120 300 17 358 = 102 942 B. t. u.
and
H, = H, T = 102 942 + 524 = 103 466 B. t. u. per mol.
(b) Thomsen's Experiments
The temperature of the combustion was 64 deg. F. and the pres
sure 14.7 lb. per sq. in. Since HO2 was the sole product of combus
tion, the water vapor at a pressure of 14.7 lb. began condensation at
212 deg. F., and at the final temperature, 64 deg. F., was entirely
condensed, the volume of water vapor remaining in the small combus
tion pipette at the end of an experiment being negligible. The heat
given up during condensation per pound of water was i, i,. For
saturated steam at 212 deg. F. i, = 1151.7; for water at 64 deg. F.
i2 32.1: hence per mol the heat due to condensation was 18 (1151.7
 32.1) =20 153 B. t. u.
The heat that would have been given up in cooling at constant
pressure from 212 deg. F. to 64 deg. F., if the water vapor had re
mained in the gaseous state, is
f 672
Sy,dT
24
which, when the proper value of y, is inserted, gives the result 1233
B. t. u. The difference H',,  H, is therefore 20 153  1233 = 18 920
B. t. u. The observed value of H', properly reduced is
H', = 68 357 X 1.8 X 0.99 842 = 122 849 B. t. u. per mol.
Hence
H, = 122 849  18 920 = 103 929 B. t. u. per mol.
The results obtained by the various investigators when reduced
to the common basis of lower heat of combustion at constant pressure
in mean B. t. u. per mol are as follows:
Year Investigator H,
1848 Andrews 104 660
1852 Favre and Silbermann 104 950
1873 Thomsen 103 830
1877 Schiiller and Wartha 103 690
1881 Than 103 980
1883 Berthelot 105 060
1903 Mixter 103 110
1907 Rumelin 103 470
ILLINOIS ENGINEERING EXPERIMENT STATION
It is obvious that in establishing the probable value of Hp different
weights should be assigned to the various experiments. Any system of
weighting is a matter of individual judgment and therefore subject
to criticism. The earlier experiments of Andrews and of Favre and
Silbermann gave values that are obviously too large in the light of
the later and more accurate experiments. These values, and also the
still higher value of Berthelot, are therefore rejected. Thomsen's
value is given a weight of 2 because of the general high accuracy of
his experiments, and because his result is the average of seven determi
nations in which a total of 18 grams of water was formed, a much
larger quantity than in the experiments of any of the other investiga
tors. Because of the comparatively recent date, the values of Mixter
and Rumelin are given the weight 3. Mixter's value is the mean of
fourteen experiments all of which are within 1.3 per cent of the aver
age, and Rumelin's value is the mean of seven experiments all of
which are within 0.7 per cent of the average. With this system of
weighting, the probable value of H, is found to be 103 530 B. t. u. per
mol at 62 deg. F.
The heat of combustion H, at constant volume is less than Hp by
the product ½RT = 0.993 X 522. Hence
H, at 62 deg. F. = 103 010 B. t. u. per mol.
The constant Ho, which is required in the equilibrium equations, is
readily derived from the value of H, or the value of H,. Thus
Ho =Hv+ Ui0 oUH  Uo0
For the standard temperature 62 deg. F. (T= 522) the values of u
per mol are as follows:
u ,o=3295, uH =2187, uo,=2587
Hence
He = 103 010 + 3295  2187  1294 = 102 824,
or with sufficient accuracy
Ho = 102 820 B. t. u. per mol
2. Carbon Monoxide (CO).The experimental results available
for the heat of combustion of carbon monoxide are given in the follow
ing table:
Temp.
Year Investigator deg. C. Value
1848 Andrews 15 3057.0 cal/liter at const. vol.
1852 Favre and Silbermann 18 2402.7 cal/gram at const. press.
1873 Thomsen 18 67 960 cal/mol at const. press.
1881 Berthelot 10 68 200 cal/mol at const. press.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 113
Andrews and Berthelot used the bomb calorimeter. Favre and
Silbermann and Thomsen used the same apparatus as they did for
hydrogen, except that the products of combustion were led out of the
combustion chamber through a coil of small pipe of considerable length
which was immersed in the water of the calorimeter, thus insuring that
the products of combustion were brought back to the initial tem
perature.
Transferring the data given in the preceding table into mean
B. t. u. per mol at constant pressure, using Callendar's specific heat
ratios, and neglecting the correction to 62 deg. F. the following values
for H, at 62 deg. F. are obtained:
Hp
Andrews ...................................... 123 800
Favre and Silbermann ..........................120 900
Thomsen ...................................... 122 130
Berthelot ........................ ........... 122 920
Andrews' value cannot be considered because of the imperfection
of his method. Hie took only one reading of the water temperature in
his calorimeter after combustion, and that one just thirty seconds
after combustion occurred. The apparatus had to be rotated in order
to keep the water at a uniform temperature. To accomplish this rota
tion it was necessary to remove the thermometer and then, to get the
temperature of the water, the rotation was stopped and the thermom
eter reinserted. Evidently such manipulation was subject to errors.
The carbon monoxide used by Favre and Silbermann in their
experiments contained about 3 per cent hydrogen by weight. The
correction for the heat of combustion of this hydrogen content
amounted to about 30 per cent of the heat resulting from the combus
tion as observed in any one determination. The possibility of error
in this correction is very great because of the methods of gas analysis
in use at that time, and also because all the errors of their determina
tion of the heat of combustion of hydrogen were automatically in
troduced.
Of the two remaining values, that of Thomsen's is chosen rather
than Berthelot's for the following reasons: First, for any given de
termination the volume of gas used by Thomsen was about six times
that used by Berthelot, this tending to reduce Thomsen's error;
second, Berthelot's result is the average of five experiments where a
total of about 1.3 liters of carbon monoxide were burned, while
Thomsen's result is the average of ten closely accordant experiments
wherein a total of about 16 liters of carbon monoxide were burned.
The first six of the experiments were performed in one calorimeter
while the last four were performed in another calorimeter. The aver
age result of the first group of experiments is exactly equal to the
average of the last group.
ILLINOIS ENGINEERING EXPERIMENT STATION
The heat of combustion of 1 mol of carbon monoxide at constant
pressure at 62 deg. F. in terms of the mean B. t. u. is therefore taken
as
H,  122 130
3. Methane (CH,).The experimental results available for the
heat of combustion of methane are:
Temp. Higher Heat of Combustion
Year Investigator deg. C. cal. per mol
1848 Andrews 15 209 728 constant vol.
1852 Favre and Silbermann 18 209 000 constant press.
1880 Thomsen 20 213 630 constant press.
1881 Berthelot 18 212 400 constant vol.
The methane used by Andrews was obtained from a stagnant
pool, and contained a large percentage of nitrogen. The Andrews
value is therefore doubtful. It is also very probable that the Favre
and Silbermann value is low because of impurities in the gas used
by them.
Thomsen generated his methane from zinc methyl and hydro
chloric acid, and purified it by bubbling through cuprous chloride
solution. The tabular result is from Thomsen's latest work on
methane, and is the average of nine experiments which show a maxi
mum variation of 1.1 per cent. The calorimeter used was of the con
stant pressure type, the products of combustion being led out through
a long tube winding around the combustion chamber, as described be
fore. Correction was made for the noncondensed vapor in the prod
ucts of combustion, this correction being very small.
Berthelot's value is the average of four determinations made with
his bomb calorimeter. These four experiments show a variation of 1.6
per cent.
The results of Thomsen and Berthelot when reduced to the com
mon basis of lower heat of combustion at constant pressure are as
follows:
Hp
Thomsen ...................................... 345 820
Berthelot ...................................... 346 030
The mean, H, = 345 920 B. t. u. per mol at 62 deg. F. is taken as the
probable value. The value of Hv is the same.
4. Acetylene (C2H2).The following are the available experi
mental values of the heat of combustion of acetylene:
Temp. Higher Heat of Combustion
Year Investigator deg. C. cal. per mol
1880 Thomsen 19 310 050 const. press.
1881 Berthelot 18 314 900 const. vol.
1906 Mixter 20 311400 const. vol.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 115
When reduced to the lower heat of combustion at constant pres
sure in mean B. t. u., these values become:
Hp
Thomsen ...................................... 538 240
Berthelot ...................................... 548 500
M ixter ........................................542 680
If Thomsen's value is given the weight 2, Berthelot's 1, and Mixter's
3, the resulting value is
H, = 542 170 B. t. u. per mol at 62 deg. F.
For constant volume,
H, = 541 650 B. t. u. per mol.
5. Ethylene (CH,).Three sets of experiments give the follow
ing numbers for the higher heat of combustion:
Temp. Higher Heat of Combustion
Year Investigator deg. C. cal. per mol
1880 Thomsen 17.9 333 350 const. press.
1881 BerthelotMatignon 16.8 340 000 const. vol.
1901 Mixter 18.8 345 080 const. vol.
The reduction to the lower I, in mean B. t. u. gives the following
values:
Hp
Thomsn ...................................... 561150
Berthelot ...................................... 576 080
M ixter ........................................584 590
Again assigning the weight of 1 to Berthelot, 2 to Thomsen, and 3 to
Mixter, the resulting value of the heat of combustion of ethylene at
constant pressure in mean B. t. u. per mol is
H, = 575 370.
6. Ethane (CH,).The data are furnished by the experiments
of Berthelot and Thomsen.
Temp. Higher Heat of Combustion
Year Investigator deg. C. cal. per mol
1893 Berthelot 13 370 900 const. vol.
1905 Thomsen 18 370 440 const. press.
The reduced values are:
Hp
Berthelot ...................................... 614 305
Thomsen ..................................... 608 360
ILLINOIS ENGINEERING EXPERIMENT STATION
and the mean, giving Thomsen's result double the weight of Berthe
lot's, is
H, = 610 340 B. t. u. per mol.
7. Benzene Vapor (CH,).For the heat of combustion of
benzene vapor the experimental value of Stohman, Rodatz, and Herz
berger, which is 10 096 cal. per gram at 17 deg. C., is chosen as the best
available. It is the average of twelve experiments. In the method
used, a current of air was passed over a wad of cotton saturated with
benzene liquid and the resulting mixture of vapor and air was burned
in a constant pressure calorimeter. The products of combustion were
led through a long spiral tube and then through absorbers to remove
the moisture in the usual way.
If the oxygen and benzene in the benzeneair mixture were present
in the theoretical proportions for combustion, the reaction equation
would be CH, + 7.50, + 28.5N, = 6CO + 3H,0 + 28.5N,. The
partial pressure of the benzene vapor in the original mixture is 1/37
X 14.7 = 0.40 lb. per sq. in. According to Young, the saturation
pressure of benzene vapor at 17 deg. C is 65 mm. of mercury or 1.25
lb. per sq. in. Since the assumed partial pressure of the benzene
vapor in the initial mixture is only about onethird of the saturation
pressure of benzene at 17 deg. C., the assumed partial pressure can be
easily attained and is reasonable.
The partial pressure of the water vapor in the products of com
bustion, if not condensed, is
3
X 14.7 = 1.176 lb. per sq. in.
37.5
The saturation temperature at this pressure is 107 deg. F.; and
the condensation of water vapor between 107 deg. and 62 deg. gives
a difference of 57 070 B. t. u. per mol between the observed higher
heat of combustion and the lower heat of combustion. The observed
higher value in mean B. t. u. per mol is
H', = 78.05 X 10 096 X 1.8 X 0.99 842 = 1416 460
Hence
H, = 1 359 400 B. t. u. per mol.
Since in the combustion the molecular volume increases by j mol, the
heat of combustion at constant volume exceeds H, by RT = T
approx.; hence
H, = 1359 920 B. t. u. per mol.
8. Gasoline.The values for the higher heat of combustion of
liquid gasoline obtained from various handbooks range from 19 000
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 117
to 21 000 B. t. u. per pound. An average value of 20 000 B. t. u. per
pound is taken. This when reduced to the lower value gives
H, = 18 675 B. t. u. per lb. of liquid gasoline, taken as at
520 deg. F. (abs.).
Wilson and Barnard give the heat of vaporization of gasoline as 70
gram calories per gram, and the molecular formula as C8H,8. Adding
in the heat of vaporization, the molecular heat of combustion is
H,p 2 143 200 B. t. u. per mol of vapor taken as at 60 deg. F.
from which
H, = 2 146 840 B. t. u. per mol.
9. Kerosene.Here also an average value is obtained from those
quoted by the handbooks and is 21 500 B. t. u. per pound for the
higher heat of combustion. The lower heat of combustion is then
H, = 20 220 B. t. u. per lb. of liquid kerosene.
For kerosene, Wilson and Barnard give the value of 60 gram calories
per gram for the heat of vaporization and the molecular formula
C12H26. The molecular heat of combustion is
H, = 3 456 100 B. t. u. per mol of vapor, taken as at 60 deg. F.;
also
1H = 3 461 800 B. t. u. per mol.
10. Amorphous Carbon.The available experimental data on
the heat of combustion of amorphous carbon are contained in the
following:
Heat of Combustion
Year Investigator cal. per gram
1848 Andrews 7678
1852 Favre and Silbermann 8080
1883 Gottlieb 8033
1889 Berthelot 8137
Andrews states in his discussion of the value given above, which
is the average of eight determinations with highly purified wood char
coal in the bomb calorimeter, that in spite of the presence of excess
oxygen, carbon monoxide was found in the products of combustion.
This fact, of course, renders his value too low.
ILLINOIS ENGINEERING EXPERIMENT STATION
Favre and Silbermann also found carbon monoxide in the prod
ucts of combustion. After determining the amount of carbon monox
ide present in any one experiment, they added to the observed
result the heat of combustion of this given amount of carbon monoxide
so that their final results give the heat of combustion of carbon to
carbon dioxide. This correction amounted to only about 3 per cent,
so that errors introduced by using an incorrect value for the heat of
combustion of carbon monoxide were insignificant. Eighteen experi
ments in three series were run, using highly purified wood charcoal.
The values of the first series consisting of five experiments showed a
maximum variation of 89 calories in the values given for the heat of
combustion per gram. The average value from the first series is 8086
calories per gram. The next seven experiments, constituting the
second series, showed a maximum variation of 31 calories. The aver
age of the second series is 8081 calories per gram. In the last six
experiments wood charcoal purified in different ways was used in
different determinations to note if the method of purification had any
effect on the results. In this last series the maximum variation be
tween any two results was 19 calories, and the average of these six was
8080 calories per gram. This is the result quoted in the preceding table.
Gottlieb used a calorimeter very similar to the one used by Favre
and Silbermann. The carbon used by Gottlieb was prepared by heat
ing a fivegram ball of cotton in a loosely covered dish, slowly at first
and then more intensely after all the combustible gases had been
driven off. Later the cotton charcoal was transferred to a tightly cov
ered platinum dish and heated to about 950 deg. C. for some hours
and then cooled in a dessicator. This carbon absorbed moisture freely.
Upon analysis the sample showed 1.5 per cent moisture. With a slight
amount of moisture present in the sample as weighed, of course, the
final result calculated on the basis of this weight will be slightly low.
Therefore, Gottlieb's result, which is the average of six experiments,
having a maximum variation of 7 calories per gram, points to the
accuracy of the Favre and Silbermann values.
Berthelot's value was obtained by burning wood charcoal, very
carefully purified and dried. Oxygen under 25 atmospheres pressure
was used in the bomb calorimeter to insure complete combustion. The
value given above is the average of six experiments, which show a
maximum variation of 10 calories per gram.
Of the values quoted, the Favre and Silbermann value is chosen
in preference to that of Berthelot.
Reducing the Favre and Silbermann value, which is in terms of
the 20degree calorie, to mean B. t. u. per mol, the heat of combustion
of amorphous carbon at 62 deg. F. is
H = 174 250 B. t. u. per mol.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 119
TABLE 16
HEATS OF COMBUSTION (LOW) IN MEAN B. T. U.
At Const. Pressure
and 62 deg. F.
Reaction
per per
mol lb.
H2+02 =H20 103 530 51 354
CO +102 =C02 122 130 4 362
H2+C02 =H20+CO
per.
cu. ft.
272
321
18 600
C (amorph.) +CO2 =2CO 70 010
C (amorph.) +0 =CO2 174 250
C (amorph.) +2H2 =CH4 35 390
CHI4+202 =CO +2HO0 345 920
C2H2 +2 02 =2C02 +H20 542 170
C2Ha +302 =2CO2 +2H20 575 370
C2HE +302 =2C02 +3H20
CI6H+7i02 =6CO2+3H20
CsHia (Gasoline Vapor) +
1 902=RCOl,+. OQTTn
610 340
1 359 400
2 143 200
14 521
21 577
20 840
20 525
20 312
17 418
18 800
C12H26 (Kerosene Vapor) +
18102=12C02+13H20 3456 100 20330
At Const. Volume
and 62 deg F.
103010 51 096
121 610 4 343
18 600
68 970
1
171 250
34350
34 350
909
1425
3572
56.31
9081
345 920
541 650
575 370
610 850
1 359 920
2 146 840
14 521
21 577
20 820
20 525
20 329
17 424 3573
18 830 5641
3461 800 20 360 9096
per
cu. ft.
271
320
At abs.
102 820
120 930
18 110
67 570
174 290
31 600
348 330
541 580
577 220
614 240
1 364 200
2 145 610
3 458 680
909
1423
1605
*In determining these values, the specific heat equation for C
NOTE.See "References on Heats of Combustion," page 156.
I
12102 =C eO 0 24 0
·· · · ·· · ·· ·~~· ·
I
ILLINOIS ENGINEERING EXPERIMENT STATION
APPENDIX III
CHEMICAL EQUILIBRIUM
1. Statement of the Problem.As shown in Chapter III, Section
17, the general form of the expression that gives the equilibrium con
stant Kp as a function of the temperature T is
H
Rlog, K,  a'loge T  er"T  ."'T2 + C
In this equation, the coefficients a', a", a'" are known from the specific
heats of the constituents involved, and Ho, a heat of combustion, is
also known. Hence, save for the constant C, the second member may
be calculated for any temperature T. In the first member the con
stant Kp is a function of the partial pressures when the system is in
equilibrium, which means that Kp is determined by the composition of
the mixture in the equilibrium state. The first member is therefore
determined by experiments on the chemical composition at various
temperatures, the second member, except for the constant C, is de
termined from known thermal data, and by subtraction the constant
C is thus found.
In this appendix the experimental evidence relating to the equilib
rium of various reactions is reviewed, the constants are determined,
and the agreement between the experiments and the theory is shown.
2. Experimental Methods.A brief outline of the methods used
in determining the equilibrium composition resulting from gaseous
reactions at high temperature is desirable. For detailed descriptions
and discussions of such experimental methods, see Haber's "Thermo
dynamics of Technical Gas Reactions" and Nernst's "Theoretical
Chemistry.''
I. Streaming Method
The gases involved in the reaction are passed through a tube, a
section of which is heated to the desired temperature and the follow
ing section kept at a low temperature. The gases are assumed to
attain equilibrium in the hot portion of the tube and to be cooled so
rapidly in the cold portion of the tube that the reaction immediately
stops. The equilibrium composition of the gas at the high temperature
thus exists in the cooled gas, which can be easily analysed.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 121
II. SemiPermeable Membrane Method
A vessel which is permeable to one constituent only of the reac
tion is evacuated, and the reacting mixture of gases at the desired
temperature is caused to circulate around the outside of the vessel. The
partial pressure of the one constituent to which the vessel is permeable
will soon exist inside the vessel and can be measured with a man
ometer. This method is applicable only to a study of the dissociation
of HO since semipermeable substances (palladium, platinum, and
iridium) are known for hydrogen only.
III. Maximum Explosion Pressure Method
The original mixture of gases is exploded in a closed vessel and
the maximum pressure of explosion measured. Since the maximum
pressure is dependent on the heat of reaction, and the heat of reac
tion on the equilibrium composition, the equilibrium composition may
be calculated from the maximum pressure. Also in those reactions
which involve a change in the number of mols, the maximum pressure
is directly influenced by the equilibrium composition.
IV. Method of the Heated Catalyst
(a) If a catalyst such as a platinum wire is heated electrically
in an atmosphere of gas the equilibrium composition of the gas, at
the temperature of the wire, will exist in the gas immediately sur
rounding the wire. Due to the circulation of the gas set up by the
heated wire the gas in contact with the wire will be swept into the
cooler regions and the reaction will thereby be "frozen." The process
is allowed to continue until the composition of the whole gas volume
becomes constant. This condition is determined by analysing samples
of gas from time to time. Temperatures are determined by the change
in the electrical resistance of the wire.
(b) A variation of this method is to heat a vessel containing the
catalyst to the desired temperature and then to lead the gases through
the vessel or enclose them in the vessel until equilibrium is established.
Samples are drawn out from time to time and analysed.
V. Iridium Dust Method
Iridium dust is produced by heating strips of iridium electrically
in various gases. The quantity of dust produced is dependent on the
nature of the gas and the temperature to which the iridium is heated.
This method is especially applicable to the measurement of the disso
ciation of carbon dioxide. Nitrogen and pure carbon monoxide pro
duce no appreciable amount of dust while oxygen produces large
quantities of dust. The amount of dust produced by carbon dioxide
ILLINOIS ENGINEERING EXPERIMENT STATION
at a given temperature of the iridium and at atmospheric pressure is
assumed to be due to the oxygen liberated by the dissociation of the
carbon dioxide. A mixture of nitrogen and oxygen is found which
will produce the same amount of dust as the carbon dioxide under
the same conditions. Assuming that the oxygen content of the two
gases is the same, the equilibrium composition of the carbon dioxide
reaction is then known.
VI. Measurement of Equilibrium in the Bunsen Flame
The inner and outer cones of the Bunsen flame are separated by
fitting a glass tube as an extension on the end of the burner, the glass
tube being of somewhat larger diameter than that of the burner. A
stopper is made to fit tightly in the annular space between the burner
and the glass tube. The inner cone then burns on the end of the
regular Bunsen burner tube which is now inside the glass tube while
the outer cone burns on the end of the glass tube. Samples of gas are
withdrawn from the space between the two cones and are assumed to
be in equilibrium. Temperatures are measured with thermocouples.
VII. Direct Determination of Equilibrium
(a) When one of the constituents in the reaction is a solid, such
as carbon, it can be placed in a porcelain tube and the whole heated
in an electric furnace. The gas constituents are then passed through
the porcelain tube at a rate sufficiently slow to give time for equilibrium
to be established. The products are then analysed.
(b) Carbon in the form of rods is heated by an electric current
in an atmosphere of the gas. Reaction occurs at the surface of the
rod and is "frozen" when the reacting gases diffuse into the colder
regions.
In either (a) or (b) a catalytic agent, such as platinum, nickel,
or cobalt, may or may not be used.
(c) A study of the watergas equilibrium is made by passing
water vapor over glowing coals and analysing the resulting gases.
VIII. Equilibrium from Density Measurement
The dissociation of carbon dioxide can be measured by electrically
heating a platinum bulb filled with carbon dioxide to a desired tem
perature and then dropping into the bulb a small piece of aluminum.
Changes of volume are measured by the movement of a short thread
of mercury in a horizontal capillary outlet tube fitted to the platinum
bulb. The following reaction occurs:
2A1 + 3CO A1,0, + 3CO
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 123
According to this reaction no change in volume will occur if the
carbon dioxide is undissociated, and if the assumption is made that
equal weights of carbon monoxide and carbon dioxide have the same
volumes under similar conditions of pressure and temperature. If
the carbon dioxide is dissociated, a change in the number of mols
will occur with the above reaction which is proportional to the amount
of dissociation and can be measured by the change of volume.
3. The Reaction CO + j0 = CO2.For the combustion of CO
with theoretical oxygen, or for the dissociation of CO, into CO and
oxygen, the expression for K, is
pco, _ o /3 o 1
K,=
p O    ~  v/iV
poo. p 1xo '1 zo /p
When the total pressure is one atmosphere,
SXo 3xo
I  Xo N 1 X
In the dissociation of CO,, 100(1Xo) is the percentage dissociated.
Table 17 gives the experimental data on the dissociation of CO,
at atmospheric pressure.
In the case of CO2 two expressions for y, are given; the first for
temperature below 2900 deg. F. (abs.) and the second for temperatures
above 2900 deg. F. (abs.). For the lower range the corresponding
equilibrium equation is
120 930
4.571 logloK,   7.4719 logioT + 1.95 103T
T
 0.13  10VT2 + C
Making suitable adjustment, the equilibrium equation for tempera
tures above 2900 deg. is
125 810
4.571 logK, 1 + 4.147 logjoT + 0.21 . 103T
T
 0.13  106T2 + C'
The value C = 0.6 brings the calculated values of K, into good
agreement with the experimental values within the range below 2900
deg. F. (abs.). The corresponding value of C' is  37.107. This value
is obtained by equating the two expressions for K, at T = 2900 deg. F.
Calculated values of 4.571 logloKp are as follows:
T = 2500 2900 3500 4000 4500 5000 5500
4.571 log,oKp = 27.644 20.989 13.902 9.641 6.337 3.693 1.526
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 17
DISSOCIATION OF CO, AT ONE ATMOSPHERE PRESSURE
Temperature
100 (12o) 4.571 logloK, Method Investigator
deg. C. (abs.) deg. F. (abs.)
1 1300 2340 0.00414
30.737
I Nernst and
von Wartenburg
2 1395 2511 0.0142 27.067 IV Langmuir
3 1400 2520 0.010.02 26.904 I Nernst and
von Wartenburg
1 i 
4 1443 2507 0.025 25.383 IV Langmuir
5 1478 2660 0.0290.035 24.648 I Nernst and
von Wartenburg
6 1481 2666 0.0281 25.036 IV Langmuir
7 1498 2696 0.0471 23.503
8 1565 2817 0.064 22.584
9 1818 3272 0.45 16.725 VIII Lowenstein
10 2243 4037 4.5 9.852 V Emich
1.436
1.674
0.195
1.020
III Bjerrum
I
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 125
'4
'6
'2
B
4
0D
q
3zOO 3600 4000 4400 4800
Temp,'era/ure /2/ aeg / ohas.
Oý
FIG. 22. AGREEMENT BETWEEN CALCULATED *AND EXPERIMENTAL VALUES FOR THE
EQUILIBRIUM CONSTANT IOR THE REACTION CO + 0,2 = CO,
The curve, Fig. 22, represents the two equilibrium equations with the
constants C and C' just quoted. The agreement with the experimen
tal points is satisfactory.
4. The Reaction H. + 02 = HO0.The expression for K, is
PHO = Yo 3yo
Kp
and when the total pressure P is one atmosphere
Yo Y /3yo
1yo ý1yo
In the dissociation of HO into H, and 0,, 100(1 yo) is the percent
age of H20 dissociated.
o0Langmi/ir
t
L
t
SN
F
S* Nernst and vo
0 Lowenste/n
5200 5600
 Em/ch
4 / iprrm,,/
+
\~
Woartenberq
I
00
£4
I


 
I
ILLINOIS ENGINEERING EXPERIMENT STATION
Table 18 gives the experimental data on the dissociation of HO0
at atmospheric pressure.
In certain cases several values for the dissociation were given for
one temperature. In such cases the mean value has been taken.
The equilibrium equation deduced from the thermal data on
specific heats and heat of combustion is
102 820
4.571 log,1oK, =
T
2.6135 logioT  0.4713  103T
+ 0.0605 , 106T2 + C
The value C  2.3 brings the curve representing this equation into
excellent agreement with the experimental points, as shown by Fig.
23. The following are the calculated values from which the curve is
drawn:
T deg. F. (abs.)
RlogeK,
= 2500 3000 3500 4000 4500 5000 5500
= 29.144 22.016 16.906 13.074 10.105 7.753 5.857
32
8 I o Langmu/r
* Nernstf ad von War/tenberg
0 Lowensfe/on
.4 + BJerrum
o yon War/enberg
+
S 
\2   ^       
t) _ _ _ _ _  _ _ _ _ _ _ _   
/ °
B _ _  ^ 
300 .3600 4000 4400
Temperalure /; , deg. / abs.
FIG. 23. AGREEMENT BETWEEN CALCULATED AND EXPERIMENTAL VALUES FOR THE
EQUILIBRIUM CONSTANT FOR THE REACTION H, + 10, = HO0
· · · · ·
f9^\/
LOUV
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OP FUELS 127
TABLE 18
DISSOCIATION OF HO AT ONE ATMOSPHERE PRESSURE
Temperature
No. . _ 100 (1Y/o 4.571 logoK, Method Investigator
deg. C. (abs.) deg. F. (abs.)
1 1325 2385 0.0033 30.892
2 1355 2439 0.0049 30.236
3 1393 2507 0.0068 29.246
4 1397 2515 0.0073 28.440
1434 2581 0.0103
1452 2614 0.0137
1458 2624 0.0147
1474 2653 0.0140
9 1480 2664 0.0189
10 1531 2756 0.0255
11 1550 2790 0.0287
12 1561 2810
0.034
28.024
27.174
26.965
27.110
IV Langmuir
I Nernst and
von Wartenburg
IV Langmuir
26.216 I Nernst and
von Wartenburg
25.324 IV
Langmuir
24.928
24.467 i I Nernst and
von Wartenburg
1 195 TT I L tei
13 1705 3069 0.102
11t 1783 3209 0 182
15 1863 3353 0.354
16 1968 3542 0.518
17 2155 3879 1.180
18 2257 4063 1.770
19 2300 4140 2.60
19.470
17.532
16.350
13.890
12.627
11.561
4.30 9.990
7.50 8.282
6.60 8.678
9.80 7.447
11.10 7.053
II von Wartenburg
III Nernst
III Bjerrum
20
21
22
23
24
owens n
128 ILLINOIS ENGINEERING EXPERIMENT STATION
5. The WaterGas Reaction, H2 + CO, H2O + CO.By a
combination of the two reactions
H, + j0, = H20
CO. + i0 = CO2
the watergas reaction is obtained. Consequently the equilibrium
equation of this reaction is obtained by subtracting the equilibrium
equation of the CO reaction from that of the H, reaction.
Two equations are thus obtained; for T<2900 deg. F. (abs.),
18 110
4.571 log,,K, = 4.8584 logoT  2.4213 103T
 0.1905 106T2  2.9
for T>2900 deg. F. (abs.),
22 990
4.571 logK, =    6.7605 logoT  0.6813 . 103T
+ 0.0905  10'2 + 34.807
The experimental data relative to the equilibrium of the water
gas reaction are given in Table 19. Fig. 24 shows the correspond
ing experimental points and the curve deduced from the preceding
equations.
It will be noted that the constants  2.9 and 34.807 are fixed by
the corresponding constants in the H, and CO reactions; hence the
I
o
'V
Temperu7Are /i? ceg . / as.
FIG. 24. AGREEMENT BETWEEN CALCULATED AND EXPERIMENTAL VALUES lOR TTIE
EQUILIBRIUM CONSTANT FOR THE WATERGAS REACTION
7
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 129
TABLE 19
EXPERIMENTAL DATA ON THE EQUILIBRIUM CONSTANT
FOR THE WATERGAS REACTION
Temperature
No.
deg. C. (abs.) deg. F. (abs.)
1 1453 2617
2 1507 2715
3 1519 2735
4 1547 2787
5 1553 2795
6 1579 2842
7 1619 2915
8 1635 2944
9 1659 2987
10 1703 3065
11 1706 3071
12 1747 3145
13 1798 3238
14 1503 2705
15 1528 2752
16 1538 2770
17 1578 2840
18 1586 2856
19 1597 2875
20 1643 2959
21 1763 3174
22 1773 3192
23 1783 3210
24 1795 3231
25 1798 3237
26 1824 3282
27 959 1726
28 1059 1906
29 1159 2086
30 1259 2266
31 1278 2300
32 1359 2446
33 1478 2660
34 1678 3020
35 1031 1856
36 1111 1920
37 1134 2041
38 1227 2209
39 1283 2309
40 1333 2399
KP 4.571 log10K Method
0.534
0.840
1.197
1.570
1.620
1.956
2.126
2.490
0.850
0.975
0.890
2.250
2.120
2.780

VI
Investigator
Allner
1.728
1.787
1.912
1.662
1.610
1.965
2.376
2.246
2.537
2.265
2.346
2.376
2.752
2.207
1.942
2.079
2.044
1.920
2.134
2.346
2.424
2.882
2.573
2.586
2.772
2.521
 1.246
 0.346
0.357
0.895
0.958
1.332
1.497
1.811
 0.323
 0.050
 0 231
1.610
1.492
2.029
VI Haber and
Richardt
IV(b) Hahn
VII(c) Harries
ILLINOIS ENGINEERING EXPERIMENT STATION
position of the curve in Fig. 24 is fixed. The good agreement between
the calculated and experimental values of K, for the watergas reac
tion confirms in a general way the accuracy of the specific heat equa
tions and the constants Ho.
6. The Reaction C + 2H, = CH,.The equilibrium constant is
given by
KPCH
P H
The constants a', a", a"' are determined from the specific heat equa
tions as follows:
For C = 0 + 4.533 • 103  0.9092 . 10T2
For 2H2 2y, = 12 + 1.3333 • 103T
12 + 5.8667 • 103T  0.9092 . 106T2
For CH, y, = 3.459 + 10.56 103T
8.541 4.6933 103T 0.9092 • 106T2
' = 8.541 o" =  4.6933 • 103 a'" =  0.9092 . 106
2.3026 a' = 19.6665  o" = 2.3467  103  a"' = 0.1515 • 10
The constant Ho is determined from a combination of the three reac
tions
C + 02= CO2 Ho = 174 290
2H + 02 = 2H20 2H = 2 X 102 820 = 205 640
CH, + 202 = 2H,O Ho = 348 330
Adding the first two and subtracting the third from the sum, the con
stant Ho for the reaction under consideration is
Ho = 174 290 + 205 640  348 330 = 31 600
The equilibrium equation is therefore
31 600
4.571 logoK, =  19.6665 logoT + 2.3467  103T
T
+ 0.1515  106T2 + C
Two sets of experiments furnish the data for the determination of the
constant C. The figures are given in Table 20.
The points in Fig. 25 represent the experimental numbers. The
curve represents the equilibrium equation with C = 37.5.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 131
TABLE 20
EXPERIMENTAL DATA ON THE EQUILIBRIUM CONSTANT FOR THE
REACTION C + 2H2 = CH4
Temperature
No. _ 4.571 logg1K, Method Investigator
deg. C.(abs.) deg. F.(abs.)
1 1373 2471 11.031
2 1493 2687 11.085
3 1548 2786 11.117
4 1623 2921 12.114 VII (b) using Pring and
5 1673 3011 11.936 amorphous Fairlie
6 123 3281 12.461 carbon rock
7 1893 3407 14.796
8 1973 3551 13.353
9 748 1346 + 3.298
10 780 1404 + 1.647 p;
11 809 1456 + 1.326 di
12 840 1512 + 0.811 sul
13 850 1530  0.522 de
14 880 1584  1.119 niel
15 898 1616  2.138 as
NOTEIn those cases where several values were given by the invesligator for one temperature
the mean value has been taken.
7. The Reaction C + CO2 = 2CO.The equilibrium constant Kp
is the ratio p2o. The constant Ho for this reaction is  67 570. To
pco"
determine the constants o', o", a"' the following specific heat equations
are available:
For C y=0 + 4.5333 103T  0.9092 106T2
For CO., , = 7.15 + 3.90 103T 0.60 10'T2
7.15 + 8.4333 • 103T  1.5092 . 106T2
For 2CO 2, = 13.86 + 0.24 106T2
 6.71 + 8.4333 103T  1.7492 • 106T2
' =  6.71 o" = 8.4333 • 103 a" 1.7492 • 101
2.3026 a' =  15.4504 ½o" = 4.2167 • 103 Ia"' =  0.2915 • 106
ILLINOIS ENGINEERING EXPERIMENT STATION
Hence the equilibrium equation is
67 570
4.571 logoKp = + 15.4504 log,,T  4.2167  10T1
+ 0.2915  106T2 + C
)
6
4
0
4
8
/2
Termper0aure /n deg. / /7gs.
FIG. 25. AGREEMENT BETWEEN CALCULATED AND EXPERIMENTAL VALUES
FOR THE EQUILIBRIUM CONSTANT FOR THE REACTION
C + 2H2, CH,
The experimental data on the equilibrium of this reaction is given in
Table 21, and the points representing the experimental results are
plotted in Fig. 26. The curve in this figure represents the preceding
equilibrium equation with the value  6.1 for the constant C. The
equation agrees satisfactorily with the experiments of Boudouard and
of Mayer and Jacoby and gives a fair compromise between the dis
cordant results of Rhead and Wheeler and Clement and Adams.
8. The Reaction CH, + 202 = CO, + 2HO.The equilibrium
equation is obtained indirectly by the addition of the equilibrium
+Pr/ng g Fa/r/e, amorphous ca'ron__
n Av~/mn AT Zirm/tn /rn,,
0
I
+
+ + 4.
'
0 U '
C
t
~IIIIIVYII
U
/fi

oo000
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 133
TABLE 21
EXPERIMENTAL DATA ON THE EQUILIBRIUM CONSTANT FOR THE
REACTION 0 + COz = 2CO
Temperature
No.
d! 4.571 10go K,
de (. C. (abs.) deg. F. (abs.)
Method
1 796 1433 11,138
2 861 1550  6.665
3 896 1613  5.378 VII (a)
4 940 1692  3.266 using sugar carbon
5 1043 1877 2.577
6 1091 1964 4.150
7 1073 1931  1.067
8 1123 2021 1.254
9 1173 2111 3.519 VII (a)
10 1198 2156 4.461 using charcoal
11 1273 2291 5.304
12 1373 2471 7.913
13 1173 2111 3.600
14 1273 2291 3.832
15 1373 2471 6.706 VII (a)
16 1473 2651 8.571 using coke
17 1573 2831 10.947
Investigator
Mayer and
Jacoby
Clement and
Adams
Clement and
Adams
18 923 1661
19 1073 1931
20 1198 2156
1931
2021
2111
2201
2291
2381
2471
2)61
 2.755
4.993
6.230
3.293
5.257
7.046
8.297
9.729
10.916
12.376
14.726
VII (a)
amorphous Boudouard
carbon
VII (a)
amorphous
carbon
Rhead and
Wheeler
equations for the following reactions:
2CO +0 = 2CO,
2H,1 +0, = 2HO
C + CO2 = 2CO
CH, = C + 2H,
CH, + 20, = CO, + 2HO
0o \ o ; "U"'
ILLINOIS ENGINEERING EXPERIMENT STATION
*16
o 
I *
8  0
__ __ _ __ _ __ _ ^____ __ __0
4 
00 Mfayer ond Jacoby__
* C/emen and Ad7ams
_ Rhead annd Whee/er
/  Bou4doi/urd
i
400
/800 2000 ZZ&O Z40o 060W
Temperaoure /,i deg. r a'bs.
FIG. 26. AGREEMENT BETWEEN CALCULATED AND EXPERIMENTAL VALUES FOR THE
EQUILIBRIUM CONSTANT FOil THE REACTION C + CO, = 2CO
The resulting equilibrium equations are as follows:
For T<2900deg. F. (abs.)
348 330
4.571 log,,K, = + 14.9462 loglT  3.606  103T
T
+ 0.001  106T2  47.0
For T>2900 deg. F. (abs.)
353 210
4.571 logioK, + 26.5651 log,,T  5.346 . 103T
T
+ 0.101  106T2  84.7071.
K,= P2 20 co
P" * P'0
0/0
®
____ ___ ____ ___ 0 O _ _ __ __ __ __ __ _ __ __ __
iWOu 3U0o
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 135
9. The Reaction N2 + 02 = 2NO.The following equation is
taken from Haber's "Thermodynamics of Technical Gas Reactions"*:
 77 400
4.571 logoKp = + 4.9 (T= abs. temp. F.)
p2o
K,=
PN, ' Po,
10. The Reaction H2 = 2H.The equilibrium constant Kp is
the ratio P . Taking the specific heat of atomic hydrogen as 5R
PH2
or 4.96 the expressions for Kp and Ho are as follows:
For H2 y, = 6.00 + 0.6667 • 103T
For 2H yp = 9.92
 3.92  0.6667 • 103T
&' =  3.92 a" = 0.6667  103 ½u" = 0.3333 . 103
Ho = Hp  T ( 3.92 + 0.3333  103T)
Ho
4.571 log,,K, =  + 9.0262 logoT  0.3333 . 103T + C
TABLE 22
EXPERIMENTAL DATA ON THE EQUILIBRIUM CONSTANT FOR THE
REACTION H2 = 2H PRESSURE UNIT = ONE ATMOSPHERE
Temperatures
deg. C. (abs.)
1800
1900
2000
2100
2200
2300
2400
2500
2600
2700
2800
2900
deg. F. (abs.)
3240
3420
3600
3780
3960
4140
4320
4500
4680
4860
5040
5220
Values of 4.571 logl0K,
Total pressure
= 0.207 mm. Hg.
28.2533
25.4084
22.8811
20.4913
18.6250
16.8062
15.0889
12.8564
11.2712
Total pressure
= 1.1 mm. Hg.
29.4084
26.1434
22.8033
20.4913
18.5048
16.4647
14.6670
12.8395
11 4718
10.0960
 8.5578
 6.2271
Total Pressure
= 4.4 mm. Hg.
27.9288
25.2278
23.2275
20.9343
18.7868
16.7692
15.0889
13.3775
11.6954
Langmuir has deduced values of K, for this reaction from a study
of heat losses by convection from heated tungsten wires sealed in
hydrogen filled bulbs. His results transformed from millimeters of
mercury as the pressure unit to atmospheres are given in Table 22.
* English edition, p. 104.
ILLINOIS ENGINEERING EXPERIMENT STATION
In this instance it is necessary to deduce values for both Ho and C
from the experimental values of K,. Langmuir chooses the value Hp
 90000 calories per mol at 3 000 deg. C. (abs.). Ho is then
 150 550 B. t. u. per mol. Giving C the value  11.7, the broken line
curve in Fig. 27 is obtained. By taking the value 1H =ý  100 00 calo
ries per mol at 3000 deg. C. abs., the value H, =  168 550 B. t. u. per
24
28
3Z00 3500
A
.4
 er mo/ o1/ 3000C aobs_ I I
,,. aerermn/reor or p/sssuro or
4000 4500
Temperatre //h deg. E bs.
FIG. 27. AGREEMENT BETWEEN CALCULATED AND EXPERIMENTAL VALUES FOR THE
EQUILIBRIUM CONSTANT FOR THE REACTION H, = 2H
mol is obtained. Taking C =  7.2, the solid curve Fig. 27 is obtained,
and there is a better agreement between the curve and the experimental
points. The equilibrium equation is then
 168 550
4.571 log,,oK =  9.0262 logoT  0.3333  103T  7.2
T
The values used in plotting the solid curve in Fig. 27 are:
T deg. F. (abs.)= 3200 3600 4000 4400 4800 5200
4.571 logoKp =29.3009 23.1194 18.1575 14.0874 10.6874 7.8053
The equilibrium equations given in this appendix are collected
in Table 23.
Usin v/ale of/OOOOO c0000 a/l 
Us/,g L /ngmui/r vio/eo
'nfl1 117I /.9 1n,/ 141 
mo/ aof 3000°C oas.
/007
440
5000













J
B^
4/gY~r
^
P
/i
ro






I
I 1 I I I ' I
^
I
#i #
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 137
TABLE 23
EQUILIBRIUM EQUATIONS
Pressure Unit = One Atmosphere pd .p
Reaction, aA + bB = dD +eE K 
T= Abs. temp. F. A B
No. Reaction Equilibrium Equation
H2+ s02=HO
102 820
4.571 logloKp = 2.6135 logloT0.4713.10T +0.0605 10OT22.3
CO +02 =C02, T<2900 deg. F. (abs.)
120 930
4.571 logloK = 30 7.4719 logioT+1.95.10 T0.130010T2+().6
CO+is0=CO2, T>2900 deg. F. (abs.)
125 810
4.571 logoKp = 15 +4.1470 logiT +0.21 103T0.0300  10T37.1070
T
II +CO =H20 +CO, T <2900 deg. F. (abs.)
18 110
4.571 logloKp = 0 +4.8584 logoT2.421310 T+0.190510'T"2.9
H2+C02 =HO2 +CO, T>2900 deg. F. (abs.)
22 990
4.571 logolKp = _6.7605 logloT0.681310T +0.0905106T2+34.8070
C+2H2 =CH4
31 600
4.571 logoKp = 19.6665 logioT +2.3467 10T+0.151510T 37.5
C +C0 =2CO, T <2900 deg. F. (abs.)
67 570
4.571 logioKp = 0 +15.4504 logloT4.216710 T +0.2915106T6.1
CH4 +202 =CO2 2H20, T<2900 deg. F. (abs.)
4.5711ogloKp=3483 +14.9462 logloT3.66060 10T+0.001010 T 47.0
CHI+202 =CO2 +2H20, T>2900 deg. F. (abs.)
4.571 logloKp = 1 +26.5651 logioT5.346.103T +0. 1010 106T84.7071
4.571 logloK, =16 50+9.0262 logoT0.3333103T7.2
T
NOTE.See "References for Experimental Data on Equilibrium," page 157,
1
2
3
4
5
6
7
8
9
ILLINOIS ENGINEERING EXPERIMENT STATION
APPENDIX IVTABLES 24 to 36
TABLE 24
THERMAL ENERGY EQUATIONS
oUT =B.t.u. required to Raise Temperature of 1 mol of Gas at constant volume
from 0 deg. F. (abs). to T deg. F. (abs.)
No.
1
2
3
4
5
6
7
8
9
10
11
12
Gas
CO2, T<2900
CO2, T>2900
H20
CO, O2, and N2
H2
CH4
C2H2
C2H4
C2Hs
CaHs
CsH1s
CeHe
(Gasoline Vapor)
Ca2112
S(Kerosene Vapor)
H (Atomic
Hydrogen)
oUT =2.978T
Thermal Energy
UT = T(5.165+1.95.103T0.210T2)
UT = T(10.211+0.21 103T)4878
oUT = T(6.3450.13810 T±+0.141.10T'2)
UT = T(4.945 +0.0400.10 6T2)
oU = T(4.015 +. 103T)
oUT =T(1.474+5.28103T)
oUT = T(4.205 +4.05 103T)
oUT = T(4.685 +3.40103T)
oUT = T(5.115+4.30103T)
oUT= T(2.015 +15.90.10 T)
our = T(36.342+19.0103T)
UrT = T(55.169 +28.333.10 T)
i
 
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 139
TABLE 25
THERMAL ENERGY OF CARBON DIOXIDE IN B. T. U. PER LB. MOL.
T  abs. temp. F.
T u
500 3 045
550 3 398
600 3 758
650 4 126
700 4 503
750 4 886
800 5 277
850 5 676
900 6 082
950 6 495
1000 6 915
1050 7 342
1100 7 775
1150 8 214
1200 8 661
1250 9 113
1300 9 571
1350 10 035
1400 10 504
1450 10 978
1500 11 460
1550 11 946
1600 12 437
1650 12 933
1700 13 434
1750 13 939
1800 14 448
1850 14 962
1900 15 481
1950 16 004
2000 16 530
2050 17 060
2100 17 594
2150 18 131
2200 18 672
2250 19 215
2300 19 761
2350 20311
2400 20 863
2450 21 418
2500 21 975
2550 22 534
2600 23 096
2650 23 659
2700
2750
2800
2850
2900
2950
3000
3050
3100
3150
3200
3250
3300
3350
3400
3450
3500
3550
3600
3650
3700
3750
3800
3850
3900
3950
4000
4050
4100
4150
4200
4250
4300
4350
4400
4450
4500
4550
4600
4650
4700
4750
4800
4850
24 224 4900
24 791 4950
25 359 5000
25 929 5050
26 500 5100
27 072 5150
27 645 5200
28 219 5250
28 794 5300
29 370 5350
29 948 5400
30 526 5450
31 105 5500
31 685 5550
32 267 5600
32 849 5650
33 433 5700
34 018 5750
34 603 5800
35 190 5850
35 778 5900
36 366 5950
36 956 6000
37 546 6050
38 138 6100
38 731 6150
39 326 6200
39 921 6250
40 517 6300
41 114 6350
41 712 6400
42 312 6450
42 912 6500
43 513 6550
44 116 6600
44 719 6650
45 324 6700
45 929 6750
46 536 6800
47 143 6850
47 752 5900
48 362 3950
48 973 7000
49 585
_~
ii
u
50 198
50 812
51 427
52 043
52 661
53 279
53 898
54 518
55 139
55761
56384
57 009
57635
58261
58 889
59 518
60 148
60 779
61 411
62 043
62 677
63 312
S63 948
64 585
65 223
65 862
66 502
67 143
67786
68 429
69 074
69 719
70 366
71 013
71 662
72 312
72963
73 614
74 267
74921
75576
76 232
76888
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 26
THERMAL ENERGY OF WATER VAPOR IN B. T.
T = abs. temp. F.
U. PER LB. MOL.
T
3 156 2700
3 471 2750
3 787 2800
4 104 2850
4 422 2900
800 5060 3000
850 5 380 3050
900 5 702 3100
950 6 024 3150
1000 6 348 3200
1050 6 673 3250
1100 7 001 3300
1150 7329 3350
1200 7659 3400
1250 7 990 3450
1300 8 325 3500
1350 8 662 3550
1400 9000 3600
1450 9 340 3650
1500 9 682 3700
1550 10 028
1600 10 376
1650 10 726
1700 11 080
1750 11 436
1800 11 796
1850 12 159
1900 12 525
1950 12 894
2000 13 266
2050 13 642
2100 14 022
2150 14 405 4350
2200 14 792 4400
2250 15 183 4450
2300 15 579 4500
2350 15 07 I 4.550
2400 16
2450 16
2500 17
2550 17
2600 18
2650 18
u
18 901
19338
19779
20 226
20 679
21 137
21 600
22069
22 544
23024
23 510
24 004
24 503
25 008
25 520
26 038
T
4900
4950
5000
5050
5100
5150
5200
5250
5300
5350
5400
5450
5500
5550
5600
5650
u
44 366
45 128
45 900
46 681
47 474
48 276
49 088
49910
50 744
51 588
52 442
53 306
54 181
55 068
55 966
56 875
57795
58 726
59 670
60 624
26 562 5700
27 094 5750
27 632 5800
28 176 5850
28 729 5900
61 590
66 600
67 639
68 689
69 753
70 829
36 595
37257
37 927
38 606
39 294
39 992
40 697
41 412
42 137
42 870
43 613
6550 75 262
6600 76 403
6650 77 557
6700 78 724
6750 79 905
6800 81 099
6850 82 308
6900 83 530
6950 84 766
7000 86 017.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 141
TABLE 27
THERMAL ENERGY OF THE DIATOMIC GASES CO, O0 AND N2 IN B. T. U. PER LB. MOL.
T = abs. temp. F.
500
550
600
650
700
750
800
850
900
950
1000
1050
1100
1150
1200
1250
1300
1350
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
1950
2000
2050
2100
2150
2200
2250
2300
2350
240')
2450
2 478
2 726
2976
3 226
3 475
3 725
3976
4227
4 480
4 732
4985
5239
5493
5 747
6003
6 259
6 516
6 774
7 033
7293
7 553
7 814
8 076
8 339
8 603
8 869
9 135
9 401
9 669
9 939
10211
10 483
10 755
11 029
11 305
11 582
11 860
12 140
12 421
12 704
12988
13 274
13 560
13 849
T
2700
2750
2800
2850
2900
2950
3000
3050
3100
3150
3200
3250
3300
3350
3400
3450
3500
3550
3600
3650
3700
3750
3800
3850
3900
3950
4000
4050
4100
4150
4200
4250
4300
4350
4400
4450
4500
4550
4600
4650
14 139
14 431
14 725
15020
15 316
15 614
15914
16217
16521
16 827
17 134
17 444
17 756
18069
18 385
18702
19 022
19 344
19 668
19995
20323
20652
20 985
21 320
21 657
21 997
22 339
22 684
23 032
23 381
23 732
24 086
24 444
24 803
25 165
25 529
25897
26 267
26 641
27 017
27 394
27 774
28 159
28 546
T
4900
4950
5000
5050
5100
5150
5200
5250
5300
5350
5400
5450
5500
5550
5600
5650
5700
5750
5800
5850
5900
5950
6000
6050
6100
6150
6200
6250
6300
6350
6400
6450
6500
6550
6600
6650
6700
6750
6800
6850
u
28 936
29 329
29 725
30 124
30525
30929
31 337
31 749
32 164
32 582
33002
33 425
33853
34 283
34 717
35 152
35 595
36038
36 486
36 936
37 390
37 849
38 310
38 774
39244
39 716
40 192
40 672
41 155
41 643
42 134
42 629
43 128
43 631
44 138
44 647
45 162
45 681
46 204
46 730
47 261
47 796
48 335
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 28
THERMAL ENERGY OF HYDROGEN IN B. T. U. PER LB. MOL.
T = abs. temp. F.
u
2091
550
600
650
700
750
800
850
900
950
1000
1050
1100
1150
1200
1250
1300
1350
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
1950
2000
2050
2100
2150
2200
2250
2300
2350
2400
2450
T
2700
2750
2800
2850
2900
2950
3000
3050
3100
3150
3200
3250
3300
3350
3400
3450
3500
3550
3600
3650
2309
2529
2 750
2974
3 199
3425
3 653
3884
4 115
4348
4583
4819
5058
5 298
5 539
5783
6 028
6279
6 522
6 772
7024
7277
7 532
7 789
8047
8307
8 568
&832
9097
9363
9 631
S902
10 173
10 446
10 721
10 998
11 276
11 556
11 838
12 121
12 406
12 693
12 981
13271
13 562
13 855
14 150
14 447
14 745
15045
15 346
15 650
15955
16261
16 570
16880
17 191
17 504
17819
18 136
18454
18 774
19 095
19 419
19 744
20071
20 399
20 729
21 060
21 393
21 728
22 065
22 403
22 743
23 084
23428
23 773
24 118
24 468
24 818
25 169
25 522
25 877
26 234
26592
26 952
27 314
T
5900
5950
6000
6050
6100
6150
6200
6250
6300
6350
6400
6450
6500
6550
6600
6650
6700
6750
6800
6850
T
4900
4950
5000
5050
5100
5150
5200
5250
5300
5350
5400
5450
5500
5550
5600
5650
5700
5750
5800
5850
500
3700
3750
3800
3850
3900
3950
4000
4050
4100
4150
4200
4250
4300
4350
4400
4450
4500
4550
4600
4650
u
27 677
28 042
28 409
28 777
29 147
29 517
29891
30 266
30643
31 021
31 401
31 782
32 166
32 551
32 938
33 325
33 715
34 107
34 500
34 895
35292
35 691
36090
36 492
36 895
37300
37 707
38 114
38 524
38 935
39 349
39 764
40 181
40 599
41 019
41 440
41 864
42 289
42 716
43 144
43 574
44005
44 439
~
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 143
TABLE 29
THERMAL ENERGY OF HYDROCARBON GASES IN B. T. U. PER LB. MOL.
T
deg. F. Methane Acetylene
(abs.) CH4 C2H1
500 2057 3115
550 2 408 3 538
600 2 785 3 981
650 3 189 4 444
700 3 619 4928
750 4076 5 432
800 4 558 5956
850 5 068 6 500
900 5 603 7 065
950 6 166 7 650
1000 6 754 8 255
1050 7 369 8 880
1100 8 010 9 526
1150 8 678 10 192
1200 9 372 10 878
1250
1300
1350
1400
1450
1500
1550
1600
10 092 11 584
10839 12 311
11 613 13058
12 412 13 825
13 239 14 612
14 091 15 420
14 970 16 248
15875 17096
Ethylene
C2H4
3 192
3 605
4035
4 482
45 n
Ethane
CzHs
3 632
4 114
4 617
5 142
5 426 6255
5924 6 844
6 439 7 454
6 971 8 086
7 519 8 740
I
8 085 9 415
8 668 10 112
9 268 10 830
9 884 11 569
Iu 010
11 169
11 836
12 521
13 223
13 942
14 678
15 430
16 200
Benzene Gasoline Kerosene
Vapor Vapor Vapor
CHR1 CsHls C12H26
4 982 22 921 34 667
5 918 25 736 38 914
6 933 28 645 43 301
8 027 31 650 47 830
9 202 34 749 52 501
10 455A i 7 4dd 57 314
11 788
13 200
14 692
16 264
17 915
19 645
21 456
23 345
25 314
13 113 27 362
13 917 29 490
14 742 31 698
15 589 33 985
16 458 36 351
17 348 38 798
18 259 41 323
19 192 43 928
41 234 62 269
44 618 67 364
48 098 72 602
51 672 77 981
55 342 83 502
59 107 89 165
62 967 94 969
66 921 100 915
70u oi
75 115
79 355
83 690
88 119
92 644
97 264
101 978
106 788
113 231
119 603
i26 116
132 770
139 565
146 502
153 581
160 803
12 330

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MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 145
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10 N  N C
OS CO (NON
00 N~t NOt
00 000 >
o oo o
o  to O 0 ~
 CO 
tO ~ 0 0
0 0 ID CO
t cc N NO
00000
CO  OO C
COSC t1No
 C N O
o COccCO
OX N N 0
00000
ccCo COcc
hCOcc 0 to
tN c C c
0 Colt O c
0 cc N NO
00000
Ci t 0 ON
CO CO cc CO 
 toO COO
00 N NO
00000
tO CO 00 CO f
OM CO OCM0cc
O0 O C
0 CO 00O
OC00 COO
0cc ON 0
00000
NO to NO
000 tOoo
S00000
d0 d d o o
OCO CO 10 to
C 000O
COONC
COS N 0 toOI'
00000
c0 0O) 10 CO
CO N~ CO CO CO
lo~o
CO N^ 00 4
00000
00m00 CO
T CO 00 N> to
CO CO 0O0 N
CO 00 NM 1~ 
0000t
00000
oo0000
0 0M CO CO
00000
ým ý c t
ON< O COS
0 x0 0O C
00^000^
CO (10 i0 m
do oo o
O ~ oo cc
00 t CO 00 '
0 to t o O C
00Cl00 *i0
Nocc oo
* CO N^ CO '
CO cc CO O to
00 t0'o CO CO
00000
to) 00 COON
00 t COOS 0
t000 CON to '
CO cc'* COO to
0£ to to CO CO
00000d
~o to to to CO
00000
COO CO 0O
ccoo lZ
00x CO
CD1~C
cOO t^ 00 O
S000cc
0n0 to to CO
00000
*^00 to  
ON1 aMC  too
to o 0Q » N
*00 tO to< CO
00000
00 N to
0 to00
030000
0  cc to
CO 00 CO
ON 0 CO 0
O .O A0 ý "t!
00000
00000
t o ON 00Soo
00000

~
ILLINOIS ENGINEERING EXPERIMENT STATION
148
o
+
o
II
+
a
0
M ^
0
mm a
mO Co CO ,
00 00
'0 0D 0 00
0^ 0 O 'I 0O
00000m
0 m 0 m0
 3000
00 £>000'iC
001CO00 0
0O 0000 MC
CO0O000'IC
000^CO000
0 0 000
0OcC00 N 0
00 0 0 0
00000d
r00000u
0000NC
00000m
Ni 6N (M'03
00 C; '0 0
0 0 0 0 M i0
0 0 00 0f ^
10 N; O 0 NS
0 0 10  9
0 orNc0 o l
SS  00='0
CO 0> 0 00 b
0 >0Nt.0 ,0 t
; 00 0 0C(MO
9 Q g i" t> m
0 MC~mOCOH
^l ^1CO 0
CO *i > >
ao ( 0 mh
0 ^t>OS r1t
t0 . . .
! 8 9 o ^
0 0 1 0N 0)
NO 00.0
S40 i0 0
N> 0 0 N N'
000 S 0T NO
' f 0 0 ^ 000
^ ^ 0 10 0o
CO 10O CO UO
0 0 0 0
0 iC 0 000
0 0i 00 C 0
S0 *0 0
COU 0 000 0
 0 00 CO0
'0 N^ 0T . 0
0 G 0 0 '0
^ ' ot0 000
§0 0 N^ 00
000(M00
*^'t 0 0 00
1du30 0' M
0 Ct0  0 00
000 N M
0000 0 < ^C
10 00 00I
Tt 0 N 000S
0D00 0 0NC
«0 0^ 0 0 0
=0^ 0 i0 '0
0000C ^ * 0
ON 000
O D 00 0
i00t 01 CM
00 *^*t 0 N
00000
0O 0 0 0
CO 0 N 0^
00000 o ~
00000
00000
0 000 0
0000 CM0Wt
'0 0 0 ON
000I10M
cN 00 0
0 0000 C
10 C N 0
Nm O 00
N'0OT 0 0
003000
> 0 NO 0* 0ff
No0 00 '
*^  t"i
00000
0 00 N
0 '0 00 N G
M0c o 1
0 0 00 C0
N 0^ O00
00000
0O0 0O0 N
00 M 0
NO(MC 00
CO0=  C 0
0NM0= 00 i
0'00CTS00
0 ON 00
0O00 0 0
i0 0 t0 00S
*.0;rioO O^cmo
ctMm N N OO O...O "
OOOCO 00"000
Co 3L1 CoC oC o Co .0 
T ~os Co o0C
F 0o o Co
Co Co*t . Co
CoCo Co0 Co Co
CE CO0C Co Co
F 0Co Co CO
F Coir Co  F
Co CoCo ' Co
Cor1 m C C
FF OCo 000C
Co Co CO0 Co F
mo FC 0o Co
o C 
Co CO CoCo
Co Co Co< CoC
o Co Co C
m T cý
mCO ^ 0CO
Codo^ <
FOoio
CO Co 0Co Co
ooco
Co t Co CoC
mccICo C C
Cso C C NCoC
CoCoCo C C
Co3o3o
F Co 3C FCOS
c0 C0 Co
mOCOCDOSML
0001
N~O
Co Co Co 0C
CO  t Co Co
FFFCo.(C
Co Co C ' Co
Co0Co0 C
Co C~ C Co Co
o Co Co O 
Co ICo C 
Co o o C C
Co o o C C
OCO CoCo
Co C Co C
CoN 3oCLO
Co o  C C
Co Co C C
Co o o C C
 CoF o C
COiOto'
00000
oCoCoo
FCo<  Co Co
Co o  Co Co
CO0 Co Co
Co Co C Co Co
Co CoCo Co Co
Co Co1^  Co
'' 00 CO 'O
0Co C  I Co
Co Co CIl Co Co
tCODCO0
00 0CO 10 
F> Cot Co Co Co
Co Co Co Co Co
CI FCoC Co
'Co Co Co Co
ýCo o Co C
F Co C
CoFCo Co C
Co Co C CoC
 Co C C C
CoCo NCo Co
SCoFCo
Fr Co1  oC
Co o o C C
I Co C oC
CoCo Co * Co
t CoCo N Co
Crio CN C Co
Coi0 ^ Co o0C
Co Co Co Co
00Co Co OCOCo
 Co r (COCo
Co o o C C
Co =Co o C
 o Co o C
co Co CoC
N 10m00 O
 Co Co Co Co
F Co0 Co Co
CO Co C t '
Co Co Co* 00 Co
 mC o C Co
00000
o0 CD o 0o Co
CD Co c Co CO
Co
om" Lý m
I
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 33
EQUATIONS FOR HEAT OF COMBUSTION
Hp =lower heat of comb., B. t. u. per mol, at const. press. at temp. T.
H =lower heat of comb., B. t. u per mol, at const. vol. at temp. T.
T =abs. temp. F.
No. Gas
Heat of Combustion
HR =102 820+T(1. 135+0.4713103T0.1210.106T2)
H = 102 820 +T(0.143+0.4713.103T0.1210.106T')
IH =120 930 +T(3.2451.9510 T+0.2600.106T2)
H = 120 930 +T(2.2531.95103T+0.2600106 T)
Hp = 125 810 +T(1.8010.2110ST+0.0600106T2)
H = 125 810 +T(2.7930.21.10T +0.0600 10 T2)
H5 =348 330+T(6.491 +3.60610 3T0.002010 T2)
S=348 330+T(6.491+3.606103T0.002010 T2)
(CsH2, T<2900 Hp=541 580+T(0.885+0.288103T+0.359010T2)
H =541 580 +T(0.107+0.288 10 T+0.3590 10BT2)
C2H4, T<2900 Hp =577 220 + T(3.500.2240103T+0.2380106T2)
H, =577 220 +T(3.500.2240 103T +0.2380 106T2)
CzHe, T <2900
C6Hc, T<2900
CsH1s, T<2900
(Gasoline Vapor)
H =614 240+T(7.935+0.8140.103T+0.1170.106T2)
H. =614240+T(6.943+0.8140103T +0.1170106T2)
Hp =1 364 200+T(11.915+4.6140103T+1.077010ST2)
H,=1 364 200+T(10.923+4.6140103T+1.0770.106T2)
H,=2 145 610 +T(7.218+4.642103T+0.831106T2)
H =2 145 610+T(0.271+4.642.103T+0.831106T2)
C1H2I6, T<2900
(Kerosene Vapor)
H,=3 458 680 +T(8.732+6.729103T+1.310106T2)
H, =3 458 680+T(2.186+6.72910aT'+1.31010ST2)
, =168 550+T(3.920 +0.3333 103T)
S,=168 550 +T(1.935 +0.3333103T)
1
2
4
CO, T<2900
CO, T>2900
CHa, T <2900
6
7
8
9
(Reaction)
(H2 =2H)

I
H2
~~
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 151
TABLE 34
HEAT OF COMBUSTION OF CARBON MONOXIDE AT CONSTANT VOLUME
IN B. T. U. PER LB. MOL.
T = abs. temp. F.
T
2400
2500
2600
2700
2800
2900
3000
3100
3200
3300
3400
3500
3600
3700
3800
3900
4000
4100
T H
H
121 602
121 636
121 641
121 618
121 568
121 493
121 395
121 274
121 135
120 976
120 800
120 608
120 402
120 183
119 955
119 716
119 470
119 217
118 960
118 700
118 438
11 1I76
117915
117 658
117408 4800
117 161 4900
116 921 5000
116 688 5100
116 462 5200
116 245 5300
116 035 5400
115 833 5500
115 640 5600
115 456 5700
115 282 5800
115 118 5900
114 964
114 820
114 688
114 566
114 456
114 358
114 274
114 201
114 141
114 095
114 063
114 045
114 041
114 052
114 078
114 121
114 179
114 253
114 344
T
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
2100
2200
4200
2300
I
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 35
LOWER HEAT OF COMBUSTION OF HYDROGEN AT CONSTANT VOLUME
IN B. T. U. PER LB. MOL.
T = abs. temp. F.
2400
2500
2600
2700
2800
2900
3000
3100
3200
3300
3400
3500
3600
3700
3800
3900
4000
4100
4200
104 205
T
4300
H,
102 528
104 232 4400 102 266
104 251 4500 101 981
104 261 4600 101 673
104 260 4700 101 341
104 247 4800 100 983
104 223 4900 100 601
104 188 5000 100 193
104 139 5100 99 957
104 076 5200 99 294
103 988 5300 98 803
103 906 5400 98 283
103 798 5500 97 732
103 672 5600 97 150
103 529 5700 96 539
103 369 5800 95 895
103 188 5900 95 218
102 989 ......
102 769
T
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
2100
2200
2300
102 994
103 049
103 110
103 174
103 242
103 313
103 386
103 461
103 537
103 612
103 687
103 760
103 830
103 899
103 963
104 024
104 078
104 127
104 170
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 153
TABLE 36
LOWER HEATS OF COMBUSTION OF HYDROCARBON GASES AT CONSTANT VOLUME
IN B. T. U. PER LB. MOL.
T = abs. temp. F.
T Methane Acetylene Ethylene
CH4 C2sH C1sH
500 345 986 541 643 574 443
600 345 733 541 697 575 091
700 345 552 541 769 574 742
800 345 444 541 862 574 399
900 345 408 541 979 574 062
1000 345 443 542 120 573 734
1100 345 551 542 289 573 416
1200 345 729 542 487 573 109
1300 345981 542 716 572 814
1400 346 304 542 979 572 534
1500 346 701 542 279 572 271
1600 347 168 542 616 572 022
I
Benzene Gasoline Kerosene
Ethane Vapor Vapor Vapor
C2aH CoHe C5His C12lH6
610 987 1 360027 2 146 739 3 461 619
610 393 1 359 540 2 147 299 3 462 697
609 819 1 359 184 2 147 980 3463 956
609 267 1 358 966 2 148 789 3 465 405
608 736 1 358 891 2 149 732 3 467 053
608 228 1 358 968 2 150 812 3 468 905
607 743 1 359 201 2 152 025 3 470 970
607 283 1 359 598 2 153 405 3 473 257
606 847 1 360 164 2 154929 3 475 772
606 436 1 360 906 2 156 609 3 478 523
606 052 1 361 832 2 158 453 3 481 521
605 694 1 362 947 2 160 464 3 484 770
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 155
REFERENCES ON SPECIFIC HEAT
YEAR AUTHOR
1912 Bjerrum
1921
TITLE AND REFERENCE
1914
1919
1905
1907
1905
1892
1911
1904
1904
1912
1884
1909
1910
1862
1909
1911
1875
1876
1904
1922
Dixon, Campbell,
and Parker
Goodenough
Heuse
Holborn and
Austen
Holborn and
Henning
Joly
Knoblauch
and Jakob
Knoblauch
and Mollier
Kunz
Langen
Lewis and
Randall
Lussana
Mallard and
Le Chatelier
Nernst
Pier
Pier
Regnault
Swann
Thiabaut
Weber, H. F.
Wiedemann
Wiedemann
Wiillner
Wilson and
Barnard
Zeit. fiir Electrochem., v. 18, p. 103.
Proc. Roy. Soc., v. 100A, p. I.
Principles of Thermodynamics, 3d ed., p. 105.
Thermal Properties of Steam, Bul. No. 75,
Eng. Exp. Sta., Univ. of Ill.
Ann. der Physik, v. 59, p. 86.
Sitzungsber. der Kgl. Preuss. Akad., p. 175.
Ann. der Physik (4), v. 23, p. 809.
Ann. der Physik (4), v. 18, p. 739.
Phil. Trans. Roy. Soc., v. 182, p. 73.
Mitteil. fiber Forshungsarbeit., v. 35, p. 109.
Zeit. der Ver. Deutch. Ing., v. 55, p. 665.
Ann. der Physik (4), v. 14, p. 309.
Mitteil. iiber Forshungsarbeit, v. 8.
Jour. Am. Chem. Soc., v. 34.
LandoltB6rnstein Tables.
Ann. des Mines, v. 4, p. 379.
Theoretical Chemistry, 4th Eng. ed., p. 253.
Zeit. fir Elektrochem., v. 15, p. 536.
Zeit. fiir Elektrochem., v. 16, p. 897.
Mem. de 1'Institute de France, v. 26, p. 167.
Proc. Roy. Soc., A., No. 82.
Ann. der Physik (4), v. 35, p. 347.
Phil. Mag. (4), v. 49, pp. 161, 276.
Pogg. Ann., v. 157, p. I.
Ann. der Physik (4), v. 14, p. 309.
LandoltBSrnstein Tables.
Jour. Soc. Aut. Engrs., v. 10, 65.
~
ILLINOIS ENGINEERING EXPERIMENT STATION
REFERENCES ON HEATS OF COMBUSTION
AUTHOR
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
TITLE AND REFERENCE
YEAR
1848
1852
1873
1880
1905
1877
1881
1881
1893
1889
1881
1893
1893
1903
1906
1901
1907
1883
1886
1913
1910
1922
Andrews
Favre and
Silbermann
Thomsen
Schiiller
and Wartha
Than
Berthelot
Mixter
Rumelin
Gottlieb
Stohman,Rodatz,
Herzberger
Callendar
Young
Wilson and
Barnard
Phil. Mag. (3) 32, p. 321.
Ann. de Chem. et de Phys. (3), 34, p. 349.
(H,) Pogg. Ann., 148, p. 368.
(CO) Thermochem. Unters., v. II, p. 284.
(CH4) Berichte d. d. Chem. Gesell. 13, p. 1323.
(C,H,) and (C,H,) Thermo. Unters., v. IV,
p. 65.
(C,H,) Zeit. Physikal. Chem., v. 51, p. 657.
Wied. Ann. 2, p. 359.
Wied. Ann. 13, p. 84.
Wied. Ann. 14, p. 422.
(H,) Compt. Rendus, 116, p. 1333.
(C) Ann. de Chem. et de Phys. (6), 18, p. 89
(CH,) Ann. de Chem. et de Phys. (5), 23,
p. 176.
(C,H2) and (C,H,) Ann. de Chem. et de Phys.
(6), v. 30, p. 556.
(CH,) Ann. de Chem. et de Phys. (6), v. 3r.
p. 547.
(H,O) Am. Jour. Sci. (4), 16, p. 214.
(C,H,) Am. Jour. Sci. (4), 22, p. 17.
(C.H,) Am. Jour. Sci. (4), 12, p. 347.
Zeit. Phys. Chem., 58, p. 456.
Jour. Prakt. Chem., 28, p. 420.
Jour. Prakt. Chem. (2), 33, p. 257.
(Specific Heat of Water) Phil. Trans., v.
212A, pp. 132.
Scientific Proc. Roy. Dublin Soc. (2), v. 12.
p. 422.
Jour. Soc. Aut. Engrs., v. 10, p. 65.
MAXIMUM TEMPERATURES ATTAINABLE IN COMBUSTION OF FUELS 157
REFERENCES FOR EXPERIMENTAL DATA ON EQUILIBRIUM
AUTHOR
TITLE AND REFERENCE
On Reaction CO + iO, = CO,
Nernst and von Zeit. Phys. Chem., 56, p. 548.
Wartenberg
Langmuir Jour. Am. Chem. Soc., 28, p. 1357.
Lowenstein Zeit. Phys. Chem., 54, p. 707.
Emich Monatshefte f. Chem., 26, p. 1011.
Bjerrum Zeit. Phys. Chem., 79, p. 537.
On Equilibrium in Reaction 1H + O 02= HO
Langmuir Jour. Am. Chem. Soc., 28, p. 1357.
Lowenstein Zeit. Phys. Chem., 54, p. 715.
von Wartenberg Zeit. Phys. Chem., 56, p. 513.
Nernst Zeit. Anorg. Chem., 45, p. 130.
Nernst and von Zeit. Phys. Chem., 56, p. 534.
Wartenberg
Bjerrum Zeit. Phys. Chem., 79, p. 513.
On Equilibrium in Reaction H, + CO, = H.O + CO
Allner
Haber and
Riehardt
Hahn
Harries
Jour. f. Gasbel., 48, pp. 1035, 1057, 1081, 1107.
Zeit. Anorg. Chem., 38, p. 5.
Zeit. Phys. Chem., 42, p. 705.
Jour. f. Gasbel., 37, p. 82.
On Reaction C + 2H, = CH4
1 1912 Pring and
Fairlie
2 1907 Mayer and
Altmeyer
3 1911 Clement
and Adams
Jour. Chem. Soc., 101, p. 91.
Berlin in Ber., 40, p. 2135.
U. S. Bureau of Mines, Bul. No. 7, p. 41.
For Reaction C + CO, = 2CO
1 1909 Mayer and
Jacoby
2 1909 Clement and
Adams
3 1900 Boudourad
4 1910 Rhead and
Wheeler
1911 lihad and
TWheeler
Jour. f. Gasbel., 52, 282, p. 305.
U. S. Bureau of Mines, Bul. No. 7.
Compt. Rendus., 130, p. 132.
Jour. Chem. Soe., 97, p. 2178.
Jour. Chem. Soc., 89, p. 1140.
No. I YEAR
1906
1906
1906
1905
1912
1905
1904
19023
1894
158 ILLINOIS ENGINEERING EXPERIMENT STATION
REFERENCES FOR EXPERIMENTAL DATA ON EQUILIBRIUM (Continued)
Equilibrium Data for Reaction N, + O = 2NO
1 1906 Nernst Zeit. f. Anorg. Chem., 49, p. 213.
Equilibrium Data for Reaction H. = 2H
1 1915 Langmuir Jour. Am. Chem. Soc., v. 37, p. 417.
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Wheels. Part II. Wheel Fit, Static Load, and Flange Pressure Strains. Ultimate
Strength of Flange, by J. M. Snodgrass and F. H. Guldner. 1922. Forty cents.
*Circular No. 10. The Grading of Earth Roads, by Wilbur M. Wilson. 1923.
Fifteen cents.
*Bulletin No. 135. An Investigation of the Properties of Chilled Iron Car
Wheels. Part III. Strains Due to Brake Application. Coefficient of Friction and
BrakeShoe Wear, by J. M. Snodgrass and F. H. Guldner. 1923. Fifty cents.
*Bulletin No. 136. An Investigation of the Fatigue of Metals. Series of
1922, by H. F. Moore and T. M. Jasper. 1923. Fifty cents.
*Bulletin No. 137. The Strength of Concrete; its Relation to the Cement,
Aggregates, and Water, by A. N. Talbot and F. E. Richart. 1923. Sixty cents.
*Bulletin No. 138. AlkaliVapor Detector Tubes, by Hugh A. Brown and
Chas. T. Knipp. 1923. Twenty cents.
*Bulletin No. 139. An Investigation of the Maximum Temperatures and Pres
sures Attainable in the Combustion of Gaseous and Liquid Fuels, by G. A.
Goodenough and G. T. Felbeck. 1923. Eighty cents.
* A limited number of copies of bulletins starred are available for free distribution.
THE UNIVERSITY OF ILLINOIS
THE STATE UNIVERSITY
Urbana
DAVID KINLEY, Ph.D., LL.D., President
THE UNIVERSITY INCLUDES THE FOLLOWING DEPARTMENTS:
The Graduate School
The College of Liberal Arts and Sciences (Ancient and Moder Languages and
Literatures; History, Economics, Political Science, Sociology; Philosophy,
Psychology, Education; Mathematics; Astronomy; Geology; Physics; Chem
istry; Botany, Zoology, Entomology; Physiology; Art and Design)
The College of Commerce and Business Administration (General Business, Bank
ing, Insurance, Accountancy, Railway Administration, Foreign Commerce;
Courses for Commercial Teachers and Commercial and Civic Secretaries)
The College of Engineering (Architecture; Architectural, Ceramic, Civil, Electrical,
Mechanical, Mining, Municipal and Sanitary, and Railway Engineering;
General Engineering Physics)
The College of Agriculture (Agronomy; Animal Husbandry; Dairy Husbandry;
Horticulture and Landscape Gardening; Agricultural Extension; Teachers'
Course; Home Economics)
The College of Law (threeyear and fouryear curriculums based on two years and
one year of college work respectively)
The College of Education (including the Bureau of Educational Research)
The Curriculum of Journalism
The Curriculums in Chemistry and Chemical Engineering
The School of Railway Engineering and Administration
The School of Music (fouryear curriculum)
The Library School (twoyear curriculum for college graduates)
The College of Medicine (in Chicago)
The College of Dentistry (in Chicago)
The School of Pharmacy (in Chicago); Ph.G. and Ph.C. curriculums
The Summer Session (eight weeks)
Experiment Stations and Scientific Bureaus: V. S. Agricultural Experiment Sta
tion; Engineering Experiment Station; State Laboratory of Natural History;
State Entomologist's Office; Biological Experiment Station on Illinois River;
State Water Survey; State Geological Survey; U. S. Bureau of Mines Experi
ment Station.
The Library collections contain (January 1, 1923) 541,127 volumes and 121,714
pamphlets.
For catalogs and information address
TSE REGISTRAR
Urbana, Illinois
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