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Ind. Eng. Chem. Fundam. 1902, 21 I 434-437
Summary of Experimental Results From the concentration fluctuation data obtained at various radial and axial positions very close to the gasliquid interface of a free-surface stirred tank, local values of concentration gradient, boudnary layer thickness, and fluctuation amplitude were obtained. Significant experimental results are summarized as follows. (a) It is the secondary flow pattern (axial flow) that affects mass transfer. Especially, the direction of the secondary flow has a strong influence on local eddy renewal rate: the local mass transfer rate is always higher on the tank wall side, where the flow is upward, compared with the tank center, where the flow is predominantly downward. As a result, the concentration boundary layer thickness decreases as the tank wall is approached. (b) Compared with a clean system, 10 ppm of BSA (bovine serum albumin) or 1000 ppm of yeast, both of which are surface active, reduces primary circulation velocity in the surface region. (c) Compared with a clean system, the surface renewal by eddies is not much affected by the presence of 1000 ppm yeast. The observed decrease in mass transfer is mainly due to a decrease in effective diffusivity. (d) Compared with a clean system, the concentration boundary layer thickness becomes thinner when 10 ppm of BSA is present. The BSA also reduces the amplitude of concentration fluctuation. These two results contradict each other and a further investigation is necessary. Acknowledgment The financial support of the National Science Foundation through the Grant 78-13282 is gratefully acknowledged.
Nomenclature c, = air saturation concentration, g/cm3 cb = concentration in liquid bulk, g/cm3 C = average normalized concentration, dimensionless Ci = ith sampled normalized concentration, dimensionless d = concentration boundary layer thickness, cm D = oxygen diffusivity in liquid, cm2/s kL = overall mass transfer coefficient, cm/s S = standard deviation or normalized concentration fluctuations, dimensionless y = depth from surface of liquid, cm Literature Cited Brauer, H. Adv. Biochem. Eng. 1079, 13, 87. Bungay, H. R.; Huang, M. Y.; Sanders, W. M. Am. Inst. Chem. Eng . j . 1073, 19, 373. Chan, W. C.; Scriven, L. E. Ind. Eng. Chem. Fundam. 1070, 9 , 114. Danckwerts, P. V. Ind. Eng. Chem. 1051, 43, 1460. Dang, N. D. P.; Karrer, D. A.; Dunn, I. J. Biotech. Bioeng. 1977, 19, 853. Davis, J. T. Am. Inst. Chem. Eng. J . 1072, 18, 169. Davis, J. T.; Lozano, F. J. Am. Inst. Chem. Eng. J . 1070, 25 405. Fortescue, G. E.; Pearson, J. R. A. Chem. f n g . Sei. 1987, 2 2 , 1163. Higbie, R. Trans. Am. Inst. Chem. Eng. 1035, 3 1 , 365. Lee, Y. H.;Tsao, G. T.; Wankat, P. C. Ind. Eng. Chem. Fundam. 1978, 17, 59. Lee, Y. H.; Tsao, G. T.; Wankat, P. C. Am. Inst. Chem. f n g . J . 1980, 26, 1008. Lozowski, D.; Langa, J.; Andrews, G. F.; Stroeve, P. Chem. Eng. Commun. 1980, 6 , 349. Luk, S.,W.D. Thesis, Drexel University, 1982. Ruckenstein, E. Int. Chem. Eng. 1087, 7 , 490. Ruckenstein, E. Chem. fng. Sei. 1966, 2 3 , 363. Ruckenstein, E. Chem. f n g . J . 1971, 2 , 1 St-Denis, C. E.; Feii, C. J. D. Can. J . Chem. Eng. 1071, 49, 885. Taylor, R., presented at Instrument Society of America Annual Meeting, Houston, Oct 1980 Tsao, G. T.; Lee, D. D. Am. Inst. Chem. Eng. J . 1075, 21, 979.
Received f o r reuiew September 10, 1981 Reuised manuscript received June 2, 1982 Accepted June 23, 1982
Lubrication Flows in Viscoelastic Liquids. 3. Approach of Parallel Surfaces Subject to Constant Loading Pradeep Shlrodkar' Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 0 1003
Alex Bravo and Stanley Middleman" Department of Applied Mechanics and Engineering Sciences, University of California, Sen Diego, La Jolia, California 92093
An isothermal incompressible viscoelastic liquid fills the space between two plane parallel rigid circular disks and retards their approach driven by a constant load. A constitutive model due to Wagner is used to predict half-times for closure of the disks. Resutts are compared to expectations for the inelastic power law model and are discussed in light of exlsting data for the phenomenon. The approach here utilizes the lubrication approximations and assumes that the normal stresses generated by the flow are of secondary importance to the transient shear stress behavior.
Lubricating fluids of unusual characteristics may be created by the addition of soluble high molecular weight polymers. There still exists considerable uncertainty over the most primitive question: does the contribution of the polymer increase or decrease the load-bearing capacity of a viscoelastic lubricating film? In a series of papers 'Mobil Oil Chemical Company, Edison, NJ.
(Shirodkar and Middleman, 1982; Shirodkar et al., 1982) and in greater detail in the thesis of Shirodkar (1981) we have performed experiments and theoretical ana yses aimed at providing an answer to this question. Of course a preexisting literature is available, reviewed thoroughly in Shirodkar (1981) and briefly in Shirodkar and Middleman (1982),also aimed toward this question. Relevant papers that provide some perspective include Brindley et al. (1976), Davies and Walters (1973), Grimm (1978),
0196-4313/82/102 1-0434$01.25/0 0 1982 American Chemical Society
f
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982
F
p
R-
I
c
Figure 1. Geometry of squeezing flow.
Harnoy and Philippoff (1977), Horowitz and Steidler (1960), Oliver (1979), Reiner et al. (1979), Rosenberg (1975), and Tichy and Winer (1978). Our own work suggests that previous analyses have used viscoelastic constitutive equations too simple to provide adequate representation of the dynamic response of real polymeric liquids. In our studies we have used Wagner’s (1977, 1979) form of the K-BKZ model (see Bird et al., 1976) with some success. From these studies we conclude that the kinematic nature of the squeezing deformation (the first class of lubrication flow we have examined so far) dictates the kind of mathematical procedure required to produce a useful model, and that in fact the answer to the question raised in the paragraph above depends to a degree upon the type of deformation. In this paper we examine squeezing flow between two plane surfaces subject to a constant loading. A straightforward procedure has been developed for calculating the rate of approach of the rigid surfaces. Details of this development are presented here, and results are presented in the form of “half-times” for squeezing, as a function of the rheological parameters of the liquid. Analysis Figure 1 shows the geometry considered here, corresponding to the approach of two identical rigid parallel disks of radius R loaded by a force Fo which is constant in time. The goal of the analysis is the history of separation of the disks, H ( t ) . The simplest analysis of any interest, beyond the trivial Newtonian liquid analysis, is that for the so-called power law fluid, defined by a constitutive equation of the general form (Middleman, 1977)
K and n are fluid parameters, IL,is the second invariant of the rate of deformation tensor i.. T is the stress tensor. Equation 1 is useful for describing purely viscous nonNewtonian behavior of typical polymeric liquids. It fails to provide a representation of two essential features of polymer flow: normal stresses developed in a simple shear flow, and transient stress behavior. For the power law fluid one finds the separation history (H(t)to obey a differential equation of the form (Bird et al., 1976)
(Note that in Bird’s notation, H = 2H.) The half-time, defined as the time required to reduce H to ‘lzH0,is found in the form
435
Experiments by Leider (1974) show the failure of this model under conditions of high loading, conditions which undoubtedly elicit strong transient viscoelastic phenomena. Analysis for any viscoelastic constitutive equation of sufficient complexity to be realistic can be quite difficult in principle. Our analysis is approximate in several respects, as discussed in more detail in Shirodkar and Middleman (1982). We begin with the Reynolds lubrication equation in the form
a~ = - -a T r z ar az
(5)
It is easily shown therefrom that the force causing the squeezing is calculable from
Thus we see that in order to find F one must know T, at the rigid boundary. In principle, it is necessary to find the deformation field, the kinematics, before T,, is known anywhere. This requires solution of the equations of motion, after introducing into those equations a constitutive equation for the fluid. For any realistic constitutive equation this leads to very complex equations. We follow the lead of Co and Bird (1974) and assume that the kinematics of this flow field are dominated by the viscous behavior of the fluid. We use the steady power law velocity field for the kinematics. We assume that a suitable consitutive equation is that of Wagner and Stephenson (1979),which we write (needing only the shear stress) as
This model has a linear viscoelastic memory function
t - t’
N k=l
thus introducing a set of moduli ah and relaxation times xk*
The strain dependent function h is taken to have the form h = ex~(-cy’[o],~) (9) introducing thereby a constant coefficient c. This specific realization of the Wagner fluid has been shown by Shirodkar and Middleman (1982) to lead to accurate predictions of squeezing flow data obtained using viscoelastic fluids. The strain function y’[olrzis related to the Finger strain tensor. Shirodkar (1981) shows that the assumption of power law kinematics leads to (we drop subscripts on y’)
H ( t ) and its previous history H’(t are unknown in the case of constant force squeezing. We seek H ( t ) as a function of Fo. Equations 6 through 10 lead to an expression implicit in H ( t ) .
where r(
-
+,)
,t’]dr
(11)
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982
436 loo0
I
100.
1000
%
$
IO0
1
F
7+ 10
10
0 01
01
10
,
I
Figure 3. Viscosity behavior of the model fluid. Data (0) are in good agreement with the prediction of the Wagner model.
I
I
where s = R/Ho
I
I
1
CFO = -U-l 21rR~s
100
IO
i.( S T
[ ( e-Y
1+ -
v = (c/n)(2n U = H(t)/Ho Y = vs(U-1 - 1)
+ 1)
and
no(t) = CakAk exp(-t/Ak) Equation 12 must be solved for the history H ( t ) that makes the right-hand side constant. This is most easily done by numerical integration over time, as a “marching solution.” A small increment At is selected, and by trial and error a value of AH is found to satisfy eq 12. The procedure is continued by “marching” forward in time to produce the function desired, H ( t ) . Several rheological parameters appear in this model and must be specified in order to perform calculations. The power law parameter n appears through the kinematic assumption, eq 10, and not through the constitutive assumption. The Wagner model introduces a constant c and a set of coefficients (a&). For the sake of illustration, an empirical set of the (ah,&) Will be introduced, based on a model fluid (1%Polyhall295 in 50/50 glycerine/water solution) studied by Shirodkar. The methods of determining these parameters is described in an earlier part of
A relaxation time A is defined here as N
N
The “zero-shear” viscosity 7jo is easily found t o be
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982
From eq 2, we see that k=l
Two asymptotic lines are shown in the figure. For sufficiently small loadings the squeezing rate will be so slow that the fluid will perform as a Newtonian fluid. Then one expects to find (from eq 3, after setting n = 1 and K = qo)
or
The other asymptote is what would be expected of purely viscous power law behavior, with no shear stress transient behavior accounted for. A t this condition one expects to find eq 3 to hold, from which it follows that
In using the approach outlined here, with these specific choices of dimensionless variables, it is necessary to keep in mind that the fluid model introduces parameter qo and A. The power law index n will enter the computation through the kinematic assumption discussed earlier, which led to eq 10. The power law parameter K never enters the computation, per se, but appears in the specific normalization used here if one wishes to examine the power law asymptote (eq 19). Thus, to produce this asymptote, one must examine the power law region of the Wagner fluid (the high shear rate region of Figure 3) and calculate specific values of n and K. In the computational procedure, using eq 12, the value of n to be used is taken to change with time as the instantaneous shear rate changes. (See Appendix for further discussion of this point.) However, in plotting the power law asymptote in Figure 4, the value of n (and K ) corresponding to the high shear rate region of Figure 3 is used. As might be expected, for fast squeezing (Tl12< 1)) deviations from purely viscous behavior are observed. If one used a power law model, with the high shear rate value of n, one would underestimate the half-time by an amount that could be an order of magnitude. This prediction is wholly consistent in direction and magnitude with Leider's experimental results. Appendix Evaluation of Shear Rate. The assumption which permits a relatively simple computation of the details of this squeezing flow is that the kinematics may be taken as those of a purely viscous power law fluid. Thus the strain measure used in the Wagner fluid model is given by eq 10. As Figure 3 shows, the Wagner fluid is not a power law fluid, except locally, i.e., over some limited range of shear rate. In manipulating eq 11 numerically, it is necessary to use some value of n. This is done in the following way. At each instant of time, in the integration displayed in eq 11,a value of n is used which corresponds to an average shear rate across the disk surface, and which is consistent with the local representation of the viscosity vs. shear rate data as a power law fluid. For a purely viscous power law fluid, the shear rate in this flow is given by 2n + 1 (-H) +Hf2 = T T r (A-1)
(-H)
-
437
fp+(l/n)
(A-2) in constant force squeezing. Thus the shear rate decreases as squeezing proceeds, since "1"
(A-3)
(Note that, by contrast, in constant speed squeezing the shear rate increases like 11112 as squeezing proceeds. Shirodkar et al. (1982) point out the important role of boundary kinematics in affecting the nature of squeezing flow.) An average shear rate, for use in calculation of n, is computed from (A-4) Using eq A-1, we find
Nomenclature ak = moduli in eq 8, dyn/(cm2 s) c = relaxation coefficient in eq 9 Fo = squeezing force, dyn 3 = dimensionless force, eq 14 h = function in the Wagner model H = separation between plates, cm K = power law coefficient, dyn/(cm2 5") mo = viscoelastic relaxation function, dyn/(cm2 s) n = power law index P = pressure dyn/cm2 r = radial coordinate, cm R = disk radius, cm s = R/Ho t = time, s T1jz= dimensionless half-time u = velocity vector, cm/s V = relative speed of disks, cm/s z = axial coordinate, cm Greek Letters ~ ' p= l codeformational strain tensor y = shear rate, s-l 1)
= shear viscosity, P
X = relaxation time T = stress tensor, dyn/cm2
Literature Cited Bird, R. B.; Armstrong, R. C.; Hassager, 0. "Dynamics of Polymeric Liquids"; Why: New York, 1976; Vof. 1, Chapter 9 and eq 5.2-50. Brindiey, G.; Davies, J. M.; Waiters, K., J. Non-Newtonian F/u/d Mech. 1978, 1 . 19. Co, A,;Bird, R. 6. Rheology Research Rept. No. 31, University of Wisconsin, Madison. 1974. Davies, M. J.; Walters, K.; Davenport, T. C., Ed. I n "The Rheology of Lubricants"; Wiiey: New York, 1973; p 65. Grimm, R. J. AIChE J. 1978, 24, 427. Harnoy, A.: Phiiippoff, W. Am. Soc. l u b r . Eng. Trans. 1977, 19, 301. Horowitz, H. H.; Steidier, F. Am. SOC. Lubr. Eng. Trans. 1980, 3, 124. Leider, P. J. Ind. Eng. Chem. Fundam. 1974, 13, 342. Middleman, S. "Fundamentals of Polymer Processing"; McGraw-Hili: New York, 1977. Oliver, D. R. Appl. Sci. Res. 1979, 35, 217. Reiner, M.; Hanin, M.; Harnoy, A. I s r . J. Techno/. 1979, 7, 273. Rosenberg. R. C. SOC.Automot. Eng. 1975, 84, 1698. Shirodkar, P. Ph.D. Thesis, University of Massachusetts, Amherst, MA, 1981. Shirodkar. P.; Middleman, S. J. Rheol. 1982, 26, 1. Shirodkar, P.; Bravo, A.; Middleman, S. Chem. Eng. Commun. 1982, 14, 151. Tichy, A. J.; Winer, W. 0. J. Lubr. Techno/. 1978, 100, 56. Wagner, M. H. Rheol. Acta 1977. 16, 43. Wagner, M. H.; Stephenson, S. E. J. Rheol. 1979, 2 3 , 489.
Receiued for reuiew September 14,1981 Accepted July 19,1982
