THE RESISTANCE OF MINE TIMBERS TO THE FLOW
OF AIR, AS DETERMINED BY MODELS
I. INTRODUCTION
1. Nature of Investigation.In underground mining it is com
monly necessary to support some of the passageways artificially.
This is usually done by installing timber supports at intervals along
the passageway, many different types of supports having been used
to meet local needs. In nearly all cases each timber partly obstructs
the passage, thereby offering a large resistance to the flow of the
ventilating current of air along the passageway.
Where a high rate of air flow must be maintained to ventilate
the workings properly, the resistance of the mine timbers to the flow
of air adds considerably to the cost of ventilation, and this investi
gation was undertaken to determine the resistance of different forms
of timbering and to compare their effectiveness and economy as
underground supports.
Smallscale models of timbers of several kinds were spaced regu
larly in a model of a mine entry, and the resistance of these model
timbers to the flow of air was measured over a suitable range in the
rate of flow. This led to a comparative evaluation of the different
kinds of timbering, and yielded indications as to the optimum choice
of timbering, to suit a given condition.
2. Prior Work.As stated in a previous bulletin* models are used
widely in studying resistance to the flow of air and some material has
been publishedt describing the use of models of parts of mines for this
purpose. However, it appears that the most comprehensive previous
work of this kind was done in this laboratory by Hsu, who reviewedt
the literature of the resistance of smooth, of rough, and of sinuous
conduits, and correlated it with his own tests of smallscale models
of timbers in a smooth duct.
3. Acknowledgments.DR. Hsu is thought to have been the first
to undertake a comprehensive study of this kind, and this investiga
*"Application of Model Tests to the Determination of Losses Resulting from the Trans
mission of Air Around a Mine ShaftBottom Bend," Univ. of Ill. Eng. Exp. Sta. Bul. 265, p.
5 (1934).
tfbid.
'"An investigation of the Flow of Air Through Mine Models and Comparison with Mine
AirFlow." PenChun Hsu, unpublished Thesis for the Doctorate in Engineering, Univ. of
Ill. (1928), pp. 624.
ILLINOIS ENGINEERING EXPERIMENT STATION
tion is an outgrowth of his work. It differs from it principally in
the use of improved apparatus with which more exhaustive tests
were made. This work has been carried on intermittently since
1930, more than eight thousand individual tests having been com
pleted. Much of the material of this report has been separately
treated in thesis form.*
The author takes pleasure in acknowledging his indebtedness to
his predecessor in the work, and to PROF. A. C. CALLEN, Head of the
Department of Mining and Metallurgical Engineering, who has given
continued guidance and assistance throughout the investigation.
This investigation has been part of the work of the Engineering
Experiment Station, of which DEAN M. L. ENGER is Director, and of
the Department of Mining and Metallurgical Engineering, of which
PROF. A. C. CALLEN is the Head.
II. EQUIPMENT AND TECHNIQUE OF TESTING
4. Duct and Auxiliaries.tA galvanizediron duct nearly 450
inches long and 10.5 inches square in cross section was used as a model
mine entry. To facilitate the installation and removal of model timbers
185 inches of it were lined with white pine boards, leaving a net cross
sectional area, 9.2 inches square. A suitably tapered approach to and
departure from the lined section were provided, and the lined part
of the duct was equipped with a removable cover. The assembly is
shown in Figs. 1 and 2.
Air was forced through the duct at the desired rates of flow by
a centrifugal fan driven by a variablespeed directcurrent motor.
The fan discharged through a 12in. round iron pipe, the lower part
of which was equipped as an airmeasuring section (t), as indicated
in Fig. 1. From this section an adapter carried the air into the square
duct.
5. Pressure Measurements.By traversing section t a center
constants was established, relating the center velocity pressure at
section t to the mean velocity in the lined portion of the duct, e.g.
at section AA, Fig. 1. It was thus possible to determine the rate of
flow for each test by means of the centervelocity pressure at section
t only, care being taken, throughout the entire system, to guard
against leakage.
*"Unpublished thesis for the Doctorate in Engineering, by the writer, Univ. of Ill. (1935).
tDimensions are given in inches to facilitate comparison of the duct with a corresponding
mine entry. For a twelvetoone enlargement the model dimensions in inches would apply
to its prototype in feet.
TFor further details see Univ. of Ill. Eng. Exp. Sta. Bul. 265, Sections 5 and 6 (1934).
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS
Ae/"a/ Coveer, 12 C7ie BS.
~~1A" 0 Wa, Tol
r B/s ac/ 7 "/,?/eri/47/y
QLa e// 7,2
os
FIG. 1. TEST DUCT
FIG. 2. TEST DUCT, COVERED
The resistance to airflow was gaged for each arrangement of
timbers and rate of flow by measuring the drop in static pressure be
tween sections U and E (Fig. 1). Inasmuch as these sections were of
equal crosssectional area, the loss in total pressure was equal to the
measured drop in static pressure between them. However, not all of
ILLINOIS ENGINEERING EXPERIMENT STATION
Sectiona/ E/lvato1 s Showr0gS Frrnig of Sets an'd ShaCapes 6/sea'
a4Plece Ses b 3Piece Sets cCross 7Bars dCen/er Praos
0 0
1 0 .. 
0v
e Tw o ,'w
e Two Rows
of Props
I07 I
i'
Same Shaes
as in t''
k ae'~~
FIG. 3. FRAMING OF SETS
this loss was due to the presence of timbers in the duct, as there was
some loss of pressure when air was forced through the untimbered, or
clear, duct. This clearduct, or tare loss was determined for each rate
of flow, and subtracted, at like rates of flow, from the gross pressure
loss which was observed between sections U and E with timbers in
place. This gave a net pressure loss which was due solely to the
presence of the timbers.
Pressures were communicated from the duct to inclined differential
draft gages* through minute holes in the duct walls, by means of
rubber tubing. These tubes were secured to the duct by slipping their
ends over hollow brass nipples which were soldered to the duct,
housing the pressure ports.
All pressures were read in inches of water. Those less than one
inch were estimated to the nearest onethousandth inch. Higher
pressures were estimated to the nearest 0.005 or 0.01 inch of water.
Suitable psychrometric and barometric readings were taken to de
termine air density, and all measured pressures were adjusted to the
basis of air of standard weightdensity, 0.075 lb. per cu. ft.
6. Model Timbers.Most of the model timbers were cut from
white pine to standard crosssectional shapes and sizes, although
*For description and illustration see "The Measurement of Air Quantities and Energy Losses
in Mine Entries," Univ. of Ill. Eng. Exp. Sta. Bul. 158, p. 9 (1926).
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS
FIG. 4. TEST DUCT, TIMBERED
some small rolled steel shapes were used. All were of length to fit
snugly into the lined duct, either upright or horizontally.
The crosssectional dimensions of several timbers of each shape
and size were measured with a micrometer, the mean value being
the accepted gage of timber size. While some irregularities were
present, deviations from the mean were moderate, averaging only a
per cent or two of the mean dimension.
The methods used for designating and installing different kinds of
supporting units are indicated in Fig. 3, and the sizes of timbers used
are listed in Tables 3 and 6. Figure 3 not only shows how the different
types of timber sets were framed and inserted in the duct, but it also
shows the shapes and orientations of the individual members con
stituting each type of timbering. The term "timber set" as used in
ILLINOIS ENGINEERING EXPERIMENT STATION
this report includes not only 4piece and 3piece sets, but single
pieces of timbering such as cross bars and props. In the case of two
rows of props a pair across the duct constitute a set.
7. Test Procedure.In preparing for a test, a suitable number of
supporting units (e.g. 3piece sets) each made up of timbers of one
size and shape were spaced at equal intervals along the duct. For
most tests, about the central twothirds of the lined duct was timbered,
although there were many exceptions to this, as will be indicated later.
Timbers were secured to the wooden lining with small nails which
were not allowed to protrude into the air stream. Figure 4 shows a
series of %in. round 3piece sets in place.
After the timbers were set, the lid was fastened down and sealed
to prevent leakage. A current of air was then forced through the duct,
discharging into the atmosphere. As previously indicated, pressure
measurements were made, from which the rate of flow and resistance
were determined. This was repeated for several rates of flow over
a three or fourfold range, for each timber setting.
III. INTERPRETATION OF DATA
8. Standard Rates of Flow.As testing was done at many differ
ent rates of flow ranging from a few hundred to more than 3000 cu.
ft. of air per minute, according to the capacity of the fan and the
resistance of the timbered duct, it was necessary to choose some
standard rate or rates at which to compare resistance for different
conditions of timbering.
Inasmuch as the mean velocity is more expressive of relative flow
conditions than the quantity (cubic feet per minute) and can better
be translated from model to mine conditions, mean velocity of flow
in the untimbered, lined duct was chosen as the mode of expression.
This corresponds with the overall mean velocity previously used.*
The alternative is to use the mean velocity in the clear area within
each timber set, but, as this area changes with changes in the kind
and size of timbering, it is difficult to make valid comparisons among
different timberings at like rates of flow. Two rows of props give a
good illustration of this. When set abreast the clear area in a tim
bered cross section is smaller than when they are staggered (see Fig.
3). This means that the cleararea velocity for props abreast is
greater than for props staggered when a given quantity of air is
*"The Measurement of Air Quantities and Energy Losses in Mine Entries," Part IV, Univ.
of Ill. Eng. Exp. Sta. Bul. 199, p. 11 (1929).
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS
TABLE 1
LIST OF PRINCIPAL SYMBOLS
Symbol Definition
a Coefficient of S in Form 1 equation log R, = aS  b
b Constant in Form 1 equation
c Coefficient of log S in Form 2 equation R, = c log S + d
d Constant in Form 2 equation
S Diameter of conduit = X crosssectional area = 9.2 inches for lined duct which is
perimeter
9.2 inches square
E Static pressure measuring section in test duct, near lower end
f, h Constants in formula n = f + 
n Exponent of V in formula PL = V"
PL Total pressure loss between sections U and E for given duct condition, inches of water
P, Meanvelocity pressure at stipulated mean velocity in duct, inches of water
R Resistance; the ratio of the net total pressure loss due to certain timbering to the mean
velocity pressure in the lined duct (standard at mean velocity of 1500 ft. per min.).
R. Net resistance per set, for a given kind, size and spacing of timbers
R, Zonal resistance; the net resistance due to a specified kind and spacing of timbers through
out an arbitrarily chosen resistance zone, ten conduit diameters in length, 92 inches
of duct
R. Zonal resistance of timbers spaced at the arbitrary standard centertocenter spacing of
one conduit diameter, 9.2 inches in duct
R, Net resistance of timbers adjusted to a mean velocity in the lined duct other than the
arbitrary standard mean velocity of 1500 feet per minute
S Centertocenter spacing of timbers, measured in duct diameters, D
St Centertocenter spacing of timbers, measured in transverse timber dimensions, T
S. Expansive spacing; the space from timber to timber in which the air stream can expand,
measured in units of T
Sma. Spacing of timbers to yield maximum zonal resistance, measured in duct diameters, D
Smi. Spacing of timbers to yield closespacing minimum zonal resistance, measured in duct
diameters, D
T The maximum dimension of the individual timbers composing a set, measured trans
versely to the direction of air flow, inches
U Upstream static pressure measuring section in duct
V Mean velocity of air flow in lined portion of duct, feet per minute
v Mean velocity of air flow in lined portion of duct, feet per second
Kinematic viscosity of air, feet2 per second
Symbol of proportionality
flowing in unit time (cubic feet per minute), and, inasmuch as the
resistance is a function of the mean velocity, the basis of comparison
changes in comparing the resistance of staggered props with that of
props abreast if cleararea velocities are used. However, it obviously
remains the same when the overallarea velocity is used. Further
more, the derivation of tare losses to be subtracted from the observed
gross losses, to give net losses due to timbers alone, is facilitated by
having the mean velocity for both gross and tare losses computed on
the common basis of the overall crosssectional area (0.588 sq. ft.) in
the lined duct.
In reducing the laboratory notes to plotted form, the gross total
pressure loss between sections U and E with timbers in the duct was
plotted logarithmically against mean velocity, as in Fig. 5. The
points are represented by a straight line having a slope of about 2.0.
The tare total pressure loss determined with no timbers in the duct
was similarly represented, check determinations of this line being
14 ILLINOIS ENGINEERING EXPERIMENT STATION
4.00
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/M'ean Ie/oc//y In L/ned DleCt In f. per 1"71/7.
FIG. 5. CURVES OF TOTAL PRESSURE Loss
made from time to time. At a given mean velocity, the difference
between the ordinates of the gross and the tare curves represented
the net total pressure loss due to the timbers.
If both lines had the ideal slope of 2.00, the net total pressure loss
would vary as the square of the mean velocity, and it would only be
necessary to express the net loss at one arbitrarilychosen mean ve
locity. Losses for other rates of flow could readily be calculated by
Newton's quadratic law of resistance which states that resistance to
fluid flow is proportional to the square of the rate of flow. However,
the slope of the lines usually differed from 2.0 by a few per cent, so
that the net total pressure loss did not vary as the square of the
mean velocity. For this reason it was derived for each of three
arbitrarilychosen rates of flow, over a range covering the range of
mean velocities to be expected in the highvelocity zone of a mine.
The mean velocities chosen were 750, 1500 and 3000 feet per minute,
as illustrated in Fig. 5.
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MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS
9. Indexes of Resistance.As formerly,* the ratio of the net
total pressure loss to the meanvelocity pressure is used as an index
of resistance, and the word "resistance" is used exclusively in that
sense throughout this report. It is, however, necessary to prorate
the total net resistance developed by the timbers in any one installa
tion to some standard amount of timbering, to permit comparisons
to be made between different timbering conditions.
Two standards were used, according to circumstances. One of
these is the net resistance of each set, Rs,t the other is the net re
sistance which was developed by a given kind of timbering within a
stipulated length, or zone, of the duct, R_. The first of these is the
quotient of the total net resistance developed by a given number of
timber sets divided by the number of sets.
To set up a suitable criterion of zonal resistance it was necessary
to establish a standard length of timbered duct for comparative pur
poses. This was taken as ten duct diameters (92 in.). A similar
zone underground (i.e. ten entry diameters) would ordinarily be of
the order of one hundred feet in length. Being square, the diameter
of the inside of the lined duct is, according to convention,$ equal to
its side, or, 9.2 in.
Hence, the index of zonal resistance R, is the ratio of the net
total pressure loss caused by timbers of a given size, shape, and kind,
set at uniform spacing throughout a conduit length of ten diameters
to the overall meanvelocity pressure. Since the model timbering
frequently extended over 100150 inches of the duct, the net total
pressure losses were prorated to ten duct diameters before R, was
calculated. A selfexplanatory example of the derivation of R, and R,
from the data of Fig. 5 is given in Table 2. There both criteria
change with the rate of flow, because either line fails to have the ideal
slope of 2.0.
The values of the resistance indexes derived for a mean velocity
of 1500 ft. per min. were used in comparing the resistances of differ
ent kinds of timbering. While such a high mean velocity is not
common underground, it was chosen in preference to a lower velocity
in view of the desirability of comparing mine and model resistances
at like values of Reynolds' Number, wherever possible. As this
would necessitate very high duct velocities in the model some ap
*Univ. of Ill. Eng. Exp. Sta. Bul. 265, p. 15.
tA list of the symbols used in this report is given in Table 1.
IThe diameter of air conduits of rectilinear cross section is commonly taken to be four
times the hydraulic radius, or four times the ratio of the area to the perimeter. In the case of
a square, this is the length of its side.
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 2
DERIVATION OF INDEXES OF RESISTANCE, R, AND R,
Data of Fig. 5
1 Size of timbers .... ...................................... 1inch
2 Shape of timbers ........................................ Square
3 Type of set. ............................................. 3piece
4 Centertocenter spacing....... ............................ 2.12 in.
5 Number of sets ............................................ 31
6 Timbered length........................................ 65.7 in.
7 Mean velocity in duct, ft. per min......................... 750 1500 3000
8 Gross total pressure loss sections U to E, in. water.......... 0. 048 0. 198 0.820
9 Tare total pressure loss sections U to E, in. water............ 0.017 0.063 0.233
10 Net total pressure loss due to timbers, in. water............. 0.031 0.135 0.587
11 Net total pressure loss per zone, in. water.................. 0.0434 0.189 0.822
12 Meanvelocity pressure, in. water ......................... 0.0351 0.1405 0.562
13 Zonal resistance, R,...................................... 1.235 1.345 1.46
14 Resistance per set, R .................................... 0.0285 0.0310 0.0337
Explanation:
Line 5 In conformity with schedule p. 18.
Line 6 = Line 4 X Line 5.
Line 7 Arbitrarily chosen for comparison.
Lines 8 and 9 Intercepts of curves of Fig. 5 at respective velocities of Line 7.
Line 10 = Line 8  Line 9.
Line 11 = Line 92 . A resistance zone is ten duct diameters (10D) or 92 in. in length.
Line 12 = 0.0000000624 (Line 7)2.
Line 11
Line 13 = Line 1
Line 14 = Line 13 X Line 4
Line 14 92
92
proach to such a comparison is obtained through the adoption of a
rather high mean velocity in the lined duct, as a standard.
10. Minimum Resistance per Set.In an indefinitely long duct,
uniformly timbered throughout its length, each timber set would de
velop a resistance equal to that of every other set, so that R, would be
a constant from zone to zone within the duct, and for zones of differ
ent length. To test for the uniformity of Rs in this experimental duct,
the number of timber sets of a given size, shape and kind of timbers,
set at uniform spacing, was varied, R, being derived from each set
of data.
There was usually a variation in R8 with the number of sets,
a minimum value being obtained with an intermediate number.
Some examples of this are shown in Fig. 6. Some decrease in the
resistance per set is to be expected when the number of sets in the
duct is increased from a very few to a moderate number. This is
because a single set, being the only obstruction to flow through the
duct, offers a considerable resistance, but a second set, put in the
wake of the first does not offer an equal additional resistance because
it is not disturbing an otherwise uninterrupted flow.* The initial
*The reduction in the resistance of a sphere which ensues from passing a loop of fine
wire about it in a transverse plane upstream from its center is thought to illustrate this
principle. See "The Generation of Vortices in Fluids of Small Viscosity," by L. Prandtl,
Aeronautical Reprints No. 20, The Royal Aeronautical Society, p. 23 (1927).
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS
Number of Sets
FIG. 6. VARIATION OF R, WITH NUMBER OF SETS
disturbance, and its incident resistance, have already been established
by the first set. Similarly, each succeeding set tends to offer less addi
tional resistance than the preceding one, until a minimum additive
resistance is reached, which is the R, for an indefinitelylong timbered
duct.
This seems to explain adequately the initial drop in R, with in
creasing number of sets, but it indicates that, having reached a mini
ILLINOIS ENGINEERING EXPERIMENT STATION
mum value, Rs should remain constant instead of increasing with
a larger number of sets, as it tended to do in these tests. It was
found that the resistance of a timber set was greater when placed
near either end of the uniform duct lining (Fig. 1) than when placed
centrally within the lined zone, and that it was this end effect, or
additive end loss, which caused the increase in R, with increased
number of sets. This was an error, which could only be avoided by
confining the timbering to the central part of the lined duct.
Having developed curves of upward concavity like those of Fig.
6, for Rs, the only acceptable value of R, is obviously the minimum,
which was used in this work. However, to develop one of these
curves meant testing a given kind of timbering at least four or five
times at each spacing, a very laborious procedure. Study of scores
of similar R, curves showed the locus of the minimum to be a function
of the spacing, so that the following schedule was used in the later
stages of the work, the timbered zone being centered longitudinally
within the lined duct.
CentertoCenter Number of Timbered
Spacing Timber Sets Length
in. Used in.
1 35 35
2 31 62
3 27 81
4 25 100
5 23 115
6 21 126
7 19 133
8 17 136
9 15 135
10 13 130
12 11 132
15 9 135
20 7 140
While it was recognized that the use of such a schedule might give
only an approximate value of the minimum Rs for a given kind and
spacing of timbering, the curves of Fig. 6 show that considerable lee
way in the number of sets used was ordinarily permissible without
incurring serious error. The use of the schedule had the advantage
of eliminating about threefourths of the testing required to procure
data for the curves of R, against number of sets used.
11. Standard Timbering Characteristics.The characteristics of
the timbering used in this investigation are spoken of in this order:
size, shape, type, orientation, and spacing; e.g., 2in. square center
props set diagonally, 4 inches centertocenter. The effect on the
zonal resistance of varying each of these characteristics was deter
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS
mined by fixing all of the variables except the one under consideration,
that one being related to some arbitrary standard condition.
For size, one inch was chosen as the base because it corresponds
with about 1foot timbers in a mine, a size that is frequently used
underground. While each timber was designated by a nominal size
(4in., 1in., etc.) the dimension actually used in computing relative
sizes was the dimension T of the installed timber measured in a
direction normal to the flow. Thus, the average width of 1in. square
timbers was 1.01 in., which is their value of T when set normally,
whereas their average diagonal, 1.43 in., was their value of T
when they were set diagonally. The standard shape, for comparison,
was round, as round timbers are probably most commonly used un
derground. For certain purposes timbers are classed according to
their shape as rectilinear or curvilinear. The former term includes
square and rectangular timbers, and the rolled steel shapes; the
latter includes round, oval, halfround and quarterround timbers.
Cross bars were chosen as the standard type of timbering, as
they give satisfactory roof support at lowest cost for transmission of
air.
Types of timbers or timber sets are grouped into internal or
peripheral timbering. The former are props which are in contact
with the duct only at their ends; they split the air stream in addition
to constricting it. The latter are those timbers which are set in con
tact with the duct walls throughout their length, constricting but not
splitting the current. Crossbars, 3piece sets, and 4piece sets are
peripheral timbers.
The standard centertocenter spacing adopted was one duct
diameter (D = 9.2 inches). This is a little wider spacing than might
appear to be most readily applicable to mine conditions but, as will
appear later (Fig. 9), the rapidity with which the zonal resistance
changes with spacing at the closer spacing of some timbering made it
advisable to choose the rather distant spacing of one duct diameter
as a basis for comparing the effect of varying the other characteristics.
This minimizes the effect of rapid changes in resistance with spacing.
The zonal resistance of a given size and kind of timbering when
set at the standard centertocenter spacing of one duct diameter
(S.= D) is represented by the symbol Re', the more general symbol
Rz representing the zonal resistance at various spacings, as specified to
meet individual situations.
Where two rows of props were used, the spacing of the rows trans
versely across the duct was an added variable which was standardized
ILLINOIS ENGINEERING EXPERIMENT STATION
O.2 04 06 08 /0 ZO
Center to Cenler Svac/l7f, (S), 1,2 Duct Di/mefers, (2
FIG. 7. VARIATION OF R, WITH SPACING
by centering each row in a vertical longitudinal plane of the duct,
equidistant from the adjacent duct wall and the other aligning plane.
Thus each plane was 3.07 in. from the duct wall.
IV. VARIATION OF RESISTANCE
A. SPACING VARIED
12. Resistance per Set, R,.As might be expected, in timbering
the duct with timbers of a given kind (e.g. 1in. round cross bars) at
successively wider spacings, the net resistance per set increases.
This is because each timber set is responsible for an increasingly long
zone of turbulent flow between it and the next downstream set. Thus,
despite the fact that the tare resistance of the untimbered duct is de
ducted from the gross resistance of the timbered duct to yield the net
resistance of timbers, the net resistance per set would be expected to
increase so long as the distance between sets increased, up to the point
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS
0.040
0.020
00/0
0.008
.0 0.006
S0.004
0.02O
Cen2ter fo Centler S1acIAg, (S6, in Duct D/' V7e/ers,
FIG. 8. VARIATION OF R, WITH SPACING AT CLOSE SPACINGS
where the interval between sets exceeded the length of the disturbance
for each set. From this standpoint, the loss per set must approach
a maximum value at some distant spacing, but the test duct was not
long enough to demonstrate satisfactorily the existence of such a max
imum loss per set.
The relation between net resistance per set and spacing is shown
for several timbering conditions in Fig. 7 where R, is plotted against
log S.* Here S is the centertocenter spacing of timber sets, meas
ured in terms of duct diameter D. While some irregularity is notice
able in the plotted positions of the points for each curve of Fig. 7,
it is evident that, for intermediate spacings, each set of data can
be represented by a straight line whose equation has the form
*In this figure and the others in which logarithmic rulings are used (Figs. 5, 7, 8, 9 and 13)
the rulings are spaced in proportion to the logarithms of the numbers they bear, with the
result that the relations indicated by straight lines on these plots are to the logarithms, not to
the antilogarithms which are shown. These are used in preference to the logarithms to indi
cate both absolute magnitudes and relative changes in the abscissas or ordinates, but in
deriving the equations of curves on such plots, the appropriate logarithms must be used
rather than the antilogarithms indicated in the charts. Unless otherwise stated the logarithms
used throughout this report are to base ten.
ILLINOIS ENGINEERING EXPERIMENT STATION
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ILLINOIS ENGINEERING EXPERIMENT STATION
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MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS
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ILLINOIS ENGINEERING EXPERIMENT STATION
R,= c log S + d where c and d are constants, and R,,= ýnet re
sistance per timber set.
The range of this equation is limited in both directions, as indi
cated in Fig. 7. Since the application of the formula to increased
spacing would lead to continually increasing Rs, whereas logic re
quires that R, reach a maximum value at some distant spacing as
already noted, the formula is not, in general, applicable to spacings
greater than about one or two duct diameters, depending on the size
and kind of timbering. The former limit is roughly applicable to
internal timbering and the latter to peripheral timbering. Some sets
o.f data represented in Fig. 7 show the failure of R, to reach the
value required by the formula at the maximum test spacing.
On the other hand, it is often the case that application of the
formula would lead to negative values of R, for very close spacings.
This is particularly true of peripheral timbering, so the formula is
limited to spacings greater than a few tenths of a duct diameter
(Table 3, col. 9). The limit for internal timbering is roughly one
half as great as that for peripheral timbering. The closespacing data
for the peripheral timbering represented in Fig. 7 have been used to
illustrate this limitation in Fig. 8, where the logarithm of R, is plotted
against S. In each case, the three or four points representing the
closest test spacings may be represented by a straight line whose
equation is of the form log R, = aS  b, a and b being constants.
This will be referred to as the Form 1 equation. It is applicable only
to very close spacings, as it soon yields values far in excess of the
observed resistances, if the spacing be increased beyond a few tenths
of a duct diameter.
The curves derived in Fig. 8 have been added to Fig. 7 as broken
lines to indicate closespacing losses which are not expressible by
equations of the form R, = c log S + d, which will be referred to as
the Form 2 equation. It is apparent from Fig. 7 that the transition
from the closespacing equation of Fig. 8 (Form 1, log R, = aSb)
to the Form 2 equation is sufficiently gradual to permit of either
being used without serious error within a range of about 0.1D in
spacing.
Table 3 lists the constants for equations of Forms 1 and 2, and
indicates the approximate range of applicability of each. This table is
essentially a catalog of the results of this investigation insofar as they
relate to the response of resistance to variations in spacing. While
an equation of Form 2 was found to be applicable at intermediate
spacings to all timbering so tested, it was not always feasible to
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS
TABLE 4
DERIVATION OF ZONAL RESISTANCE FROM EQUATIONS OF FORM 1 AND 2
Equation (3), R, =
For 4inch Round Cross Bars
From Table 3, Cols. (5) and (6) Form 1 gives
log R. = 2.73S  2.82
Applying Equation (3)
log R, = log 10 + 2.73S  2.82  log S
log R, = 2.73S  log S  1.82
(1) (2) (3) (4) (5)
S 2.738 log S log R, R,
0.08 0.218 1.097 0.505 0.313
0.10 0.273 1.000 0.547 0.284
0.12 0.328 0.921 0.571 0.268
0.14 0.382 0.854 0.584 0.261
0.159* 0.434 0.799 0.587 0.259
0.18 0.492 0.744 0.584 0.261
0.20 0.546 0.699 0.575 0.266
0.25 0.683 0.602 0.535 0.292
0.30 0.819 0.523 0.478 0.333
0.35 0.956 0.456 0.408 0.391
Explanation:
Col. (1) arbitrarily chosen spacings
Col. (2) = 2.73 X Col. (1)
Col. (3) = log Col. (1)
Col. (4) = Col. (2)  Col. (3)  1.82
Col. (5) = antilog Col. (4)
*locus of minimum R, = 2.73 = 0.159D
=. 17A Olý
From Table 3, Cols. (7) and (8) Form 2 gives
R, = 0.063 log S + 0.039
Applying Equation (3)
0.63 log S + 0.39
S
(i) (ii) (iii) (iv) (v)
S logS S +0.39 R,
0.35 0.456 0.287 0.103 0.294
0.40 0.398 0.250 0.140 0.350
0.50 0.301 0.190 0.200 0.400
0.653** 0.185 0.116 0.274 0.419
0.80 0.097 0.061 0.329 0.411
1.00 0.0 0.0 0.390 0.390
1.2 1.079 0.068 0.458 0.381
1.4 1.146 0.072 0.462 0.330
Explanation:
Col. (i) arbitrarily chosen spacings
Col. (ii) = log Col. (i)
Col. (iii) = 0.63 Col. (ii)
Col. (iv) = Col. (iii) + 0.39
Col. (iv)
Col. (v) Col. (i)
**locus of maximum R, =
antilog (0.434  0.063 = 0.653D
derive an equation to fit the data for close spacings. This was fre
quently the case with internal timbering, in which case the applica
bility of Form 2 equations often extends to very close spacings (e.g.
0.1D for /2in. square center props, set normally).
The coefficient a of Form 1 is highly variable, having a 7fold
range (0.84 to 6.01) for the timbers tested at different spacings.
Neither it nor the constant b seems to vary consistently with
differences in the size or kind of timbering. The approximate range
of b is from 1.2 to 3.1. However, both the coefficient c and the ad
ditive constant d of the Form 2 equation seem to be fairly well re
lated to size and kind of timbering. The former ranges from 0.03 to
1.12, and the latter from 0.021 to 0.601, for the timbering represented
in Table 3.
13. Zonal Resistance, Rz.In considering the resistance of differ
ent kinds of timbering for a mine entry, the item desired is not the
ILLINOIS ENGINEERING EXPERIMENT STATION
Center to Center SVpaci,?, (S) i/, D194t D0iameers, (49
FIG. 9. VARIATION OF Rz WITH SPACING
resistance per timber set, but the resistance for a given length of
entry; i.e. the zonal resistance. For this reason the criterion R. was
set up. It has been defined as the resistance of timbers in a zone ten
10
duct diameters long. With a spacing of S diameters, there are
S
timber sets in such a zone, so the zonal resistance is related to the
resistance per set by the equation R, 10 SR, (3). Table 4 illus
S
trates the use of Equation (3) in deriving zonal resistances from equa
tions of Form 1 and 2. The zonal resistance of the timbering repre
sented in Figs. 7 and 8 is shown in Fig. 9, where R. is plotted semi
logarithmically against S. Semilogarithmic plotting is used to bring
out the relative behavior of the points on each curve as well as a com
parison of the absolute values of zonal resistance. Each point repre
sents an observed resistance, while the curves were derived by calcula
tion, as shown in Table 4. The three lower curves of Fig. 9 are each
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS
Top o/ DAcfi
rTr_,riT TT'_T3T 4  T 5S 
4 6 51.___.I 
Top, rTl  8T T 9 TT"1f
 ý
Square Cross Bars are Represened at V~aried" /oafc/ng
'n4iform S4oacwg on7/ was Useld k te Tests
FIG. 10. POSTULATED LINES OF FLOW ACROSS TIMBERS
divided in two parts, that to the left being concave upward, that to
the right having the reverse concavity. The left part is derived from
the Form 1 equation, and the part to the right from the Form 2 equa
tion of Table 3. A Form 2 equation was developed for every kind of
timbering tested at varied spacing, which means that for each size
and kind of timbering there is a spacing which yields a maximum
zonal* resistance. For those kinds of timbering to which a Form 1
equation applies, there is a close spacing which yields a minimum
zonal resistance. Form 1 equations were developed for more than
threefourths of the peripheral timbering, but for only onehalf of
the internal timbering represented in Table 3.
14. Extreme Zonal Resistances.
(a) Tentative Explanation
The spacing of maximum zonal resistance usually lies within the
range of the Form 2 equation, and such a maximum resistance is
thought to exist for all kinds of timbering, although it was not
demonstrated for some props. The existence of the closespacing
minimum resistance was demonstrated for %in. round cross bars
(Fig. 9) and other peripheral timbering, not illustrated. The exist
ence of these extremes of zonal resistance within a narrow range of
spacing may be tentatively explained with the aid of Fig. 10, which
*Hsu is thought to have been the first to demonstrate the existence of this phenomenon.
ILLINOIS ENGINEERING EXPERIMENT STATION
represents square crossbars placed at spacings which increase in
the direction of airflow. This is for purposes of illustration only, as
uniform spacing was always used in testing. Where air is flowing
across bars in contact, as are bars 1 and 2, its speed is greater than
it would be in the clear duct, and since it is everywhere in contact
with solid surfaces along its upper surface it presumably encounters
a considerable frictional resistance which corresponds with the re
sistance of the closest spacing indicated for peripheral timbering in
Fig. 9. Where there is a small gap between successive bars, as between
bars 2 and 3, the air speed remains essentially as before, but there is
less rubbing surface exposed to the stream because its momentum
carries it directly across such gaps. This may be true for wider
spacings, as between bars 3 and 4, the amount of solid rubbing sur
face per unit length of duct decreasing with increasing spacing. This
decrease in rubbing surface is apparently accompanied by a lowered
resistance to flow, with the result that the zonal resistance is less
with a short gap between successive timbers than it is with the
timbers in contact with each other. However, the air stream soon
tends to expand into the gaps, as shown for the spaces beyond bar
4, Fig. 10. This induces eddying in the interbar spaces with a con
sequent increase in resistance, which offsets the decrease due to less
ened rubbing surface. Thus the decline in zonal resistance is not
only stopped, but the eddy losses mount rapidly as the expanding
stream tends to fill the gap. The result is the development of a close
spacing minimum resistance, as illustrated by the Form 1 curves of
Fig. 9.
A new countertendency appears when the stream fills the entire
crosssection of the duct. The mean velocity is less in this enlarged
stream section than it is abreast a timber set, so that a given quantity
of air can be transmitted with lower losses, resistance to flow being
nearly proportional to the square of the velocity. Increasing benefit
is received from this so long as the length of the interspace between
sets is increased, with the result that the zonal resistance of the tim
bered duct declines with increasing spacing, once the spacing of
maximum resistance has been exceeded. It is probable that the maxi
mum resistance is developed when the clear space between sets is just
great enough for the expanding stream to reach the duct wall and a
timber set, simultaneously, as between bars 8 and 9, because with
wider spacing the stream completely fills the duct for some distance
above each timber set, thereby moving at lower mean velocity and
lower zonal resistance, as just indicated. At spacings closer than
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS 31
TABLE 5
EXPANSIVE SPACING
The air stream has an opportunity to expand in the space between the downstream (to right)
limit of the maximum transverse dimension, T, of one timber set and the upstream (to left) ex
tremity of the next set.
Thus the nonexpansive space which exists abreast each timber set between its upstream extremity
and the downstream limit of its dimension T is to be subtracted from the centertocenter spacing to
give the expansive spacing.
To conform with the expansive theory of flow and to allow for differences in size of timbers of a
given ,hape, the expansive spacing, Se, is measured in units of T. It is related to St* which is the
centertocenter spacing in units of T by the expressions listed below.
Cirti//'r7eer Shgcpes Recf/'/iXeai S/2cpaes
Sh/2aet .S Shape Se S'/e7pet 5e S/26?P t Se
S s,..5 K S S7079
tOrientation is also indicated; air assumed to flow from left to right.
tThis formula also applicable to channels set with long axis vertical.
¶This formula also applicable to channels set with long axis horizontal.
that of maximum resistance less of the stream is intercepted to eddy
in the interspaces. This results in increasingly lowered resistance,
down to the spacing of minimum resistance.
(b) Expansive Spacing
It is clear from Fig. 9 that the loci of the extreme resistances are
quite variable, the maximum zonal resistance of 1in. square center
props, set diagonally, occurring at a spacing of 0.24D while that for
1in. square cross bars occurred at a spacing of 0.96D. If the loci
of the extremes are conditioned by the length and depth of the inter
spaces, as indicated in Fig. 10, interpretation of the data should be
facilitated by expressing spacing in terms of these dimensions, rather
than in terms of duct diameters. This is done by using the transverse
dimension T of the timbers as the unit of measurement, and measuring
in terms of this unit the space between sets into which the air stream
can expand. This expression of spacing is known as the expansive
ILLINOIS ENGINEERING EXPERIMENT STATION
spacing, S,. Thus, in Fig. 10, the expansive spacing between bars 4
and 5 is three times the timber dimension T, or S, = 3T, for a series
of square cross bars spaced uniformly along the duct in this way.
Were round cross bars to be substituted for the square ones, the air
stream could begin to expand at the midsection of each bar instead of
at its downstream extremity, with the result that the expansive spac
ing would be greater by 0.5T, or S. = 3.5T in that case. Similarly,
for each shape and orientation of timbers there is a simple relation
between the expansive spacing and the centertocenter spacing ex
pressed in terms of T. Table 5 lists these relations for the shapes
and orientations of timbers used in this investigation.
(c) Loci
Analytical methods were used to locate and evaluate the extreme
resistances.
The minimum zonal resistance falls within the range of the Form
1 equation, log R. = aS  b, or logeRs = a'S  b' where a'= 2.303a
and b'= 2.303b. This may be written R, ea'8b'. By Equation
10R lOea'Sb'
(3) R7, = , so Rz=  The first derivative of this with re
S S
1 0.434
spect to S, equated to zero, gives Smin , , or Smin ý where
a' a
Smin is that centertocenter spacing of timbers, measured in duct di
ameters, at which the zonal resistance of the timbers is a minimum.
It is confined to the closespacing range of applicability of the Form
1 equations. Three minima of this kind are illustrated in Fig. 9, and
all determined values with their corresponding zonal resistances are
listed in cols. 10 and 11 of Table 3.
As the maximum zonal resistance falls within the range of the
Form 2 equation, R. = c log S + d, the equation for the zonal re
sistance is R. = 10(clogs  . Differentiating with respect to S,
S
and equating the first derivative to zero to locate the maximum, gives
log Smax = 0.434   where Smax is the locus of maximum zonal resist
c
ance. Solutions of this equation are also indicated in Fig. 9, and listed
in Table 3, where col. 13 gives the spacing of maximum zonal resist
ance, col. 14 lists the maximum resistances themselves, and col. 15 re
lates them to the zonal resistance at the standard spacing of one duct
diameter.
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS
The mean of the loci for the closespacing minimum resistance
(col. 10, Table 3) when expressed in terms of expansive spacing Se was
1.3T, with individual values ranging from 0.3 to 2.9T. The locus does
not seem to vary systematically with size or kind of timbering.
The locus of the maximum zonal resistance was better defined than
that of the minimum resistance, due in part to the larger number of
determinations. To analyze this locus, timbering was divided into two
types, peripheral and internal, and timbers into two shape groups,
those of rectilinear and those of curvilinear crosssections (see p. 19,
and Table 5).
The mean locus of the maximum zonal resistance varied as follows:
Type
Item Shape Peripheral Internal
1 Curvilinear.................. 6.1 0.9
2 Rectilinear.................. 7.5 2.5*
3 Square props set diagonally.... ... 1.1
These loci are expansive spacings, measured in terms of T. The
schedule shows that there is a marked difference between the locus of
maximum zonal resistance for peripheral and internal timbering, a
difference which is qualitatively in harmony with the expanding
stream theory which has been advanced (p. 30) to explain the
existence of a maximum resistance. The tentative postulate of that
theory is that, in the case of peripheral timbering, the maximum re
sistance is developed when, after clearing one timber set, the stream
expands to reach the duct wall just as it encounters the next timber
set.
Applied to internal timbering this would mean that the schism set
up in the stream by a given prop would just be closed by the combined
expansion of the two streams, as they encountered the next prop. If
the expansions were linear with respect to travel along the duct, and
at the same rate as with peripheral timbering, the two streams would
obviously unite in half the travel with internal as with peripheral tim
bering. Hence the locus of maximum zonal resistance would be ex
pected to be onehalf as great for internal timbering as for peripheral
timbering. Actually it lies at spacings from /7 (for curvilinear tim
bers) to 1/3 (for rectilinear timbers) as great, so that other factors
than those indicated enter into the determination of the locus of
maximum zonal resistance for internal timbering.
Experiments with water flowing past a circular cylinder have
shownt the point of confluence of the two streams to lie about one
*Exclusive of item 3.
f"Applied Hydro and Aeromechanics." L. Prandtl and 0. G. Tietjens. Plate 1, Fig.
6, p. 279 (1934).
ILLINOIS ENGINEERING EXPERIMENT STATION
cylinder diameter behind the cylinder. If this were true of the turbu
lent flow of air past a cylinder, round props would, according to the
expandingstream theory, be expected to develop their maximum re
sistance at an expansive spacing of 1.5T, rather than at 0.9T as shown
in the schedule. There is no apparent explanation for the closeness of
spacing at which such props develop their maximum zonal resistance.
Item 3 of the foregoing schedule shows that square props when set
diagonally behave more like curvilinear than rectilinear props in this
respect.
In peripheral timbering, the curvilinear timbers developed their
maximum resistance at an expansive spacing of 1.4T less than those
of the rectilinear timbering, on the average. This unexplained differ
ence is in harmony with the tendency in internal timbering for the
curvilinear timbers to develop their maximum zonal resistance at
closer spacings than the rectilinear timbers, and at closer spacings
than would be expected from the simple expandingstream theory.
(d) Magnitudes
For a given size and kind of timbering the minimum zonal re
sistance is roughly onehalf to twothirds of the maximum resistance
which is developed at about twice the spacing of the minimum resist
ance. Individual values of the minimum zonal resistance (col. 11,
Table 3) range from 0.14 for % x 1in. cross bars lying flat to 3.40
for %in. square center props set normally. However, many kinds of
timbering tested failed to yield a Form 1 equation with the ensuing
minimum zonal resistance. The ratio of the closespacing minimum
resistance to the zonal resistance at standard spacing (col. 12, Table 3)
averages 1.24 for peripheral timbers, and 0.62 for internal timbering.
On the other hand, the ratio of maximum zonal resistance to that
at standard spacing (col. 15, Table 3) for internal timbering averaged
2.4, just twice the corresponding mean of 1.2 for peripheral timbering.
This is expressive of the steep slope of the Form 2 curves representing
internal timbering, in Fig. 9. Individual values of the maximum zonal
resistance (col. 14, Table 3) range from 0.25 for the % x 1in. cross
bars to 6.32 for two rows of 1in. round props set abreast.
15. Varied Transverse Spacing. The term "transverse spacing"
is applicable only to multiple rows of props. It refers to the separa
tion of the rows from each other and from the duct walls. Only two
rows of props were used in testing, each row ordinarily being centered
along a vertical longitudinal plane of the duct, 3.07 in. from the side.
The effect of varying the transverse spacing was tested for both ar
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS
CenOra/ G Tap N2 inches
FIG. 11. VARIATION OF R, WITH TRANSVERSE SPACING
rangements of two rows of props shown in Fig. 3, i.e. props set abreast,
or staggered. In each case, oneinch square props, set normally, were
spaced twelve inches, centertocenter. The results are shown in
Fig. 11, where the ratio* of the zonal resistance of the props at each
transverse spacing to their resistance at standard transverse spacing is
plotted against a function of the transverse spacing. This function is
called the central gap. It is the distance between the planes of the
inboard faces of the two rows of props, the inboard faces being those
surfaces of the props which are parallel to and nearest to the center
line of the duct. At the standard transverse spacing, oneinch square
props set normally leave a central gap of 2.06 inches.
(a) Rows Abreast
In testing for the effect of varying the transverse spacing of two
rows of props set abreast, they were first placed in contact with each
*Later referred to as the relative resistance. Since there is little difference between the
zonal resistances of two rows of props set abreast or staggered, curves of absolute rather than
relative resistance for these two timbering conditions would be like those of Fig. 11 in all
essential respects.
ILLINOIS ENGINEERING EXPERIMENT STATION
other along the center line of the duct, as illustrated by timbering
condition B, Fig. 11. After their resistance to the flow of air had
been determined in that position over a suitable range in the rate of
flow, a small central gap was made between the rows, and a new
series of tests was run. The central gap was successively increased
from series to series of tests until a row of props was set against each
duct wall (condition D, Fig. 11).
The results are shown in Fig. 11 by the solid curve which has a
surprising similarity to the curves of Fig. 9, relating zonal resistance
to the longitudinal spacing of timbers. A maximum resistance (point
B, Fig. 11) was developed with no central gap between the rows. The
least separation tried (%in.) permitted the resistance to drop 40
per cent, whence it reached a narrowgap minimum at a central gap of
virtually one inch. Here the resistance was 82 per cent of that at
standard transverse spacing. As the central gap increased the re
sistance reached a maximum value at a gap of about three inches.
At this transverse spacing the resistance was eight per cent greater
than at the standard spacing. It dropped rapidly from this level until
it reached a widegap minimum with the props set against the duct
walls (point D, Fig. 11).
(b) Rows Staggered
With the rows staggered, the relative resistance is in essential
agreement with the resistance of props abreast for central gaps greater
than one inch, so that in this range the solid curve of Fig. 11 repre
sents the relative resistance of either timbering condition, abreast or
staggered. However, for closer gaps there is a marked variance be
tween the two resistances. With staggered rows the relative resistance
continued to decline from the point of maximum resistance beyond
the point where there was no central gap (point B', Fig. 11). Its
behavior in this range is illustrated by the broken curve.
When two staggered rows of props are brought together beyond
the setting of no central gap (condition B') so that each is centered
along the center line of the duct (condition A') they constitute a
single row of center props set at half the centertocenter spacing
which exists in each of two separate staggered rows. In the singlerow
arrangement they offer their minimum resistance (point A') as is to
be expected from the fact that they thus present their minimum ex
posure to the air stream. By exposure is meant the area of both
props of a given pair (one from each row) projected on the cross
section of the duct. Its minimum width was one inch in these tests.
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS 37
As the two rows are removed from their central position their ex
posure increases to a maximum (width two inches in these tests) at
no central gap (condition B'), and this exposure is retained for all
wider central gaps.
Since the curve of relative resistance for staggered rows passes
through point B' (Fig. 11) without inflection, it is apparent that ex
posure alone does not play a highly significant role in relating re
sistance to the transverse spacing of two rows of props, when stag
gered. A similar statement applies when the props are set abreast, for
then the exposure is constant for all transverse spacings.
Neither can the number of streams flowing through a timbered
crosssection of the duct be a major determinant in the relation of
resistance to transverse spacing, for, in general, there are two streams
with staggered rows and three streams with props abreast, yet the
resistances for these two timbering conditions are in essential agree
ment, except at small central gaps.
(c) Postulated Conditions of Flow
It is possible that the major features of the resistance curves of
Fig. 11 may be explained as follows:
With props set abreast against the duct walls the central gap is a
maximum and the resistance is comparatively low because the flow
is in a single central stream. As the props are removed from the wall,
the central gap is restricted with the result that. the central stream is
narrowed and its resistance is increased accordingly. At the same
time, two subsidiary streams are opened along the duct walls and they
take an increasing proportion of the flow, thereby moderating the
duty and the resistance of the central stream. Part of their ability
to do this may be attributable to the fact that each side stream is
bounded by three smooth duct surfaces, roof, wall, and floor, whereas
the central stream is so bounded by the roof and floor only.
Nevertheless, as the central gap is decreased the total resistance
of the system increases until the gap is about onethird of the width
of the duct. As the gap is further narrowed the side currents are
widened to such an extent that their favorable performance leads to
a reduction in the total resistance. This reduction is increasingly effec
tive until resistance to the flow of the central stream begins to be
doubly augmented, first by continued narrowing, and second by a
mutual influence which is exerted between each pair of props it en
counters. As the props approach each other across the center line
their individual eddies begin to merge and thereby add to the resist
ILLINOIS ENGINEERING EXPERIMENT STATION
ance which is offered to the flow of the central stream between the
props. This effect increases until the gap is closed by putting the props
in contact with each other along the center line (condition B, Fig. 11).
At this setting, there is no longer a central current, and the two side
streams are forced to carry the entire flow. It seems likely that the
highest velocities are developed under these conditions, and these
contribute to the development of the maximum relative resistance for
two rows of props.
Since staggered rows produce essentially the same resistance as
props abreast for transverse spacings in excess of one inch, it seems
probable that much the same explanation applies to their flow con
ditions. At narrow central gaps the mutual interference of the props
as they approach each other across the center line is avoided by
staggering, and, at the same time the two streams which flow past
each prop are more nearly balanced as the rows become more nearly
centered along the center line. The final result is that the minimum
resistance is offered when two staggered rows have been merged into
a single row of center props (condition A', Fig. 11).
Point E of Fig. 11 furnishes an interesting comparison with
point D, the latter representing the relative resistance of two rows of
props set abreast against the duct walls. This was modified in timber
ing condition E by removing the west row of props from the west wall,
and setting each prop against the corresponding one in the east row,
which in turn was against the east duct wall. Thus, only one stream
flowed through a timbered cross section in each case, and the net
crosssectional area was unaffected. Nevertheless, the resistance was
appreciably lower for condition E than for condition D. Under the
latter condition the stream was constricted on both sides, but in the
former the same total constriction was applied on only one side, with
a consequent reduction in resistance. Apparently the saving which
resulted from clearing the flow along one duct wall was in excess of the
additional loss which resulted from simultaneously doubling the ob
struction on the opposite side.
As further testing at varied transverse spacing was not completed,
it is impossible to say what effect differences in size, shape, orienta
tion, and longitudinal spacing would have on the resistance of multiple
rows of props when tested at varied transverse spacings.
B. AT STANDARD SPACING
16. Size of Timbers.The zonal resistances Rz' of the different
sizes of timber used in each shape, type, and orientation, at standard
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS
TABLE 6
ZONAL RESISTANCE OF TIMBERS AT STANDARD SPACING*
Round T/'wzbers
/Vomine/ S/ze, (hn.)  / /
Raio.: T7D 0.056 0.069 0.082 0.//0
Cross Bars 0.25 0.35 0.39 0.56
3P/ece Sets 1./3 1.43 1.86 Z.71
4P/ece Sets .56 2.3/ 3.04 4.55
Center Props 0 55 0.72 0.89 /.05
2 Rows Abreast /.3 1.70 1.82 2.57
SRows Staaered /.29 1.56 2./0 2.37
Oval 7Tnbers
/Vomina/ S/ze /
Or/,entat/on t 0 0
Rato. T+D 0.107 0.//115
Cross Bars 0.44 0.6/
3Piece Sets .5/ 
Center Props 0.89 
Z Rows Abreast 2./17
Ha/f?ound T/'mbers
/omina/ Size z /
Rati/o: TD 003Z 0.043 0.059
Cross Bars 0.024 0.18 00.6
3P/ece Sets 046 066 /.09
4Piece Sets 0.6 e 099 /.58
/tnch quarterRouna' Timbers
Oriertfati/on _ J U 0 <0
Cross Bars 033 0.50 
Center Props   /1.64 1.73
E Rows Abreast   4.59 438
2 Rows StagYered   4.26 40/
Square Timbers
/omi/na Si/e, (2n.) g A z J i if / ~/
Orientatlon D 0 D 0 O O O O O
Rat/o.: 7'T+D 0.043 0.055 0.077 0.072 0098 0.85 0./ 0.//0 0.155
Cross Bars 0.1/9 0.31  0.41  0.49  0.62 
3Piece Sets 0.84 /.9  1.94  2.57  3.57 
4P/ece Sets 1.30 1.91  3.06  99  60/ 
Center Props  0.6 0.90 0.83 1.26 .07 1.65 1.3Z 2.37
2 Rows Abreast  1.70 2.30 2.18 3.53 250 4./6 3.6/ 646
2 Rows Staggered  1/.70 2.5/ Z./ 3.0 2.70 4.12 3.57 6.13
Ro//led Steel Bars
A/omAna/ Sze, (iz.) I  / / / I
Shfape Or/enltat/ot7 ] E I 3 E I H
Ratio: T.D 0.50 03 0.503 0.503 0.998 0.998 0.998 0998
Cross Bars.5 0.8 26 0.25 0.70 0.66 0.60 0.44
* Tabular values R are. e total pressure /oss per /ODL, (92i) at mean
Heai7nveoc/ry ^nprssure
veloci'ly of /JO mer /wo;. n // l/fe al' , with '/mber spaced oD cenmerlocenter.
t Air assnme to fow' from /eft to ri/gh
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 7
RELATIVE RESISTANCE OF TIMBERS ACCORDING TO SIZE*
Round T/mbers
A./omna/ Size, C/'.})  f /
Rai/o T+D 0.056 0.069 0.08Z 0.110
Cross Bars 0.45 0.6Z 0.71 1.00
3P/ece Set 0.41 0.53 0.72 /.00
4P/ece Sets 0.34 0.5/ 067 1.00
Cenfter Props 0.52 0.68 0.84 .00
2 Rows Abreast 0.52 0.66 0.7/ /00
2 Rows Staggered' 0.57 0.66 0.89 /.00
Sqcuare Timbers
Or/en7tat/on D 1 D0 0 0 0 0 D
Rai/zo; T+D 0.043 0.055 0.077 0.072 0.098 0.085 0.120 0.11/0 0.155
Cross Bars 0.31 0.50  0.65  0.79  1.00 
3P/ece Sets 0.23 0.36  0.53  0.75  1.00 
4P/ece Sets O.2Z 0.32  0.50 0.71  1.00
Ce7nter Props  0.5/ 0.38 0.57 0.53 0.87 0.70 00 1.00
2 Rows Abreast  047  0.6/ 0.70  .00 
Z Rows Sfagered'  048 0.41 0.59 049 0.76 0.67 00 /.00
Ro//ead Stee/ Bars
/ominal/ /ze, (/'.) 5 1 / /,
Sha'ee4Ori/entat/onr t 3 C I 3 L I
Rai'1: T+D 70.S 0.055 0.055 0.0 108 0./08 0.08 0.108
Cross Bars 0.41 0.38 0.43 1.00 1.00 1.00
* Tabular vlaes are,. f' g sl:"e for ea/" 01r 1'oimhrmi.
Sfor oe/nch1 (/Voimna/) (oreach /id o/hmbe
tO r/er.'ta/on is incdicated b9 crosssectiona/ d/agfrars, airf/ow assumed'
to be from /eft to rght
spacing are listed in Table 6. It shows that, for all timbering so tested,
there is a rapid increase in resistance with increasing timber size, as is
to be expected. To better judge the nature of the relation between
timber size and resistance, Table 7 has been derived from Table 6 by
treating the zonal resistance of oneinch timbers of each kind as the
base (unity) and relating to it the resistance of smaller timbers of
that kind. The variations in the resulting ratio from line to line in
Table 7 indicate that, to a considerable extent, the relation of resist
ance to size differs according to the type and shape of timbering. For
example, ratios for onehalfinch cross bars range from 0.38 for chan
nels set with their flanges downstream to 0.50 for square bars. Such
variations mean that type and shape factors cannot be wholly elimi
nated in determining the effect of the size of timbers on their relative
resistance.
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS 41
TABLE 8
RELATIVE RESISTANCE OF TIMBER SETS ACCORDING TO SHAPE
AND ORIENTATION OF TIMBERS*
ap Cross 3PAce 4Pece Cener 2 ws 2 1 Aows e
Shae Bars Sets e Props Abreast Sta erea' r
SInch T/7mbers
S /.o00 /.00 .00 /.00 /.00 1.0o /.0
Q 0.88 0.8e6 096    g090
O 1.Z5 1.17 /.4 1.16/ /.e /.26f /.z3
1.14     
E 1.08  . 
/ /nc1y T/'mbers
o /I.OO /.OO .OO i.oo .oO /.OO /.OO
o /.// 1.31/ /.27 /./O /.23 /.7 /.2
fi/nch T/mbers
o /.00 /1.00 /.OO /. 1.00 /.00 /.00
Q 0.8     
1.24      
O /.2/ 1.36 1.38 /.8 1/.33 1.24 /. 2
0    1.08 /.38 1.26 /.24
//nch T/mbers
0 /.00 1.00 OO 1.00 00 1. 00 1.00
0 0.79 0.97  0.87 0.87  0.88
0 .02      
0    1.55 1.70 1.69 1.65
0    1.48 /.62 /59 /.56
S /.e 1.34 1.33 /.16 1.39 /1.48 /.30
0    1.38 /.50 1.48 1.45
I 1.06     
3 1.23   
E 1Z6      
S 1.18     
* Tabular va/ues are
RA,' for nad/cated s/e,, sheve,
or/ental'No anad Alve
Az' for rouna' of/that s1e ano' toe
t OrientatI'n /1s ndCicateda bye crosssectiona/ da'/grams, airf/ow assurned
to be from left to ri'hf.
/nprepar/iy th tabl/e, the R "s of Table 6 were ad5'asted to nomnal/
Ts in cdiect propor/ton to the actuca T. , e.y. For 4." round' the actual
T Is 05//1 in., or vwrtcar// 2 per cent greater than the nom/na/ Tof .00;/7.
Hence, ii7 prearAy' tis 7tab6/e al// a/les of Tabl/e 6 re/a'g to P1. round
timbers were arbitrar//y re'ducea' per cent. S/m/acrr/y for other sizes and
shapes.
One partial generalization that results from Table 7 is that the
resistance relative to size increases more rapidly than the size, e.g.
doubling the size of round fourpiece sets (from onehalf to one inch)
nearly triples the relative resistance (0.34 to 1.00). This is true for
most other shapes as well, although, for some of the internal timbering,
the relative resistance is nearly proportional to the size.
ILLINOIS ENGINEERING EXPERIMENT STATION
17. Shape and Orientation.An attempt was made to bring out the
effect of differences in shape and orientation of timbers by relating
the zonal resistance of timbers of a given size, shape, and orientation
to that of round timbers of the same kind. This comparison is made
in Table 8, which lists the ratio of Rz' for the indicated size, shape,
orientation, and kind of timbering to the R,' of comparable round
timbers.
Halfround, quarterround, and oval timbers have less resistance
than the corresponding round timbers in certain orientations. This is
true of the oval timbers only when their major axis is parallel to the
direction of airflow. It is probably true of the halfround timbers only
when their chord is against the duct wall, the single position in which
they were used. For %in. quarterround cross bars, the relative
resistance was 0.82 when the bars were set with the arc upstream,
but 1.24 when set with the arc downstream. This is an increase of
50 per cent in resistance, with no change in supporting capacity. Set
as props, with the midradius in the direction of flow, the quarter
round timbers had very high relative resistances, from 50 to 70
per cent more than for corresponding round props. Their resistance
was greater with the arc set upstream than with it set downstream,
the reverse of what occurred when they were used peripherally as
cross bars. The comparison is not direct, however, as the midradius
was inclined to the direction of flow in the latter case, but parallel
to it in the former.
The rolled shapes as cross bars have up to 26 per cent more re
sistance than round cross bars of the same T, their relative resistance
increasing with their size.
Square timbers, whether set normally or diagonally, have from
8 to 50 per cent more resistance than the corresponding round timbers,
a rough average being 25 per cent excess resistance. Square timbers
are set diagonally only as props, and the results for the diagonal and
the normal orientation of such props are inconsistent. In some sizes
the diagonal orientation gave higher and in other cases lower resist
ance than did the normal, comparison being made at the same T, of
course. To turn a given square prop from the normal to the diagonal
orientation increases its T 41 per cent and its resistance accordingly,
but the data are not clear as to the general effect of replacing a given
square prop set normally with one having the same T when set
diagonally.
18. Type of Set.The relative resistance of different types of
timber sets is brought out in Table 9 which is patterned after Table 8.
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS 43
TABLE 9
RELATIVE RESISTANCE OF TIMBER SETS ACCORDING TO TYPE*
Cross 3PL'ce 4P/ece CeInter 2 Rows 2 Rows
Shape 8r Sets Sets Props Abreast Sftaqered
/n7cT 777?ber~s
0 .0 67 9.3 3.3 7.9 8.1
C 1.0 43 6.6
O /.0 4.2 6.1 2.0 5.4 5.4
[/nch T/mrbers
o0 . 4.3 7.0 2.2 5.2 4.7
0 /.0 4.8 7.5 2.0 5.3 5.
j /Inch T/mbers
O 0.0 4.8 7.4 2./ 4.4 51
O /.0 5.6 6.8 2.2 5.1 5.5
//nch7 T/mrbers
0 IO. 4.8 8.0 /.9 4.5 4.1
0 1/.0 5.9 2.0 5.0
0 /.0 5.9 9.8 2.0 .5.8 5.8
Average /.0 5.1 7.7 2.2 5.5
. 7~ v,& a (? for 'nad/caf ed tfOe, sIze a7?d sba, e
"blar /' for cross bars of thoa s/2e ad sqhape
\ Orietati/on' is a/so indico/eda' 1, crosssecf/ona/ d'o~'raoms, a/rflow
asseume&d o lbe froaw left ;o r/Igqt.
In Table 9 the resistance of a cross bar of a given size, shape, and
orientation is regarded as unity, and the resistances of other types
of sets of that class are related to it. The ranking of the types in the
order of increasing resistance, is cross bars, center props, 3piece sets,
2 rows of props, and 4piece sets, their mean relative resistances being
1.0 for cross bars, 2.2 for center props, 5.1 for 3piece sets, 5.5 for
2 rows of props and 7.7 for 4piece sets. The cross bar is by far the
most desirable form of timbering insofar as the economy of trans
mitting air through the passageways is concerned.
19. Effect of Lining.In an underground test,* a zone of 3piece
sets was sheathed with boards along the three faces of the sets to give
a smoothlined test length in which the resistance was found to be
much less than it was in the unsheathed zone. A somewhat comparable
test was made in the laboratory by lining a zone of oneinch oval
3piece sets, spaced 6 inches centertocenter, with muslin stretched
tightly along the three faces established by the inside surfaces of the
timber sets. Although such a lining reduces the available cross sec
*"The Measurement of Air Quantities and Energy Losses in Mine Entries," Part IV,
Univ. of Ill. Eng. Exp. Sta. Bul. 199, pp. 22 and 35 (1929).
ILLINOIS ENGINEERING EXPERIMENT STATION
tion between timber sets, thereby increasing the mean velocity of
flow through the zone, it reduced the zonal resistance of the timbers
from 1.26 to 0.50. Evidently, the reduction in turbulence which ensued
from smoothing the passageway more than offset the increased re
sistance due to the higher rate of flow.
The responsiveness of resistance to the nature of the rubbing sur
face was indicated by two tests which were made with all four surfaces
of the untimbered duct lined with coarse (No. 31/2) sandpaper, which
averaged about %6 inch in thickness. In the first test the paper was
turned with its smooth side to the air, and in this position its net zonal
resistance Rz was 0.05. In the second test the paper was turned with
the coarse, sanded side to the air, and its Rz rose to 0.21, which is
more than four times as great as when the smooth surface of the paper
was exposed to the air stream.
C. RATE OF FLOW VARIED
20. Pressure Losses.In dealing with the resistance of fluids in
turbulent flow it is commonly assumed that the resistance is pro
portional to the square of the rate of flow, but scores of totalpressure
loss curves, plotted logarithmically like those of Fig. 5, show that the
loss was frequently proportional to a power of the mean velocity ap
preciably greater or less than 2.0. For each timber condition the total
pressure losses were represented on logarithmic ruling by a straight
line whose slope is the power of the mean velocity to which the losses
are proportional. In many instances careful logarithmic plotting of
test data to large scale indicated a line of moderate curvature, usually
of upward concavity, but this curvature was not of a degree to in
validate the straightline approximation, which was applied to each
set of test data.
Accordingly, in the logarithmic plot of Fig. 5, the derived net total
pressure loss is represented by a straight line, although analysis shows
that if the gross and tareloss curves are straight lines of different
slopes, as they usually were in this work, the netloss curve cannot be
a straight line. The deviation from linearity is, however, negligible
over the range in rate of flow which was considered in this work. The
result is that net pressure losses, i.e. the losses due solely to the
presence of timbers in the duct, are also proportional to a power of
the mean velocity, or PL oc V", where n is within a few per cent of
2.0. V is the mean velocity of air flow in the lined duct, and PL is
the net total pressure loss due to timbers, in a given test.
21. Resistance.Were the quadratic law to hold for net total pres
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS
sure losses, resistance as here used would be independent of the rate
of flow. This is because resistance is the ratio of the total pressure loss
to meanvelocity pressure, which in turn is proportional to V2, so that
if the total pressure losses were also proportional to V2 the resistance
would be independent of the rate of flow. Inasmuch as the pressure
losses due to timbers frequently are not proportional to the square of
the mean velocity it is necessary to adjust the corresponding resist
ance for variations in rate of flow for complete accuracy. This may be
done as follows: PL cc Vn and P, cc V2 where P, = meanvelocity
PL V"
pressure of air flowing through the duct. Since R = , R cr cc V"2.
In this work, R has been determined at a standard mean velocity of
1500 feet per minute in the lined duct, its most common form being
Rs, the zonal resistance of certain timbers at a specified centerto
center spacing. Having determined n experimentally, R, may be
adjusted to a mean velocity other than 1500 feet per minute by means
of the relation noted. Representing the resistance adjusted to the
velocity V by R,, R, = R, (
If, for the data of Fig. 5 and Table 2, R, were known to be 1.345
with n = 2.12, and it were desired to derive the zonal resistance at a
mean velocity of 750 feet per minute, Ro  1.345 X 0.50°12 = 1.238.
This is in close agreement with the values derived by graphical
solution (Fig. 5 and Table 2). It represents nearly an 8 per cent re
duction in R, for a 50 per cent reduction in mean velocity, which em
phasizes the significance of deviations from the quadratic law.
In general, a deviation in n of a few per cent from 2.0 will produce
a relative deviation in R, or Rz about onethird larger, if the rate of
flow be halved or doubled. The resistance increases with the mean
velocity when n is greater than 2.0. The foregoing example illustrates
this principle, for an excess over 2.0 of 6 per cent in n (n = 2.12)
yielded an 8 per cent decrease in resistance (R, or Rz) when the rate
of flow was halved. Had the excess in n been twice as great, or 12 per
cent (n = 2.24), the decrease in resistance would have been nearly
16 per cent, when the rate of flow was halved. Similarly, there would
be an increase in resistance of nearly 16 per cent in doubling the flow.
For changes in flow other than halving or doubling, the changes in
resistance are proportional to the relative changes in flow as indicated
by the formula R,, = Rz 5 1 5
1 1500/
ILLINOIS ENGINEERING EXPERIMENT STATION
I I I , I I ,
Genera/ Formul/af, n= f+
0
Type
3P/ece Sets
CrossBarys
Center Props
f Rows Abrea'st
I I.,, II
o, \4
b 2.24
.3
I
Z o
S1
2.0
ii
t^e.
Shapoe S"e. (g.) f
0 z Z.05
o 0 1.85
o / /190
0 Z e.o
0 I /.85
o f /.o
o T1.70
0 / 180
0 2 2.00
O / 2.00
0 ~ /80
0 f /?0
0 § 1.75
oinrn
a,,
~ 4Vcze/er 9oc't2d Cet2fer Props
P/e7Ced.~~c~
0020
0./00
0.067
0.0/7
o967
O.0/5
002oe5
0008
0.0/7
0.0/7
0.01/6
U Of 0.4 ua 0 8 . /.A Z9 ,q 1. 1 18
Center /o Cew/er Spac/ngk, (5), 1n D7 cf D/ie/ers, (D)
FIG. 12. VARIATION OF n WITH SPACING
22. Behavior of n.For a given size and kind of timbering, n
usually tends to decrease with increased spacing although in some
instances the opposite trend was evidenced, and in others the slope
was so erratic as not to indicate any systematic response to the varied
spacing of timbers.
A set of data illustrating a decreasing n and one illustrating an
increasing n are shown in Fig. 12, where n is plotted against S. In each
case the data have been roughly approximated by a curve whose
h
formula is n = f + where f and h are constants, and S is the center
S
tocenter spacing of the timbers in duct diameters; h may be positive
or negative. Such an equation represents an hyperbola, asymptotic on
the n axis and on the line n = f, parallel to the S axis. Figure 12 lists
the constants for formulas of this type which were determined in this
investigation.
. = 1/ + . I
/
0
*
o for Rouno', 3P/ece Sets
\ I/0^g  ~ '"'
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS
At the standard spacing of one duct diameter, there is a slight
tendency for n to decrease with increasing size of a given kind of
timbering, but the trend was too indefinite to permit evaluation. There
are, however, marked differences in the slopes of curves representing
the losses of peripheral and internal timbering and of rectilinear and
curvilinear timbers, the former exceeding the latter by three or four
per cent in each case. This is brought out in the following table:
Mean Values of n for Timbers at Standard Spacing
Type of Timbering
Shape of Timber Peripheral Internal
Rectilinear .................. 2.02 1.95
Curvilinear .................. 1.95 1.87
V. APPLICATION OF RESULTS
23. Extension of Data.Since the Form 2 equation is applicable
over the range of spacings most likely to be used underground it is
useful in estimating the resistance of timbering of a given kind at
various spacings. If the resistance at one spacing is known, this
locates a point on a plot of R, against log S such as Fig. 7. The co
efficient c of the Form 2 equation is proportional to the slope of the
line representing the formula, so the relation between R, and S can be
determined from one known resistance and the coefficient. For this
reason the values of c (col. 7, Table 3) which were determined in this
investigation are useful in estimating the resistance of timbers at
various spacings, when tests have been made at only one spacing.
The additive constant d (col. 8, Table 3) is similarly useful in
estimating the resistance of sizes, or of types and shapes of timbering
which have not been tested at the standard spacing. Both it and the
coefficient c increase rather regularly with increased timber size, and
respond to changes in kind of timbering with some consistency.
A useful peculiarity of the constant d is that it is the resistance
per set at the standard spacing of one duct diameter, for log S is zero
at that spacing, and the Form 2 equation becomes R8, d. Further
more, d is onetenth of the zonal resistance at standard spacing Rz'
because there are ten sets in a resistance zone ten duct diameters in
length, at that spacing.
To illustrate the use of the Form 2 constants, c and d, it may be
assumed that the only information available concerning the resistance
of square center props set normally is that given in Table 3 for the
%in. and %in. sizes, whereas the resistance of the corresponding
1in. square props is needed. The first step in estimating the Form 2
ILLINOIS ENGINEERING EXPERIMENT STATION
equation for the 1in. props is to estimate d from the values of d for
1/2in. and /4in. props. These values are 0.062 and 0.107, respectively,
and straightline extrapolation to 1 inch gives 0.144 for d. This is
nearly ten per cent above the observed value of 0.132.
Similar treatment yields an estimated value of 0.164 for c, which
is about five per cent above its observed value of 0.156. The resulting
estimated Form 2 equation for 1in. square center props set normally
is R, = 0.164 log S + 0.144. The line representing this equation has
been added to Fig. 7. Estimates of resistance taken from this line are
higher than those from the accepted mean line as determined by
observation, largely due to the excess in the estimated d over the
observed d.
An estimated line of this kind is drawn by first plotting the value
of d at its proper ordinate and the abscissa of standard spacing, D.
For the estimation just made, it is shown as point d in Fig. 7. The
slope of the line which is to pass through this point is proportional
to c, the constant of proportionality being 5.0 for drawings to the
scale of Fig. 7. Hence the representation of the estimated Rs of 1in.
square center props set normally is a line through point d having a
slope of 5c or 0.82, as shown.
The method of procedure outlined for estimating the resistance of
1in. square props can generally be used with a fair degree of accuracy
to extend the data of the investigation, where a specific timbering
condition has not been reported.
24. Composition of Resistances.If to the zonal resistance of two
rows of 1in. square props set 12 inches centertocenter along the
duct walls (condition D, Fig. 11) be added the resistance of 1in.
square cross bars equally spaced, the result might be expected virtu
ally to equal that of 1in. square 3piece sets at that spacing. A
calculation comparing the expected and observed resistances follows:
1in. square timbering set normally twelve inches
centertocenter
Type of Set R,
1 row of props along each duct wall (observed) .... 2.2
Cross bars (observed).......................... 0.6
3piece sets (estimated)........................ 2.8
3piece sets (observed)......................... 3.5
Thus the actual resistance of 3piece sets far exceeds the sum of
the resistances of their component parts, a result which is in harmony
with the fact that the air stream is more restricted with 3piece sets
than with either of the component types of timbering, so that the
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS 49
resistance should be expected to increase more rapidly, to the extent
that it varies as the square of the velocity. It thus appears that each
type of timber set has its own peculiarities in the development of re
sistance, and must be studied individually. A further illustration of
this is had by adding the resistance of cross bars to 3piece sets:
Type of Set R,
Cross bars (observed).......................... 0.6
3piece (observed) ............................ 3.5
4piece (estim ated) ............................ 4.1
4piece (observed) .......................... 5.8
Again the resistance of the actual sets is much greater than the
combined resistances of its component parts. Were this not the case,
the resistance of 3piece sets at standard spacing should be three
times that of cross bars, and the resistance of 4piece sets should be
four times that of cross bars, instead of 5.1 and 7.7 times, respectively,
as shown by Table 9.
25. Application to Mines.As has been previously explained,* the
laws of dynamic similarity require the results of model tests of
this kind to be applied to a prototype at like values of Reynolds'
vD
Number, v .t For a mine entry 7 X 7 feet in cross section, Reynolds'
V
Nutmber is 43 800v, or 1 095 000 at a mean velocity of 25 feet per
second (or 1500 feet per minute), the standard mean velocity at
which resistances of the model timbers were computed. For the lined
duct, Reynolds' Number is 4800v so that a mean velocity in the duct
of 228 feet per second or 13 700 feet per minute would be required
to bring the Reynolds' Number to that in the mine entry with air
flowing at 1500 feet per minute. Since the maximum test velocity
ranged from '50 to 80 feet per second, according to the resistance of the
timbers in the duct, about a 4fold extrapolation of the test results
would be required to cover a mine Reynolds' Number of about one
million.
This is illustrated in Fig. 13, where the relation of zonal resistance
to Reynolds' Number is represented for several timbering conditions.
It shows that, if the values of n determined in the model tests for a
given test condition hold for a multiplication of mean velocities
sufficient to yield a Reynolds' Number for the duct of one million or
*Univ. of Ill. Eng. Exp. Sta. Bul. 265, p. 14 (1934).
tHere v = mean velocity of air flow, (feet per second) D = diameter of conduit, (ft.)
and v = kinematic viscosity of air (mean value 0.00016 ft.2 per second.) For a lucid explana
tion of the application of the principles of similarity to model tests see, "Use of Models in
Aerodynamics and Hydrodynamics," 0. G. Tietjens, Trans. A.S.M.E., Vol. 54, No. 8, pp.
22533, Sept. 30, 1932.
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 13. VARIATION OF Rs WITH REYNOLDS' NUMBER
more, the zonal resistance of certain timbers in a mine entry may
differ considerably from that of their models. The extreme case il
lustrated is that for two rows of 1in. round props, set abreast. Their
mean R, in the test range was 4.0, whereas the value indicated for
the mine range is but 2.2, n being 1.75. On the other hand, the zonal
resistance of the 1in. square 3piece sets illustrated in Fig. 5, rose
correspondingly from 1.35 to 1.75, n being 2.12, while the resistance
of %in. square cross bars at standard spacing remained nearly con
stant at 0.48, n being 2.01. Slopes corresponding to the mean values
R
N
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS
of n for four types and shapes of timbers are also illustrated in Fig. 13.
Although the resistances computed in this investigation are net
resistances attributable solely to timbers, Reynolds' Number has
been computed for the conduit (duct or entry) rather than for the
timbers, because the mean velocity in the duct was measured with
accuracy, whereas the velocity to which timbers of a given size and
type were subjected is conjectural, no determinations of velocity
distribution having been made within the timbered zone. It is sup
posed that peripheral timbers were subjected to velocities less than
and internal timbers to velocities greater than the mean velocity.
Furthermore, it is probable that no single timber was in a uniform
velocity field.
Where geometrical similarity is maintained between the conduit
and the timbering in both the model and the prototype, the scale
factor is the same for the timbering as for the conduit, and the rela
tive extrapolation of Reynolds' Number. from the model to the proto
type would be unaffected by the basis of computing Reynolds' Num
ber. In general, the latter would be about 1/20 to 10 as great for
timbers as for the conduit, depending on the size of timbers used.
While it is clear, theoretically, that changes in zonal resistance with
Reynolds' Number should be taken into account in applying the
data of this investigation to mines, two points of difficulty make it
impractical to attempt this at present.
A minor point is that while the extrapolations of Fig. 13 are ap
parently acceptable, much greater extrapolations commonly being
used in conjunction with model testing, it has been demonstrated that
plots of resistance against Reynolds' Number sometimes develop
marked irregularities,* which render extrapolation uncertain. This
casts some doubt on the validity of such extrapolations as are illus
trated in Fig. 13 until their accuracy can be demonstrated.
A further difficulty in applying such extrapolations in this inves
tigation is the high sensitivity of n to changes in size, shape, type,
and spacing of timbers. This is illustrated in Fig. 12. Even where a
formula like those listed in that figure can be used to relate n approx
imately to spacing, it would be an exceedingly tedious task to adjust
each zonal resistance as determined from these data to equivalence
with mine Reynolds' Numbers, for every assumption as to size, shape,
orientation, type, and spacing of timbers, taking into account changes
in the estimated values of n for each condition.
In view of the uncertainty that extrapolation would enhance the
*Prandtl and Tietjens, op. cit., pp. 91101, Figs. 5055.
ILLINOIS ENGINEERING EXPERIMENT STATION
applicability of the test data, and the difficulty of making such ex
trapolations, mine conditions will be estimated by making direct
comparisons between the model and the mine at like mean velocities
in each, as was done in a previous bulletin.*
26. Timbering and Ventilating Economies.Inasmuch as the func
tion of timbers is to support the passageways in which they are
installed, they must be chosen primarily from the standpoint of their
supporting capacity, and only secondarily from the standpoint of low
resistance to air flow. However, the comparative merits of the various
types of sets will be discussed from both points of view.
Fourpiece sets are ordinarily used only in shafts or in moving
ground which must be heavily timbered. As no other form of support
would serve, their high resistance to the flow of air (Table 9) can
only be offset by smooth lining the timbered passage.
On the other hand, threepiece sets, which is probably the type
most commonly used, ordinarily support only the roof of haulageways,
a purpose which could be as well accomplished by the crossbar
member of the set, were it properly supported by the coal or rock
walls. As longer cross bars would often be required to reach ade
quate support by the ribs, it is probable that the substitution of
cross bars for threepiece sets would not reduce the timbering costs
in full proportion to the visible reduction in the amount of timber
required, but that it would cut the cost of timbering an entry of
square cross section about onehalf.t Table 9 indicates that it would
reduce the air resistance of the timbers in such an entry about 80
per cent, on the average. This double advantage of decreasing both
the timbering and ventilating costs emphasizes the importance of
using cross bars instead of threepiece sets for roof support, wherever
possible.
Due, in part, to their ease of placement, props in either one or
two rows are commonly used for support in air courses where there
is no track. Since the resistance of a single row of center props
is twice that of the corresponding cross bars, the latter are prefer
able from the standpoint of ventilating costs. However, as the
comparative timbering costs are difficult to estimate, careful analysis
would be required in an individual case to determine whether a
lower total cost for support and ventilation could be attained by
timbering with cross bars or center props. Nevertheless, where two
*Univ. of Ill. Eng. Exp. Sta. Bul. 265, p. 12 (1934).
tSee also "Low Cost and Safety Feature Entry Timbering with the Hitch Drill at Indiana
CoalMining Operations," Coal Age, Vol. 40, No. 6, p. 237, June 1935.
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS
TABLE 10
ESTIMATION OF COMPARATIVE VENTILATION COSTS OF TIMBERING
Assumed length of entry.................................. 700 ft. = 10 resistance zones
Assumed crosssection.................................... 7 X 7 ft. sq., D = 7 ft.
Assumed centertocenter spacing........................... 3.5 ft. = 0.5D
Assumed number of sets............................ ..... 200
Assumed diameter and shape of timbers.................... 91Y in., round, T/D = 0.11
Corresponding size of model timbers ....................... 1 in. = (9.2 X 0.11)
Assumed fandrive efficiency. ............................. 66 % per cent
Assumed cost of electrical energy. ......................... $100 per h.p. year
Type of timbering ...................................... Cross Center 2 rows 3piece
bars props
Zonal resistance, R. from Table 6......................... 0.56 1.05 2.57 2.71
d of Form 2 equation from Table 3 ....................... 0.056 0.105 0.257 0.271
c of Form 2 equation from Table 3 ........................ 0.096* 0.081 0.197 0.400
B, = clog S +d = d 0.301c .......................... 0.028 0.081 0.198 0.151
Total resistance of timbers = 200 R,....................... 5.6 16.2 39.6 30.2
(1) At mean velocity of 500 ft. per min.
Quantity Q = 24 500 cu. ft. per min.
Meanvelocity pressure = 0.0156 in. water
Total pressure loss i, in. water (0.0156X 200 R)..... 0.087 0.253 0.618 0.471
Air horsepower 53 ............................ 0.34 0.98 2.39 1.82
Electrical horsepowert ........................... 0.51 1.47 3.60 2.73
Annual cost of power to overcome resistance of timbers $51 $147 $360 $273
(2) At mean velocity of 1000 ft. per min.
Quantity Q = 49 000 cu. ft. per min.
Meanvelocity pressure = 0.0624 in. water
Total pressure loss i, in. water (0.0624 X 200 R,)...... 0.35 1.01 2.47 1.88
Air horsepower.................................. 2.70 7.30 19.0 14.5
Electrical horsepowert............................ 4.1 11.0 28.5 21.7
Annual cost of power to overcome resistance of timbers $410 $1100 $2850 $2170
*By extrapolation for size.
fElectrical horse power = 1.5 air horsepower at assumed fandrive efficiency of 66 % per cent.
or more rows of props would be required for support, it is probable
that cross bars could be installed at nearly the same cost, and at a
great saving (Table 9) in the cost of transmitting air through the
entry.
As the cost of timbering is almost a nonrecurring item, while the
ventilating cost is continuous, considerable savings can be expected
from the substitution of cross bars for other forms of timbering,
wherever air is to be moved at considerable velocities, say at more
than a few hundred feet per minute for a long time.
An estimate of the cost of transmitting air through an entry sup
ported by different types of timbering will illustrate the relative merits
of cross bars. Such a comparison is made in Table 10, which develops
the probable savings in ventilating costs which may be expected from
the use of cross bars instead of props or 3piece sets, where timbers
are necessary. Bars are particularly advantageous at the higher rate
of flow, showing a saving of nearly $700 per year over center props
and of $2400 per year over 3piece sets where 700 feet of 7 X 7 ft. en
ILLINOIS ENGINEERING EXPERIMENT STATION
try timbered with 9%in. round timbers, 3.5 ft. centertocenter, is to
transmit air at a mean velocity of 1000 feet per minute. A calcula
tion at 500 feet per minute has been included to show that the cost
of transmitting air through timbered entries at comparatively low
velocities is a considerable item. The estimates in Table 10 repre
sent only the resistance due to the timbers. To this must be added
a comparatively small amount to represent the resistance of the un
timbered entry.
Owing to their general availability and favorable characteristics,
round cross bars may be regarded as the normal form of roof support.
A question arises as to the desirable mutual relations between their
diameter and spacing, an increase in diameter obviously permitting
an increase in spacing. Knowing that the entry described in Table
10 could be timbered as indicated, would it pay to timber it with
larger cross bars at wider spacing, say D, centertocenter, provided
the roof would still be properly supported? As the stiffness of beams
is virtually proportional to the fourth* power of their diameters and
each cross bar would have twice the load at a spacing of D as at
onehalf D, the new timbers would need to be nearly 20t per cent
larger than those of Table 10, so that their ratio T/D would be 0.13.
If the cost of timbering were in proportion to the amount of timber
used, it would be decreased about 30t per cent. The new timbers
would correspond nearly with 1%in. timbers in the model, and ex
trapolation in Table 6 indicates that their zonal resistance would be
0.72, nearly 30 per cent greater than the resistance of the 9%in.
timbers of Table 10. Thus, while the single timbering cost would be
reduced 30 per cent at the wider spacing, the continuing ventilating
cost would be increased 30 per cent, or $123 annually for 700 feet of
entry with air flowing at 1000 feet per minute. Thus it seems clear
that in this case the wider spacing of timbers would be undesirable at
any substantial rate of air flow.
This suggests the desirability of decreasing the size of timbers
and spacing, say to 0.25D. This would permit the use of 71in.
round cross bars at an estimated increase in timbering cost of about
40 per cent over that for the 9¼in. cross bars of Table 10. These
cross bars would have a T/D ratio of 0.92, corresponding nearly with
*The maximum deflection of a beam supported at both ends is inversely proportional to the
moment of inertia of its cross section so it may be said that the stiffness of the beam is
directly proportional to the moment of inertia, which, in turn, is proportional to the fourth
power of the diameter. Hence, the stiffness of a beam is considered to be proportional to the
fourth power of its diameter.
f 1 = 1.19.
tHalf as many timbers each 1.4 (1.19 X 1.19) times as heavy gives 0.7 as much timber, or
30 per cent less timber.
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS 55
75in. round cross bars in the duct. Their estimated R,' is 0.49 and
coefficient c is 0.077, so that at S = 0.25D, R, = 0.077 log 0.25 +
0.049 = 0.0026. With 400 sets in the 100 diameters or 700 feet of
entry, the total resistance would be 1.04, twice that estimated in Table
10 for round cross bars 0.5D, centertocenter. Thus, in this case,
closer spacing would carry a double penalty in that it would increase
both the timbering costs and the air resistance. As the resistance
was higher under both assumptions S = D and S = %D than for
S  1/2D it appears that there is an optimum combination of diameter
and spacing of cross bars, for a given requirement for roof support.
Once this requirement is known a solution can be worked out as indi
cated, to give the lowest combined timbering and ventilating costs
consistent with safety.
27. Mine Resistances.A few data on the actual zonal resistance
of timbered passageways in mines are available for comparison with
analogous model determinations. The comparison is made in Table
11, which shows estimates of the zonal resistance of timbers as deter
mined by models to vary from 46 per cent less than to 74 per cent
greater than the corresponding mine resistances. However, in each
case there were present several points of dissimilarity between the
mine and the model, perhaps the chief one of these being dissimilarity
in the crosssectional shape of the conduits. While the duct was
square the mine passageways were rectangular or trapezoidal, the
width exceeding the height in three out of the five cases. Further
more, the shape of the mine entries is known to have varied, in one
case quite markedly (note ***, Table 11) along the test zone. Other
sources of variability and dissimilarity were irregularities in the sizes
and shapes and alignment of timbers, and in the nature of the rubbing
surfaces of the entries themselves.
One point of difficulty with the comparisons is that, in several
instances, the model resistance exceeded the mine resistance by rela
tively large amounts. This is hardly to be expected in view of the
greater irregularity of the mine conditions, and may result from faulty
estimates of the overall dimensions of the entries, size of timbers,
spacing, etc., or from error in interpolating or extrapolating from
Table 3 to bring the model determinations into correspondence with
the estimated mine conditions. It cannot be attributed solely to the
discrepancy between the shape of the entries and the duct because in
some estimates for which this dissimilarity existed the model resistance
was less than the mine resistance.
56 ILLINOIS ENGINEERING EXPERIMENT STATION
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MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS 57
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ILLINOIS ENGINEERING EXPERIMENT STATION
Another difficulty is indicated by note ¶, Table 11, to the effect
that gross mine resistances are shown rather than net resistances.
The reason for this is that only gross losses were reported, so that
any tare loss to be deducted to yield an estimated net loss due to
timbers would have to be estimated. Estimates showed these postu
lated tare resistances to be so high in comparison with the remaining
net resistance of timbers as to make them of questionable value.
For example, assuming the coefficient of friction kc* to be 75 X 1010
for the rough rock surfaces which existed between timber sets and
along the floor of the air course of col. 13, Table 11, the tare zonal
resistance would be 0.9, which is nearly equal to the remaining net
zonal resistance of timbers (1.0). Such a result is hardly to be ex
pected in a heavilytimbered entry of this kind,t it being difficult to
understand how the entry surfaces could contribute nearly half the
resistance in an entry so closely timbered with large timbers. For
this reason, where indicated by note ff, comparison has arbitrarily
been made between the gross mine resistance and the net resistance
due to the model timbers.
Items J9 and J12 of Table 11 show the mine resistance of cross
bars to exceed that of center props of the same size and spacing,
whereas the model determinations indicate the reverse relationship.
It is probable that the shape of the conduits plays a major part in
this, for the length of each model cross bar was equal to that of
each prop, whereas in this mine entry the cross bars were nearly 50
per cent longer than the props. Hence, per unit length of timber,
the mine cross bars developed appreciably less resistance than did the
props. This is in harmony with the model indications, and this point
illustrates the importance of the shape of the conduit in attempting
comparisons between the model and a mine passage.
Similarly, items K12 and K9 show the resistance of model center
props to be nearly three times that of the corresponding cross bars
at a spacing of 0.68D, whereas this ratio at the standard spacing of D
was but 2.2. The increase in the ratio with closer spacing is due to
the greater rapidity with which the zonal resistance curves (Fig. 9)
of props approach their maxima than do those of cross bars, as the
spacing is reduced below its standard value. In general, this means
that tables like Tables 69, drawn up for any common spacing other
than D would differ, in 'some cases appreciably, from Tables 69.
This emphasizes the necessity of keeping in mind the fact that all
*Univ. of Ill. Eng. Exp. Sta. Bul. 199, p. 10.
tSee Bul. 199, Fig. 2.
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS
general statements as to the comparative resistance of timbers in
different sizes, shapes, and types must be based on a specified spacing,
and that they may not be valid for another spacing.
VI. SUMMARY AND CONCLUSIONS
28. Summary and Conclusions.The resistance of models of mine
timbers in several sizes, shapes, and types of sets, at different spacings,
was measured in a duct 9.2 inches square in cross section by forcing
air through the timbered duct at different rates of flow, and noting
the resultant losses in pressure. To determine the net loss due to
timbers, a tare loss, measured without timbers in the duct, was de
ducted from each gross loss observed with timbers at a like rate of
flow.
The model timbers were about onetenth scale, ranging in size
from % inch to one inch. Most were made of soft pine in round,
halfround, quarterround, or square cross sections, although some
rolled steel shapes were used as cross bars. Other types of sets used
were 3piece and 4piece sets, center props, and two rows of props
set abreast or staggered.
Many of the timbers were tested at different centertocenter spac
ings, ranging from onetenth of a duct diameter to three or four duct
diameters. All were tested at an arbitrarily chosen standard spacing
of one duct diameter (D = 9.2 inches).
Resistance of timbers was expressed as the ratio of the net total
pressure loss due to a specified amount and kind of timbering to the
meanvelocity pressure of the air flowing through the duct. Two
resistance indexes of this kind were established, one expressing the
resistance of a single timber set, R,, the other expressing the resistance
of a length of timbered duct equal to ten duct diameters (92 inches).
This is known as the zonal resistance, R,.
Tests showed that, due to end effects, R8 was not independent of
the number of sets of timbers used at a given spacing, but that an
intermediate number of sets gave a minimum R, which was in most
cases but a few per cent less than the maximum R,. The minimum
R, for each kind of timbering at each spacing tested was accepted as
the appropriate value, and a schedule was developed relating to the
spacing, the number of sets to be used, to yield as nearly as possible
the minimum R. for each timbering condition.
For intermediate spacings, R, was found to vary as the logarithm
of the spacing, and a formula of the type R, = c log S + d was
ILLINOIS ENGINEERING EXPERIMENT STATION
derived for each kind of timbering tested at varied spacing, S being
the centertocenter spacing of timbers in terms of D. The range in
spacing over which this formula is applicable varies widely among
the sizes and types of timbering, but it is roughly a sevenfold range
for a given kind, beginning at a spacing of from 0.1 to 0.3D. In
several cases log R, proved to be proportional to the spacing at
closer spacings than the lower limit for the foregoing formula (Form
2), so that the closespacing equation is log R, = aS  b, (Form 1).
The zonal resistance is related to the resistance per set by the
equation Rz = OR. Where a Form 1 equation is applicable, solution
S
for R, in terms of a, b and S leads to an expression which has a min
0.434*
imum value of Rz at S = . The closespacing minimum Rz usu
a
ally falls at an expansive spacing of one or two transverse timber
dimensions, T. Expansive spacing is the length of the interval be
tween successive sets, in which the air stream can expand. For
peripheral timbers the minimum zonal resistance averaged about 1.2
times the zonal resistance at standard spacing, R,', and for internal
timbers its average was 0.6 Re'.
In the range of the Form 2 equation, solution for R, in terms of
c, d, and S gives an expression which has a maximum value at a spac
ing such that log Smax = 0.434  . For peripheral timbering the max
c
imum fell at an expansive spacing of about six or seven times T, but
for internal timbering it fell at a spacing of one or two times T, as a
rule. The mean ratio of maximum zonal resistance to Rz' was 2.4
for internal timbering and 1.2 for peripheral timbering.
The existence of the extremes of zonal resistance is tentatively
explained by assuming the minimum resistance to be developed at
such a close spacing that the air stream flows smoothly across the gaps
between timbers. While the mean velocity of such a stream would
be the same as it would with the timbers in contact with each other,
it would encounter less solid rubbing surface, and hence have a lower
resistance. As the gap is widened beyond one or two times T the
stream begins to expand. The expanded portion is obstructed by the
next timber and part of it is deflected into eddies or back flow in the
interspaces. The zonal resistance thus increases with the spacing
*Throughout this report, logarithms are to base 10, 0.434 being the ratio of the logarithm
of a number to base 10 to its logarithm to base e.
MODEL DETERMINATION OF RESISTANCE OF MINE TIMBERS
until a spacing has been reached such that the stream, as it ex
pands on passing one timber set, encounters the next set at its base
on the duct wall. If the spacing be increased beyond this, the
stream occupies the entire duct from the point at which it expands to
fill the duct, until it encounters the next timber. It is then flowing at
its lowest velocity and minimum resistance, with the result that the
zonal resistance continues to decrease as the spacing is increased be
yond that of maximum R,.
In testing the resistance of two rows of props they were normally
centered along planes which were equidistant from each other and
from the adjacent duct wall. This was their standard transverse
spacing. In one series of tests with rows abreast the two rows were
set against each other along the center line of the duct, then a
central gap was made between them, and increased from test to test
until each row was along a duct wall. Zonal resistance responded to
increase in the central gap much as it did to increase in longitudinal
spacing, developing similar minimum and maximum values. With
the rows staggered the resistance with no central gap was less than
half the corresponding resistance with props abreast. At central gaps
in excess of one inch the resistances for the two timbering conditions
were in harmony. A tentative explanation for the response of re
sistance to variation in transverse spacing is offered in the text (p. 37).
At standard longitudinal spacing of timbers (D, centertocenter)
the resistance increases sharply with the size of the timbers. Their
shape and orientation are also important factors in the resistance,
round timbers standing between square ones and such curvilinear
shapes as oval, halfround and quarterround timbers in certain
orientations.
Cross bars were found to be the best type of timbering from the
standpoint of air resistance, other comparable types developing from
two to nearly six times the resistance of cross bars. Since, in many
instances, the use of cross bars for roof support would effect a saving
in timbering as well as in ventilating costs, they should be installed
wherever feasible, in preference to props or 3piece sets. Where air
must be transmitted at considerable velocities through timbered en
tries, substantial savings will result from the use of peripheral tim
bering and smooth lining the passageway, inside the timbers.
Ideally, as here defined, the resistance should be independent of
the rate of flow, but it was found that it generally varied as some
small fractional power (e.g. ± 0.10) of the mean velocity, because
the pressure losses varied by a power of the mean velocity differing
ILLINOIS ENGINEERING EXPERIMENT STATION
from 2.0 by a few per cent. This variation was highly sensitive to
changes in size, shape, orientation, type, and spacing of timbers,
which makes it difficult to extrapolate from the model tests to values
of Reynolds' Number in the mine range and impairs the dependability
of such extrapolations. As it is theoretically necessary to compare
model results with mine conditions at mine values of Reynolds' Num
ber (about one million) and it was impossible to reach the mine range
with the model, extrapolation of the model results would be necessary
to make a theoretically valid comparison. However, the uncertain
ties and difficulties involved in extrapolation make it preferable to
compare model and mine conditions directly, i.e. at like mean
velocities.
No data are available on the resistance of mine timbers which are
strictly comparable to the model timbers. This is chiefly due to
differences in crosssectional shapes of the conduits and to irregular
ities and uncertainties in the size and shapes of the mine passageways
and timbers for which data are available. However, several com
parisons were attempted between the resistances of timbered rectangu
lar and trapezoidal entries and of the model duct when timbered in
the most nearly comparable manner. Good checks were obtained in
some cases, but in others the discrepancy between the mine resistance
and the model estimate was many per cent of the former. Never
theless, it seems evident that estimates of the resistance of mine tim
bers made from the model tests are dependable under comparable
conditions, and that the essential characteristics of the flow of air in
timbered entries and the mutual relations among the resistances of
mine timbers of different sizes, shapes, types, and spacings are clearly
defined by the model tests.