Ind. Eng. Chem. Res. 1994,33, 871-877
871
PROCESS DESIGN AND CONTROL A Model for Investigating Mixing Times in Circulation Mixers Arthur Mazer Department of Mathematics and Science, Singapore Polytechnic, Singapore 0513
A model is presented for calculating tracer concentrations in a mixer with a well-defined circulation path. T h e model is applied to the airlift reactor. It is shown that the outcome corresponds well with the experimental data given in the literature. However, the model points t o a discrepancy in the interpretation of the data. This discrepancy is the time interval between peak concentrations. 1. Introduction
There are many mixing processes in which there is a well-defined circulation path in the flow pattern. The chemical engineering literature has numerous studies of such processes on a case by case basis. In Nagata (1975a) one can find an analysis of helical ribbon agitators which correlates mixing times to circulation times. The investigation of Fields and Slater (1983) correlates circulation times to mixing times in an airlift reactor. There are other mixers in which a circulation pathway exists such as stirredtank reactors. It is the intention of this paper to present a generic model for any system that has a well-defined circulation loop and apply the model to the airlift reactor. The airlift reactor is chosen because it is possible to analytically calculate the circulation time profile which is fundamental to the model. Nevertheless because of the generic nature of the model, much can be learned about other circulation mixing processes as well. The work of Fields and Slater is influenced by the papers of Taylor (1953) and Lievenspiel (1958). Taylor first introduced a model for longitudinal dispersion in a pipe in which concentrations of a tracer are distributed convectively by the flow of the fluid in the pipe and by diffusion as well. A diffusion equation is derived to describe longitudinal dispersion. The diffusion constant is not the usual molecular diffusion constant that is found in other diffusion processes. I t is a quantity that depends upon the fluid‘s velocity profile, the pipe’s radius and the molecular diffusion. For this reason Taylor calls it the virtual diffusivity. Following Taylor’s work, Levenspiel (1958) examines mixing in a pipe that is closed up into a loop. He applies Taylor’s analysis to derive a diffusion equation in the longitudinal direction along the circulation pathway. The solution of the diffusion equation represents the longitudinal concentration of a tracer. Solutions for the resulting partial differential equation are presented; however, no expressionis presented for the virtual diffusion term. Fields and Slater (1983) apply Levenspiel’s model to an airlift reactor. They determine the virtual diffusion constant of Levenspiel’s model by matching experimental output from an airlift reactor with Levenspiel’s solution. In their paper the virtual diffusion constant is called a dispersion constant in order to draw a clear distinction between it and molecular diffusion. A classical text which presents results for mixing in helical ribbon agitators is one by Nagata (1975a). Fluid
is forced to circulate from the top of the tank to the bottom and then back to the top, but the path is not a simple loop. Nagata presents an argument which states that the ratio of the mixing time to the average circulation time is around 3. Experimental results usually, but not always, confirm Nagata’s argument. Most mixing times reported for a helical ribbon agitator lie between 2.6 and 4 times the average circulation time (Nagata et al., 1975b;Coyle et al., 1970; Carreau et al., 1972; Takahashi et al., 1988). These approaches to the problem of mixing in circulation mixers have some drawbacks. One drawback is that there is no unifying theory to treat the general case of circulation mixing processes; instead the problems are tackled for specific situations. Within the special situations there are inadequacies. Levenspiel’s dispersion constant has no analytical expression and can only be arrived at experimentally. Furthermore when the model is applied to the airlift reactor there is a discrepancy which has been overlooked by Fields and Slater. This discrepancy will be discussed below. Nagata’s argument while often useful is not always true, and his methods do no indicate why this is so. The approach in this paper overcomes these drawbacks. Mathematicians have adopted an axiomatic framework for investigating circulation mixing processes. A rigorous foundatioq for studying such systems has been developed by Bowen and Ruelle (1975). At present, systems must satisfy an extreme set of conditions in order to apply the approach of Bowen and Ruelle. Even with these conditions the analysis is very difficult. Current research is focused on examining the asymptotic behavior of such systems; this limited investigation is still far from being complete. Nevertheless the approach is a good starting point for developing a model which is easier to analyze. In Mazer (1992) a model for investigating mixing processes in circulation flows is presented. This model assumes a hypothesis which is even more severe than the conditions set by Bowen and Ruelle. The advantage is that the model becomes much simpler and is amenable to analysis. Questions which are presently unanswerable using Bowen and Ruelle’s approach can be addressed using the model. In particular the model can be solved to yield mixing times. This model will be referred to as the loop model. (In Mazer it is called the circulation feedback model.) The objective of this paper is to present a framework for incorporating actual mixing systems into the loop model and applying the result to the airlift reactor. This is carried out as follows. In section 2 the loop model is presented.
0 1994 American Chemical Society osss-5aa5~~4~2~33-oa7i~o4.5~1o
872 Ind. Eng. Chem. Res., Vol. 33, No. 4, 1994
In section 3 it is shown by example how to incorporate a specificflow pattern into the loop model. The flow pattern in the airlift reactor is also discussed and incorporated into the model. The model's results for the airlift reactor are presented in section 4. The discussion in section 5 indicates the disagreement in the results of our model and the Fields and Slater approach. The disagreement lies in the time interval between successive peaks of tracer concentration measured at a designated location in the loop. 2. The Loop Model
The loop model can be used to calculate concentrations of a substance within the mixer when the initial concentration is known. The model itself is a simplification of mixing systems known as suspended flows. A brief description of the suspended flow is presented before introducing the loop model. Assume that a mixing process has a well defined circulation path. By this it is meant that there is a cross section in the mixer through which all the fluid flows in one direction. The fluid is perpetually recirculated through the cross section, and the circulation time profile over the cross section does not change. The cross section is called the base space. In general the flow pattern of a fluid as it recirculates to the base space is quite complicated. Trajectories can twist about one another, and the fluid velocities are not constant. The suspended flow is an imaginary flow that is derived from the actual flow. The trajectories of the actual flow are straigtened out, and velocities are set at a constant in such a way as to preserve the characteristics of the actual flow. This is done as follows. Assume that the base space is a flat two-dimensional surface that has a planar coordinate system on it denoted by (x1,xz). Associate a circulation time to each point on the base space, ct(x1,xz). The function ct(x1,xz)has been coined the roof function, or the roof. Its physical meaning is that a fluid element which crosses the base space at the point (~1~x2) returns to the base space in return time ct(x1,xz). The motion of a fluid element in the suspended flow is described using the roof function. First one graphs the roof function above the base space. A fluid element which startsat the base space at a point (x1,x2)proceedsvertically toward the roof with unit speed. It then strikes the roof at the point (x1,x2,ct(x1,x2))and is instantaneouslyreturned to the base space at a new location which is dependent upon the original flow. This cycle is repeated indefinitely. See Figure 1. In what follows, the vertical coordinate above the base space is denoted by y. One point to address is the relationship between the y coordinate in the suspended flow and time. They value of the suspended flow indicates the time that has elapsed since a fluid element has passed through the base space. There is a natural correlation between sets in the suspended flow and sets in the actual mixer, which is helpful in interpreting the loop model. Notice that for any constant, k, the set y = k in the suspended flow corresponds to the set of all points which take k units of time to reach when starting from the base space. Henceforth it is assumed that volume measurements are taken in units so that the volume of the all the fluid in the mixing tank is 1. Let w(y) denote the volume of fluid that circulates through the base space and returns to the base space in time greater than y. This function will be of interest to us later on. Notice that w ( y ) is proportional to the volume flux of fluid through the set
Figure 1. (a) The trajectory of a fluid element aa it is recirculated in an actual mixing process. (b)The correspondingtrajectory for the suspended flow.
on the base space in which ct(x1,xz) is greater than y. If time units are measured in such a way that the average circulation time is 1, then w(y) is equal to the volume flux of fluid through the set on the base space in which ct(x1,x2)is greater than y. This provides a means to calculate W(y).
A simplification of the suspended flow which makes it amenable to some analysis is to assume that upon striking the roof, a fluid element is uniformly distributed over the base space. Since the flow proceeds vertically toward the roof at uniform speed, there is perfect homogeneity in every cross section, y = constant, and inhomogeneities are present only in the y direction. Concentrations of any substance can then be considered only as functions of y, and a one-dimensional model which determines the concentrations can be derived. This derivation was carried out in Mazer (1992). We present a description of the model. A consequence of assuming that the concentration of a tracer is uniform at the base space is that points in the base space cannot be distinguished by tracer concentration. Therefore the geometry of the base space becomes irrelevant. In particular the base space can be reduced to a single dimension. Henceforth, the base space is considered to be the unit interval, [0,11. A point in the base space is denoted by x . A suspended flow can be created by assigning a roof function, ct(x), to the base space. Later it will be shown how to correlate a roof function to return times in a mixer, but for now we assume that the roof is given and satisfies the following properties: (i) ct(x) is differentiable, and ct'(x) is not equal to 0. (ii) ct(x) is strictly decreasing (if x 2 > xl, then ct(x2) < ct(x,)). (iii) Jolct(x) dx = 1. Property i can be relaxed to piecewise differentiabilty,
Ind. Eng. Chem. Res., Vol. 33, No.4, 1994 873 and ct'(x) can be 0; however, this necessitates complicating the notation and obscuring the main features of the model, so these cases will not be addressed. Property ii does not restrict the type of circulation time profiles over the base space, since this can be satisfied by labeling the x's in a manner so that the function is decreasing. Property iii is satisfied provided that the time unit is given in terms of the average circulation time (Le. 1 time unit = 1 average circulation time). In this paper one time is equal to the average circulation time. The model describes the evolution of the concentration of a substance that flows between the base space and the roof in a manner similar to that of the suspended flow. However, in the loop model upon striking the roof any substance is returned to the base space and uniformly distributed over the base space. Let f(xy,t) be the concentration of a substance at the position (x,y) and at time t. Then the evolution off under the above assumptions is as follows.
-
f(x,y,t) = f(x,O,t - y) for t y 1 0
-
(1) (2)
f(x,O,t) = S,lf(s,cW,t) ds
(3)
Equations 1and 2 result from the vertical transport of the substance between the base space and the roof. Equation 3 shows that the concentration a t the base space is an average of the concentrations over the roof. This occurs because of the assumption that the substance is homogeneously distributed over the base space upon striking the roof. Notice that the concentration at the base space is independent of x as shown by eq 3. Also, the concentration at any point in the domain comes up from the base space as shown by eq 1. So, inhomogeneities in x are eliminated at the base space and then the concentrations which are independent of x are propagated vertically upward. Since this is the case, it is reasonable to disregard inhomogeneities in x altogether and consider initial concentrations that vary only vertically. Because the concentrations are independent of x , the two-dimensional model can be collapsed onto the y axis. Let f@,t) be the concentration of a substance. The following equations are the one-dimensionalanalogs of eq 1-3. These equations are the loop model. They are derived by a direct application of the inverse function theorem and a change of variables on eq 3. f(u,t)= f(0,t-Y)
t -Y 2 0
(4)
f(y,t) = f(u - t,O)
y - t 10
(5)
dy
(6)
The function m(y)is referred to as the feedback measure. It is the negative of the derivative of the inverse of the roof function. More formally we have w(y) = ct-'(y)
=1
m(y) = -dw/dy
Y 1r(1)
y I r(1) r(1) Iy Ir(0)
4
x
Q Q
0: 8 0 ' ' 7.00' ' '1' 1 h0
' -d x
f(x,y,t) = f(x,y t,O) for y - t 1 0
f(o,t)= J$rf(u,O
-
(7)
(8)
where ct-'(y) denotes the inverse of the roof function (i.e., ct(x) = y implies ct-l(u) = x ) . The derivation of (6) is given in Appendix 1. There it is shown that the function w(y) is the same as was presented above. Figures 2 and
= u(y)
= 1/4yz
Figure 2. Plot of the roof function for the hypothetical mixer discussed in section 3.
Q Q
Y
Figure 3. The corresponding weight function for Figure 2.
3 are sketches of the roof function and weight function for a process that is later described. Before proceeding, we consider the circumstances for which the model is expected to be accurate. No mixer can perfectly homogenize the base space. However, if mixing in the base space is much faster than around the circulation loop, the simplification is reasonable and the model can be applied. This is true whether the mixer is a stirredtank reactor, a helical ribbon agitator, an airlift reactor, or any other process in which a circulation path is evident.
3. Calculation of the Weight Function, Two Examples In this section, it is shown how to calculate the weight function and feedback measure for a hypothetical mixing process and the airlift reactor. In the hypothetical mixer, fluid circulates through a loop with a circle for a cross section. The cross-sectional radius of the circle is much smaller than the path length along the central longitudinal axis, so variations in path length for fluid flowing around the loop is negligible and will be ignored. With this assumption we consider the loop to be of length k. Also at a given cross section of the loop there is a propeller which rotates along the loop's
874 Ind. Eng. Chem. Res., Vol. 33, No. 4, 1994
longitudinal axis. The propeller is assumed to scramble the fluid in a completely random manner such that concentrations of a tracer are uniform at the cross section where the propeller lies. Away from the propeller the flow is laminar. It is assumed that the time that a fluid element spends near the propeller is negligible in comparison to the time in the laminar region, so the roof function can be calculated from the laminar flow. The base space is taken to be the cross section where the propeller lies. The radius of a point in the cross section is denoted by r. The circulation time is a function of the radius and is denoted by either ct(r) or ct(x1,xz). The velocity profile through the laminar region is u(r) = (Urn/ 2)(1 - (r/R)2),where R is the radius of the tube and urn is the mean fluid velocity. Time and spatial measurements are taken in units so that the average circulation time = 1 and the volume of all the fluid in the mixer = 1. A consequence of this normalization is that k / U m = 1. The weight function is calculated according to the formula given in Appendix 1. To use the formula some notation must first be presented. Let A, = ( ( x 1 , x z ) in the base space such that ct(xl,x2) > y). The weight function w(y) is the volume flux of fluid through the set A, in one time unit. Since it is assumed that the fluid flows through the base space perpendicularly this becomes w(y)
= JAyu(x1,x2)dx, dx2 = Q(r)2nr
dr
The value r, denotes the radius that satisfies the following relationship. ct(ry)= y where y > min, ct(r) = ct(0) = ' I 2 (10) ry = o
y Ict(O) =
(11)
To evaluate the integral in (9), it is necessary to calculate ru. For the case that y Il / 2 , r(y) is given by (11). For the case that that y I 112, ry is found by inverting the roof function in (10). ct(ry) = k/u(r,) = k/[2urn(l- (ry/R)2)l = 1/[2(1- (ry/R)2)l= Y
= 1,y I
(16)
Figure 3 is a graph of the weight function of (16). Figure 2 is a graph of the corresponding roof function. The feedback measure is obtained by differentiating the weight function in the region y > r(0). This is defined in eq 8.
Mixing in an airlift reactor occurs in a U shaped pipe and a channel which connects the top of the U, so the system is closed. Air is continuously pumped into a fluid a t a point about halfway down one side of the U. The air forms bubbles which percolate to the channel at the top of the U and escape into the atmosphere. The rising bubbles cause the fluid to circulate into the channel and back through the U in a well-defined circulation path. Away from the bubbles, the fluid flow is laminar. Around the bubbles the flow is very complicated. We find the weight function for an airlift reactor with the following properties. i. All the fluid passes through the bubble region in a fixed time which is designated by s. Because the total circulation time is normalized, s must be less than 1. ii. The transition zone between laminar flow and the bubble region is negligible and will not affect the weight function. iii. The channel across the top of the U has a cross section that is almost a circle. A slit in the top allows the air to escape. iv. Concentrations of any tracer are uniform in a cross section of the channel just beyond the bubble region. We take this cross section as our base space. With these assumptions, it is only necessary to slightly modify the weight function calculated for the hypothetical mixer. The modification is in eq 12 which becomes ct(r) = k/(u(r) + s)
(18)
Carrying out the above calculations with this modification gives the following weight function. (12)
w(y) = (1- s)2/4(y - s)2
Inverting (12) so that ry can be expressed as a function of y gives
(
r , = R l - - 2ty)li2 Placing the values of ruand u(r) into the integral of (9) gives the following expression for w(y).
=1
y 1 ( 1+s)/2
y L (1+ s)/2 (19)
The plots in Figure 4 are of the roof function corresponding to w(y) for s = and s = 3/4. Notice that as s becomes larger the roof function flattens indicating more uniform circulation times. Another feature to notice is that for smaller s the roof approaches the y axis much slower yielding a wider boundary layer. Similarly the boundary layer when s = 0 is wider than the boundary layer when s = 114.
= nurnR2, y 5 ' I 2
(14)
The integral (9) when ry = 0 is also the total volume of fluid flowingthrough the base space in one time unit. Since measurements are taken in units that normalize the volume of fluid as well as the average circulation time, this gives Jocu(r)2nr dr = nv,R2 = 1 Substituting the identity 15 back into (14) gives
(15)
4. Results
This section presents numerical solutions to the loop model with w(y) as given in (19). The numerical solutions are obtained using the discretized version of the loop model which is presented in Appendix 2. Figures 5-7 are plots of the normalized concentration of a tracer at the base space vs time for the three different cases, s = 0, s = l/4, and s = 314. By normalized concentration it is meant the concentration divided by the steady-state concentration; so the values in each plot
Ind. Eng. Chem. Res., Vol. 33, No. 4,1994 875
= a
.: % 08
6.00
0.20
0.40
0.60
0.80
1.00
1.20
X
4.00 ttrno,
4.00
ttrno.
Figure 4. The roof functions for the airlift reactor with s = l / d and s = 3/4. Notice ass becomes larger, the roof function flattens out and the boundary layer becomes thinner.
2.00
2.00
E
6.00
E
6.00
8.00
= 1/4
Figure 6. Plot of the normalized concentrations at the base space for the model of the airlift reactor with s = I/,.
8.00
= 0
ttrno.
E
= 3/4
Figure 5. Plot of the normalized concentrations at the base space for the model of the airlift reactor with s = 0.
Figure 7. Plot of the normalized concentrations at the base space for the model of the airlift reactor with s = 3/4.
converge to 1. Each plot is with respect to time in which one unit of time represents the average circulation time. The initial conditions for the plots are
Figures 5-7 only indicate the degree of mixing at the base space. To indicate the degree of mixing throughout the mixing tank, the following function is used.
f(y,O) = 1, 0 < y 50.3
= 0, y = 0 and 0.3 < y
There is an obvious effect on the outcome as the time spent in the bubble region becomes longer. As s becomes larger the roof function flattens and mixing at the base space is slowed down. For s = 0, the concentration is within 95% of its final value at t = 2.4. For s = V 4 , the corresponding time is 3.8. For s = 314 the corresponding time is beyond 6. Also, notice that the time between peaks changes as s changes. The time between peaks for s = 0 is 0.6; for s = I/, it is 0.8 and for s = 314 it is 0.9. The shape of these plots is strikingly similar to the data that is shown in the paper of Fields and Slater. In fact the shapes are nearly indistinguishable. The same goes for Levenspiel's model with one major difference; the distance between peaks for Levenspiel's model is always one.
In the above equation, feeindicates the steady state value of the concentration, f. The initials GM are chosen to indicate global mixing throughout the tank. GM(t) approaches 0 as time gets longer, and a value close to 0 indicates near homogeneity throughout the tank. The above integral can be approximated by discretization. The values f(y,t) are obtained by using eq 4 and 5. Figures 8 and 9 are plots of GM(t) versus time for the cases s = 0, s = l/4, and s = 3/4. The curves for the cases s = 0 and s = are nearly indistinguishable. For both cases GM(t) first reaches 0.05 at t = 2.9,indicating that mixing is 95% complete at t = 2.9. By coincidence, t = 2.9 is also the cross-over point between the two curves. When t < 2.9,GM(t) with s = 0 is smaller than GM(t) with s = l / 4 , however, for t > 2.9,the opposite is true. For the case s = V 4 , the value GM(t) is still quite large after 6 circulations, GM(6) = 0.16.
876 Ind. Eng. Chem. Res., Vol. 33, No. 4, 1994
case s = 0 than the case s = l/4, the global mixing is not faster. In fact after t = 2.9 the cases = l/4 is better globally mixed than the case s = 0. This can be explained by the wider boundary layer when s = 0. The model shows that a large boundary layer slows mixing. The results of this model in a pipe which forms a loop are different from those predicted by Levenspiel's model. The peaks in concentration according to Levenspiel's model are always 1 circulation time unit away from one another. Fields and Slater (1982) accept this result and use the distance between peaks as a measurement of the circulation time. Our model predicts that the time between peaks is dependent on the velocity profile. Relatively uniform circulation times with their corresponding flat roof functions result in long intervals between peaks; whereas large variations in the circulation time result in shorter intervals between the peaks. 6. Conclusion t Lme
Figure 8. Two plots of the global mixing time in the model of the airlift reactor with s = 0 and s = l/A.
i\
The loop model has been presented along with a method to calculate the weight function so that the model can be applied to actual mixing processes. Results from this model disagree with those presented in Fields and Slater with regard to the time between peakconcentrations. Since this parameter is easy to measure, it along with the mixing time makes a good test case for the model. Acknowledgment The author would like to thank Dr.Reginald Tan of the chemical engineering department at National University of Singapore for introducing him to previous work that has been done in the study of mixing in circulation flows. In addition, his comments in making this paper suitable and relevant to the chemical engineering community are appreciated.
?m
Appendix 1
8 6.00
4.00
2.00 tlrne,
a
6.00
8
8.00
= 3/4
Figure 9. Plot of the global mixing time in the model of the airlift reactor with s = 3/4.
It is first shown that (3) yields (6). The function f in (3) is independent of the variable x , so this variable is eliminated. Performing a change of variables on (3)with y = ct(s) and using the inverse function theorem give the following.
5. Discussion
The loop model is different from other models that describe mixing in circulation mixers in several respects. First it is a generic model which describes mixing in any circulation mixer regardless of the physical attributes of the mixer. As long as a weight function can be extracted the model can be used. This is in contrast to other approaches which only apply to a specific mixer. The reason that a generic description is feasible is that mixing in all circulation mixers is caused by a generic mechanism; a velocity gradient causes the tracer to spread out along the circulation path. The loop model focuses on this mechanism. This generic mechanism for mixing in circulation mixers causes a generic mixing time of around 3 for many circulation mixers. This explains why the result for the cases s = 0 and s = l/4 are in good agreement with Nagata's result, even though Nagata presents his argument for a different setup. The case s = 3/4provides an explanation for why Nagata's rule fails even when it is appliedtomixing processes for which it is intended. A flat roof function slows down mixing considerably. Another point to note is that while the concentration at the base space approaches homogeneity faster for the
It is next shown how to calculate the weight function. If y < ct(l), it is obvious from Figure 3 that w ( y ) = 1.We consider only the case when y 2 ct(1). First some notation is given. As before let A, be the set on the base space described as follows: A, = ( ( x 1 , x z ) such that ct(x1,xz)> y). There is a natural correspondence between any line, y = constant and sets in the mixer such that the area of a region in the two-dimensional model corresponds to a volume of fluid in the mixer. This correspondence is as follows. Let y = constant correspond to the set of all the points in the mixer that the fluid passing through the base space a t time t = 0 would reach at time t = constant, prior to returning to the base space. With this correspondence, the volume of fluid passing through the set A, in one unit of time is given as follows.
Ind. Eng. Chem. Res., Vol. 33, No. 4,1994 877
necessitates truncating the actual weight function when return times are arbitrarily long. =
E&
1
dP
Appendix 2 A solution to the loop model for a given initial concentration f(y,O) is a function f(u,t) that satisfies eqs 4-6. Analytic solutions to the model are not easily obtained even for the simplest of weight functions. Numerical mfthods must be used. One approach for developing a numerical solution is to approximate the loop model by a discrete analog and generate a computer solution. In this appendix, the discrete analog to the loop model is presented. Let { y j = j d ) be a family of points in which j assumes integer values from 0 to T and d is a small constant. Also, let f ( y j j d ) be the concentration of a tracer a t the point yj and at the time j d . Then the discrete analog to the loop model is given by the following equations
Literature Cited Arnold, V. I.; Avez, A. Ergodic Problems of Classical Mechanics; W. A. Benjamin: New York, 1968. Bowen, R.; Ruelle, D. The Ergodic Theory of Axiom A Flows. Invent. Math. 1975,79,181-202. Carreau, P. J.;Patterson, I.; Yap, C. Y. Mixing of ViscoelasticFluids with Helical Ribbon Agitators. Can. J. Chem. Eng. 1972,54(31, 136-142. Coyle, C. K.; Hirschland, H. E.; Michel, B. J.; Oldshue, Y. J. Mixing in Viscous Liquids. AZChE J. 1970,16,903-906. Fields, P. R.; Slater, N. K. H. Tracer Dispersion in a Laboratory Air Lift Reactor. Chem. Eng. Sci. 1983,s (4),647-653. Levenspiel, 0.Longitudinal Mixing of Fluids Flowing in Circular Pipes. Znd. Eng. Chem. 1958,50 (3),343-346. Mazer, A. A Model for Investigating Mixing Rates in Suspended Flows. Physica D 1992,57(122),226-237. Nagata, S.Mixing, Principles and Applications; Wiley: New York, 1975a. Nagata, S.; Yanagimoto, M.; Yokoyama, T. A Study of the Mixing of High Viscosity Liquid. Kagaku Kogaku 1975b,21,276286(in Japanese). Ottino, J. M.;Leong, C. W.; Rising, H.; Swanson,P. D. Morphological Structures Produced by Mixing- in Chaotic Flows. Nature 1988, 33 (6172),419-425. Ruelle. D. Resonances for Axiom A Flows. J.Differential Geometrv 1987,25,99-116. Takahashi, K.; Iwaki, M.; Yokota, T.;Konno, H. A Study of Circulation Time For Pseudoplastic Liquids in a Vessel Agitated by Helical Ribbon Impellers. Proceedings of the 6th European Conference on Mixing; Pavia, Italy, 1988. Taylor, G . I. Proc. R. SOC.London Ser. A 1953,219, 186.
Received for review June 11, 1993 Revised manuscript received November 17, 1993 Accepted November 29, 1993.
Computer code can be written to directly solve the discrete model provided that T is a finite value. This
8 Abstract published in Advance ACS Abstracts, February 1, 1994.
