Flow of Muds, Sludges, and Suspensions in Circular Pipe D. H. CALDWELL AND H. E. BABBITT University of Illinois, Urbana, Ill. flow or deformation of a fluid. HEMICAL, sanitary, peA theoretical analysis of the flow of sludges The rate of deformation is a troleum, and mechanical in circular pipe is presented. It was found linear function of the deforming engineers may be frethat two distinct types of flow occurred, force. The coefficient of visquently faced with the problem depending upon the velocity of flow. These cosity of a fluid is equal to the of designing pumping equipment two types have been termed “plastic flow” tangential force on a unit area of and pipe lines for conveying either of two horizontal planes a t muds, sludges, or suspensions. and “turbulent flow”. The velocity at a unit distance apart required to Among the industries which which plastic flow changes to turbulent flow move one plane with a unit vemight profit from a knowledge is called the “critical velocity”. locity with reference t o the other of the flow characteristics of Equations are developed and verified explane, the space between being muds, sludges, or suspensionsare perimentally for determining friction losses filled with the viscous substance. the following: oil well drilling, It follows, then, that papermaking, water treatment, for plastic flow and for turbulent flow in sewage treatment, and dredging. circular pipe. An equation for the deter,LLt = 7 x / v (1) At present, pipe friction loss demination.of the critical velocity is also preWhen the centimeter-gramterminations for the flow of sented, together with experimental verificasecond system is used, the name sludges or suspensions are based tion. given to the coefficient of visupon empirical formulas or arbicosity is the poise. I n the foottrary assumptions as the result of Methods of determining the significant pound-second system no name past experience in conveying the constants in the plastic flow and critical is given to the coefficient. particular sludge or suspension velocity equations are described. Plasticity is the property of a in question. substance which enables it to Previous authors @,S, 6,6,11) be continuously and permanently deformed in any direction have noted that for a sludge two types of flow exist similar to without rupture under a stress exceeding the yield value. the laminar and the turbulent states of flow in the case of a After deformation has started, equal increments of stress will trueliquid. I n laminar flow Poiseuille’s equation, used for true produce equal increments in velocity. Reverting to the funfluids, did not apply to the flow of sludge, but for turbulent damental conception of flow between two parallel planes, since flow the same laws applied t o sludge flow as to true fluid flow. a part of the applied force 7 is used up in overcoming the yield I n the case of sludges plastic or laminar flow changed to turbuvalue T ~the , equation for plastic flow becomes lent flow a t a more or less clearly defined velocity, called the “critical velocity”. For the purpose of the present article the type of flow which occurs below the critical velocity is termed “plastic flow” and the type occurring above the critical where ?I‘is the coefficient of rigidity of the material, analogous velocity is called “turbulent flow”. to the coefficient of viscosity of a true fluid. There has been a marked scarcity of data in the literature Figure 1 represents viscous flow and plastic flow. Curve I dealing with the flow of sludges in pipes from which formulas represents the flow of a true liquid; the slope of the line is applicable to all types of sludge flow problems might be deproportional to the coefficient of viscosity. Curve I1 repreduced. Bingham (4)developed an equation for plastic flow in sents the flow of a true plastic and is a graphical representacapillary tubes. Although the flow of sludge in pipes larger tion of Equation 2. The apparent viscosity of the plastic a t than capillary tubes has been found t o follow Bingham’s equaany point A on curve 11, if measured in the usual way for tion over certain ranges, in turbulent flow the Bingham liquids, is proportional to the slope of line OA. It is evident, equation no longer applies. therefore, that the apparent viscosity is not constant for The purpose of this article is to present equations from different velocities and stresses. Two different velocities which pipe friction losses may be computed for the flow of all such as A and B in Figure 1 correspond to entirely different types of sludges commonly encountered in practice. The viscosity lines OA and OB, the slopes of which are proporequation for plastic flow is not original with us, but the equational to the apparent viscosity. It has been found in this intions for critical velocity and for turbulent flow are presented vestigation that the flow of sludge follows the type of flow for the first time, so far as we know. All equations have been illustrated by curve 11. We conclude, therefore, that sewage verified by tests performed by us or by tests reported in the sludge and clay slurries are true plastics. literature. I n an attempt to formulate the factors affecting the friction Plastic Flow resulting from the steady uniform flow of sludge in a circular pipe, certain assumptions will be made and the resulting forTHEORY. For a clear understanding of the characteristics mulations checked by tests. Most of the following assumpof plastic flow it is necessary to distinguish between viscosity tions were made by Bingham (4) who was the first to develop and plasticity. Viscosity is the measure of the resistance to
C
249
INDUSTRIAL AND ENGINEERING CHEMISTRY
250
the plastic flow equation. It will be assumed that the conditions affecting the friction resulting from the plastic flow of sludge in a circular pipe are the velocity of flow, 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 the temperature 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. However, it is known that in the laminar flow of fluids pipe-wall roughness does not affect the friction loss. We have assumed, theref or e, that pipe-wall FIGURE1. REPRESENTATION OF VISroughness will not COUS FLOW AND PLASTIC FLOW affect the friction loss in the plastic flow of sludges. The friction loss will be assumed t o result only from the rubbing of the sludge layers against one another and not from kinetic energy losses. The following mathematical analysis was first presented by Bingham ( 4 ) . The total force producing flow in a pipe between sections l and 2 (Figure 2) is TR*(AP)where AP is the difference in pressure between sections 1 and 2, and R is the radius of the pipe. Since there is no acceleration in steady uniform flow, this force is opposed by an equal force 2 nRLr,, hence rp = R(AP)/2L
Substituting in Equation 6 the value of
T, from Equation
4,
(7)
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 = T :
The velocity of the solid plug is obtained by making r = TO and substituting in Equation 8 TO = ~ , 2 L / f l pfrom Equation 5: (9)
The total rate of flow, &, is made up of two quantities. The first is the rate of flow due to the plug, and the second is the rate of flow due t o the material between the plug and the pipe wall. Total flow Q is, therefore:
Q
=
+ 2 r LRrv,dr
nro2vo
where v, is the velocity a t any radius in the pipe between ra and R.
(3)
where T~ is shear a t the pipe wall and L is the length between the sections. Then T , = r ( AP)/2L (4) 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 a t the center of the pipe t o a maximum a t the pipe wall, as shown by Equation 4 and Figure 3; and a t some distance between the center of the pipe and the wall the shear will be equal to the yield value, T ~ of, the sludge. Therefore
T~
Vol. 33, No. 2
= TO( A P ) / 2 L
+i--L&.
per Sq Fi
-
FIGURE3. DISTRIBUTION OF SHEARING FORCES IN CIRCULAR PIPE
A
The plug rate of flow may be evaluated by substituting in the first term of Equation 10 the value of vo from Equation 9 and the value of TOfrom Equation 5 :
(5)
The rate of flow of the material between the plug of radius
ro and the pipe wall of radius R may be evaluated as follows:
FIGURE 2. DI.4GRAM
OF
FLOWBETWEEN PIPE
TWO
SECTIONS
OF
Substituting the value of ro from Equation 5 :
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. For a circular pipe, Equation 2 becomes The total flow Q is therefore the sum of Equations 11 and 12 and is:
INDUSTRIAL AND ENGINEERING CHEMISTRY
February, 1941
251
0.8 b
\
Substituting the value of T~ from Equation 3:
Q
TRS
= -
41
Q a4 F
4
- 3ry+
D
TP
02
The mean velocity of flow is: ' 0
P
4
6
B
/O
/Z
/4
/6
/8
FIGURE4. GRAPHICAL REPRESENTATION OF EQUATIONS 15 AND 17 I n Equation 14 the unit of the coefficient of rigidity @ is 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:
Equation 15 may be written:
It can be shown that the last term may be omitted with little error when the ratio of the yield value, in pounds per square foot, to the shear at the pipe wall, also in ,pounds per square foot, is less than 0.5 (i. e., when ru/rp< 0.5). The error will be 5.9 per cent when r,/rP = 0.5 and 1.8 per cent when ry/rl, = 0.4. Omitting the last term, Equation 16 reduces to:
v
=
g (%- 2 4
Bingham (4) showed that coefficient of rigidity q and yield value rg are independent of the characteristics of measuring apparatus and depend only on the nature of the sludge. Both of these facts have been corroborated in this investigation. (A discussion of the various factors affecting the yield value and the rigidity has been published, 3.) If a graph is plotted with the shear a t pipe wall rPas ordinate against 8 times the velocity divided by g times the diameter, 8V/gD, as abscissa, the slope of the resulting line will represent the coefficient of rigidity 9; and the intercept of the line on the ru axis will be 4/3 the yield value. The one line, therefore, represents the flow of a given sludge in a pipe of any diameter. A graph of this type, illustrating Equations 15 and 17, is shown in Figure 4. The error in neglecting the last term of Equation 15 is thus shown graphically. For industrial piping with sewage sludges, clay slurries, and drilling muds as the flowing material, Equation 17 will yield accurate results within the limits of experimental error in determining the yield value and the rigidity of the sludge: since
then
HprRZ = 2rPuRL ru = HpR/2L
and Equation 17 can be written in another form which may be more convenient in certain cases:
Equation 18 was checked by experiments in this investigation and by the use of tests reported in the literature. Table I shows a few comparisons of the observed and the computed values of head loss, together with the percentage variation, using Equation 18. It is evident that the agreement between observed and computed values of friction head loss is sufficiently precise for practical purposes. EXPERIMENTAL. In order to verify Equation 17 the apparatus shown in Figure 5 was constructed to pump sludge through four sizes of pipes. Measurements of pressure loss, velocity of flow, density, and temperature of the sludge, and of solids concentration were made. The pressure loss and velocity measurements were converted to terms of shear a t the pipe wall and 8V/gD, respectively, and plotted on graphs similar to Figure 4, for the purpose of studying the effect of the diameter of the pipe on the yield value and rigidity. Figure 6 shows the results obtained for some of the sludges tested.
TABLEI. COMPARISON OF OBSERVED AND COMPUTED VALUESOFHEAD Loss IN PIPESOF VARIOUS SIZES Obsvd. Computed VariaHead Loa8 Head Loss tion, Ft./100 Si. Ft./100 Ft: % 1.32 1.34 1.5 1.42 1.43 0.7 1.52 1.53 0.7 1.63 2.0 0.6 1.62 1.72 1.73 2.5 0.6 1.92 1.92 3.5 0.0 .0.5 2.18 2.17 4.5) Sewage sludge: 90 12 0.5 0.144 0.150 4.2 Calumet treat0.7 0.152 0.161 5.9 men: 0.9 0.162 0.176 8.6 Tu 1.0 0.169 0.178 5.3 = 0.035 1.5 0.200 0.214 7.0 2.0 0.242 0.234 -3.3 2.6) 0.300 0.268 -10.6 Sewage sludge. 90 8 0.5 0.260 0.265 1.9 Calumet treat: 0.7 0.275 0.287 4.4 ment plant: 1.0 0.300 0.315 5.0 0.361 -3.0 Q 0.017, q 1.5 0.372 = 0.026 2.06 0.480 0.408 -15.0 Clay slurry from 86.4 4 0.5 0.30 0.32 6.7 water urifioa0.7 0.31 0.34 9.7 tion pPant; r y 0.9 0.33 0.35 6.1 = 0.011, q = 1.5 0.37 0.40 8.0 0.005 1.86 0.41 0.41 0.0 Sewage sludge, 90 7.9 1.0 1.50 1.57 4.7 digested.; sew1.5 1.62 1.72 6.2 age disposal 2.0 1.82 1.86 2.2 plant Stutt2.5 2.00 2.00 0.0 gart, hermany; 3.0 2.20 2.14 -2.7 4.0 2.47 2.33 -5.7 ;yotb;70.'o' v -1.5 5.Ob 2.75 2.71 Illinois yellow 52 26.3 0.8 3 2.0 26.1 27.1 -0.4 27.2 clay suspendon. 3.0 0.4 tests made here: 28.0 27.9 4.0 -0.3 28.8 28.9 5.0 20;80.72. 7 29.7 0.0 29.7 6.0 0.0 30.5 30.5 7.0b a The valuea of ry.and rl were determined by plotting apprnpriate data similarly to Figure 4 and readin intercepts and slopes. b Observed criticaf velocity.
Type of Nudge and Source" Sewage sludge; Calumet treatment plant, Chicago. r y 0.060, q 2 0.021
-
g!;~;;
-
-
Moisture, Diameter of Pipe, Velocity In. Ft./Sec: 88 5 0.5 1 .o 1.5
ge$gt
INDUSTRIAL AND E N G I N E E R I N G C H E M I S T R Y
252
Vol. 33, No. 2
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 a t a minimum, the cup illustrated in Figure 8 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.
L -)
FIGURE5. DIAGRAM OF SLUDGE FLOW APPARATUS
It is evident from Figure 6 that the flow of sewage sludge and of clay suspensions follows closely the theoretical equation for the flow of a true plastic in circular pipe. Figure 6 also shows that both the yield value and the rigidity of the sludge are independent of the pipe diameter. This fact corroborates the assumption made a t the beginning of the development of the plastic flow equation. Yield value T~ and coefficient of rigidity 9 may be measured in a variety of ways. Any existing pipe line through which the sludge can be pumped may be used in their measurement. The observations 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 LI diagram with the shear at the pipe wall, T ~as , ordinate, and 8 times the velocity divided by g times the diameter, 8V/gD, as abscissa. The slope of the line formed by connecting the points is the coefficient of rigidity, and the intercept of the line on the 7, axis is 4/3 the yield value, as Equation 17 shows. The values of T~ are obtained from the head losses, expressed in feet of flowing material, by means of the following equation: rp = HpD/4L
(19)
where H is the difference in static head of the flowing substance between two points in a pipe, a distance of L feet apart, and p is the density of flowing substance. The Stormer viscometer (Figure 7) can be adapted to the measurement of yield value T~ and coefficient of rigidity 9 of a sludge by slight modifications. Sewage sludges in particular
S v 9D FIGURE
6.
&Ve/ocify Dlomefer
gx
EXPEHIMEXTAL VERIEIC.4TIQN O F
When using the modified Stormer viscometer for the measurement of yield value T~ and rigidity 7,the procedure is as follows : The material to be measured is poured into the cup to exactly 0.25 inch from the top, and the rotating cylinder is inserted. A known weight t o act as a driving force is attached to the string wound on the drum; the brake is released, and the time for the cylinder t o make a hundred 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 t o thixotropy. Thixotropy is the property of some gels of becoming fluid when shaken or agitated. The phenomenon is also reversible.
Plots of observations made with a modified Stormer viscometer for four clav sludges are shown in Figure 9. The drivrng force, W , is plotted against the speed of revolution, N . Figure 10 is a plot of the pipe flow characteristics of the same clay sludges with the shear a t the pipe wall, T ~ plotted , against 8 times the velocity of flow divided by g times the diame&ark Steel ter, 8V/gO. The similarity of the graphs is apparent and is the basis for converting the Stormer data to the pipe flow constants T~ and 7. I n Figure 9 the intercept W , on the W axis of a line connecting the points representing driving force W and speed of revolution N is proportional to yield value T#; and the slope, A, of the same line, is proportional to the coefficient of rigidity 7. Table 11 PLASTIC FLOW EQUATION
INDUSTRIAL A
February, 1941
ENGINEERING CHEMISTRY
253
presents a comparison of values of W , and r y and of X and 7, and demonstrates that the modified Stormer viscometer may be used in measuring yield value 7, and rigidity q of a sludge.
the plastic flow equation and above which the friction loss is directly proportional to some power of the velocity between 1.7 and 2.0. The Reynolds number, D V p / p , may be used as a criterion (8) for determining the critical velocity when a fluid flows in a pipe. For circular pipes the flow of a fluid will be laminar when the Reynolds TABLE 11. RELATION OF rY TO W , AND OF q TO X Coefficient Ratio number is less than 2000 (9). Sometimes laminar Ratio Test Yield NO. Substance Sp.Gr. SoBdds Value, rv W V T ~ W Vof Rigidity,? X n/h flow persists to higher Reynolds numbers, but I c l a y SUB1.49 52.0 0.72 355 0.0020 0.030 8.0 0.0037 in industrial piping installations the flow will pension 0 036 9.5 o 0038 usually be turbulent above a Reynolds number of 1.49 62 3 0 74 370 0 0020 11" Same IIIa Same 442; 2 $.o:E 139 66 go g$:z g.0.013 !?: 47:4.0: 0.0033 ~:~~~~ number 3000 (10). Between these values of the Reynolds IVQ Same VO Same 136 0021 there is uncertainty as to the type of VI0 Cylinder 0.88 0 0 00 0 0 0020 0.074 21.2 0.0035 flow which may occur. The velocity correspondoil Average 0.0020 0.0035 ing to a Reynolds number of 2000 is usually desiga See Figures 9 and 10. nated as the lower critical velocity while the velocity correwonding t o a Reynolds number of 3000 may be designated as the upper critical veAny Stormer viSCOn~tercan be adapted to the measurelocity. The value of the Reynolds number a t the critical ment of T~ and 1 without calibration if the ~-~odified CUP develocity for a given apparatus is usually constant, the value of scribed in the foregoing is employed. The following equations D v p / p = 2500 sometimes being used. can be used for converting Stormer data to pipe flow conSince the Reynolds number involves the viscosity, 1.1, of stants: the flowing material, it is necessary in evaluating the criterion r y = 0.0020u7, (20) to know the apparent viscosity of the sludge a t the particular q = 0.0035X (21) velocity for which the Reynolds number is desired. HOWAt low rates of shear in the modified Stormer viscometer ever, the apparent viscosity of a sludge is a variable, depending upon the velocity of flow as PreViOUSlY shown. From corresponding to small speeds of revolution, the relation beFigure 4 it is evident that the apparent viscosity of any partween the driving force and the speed of revolution is not ticular value of 8Tr/gD is the ratio of the ordinate to the ablinear; the result is that the line connecting the points repreSciBsa, or senting driving force and the corresponding speed of revolution bend toward the origin. The explanation is that at low IJ. = gr,D/SV (22) rates of shear the material shears first a t the point of greatest which is an expression of Poiseuille's equation. stress, which is in the layers of mater-? S/udge rial 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 Nofee.'surfuces in confort with s/udge must nof be h/ah/ypo//&ed. sides. A t t h i s FIGURE 8. MODIFIEDSTORMER CUP point the 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 in the area bounded by lines AB and CD in Figure 9, because only between these lines is the relation between the increase in driving force and the increase in speed of revolution linear. Viscometers suitable for measuring yield value rv and rigidity 7 of a sludge include (a) rotating cylinder viscometers, of which the Stormer and Kampf are examples, (b) falling ball viscometers, (c) swinging pendulum viscometers, and (d) capillary tube viscometers. Of these four only the last will measure the true values of constants rV and 7. The underlying principles of some of these viscometers have been given in detail (1).
;::
i:
.
Critical Velocity The critical velocity for sludge flow will be conTHEORY. sidered as that velocity below which the friction loss follows
Rufe o f Shear, N, Revo/u t tons p e r Second
FIGURE 9.
FLOWOF CLAYSLUDGE IN STORMER VISCOMETER
INDUSTRIAL AND ENGINEERING CHEMISTRY
254
Vol. 33, No. 2
Equation 24 is an expression for the apparent viscosity of a sludge flowing a t any velocity in any pipe size. Substituting Equation 24 in the Reynolds criterion for the lower critical velocity and solving for V,,,
vzc
=
DP
and for the upper critical velocity
The range between the lower critical velocity and the upper critical velocity represents a region of uncertainty as to whether plastic flow or turbulent flow will exist. Apparently the exact value of the critical velocity is controlled by the roughness of the pipe. Recently an equation ( S A ) for determining the exact value of the critical velocity was developed and checked experimentally :
8V &Ve/ocitu ' g x Diameter
9D
v,= 87 + 8
IN PIPES FIGURE10. FLOWOF CLAYSLUDGE
+
fr,pD2p 24
fPD
where f = friction factor in Fanning's formula, H = 2fLV2/gD (solution of this equation must be by trial and error).
I n Equation 22 rp is a variable, and since an expression for V,, the critical velocity, is desired which involves only the known quantities pipe diameter D, yield value T ~rigidity , 7, and density p, rpmust be expressed in terms of these quantities. From Equation 17 it is noted that
which involves only the desired quantities. value of rpin Equation 22:
+
The comparison between observed and computed values of the critical velocity shows close agreement when the roughness of the pipe is accurately known. EXPERIMENTAL. To check the validity of Equations 25 and 26, tests were made with wrious types of sludges in the apparatus already described, to observe the critical velocities together with yield value T~ and rigidity 7. The results of these tests are given in Table 111, together with observed and computed values of critical velocity taken from various tests reported in the literature. These results show a high degree of correlation between observed and computed values.
Substituting this
(24)
Reynolds N u m b e r , D Y p / j , i n Thousrmds
FIGURE11. FRICTION-FACTOR CHART(7) Relative Roughness of Pipes Tubinga A B
C
D
E
F a b
-
1
2
0 . 3 5 up 72
48166
...
... ... ...
... ... ... ...
3
iili2 30 48-96 96 220
4
5
6
Alii
4-5
i-8
10-24 6-8 20-48 12-16 42-96 24-36 84-204 48-72
3-5 5-10 10-20 20-42
Curve Kumber7 S 9 Diameterb, Inches iijl jjl 1-11/&
.......
21/n
3-4
6-8
16-18
1'/2-2 2-21/2
4-5 10-14
l'/a 1,'s 3 8
10
11
12
13
. ij) . . . . '/a. . . . 318 314
'/8
14
15
16
17
18
. . . . . . 0.0625 . . . . . . . . . . . . .
0,125
.. . . '/a . . 1 . . . . . . . . . . . . . . . . . . ........................
1
11%
3/s
1/8
11/4
6
4
3
D = best cast iron, cement, light riveted sheet ducts. A = drawn tubing, brass, tin, lead, glass. E clean steel, wrought iron. E = average cast iron, rough-formed concrete. F 3 first class brick, heavy riveted steel. C olean galvanized iron. I n drawn tube, aotual diameter is given; in pipe, nominal size of standard weight is given.
. . . . . . . . . . . . . . .
I N.DU S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y
February, 1941
the material but in no way affects the viscosity of the dispersion medium. It is this viscosity which causes a relatively thin layer of material to move in laminar flow along the walls of the pipe while the material a t the center of the pipe is flowing turbulently. The thin layer a t the pipe is called the “boundary layer”. It is due to the boundary layer that the exponent of the velocity term in the common hydraulic formulas is not 2, as would be expected if the flow were wholly turbulent. It has been found that for very rough pipes or in smooth pipes a t very high velocities the exponent does become 2; this indicates that the flow is wholly turbulent and that frictional losses are due only to kinetic energy impact losses, the viscosity of the material having no effect on the losses. This fact is shown graphically on the friction factor chart (Figure 11). At high Reynolds numbers the lines become nearly horizontal, which indicates friction losses varying as the square of the velocities. Figure 11 represents the latest available data for flow in various types of pipes. It can be used for solving sludge flow problems with the restrictions noted above. EXPERIMENTAL. The tests, from the results of which the above conclusions were drawn, are summarized in Figure 12. About nine hundred tests on eight different sludges were made. The range of the characteristics of the sludges tested is shown in Table IV. Velocities of flow as high as 40 feet per second were used to check the relations a t velocities higher than those ordinarily encountered in practice. Pipes tested, all new black steel taken directly from stock, ranged from to 3 inches in diameter. It is apparent from Figure 12 that as long as the flow of a sludge is turbulent, as determined by the critical velocity, all experimental points lie on the same line as given for new black steel pipe.
OF COMPUTED AND OBSERVED VALUES TABLE 111. COMPARISON OF CRITICAL VELOCITIES Critical Velocity
Diam. of Pipe, In.
Yield Value,
8
0.057 0.043 0.033 0.024 0.017 0.013 0.006 0.003 0.0015 0.020 0.014
12
TU
0.009
3 2 1 1
0.005 0.0026 0.0015 0,115 0.091 0,075 0.065
4
Coefficient of Rigidity, Computed n vio vue Obsvd. Imhoff-Tank Sewage Sludge 0.044 4.2 6.7 4.0 0.040 3.8 5.1 3.7 3.3 4.5 3.6 0.036 0.034 2.9 3.8 3.0 0.026 2.4 3.2 2.7 1.9 2.6 2.2 0.018 0.016 1.1 1.6 2.0 1.6 1.3 1.8 0.016 0.014 0.9 1.3 1.0 0.0514 2.8 3.8 3.0 0.035 2.1 3.0 2.6 0.032 1.8 2.5 2.3 0.036 1.6 2.2 2.1 0.025 1.1 1.6 1.7 0.024 1.0 1.4 1.4 0.0163 5.4 7.3 6.5 7.0 6.6 7.7 0.0165 7.0 7.6 11.1 0,0165 7.0 7.4 10.8 0.0165
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Clay Slurry from Sedimentation-Tank Sludge 0.020 0.005 2.0 2.7 2.2 0.056 0.005 3.3 4.2 3.6 0.117 0.005 4.6 5.8 6.0 0.158 0.005 5.3 6.8 7.0 0.214 0.005 6.1 7.6 8.5 0.012 0.005 1.5 2.1 1.8
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Turbulent Flow THEORY. Results of tests made a t the University of
Illinois show that the familiar friction factor-Reynolds number chart (such as Figure 11) may be used, with certain limitations, for the turbulent flow of sludge. Figure 11 is taken from an article by Pigott (7). If the viscosity of the dispersion medium of the sludge is used in determining Reynolds OF SLUDGES TESTED number, the friction factor corresponding to the TABLE IV. CHARACTERISTICS Reynolds number is the same as if a true fluid Sludge Lb./ p , Lb./ % Moisture % ’ Solids such as water were the flowing BeNo. %.”,”;/ $o.-Ft. Sp. Gr. Cu. Ft. by Wt. b y Wt. Type of Sludge 1
cause of this fact yield value T~ and rigidity 2 3 7 of the sludge have no effect on the fric4 tion loss in turbulent flow. It is to be emphasized 5 that thisstatement is true onlywhenthevelocity of flow is greater than critical-i. e., when the 7 flow is turbulent. I n nonturbulent or plastic flow the conditions of Equation 18 hold. The use of the viscosity of the dispersion medium in the Reynolds number in place of the viscosity of a true liquid is reasonable when it is recalled that in turbulent flow the friction loss is due essentially to impact kinetic energy loss which depends on the density of the material and velocity of flow. The presence of suspended particles increases the density of
0.90 0.60 0.44 0.29 0.065
0.015 0.015 0.011 0.010 0.0165
o.lg 0,082
o.oo8 0.006
ff
’
72,0 1.12
70.0
79.7 83.9
20.3 16.1
Clay suspension Same Same Same Sewage sludge (Imho&-tank) Clay suspension Same
ft. of flowing substance
&2m3
4
4Q0
Thousands
OF TURBULENT FLOW TESTS FIGURE 12. RESULTS
ft./sec./sec.
= length Of pipe, ft*
‘K-
is the viscosity of the dispersion medium.
29.8 26.5 26.0 23.3 14
= difference in static head tetween two points in a pipe,
8s a m
For sludge,
70.2 73.5 75.0 76.7 86
Nomenclature
,&Q@8
zoo
75.6 75.0 73.8 72.6 66.3
= diameter of pipe, ft. = acceleration due t o gravit
$ am
@.c, QOa220 40 60 Bo 100 4 Regnolds Number, DVph, ri-/
1.21 1.20 1.18 1.16 1.06
m
N
=
speed of revolution of cylinder in modified Stormer viscometer, revolutions/ sec. Q = rate of flow, cu. ft./sec. r = distance from any point within a pipe to the center of the pipe, ft. P = pressure, lb./sq. ft. R = radius of pipe, ft. Re = Reynolds number D V ~ / Mdimensionless , TO = radius within a pi e at which the shearing stress equai the yield value, ft. (Fig. 3) V = meaq velocity of flow in pipe, ft./sec. u = velocity of one plane with respect t o the otxer, ft./sec. yo = velocity of plug of radius TO, ft./sec. u, = velocity at any distance r, from the center of a pipe, ft./sec. Vi, = lower critical velocity, ft./sec. (Re =
Vu,
= =
x
W
2000)
=
upper criticalvelocity,ft./sec. (Re = 3000) distance between planes, ft. driving force in Stormer viscometer, grams
256
INDUSTRIAL AND ENGINEERING CHEMISTRY
intercept on W axis of line connecting points representing driving force W and corresponding speed of revolution N in Stormer viscometer (Fig. 9) = coefficient of rigidity, slugs/sec.-ft. = coefficient of rigidity, lb./sec.-ft. = slo e of line connecting points representing driving force and corresponding speed of revolution N in Stormer viscometer (Fig. 9) = coefficient of viscosity, slugs/sec.-ft. = coefficient of viscosity, lb./sec.-it. = density of flowing substance, lb./cu. ft. = tangential unit shearing stress, lb./sq. ft. = shearing stress in a flowing material at the boundary or pipe wall, lb./sq. ft. = shearing stress in a circular pipe a t distance r from the center, lb./sq. ft. = shearing stress at yield point of plastic material, called “yield value”, lb./sq. f t .
=
6
Literature Cited ‘4cad. of Sciences of Amsterdam, Rept. on Viscosity and Plasticity, 1st Rept 1935. 2nd Rept. 1938.
Vol. 33, No. 2
(2) Am. Soo. Civil Engrs., Transactions, 55, 1773 (1929). (3) Babbitt, H . E., and Caldwell. D. H.. Univ. Illinois Ene. Expt. Sta., Bull. 319 (1939). (3A ) I b i d . , 3 2 3 (1940). (4) Bingham, E . C., “Fluidity and Plasticity”, New York, McGrawHill Book Co , 1922. Gregory, W. B., Mech. Eng., 49, GO9 (1927).
I
Merkel, Wilhelm, “Die Fliesseigensohaften von Abwassersohlamm und anderen Dickstoffen”, Beihefte 1 4 rum Gesundheits-Ine.. Reihe 11. Munich. R. Oldenboure. 1934. Pigott, R.7. S., Mech. Eng., 55, 497 (1933). Reynolds, Osborn, Trans. Roy. SOC.(London), 174, pt. 3 . 935
-
(1883).
Rouse, Hunter, “Fluid Mechanics for Hydraulic Engineers”. New York, McGraw-Hill Book Co., 1937. Walker, W. H., Fewis, W. K., McAdams, W. H., and Gilliland, E. R., Principles of Chemical Engineering”, New York, McGraw-Hill Book Co., 1937. Wilhelm, R. H., Wroughton, D. M . , and Loeffel, W. L . , IND.
ENCI. CHESS., 3 1 , 622 (1939). PRESENTED before the meeting of t h e hrnericnn Institute
of ChPmica
Engineers, N e w Orleans, 1 , ~ .
CHINESE ALCHEMIST By Y . Y . Ts’ao
No.
122 in the Berolzheimer series of Alchemical and Historical Reproductions takes us back to a time before European alchemy. Chinese alchemy is an expression of Taoism, founded and expounded by Lao Tzu, who was born about the Third Century, B.C. Its main purpose was not “transmutation’’, but to make the “pill of immortality”, which was later called the “elixir of life”. When or why the change from a solid (pill) to a liquid (elixir) came about is not known. Possibly the “Philosopher’s Stone” is the direct descendant of the “Pill”, one of its uses being to impart to the “Elixir” its life-prolonging power. On the other hand, gold because of its stability, should be “good medicine”, and was so considered. Hence the Chinese alchemists attempted to make it artificially. Note the written “charms” on the wall of the cave, which was the alchemist’s workshop, to ward off evil spirits. The purpose of the sword sticking in the furnace is not known. The original was drawn in color in 1932 b y Y . Y. Ts’ao and is in the possession of the Science Society of China in Shanghai.
D. D. BEROLZHEIMER 50 East 41st Street
New York, N. Y.
The lists of reproductions appear as follows: 1 to 96 January 1939 issue page 125. 97 to 108 January 1940, p&e 134; 104 to 120, January. 1941,’page 114: An additional reproductlon appears each month.
