Dispersion and Ultimate Separation in the Parametric Pump Richard G. Rice Department of Chemical Engineering, liniversity of Queensland,St. Lucia, Queensland, Australia
A steady-state diffusion model for the closed, direct-thermal mode parametric pump i s presented. The model considers the ultimate separation obtainable following the transient when the time-average flux tends to zero. The theory was developed for uniform pores or tubes and shows that the maximum concentration difference between reservoirs i s directly proportional to pore length and equilibrium concentration difference and i s a quadratic function of velocity amplitude. The theoretical dependence of Peclet number indicates the existence of an optimum frequency. For sufficiently large Valensi and Schmidt numbers, the ultimate separation i s shown to depend on the square root of frequency and i s independent of Schmidt number.
I t is rare indeed when an entirely novel separation device enters the literature of chemical engineering, but the concept of parametric pumping developed by Wilhelm and coworkers (1966) is just such a device. Since then, research in this area has proceeded along historically predictable lines: the technique has been experimentally verified but t’he theory is inadequate for design purposes. The effort in the current work is to present a theory which allows prediction of final steadystate concent’ration differences (ultimate separation) in a parametric pump. Essentially, parametric pumping is a dynamic separation principle which couples adsorption-desorption mass transfer with a purely periodic fluid motion. I n the simplest case, we imagine a vertical adsorbent tube containing a fluid, one component of which can adsorb onto the wall. Xow suppose the fluid is caused to move upward and, simultaneously, the adsorbent capability of the solid surface is increased (for example, by decreasing the surface temperature). The adsorbed solute will be selectively retarded, while the solvent moves upward unimpeded. When the fluid changes direction and moves downward, the adsorbing affinity of the wall is simultaneously decreased (by increasing the temperature), thus releasing the solute and allowing i t t o move downward with the fluid motion. When reservoirs are attached to t h e ends of t,he adsorbing column, enrichment of the solute will occur in the lower reservoir, while depletion occurs in the upper reservoir. Although temperature is an obvious choice for accomplishing the adsorption-desorption step, the technique is not limited to this driving force alone. Any method for producing a synchronous adsorption-desorption step will do, including the variations of liquid p H and application of electrical or magnetic fields. The current theory does not preclude any of the above methods of producing a synchronous interphase flux, but simply notes there exists a periodic conceiitration boundary condition between the solid adsorbent and the fluid. Most operating parametric pumps have been of the packed bed configuration. The current work considers a single, continuous uniform pore in this bed. The pore is modeled as a simple tube with adsorption-desorption occurring a t the tube wall, synchronous with a purely periodic fluid motion. For a vertical system with aii adsorbing tube connecting two reservoirs, the phase difference (180’ or 0’) between interphase flux and velocity fields becomes important 406 Ind.
Eng. Chem. Fundam., Vol. 12, No. 4, 1973
if density driven flow is substantial or if the velocity profile seriously lags the applied fluid driving force. Aimore complete overview of the direct thermal and recuperative modes has been presented (Wilhelm, e f al., 1966; 1968). Separations of toluene from n-heptane have been reported (Sweed and Wilhelm, 1969; JF‘ilhelm, et al., 1968) using the direct thermal mode, and separations of sodium chloride from water have been accomplished using both the recuperative mode (Rolke and Wilhelm, 1969) and the direct mode (Sweed and Gregory, 1971). Separation of gas mixtures in a packed bed of adsorbent has also been reported (Jenczewski and llyers, 1970). The use by Wilhelni and coworkers of a packed bed configuration for the parametric pump has led to fundamental modeling difficulties. For example, Sweed arid Gregory (1971) have presented numerical solutions (using the so-called STOP-GO algorithm) to a set of transient hyperbolic transport equations, but they essentially curve-fitted the critical (and unknown) parameter, mass transfer coefficient, to match the transient data. Although this effort s h o w a level of qualitative and quantitative correctness, the lumping of all destructive effects into a single transport, coefficient may not be justifiable, especially with regard t o induced dispersion (Horn and Kipp, 1967). Pigford, el al. (1969), developed a very interestiiig “equilibrium theory” for the parametric pump which is compact and easy to use. This model, however, does not include interphase resistance or axial dispersion, both of which may be considered destructive effects. hlthough their comparison of theory and experiment was very good for short times, the equilibrium theory cannot predict the ultimate steady-state separation obtainable. The current work analyzes the situation when the ultimate separation has been reached, that, is, when the time-average concentrations in the reservoirs no longer change. Contrary to the assumptions of Svieed, molecular diffusion can be shown to be of crucial importaiice for long times (ultimate separation), since it determines a crit,ical operating frequency to maximize separation. Formulation of the Problem
This work will be concerned with a cloaed, direct-mode parametric pump. .Ispreviously mentioned, the coupling of
oscillating thermal-velocity fields can lead t o separation of chemical species. A solvent containing a single adsorbingdesorbing species is made to oscillate in a tube connecting two reservoirs. The shape of the velocity profile through a halfperiod is indicated in Figure 1. The tube connecting the reservoirs has its inside walls coated with a n adsorbent or an ion-exchange resin. The coated-wall configuration is mathematically tractable and, more importantly, can lead to a fundamental understanding of the underlying separation mechanism in the pore structure of the commonly used packed bed configuration. The tube is surrounded by a jacket through which a heat transfer medium is pumped. The jacket fluid temperature is varied in a periodic fashion and it will be assumed that the adsorbent temperature responds instantaneously to these variations. For the current model, we may consider t h a t the temperature is caused to decrease when the fluid moves upward and increase when the fluid moves downward. Experimental transient data (Sweed and Wilhelm, 1969) have shown that reservoir concentrations, after a long time, appear to tend toward a steady value; hence, t h e timeaverage flux teiids to zero. It is this zero-flus condition which will provide the basis for calculating ultimate separation in the parametric pump. Transient models which include a destructive effect have also predicted that a n ultimate steady state should be attained (Sweed and Wilhelm, 1969). The fluid motion is caused by a purely sinusoidal pressure gradient imposed by way of a piston as shown in Figure 1. Under these conditions the complex domain rectilinear velocity can be shown (Harris and Goren, 1967) to be given by
I Upper Reservoir At Concentration I
1
Adsorbent
Alternating
Tube
Hot a n d C o l d Jacket
Coated
Wall
Fluid
Piston
~
I
Pulser
L o w e r Reservoir A t Concentration
cH
Figure 1. Schematic showing a simulation of the uniform pore model
where
p+l
= 1/-ii~Vn;
5
=
r/ro
As usual, physical significance can only be attached to the real part of this velocity. Since the sum of a function and its complex conjugate is real, the real velocity can be represented by Re(v) = C+l(€)e f W 1 U-l(t)e-"f (2)
+
The Mass Transfer Theory
where
C+,(~)/AW= M+l[%
ture is used to cause interphase flux. Furthermore, we shall invoke the n-ell-known esperimental property that the timeaverage compositions a t the ends of the tube (pore) are constant (steady-state). Hence, we simply state that the masimum separation has been reached, and it is the purpose of this analysis to predict the ultimate concentration difference between upper and lower reservoirs.
- 11:
The convective-diff usion equation will be solved in the complex domain, attaching physical significance only to the real part of the composition and axial flus. For large time, we assume that the form of the solution is composed of a steady and a purely siiiusoidal unsteady part. 1somewhat general functioiiai form of this type would be
C(z,r,t)
=
C,(x,r)
+ C+l(x,t)e"t + C-l(xjr)eiw'
(6)
We select the simplest form of this type, namely
C(x,r,t) = C,(x)-tC+l(r)e'"'
It is seen that ~ 1 1 , depends ~ only on dimensionless frequency (Valensi number). For constant physical properties and simple rectilinear motion, the convective-diffusion equation for the adsorbingdesorbing species is
Average physical properties will be assumed, even though the fluid can sustain sizeable temperature variations, if tempera-
+ C-l(r)e-Lw'
(7)
The last two terms represent the real, purely sinusoidal contribution to composition and C - l , C+l are complex conjugates. Higher harmonics are not included and it will be shown later that these make no net contribution to the expression for average steady-state flus aloiig the tube asis, provided velocity is purely sinusoidal. The solution of the convective-diffusion equation using a form esactly aiialogous to eq 7 has been experimentally verified by Harris and Goreii (1967) and by Rice and Eagleton (1970). These researchers studied mass transfer in a tube connecting two reservoirs of different concentration in the presence of laminar flow oscillations. 111 these works, no mass transfer occurred a t t'he n-all as Ind. Eng. Chem. Fundam., Vol. 12, No.
4, 1973
407
distinguished from the current problem. When eq 7 along with the velocity from eq 2 are inserted in eq 5, there results
Differentiating the expression for C,(z) along with (3) into (8) gives
-+--
(9)
+
Equation 9 is linear with the integral given by C,(x) = ax b. The boundary condition a t the wall must be compatible with the functional form of eq 7 , and in addition, must reflect a forced periodic adsorption taken about a n average steady concentration. The linear functional form of C,(x) must be equally valid at the pore wall. Considering these circumstances, the wall condition is taken as
d2C+i dtZ
C + I ( ~= ) AC*/2
+b
The amplitude of the average difference in equilibrium adsorption at the maximum and minimum wall temperatures is designated by AC*. It is seen that the pore wall boundary condition depends linearly on axial distance, and superimposed on this is a purely sinusoidal driving force of amplitude AC*. The driving force, AC*, is taken as constant or more specifically as an average driving force. This driving force for separation, AC*, can only be defined exactly from a knowledge of the adsorption isotherm, i.e.
c* = f(q,T) where C* is the fluid composition in equilibrium with the composition q a t temperature T. For a periodic temperature amplitude AT, the equilibrium driving force can be obtained from the isotherm as
where Aq is the change in solids composition owing to the change in temperature AT. I n the current work, qo is taken as the solids composition in equilibrium with the initial charge of fluid a t the steady system temperature T,.A more exact representation would be to use the solids composition in equilibrium with C,(s) in the above expression hence requiring AC* to depend on asial distance s. If this were done, the functional form given by eq 7 would no longer be valid and one must then attempt a solution using the more general form, eq 6. As a first approximation, we define AC* as a n average system parameter which depends on a suitable equilibrium relationship. The constants a and b can be determined b y recalling the condition t h a t the time-average compositions at the pore ends are constant
(13)
(14)
and in addition symmetry requires
dE
(C(Z,TO,~))= ax
1 dC+i d€
E
A similar expression for C-I(€) can be deduced, but is superfluous since this function is simply the complex conjugate of C + I ( ~ )By . comparing eq 7 with the wall condition (10) the boundary value for C + l ( f )is deduced to be
-dC+1 =o the time average of which is simply
and inserting this
(E=0)
The solution to eq 13 subject to boundary conditions (14) and (15) is
The superposition of solutions used here will obviously not predict the exact concentration profile existing in a parametric pump. It is well known that the adsorption-desorption step is highly nonlinear and depends strongly on bulk fluid compositions, not t o mention the complicated temperature dependence. It should be emphasized t h a t the absolute value of the concentration is not considered as important as the mean axial flus computed from these assumed forms. We have invoked the notion of Pigford, et al. (1969), t h a t a n equilibrium condition exists a t the solid-fluid boundary. For the large cycle times used in practice, this appears to be a reasonable assumption and has some experimental justification. I n addition, the current model has a built-in destructive effect, often called “induced dispersion” (Horn and Kipp, 1967), caused by the coupling of velocity and concentration profiles. For example, Rice and Eagleton (1970) reported experimental results which show effective axial diffusion increases up to 1100 times that of simple molecular diffusion; a model for the periodic convection-diffusion exactly like eq 7 agreed very well with experimental data. Before proceeding to the calculation of asial flus, me note t h a t both C,(x) and C,1(E) contain the unknown quantity AC. The Axial Flux Expression
At the outset, i t was proposed t h a t the time-average flux following the transient would be zero, and this information should allow computation of the ultimate separation, AC. The flux along the axis is given by the expression
hence and for the current problem, the time-average flux can be easily seen to be
408
Ind. Eng. Chem. Fundam., Vol. 12, No.
4, 1973
I
I
'
'
" ' I
"
I)
"
1
I
\
I
0,
IO
IO
PECLET
I
\
. 1
I l l 1
I00
NUMBER,
Figure 2. Ultimate separation A/ro = 1.0
I
1
u
. I l l 1
10,mo
1.000
':w/o
for
small
amplitudes,
-0 6
I
'
u
\ i
-0 5
" " "
1
I
08
01
IO
10
PECLET NUMBER, r,'
1
1,000
I00 Y
/
10,000
~
Figure 4. Effect of amplitude on ultimate separation; Nso =
3 75 where a+ is the Fourier velocity coefficient. The total flus Q, is obtained by integrating over the crosssection
-0 I
01
I IO
1
,
,,#,,I
P E C L E T NUMBER
IO
,
, 8 8
, 100
,
1
1
1
1
1
loo0
'E"/D
Figure 3. Ultimate separation for large amplitudes, A/ro =
10.0
Since a t the steady state Q is exactly zero, the ulitmate separation, AC/AC*, can be estracted from eq 19. After performing the integration, and some straightforward but tedious algebra, the dimensionless expression AC/ AC* becomes
It was alluded earlier t h a t higher harmonics in the unsteady concentration function would make no net contribution t o the time average axial flus, provided velocity was purely sinusoidal. This can be easily shown if one takes the time average of a product such as
where it is seen t h a t the only nonzero terms are C + l u - ~and C-lu+t. However, if the velocity were a square wave and if the square wave can be adequately modeled by a Fourier series, then one should extract the time average of
Under these circumstances, higher harmonics in concentration produce a significant effect on the average axial flux and subsequently on the computed ultimate separation. It is seen t h a t the additional terms owing to a square velocity potential would include U + ~ C - ? ,U - ~ C +etc. ~ The analysis proceeds exactly as for the siriusodial velocity except t h a t a generai nth harmonic diffusion equation is solved
I t is seen that' even a simple model which does not include the iniportant nonlinear boundary effects leads to a very complicated frequency dependence. Asymptotic results are discussed later. The dependence of ultimate separation on Schmidt number, dimensionless amplitude, and frequency (Valensi number) is presented in Figures 2 and 3. It is seen t h a t there exists a critical frequency for masimum separat,ion;this frequency depends on the value of amplitude ratio as seen in Figure 4. I n passing, the curves in Figure 4 apply to the aqueous sodium chloride system for a n average temperature of 35"C, giving a Schmidt number of 375. It is important to note in this figure that the cross-over Valensi number, L e . , the frequency which produces no separation, is independent of amplitude and depends only on Schmidt number. I n all cases studied, the funcbional dependence of concentration diff ereiice on frequency (Valensi or Peclet numbers) Ind. Eng. Chem. Fundam., Vol. 12, No. 4, 1973
409
produced a local maximum. Further, if amplitude ratio is small, a reversed separation occurred and this interesting anomaly will be discussed shortly. The physical explanation of the existence of t!ie local maximum can be interpreted as a consequence of induced dispersion (Horn and Kipp, 1967 ; Harris and Goren, 1967). It was mentioned earlier t h a t flow oscillations can increase axial flux by as much as 1100 times (Rice and Eagleton, 1970) that obtainable by molecular diffusion, depending on t’he magnitude of fluid amplitude and frequency. Flux along the axis tends to zero when the concentration difference between reservoirs is such that the combination of axial molecular diffusion and velocity induced dispersion are just balanced by mass interchange between fluid and solid. Mass is transferred by the bucket-brigade concept mentioned previously, and the fluid velocity not only provides the axial transport mechanism but also partially destroys (disperses) the gradient’. As frequency is increased, the periodic mass exchange a t the pore wall becomes more out of phase with the velocity, thus diminishing the convective mode of transport along the axis. Initially, the velocity assists the axial mode of transport, producing large concentration gradients. However, as the phase difference increases, the adsorbing species lose mobility and remain close t’o the pore wall. This requires a smaller and smaller concentration difference to nullify the ever increasing velocity induced dispersion effect. Eventually, as frequency is increased, a mode of reenforcement occurs, causing the concent’ration difference to build up again, provided a small amplitude ratio exists (see Figure 2). Qualitatively, the theory predicts the experimental behavior observed by Sweed and Gregory (1971). For example, these workers have shown t h a t a t a given frequency, a n increase in amplitude reduces t,he separation. This is exactly what should occur according t o the current theory (Figures 2 and 3), provided the frequency is such that operation occurs in the region to the right of the peak. The figures show that a decade increase in amplitude causes the peak position to shift approximately a decade lower in Peclet number. Xoreover, in the region to the right of the peak, a n increase in frequency
For liquid systems (large Schmidt number) it may be observed that a small frequency or pore size (Valensi number) is required for optimum ultimate separation and this condition has always been observedin practice. On the other hand, gaseous separations require a high frequency; this has been shown experimentally to be the case by Turnock and Kadlec (1971) for the separation of nitrogen from methane. There appears to be a n anomaly in these results for small amplitudes, since the separation becomes opposite to t h e assumed direction at large frequencies (See Figure 2), that is, a negative AC/AC* results and the reservoirs reverse position. This phenomenon is not a t all unexpected owing to the large phase lag between applied pressure and velocity which occurs a t high frequency. I n fact, a t high frequencies the rootmean-square velocity maximum occurs very near the wall where adsorption takes place and hence reverses the direction of separation. This interesting situation, called the “annular effect” by Richardson and Tyler (1929), is illustrated and discussed by Schlicting (1960). One of the results of this study (see Figures 2 and 4) shows t h a t the largest absolute separation should be obtained using high frequency and small amplitudes. I n practice, this type of operation has severe drawbacks. A small amplitude ratio is difficult to obtain in a packed bed. Noreover, high-frequency operation would probably cause turbulence, destroying the concentration gradient more than predicted by the simple laminar model proposed here. Hence, for optimum ultimate separation, we conclude that the “kinematic Peclet number” (Arow/D) takes a value of approximately 3.0.
decreases separation; this effect has also been reported by Sweed and Gregory. Inspection of Figure 3 shows that (for large amplitude) in the important region surrounding the peak, the curves for different Schmidt’ numbers merge into one. Indeed, when the product of amplitude ratio and Peclet number is plotted us. separation factor, as in Figure 5, a single design curve results. This holds strictly when A / ~ o> 10 and Nsc > 100. I n actual practice (Sweed and Gregory, 1971) the computed ratio of fluid amplitude to pore radius can exceed lo3. For gaseous separations, a comparison of Figures 2, 3, and 5 shows a single curve exists up to a value of 100 for the product of amplitude ratio and Peclet’ number. Hence, for both liquid and gaseous systems, the optimum velocity amplitudefrequency product is seen in Figure 5 to be obtained from Arow/D N 3.0. This is one of the more important results of the current work, since in practice a packed bed configuration always gives a small pore size, and hence A / r o >> 1. The single design curve, Figure 5 , would appear to have wide general application.
This expression shows that anonzero asymptote exists for very large frequency; for large Schmidt number, this asymptote takes the value
410
Ind. Eng. Chem. Fundam., Vol. 1 2 , No. 4, 1973
Asymptotic Expression for Large Arguments
A simple expression for calculating the ultimate separation is obtainable when the arguments cy+1 and Pt1 become large (ie., .Vva >> 1, Np, >> 1). This does not preclude Schmidt numbers of order unity. Under these conditions, the ratio of the complex Bessel functions can be shown to take the limiting values f With this approximation, the real part of eq 20 becomes
d;,
provided A / ? o is finite. Equation 21 also shows that for large Schmidt and Valensi numbers, the ultimate separation has a simple square root dependence on frequency (Valensi number), that is
Asymptotic Expression for Small Frequency
Most of the experimental work on liquid separations by parametric pumping was performed a t very low frequencies,
transfer theory described here, a n accurate estimate of pore size and pore length is absolutely essential in calciilating ultimate separation in a packed bed parametric pump. S e w experimental results over a broad frequency spectrum are required to test the theory described herein. In particular, the conclusion that optimum ultimate separation occurs a t a kinematic Peclet number ( d r o w / D ) of around 3.0 should be experimentally verified. Current literature data are riot in a form suitable for checking this important result. The effects of thermal lag, especially in the radical direct’ion, and t’he capacitance effect of the solid were not considered in this work and are probably important. Iri addition, mass transfer resistance a t the fluid-solid boundary could introduce another important rate-limiting step. 01
, ,
I
l
l
,
I
,
io
,
, , , I
, ,
IO
I00
J
1,000
Figure 5. Ultimate separation for large amplitude as function of single variable, Arow/D
typically with cycle times in the range 10-60 min/cycle. Hence, a simplified expression in the region of very small Valensi number could be useful. To do this, the Bessel functions in the general expression, eq 20, are expanded in series, keeping the first two terms for Jo(z)and the first term for J l ( x ) . For Nsc >> 1, only the fifth-order term in the denominator is significant. With these approximations, eq 20 reduces to
Acknowledgments
The computer calculations in this study were performed a t the University of Calgary, Calgary, Alberta, Canada. The assistance of Dr. R . H. Weiland on the computer work is gratefully acknowledged. Nomenclature
constant in eq 10 amplitude of oscillation constarit in eq 10 concentration of adsorbing species concentration in upper reservoir concentration in lower reservoir steady component of concentration, defined by eq 7 unsteady components of concentration, defined by eq 7 diff usivity
4-1
complex Bessel function of first’ kind, order n length of tube or pore coefficients defined by eq 3 and 4 local axial mass flus Peclet number ( d V ~ c S ~ a ) Schmidt number ( Y I D ) Valeiisi number ( r o 2 w / Y ) total axial mass flux radial coordinate tube or pore radius operator (real) time components of complex velocity, defined by eq 3 and 4 complex velocity axial coordinate time average
When the Schmidt number is very large and the amplitude is finite, this can be written as
Comments and Conclusions
In the early literature of chemical engineering, the problem of calculating pressure drop through packed beds was confronted. Several approaches were considered, but the one that has been most successful, called the “tube-bundle theory,” was perhaps the simplest (Bird, et nl., 1960). I n this theory the packed column is regarded as a bundle of tangled tubes of weird cross section; the theory was then developed by applying the friction factor results for single straight tubes to the collection of crooked tubes. This notion can also be applied to mass transfer in the pore structure of a packed bed, provided the behavior in single uniform pores or tubes is known. In this way the mass transfer theory presented here may be applicable to some of the current parametric pump data. I n order to use the “tube-bundle theory,” a technique of deducing average pore size which eliminates “dead end” pores must be applied, since only pores with channels for flow will contribute to mass transfer between the reservoirs. -1recently proposed method by Felch and Shuck (1971) for deducing pore-size distribution shows considerable promise. Using simultaneous flow and solute diffusion, these workers appear to have eliminated the effects of “dead end” pores in estimating effective mass transfer pore size. T o apply the mass
GREEKSYMBOLS a+1
=
p+l
=
13-1
E
= = = =
w
=
e Y
4-iisp, 4-i.Yva
4L+Ya angle around tube (pore) axis kiiirmatic viscosity din-,ensionless radial coordinate (r/ro) f r e ~ ~ u e n cofy oscillat,ion
Literature Cited
Bird, R. B., Steward, W. E., Lightfoot, E. N., “Transport Phenomena,” p 196, Wiley, New York, N. Y., 1960. 10, 299 Folch, D. E., Shuck, F. O., I N D .ENG.CHEM.,FUNDAM. (1971). Harris, H. G., Goren, S.L., Chem. Eng. Sci. 2 2 , 1371 (1967). Horn, F. J. AI., Kipp, K. L., Chem. Eng. Scz. 2 2 , 1879 (1967). Jenczewski, T. J., Myers, A. L., IKD.ENG.CHLM.,FUSDAM. 9, 216 (1970). Ind. Eng. Chem. Fundam., Vol. 12, No. 4, 1973
41 1
Pigford, R . L., Baker, B., Blum, D. E., IND.ENG.CHEM.,FUNDAM. 8, 144 (1969). Rice, 11 (> Eagleton, L C , Can. J . Chem. Eng , 48, 46 (1970). Ilichardson, E. G., Tyler, E., Proc. Phys. Soc. London 42, 1 (1929). Iiolke, R . IT.,m'ilhelm, R. H., IND.ESG. CHEII., FVNDAX. 8, 23.5 (1969). Schlicting, H., "Boundary Layer Theory," 4th ed, p 230, Me(iraw-Hill, New York, ?;. Y., 1960. Sweed, ?;. G., Gregory, R. A,, A.I.Ch.E. J . 17, 171 (1971). ~
Sweed, N. G., Wilhelm, R. H., IND.ENG.CHEM.,FUNDAM. 8, 221 (1969). Turnock, P. H., Kadlec, R. H., A.I.Ch.E. J . 17, 334 (1971). 'Wilhelm, R. H., Rice, A. W., Benedlius, A. R., IND. ENG.CHEM., FCNDAM. 5 , 141 (1966). Wilhelm, R. H., Rice, A. W., Rolke, R. W., Sweed, N. H., IND. ENG.CHEM.,FUNDAN. 7, 337 (1968). RECEIVED for review August 15, 1972 ACCEPTEDJune 8, 1973
Axial Dispersion in Nonisothermal Packed Bed Chemical Reactors Larry C. Young and Bruce A. Finlayson* Department of Chemical Engineering, University of Washington, Seattle, Vash. 98196
A criterion i s developed to predict when axial dispersion is important in nonisothermal packed-bed reactors with cooling or heating a t the walls. In contrast to the isothermal problem, the criterion does not depend on the length of the reactor, so that the importance of axial dispersion cannot b e minimized by increasing the length of the reactor. An increase in flow rate does decrease the importance of axial dispersion. The criterion is applied to the experimental data presented by Schuler, et a/. ( 1 954)) for SO2 oxidation on an alumina catalyst impregnated with platinum, and the criterion suggests that axial dispersion is important. The experimental data apparently cannot be reconciled with a model excluding axial dispersion, but a model including both axial and radial dispersion correctly predicts the data.
M a t h e m a t i c a l models of chemical reactors are useful for predicting the conversion and temperature profiles in packed bed reactors. TT'hile very general models can be written donx, these are not often used either because of the computational complexity or because it, is difficult' or impossible to estimate the parameters in the model. We are concerned here with a cylindrical tube, which is cooled or heated a t the walls, and which is packed with catalyst. We wish to determine when axial dispersion is important in such a reactor and relate the results to experimental data. The importance of axial dispersion is shown below to depend on the type of reactor, and reactors with cooling or heating have a different criterion than isothermal or adiabatic reactors. To understand t'liis difference we first examine the known information about the importance of axial dispersion in isothermal and adiabatic reactors. For a first,-order reaction in a n isothermal reactor, Carberry (1958), Epstein (1958) and Levenspiel and Bischoff (1963) provide criteria for the neglect of axial dispersion effects. Levenspiel and Bischoff give, in the notation of this paper
where C is the outlet concentration from a reactor of length L with axial dispersion and C p is the concentration a t length L in a reactor without axial dispersion. Levenspiel and Bischoff and Carberry present the above result in terms of the nominal residence time. This notational convention is not followed here since the residence time is proportional to the reactor length, and n e nish to display the reactor length directly. We can 412
Ind. Eng. Chem. Fundam., Vol. 12, No. 4, 1973
rearrange this result to be expressed in terms of the concentration difference.
cap[ - (F)]
