LAMINAR FLOW OF SLUDGES IN PIPES
I. INTRODUCTION
1. The Problem.-In the design of pipe lines through which sludge
will flow, or must be pumped, insufficient information is available to
make possible precise estimates of the head losses due to friction. In
the pumping of sewage sludges the problem has usually been solved
by assuming that the common hydraulic formulas applicable to the
flow of water may be used in the flow of a sludge, provided the velocity
of flow of the sludge is great enough to create turbulence. Measure-
ments of head losses resulting from the flow of sludge in a pipe, as
shown in Fig. 1, indicate that such an assumption is only an approxi-
mation of the true conditions.
Problems involving the laminar flow of fluids in all sizes of pipes
can be solved by the use of Poiscuille's equation. Turbulent flow
frictional losses are evaluated by means of the Reynolds-Stanton
diagram, Fig. 6. At present the only methods available for determining
laminar flow frictional losses for plastic and pseudo-plastic materials
flowing in circular pipes have been the use of empirical formulas and
information from reports of a few special instances of the flow of
sewage sludges, clay slurries, etc., in the literature. Such materials as
sewage sludge, aqueous suspensions of clay and sand in dredging,
wood pulp suspensions in the paper making industry, and drilling
mud in well drilling operations may be required to be pumped through
pipes, and for design purposes the frictional losses must be evaluated.
2. Purposes of Investigation.-The purposes of this investigation
were to formulate the various factors that influence the frictional losses
when a sludge is pumped through a circular pipe, to formulate the
variable factors that affect the critical velocity, to verify the formulas
experimentally, and to present a simple method for measuring the
characteristics of individual sludges. For a clear understanding of
the characteristics of flow it is necessary to distinguish between vis-
cosity and plasticity.
3. Viscosity and Viscous Flow.-Viscosity is the measure of the
resistance to flow or deformation of a fluid. The rate of deformation
is a linear function of the deforming force. The coefficient of viscosity
of a fluid is equal to the tangential force on a unit area of either of
two horizontal planes at a unit distance apart required to move one
ILLINOIS ENGINEERING EXPERIMENT STATION
/ ~ C -5 ^ 00 0 /&/ 46/ 1 1 /* --/U L/ 01 o i/4/
Ve/ocif~/ /n '-9f p'r SPcod
FIG. 1. FRICTION LOSSES OF VARIOUS SLUDGES FLOWING IN A %-IN. PIPE
plane with a unit velocity with reference to the other plane, the space
between being filled with the viscous substance. From which it
follows that
Sx
' = -- (1)
v
Where
p' = coefficient of viscosity
S = tangential unit shearing force
x = distance between planes
v = velocity of one plane with respect to the other
When the C.G.S. system is used, the name given to the coefficient
LAMINAR FLOW OF SLUDGES IN PIPES
0 Ve/oc/fy, of //ow
FIG. 2. FLOW CHARACTERISTICS OF DIFFERENT
CLASSES OF MATERIALS
of viscosity is the poise. The centipoise is one one-hundredth of a
poise. In the F.P.S. system no name is given to the coefficient.
4. Plasticity and Plastic Flow.-Plasticity is the property of a
substance which enables it to be continuously and permanently de-
formed in any direction without rupture under a stress exceeding the
yield value. After deformation has started, equal increments of stress
will produce equal increments in velocity. Reverting to the funda-
mental conception of flow between two parallel planes, since a part
of the applied force S is used up in overcoming the yield value Sy,
the equation for plastic flow becomes
(S - S) x (2)
V
where y' is the coefficient of rigidity of the material, analogous to the
coefficient of viscosity of a true fluid.
Figure 2 represents all the recognized types of flow. Curve I
represents the flow of a true liquid, the slope of the line is proportional
to the coefficient of viscosity. Curve II represents the flow of a
pseudoplastic material. It can be seen that this curve does not obey
the fundamental equation of plastic flow, because the line bends
ILLINOIS ENGINEERING EXPERIMENT STATION
towards the origin at low rates of flow. Curve III represents a true
plastic, and is a graphical representation of Equation (2). The ap-
parent viscosity of the plastic at any point A on curve III, if measured
in the usual way for liquids, is proportional to the slope of the
line OA. It is evident, therefore, that the apparent viscosity is not
constant for different velocities and stresses. It is seen that two
different velocities such as A and B in the figure correspond to entirely
different viscosity lines OA and OB, the slopes of which are propor-
tional to the apparent viscosity. Curve IV represents the flow of an
inverted plastic substance. This material is thin at low rates of flow
but becomes increasingly thicker as the force increases. It has been
found in this investigation that the flow of sludge follows the type
of flow illustrated by curve III. It is concluded, therefore, that
sewage sludge and clay slurries are true plastics.
5. Procedure and Nomenclature.-The experimental procedure has
been to compare the theoretically developed formulas with measured
friction losses in various sizes of pipes resulting from the flow of
sludges from different sources.
The nomenclature used is as follows:
D = Diameter of pipe, feet
g = Acceleration due to gravity, feet per second per second
H = Difference in static head between two points in a pipe, feet
of flowing substance
H, = Difference in static head between two points in a pipe, feet
of water
L = Length of pipe, feet
n = Speed of revolution of cylinder in modified Stormer vis-
cometer, revolutions per second
Q = Rate of flow, cubic feet per second
r = Distance from any point within a pipe to the center of the
pipe, feet
R = Radius of pipe, feet
Re = Reynolds number DVp/p, dimensionless
ro = Radius within a pipe at which the shearing stress equals
the yield value, feet (see Fig. 4)
S = Shearing stress, pounds per square foot
Sp = Shearing stress in a flowing material at the boundary or
pipe wall, pounds per square foot
Sr = Shearing stress in a circular pipe at distance r from the
center, pounds per square foot
LAMINAR FLOW OF SLUDGES IN PIPES
Sy = Shearing stress at the yield point of a plastic material, called
yield value, pounds per square foot
u, = Intercept on the u axis of line connecting points representing
driving force, u, and corresponding speed of revolution,
n, in the Stormer viscometer (see Fig. 10)
V = Mean velocity of flow in a pipe, feet per second
vo = Velocity of plug of radius ro, feet per second
Vr = Velocity at any distance r, from the center of the pipe, feet
per second
V1e = Lower critical velocity (Re = 2000), feet per second
Vu, = Upper critical velocity (Re = 3000), feet per second
A = Coefficient of viscosity, pounds per foot per second
-' = Coefficient of viscosity, slugs per foot per second
-' = Coefficient of rigidity, slugs per foot per second
S= Coefficient of rigidity, pounds per foot per second
X = Slope of line connecting points representing driving force,
u, and corresponding speed of revolution, n, in the
modified Stormer viscometer (see Fig. 10)
p = Density of flowing substance, pounds per cubic foot
6. Acknowledgments.-The tests herein described were made as
part of the work of the Engineering Experiment Station of the Uni-
versity of Illinois, of which DEAN M. L. ENGER is the Director, and
of the Department of Civil Engineering, of which PROFESSOR W. C.
HUNTINGTON is the head. MR. LUDWIG STOYKE, a graduate student in
the College of Engineering, devoted part of his time to the performance
of the preliminary tests and to a search of the literature. Some of the
routine work, the setting up of apparatus and similar work, was done
by undergraduate students employed through the National Youth Ad-
ministration. The authors are grateful to DEAN ENGER for his careful
analysis of the manuscript and suggestions for improvement. A debt
of gratitude is due to DR. W. D. HATFIELD for his advice, assistance,
and unfailing interests in the progress of the investigation, and for
the use of the facilities of the sewage treatment plant at Decatur,
Illinois, in the measurement of sludge. The cooperation and interest
of sewage treatment plant operators at Indianapolis and at the larger
plants in central Illinois was encouraging and is appreciated.
II. PRINCIPLES OF FLOW
7. Factors Affecting Friction.-In an attempt to formulate the
factors affecting the friction resulting from the steady uniform flow
ILLINOIS ENGINEERING EXPERIMENT STATION
Sect/on? Section
< - L I
Pressure= I -ec-/o-7 V =R Press"ure=
FIG. 3. DIAGRAM OF FLOW BETWEEN Two SECTIONS OF PIPE
of sludge in a circular pipe certain assumptions will be made and the
resulting formulation checked by tests. It will be assumed that the
conditions affecting the friction resulting from the laminar flow of
sludge in a circular pipe are the velocity of the sludge, the diameter of
the pipe, the length of the pipe, and the characteristics of the sludge
such as density, rigidity, and yield value. The pressure and tempera-
ture will be assumed to affect the friction only through their effect
on the characteristics of the sludge. Another factor that might be
assumed to affect the friction is the roughness of the pipe walls. How-
ever, it is known that, in the laminar flow of fluids, pipe wall roughness
does not affect the friction loss. It has been assumed, therefore, that
pipe wall roughness will not affect friction loss in the laminar flow of
sludge. The friction loss will be assumed to result only from the
rubbing of the sludge layers past each other and not from kinetic
energy losses.
8. Theoretical Formulation of Factors Affecting Flow.-The fol-
lowing mathematical analysis was first presented by Bingham.'* The
total force producing flow in a pipe between sections 1 and 2, Fig. 3, is
7rR2 (AP) where AP is the difference in unit pressure between sections
1 and 2. Since there is no acceleration in steady, uniform flow this
force is opposed by an equal force 2irRLS,
Hence
R(AP)
2L
and
r(AP)
S, =nb (4)
2L
*Index numbers refer to items in bibliography.
LAMINAR FLOW OF SLUDGES IN PIPES
FIG. 4. DISTRIBUTION OF SHEARING FORCES IN A CIRCULAR PIPE
The sludge flowing in the center of the pipe moves as a solid plug,
with a radius of ro. This phenomenon results from the fact that the
shear between the moving layers increases from zero at the center of
the pipe to a maximum at the pipe wall, as shown by Equation (4),
and in Fig. 4, and at some distance between the center of the pipe and
the wall the shear will be equal to the yield value of the plastic
sludge. Where the shear is less than the yield value there will be no
relative motion between adjacent particles which will, therefore, flow
together as a solid plug.
If Ap is the unit pressure used in overcoming the friction due to
the yield value, the yield value S, is
R(Ap)
S, = (5)
2L
It also follows that
rAP
S, = - (6)
2L
For a circular pipe Equation (2) becomes
1
dv =- (S, - S,) dr
or
1 r(AP)
dv - -- - - S, dr
,qL 2 j
ILLINOIS ENGINEERING EXPERIMENT STATION
The velocity, at any distance r from the center of the pipe, in
the region between the plug of radius ro and the pipe wall, is obtained
by integrating Equation (7) from r = R to r = r
1 (AP)r 1 (AP)r2 R
v, - r " - S, dr = - Sr
n' 2L ' 4L ,I
1 AP P]
Vr = --- L-(R2 - r2) - S,(R - r)
7 L 1 4L I
The velocity of the solid plug is obtained by making r = ro
in Equation (8)
1 [ (AP)R2
4vt L
Hence the flow Q is
LS2 P R
+ A"- SR
AP J
R
Q = 7rrvo + 27r rvdr
J ro
LS'
+
AP
2 r fR (AP)
,-r/ (P (Rr - 3) - S_(R-
i'JroL 4L
27R4(AP) R3S, AP (R22 4 Ro2
T'L 16L 6 4L f 2 4 ES 2
Substituting the value of ro from Equation (6)
R 2r- / R4 (AP)
27r rvdr=-
Jro ' \ 16L
R3S R2LS2 2RL2S3
6 2(AP) (AP)2
Substituting these values in Equation (10)
_ R (AP) R8Sy 2 L(SA 1
Q 8L 3 + 3 (AP)J
- rR4 4 2LS 1 2LS
8L' AP) -3 R 3(AP)3 R
- r4SL2 r (AP)R2
(rrio =2 o
(AP)2 L 4L
27 f rvrdr =
J ro
SyR]
r2) ] dr
- r)]
5 L3S\
3 (AP)3
LAMINAR FLOW OF SLUDGES IN PIPES
Substituting from Equation (5)
1r)4[- 4 (AP)4 1
Q- --4, [(AP) - - (Ap) + AP)
8Lr' 3 3(AP)3
The mean velocity of flow is
Q R2 [ 4 (Ap)4
V = - = -- AP - --Ap + --
7rR2 8Lq' 3 3(AP)3
or, changing AP and Ap to terms of shear by Equations (3) and (5),
R [ 4 -s± ]
V - SP - -Sy +
4' L 3 S3,
In the foregoing equation the units of q' are slugs per second foot.
If it is desired to express the coefficient of rigidity in pounds per
second foot, 7, the left hand member of the equation must be divided
by g. Using 77 instead of 7', and taking g as 32 feet per second per
second, the equation reduces to
4D 4 1 St
V = - S - -Sy + (11)
L 3 3 SP
Equation (11) may be written
4DSp 4 S 1 S 4
V L. 1-3 S,+ 3 S, )\
from which it can be shown that the last term may be omitted with
Sy
little error when -- < 0.5. The error will be 5.9 per cent when
S P SS
-= 0.5 and 1.8 per cent when -- = 0.4. Omitting the last term,
Sp Sp
Equation (11) reduces to
4D 4
V (= S- S) (12)
Bingham' has shown that the coefficient of rigidity 7 and the
yield value Sy, are independent of the characteristics of the measuring
16 ILLINOIS ENGINEERING EXPERIMENT STATION
1.0
0.8
i 0.8
0'a
/9
^4
,-i i:
0 2 4 6 8 /0 /2 /4 /6 /8 20
4D
FIG. 5. GRAPHICAL REPRESENTATION OF EQUATIONS 11 AND 12
apparatus and depend only on the nature of the sludge. Both these
facts have been corroborated in this investigation. On this basis,
if a graph is plotted with Sp as ordinate and V/4D as abscissa, the
slope of the resulting line will represent the coefficient of rigidity, 7;
and, for a given sludge, the same line will represent the flow of that
sludge in a pipe of any diameter. Graphs of this type, illustrating
Equations (11) and (12), are shown in Fig. 5. The error in neg-
lecting the last term of Equation (11) is thus shown graphically.
For industrial piping, and with sewage sludges, clay slurries, and
drilling muds as the flowing material, Equation (12) will yield
results within the limits of experimental error in determining the
constants S, and n of the sludge.
Since HpirR2 = 2SpRL
then
HpR
2L
and Equation (12) can be written in another form which may be
more convenient in certain cases, as follows:
H 16S, 77V
- = - + (13)
L 3pD pD 2
/-/qsr terrodced /n teo/ec//-
l ast teprm7 of Eq'C1/c/'/ot? Xl! ~
_7
,..t Yl
LAMINAR FLOW OF SLUDGES IN PIPES
TABLE 1
COMPARISON OF COMPUTED AND OBSERVED VALUES OF HEAD LOSS IN
PIPES OF VARIOUS SIZES
Type of Sludge
and Source
Sewage-Imhoff Tank, Calumet
Treatment Plant, Chicago, Ill.
S, = 0.060, 7 = 0.021:
Sewage-Imhoff Tank, Calumet
Treatment Plant, Chicago, Ill.
S. = 0.014, , = 0.0351
Sewage-Imhoff Tank, Calumet
Treatment Plant, Chicago, Ill.
8, = 0.017, ) = 0.0261
Clay Slurry from Water Purifica-
tion Plant
S, = 0.011, v = 0.0051
Sewage-Digested, Sewage Dis-
posal Plant, Stuttgart, Ger-
many. S5 = 0.10, n = 0.0771
Illinois Yellow Clay suspension-
Tests made in this investiga-
tion. S, = 0.72, 7 0.0281
Test*
Refer-
ence
No.
90
36
30
50
73
Per
Cent
Mois-
ture
(by
weight)
88
90
90
86.4
90
52
Diame-
ter of
Pipe
in.
5
12
8
4
7.9
3
Velocity
ft. per
see.
0.5
1.0
1.5
2.0
2.5
3.5
4. 5t
0.5
0.7
0.9
1.0
1.5
2.0
2. 6t
0.5
0.7
1.0
1.5
2.0 t
0.5
0.7
0.9
1.5
1. 8t
1.0
1.5
2.0
2.5
3.0
4.0
5. 0
2.0
3.0
4.0
5.0
6.0
7. Of
Ob-
served
Head
Loss
ft. per
100 ft.
1.32
1.42
1.52
1.62
1.72
1.92
2.18
0.144
0.152
0.162
0.169
0.200
0.242
0.300
0.260
0.275
0.300
0.372
0.480
0.30
0.31
0.33
0.37
0.41
1.50
1.62
1.82
2.00
2.20
2.47
2.75
26.1
27.2
27.9
28.9
29.7
30.5
Com-
puted
Head
Loss
ft. per
100 ft.
1.34
1.43
1.53
1.63
1.73
1.92
2.17
0.150
0.161
0.176
0.178
0.214
0.234
0.268
0.265
0.287
0.315
0.361
0.408
0.32
0.34
0.35
0.40
0.41
1.57
1.72
1.86
2.00
2.14
2.33
2.71
26.3
27.1
28.0
28.8
29.7
30.5
*Refers to numbers in Table 8.
tObserved critical velocity.
IThese values of S, and ,1 have been determined by plotting appropriate data similarly to Fig. 5,
and reading intercepts and slopes as described on page 16.
or
H. 16S, ?IV
- = - +-- (14)
L 3WD WD2
Equation (14) has been checked by experiments in this investiga-
tion, and by using tests reported in the literature.2 3, 4 Table 1
shows a few comparisons of the observed and the computed values
of head loss, together with the percentage variation, using Equa-
tion (14). It is evident that the agreement between observed and
Per
Cent
Varia-
tion
+1.5
+0.7
+0.7
+0.6
+0.6
0.0
-0.5
+4.2
+5.9
+8.6
+5.3
+7.0
-3.3
-10.6
+1.9
+4.4
+5.0
-3.0
-15.0
+6.7
+9.7
+6.1
+8.0
0.0
+4.7
+6.2
+2.2
0.0
-2.7
-5.7
-1.5
+0.8
-0.4
+0.4
-0.3
0.0
0.0
ILLINOIS ENGINEERING EXPERIMENT STATION
A/Vub 117r, 7-ho&saco-7ds
FIG. 6. THE REYNOLDS-STANTON DIAGRAM
computed values of friction head loss is sufficiently precise for
practical purposes.
9. The Critical Velocity.-The critical velocity will be considered
as that velocity below which the friction loss is directly proportional
to the velocity and above which the friction loss is directly propor-
tional to some power of the velocity between 1.7 and 2.0. The fric-
tion loss resulting from the flow of a liquid through a pipe may be
determined by means of Reynolds-Stanton diagram, 5'6,7, Fig. 6.
Such a diagram is constructed by plotting as ordinates values of
L V2
the friction factor f in Fanning's formula H = f--- and as
D 2g
abscissas the corresponding Reynolds number, DVp/u.
Reynolds9 showed that the critical velocity occurred at a definite
value of the Reynolds number. Recent work10 has shown that in
industrial piping the value of the Reynolds number at the critical
velocity is approximately 2300.
For circular pipes the flow will be laminar when the Reynolds
number is less than 2000.11 Sometimes laminar flow persists to
higher Reynolds numbers, but in industrial piping installations the
flow will usually be turbulent above a Reynolds number of 3000.12
Between these values of the Reynolds number there is uncertainty
as to the type of flow which may occur. In order to distinguish
between these two values the velocity corresponding to a Reynolds
number of 2000 will be designated as Vze, the lower critical velocity,
Relol/w/'s
LAMINAR FLOW OF SLUDGES IN PIPES
and the velocity corresponding to a Reynolds number of 3000 will
be designated as Vuc, the upper critical velocity.
In order to determine Reynolds number it is necessary to know
the viscosity of the flowing material. Since sludge possesses no
definite viscosity, but a varying apparent viscosity, as described on
page 10, Reynolds criterion for critical velocity cannot be directly
evaluated for sludge. However, if reference is made to Fig. 5, it
can be seen from Poiseuille's expression of viscosity
4DSp
4 = (15)
V
that the apparent viscosity for any particular value of V/4D is the
ratio of the ordinate to the abscissa or 4DSiV.
Poiseuille's equation can be used in this case in determining the
apparent viscosity of a sludge at any particular rate of flow by
merely substituting the known values of D, Sp, and V. The equa-
tion is not good for solving for the friction loss at any other velocity
when a sludge is the flowing material, because the apparent viscosity
will have changed. The term Sp in Equation (15) can be expressed
in terms of V14D and S, so that the only variable is V.
Referring to Fig. 5,
4 V
3 4D
and from Equation (15),
16DSy
S=- + 77 (16)
3V
This is an expression for the viscosity at any value of V/4D in
Fig. 5. Substituting this value for u into Reynolds' criterion for
critical velocity, and solving for V it is found that for the lower
critical velocity
v 1000 + 103vS944n + DSp (17)
Vie = (17)
Dp
and for the upper critical velocity
150071 + 127 140,?2 + D2Syp
V,.c = -- (18)
ILLINOIS ENGINEERING EXPERIMENT STATION
To check the validity of Equations (17) and (18), tests were
made in various sizes of pipes and with various types of sludges to
observe the critical velocities together with the values of S, and 7q.
The results of these tests are shown in Table 2, together with
observed and computed values of critical velocity taken from various
tests reported in the literature.2'3 These results show a high degree
of correlation between observed and computed values.
III. FACTORS AFFECTING YIELD VALUE AND COEFFICIENT
OF RIGIDITY
10. Yield Value, Sy, and Coefficient of Rigidity, i.-The yield value
and the coefficient of rigidity of a sludge are independent of the dimen-
sions of the pipe through which flow is taking place and are independ-
ent of the velocity of flow. Hence, from Equations (13) and (14) it
is evident that it is necessary to determine only the values of S, and q
for a given sludge in order to predict friction losses in the laminar flow
of the sludge through a pipe. Among the important factors affecting
the yield value and the coefficient of rigidity may be included (a) con-
centration of suspended matter, (b) size and character of particles of
suspended matter, (c) nature of the continuous phase, (d) tempera-
ture, (e) thixotropy, (f) slippage and seepage, (g) agitation, and
(h) gas content.
11. Concentration of Suspended Matter.-In this investigation the
concentration of suspended matter is taken as the ratio of the weight
of dry solids to the weight of the mixture of dry solids and liquid. In
the mixture the fine particles suspended in the liquid are termed the
dispersed phase and the liquid in which the particles are suspended is
termed the continuous phase. The tests made in this investigation
show, as in Fig. 11, that the concentration of suspended matter greatly
affects the yield value, and affects the coefficient of rigidity to a lesser
degree. Bingham' (p. 220) has shown that when the concentration of
suspended matter is low the material may exhibit no measurable
yield value, but as the concentration of suspended matter is increased
a measurable yield value will appear and will increase almost in
direct proportion to the increase in concentration of suspended matter.
The absence of a yield point at low concentrations of suspended mat-
ter may be explained on the supposition that the suspended particles
are not in contact. When the concentration becomes sufficiently great
to force the particles into contact with each other a measurable force
LAMINAR FLOW OF SLUDGES IN PIPES
*0
.0
C
C) ))
0
0
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C
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CO
'0
0
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CII
00C.-0^'00COC)COO tOCO~' NCO0 0 50C0~ '0000
t^-'OtC~ DOO ''C^ t-( 00 (0N'1 COt 0 .N I00
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ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 3
EFFECT OF PERCENTAGE OF SOLIDS ON FLOW OF SLUDGE
Per Cent H,* H,*
Character of Sludge Solids S, 7 V = 2 V = 4
(by weight) ft. per sec. ft. per sec.
Digested Sewage Sludge 8 0.236 0.032 4.44 4.85
14 0.339 0.040 6.31 6.82
8 0.184 0.030 3.52 3.91
7 0.136 0.029 2.72 3.11
Tennessee Ball Clay 15.1 0.059 0.011 1.15 1.29
20,3 0.201 0.014 3.62 3.80
22.8 0.340 0.015 6.00 6.19
23.2 0.431 0.017 7.58 7.80
24.8 0.532 0.019 9.34 9.59
Illinois Yellow Clay 34.0 0.030 0.011 0.65 0.79
45.0 0.133 0.017 2.49 2.71
48.5 0.208 0.024 3.86 4.17
49.0 0.402 0.023 7.15 7.45
52.0 0.709 0.028 12.46 12.82
*Head loss in feet of water per 100 feet of 6-in. pipe, computed by Equation (14).
is required to cause them to slide over one another, thus exhibiting a
yield value. When the concentration of suspended matter is less than
the concentration at which a yield value exists the material will, in
most cases, exhibit the properties of the continuous phase, the coeffi-
cient of rigidity becoming equal to the coefficient of viscosity of the
continuous phase. As the concentration of suspended matter is in-
creased the yield value appears, and the coefficient of rigidity is in-
creased over the corresponding viscosity coefficient of the continuous
phase. The net effect of an increase in solids concentration is to
increase the resistance to flow of the material. Results of tests that
show this effect are recorded in Table 3, and are shown in Fig. 7.
12. ,Size and Character of Suspended Particles.-That the size of
particles in suspension will affect the flow characteristics of a sludge
has been found by other investigators. Bingham1 conveys the impres-
sion that resistance to flow increases as the size of the particle is
decreased when he states: "There is abundant evidence that as the
diameter of the particles is decreased, the opportunity for the particles
touching is increased, which enhances the friction (yield value), but
this effect reaches a limit eventually when the particles are so small
that their Brownian movement becomes appreciable and strains in the
material are not permanent."
Tests made in this investigation corroborate this statement by
Bingham. In a mixture of coarse yellow clay with water the yield
value found for a concentration of 50 per cent by weight was the same
LAMINAR FLOW OF SLUDGES IN PIPES
T,-r-7}es<;ee Ball C/laj
I I I y//ow
1111"701/5 celayClb,
Coarse Par,'c/es- I
1
PYre Part/c/es
0
0
7/
I
0 8 I6 21 32 40
Per Cent So//ias by Wegh2
FIG. 7. EFFECT OF PARTICLE SIZE AND CONCENTRATION ON THE
FLOW CHARACTERISTICS OF SLUDGES
as the yield value found for a fine Tennessee ball clay at a concentra-
tion of 25 per cent. Figure 7 shows the effect of particle size and
solids concentration on the frictional resistance to flow of sludge in a
six-inch pipe. A sewage sludge of 15 to 20 per cent solids is almost
thick enough to handle with a pitch fork and thus would be almost
impossible to pump. This characteristic may be due to the gelatinous,
water-binding properties of the solids present.
In general, the sizes of particles in the sludges studied in this
investigation were small, or even colloidal, although they sometimes
gathered together in flocculent masses, so that they either could not
be measured by ordinary means, or their measurement would give a
dimension of no practical value. Although agitation, temperature,
chemical reactions, and other factors may affect the size of the particles
in a sludge so as to cause changes in the constants S, and q there is
insufficient information available at the present time to predict the
effect of such changes.
Tnnes.ee BII Clau
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 4
EFFECT OF CONTINUOUS PHASE ON FLOW OF SLUDGE
Characteristics Per Cent Value of Value of VH*3 V* 5
of Sludge (by weight) S 7 ft per sec. ft. per see.
Tennessee Ball 15 0.087 0.0104 1.69 1.83
Clay and Water 18 0.129 0.0124 2.44 2.60
20 0.210 0.0140 3.86 4.04
Tennessee Ball 15 0.087 0.0300 2.07 2.45
Clay and Glycerine 18 0.129 0.0322 2.82 3.23
Diluted with Water 20 0.210 0.0340 4.24 4.68
*Head loss in feet of water per 100 feet of 6-in. pipe, computed by Equation (14).
13. Nature of the Continuous Phase.-Most of the sludges con-
sidered in this investigation were mixtures of solids and water, the
particles of solids being suspended in the water. Sludges may, how-
ever be composed of a mixture of solids suspended in some fluid other
than water. Since the viscosity of all fluids cannot be expected to be
the same it leads to the conclusion that the coefficient of rigidity of a
sludge, and hence its resistance to flow, depends on the viscosity of the
continuous phase. In order to demonstrate that the continuous phase
is of importance in affecting the coefficient of rigidity, measurements
have been made in this investigation of the coefficient of rigidity and
the yield value of sludges composed of water and of sludges composed
of glycerine as the continuous phase. Some results are reported in
Table 4. It will be noted from the table that the continuous phase has
no effect on the yield value provided the percentage of suspended
matter remains constant. Because of the fact that the specific gravities
of the continuous phases in the two sludges shown in Table 4 are
almost the same, the percentages by volume of solids in the two sludges
were closely alike. The equality of the yield value of these two sludges
bears out Bingham's hypothesis to the effect that as long as the per-
centage by volume of the suspended matter in different mixtures is the
same, the yield value is the same, and is independent of the character
of the continuous phase, as long as the continuous phase is inert with
respect to the suspended particles. This is an interesting fact and
should prove useful in evaluating the effect of particle size on yield
value since the effect of a change in the continuous phase does not have
to be considered. In this investigation no attempt has been made to
correlate particle size and yield value.
14. Temperature.-Temperature has a marked effect on the vis-
cosity of fluids. In the case of liquids a rise in temperature lowers the
LAMINAR FLOW OF SLUDGES IN PIPES
viscosity while in the case of gases the reverse is true. The relations
between temperature and viscosity have been determined for water
and many other liquids, and since the coefficient of rigidity of sludge
depends to a great extent on the viscosity of the continuous phase, an
estimate of the effect of temperature on the coefficient of rigidity may
be made. No attempt was made to formulate the effect of tempera-
ture on the yield value other than to show that an increase in tempera-
ture lowered the yield value in the case of sewage sludge. Hatfield13
also found a decrease in resistance to flow of sewage sludge with
increase in temperature.
The effect on a sewage sludge of a rise in temperature is to reduce
the yield value and the coefficient of rigidity, as is shown in Table 5.
According to Bingham the temperature and viscosity of the continuous
phase is without effect on the yield value. This apparent disagreement
may be explained by the fact that Bingham's hypothesis refers only
to a mixture in which the character of the dispersed phase is not
affected by the temperature, whereas, in sewage sludge, the character
of the dispersed phase may be affected by temperature.
15. Thixotropy.-Thixotropy is the property, or phenomenon, ex-
hibited by some gels of becoming fluid when shaken. The change is
also reversible. Such properties have been found in sewage sludges by
other investigators and also in this investigation. In measuring the
coefficient of rigidity and the yield value of a sludge it is essential that
their changes due to agitation be controlled, otherwise a true measure-
ment cannot be made. Hatfield1" states: "This thixotropic property,
first mentioned by Merkel14 is very important and must be considered
both in the determination of the viscosity of the sludges, and in the
application of viscosity data to engineering problems, because (1) it
is impossible to obtain a sample of quiescent sludge and get it into the
viscometer beaker without some agitation or stirring, and (2) during
each 100 revolutions of the viscometer cylinder the velocity of revolu-
tion has accelerated with each revolution. This acceleration is par-
ticularly noticeable at very low velocities on thick sludges, in which
case only 10 to 20 revolutions are timed." It was realized that meas-
urements of head loss made by recirculating sewage sludge through
any form of apparatus led to erroneous observations due to thixotropy.
It is believed the errors due to thixotropic properties are avoided
where desired in this investigation through the use of the long tube
viscometer, through the absolute measurements made in the apparatus
described on pages 47-49, or through direct measurements of flow made
with the apparatus at Decatur, described on pages 45-47.
ILLINOIS ENGINEERING EXPERIMENT STATION
H
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LAMINAR FLOW OF SLUDGES IN PIPES
It is possible that the acceleration of the cylinder in the Stormer
viscometer, observed by Hatfield and the present authors, is due partly
to a combination of "slippage" and "seepage," as a result of which a
layer of water has tnrmed between the revolving cylinder and adjacent
sludge, causing slippage. It was found that by smartly pulling the
string attached to the drum, thereby causing the cylinder to revolve
at a high rate, much of the original resistance could be restored. It is
probable, therefore, that some of the thixotropy of sewage sludge is
only apparent, and is actually due to seepage and slippage. A similar
condition was observed in the swinging pendulum viscometer described
on pages 34 and 35. It was found that by stopping the motion of the
viscometer and tapping it or gently stirring the sludge in it, much of
the original resistance to the swinging of the pendulum was restored.
16. Agitation.-Agitation may change the resistance to flow of a
sludge in a given pipe line by changing both the yield value and the
coefficient of rigidity. Agitation may change the size of particles in
the mixture, rearrange or redistribute the particles, or produce mani-
festations of thixotropy. The pumping of sludge through reciprocating,
centrifugal, or rotary pumps or by other mechanical means, is a com-
mon cause of agitation, which affects its flow characteristics. It is
also possible that merely flowing through a long pipe line may so
change the values of q and S, of a sludge that the resistance to flow
near the end of the line is less than at the beginning. It is essential
to make note of the effects of this phenomenon in measurements of the
factors affecting the flow of sludge, and, where the effect of agitation
is of importance, to report it in connection with the measured factors.
Results of tests of the flow of sludges which show the effect of
agitation are reported in Table 6. It is to be noted that the resistance
to the flow of yellow clay was increased by agitation, probably by the
breaking up of the particles, and the resistance to the flow of sewage
sludge was decreased by agitation, probably as a result of thixotropy.
17. Slippage and Seepage.-One of the assumptions made in the
development of the laminar flow equation for sludges was that no
slippage occurred at the pipe walls. That slippage is possible, espe-
cially when the continuous phase seeps out toward the pipe wall,
thereby causing the solids concentration to be less in this region, is ap-
parent, and has been the subject of investigation by others." 14 Slip-
page is a rare phenomenon in the laminar flow of sludges in pipes,
but may occasionally be noticed where the wall with which the sludge
is in contact is glass smooth and the yield value of the sludge is high.
ILLINOIS ENGINEERING EXPERIMENT STATION
An interesting use of slippage is made in a few long pipe lines designed
to carry crude petroleum which exhibits a yield point. The pipe line
is designed with spiral ridges on the inside of the pipe. About 10 per
cent of water is mixed with the crude oil, and the mixture pumped into
the pipe. The ridges cause the mixture to revolve like a bullet fired
from a rifle. The water, being heavier than the oil, is thrown to the
outside of the pipe, where it lubricates the flow of the more viscous
oil in the center of the pipe. The phenomenon of slippage is commonly
easy to recognize as its occurrence will cause a drop in the friction
loss as the velocity of flow is increased. In measuring the resistance
offered by a sludge to flow through a pipe, or in a viscometer, errors
due to slippage may be overcome by avoiding the use of too smooth
a surface in contact with the sludge.
Seepage is the flow of the continuous phase through the dispersed
phase, and occurs only when the shearing forces tending to produce
flow are less than the yield value. The medium in effect filters through
the mixture. Seepage is relatively unimportant in pipes larger than
capillary tubes.
18. Gas Content.-Bubbles of gas so finely divided as to be unable
to rise and escape may occur in a sludge as a result of bacterial
fermentation or as a result of mechanical stirring. Because of this the
density of the mixture is lowered. Sewage sludges have been observed
in this investigation with 15 per cent solids whose density is materially
less than that of water. Density theoretically has no effect on the
laminar flow of sludges in pipes, but it has been observed that, when
the velocity of flow is large enough to cause turbulence, the head loss
due to friction is proportional to the density, as is shown in Fig. 1,
where the lines representing the flow of sewage sludges and one test
on Illinois yellow clay are displaced from the line representing the
flow of water by amounts approximately proportional to their densities.
Investigations into turbulent flow relationships of sludges should take
account of the density factor.
IV. MEASUREMENT OF YIELD VALUE AND COEFFICIENT
OF RIGIDITY
19. Measurements in Pipe Lines.-Values of S, and y7 may be
measured in a variety of ways. For sewage sludges and for sludges
commonly encountered in industry the simplest means is probably the
best. Any existing pipe line through which the sludge may be pumped
LAMINAR FLOW OF SLUDGES IN PIPES
FIG. 8. THE STORMER VISCOMETER
$ sudge
Non-rotat'ng I I
Cup ---: - -_
Rotating I-1-- I-I-- .
C/o? ea - - - - - -_I _ - - "
Cy//de-- -
/ofe : Surfaces in contact wit/? s/uodge
must not /be h/ih// po/s/~ed.
FIG. 9. MODIFIED STORMER VISCOMETER CUP
ILLINOIS ENGINEERING EXPERIMENT STATION
can be used in the measurement of the values of S, and q. The ob-
servations to be made under such conditions are the friction losses
between two points on the pipe line, at two or more velocities below
critical. These observations are plotted on a diagram with S, as
ordinate and V/4D as abscissa. The slope of the line formed by con-
necting the plotted points is the coefficient of rigidity, and the intercept
of the line on the Sp axis is 4/3 the yield value, as is seen from
Equation (13). The values of S, are obtained from the head losses,
expressed in feet of water, by means of the following equation:
15.5H,,D
S, - (19)
L
20. Measurements with a Modified Stormer Viscometer.-The
Stormer viscometer, illustrated in Fig. 8, can be adapted to the meas-
urement of the yield value and the coefficient of rigidity of a sludge
by slight modifications. Sewage sludges in particular have numerous
large particles that render the standard Stormer cup useless by binding
the rotating cylinder. To overcome this difficulty, and yet to maintain
the clearances at a minimum, the cup illustrated in Fig. 9 was designed.
Although turbulence occurs sooner with the modified cup than with the
standard cup, interference of the particles with the rotating cylinder is
believed to have been eliminated.
The procedure when using the modified Stormer viscometer for the
measurement of S, and q is as follows: The material to be measured
is poured into the cup to exactly /4 inch from the top, and the
rotating cylinder is inserted. A known weight to act as a driving force
is attached to the string wound on the drum; the brake is released and
the time for the cylinder to make 100 revolutions is noted, and recorded
as revolutions per second, together with the corresponding driving
force in grams. After several readings have been taken at various
values of driving force a new cupful of material is taken in order to
eliminate, as much as possible, errors due to thixotropy.
Plots of observations made with a modified Stormer viscometer for
four clay sludges are shown in Fig. 10. The driving force in grams is
plotted as ordinate using the symbol u. The speed of revolution is
plotted as abscissa using n as the symbol. Figure 10 is a plot of the
flow characteristics of an Illinois yellow clay of four different concen-
trations of suspended matter. In Fig. 11 the pipe flow characteristics
of the same four sludges are plotted with S, as ordinate and V/4D as
abscissa. The similarity of the graphs is apparent, and is the basis for
LAMINAR FLOW OF SLUDGES IN PIPES
Revo/u/,ons per Second, '1"
FIG. 10. PLOT OF U AGAINST U FOR CLAY SLURRIES USED FOR
CAI3BRATION OF THE STORMER VISCOMETER
converting the Stormer data to the pipe flow constants S, and q. It has
been found that in Fig. 10, the intercept u, on the u axis of a line
connecting the points representing driving force, u, and speed of revo-
lution, n, is proportional to the yield value Sy, and that the slope, A, of
the same line, is proportional to the coefficient of rigidity, q. Table 7
presents a comparison of values of u, and S, and A and q, and demon-
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 11. PLOT OF S, AGAINST V/4D FOR CLAY SLURRIES USED FOR
CALIBRATION OF THE STORMER VISCOMETER
strates that the modified Stormer viscometer may be used in measuring
the constants S, and q.
Any Stormer viscometer can be adapted to the measurement of S,
and q without calibration if the modified cup described in the fore-
going is used. The following equations can be used for converting
Stormer data to pipe flow constants:
S2 = 0.0020u, [(20)
7 = 0.0035X (21)
At low rates of shear in the modified Stormer viscometer cor-
responding to small speeds of revolution the relationship between the
driving force and the speed of revolution is not linear, which results
in the line, connecting the points representing driving force and
LAMINAR FLOW OF SLUDGES IN PIPES 33
E<
ZS
0
Ei
0
a
2l
0
0
CCC
0
0 ~-
2<
0
12~
001.~
01
020
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00
0
0
~0
0
01
12~
0- 0 0 0 0 C 0 i0
0 05 0 0^ 0 00
00 0 0^ - 0 01'
0 0ff C- ^j M -^
0 0 01 0 01 0
0 0 0 0i 0- 0
0N 0N 0N 0- 0 (
00 0 >0 0 0
0> 0' 01 .CM 0 0
00-i 001 '0
1 0 0^ ^ j 0
01 0 -i ^ 0d
2* 0- 0 01 = 0
03 o3 o3C oi o 0
'0 '0 '0 '0 01 t
0 0 0 0 0 ,
'3 230. "3'3
0x 0 0 0H 04
0
CC
01
ILLINOIS ENGINEERING EXPERIMENT STATION
corresponding speed of revolution, bending toward the origin. The
explanation of this is that, at low rates of shear, the material shears
first at the point of greatest stress, which is in the layers of material
next to the rotating cylinder. As the rate of shear is increased, shear
takes place progressively further from the rotating cylinder until, at
some point, shear is taking place from the rotating cylinder to the cup
sides. At this point the relation between the increase in driving force
and the increase in speed of revolution becomes linear, and continues
in this manner until turbulence is introduced by the high rates of
shear. It is essential that the driving force and speed of revolution be
measured between the areas bounded by lines AB and CD in Fig. 10,
as only between these lines is the relationship between the increase in
driving force and the increase in speed of revolution linear.
21. Other Viscometers Useful in Measuring Sy and q.-Viscometers
suitable for measuring the values of S, and q of a sludge include
(1) rotating cylinder viscometers, of which the Stormer and Kampf
are examples, (2) falling ball viscometers, (3) swinging pendulum
viscometers, and (4) capillary tube viscometers. Of these four only
the last one will measure the absolute values of the constants S, and
7. The underlying principles of some of these viscometers are explained
in detail in the First and Second Reports on Viscosity and Plasticity
of the Academy of Sciences of Amsterdam.15
22. Rotating-cylinder Viscometers.-Most standard rotating-cylin-
der viscometers are suitable for measuring the significant constants of
a sludge. The Stormer viscometer was used in this investigation
because it is a commonly-used type in industry. The method of using
the Stormer viscometer is outlined in Section 20.
23. Falling-ball Viscometers.-Viscosity can be measured by apply-
ing Stokes Law to the observations made on a sphere dropping through
the viscous substance. A falling-ball viscometer was tried in this
investigation and discarded as impracticable because the falling
sphere tended to travel in a helical path instead of in a straight line
when the sphere fell sufficiently rapidly to cause turbulence.
24. Swinging-pendulum Viscometers.-The swinging-pendulum vis-
cometer operates on the principle that the rate of damping of a
swinging arm attached to a heavy pendulum is proportional to the
viscosity of the material in which the swinging arm is immersed. A
viscometer of this type was constructed and used on some measure-
LAMINAR FLOW OF SLUDGES IN PIPES
ments in this investigation. The instrument was not used extensively,
however, because of the greater availability and greater simplicity of
operation of the Stormer viscometer. It was concluded that measure-
ments of the constants in plastic flow are possible with swinging-
pendulum viscometers after calibration.
25. Capillary-tube Viscometers.-Capillary-tube viscometers are
not satisfactory for measuring the pipe flow constants of the type of
sludges studied in this investigation, because of trouble from clogging
of the capillary. An outstanding advantage of the capillary-tube
viscometer, where the material to be studied will flow through it
without clogging the tube, is the possibility of computing Sy and q
directly by applying the proper observations to Poiscuille's equation.
This type of viscometer is described by Herrick16 who used it to
measure the flow characteristics of rotary drilling mud. Bingham'
advocates this type of viscometer, which he calls a "plastometer." It
was not used in this investigation because of difficulty in attempting to
cause sewage sludge to flow through a capillary tube.
V. APPARATUS AND TESTS
26. Preliminary.-Due to the fact that sufficient sludge was not
available from the Sewage Testing Plant operated by the Engineering
Experiment Station to permit tests on a large scale, the preliminary
apparatus was set up in the Sanitary Engineering Laboratory to per-
mit recirculation of sludge through glass and rubber tubes about 14 in.
in diameter. The sludge in this apparatus was circulated by gravity
from a reservoir near the ceiling of the room. This reservoir was
intermittently refilled by catching the discharge from the apparatus
in containers which were lifted and emptied into the overhead tank.
The apparatus was later supplemented by similar equipment, set up in
the Sewage Testing Plant, using a section of smooth brass pipe one
inch in diameter, in which friction losses were measured, and through
which the sludge was recirculated by means of a rotary pump that
returned the used sludge to a constant-level tank at a fixed elevation,
from which the sludge flowed through the apparatus, the rate of flow
being controlled by adjustments of a gate valve.
Observations of friction losses as sludge flowed through such an
apparatus proved of little value because of various difficulties. The
apparatus served, however, to indicate methods of measuring heads or
pressures, and how to overcome clogging by avoiding the slightest
ILLINOIS ENGINEERING EXPERIMENT STATION
Flush Water
FIG. 12. GAGE CONNECTION AND MUD TRAP
constriction in the sludge pipe, and to draw attention to the impossi-
bility of using partly-open valves to control the flow of sludge, and to
other difficulties.
27. Difficulties in Measurements.-Tests with the first constant-
level apparatus showed that difficulties were due, in part, to the follow-
ing causes: (1) unsuitable gage connections to the test pipe,
(2) entrance of sludge into the gages, (3) attempts to regulate the rate
of flow by means of a valve and (4) the recirculation of sludge. All
of these difficulties were satisfactorily overcome in subsequent arrange-
ments of the apparatus.
28. Gage Connections.-Sludges have been found to give trouble
by clogging gage connections unless precautions are taken to avoid the
difficulty. An easily-made and satisfactory connection is shown in
Fig. 12. This is the type of connection used in the final form of
apparatus. The plug, shown at A in the figure, was inserted in one
of the openings of a cross and the opposite opening was connected, by
means of standard pipe-fittings, to the gage. In case of clogging the
plug can be removed and the connection cleaned. For pipes 3 inches
or larger it is desirable to use a cross with 11/-in. openings on one run.
Sect/on
al Y-Y
LAMINAR FLOW OF SLUDGES IN PIPES
29. Clogging of Gages.-To keep sludge from entering the gages
through the gage lines a trap was provided at the gage connection, as
shown in Fig. 12. The top of the trap was capped and a /s-in. air cock
was tapped into the center of the cap. It has been found that the gage
take-off B should be located at least 4 inches below the top of the trap
in order that entrapped air may not enter the gage lines. Frequent
flushings through valves C and D are necessary during operation in
order to expel accumulated sludge and air. Putting a valve between
the bottom of the trap and the gage connection on the test pipe, as
shown at E in the figure, makes it possible to flush the gage lines
without shutting down the flow in the test pipe.
30. Velocity Control.-The initial set-up contemplated the control
of the velocity in the test pipe by means of a valve. It was soon found
that this means of velocity control was impossible in the case of
several sludges. Such articles as wads of chewing gum, match sticks,
etc. found in sewage sludge would clog the valve at low flows so that
constant velocity for even a short time could not be maintained. It
was found possible to control velocity of flow by either one of two
methods: (1) by the use of a constant-level tank in which the level
could be changed manually as desired, and (2) by means of a rotary
pump with controlled speed. The latter method proved the more
successful, particularly at low speeds.
A constant-level tank will not assure constant velocity in the
neighborhood of the critical velocity because of the unbalanced con-
ditions in the transition region. On the other hand, a rotary pump
driven at constant speed may not maintain a constant head near the
transition region. As a result of experience with the preliminary tests,
methods of measuring pressures and of catching and weighing sludge
passing through the apparatus were so improved as to facilitate the
work done later with the long tube viscometer, and with the bank of
three pipes set up to measure viscosity directly. The only satisfactory
method of forcing sludge through such pipes was found to be with a
rotary pump with a closely controllable speed. Any other method of
pumping was found to give sudden and frequently apparently incon-
sistent variations in head loss and velocity of flow.
31. Recirculation.-Due to the uncertainty of maintaining constant
physical characteristics of a plastic with thixotropic properties on
prolonged agitation, recirculation of thixotropic sludges is undesirable.
In tests of friction losses in such sludges it is necessary to provide a
reservoir of sludge of sufficient capacity to make a number of tests
ILLINOIS ENGINEERING EXPERIMENT STATION
Compressed A/'r /n/7ef-
Mercury
Gaye
FiG. 13. DIAGRAM OF LONG TUBE VISCOMETER
without reusing the sludge. Where the sludge does not possess thixo-
tropic properties there is no objection to its recirculation through the
apparatus when friction losses are to be measured.
32. Sludge Analyses.-Analyses of the sludges were made through-
out the entire investigation, the analyses including determinations of
the total solids, ash, specific gravity, and in some cases the pH value.
In all tests, except that for specific gravity, the recommendations of
"Standard Methods for Water and Sewage Analyses" of the American
Public Health Association were used.
33. Specific Gravity Measurements.-In the preliminary tests
specific gravities were determined by placing a known volume of
sludge in a weighed and calibrated, 500-ml. volumetric flask, and
filling up the flask with distilled water, taking care not to entrain air.
The flask containing the sludge and the water was then weighed, and
the volume of added distilled water was computed. The density of the
sludge was then found as the ratio of the weight of the sludge to the
Receiving
Tank and
Sca/es;
-0u/1C/ Openim, Val1e>
LAMINAR FLOW OF SLUDGES IN PIPES
FIG. 14. LONG TUBE VISCOMETER
weight of an equivalent volume of distilled water. A more satisfactory
and accurate method was by means of a pyenometer.
In certain regions of flow the specific gravity of the flowing sludge
plays an important part (see pages 8 and 28). It is necessary, there-
fore, to measure the specific gravity with precision. The use of a
hydrometer is not feasible for this purpose in relatively thick sludges,
because the hydrometer is not free to come to equilibrium. The most
satisfactory method of measurement of specific gravity is by means of
the pyenometer. In all of these tests specific gravity was measured by
means of a calibrated, small-mouth gallon jug which served as the
pycnometer. The jug was weighed to the nearest one hundredth of a
pound and carefully filled with the sludge to be measured, in such
a manner as to avoid the entraining of air, since bubbles entrained in
the sludge when pouring into the measuring bottle cannot rise in the
sludge. Specific gravity tests should be made immediately after the
ILLINOIS ENGINEERING EXPERIMENT STATION
test runs so that finely divided gas bubbles which affect certain regions
of flow do not escape.
34. Long Tube Viscometer.-The apparatus illustrated in Figs.
13 and 14 has been designated as a long tube viscometer. In its
operation about fifty gallons of the sludge to be tested was poured
into the tank through a %-inch mesh wire strainer at A, to remove
large particles, such as chewing gum, leaves, etc. After quiescent
conditions had existed long enough to avoid errors through thixotropic
properties of the sludge, air, under pressure, was admitted into the
tank. Pressures less than about 25 lb. per sq. in. were observed by
means of a mercury gage. Higher pressures, up to a maximum of
60 lb. per sq. in., were measured on a calibrated Bourdon gage. The
pressure datum was taken at the elevation of the horizontal discharge
tube, corrections being applied to the gage readings, making allowance
for the depth of sludge in the tank during a run. The amount of
sludge removed during any run was relatively so small that no allow-
ance was necessary for the small amount of change of depth of sludge
in the tank during a run.
When the desired pressure was attained, the quick-opening valve
was snapped open and sludge was allowed to flow into the container
shown on the scales in Fig. 13. During a run the air pressure was
maintained constant by the manual adjustment of a needle valve on
the air pipe. As the scale beam rose the time was observed, a known
weight was added to the scale arm, and the time taken when the
beam arose again. The observations thus gave information from
which friction loss and the velocity of flow could be computed. A
summary of all results is given in Table 8.
Advantages found with this type of viscometer were quick measure-
ments and avoidance of interference by thixotropic properties of the
sludge. Occasionally runs on the same sludge were repeated to check
this fact. Unfortunately, as designed for these tests, the short section
of 2-in. pipe shown at D in Fig. 13 precludes the possibility of
determining the yield value. Although the friction loss in the 2-in.
pipe is negligible when compared with the friction loss in the 3-ft.
length of %-in. drawn brass tube, the larger pipe greatly changes the
apparent value of the yield value as calculated from the area of the
%-in. test pipe. It was necessary, therefore, to calibrate the appara-
tus with a material of known rigidity and yield value in order to de-
termine the absolute rigidity and yield point of an unknown material.
This was done with material measured in the circulating apparatus
described on pages 47-49.
LAMINAR FLOW OF SLUDGES IN PIPES 41
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ILLINOIS ENGINEERING EXPERIMENT STATION
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ILLINOIS ENGINEERING EXPERIMENT STATION
FI-7 LJ [ 2 "I4'r'// /p/o e I
FIG. 15. DIAGRAM OF EQUIPMENT AND APPARATUS USED AT
DECATUR, ILLINOIS
FIG. 16. EQUIPMENT AND APPARATUS USED AT DECATUR, ILLINOIS
Among the important findings resulting from the tests with this
viscometer were (a) the true plastic flow properties of sludges, par-
ticularly sewage sludges, and (b), by computation after the calibration
of the viscometer, the values of the yield value and of the coefficient
of rigidity of the sludges tested.
LAMINAR FLOW OF SLUDGES IN PIPES
30
*a
'I.
"4
/
Velocity in ft per sec.
FIG. 17. FRICTION Loss OF DIGESTED SEWAGE SLUDGE FLOWING IN
2-IN. IRON PIPE AT DECATUR, ILLINOIS
35. Tests and Apparatus at Decatur, Illinois.-Conditions at the
sewage treatment plant of the Sanitary District at Decatur, Illinois,
were unusually favorable for conducting tests on the friction losses
for the flow of sewage sludge in pipes without the necessity for recircu-
lating the sludge. The test installation is illustrated in Figs. 15 and 16.
The separate sludge digestion and storage tank from which the sludge
was drawn is shown at the left of the illustration. Sludge was allowed
to run from the tank through a 41-ft. length of 2-in. pipe, discharging,
by gravity, into a container resting on a beam scales. The pipe ar-
rangement was such that sludge could be drawn from two different
levels in the storage tank. Gage connections and gage-line equipment
similar to that finally adopted on the preliminary apparatus were used
on this equipment, the upper gage connection being placed on the pipe
following a straight run of 10 ft. The rate of flow through the pipe
was controlled by raising or lowering the outlet by adding upright
sections of straight pipe. Such a method of control was not entirely
satisfactory, as clogging of the entrance to the test pipe interfered
with the control of the rate of flow in the pipe and with the head-loss
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 18. DIAGRAM OF CIRCULATING APPARATUS USED FOR MEASURING Sy AND 1
FIG. 19. APPARATUS USED FOR MEASURING S, AND 77
LAMINAR FLOW OF SLUDGES IN PIPES
observations in the gage. The maximum head available was about
20 ft. between the top of the sludge in the tank and the lowest point
to which the end of the test pipe could be lowered.
Some of the observations made during these tests are shown
graphically in Fig. 17. Using the values of the yield value S,, ob-
tained from the tests and assuming values of the coefficient of rigidity,
,, of 0.02 and 0.01 for sludge drawn from the lower and higher levels
in the sludge tank, respectively, the computed critical velocities are
7 ft. per sec. and 3.5 ft. per see., respectively.
36. Final Recirculating Equipment.-A special circulating appa-
ratus, consisting of lengths of three different diameters of pipe for
measurement of values of S, and of q was constructed at the Univer-
sity Sewage Treating Plant. The purposes of the apparatus were (1) to
demonstrate that the values of S, and of qj are independent of the di-
ameter of the pipe, (2) to correlate the data from the %3-in. pipe vis-
cometer with known absolute values of S, and of q, and (3) to correlate
the data obtained by the Stormer viscometer with absolute values of S,
and of -q.
The apparatus, illustrated in Figs. 18 and 19, consisted of a tank
for holding the sludge, a rotary pump for circulating it, a variable
speed motor for driving the pump which permitted quick adjustment
to any speed between about 30 r.p.m. and 400 r.p.m., and lengths of
1-in., 2-in. and 3-in. pipe. The sludge could be diverted into any one
of the pipes by means of gate valves. Gage connections and gage-line
equipment were similar to those used in the Decatur tests.
Loss of head was measured in a mercury U tube. Unsteadiness
of level of mercury in the U-tube gage, due to pulsations caused by
the rotary pump, were overcome by inserting a small bore tip in each
gage line where it entered the gage. This expedient reduced the pul-
sations of the mercury levels to a negligible amount and gave ap-
parently very precise readings. Velocity of flow was measured by
weighing the quantity of sludge discharged through the pipe in a
known time.
Figure 20 shows, for comparative purposes, a plot of head loss
against velocity in all three pipes in the special circulating apparatus
for a well-digested sludge taken from an Imhoff tank at Decatur,
Illinois. Before running these tests the sludge was circulated through
the pipes at a high velocity for two hours, and the observations were
then made as rapidly as possible.
ILLINOIS ENGINEERING EXPERIMENT STATION
Velocit'y in ft per sec
FIG. 20. FRICTION LOSSES OF DECATUR IMHOFF SLUDGE FLOWING IN
1-IN., 2-IN., AND 3-IN. PIPES
The observations in these tests are plotted in Fig. 21 with S, as
ordinate and V/4D as abscissa for the purpose of studying the relation
between S, and q and the pipe diameter. Since the points all fall on
the same straight line when the flow is laminar, within the limits of
experimental error, it is concluded that Sy and q are independent of
the diameter of the pipe in which they are measured.
Tests on a thick yellow clay were also run in the circulating ap-
paratus. The clay, obtained locally, had about 52 per cent solids by
weight. Results of the tests are shown in Fig. 21. It is apparent that
these results further corroborate the theory that S, and 7 are inde-
pendent of pipe diameter.
In order to correlate the measurements made in the long tube vis-
cometer and the Stormer viscometer with the absolute values of S,
LAMINAR FLOW OF SLUDGES IN PIPES
C~)
K.
2.0 - I I I I I I
/rocuqhf /ron P/'e I
-- - o- -- -C-
L '- 9------ ----- -
iro -_ _ 3- -/c/7 __ __ __ _ o, .- - _
I III04re
0. - - -- -- -- "-- -- -- '-- -- - "--==7
S 4 8 /0
4Z
FIG. 21. EFFECT OF PIPE DIAMETER ON FLOW CHARACTERISTICS OF SLUDGE
and 7 determined in the circulating apparatus, the sludge used in the
preceding two tests was run also in the long tube and the Stormer
viscometers immediately after the tests in the circulating apparatus
had been completed. From the observations with the same sludge in
the various viscometers conversion factors were computed by which
the relative values of S, and ] found in the viscometers could be
converted to absolute values of these factors.
VI. RESULTS AND CONCLUSIONS
37. Summary.-It has been shown that sludges such as mixtures
of clay and water used in deep-well boring, sewage sludges, sludges
from water softening plants, and other similar aqueous suspensions
of fine particles, obey the fundamental formula for the flow of a true
plastic.
The theoretically developed formula for friction loss when a sludge
flows through a straight circular pipe has been experimentally verified
for 1-inch, 2-inch, and 3-inch pipes using several different sludges
as the flowing material. The theoretical formula applying only to
laminar flow is:
He 16S, 77V
L 3 WD WD2
*Nomenclature not given here will be found on pages 10 and 11.
(14)*
/Z /4 /6" /8 Z0
ILLINOIS ENGINEERING EXPERIMENT STATION
It has been shown that for a sludge flowing in a circular pipe a
critical velocity is encountered as the velocity of flow is increased.
Below the critical velocity the flow is laminar, while above this velocity
the flow is turbulent. The critical velocity does not always occur at
the same velocity in a given pipe with a given sludge. Based on the
known behavior of fluids in this region two formulas for critical
velocity have been developed and verified experimentally. One formula
gives a velocity that has been called the lower critical velocity, below
which the flow will be laminar, and the other formula gives a velocity
that has been called the upper critical velocity, above which the
flow will be turbulent. Between these two velocities the flow may be
either laminar or turbulent or a combination of the two, depending on
various factors. The formula for the lower critical velocity is
V 10007 + 103-/ 942 + D2Syp
Vy, = (17)
Dp
The formula for the upper critical velocity
1500i7 + 127V 14072 + D2Syp
Vuo = (18)
Dp
The yield value, Sy, and the coefficient of rigidity, 7, have been
shown to be independent of the diameter and roughness of the pipe
in which they are measured.
A simple method for the measurement of S, and y by means of a
modified Stormer viscometer has been given, and it has been shown that
S, = 0.0020 (20)
and
- = 0.0035X (21)
Table 8 lists the yield value and the coefficient of rigidity for
sludges measured in this investigation, together with similar values
computed from tests reported in the literature. The table also gives
important data in regard to the other characteristics of the sludges.
Heretofore no formulation of the factors affecting the friction loss
in the flow of sludge through straight circular pipes has been avail-
able. Further research as to the factors affecting the turbulent-flow
frictional losses of sludges flowing in circular pipes is desirable, as
indications are that the common hydraulic formulas for the flow of
water are not applicable to the flow of sludges.
APPENDIX
OTHER INVESTIGATIONS
Investigations into the flow characteristics of fluids have been
extensive and have resulted in the development of the familiar hy-
draulic formulas, such as that of Hazen and Williams, and in the
universal flow formula of Weisbach which, by means of Stanton's'
expansion of Reynold's9 studies, makes possible the solution of prob-
lems involving head-loss-velocity relations for any fluid, provided
its specific gravity and viscosity are known and constant. Many in-
vestigations have been made of the flow characteristics of plastic
solids in capillary tubes, an exhaustive treatise on this subject being
presented by Bingham' in 1922. However, in the words of Bingham,
in referring to "hydraulic flow in the plastic state," "So far as known
to the author no one has yet used rates of flow high enough to bring
about eddy currents which are so troublesome in the case of fluids."
None of the reports of investigations mentioned by Bingham nor
found in the literature subsequent to 1922 has attempted to present
a formula applicable to the flow of thin sludges in either the laminar
or the turbulent region.
A brief summary of references to some pertinent investigations into
the flow of fluids and plastics is contained in the bibliography on
pages 57 and 58. The following extracts have been taken from reports
of investigations into the flow of thin sludges published subsequent to
1919:
1. Nevitt, T. H.17 The tests were made on a 12-in. cast-iron pipe
240.5 feet long. They gave a variation for Kutter's n from 0.0168 to
0.0181 after five years of service. The sludge was obtained from sedi-
inentation tanks, the age being not more than ten days. Its density
was 1.01, moisture content 95.9 per cent, ash, 49.5 per cent, and
temperature, 54 deg. F.
2. Clifford, W. E.18 The author attempted to base the flow of
sewage sludge on the laws dealing with viscous fluids. He determined
the kinematic viscosity by two methods:
(a) By comparing the flow of sludge with the flow of a homogene-
ous fluid of known viscosity, measuring the time of efflux through a
tube 4e6 in. in internal diameter and 51/4 in. long.
(b) By determining the viscosity directly by measuring the fric-
tion loss in a pipe, converting to kinematic viscosity, and applying
ILLINOIS ENGINEERING EXPERIMENT STATION
the general equation of the flow of liquid through a pipe. From a com-
parison of the flow of a sludge containing 90 per cent moisture with
the flow of glycerine with known kinematic viscosity, Clifford found
the kinematic viscosity of his sludge to be 0.001055 lb. per ft. per sec.
By applying the assumed viscosity to the data for sludges containing
90 per cent moisture flowing in an 8-in. pipe at Calumet he calculated
theoretical friction losses which checked closely with observed fric-
tion losses.
3. Gregory, W. B.3 studied the flow of a clay slurry, containing
about 85 per cent moisture, through a pipe of 4-in. diameter. The
viscosity of the clay slurry was computed by Poiseuille's law, with the
conclusion that the substance has no viscosity comparable to that
of a homogeneous fluid similar to oil or water. Gregory found that in
the viscous or stream-line zone the head loss was almost constant,
and explained this by stating that, since the viscous or stream-line
resistance in a true liquid varies as the velocity, then the presence
of solids suspended in the liquid will cause a variation in the opposite
direction, so that the sum of the two is a constant below the critical
velocity. "It may be expected that with increase in concentration of
solid particles the starting resistance will increase. Also that with
equal concentration, a suspension of a finer material will have a
smaller starting resistance and, hence, a lower critical velocity." He
concluded that
(a) The most economical velocity for pumping is the critical
velocity, and
(b) The apparent viscosity of slurry at the critical velocity varied
from 24 to 85 times that of water, depending on the concentration of
solids.
4. A Committee of the American Society of Civil Engineers2 re-
ported the results of extensive research and study of the flow of sewage
sludge, including valuable data on flow tests in an 8-in. pipe at the
Calumet Sewage Disposal Plant in Chicago. The sludges tested varied
in moisture content from 89 to 97 per cent, and head loss measure-
ments were made at various velocities in both laminar and turbulent
flow. The conclusions were summarized as follows:
(a) Sludge is neither a viscous nor a homogeneous material but
is variable in character.
(b) The usual analytical tests do not define its physical qualities,
but it seems to behave more like suspended material.
LAMINAR FLOW OF SLUDGES IN PIPES
(c) Below the critical velocity sludges have a different friction
factor from that found above the critical velocity. As yet the co-
efficient of flow below the critical velocity cannot be concisely stated;
above the critical velocity it can only be expressed in ranges.
(d) Sludge friction losses increase with decrease of moisture
content.
(e) Sludge friction losses tend to increase with lower temperature.
(f) Sludge friction losses for high velocities, from about 5 to 6
feet per second or more, tend to follow more closely the characteristic
law for the flow of water.
(g) Friction losses for fresh or undigested sludge and for sludge
from combined sewage are more erratic and the determination of the
friction factor is correspondingly more difficult.
(h) Within the limits of the investigation no law of flow was
found.
5. Hubell, G. E.19 Sewage sludge obtained by sedimentation of
sewage from a combined sewer was pumped through 20 466 feet of
8-in., cast-iron pipe. The sludge, which contained 99.1 per cent mois-
ture, gave Hazen and Williams coefficients of 140 and 137 at velocities
of 3.15 and 4.06 feet per second, respectively.
6. Woolgar, C. A.20 The article deals with the measurement of
viscosity of drilling fluids by means of a Stormer type of rotating vis-
cometer. It is recommended that such an instrument be not used for
liquids with viscosities below 6 centipoises. The flow characteristics
of a clay suspension and the relation between the yield point, the
viscosity, and the percentage clay content of a well-drilling fluid are
discussed.
7. Evans, P. and Reid, A.21 The article deals with the meaning
of the viscosity of a suspension and its measurement. The authors
state that unless the conditions of measurement are specifically
defined the viscosity is meaningless, since the apparent viscosity de-
creases with increasing flow. For many drilling fluids (mud) the graph
connecting the points representing the applied pressure and the dis-
charge from a long narrow pipe approximates a straight line which,
if produced, cuts the pressure axis at a definite point. At low velocities
the graph is not straight but bends to the origin, intersecting the
pressure axis at a point called the yield point. The slope of the straight
line gives the mobility of the mud. The authors give also a method
of measurement of the mobility and the yield point of the drilling
ILLINOIS ENGINEERING EXPERIMENT STATION
mud. They discuss the phenomena of a mud being forced through a
small pipe, changes in viscosity, and the effect of thixotropy on the
flow of mud.
8. G. D. Hobson.22 This article discusses the value of various
types of viscometers in the measurement of the viscosity of mud or
clay-water suspensions. The author points out that the viscosity of
such mixtures can be altered by the addition of certain chemicals,
and that the effect of entrained air is to give a viscosity value far
below that necessary to account for the volume of air present.
9. Andrews, C. A.23 In discussing the viscosity characteristics of
a chemically-doctored drilling fluid the author defines the viscosity of
a colloidal suspension and points out the factors which tend to change
the viscosity. He discusses also the measurements of the viscosities
of drilling fluids under turbulent conditions by means of the Dallwitz
and Wegner viscometer. Precise results will not be given by this
instrument, if used as described, but reliable and reproducible results
can be obtained quickly for comparative purposes.
10. Merkel, W.4 This article is a report on an experimental and
theoretical study of the flow of sewage and plastic materials, largely
based on the more general work on plastic materials reported by Bing-
ham,' Oswald,24 and Reiner.25 The author discusses the various
factors affecting the flow of sludge. He made tests of the viscosity of
sewage sludge flowing through 20-cm. pipes at the Stuttgart Sewage
Disposal Plant. As a result he suggests using a modification of Poi-
seuille's formula developed by Reiner to estimate the flow of sludge
containing 80 to 85 per cent moisture. His tests were made on raw,
separate digested, and activated sludge containing from 80 to 94
per cent of moisture.
11. Hatfield, W. D.11 studied the pseudo-plastic properties and
thixotropic properties of sewage sludges. The author measured the
viscosities of sludges containing from 90 to 99 per cent moisture by
means of the Stormer viscometer, the instrument having been cali-
brated for absolute viscosities by measuring in it the viscosities of
fluids with known viscosities. By plotting shearing stresses against
rate of shear he obtained flow curves similar to those obtained for a
semi-plastic material. In his first article" the author concludes
(a) A rotational viscometric method of studying the viscous proper-
ties of sewage sludge has been described.
LAMINAR FLOW OF SLUDGES IN PIPES
(b) The viscous properties of sewage sludge have been shown to
be pseudo-plastic, that is, the apparent viscosity decreases as the rate
of shear and the shearing stress increase.
(c) The sludge is thixotropic, the pseudo-plastic resistance and,
therefore, the apparent viscosity being greatly reduced by stirring or
shaking.
(d) The apparent viscosity when plotted against the rate of flow
or the percentage of solids produces a straight line on logarithmic
coordinates.
(e) Application of the laws of fluid flow to sewage sludge gives
valuable information, even though these laws are only approximate
when applied to pseudo-plastic flow. Further studies should be made
to correct the equations of flow for the pseudo-plastic properties of
sludge flow.
In his second article"1 the author compares his results with those
of Merkel,4 and shows that, in general, the viscosities of sewage
sludges range from 1000 to 5000 centipoises at low velocities, down
to 25 to 100 centipoises at turbulent velocities. He shows that his
shearing curves for partly-digested sludges containing 84 to 91 per
cent moisture are similar to those of Merkel's in 20-cm. diameter
pipes. By computing the viscosities in this study and plotting these
values against their corresponding velocities the author shows that
these values will plot in a straight line on logarithmic coordinates,
giving a slope of 1.4 to 1.6, which checked those given by sludges
used in Springfield, Illinois.
12. Herrick, H. N.1" This is a report of a series of experiments in-
dicating the relationship between various characteristics of plastic
solids used in the oil fields. The experiments were made on California
muds. The author used a capillary-tube viscometer and gives mathe-
matical formulas for yield point and viscosity values of sludges,
together with methods for measuring the constants and variables in the
formulas.
13. Traxler, R. N.26 It is pointed out that the flow properties of
dilute suspensions of clay and other minerals are of interest to the
pulp and paper technologist. The various concentrations of solids are
discussed. The instruments for evaluating the flow properties in such
systems are mentioned. It is pointed out that particle size, size distri-
bution, and shape are primary properties of the particles of a mineral
ILLINOIS ENGINEERING EXPERIMENT STATION
filler and that, dependent on these, are the secondary properties of the
compacted powder, such as per cent voids and average void size. The
methods for determining both primary and secondary properties are
briefly discussed. A simple relationship has been established between
concentration of the solids present and the viscosity of the suspension.
This relationship makes possible an accurate evaluation of the in-
fluence of a dispersed solid on the flow properties of slurries of which
they are a part.
14. Report on Viscosity and Plasticity prepared by the Committee
for the Study of Viscosity of the Academy of Sciences at Amsterdam.
First Report in 1935,1' Second Report in 1938.1' This is an exhaustive
treatise of the subject, printed in English, in two volumes. The ob-
jects of the committee were: (1) "To gather information regarding
the phenomena of viscous and plastic deformations as they present
themselves in various domains of physics, chemistry, technology, and
biology; . . .. (4) To study the methods used for the measurement
of viscosity and of related properties of matter, to interpret the mean-
ing of the results given by various technical instruments, and, where
possible, to indicate instruments which allow an unambiguous measure-
ment of scientifically well defined quantities."
15. Wilhelm, R. H., Wroughton, D. M. and Loeffel, W. L.27 This
article is a report of experiments on pumping filter-cell and cement
rock suspensions through pipes. Head losses were measured in three
sizes of pipes at velocities ranging from 0.3 to 14 feet per second.
Apparent viscosity characteristics of all suspensions were determined
by means of a rotating viscometer.
Flow was found to exist in two states which were termed "plug"
and "turbulent" flow. Separate correlations, involving pipe and vis-
cometer data, are presented for each type of flow.