March 1952
INDUSTRIAL AND ENGINEERING CHEMISTRY
(31) Sillen, L. G., and Ekedahl, E., Arkiv Kemi Mineral. Geol., 22A, No. 16 (1946); 25A, No. 4 (1947). (32) Spedding, F. H., Fulmer, E. I., Butler, T. A., and Powell, J. E., J . Am. Chem. SOC.,72,2349, 2354 (1950). (33) Stene, S., Arkiv Kemi Mineral. Geol., 18, No. 18 (1945). (34) Thomas, H. C., J . Am. Chem. soc., 6 6 , 1 6 6 4 (1944); Annals N.Y. Acad. Sei., 49, 161 (1948). ’ (35) Tompkins, E. R., J. Am. Chem. Soc., 70, 3520 (1948). (36) Tompkins, E. R., private communication. (37) Tompkins, E. R., Harris, D. H., and Khym, J. X., J . Am. Chem. SOC., 71,2304 (1949).
65 1
(38) Trueblood, K. N., and Malmberg, E. W., J. Am. Chem. SOC.,72, 4112 (1950). (39) Walter, J. E., J . Chem. Phys., 13, 229, 332 (1945). (40) Wilke, C. R., and Hougen, 0. A., Trans. Am. Inst. Chem. Engrs., 41, 445 (1945). (41) Wilson, J. N., J . Am. Chem. Soc., 62, 1583 (1940). (42) Woods, F. S., “Advanced Calculus,” Boston, Ginn and Co., 1934. RECEIVED for review May 8, 1950. ACCEPTED November 19, 1951. Presented in part before the Division of Physioal and Inorganic Chemistry CHEMICAL SOCIETY, Atlantic City, N. J. at the 113th Meeting of the AMERICAN
EngFnyri ng
Flow of Plastics Materials in Pipes
pocess development I
BENGT 0.A. HEDSTROM
.
D E P A R T M E N T OF I N O R G A N I C CHEMISTRY, R O Y A L INSTITUTE OF T E C H N O L O G Y , S T O C K H O L M , S W E D E N
HEMICAL, mechanical, and civil engineers are often faced with the problem of predicting the pressure drop for nonNewtonian materials flowing through pipes. Hitherto, the common practice has been to determine from experiments the volume rate of flow as a function of pipe diameter and applied pressure drop, without any accompaning critical analysis of the data obtained in terms of the flow-curve-Le., the curve combining the shear stress and the rate of shear when the material is flowing laminar1y. I n the author’s opinion, not one of the very few investigations where this combined study has been undertaken will stand a critical analysis, in most cases because in the viscometers used the velocity gradient was kept far from constant or was for other reasons ill-defined. At the present time, the rotational viscometer of the Couette type (two concentric cylinders), especially that designed by Green and
C
Pa
=
7
dv dr
is not a constant, but a function of r. In most cases, Newtonians are one-phase systems and therefore often well defined and easily reproducible. Non-Newtonians, on the other hand, are generally made up of two or more phases, and their flow behavior is therefore influenced by a number of factors-e.g., particle shape, size, weight, distribution, and surface properties of the different phases. The big problem in theoretical and applied rheology is t o interpret the flow curve in terms of such factors. The engineering problem, however, should be t o predict quantitatively the characteristics of flow in engineering equipment from a knowledge of the flow curve.
most cases the Weltmann ( 4 )only , is inin strument which can be recommended (7‘). The existing situaf - 9 y a :t g w . tion is an unfortunate PP”W = a function o f C f4 a constant one, because the flow p,, tgu = a constant ‘ W curve is of equal imc r portance to both NewFigure 2. Flow Curve of a tonians and non-NewFigure 1. Flow Curve of General Non-Newtonian MaFigure 3. Flow Curve of an tonians. One reason a Newtonian Fluid terial Ideal Plastic Material a great deal of both laminar and turbulent Given the flow curve, it is always possible, a t least in principle, Newtonian flow is known is that the flow curve in this case is the to determine the characteristics of laminar flow in a certain piece simplest possible-a straight line passing through the origin, as of equipment, for instance a pipe, by means of integration shown in Figure 1. This implies t h a t only one material constant methods. At the present time, two ideal materials have been -viscosity, which is defined by defined, the ideal Newtonian and the ideal plastic. For both, the 7 respective flow curve has been integrated for the case of isother(I ) PN = dv .mal, laminar flow in long, cylindrical pipes. Integration of dr Equation 1 leads t o Poiseuille’s formula is needed to characterize laminar flow. In the case of a general O2 32 (3) non-Newtonian on the other hand, the flow curve, as shown in 1 PNV Figure 2, starts at some point on the 7 axis, corresponding to the valid for Newtonians. Plastics are defined by Bingham’s equayield stress, and is not a straight line. Consequently, the apparent viscosity, defined by tion i
T kflj/-T -.-=
652
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY PP =
-.f __ 7
(4)
d-r dr
whew ,UP and f are constants, giving the flow curve of Figure 3. Integration leads to Buckinghani’s equation (3) ~0 2 * ( P - 4 . p
; g)
+ .
= 32
VPPl
If B certain non-Kewtonian material under consideration cannot hc approximated as a Ringham body, one must determine empirically the flow-curve equation and integrate it. This problem has been discussed by Reiner (6).
Vol. 44, No. 3
flow, end effects, and kinetic energy effects are assumed to be absent. Furthermore, the discussion will be restricted to stationary flow, thereby eliminating any thixotropic effects. In practice, this is reasonable, as long as one deals with flow in long and not very thin conduits. By introducing a few dimensionless quantities, McMillen (6) has considerably simplified the pressure drop calculation for laniinar flow: The pressure gradient is obtained from the simple equation 4.f
p
- 3 -
Dc
1
c, the ratio of plug diameter to pipe diameter, is in turn obtained graphically from a =
- 4c
c4
$-
a
12c
where D(
V P P= 2 _
I?r Equations 6, 7 , and 8 can easily he derived from Buckingham’s Equation 5 , which can be written
-.:>
! ? . E(1. - 4- c + 1 1
3
VPP
3
= 32
where
.A
3
.I
r. ov II’ r 0 3
Figure 4.
1
f0
rd
70’
(00
io00
has the same meaning as previously. Equation 6il is identical to Equation 6. NOR, Equation 5R can be written
-1. - Vos .4c. kP
Laminar Flow of Plastics in Pipes
The differentiation method, Le., that of deriving the flow curve from pipe flow data in the laminar region, is, in general, an extremely complicated one, as has been stressed by Reiner ( 6 ) , Green ( 4 ) ,and others. There is, however, one exception-that of ideal plastics. But since real materials always differ from ideal ones to a certain extent, the flow curve should always be made the basis of an interpretation of floiv data, in this author’s opinion. If this method is accepted, one should be able to predict flow characteristics, in the future, on the basis of laboratory viscometer data only, as is the case for Newtonians in a number of different kinds of conduits. The whole problem is complicated by the fact that the possibility of turbulent flow must be considered. At the present time extremely little is known of turbulent flow in terms of the flow curve. Therefore, a big field is open for investigations, which should he of greatest importance to the development of this branch of fluid mechanics. In this paper, flow of plastics in pipes is discussed. On the basis of the limited data reported in the literature, a simple criterion, which distinguishes between laminar and turbulent flow, i s proposed. Furthermore, it is shown that in the case of turbulent flow, the relation between the Fanning friction factor and the Reynolds number, valid for Newtonians, is valid also for plastics, at least as a good approximation, if the Reynolds number is defined as Re = DV p / p P . It is suggested that the theory developed in the paper will form the basis of future investigations on the turbulent flow of materials, which, from a knowledge of the flow curve, can be approximated as ideal plastics. LAMINAR FLOW O F PL4STICS
In the following discussion the same basic assumptions are made as were made for Equation 5 . This implies that slippage
c4
-
4c
+ 3 -- 32
(X)
12c
By combining Equations 6A and 5C there is obtained
in accordance m-ith Mc1:illen’s equations (Equations 7 and 8). The variables involved should also be seen from a dimensionalanalytical viewpoint. From Equation 5 it appears that the following quantities will enter the general function looked for : P, 1, D , PP, f, and V . Following the usual procedure, see for instance Bridgman ( 2 ) , the equation
PD =
PPV
$
1
fu -)
PP V
is arrived a t , or, since P will be proportional to 1
Thus
P
.
-
P
D2
1
PPV
1
D2 ~
is
a
unique
function
of
S,
i.e.,
if
PPV
,
is plotted against S, the points will fall on a single
curve. From Equations 8 and 9 it follows that
and a(S)
4S = -8 c ac
Equations 7, 10, and 11 make it possible to determine numerically -P . - Dz as a function of S. The solution is given graphically 1 PPV in Figure 4. In constructing the curve, use was made of the
INDUSTRIAL AND ENGINEERING CHEMISTRY
March 1952
w tx
e
2.20 C
x
C
0,980
o(
C
o c c
C
o(
6.0
-$-0.100
653
0,910 5.5 0.011
6
0.110 1.90
5.0
Y
c
0.012 20.0
0,920
1.a o
0,120 1.70
2500 0!
4.5
4
0,014
a984
0,015 0.016 0.060
150
0,017
0.140
0*0°’
i
a002
0.018 3.5
0,987
0,150
a004
0.988
3.0
0.160
0,989 0,990
a003
0,020
0,940
0,080
1.20
0,950
0.010
0,170 1.10 0,180
0,960
0.035
0.190
3%.
0,995
0,040
0,200
Figure 5. Nomogram (Continued) c4 - 4c + 3 a =
s
c o(
C 0,310 9,O.ib’
0,320 8.5.16‘
0.330 8,0.1tJ-’
0,340
7,5.10-‘
C 0,200
0,210
0,220
0,230
7,O.1Om1
0,250 6,5.10-l 0,260 0.380
With a rotational viscometer the yield stress and EXAMPLE. the plastic viscosity of a cement rock suspension were determined at 100 dynes per cm2. and 0.2 poise, respectively. The density, p , was 2 grams per Compute the pressure drop due to friction for this plastic flowing with a n average velocity of 1 foot per second in a long, straight smooth pipe with an inner diameter of 1 inch.
SOLUTION.To ensure that the flow is laminar, Reynolds number is first calculated
0,24 0
0,360 0,370
double scale in Figure 5. In order to demonstrate the use of the curve, a pressure drop calculation is carried out in the following example.
DVp
0,350
6.0.16’
- 2.54
P
-* 1
0,290 0,410 5.0.16’
0,300
0.310
u =
12c
D 2 - 230 WPV -
Thus
- 230 -
0,420
Nomogram
-
and from Figure 4
0p oo 5, 5.li1
- 4c + 3
- 770
This low value of R e means t h a t the flow is laminar. As will be shown below, turbulence does not set in until R e 18,000 for the S value under consideration. See Figure 8. There is obtained
0.280
c4
X 30.5 X 2
0.2
MP
0,270
0,390
Figure 5.
12c
1
o‘2
2.542
30.5 X 0.0638 = 13.9 lb./cu. f t .
RELATION BETWEEN u AND r
A glance at McMillen’s equations (Equations 6, 7, and 8),will show that if c could be expressed explicitly as a function of a,
INDUSTRIAL AND ENGINEERING CHEMISTRY
654
then Equation 6 would result in an explicit expression for PI2 as a function of the rheological material constants, the pipe diameter, and the average velocity. As will be shown, however, this cannot be done in a manner suitable for technical calculations. Equation 7 can be transfoimed to an equation of the fourth degree en
+ Ac + 3 = 0
(12)
where
&4 = -4 - 12a
(13)
Equation 12 has the folloving roots:
Vol. 44, No. 3
lim p(S) = 45’
S-.
m
TURBULENT FLOW OF PLASTICS
T h i l e the turbulent flow of NeiTtonian fluids has been extensively investigated, the literature of turbulent f l o of ~ plastics is meager ( 1 ) . Turbulent flow of plastics is here taken up from a dimensional-analytical vielypoint, as far as knon n to the author, for the first time. Only flow in smooth pipes --ill be discussed. P , 1, D,p p , f, p, and V will all certainly enter the function looked for. Since there are seven variables and three fundamental dimensions-mass, length, and time-it may be expected that 7 - 3 = 4 dimensionless ratios will suffice t o characterize the turbulent flow of plastics. Postulating that P , 1, ,up, andfshall appear only once, the following dimensionless ratios are obtained: Eulers number,
Eu =
Reynolds number, Re =
T z,=
#B
+ QZ
I
P/pVS DT‘p/pp
=
pVz/f
and
1/D T has the property of being an Euler number. portional to 1
A‘
A2 128
-1. -p v = = p ’ ( R e . T )
1
B=-+dma-1 is obtained.
A2 1 t’=--d&-64 128 c1 and c2 are real roots, while cy and
e4 are conjugated complex. to 03 j as c goes from 1 to 0. T h a t root of c1 and c2 which satisfies the condition 1 > c > 0 is the solution of Equation 12. It is obvious that the solution given is too troublesome for technical calculations. McMillen’s graphical solution gives c with an accuracy of two figures in the interval c = 0.02-0.98. In order to increase the usefulness of ilIcMillen’s equations, Equation 7 has been converted to a double scale (Figure 5j, covering the interval c = 0.001 to 0.999. From Equation 7 it can be seen that for high values of a (low c values) the following formula can be used:
A goes from -4 to -
+
( a from 0
1 E-
4c
On substituting the c value given by Equation 15 in Equation 6, there is obtained
P 1
45. 4a D
Assuming P pro-
Since furthermore
then
-.1
= (p”(Re, S) pv2
or
DISCUSSION OF
‘p
FUNCTIONS
While @ (see Table I ) is graphically represented by a single curve, when plotted against Re, (p” will be represented by a family of curves, each curve corresponding to a certain value of S. In the laminar region, (P” is completely known and ident,ical to p/Re. In Figure 3 the corresponding family of straight lines has been drawn. For (P = 32, corresponding to S = 0, p ” will be identical to @. Curves corresponding to
32T’pp 0 2
have also been put into the Fanning friction factor-Reynolds number diagram. Such a curre tells how F varies with V,multiDP plied with the constant factor --. In the laminar range
or
PP
In the limiting case whei ef = 0 and hence c = 0, %quation 16 will be identical to Poiseuille’s equation; listed as Equation 3, which is, of course, also directly obtainable from Equation 5 by letting p = 0. On the other hand, in the case of complete Plug flow, the following formula is valid
P-8f
(I7)
1 - D w-hich follows directly from the definition cussion given i t folloas that lim p(S) = 32 8-0
Of
p.
From the dis(18)
the slope of the U curves approaches the value -2 as S approaches infinity. With decreasing values, the slope approaches -1. ti^^ will show, that in most of the diagram the U curves are approximately straight lines with a slope of about -1.8. Experimentally, a U curVe and one point on every X line will be obtained if T’ is varied in a given pipe with a given material. Now, the following interesting question arises: At what point on the U curve will turbulence set in? Furthermore, what rela~ tionship exists between the Fanning friction factor and the R olds number in turbulent flow? As long as no complete theoretical investigation of the true nature of @ has been undertaken, the exact nature of p”,,,~, will
~
~
March 1952
655
INDUSTRIAL AND ENGINEERING CHEMISTRY
and f can be derived from experiments in one pipe and can be used in predicting the pressure drop in the two others. If the c a l c u l a t e d and experimentally found pressure drops coincide, it may be concluded t h a t the basic assumption was right-Le., that the suspension behaves as an ideal plastic. As will be shown, this is found to be the case. pp
TABLE I. SURVEY OF NEWTONIAN AND PLASTIC FLOW IN PIPES Flow in pipes: Re
E’
R e = S X T
BN.P
S
=
E
F = , ! X E X A z 1 ’va
PP V
T
=
ea i
Turbulent Flow Newtonians Plastics
Laminar flow
New tonians
P Dz -.I
pNTi
= 32
Plastics
.......I
F=16
Re.....II
E.E Ppv
= p(S) . . . .I11
2F =
@ . ,. . I V
Re limp(8) = 32..
s+o
2F
... .. V
-
.
@(Re). . V I
DETERMINATION O F NP AND f FROM PIPE FLOW DATA
2 F = p ” ( R e , S ) . ..VI1
2F = p”’(T.
As). . VI11 . IX ,
2 F = p‘(Re, 7‘).
From Equations 6 and 8 it can be seen t h a t for two pairs of PI1 and V the following- equations hold:
,
(26)
(P/l)l/(P/l)?= c2/n
f
f
0,f
Vl/ V2 = aJa2 (27) These equations were derived by McMillen, who states “there is only one set of c values related thus which possess a values in the same ratio as the experimentally observed flow rates in the two tests.” Therefore, CI, CZ, a:1, and a2 can be computed accurately from the double scale of Figure 5 by means of a simple trial-anderror procedure. When a: and c are known, wP and f can be calculated from Equations 6 and 8.
0.01
EXPERIMENTAL DATA
OWI 102
703
Figure 6.
i O*
iOJ
F us. Re Diagram for Plastics
probably remain unknown. Therefore, in analogy with the case of Newtonian turbulent flow, p ” t u r b . must be experimentally determined. Since two dimensionless ratios are encountered (Re and S or U ) , different curves for different values of S (or U ) may be expected. However, with increasing R e these different curves should approach the usual F-Re curve of Newtonians, since at very high velocities S will approach zero. To put it into mathematical symbols, the following relationship should hold:
Of the pressure drops reported by Wilhelm et al. (8) only those referring to 54.3% rock by weight can be used in the calculation, because in the other series of experiments the concentration was not kept constant a t different pipe diameters. I n Table I1 the following quantities, which are based on the data of Wilhelm et al. have been computed: V in cm. per second; P/t in dynes per square cm.; Re; U; and F. From run 23, PP and f were calculated and found to be 0.0686 poise and 38.0 dynes per square cm., respectively. Individual runs 7 and 9 were used in this calculation. From the values thus obtained, Re and U have been computed. In Figure 7, F has been plotted against Re. Curves corresponding to the calculated U values and the usual friction factor curve of Newtonians have also been put into the diagram. The U curves in all three cases follow closely the experimentally found curves in the laminar region; consequently, a Bingham F
lim p‘’turb. = lim @ = constant S+O Re+ m Re+ m
(25)
Another way to arrive at the same conclusion is to note that a t very high velocities F will depend upon kinetic forces only. After a detailed study of investigations reported in the literature, the author has come to the conclusion t h a t only the data given by Wilhelm, Wroughton, and Loeffel (8) on cement rock suspensions are detailed enough to permit a determination of
i
0.i
P”turb.
Unfortunately, the viscometer data reported by these authors are of little value in determining the flow curve, since the viscometer was run in the turbulent region. However, it seems highly probable t h a t for practical purposes the thick suspensions used can be approximated as ideal plastics. A great many other solidliquid systems, such as paints and bentonite suspensions, studied by Green and others, have been shown to approximate very closely the Bingham body. Fortunately, Wilhelm, Wroughton, and Loeffel (8) have made experiments on the same material flowing through pipes of three different diameters. Therefore,
001
0 001
foo
Figure 7.
i,ooo
iaooo
Rr
mmo
F us. Re Diagram for Cement Rock Suspensions in Pipes
656
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 44, No. 3
TABLE 11. FLOWD A T A O N CERIENT ROCK SU~PENSIONS I N PIPES R u n 23. D = 0.812 inch = 2.06 cm.; LIP = 0.0686 poise: f = 38.0 dynes per square om.: p = 1.52 grams per cubic om.; C = 52,100 Individual run V , Cm./Gcc. PIZ, Dynea/Sq. Cm. Re F 1 347 1106 1.5800 0.00f.iP 2 323 1001 14700 0.0065 3 297 843 13500 0 0066 4 226 540 10300 0.0072 2 181 360 8250 0.0072 6 137 204 6250 0.0074 7 89.3 141 4070 0.0120 8 120 159 5470 0.0075 9 36.0 110 1640 0.0374 10 44.2 127 2020 0.0439 R u n 22. D = 1.59 inches = 4.04 em.: p p = 0.0686 poise; f = 38.0 dynes per square c m . : p = 1.52 grama per cubio CIKI.; I - = 200,000 287 349 25700 0.0057 241 257 21600 0.0059 186 165 16600 0 0064 162 113 13600 0.0066 109 62.7 9790 0.0070 69.8 50.0 6250 0.0136 51.2 56.7 4580 0.0288 37.3 58.2 33.50 0.0661 12.2 50.0 1090 0.447 R u n 21. D = 3.05 inch = 7.75 cm: PP = 0.0686 poise; f = 38.0 d y ~ ~ e s per square c m . ; p = 1.52 grams per cubic c m . ; L: = 737,000 83.8 34.3 14400 0 0124 70.4 29.9 12100 0 0154 50.0 34.3 8580 0 0349 26.5 34.3 4550 0 124 14.3 31.4 2460 0.392 6.7 20.9 1150 1 18 9.8 26.3 1670 0.68 I
7
70
Figure 8 .
$02
to3
Critical Reynolds Numbers for Plastics
body is being dealt with. In the turbulent region, the experimental points fall very close to the Newtonian F curve. This seems t o indicate that the same F curve is applicable for both Xewtonians and plastics, at least as a good approximation. Furthermore, turbulence sets in when the U curves and the Kentonian F curve intersect. The critical Reynolds number is thrrefore a function of S only. This function is shomn grnphically in Figure 8. RECOMMENDED METHOD OF CQ3IPUTING PRESSURE DROPS
The facts given result in a very simple method of computing the pressure drop: Re and S are first calculated. The curve of Figure 8 then tells if turbulence is to he expected. If so, the usual F-Re curve of Newtonians should be used. As a consequence, f does not enter the pressure drop calculation if turbulence reigns. I n the case of laminar flow, the pressure drop can he calP D2 culated from the curve of Figure 4, combining S and -- -. 1 UPV Alternatively, McMillen’s equations and the double scale of Figure 5 can be used, especially when very high accuracj- is needed. SUMMARY
.
The importance of the flow curve (the rate of shear-shear stress relationship) in interpreting non-Newtonian flow data is stressed. A simple criterion is proposed, distinguishing between laminar and turbulent flow of plastics-e.g., thick suspensions-flowing isothermally in long, straight and smooch, cylindrical pipes. On the basis of experiments reported in the litreratwe, it is found that. for turbulent flow, t.he usual Fanning friction factor curve of Newtonians is a t least approximately applicable also for plastics, if the Reynolds number is defined as D V p l p p , ,UP,being t,lie plastic viscosity. Curves and a nomogram are given, permitt.ing accurate calculntions of pressure drops in a straightforward, simple manner. It is suggested that the theory be Gested further by combined viscometer and pipe flow tests.
E
=
X
=
c
=
11 Eu .f
= = =
F
=
7
=
1 P P I p
NOMENCLATURE
Dimensions are indicated in terms of M = mass; L = length; and T = time.
A B
- 12p, dimensionless 1 = 128 + - 64, see Equation 14, dimensionlesv = -4 A4
d&M
d-1
flow, M L-1 T-2 distance, a t right angle t o direction of flow, L Reynolds number, D V p / p , dimensionless f D / p p V , dimensionless pV2/f, dimensionless pf D ~ / M dimensionless P~, local velocity, L T-’ = rate of shear or velocity gradient, T-1 = average linear velocity,.L T-I c 4 - 4c 4= = ,: dimensionless = = = = = =
I’
Re S
Y’
c‘ 2
‘2 dr V L
0
i , ip, p pa p . ~ i”p
p
ACKNOWLEDGMENT
The author is pleased to acknowledge the stimulat’ing interest of Laborator K.-I. Skarblom, a t the Research Institute of National Defense, Stockholm, and the criticism of this paper by L. G. Sill&, Royal Institute of Technology, Stockholm. Lennart Andersson, of ILIessrs. Nomogramkonstruktioner, Stockholm, spent much time and effort in constructing the double scale of Figure 5 and the other drawings.
1
= = length of pipe, L = pressure, M L-1 T-* = pressure gradient along tube or pipe, M L - 2 T-3 = 4 Zf/D = minimum (yield) pressure, required to start
2
o
dm4
A2 A4 128 - &, see Equation 14, dimensionless function of B and E , defined bv Equation 14, dimen” _ sionless ratio of plug diameter to pipe diameter, p / P , dimensionless pipediameter, L Eulers number, P / p V 2 ,diniensionless yield stress, ML-IT-2 Fanning friction factor, dimensionless 2 1 pV2’ shear stress, M L-1 T-2
functions, defined in the text, diniensionless = viscosity, general term, M L-1 T-1 = apparent viscosity, M L-1 T-1 = Newtonian viscosity, M L-1 T-1 = plast,ic viscosity, M 1,-1 T-1 = density, M L-3
p’, q”, q”‘,
C$
=
LITERATURE CITED
Alves, G. E., Chem. EuQ., 56, KO.5, p. 107 (1949). Bridgman, P. W., “Dimensional dnalysis,” New Haven, Conn.,. Yale University Press, 1931. (3) Buokingham, E., Am. SOC.Testing Materials, PTUC., 21, 1154. (1921). Green, H., “Industrial Rheology and Rheological Structures,’’ New York, John Wiley and Sons, 1949. ?rIcMillen,E. L., Chem. Eng. Progress, 44,537 (1948). Reiner, M., “Deformation and Flow,” London, Interscience Pubs., 1949. (7) Xeltmann, R. N., J . Colloid Sci., 5 , 295 (1950). (8) Wilhelm, R. H., Wroughton, D. M., and Loeffel, W. F., IND. ENG.CHEX.,31,622 (1939). (1) (2)
RECEIVED for review March 13, 1951.
ACCEPTED October 8 , 1951.
