VISCOUS HEATING OF A POWER-LAW LIQUID IN PLANE FLOW The solutions of the equations of motion and energy for a power-law liquid with temperature-dependent consistency in plane flow in the absence of a pressure gradient are obtained. For an exponential temperature dependence of the consistency the problem is formally the same as that for Newtonian liquids, provided the Brinkman number is properly defined. The temperatures developed a t corresponding values of the Brinkman number differ, depending upon the power-law index, n.
GAVIS and Laurence
(1968) presented the solution of the equations of motion and energy for the nonisothermal flow of a n incompressible: Newtonian liquid with temperaturedependent viscosity between moving parallel surfaces in the absence of a pressure gradient. Although nonlinear, the equations admitted solutions in terms of known elementary functions for exponential dependence of viscosity on temperature. The most interesting feature of the solutions \vas their double-valuedness in the applied shear stress. The work has been extl-nded to include a particularly simple yet important type of non-Newtonian behavior of liquids. The equations of motion and energy are considered for plane shear flow of a power-law liquid whose consistency depends exponentially on temperature. Although not generally applicable, the po\ver-la\v description of many non-Newtonian liquids is useful in one-dimensional steady-flow situations, and investigators often determine the po\.r-er-law parameters, m and n, using Couette-type viscometers. How the temperature and velocity profiles are affected by viscous dissipation in such liquids is of some interest, therefore. The type of liquid to be considered is one for which the shear stress-rate of shear relationship is of the form
The problem is considered for plane shear flow only, since, when non-Newtonian liquids are under investigation, as narrow a gap as is feasible will be used in Couette-type viscometers in order to approximate a uniform rate of shear across the gap. Flow in such narrow gaps may be approximated as plane flow. The equation of motion, (4)
becomes, with Equation 1, for a power-law liquid Tzu
=
mi-^
dv , It-‘ dv - -dz dz
u ( 0 ) = 0; v(h) =
=
m , exp
- p(T - T , ) / T ,
(2)
(7)
-ta ( T -
T,)/T,
=
-1
-[vo/J
dz
0
miln
The energy equation
becomes, with Equations 2 and 5 ,
H e represents the variation of n in the form
n = no
v,
to give, since n is considered constant,
70
m
(5)
Then Equation 5 may be integrated subject to the boundary conditions
with
Both the consistency, m , and the index, n, are functions of temperature. Turian (1965) has indicated, however, that m is a much more sensitive function of temperature than is n and can, in fact, be represented by
r o (constant)
n+l
(3)
-+-
d2T dt2
km,lln ex’
( 7 = )
The boundary conditions are and states that he found the constant (Y to be small. In this note the temperature variation of n is neglected. With this assumption the equations are readily soluble, yielding solutions identical to those for Newtonian liquids. These solutions should describe I-eality reasonably well when temperature changes in the liquid are moderate, and provide insight into what may be expected when temperature changes are high.
T(0) = T ( h ) = To for isothermal Malls, and
for one adiabatic wall. VOL. 7
NO. 3 A U G U S T 1 9 6 8
525
Introduction of the dimensionless variables
(:) YoTo -
or
___
and
and {
G
r/h
gives the following boundary value problems for the temperature and velocity fields
BrlW) =
hi-nv oi + n m o
k To
- Brinkman number law liquids
Figure 1 . 526
l&EC FUNDAMENTALS
for power-
This is formally identical to the problem for Newtonian liquids. The solutions given by Gavis and Laurence (1968) are, therefore. applicable here, provided the Brinkman number is defined by Equation 15 and the relationship between X I and the Brinkman number is given by Equation 14. There are, as in the Newtonian case, two solutions for each value of XI below a maximum value, a t which there is one solution, and beyond which no solutions exist. This means that there is a maximum shear stress Lvhich can be applied and that two different temperature and velocity fields can exist a t each applied shear stress below the maximum. Figure 1 is a plot of (/3/n)Br1(~) against X I for three values of n: for a shear-thinning liquid (n = l / 2 ) , a Sewtonian liquid ( n = I), and a shear-thickening liquid (n = 2). For n = 1 the plot coincides with that of Figure 5 of Gavis and Laurence (1968). The plots were obtained from the solutions presented in the previous paper and Equation 14 by direct computation.
Plot of (P/n)Brl(") against X I for power-law liquids
= = = =
Table I. AT, for Newtonian and Power-law liquids
0.025
0.5 1.o 2.0
0.24 0.24 0.24
0.04 0.04 0.04
=
0.2 0.4 0.6
= = =
= = =
200
0.5 1 .o 2.0
0.45 1.7 3.0
5.2 3.25 2.0
26. 32.5 40.
= = = =
As an example of the differences in the behavior of powerlaw liquids depending upon n, consider a liquid for which /3 = 30 with isothermal walls a t 300' K. At a given value of @ / n ) Brl(n) the solutions, depend upon n; a t low values the solutions coincide. I n Table I the solutions for the maximum temperature, Om, a t j- = '/2 are compared for a low, an intermediate, and a high value of ( P i n ) Brl(n), for the shearthinning, the Newtonian, and the shear-thickening liquids. The maximum temperature rise, AT,, increases with increasing n, even when 8, is the same for all values of n.
literature Cited
Gavis, J., Laurence, R . L., IND. END. CHEM.FUNDAMENTALS 7, 232 (1968). Turian, R . M., Chem. Eng. Sci.20, 771 (1965). JEROiME GAVIS R . L . LAUREXCE
The Johns Hopkzns Uniwrsitj Baltimore, M d . 2 72 18
Nomenclature
Brl(n) = Brinkman number for power-law liquid, h k m
power-law index: dimensionless local velocity, cm./sec. velocity of moving wall, cm./sec. temperature, ' K. maximum temperature, a t { = l / 2 , K. coordinate in direction of flow, cm. position coordinate, cm. temperature coefficient for n, dimensionless temperature coefficient for rn, dimensionless reduced position coordinate, z/h: dimensionless reduced temperature ( P i n ) ( T - T o /) T o ,dimensionless maximum reduced temperature, at { = I / * , dimensionless constant: dimensionless shear stress, dynes/sq. cm. (subscript) value of a quantity at an isothermal surface
hl-nVol+nmo, k To
dimensionless = distance between walls, cm. = thermal conductivity of liquid, g. c m . / ~ e c ' . ~K . = liquid consistency, g./cm. seen
RECEIVED for review December 27, 1967 ACCEPTED May 29, 1968 Work supported by grants from the National Science Foundation, NSF GK 1714 and GK 839.
CORRESPON DEINCE HYPER F I L T R A T ION Beclouding the issue of the second test with a demonstration SIR: We take exception to the conclusion presented in the article "Hyperfiltration" [IND.ENG.CHEM.FUNDAMENTALS of the ability to adjust a four-constant equation so as to arrive at a concentration polarization expression which maintains 7 , 4 4 (1968)l. The test of the theory embodied in Equation 10 is that a a Colburn parameter value of 25 does a disservice to the data. plot of In (1 - Robed/Robsd)as a function of v / i i O , 7 6 should The important conclusion upon which the model converges yield a straight line for each value of feed concentration, and \vhich the data support is that the ratio c,/G, is a constant c,, and give a slope of for each value of cf and that this ratio does not depend on k ;)2i3] a. The parameter R is: after all: defined in terms of c, and
[(' 7" ( ,Bk
The first part of the test shows good agreement of the theory with the data. With regard to the second test, visual inspection of Figures 1 and 2 indicates a reasonably constant slope and probable random variations of slope with .6,
.6 ,
J . L. Cooney A . E. Rabe E. I du Pant de .\'emours Tb7i1mington, Del.
& Co., Inc.
VOL.
7 NO. 3 A U G U S T 1 9 6 8
527
