I
ARNOLD G. FREDRICKSON and R. BYRON BIRD Department of Chemical Engineering, University of Wisconsin, Madison, Wis.
Non-Newtonian Flow in Annuli Extrusion of molten plastics and Row of drilling muds in annular space are typical problems to which this study can be applied
To
MAKE deductions from flow data concerning the range of applicability of various empirical non-Newtonian flow models, it is necessary to have good experimental flow data and accurate solutions to the equations of motion in various geometrical arrangements. Experimental data for a few fluids and analytical solutions are available for axial flow in tubes and for tangential flow in cylindrical annuli (3, 7, 77, 72). This discussion is concerned with the analytical solution of the equation of motion for the steadystate axial flow of a n incompressible, non-Newtonian fluid in a long cylindrical annulus. This problem is of importance in connection with heat transfer to and from fluids flowing in annular spaces, flow of molten plastics in extrusion apparatus, and flow of drilling muds in annular spaces. T h e equations describing the flow of a compressible, isothermal fluid are equations of continuity and motion (2, 4) :
dr/dt
r[bulbt
+
+ (V.YV)
(U.V)Vl
= 0
=
-VP - (V.7)
+ Yg
(11 (2)
Assumption of isothermal flow implies not only that there is no impressed temperature field (6) but that the viscous dissipation term (7:Vu) in the energy balance equation is negligible (7, 2). In the developments which follow, the flow between two coaxial cylinders (Figure 1) is considered. T h e following assumptions are made: The fluid is incompressible (y = constant). T h e flow is in steady-state-Le., timeindependent. T h e flow is laminar. T h e cylinders are sufficiently long that end effects may be neglected. For the specific system under consideration Equations 1 and 2 may be written in cylindrical coordinates and combined and simplified to :
in whichpo and p L are the static pressures a t z = 0 and z = L, respectively, and gs is the component of gravitational acceleration g in the direction of flow. P designates the sum of forces per unit volume on the right side of Equation 1. This first-order differential equation, valid over the entire annular region for
any kind of fluid, may be integrated to give :
(4) in which X is the constant of integration. T h e radial distance r = XR represents that position a t which r,, = 0. Equation 4 is taken as the starting point for the derivations for the Bingham plastic and the power law models.
Solution for Bingham Plastic Model V a n Olphen (70) has presented a n approximate solution; Mori and Ototake (9) have given the complete solution, b u t their analytical and graphical results are in error; Laird's solution ( 5 ) is correct. However, Laird's expression for the volume throughput is more complicated than that given here, and he has not presented his results in terms of a dimensionless correlation for general use. For this model the local shear stress, T,#, is related to the local shear rate, dv,/dr, according to the formula :
+
wherein is used when momentum is being transported in the +r direction and - when transport is in the -rdirection. The meaning of T O and pa is given in Figure 2 , where the Bingham model is compared with the Newtonian model. The introduction of the following dimensionless variables is useful :
Figure 1. Shear stress distribution for axial annular flow corresponding to Equation 4 and characteristic velocity distributions for power law fluids (Equations 22 and 23) and Bingham plastic fluid (Equations 1 1, 12, and 13)
Actually it is convenient to express all the final results not in terms of X but rather in terms of either A+ or X- ; let us choose k+ to be consistent with Mori and Ototake (9). From Equation 9 it follows that: Xa = X+(X+ h = (A+
- To) - To)
(10)
Hence X is just the geometric mean of A+ and X-.
T = 2r,,/PR = dimensionless shear stress
To = 2ro/PR
= dimensionless limiting
shear stress
9 = (2~0/PR~)p~ p
= dimensionless velocity = r/R = dimensionless radial distance
T h e equations describing the system are
Figure 2. Shear stress vs. shear late for several types of fluids
X+ and 1- represent the bounds on the plug flow region. Clearly they are those values of p for which TI = TO
1
N.
Newtonian, with slope p
B. Bingham plastic, slope po and intercepts 1 r p P. Pseudoplastic, n < 1 in Equation 21 D. Dilatant, with n > 1 in Equation 21 In annular flow rpSis negative for r / R X and positive for r / R A (see Figure 1 )
>
VOL. 50,
NO. 3
(1 - K ) ; there is no flow if TO5 (1 - K ) . T h e dimensionless volume rate of flow OB =
0.9015b 0. 9022b 0.9027b 0.9032b 0.9036b 0.9041b 0.9045h 0.9050b 0.9054b
2 * 000 1.802 1.752 1.670 1.505 1.340 1.257 1.208 1. 177c 1. 145c 1. 126c 1.1120 1.1000 1.091c 1.000
2.000 1.802 1.752 1.669 1.504 1.3370 1. 254c 1.205" 1. 174c 1. 144c 1.126O 1.111c 1.lOOC 1.091c 1.000
2 * 000 1.800 1.751 1.668 1.502 1. 336c 1. 252c 1. 203c 1. 17lC 1. 143' 1.125' 1.111c 1.1OOG 1.0910 1.000
1
0.9000
0.9002 0.9003 0.9004 0.9008
0.9500 0.9500 0.9501 0.9501 0.95OZb 0.9504b 0. 9506b O.950Sb 0.951ob O.951lb 0.9513b 0.9515b 0.9517b 0.951Qb 2.000 1.800 1.750 1.667 1.501. 1,334= 1.2518 1.201s 1. 16QC 1. 143c 1. 125c 1.111' 1* low
1* 0910
1,000
ass+m
rate of flow for a given pressure drop when the dimensions of R and K and po and TOare known. A plot of ClB/To (Figure 5 ) is useful for calculating the
,021
5p 5
1.0
1 (13)
0.8
I
-I- 2To(l
0.8500 0.8504 0.8506 0.8509 0.8517 0.8533 0.8546 0.8551 0.856Sb 0.8577b 0.8585b 0.8594b 0.8602b 0.8611b
0.9
+
where use has been made of the boundary conditions that 4 = 0 at p = K and p = 1. T h e determining equation for A, is just the statement that the velocity 4- (A-) be the same as q5+ (A,) :
) ~
0.8000 0.8009
0.8011 0.8018 0.8034 0.8064 0. SO81 0.8107 O.812Sb 0.8144b 0.8160b 0.816Sb 0.8176* 0.8184b
0.8
Q/(rR4P/8pO) is plotted in Figure 4 as a function of K and TO. This graph enables one to compute easily the volume
v,pdp
5 P 5 h (11) A-
X a l n p ; A+
(To4-K
-
1
Q = 2rR2
0.7
= y(S, K )
2.000
Obtained by graphical interpolation of
0.6
0.5
-
- A+)
= 0
2
(14)
From this equation X+ has been determined as a function of K and TOand is plotted in Figure 3. The volume rate of flow for the Bingham plastic is obtained by integrating the velocity distribution in Equations 11, 12, and 13 over the annular region and simplifying the result with the help of Equation 14 :
06
04
a2
Figure 3. X, and &, for Bingham plastic flow in an annulus
00
VOL. 50, NO. 3
MARCH 1958
349
4 Figure 4.
for Bingham plastic flow through an annulus
YO
10
09
09
08
08
07
07
Figure 6. X for power law flow through an annulus
06
-
b
06
-
i 50s
%05
I-0
P O4
04
03
03
02
02
01
01
00 00 00
01
02
a3
04
05
07
06
08
09
io
K
Bingham Flow in Circular Tubes 0,X- = 0,X+ = To)
pressure drop when the volume rate of flow is known. T h e use of Figures 4 and 5 is discussed in Examples 1 and 2. T h e numerical results for the Bingham flow calculations are presented in Table I. For high rates of flow when the plug flow region is small compared with the dimensions of the annulus, Equation 15 may be simplified by using the Newtonian expression for X (see Equation 18) and making the further assumption that (A, X-) F= 2X. Then:
(K =
TR4P Q = __ [ I $PO
Q
=e[
- .4) - (M 1n (1/.)
(1
1
+ .3) To
Q
8~
Newtonian Flow in a Very Thin
(15a)
'lit
(To = 0,Lo./ ?rR4P =7 (1 UP
(16)
Q
rR4P
- (I GPO
~1~
Equations 21 and 4 may be combined and integrated to give the velocity distribution:
R(PR/2m)s
f:(
= p, K w 1)
-
(P
(19)
in a Very Thin K)3
[I -
- p)'dp;
ti
l1
v, = R(PR)/2m)e
-
Bingham Iar Slit ( K = 1 )
= 0, Po = P )
rR4P
(21 1
1-(
Newtonian Flow in Circular Tubes Q=-
W ] (17)
For this model the local shear stress depends on the local shear rate as follows:
in which m and n are the rheological con( T , = O , P ~ = ~ , X + ~ = X _ ~stants, = ( ~obtainable by viscometric tech21n (1;;;) niques (7, I I , 12). Neivtonian fluids are TR4P a special case of Equation 21 with rn = p Q = __ [(l - ti*) (18) 8P In ( 1 / K ) a n d n = 1.
This expression differs slightly from that given by Laird (5, Equation 30). Five important limiting cases of Eauation 15 are:
(To= 0,K
+
1
Newtonian Flow in Circular Annulus
+
(1
-
4 -3 T~
Solution for the Power l a w Model
5
p
5
-
5
P
(22)
X
5
1 (23)
in which s = l / n ; in the integrations, the boundary conditions u, = 0 at p = K and p = 1 have been used. Clcarly both Equations 22 and 23 must give the same value of velocity at p = X:
LA:(
- p)"
dp =
L1
(P
-
(24)
?)'dp
This is the determining equation for A, which is a function of K and s. Finally the volume rate of flow is obtained by substituting Equations 22 and 23 into the first line of Equation 15. T h e order of integration may then be interchanged and one integration performed to give: Q =
%'b
350
INDUSTRIAL A N D ENGINEERING CHEMISTRY
L1
T ~ 3 ( ~ ~ / 2 m ) h
I ~2 -
6
+lp-a
dP
(25)
which can easily be integrated once X has been determined from Equation 24. The integrals appearing in the general results in Equations 24 and 25 may
WON-NEWTONIAN FLOW Figure 7. vmnx/vav for power law flow through an annulus
IO
09
4 on
07
b 8.
Figure
T(s,
for power law flow through an annulus K)
K
0
01
02
03
04
05
06
01
I O
09
08
P o w e r L a w Flow in a V e r y Thin Annular Slit
K
easily be integrated when s is a positive integer (pseudoplastic materials), by expanding the integrands in a binomial expansion and interchanging the order of summation and integration. Equation 24, which determines X, then becomes:
= rR3(PR/2rn)8 n p ( s ,
(30)
K)
in which k A s g hac
+
&Asf1
=0
s = 1 , 2 , 3 . . .(26)
i=O
i
( -l)a+lKI--2i+8
#s + lT
in which
(;) A,I =
1
(31)
s-1
(-1) (-1)ifl
s-2i+1
[1 + (-1)8
In ( 1 / ~ ) [sodd] (32)
K8--2it1
1 [seven] (33) i=o
2i - s + l
+
K ) , defined by 'T (s, E ) = (s 2) QP/(l K ) ' + ~ , is tabulated in Table 111 as a function of s and X . O n e can
T(s,
A$ = 2
2 (:)i=O s
(-')' [seven] 2i 1
+
(29)
Equations 26 to 29 a r e polynomial equations which can be solved to get X as a function of s and K , although this method breaks down for high values of s and K . Values of X so computed were used to prepare Table I1 and Figure 6. The limiting values of h are h = (1 K ) / 2 a t s = 0 and X = a t s = co. T h e latter may be shown by expanding the integrand of the left side of Equation 24 in a Taylor series about p = K and the right side about p = 1, and taking the first term in both expansions. The expression for the throughput rate given in Equation 25 may be expanded in a similar fashion to get:
4
+
-
easily compute the throughput for any pressure drop once the dimensions of R and K and m and s are known. Table 111 may also be used to deduce the rheological constants from annular flow data (Examples 3 and 4). The various limiting cases of Equation 30 may be tabulated : Newtonian Flow in a Circular T u b e (s = 1, m = p, K = 0)-Equation 16 Power-Law Flow in a Circular T u b e (S # 1, K = 0)
Newtonian Flow in a Circular An-
nulus 18
(s =
= kh
* o)-Equation
Newtonian Flow in a V e r v Thin Ann u l a r Slit (s = 1, m = p, K = 1)-Equation 19
Applications Example 1. Calculation of Pressure Drop for Annular Flow of a Bingham Plastic Material. A mud having a density of 1.69 grams per cc. flows at 5 feet per second average velocity through an annulus made from 0.5-inch standard pipe (outside diameter = 0.840 inch = 0.0700 foot) and 2-inch standard pipe (inside diameter = 2.067 inches = 0.1726 foot). The Bingham plastic constants for this solution are 70 = 0.554 pound/ per square foot and po = 0.000582 pound, second per square foot. Compute the pressure drop per unit length required. SOLUTION. From the dimensions of the annulus, K = 0.840/2.067 = 0.406 and R = 0.1726/2 = 0.0863 foot. The volume throughput is given by: Q = nR2(1 -
K')U,,.
~ ( 0 . 0 8 6 3 )[~l
- (0.406)2]
= 0.09775 cubic foot per second.
The quantity &/To = 4fi0Q/nR3~o is then:
OBITO= 4(0.000582) (0.09775)/ ~ ( 0 . 0 8 6 3(0.554) )~ = 0.204 From Figure 5, T O = 0.295. pressure drop per unit length:
Thus, the
P = 270/ToR = 2(0.554)/(0.295) (0.0863) = 43.5 poundf per square foot per foot = 0.30 pound/ per square inch per foot Example 2. Deduction of Bingham Plastic Rheological Constants from Annular Flow Data, A Bingham plastic material flows through the annulus described in Example 1. The following data are obtained; For v, = 5 feet p& second, P = 16.8 pound, square foot per foot; for 0," 10 feet per second, P = 28.3 VOL. 50, NO. 3
MARCH 1958
351
pound, per square foot per foot. Calculate T O and P Ofor the substance. SOLUTION.By using Equation 15 we can calculate the ratio of 0, (5 feet per second) to C ~ B(10 feet per second) :
pound, per square foot per foot and at Q = 0.19550 cubic foot per second, P = 460 pound, per square foot per foot. It is desired to calculate the power-law constants of this fluid.
Q B ( ~ft./sec.) Q ~ ( 1 ft./sec.) 0 Q ( 5 ft./sec.) P(10 ft. /sec.) Q(l0 ft./sec.) P(5 ft./sec.)
0.09775 = ?iR3(326R/2m)s Qp(s, 0.406)
P
= Xewtonian viscosity = “plastic viscositv” of the ABingliarn plastic (Equation 5) = expansion coefficients in expression for volume rate of flow of a power-law fluid through a n annulus (defined i n Equations 31, 32, a n d 33) = 3.1416 = r,’R = dimensionless radial. coordinate = shear stress tensor = limiting shear stress of Bingh a m fluid = rt-component of shear stress tensor = function defined just after Equation 33 = dimensionless velociry for Bingham plastic (Equation
40
= dimensionless maximum ve-
SOLUTION.From Equation 30 we have 0.19550 = .rrR3((460R/2m)’Qp(~, 0.406) Division of these two expressions gives then: (112) = (326/460)s
Similarly, from Equation 6, we find: Ta(5 ft./sec.) - P(10 ft./sec.) To(1O ft./sec.) ~ ( ft. 5 /sec.) -
’$
= 1.685
0.09775
I n order to calculate the value of TO at either flow rate, a trial-and-error method is used. This can be done by assuming a value of TOat one velocity from which one can get To a t the other velocity. Values of fie corresponding to these two TOvalues may be found from Fi ure 4. T h e ratio of ne (5 feet per second7 to Q B (IO feet per second) is then computed; the procedure is repeated until this ratio is as near to 0.843 as is desired. Such a trial-and-error procedure takes the form : To
(5 Ft./Sec.)
0.100 0.200 0.138
To(l0 Ft./Sec.) 0.060 0.119
QB(5
Ft./Sec.)/
Re(10 Ft./Sec.) 0.868 0.730 0.837
0.082
Hence, Equation 6 gives T O = ToPR.!2 = (0.138)(16.8)(0.0863)/2 = 0.10 poundf per square foot. Then from Equation 15
Q = 0.09775 = ~ ( 0 . 0 8 6 3 )(16.8) ~ (0.133)/8~0 whence it is found that po = 0.0005 pound? second per square foot. Example 3. Calculation of Pressure Drop for Power-Law Flow through an Annulus. A 0.67% aqueous solution of carboxymethyl cellulose flows a t 5 feet per second average velocity through the annulus described in Example 1. T h e power law constants for this solution (8) are s = 1.398 and rn = 0.00635 pound/ (second)O.7’6 per square foot. Compute the pressure drop per unit length required. SOLUTION.As in Example 1, K = 0.406 and R = 0.0863 foot. The volume throughput is
Q = .rrR’(I - K~)U,\.. ~(0.0863)’[l - (0.406)’] (5) = 0.09775 cubic foot per second For the values of K and s given above, we find by interpolation from Table I11 that the dimensionless function T(s, K ) is 0.7155. Hence the dimensionless throughput is:
+
2)-’T(S, fi p = (1 - K ) ’ + ’ ( S (1 - 0.406)’.’9*+2(1.398 (0.7155) = 0.0359
+ 2)-’
K)
From Equation 30 0.09775 = ~ ( 0 . 0 8 6 3(0.0863 )~ P/ (2) (0.00635))’,3@5 (0.0359) whence P = 25.5 pound, per cubic foot or 0.177 pound, per square inch per foot. Example 4. Deduction of Power Law Constants from Flow through a n Annulus. A polymer solution known to be of the power-law type flows through the annulus described in Example 1. T h e following data are obtained: at Q = 0.09775 cubic foot per second, P = 326
352
when s = 2.0 (or n = 0.50). Interpolating from Table 111, we find T(2.0, 0.406) = 0.7205, whence Q p = 0.0224. Hence Equation 30 gives (for Q = 0.09775 cubic foot per second). =
~10.0863)’[(326) (0.0863)/ (2m)]*.O(0.0224)
from which m = 0.30 pound, (second)0,50 per square foot.
Acknowledgment
6) locity for Bingham plastic (Equation 12) = dimensionless velocity for Bingham plastic outside plug flow region (defined in Equations 11 a n d 13) = dimensionless flow rates for Bingham a n d power-law models, respectively = “del” o r “nabla” operator
T h e authors are greatly indebted to the computing staff of the University of Wisconsin Naval Research Laboratory, under the direction of Elaine Gessert, for assistance with the computational work. They wish to thank J. 0. Hirschfelder for making these arrangements possible.
+-, ++
Nomenclature
literature Cited
= external body force per unit mass = dummy index used in sum2 mations = length of annular region L = parameters in power law m, n model (Equation 21) = static pressure P = static pressure a t entrance Po to annulus (z = 0) = static pressure a t exit to PL annulus ( z = L ) = (Po - P d / L P Ygz = volume rate of flow through 0 annulus = radial coordinate, measured r from common axis of cylinders forming annulus = radius of outer cylinder of R annulus = reciprocal of n S = time t = dimensionless shear-stress for T Bingham flow (Equation 6) = dimensionless limiting shearTO stress for Bingham flow (Equation 6) = velocity vector V = z-component of velocity vecVZ tor = axial coordinate, measured 2 from entrance of annulus = mass density of fluid Y = ratio of radius of inner cylK inder to that of outer cvlinder = value of dimensionless radial coordinate for which shear stress is zero = limits of plug flow region i n L, Bingham flow AS, ASZ = coefficients defined in Equations 27, 28, a n d 29
(1) Bird, R. B., SPE Journal 11, 35-40 (1955). (2) Bird, R. B., “Theory of Diffusion,” Chau. in “Advances in Chemical Engineering,” vol. 1, pp. 155-239, Academic Press, New York, 1956. (3) Christiansen, E. B., Ryan, N. W., Stevens, Lt’. E., A.I.Ch.E. Journal 1, 544-9 (1955). (4) Hirschfelder, J . O., Curtiss, C. F., Bird, R. B., “Molecular Theory of Gases and Liquids,” Chap. 11, Wiley, New York, 1954. (5) Laird, h’. M., ~ N D .ENG.CHEM. 49, 138-41 (1957). Lyche, B. k., Bird, R . B., Chem. Eng. Sci. 6, 35-41 (1956). Metzner, A. B., “Yon-Newtonian Technology : Fluid Mechanics, Mixing, and Heat Transfer,” in “Advances in Chemical Engineering,” vol. I, pp. 77-153, Academic Press, New York, 1956. Metzner, A. B., Reed, J. C., A.I.Ch.E. Journal 1, 434-41 (1955). Mori, Y., Ototake, N., Chem. Eng. (Jupun) 17, 224-9 (1953). Olphen, H. van, J . Znst. Petroleum 36, 223-34 (1950). Philippoff, W., “Viskositat der Kolloide,” Steinkopff, 1942; Edwards Brothers, Ann Arbor, Mich., 1944. (12) Reiner, M., “Deformation and Flow,” H. K . Lewis and Co., London, 1949.
9
+
x
INDUSTRIAL AND ENGINEERING CHEMISTRY
x,
OB: Q p
A
RECEIVED for review February 13, 1957 h C E P T E D June 10, 1957 Division of Industrial and Engineering Chemistry, Symposium on Fluid Mechanics in Chemical Engineering, Purdue University, Lafayette, Ind., December 1956. Presented in part, Society of Rheology, Pittsburgh, Pa., November 1956. Work supported by fellowship from National Science Foundation and grant from Wisconsin Alumni Research Foundation.
