Ind. Eng. Chem. Fundam., Vol. 18, No.
2, 1979
117
Limiting Separations in Parametric Pumps Richard G. Rice,' Swee C. Foo, and Gary G. Gough Department of Chemical Engineering, University of Queensland, St. Lucia, Queensland, Australia
I t has been predicted theoretically that the proper selection of frequency and fluid displacement is critical in maximizing separation in batch, direct-thermal-mode parametric pumps. New data are presented to verify these earlier theoretical predictions. Limiting (ultimate) separation factors exceeding 3000 are reported on the system aqueous sodium chloride on BioRadAGllA8 resin for which, heretofore, researchers were able to obtain only order 10 using equivalent fluid displacements. A new.parameter estimation technique is presented which allows determination of key dissipation factors. These, in turn, can then be used to estimate conditions under which maximum separation is likely to occur. For the first time, a simple theoretically based expression is uncovered to estimate optimum velocities in parapumps. I t is shown that the optimum velocity is approximately equal to the square root of the product of the coefficients of mass transfer and of molecular diffusion.
Introduction Parametric pumping is a novel chromatographic separation technique whereby cyclic sorption is tuned in phase with a cyclic velocity field so as to build a solute concentration gradient in a closed packed column of sorbent, solvent, and solute. The cyclic sorption is accomplished most easily by cycling the entire bed temperature; this is called the direct-thermal mode (Wilhelm et al., 1968). The basic separation depends on a retard-release mechanism which can be synchronized with fluid displacement. A review covering fundamental research on the subject has recently appeared (Rice, 1976). In the recent series of papers (Rice, 1975; Foo and Rice, 1975, 1977), research was undertaken to explain in a fundamental way the effect of frequency and fluid displacement on separation potential in a batch-operated, direct-thermal-mode parametric pump. In this mode of operation, heat is added or extracted directly through the walls of the chromatographic column. Two types of rate models for the uptake or release of solute by the immobile phase were used, viz. (1)a distributed parameter or diffusion model and (2) a lumped parameter model which incorporates solid (or pore) diffusion into an effective mass transfer coefficient. It was shown (Foo and Rice, 19751, for typical parapump operating conditions, that the lumped model follows the distributed model very closely over several frequency decades. Significant deviations between the models occur only a t high frequency. However, low separations were shown to occur in this region; hence, for practical situations, the simple lumped model will be taken as entirely adequate. The basic transport equations are now taken to be: mobile phase mass balance aC a2C u(t)- + kpap(C- C*) = E at dX ax2 and immobile phase mass balance aC
-
+
(1)
The key dissipative terms are the effective mass transfer coefficient, kpap (which depends on particle or pore diffusion coefficient), and eddy dispersion coefficient, E. Here, C is the bulk (mobile) phase composition, qp is the particle (immobile) phase composition (including any possible pore fluid), and C* is the equilibrium composition which in the present case is taken to be related to qp in a linear way 0019-7874/79/1018-0117$01.00/0
qp = a ( T ) + k(T)C*
(3)
The linear equilibrium relationship has a slope and intercept which, in general, depends on temperature. Thus, should a linearization of a nonlinear equilibrium expression be taken at higher concentrations (where ~(2') # 0), it has been shown (Foo and Rice, 1977) that the temperature dependence of the intercept can contribute significantly to the ultimate separation obtainable. In the direct-thermal mode of operation, bed temperature response (to changes in jacket temperature, for example) can be significant (Rice and Mackenzie, 1973; Foo and Rice, 1975). The characteristics of this response can be adequately modeled by a first-order time lag plus delay, as follows
(4) which in the Laplace domain has the familiar representation
While difficult to estimate from first principles, the time constants rB (first-order lag) and T D (time delay) can be easily determined from experimental temperature response curves. However, as we show later, temperature response characteristics have very little effect in the region of maximum separation where cycle times are predicted to be quite large. Theoretical Development The purpose of this research effort is to show in a fundamental way how separation potential depends on applied frequency and fluid displacement. Figure 1 illustrates the basic features of an operating direct-mode parapump. While the dual-column arrangement shown is atypical, the steady state obtainable is identical with the single column arrangement (of equal length), being different only in the rate of approach to steady state. The dual-column system is also convenient experimentally in handling expansion and contraction of fluids. In our earlier theoretical developments, it was shown that a poor selection of amplitude-frequency can lead to low separations (Foo and Rice, 1977); this forecast was compared with the low separations obtained by Sweed and Gregory (1971). Furthermore, these earlier models predicted there exists a maximum ultimate separation factor for a particular system. Separation factor is defined as the 1979 American Chemical Society
118
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
*& We also put C*(x) = a x )
and note and C1are complex conjugates (similarly for C*+l and C*_J. The applied velocity takes the form (cf. eq 6) AU e i ~ t+ Aw e-iwt = 4 4
@ Cold @Hot
@ Flow
@ 3 - w a y solenoid valves @ H o t - w a t e r recycle @ Cold-water recycle
bath bath conductance Cell
@ ~J~~on-wlthdrawal
8 Sampling
point
0Circulating pump
The structure of the assumed solutions is such that C, C*, and u are real functions. The purely sinusoidal applied thermal oscillation which satisfies eq 5 is of the form T(t) = T - '/ATbei(Ut+r) - 1/,ATbe-i(wt+r) (11) where
Figure 1. Schematic of dual-column parapump.
ratio of the compositions a t the column ends (rich/lean). The ultimate separation occurs following the transient buildup to steady state and is the limiting separation. Actually, one should speak of a quasi-steady state since at the ultimate separation, reservoir compositions will cycle around a steady mean value. A square wave velocity-temperature periodicity was used in the earlier experiments (and as well in the present case). The square wave is tedious to model, owing to the many harmonics present. We model the velocity, temperature, and composition with a purely sinusoidal periodicity, using the same frequency as that occurring under conditions of a square wave potential. Purely sinusoidal functions are tractable and easy to manipulate, and if one ensures the total volume displaced per unit time is the same for both model and experiment, it would seem such an approach closely approximates square wave operation. It is easy to ensure that fluid displacement per unit time is the same in model (sinusoidal wave) and experiment (square wave) as follows; taking one-half cycle as a basis, we require
and the phase angle is y = tan-'
(-wb)
- TDW
(13)
Here, AT is the amplitude of the applied jacket temperature (one-half the applied temperature difference) and T is the mean temperature of the packed bed. Equations 10 and 11 suggest that during upflow, the bed is cooled, while during downflow, heat is applied to the packed bed. This picture applies strictly to the case of zero phase lags in the mass or heat exchange process. As we shall see, inherent resistances and capacitances in the coupled exchange process can lead to considerable internal phase differences, even though the external potentials (velocity and jacket temperature), are adjusted to be in phase. When the assumed solutions, eq 7 and 8, are inserted into the transport equations (eq 1 and 21, and products of harmonics are neglected, then eq 1 gives
J":'uSinesin ut d t = LTcJ2 uSq dt and since usq over one-half cycle is constant, this gives, with T, = ~ H / U 5-
usme
=
2usq
Now usq is the distance moved (amplitude A ) during the time of a half-cycle, so
hence which ensures equivalent volumetric displacement per unit time. Throughout this work, A is the half-cycle displacement for square waue velocity operation. We wish to explore conditions around the ultimate steady state such that the time-average reservoir compositions are unchanging; hence the time-average flux along the packed column axis is nil. The unknown compositions are assumed to be comprised of a steady part plus a purely sinusoidal part; hence
e
where we take = C*. The perturbation functions are determined from eq 1 and 2 by equating like coefficients of exp(iwt) or exp(-id). The experiments presented later in this report are such that a(7') is constant (see Rice and Foo, 1979). The effect on separation of linearized isotherms with temperaturedependent intercept has been treated previously (Foo and Rice, 1975). For temperature oscillations small relative to the mean, the temperature dependence of k()'2 can be taken to be linear (Rice and Foo, 1979) so that qp = ( k , - klT)C* + u (15) where ho, h l are positive numbers for normal isotherm behavior. In forming the partial derivative in eq 2, i.e. 8% - _ a q p aT _ -
aqp aC* (16) at aT at aC* at products of perturbation variables are neglected so that
+--
aq, 1w - = c(.X)klhTb-[eXp(i(ic't -b y)) - eXp(-i(wt + at 2 ?))I + h ( n io[C*+,exp(iot) - C*-' exp(-iwt)] (17)
Ind. Eng. Chem. Fundam., Vol. 18, No.
Thus eq 2 relates C+l to C*+, as follows AT&w pkl e i y C(x) + i w ~ k ( n C * + = ~k,a,(C+l
7
-
C*+J
where we have neglected second derivatives of the perturbation function, being considered small in the local sense, and more properly represented in the global sense by association with the steady component C(x), viz. eq 14. Our line of reasoning so far is to associate dissipation !rising from axial dispersion with the steady component C(x) while dissipation arising from periodic transfer between particle and fluid phases (the effective transfer coefficient) is included in the equation for C+l and E,. Here we distinguish between dissipation in the large ( E ) and dissipation in the small (kp,). After some tedious but straightforward algebra, the complex perturbation functions are found to be 4 dx which is used to obtain C + l ( x ) in eq 19 as
Hence, the complex functions (&, 4.) which determine amplification or attenuation of separation potential are represented by
+ Pk(n X (iw + ~ , ~ , ) / ~ , a , l(23) l
At this point in the analysis, we note that if the linear isotherm contains an intercept which is strongly temperature dependent (47‘);see eq 3) an additional constant term arises in eq 21. A derivation to account for this small correction factor has recently appeared in the literature (Foo and Rice, 1977), and is discussed more fully elsewhere (Foo, 1977). The above result allows the steady concentration profile to be determined from the solution of eq 14 to give
where the boundary conditions are taken to be c(0) = CB (steady, rich bottom reservoir composition) and C(L) = CT (steady, lean top reservoir composition), and the dissipation function, u , is . ,A ;(
u=
E
+
-
119
the key destructive terms, eddy mixing ( E ) and particle transfer coefficient (k,a,), are suitably defined.
(18) The intimate coupling of perturbation functions and steady components should be carefully noted. The final relationship is obtained by extracting the relevant perturbation functions from the convective diffusion relationship, eq 1, to give Aw dC i&+l + -- - + k,a,(C+1 - C*+J = 0 (19) 4 dx
42 = l/Z[pk(niw+ k,a,]/(iwk,a,[l
2, 1979
Ultimate Separation The ultimate or limiting separation occurs as the transient buildup of concentration profile approaches the steady value represented by C(x) in eq 24. For a closed (batch) system, the limiting separation occurs when solute pumping ceases (averaged over a cycle). Solute pumping ceases when the time-average axial flux tends to zero; hence the necessary condition for limiting separation is
(
( J )= - E
:+
-
u.c
) -0
In view of the structure of the assumed solution (eq 7), the axial flux reduces to Aw - [C,(1 - exp(-uL)) - (CB - CT)] Real [ $ I ] = 0 (27) 2 The only nontrivial solution results when the expression containing compositions is identically zero since Aw Real
>0
(28)
therefore CB/CT = a , = exp(uL)
(29)
It is interesting to compare this functional relationship with classical separation theory. Thus one would take In cy, = N T U and the inverse of the present dissipation function as equivalent to the height of a transfer unit, i.e., l / u = HTU; hence L = NTU X H T U . This parallel is not surprising, and the similarity becomes even more clear when the asymptotic form for u is presented later on in the work. That the separation factor can be extracted directly, without recourse to the overall conservation relation, is fortuitous. It was shown in a previous publication (Foo and Rice, 1977) that a more general linear isotherm representation (temperature dependent intercept) requires the simultaneous solution of the “zero flux equation” with the “constant mass equation” in order to obtain the separation factor. A nonzero intercept can arise owing to a linearization of the sorption isotherm a t intermediate compositions. Fundamentally, however, Henry’s law requires the intercept to be zero as compositions become diminishingly small. For the particular experimental system considered in the present work, it has been shown (Rice and Foo, 1979; Foo, 1977) that the linear representation of the sorption isotherm takes an intercept which is practically independent of temperature. The expression for separation factor in eq 29 has only one additional constraint, that being the solubility limit for the rich reservoir (CB). The consequences arising from this limit were first analyzed by Foo and Rice (1977) and are discussed in depth elsewhere (Foo, 1977). Representation of Dissipation Forces At the outset, it was suggested (based on previous studies, see Foo and Rice, 1975) that a lumped model could be used to represent particle diffusion resistance. A lumped mass transfer coefficient which represents effective particle diffusion was used by Gueckauf and Coates (1947) and is expressed as
Real
It is convenient to express the above results in dimensional form, until such time that the geometrical dependence of
Because of the critical importance of mass transfer re-
120
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
sistance, we determine this parameter directly from experimental data using an asymptotic form of the separation theory, which is derived shortly. While axial dispersion is expected to have very little effect on separation potential under practical operating conditions, it will be shown later the value and character of this parameter is critical in defining the maximum possible separation (at steady state). Very little has been published on the subject of axial dispersion, under purely sinusoidal flow conditions, except for the work of Harris and Goren (1967) and Rice and Eagleton (1970). These researchers suggest that axial mixing in smooth bore tubes is directly proportional to the square of RMS velocity. In a packed column arrangement, the situation is quite different. Analysis of the data for steady flow through packed beds in the comprehensive review by Gunn (1968) shows a linear dependence on velocity, and for liquids there results E = 2.2dPu 0.02 5 Re 5 20; Re = p d p t p / p (31) Such linearity is not unexpected, since Klinkenberg and Sjentzer (1956) predicted theoretically that eddy mixing follows E = qldpu (32) The pre-coefficient ( T ~ has ) been experimentally measured by Klinkenberg and was found to vary between 2 and 4 for packing of 50-100 mesh particles. Indeed, experimental results for identical packing size as that used in the separations presented later in this work (see Foo and Rice, 1979) gave a value of vi = 4.2, which was practically independent of temperature. For the reasons outlined above, axial dispersion in the present study was represented by
E =D
+ vldp(u)
(33)
where ( u ) is the RMS velocity and for uniform sinusoidal oscillations (say, u = ( A w / 2 ) sin wt) gives Aw (u) = (34) 2 f i
The pre-coefficient, vl, can be determined from experimental data taken from an operating parapump. Asympotic Theory The complete separation theory embodied by eq 29 in conjunction with eq 25 is presented in Figure 3 using parameters given in Table I. A maximum in separation factor is predicted to occur, followed by an exponential decay. Experiments conducted in the exponential decay region could be used to facilitate parameter estimation. Assuming fast temperature response, it is easy to show that the Real parts of the complex functions 41,4~~reduce to Real
[&I
MPklAr)/(l
+Pk(n)
(35)
provided w > wopt and kpa, > w . Here coopt is the frequency which gives the maximum separation (see Figure 3 ) . In passing, we note that when Pk(T) >> 1 then Real [&]
-
(2k,aP)-'
(37)
This last result will be used later in uncovering an approximate analytical relationship to estimate wept. When the above simplified expressions, eq 35 and 36, are inserted in eq 29, along with the axial dispersion re-
Table I. Physical Properties for t h e ParaDumD SeDarations"
p i = p s ( l - E,) e, = 0.40 p , = 1.1g/mL
= 0.66 g/mL
k , = 2.26 mL/g k = 0.0112 m&/g " C k @ l = k , - k , T = 1.924 mL/g (30 " C ) p k ( T ) = 2.359 d, = 0.03 c m ; L = 77 c m ; d, = 2 cm T c o =~ 10 " C That = 50 " C D = 1.8 x cm'/s (30 " C )
D , x lo', cm2/s
T, " C
0.34 0.65 1.0 3.7
10 20 30 50
See Rice and Foo (1979); Foo and Rice (1979); Foo (1977).
lationship (eq 33 neglecting for the moment the contribution of molecular diffusion, D ) and following a simple rearrangement, there results 1 -
In
f i i d p [ l + Pk(r)l
E
PklATL
cy,
1/2pk2(n(A/L)w
+ k l A T [ l + Pk(T)]k,a, (38)
Thus a plot l / l n a , vs. w (with constant amplitude A ) should produce a straight line with slope proportional to ( k p a J 1 and intercept proportional to vi. It is convenient to express this result in dimensionless form by introducing the Sherwood (dimensionless transfer coefficient) and Peclet (dimensionless frequency) numbers; thus eq 38 becomes
--
In a ,
PklATL
+
PkiAT[1 + P k ( n 1 S h ~ (39)
where PeL = L 2 w / D and ShL = k,a&'/D. The dimensionless form of the complete separation theory is computed from eq 29, which now becomes In a , = u L = ' / , ( A / L ) P e LReal 41
(40) The behavior of this theoretical separation factor is shown in Figure 3, using parameters given in Table I. The contribution to axial mixing by molecular diffusion (unity in denominator of eq 40) is highly significant in the region of the maximum as we show later. The dimensionless functions $, and now take the form '/2pklATbeiY ShL (41) " = S h L + p k ( n ( S h L+ iPeL) ShL + pk(nPeLi
" = 1/2iPeL[ShL+ P k ( n ( S h , + i P e L ) ]
(42)
During the computer explorations to generate the theoretical curves presented in Figure 3, it was observed that in the region of the maximum separation the complex functions Real [ql]and Real [b2]were invariant as the
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979 1
Peclet number was changed, as shown in Figure 4. Thus, in the region of the maximum, the separation theory, eq 40 reduces to
1
1
O,4sL
I
I
I
,’
,’
,’’
121
?,
..,’
0 40C 0.35-
0.30
where
025
bo = Real
[&I
[Inrr-]-’
= constant
?I
~
bl = - d , / L , constant
fi
b2 = Real
Regression Line Ilna,l~’=0.113(t +13)+1,54 (*43)xlO-’Pe,
i
...... 9 5 % Confidence Bound
0.10
[&I
N
constant
Inverting this expression and solving for the minimum (In with respect to ( A / L ) P e Lgives
c d 1
L’W/D=PeL
Figure 2. Verification of asymptotic theory (eq 39).
Thus the maximum corresponding to the complete theory can be found from Figure 3 to occur when 1 / 2 ( A / L PeL ) 1.0 x lo4,while the above approximate expression (using Figure 4) gives 1.01 X lo4, which is in remarkably close agreement. When ShL >> PeL, it is easy to see from eq 42 the approximate result ‘ / ( P k ) 2 / S h L (+ 1 PkI2
Real ij2
(45)
which is the dimensionless equivalent to eq 36. Furthermore, when Pk >> l , this reduces to Real
q2
-
1/2ShL-l
(46)
Inserting this into eq 44 gives the useful result for the optimum (47) or in dimensional form, this result is
The approximate result shows that optimum velocity is independent of column length. Furthermore, molecular diffusion enters the picture by way of the assumed axial mixing expression
and is seen to be very important in the region of optimum separation. It should be stressed here that D in eq 48 is a dispersive force for mixing along the column axis. Had we ignored the molecular diffusion component of mixing, a maximum would not exist. We illustrate the very small velocities necessary to obtain maximum separation by way of an example. Using the Gueckauf-Coates expression (eq 30) to estimate kpapwith 0.03 cm, pp = 0.66 g/mL, k ( h 1.92 mL/g, D, N d, cm2/s, and t b E 0.35, the equivalent transfer coefficient is calculated to be k,a,
1.57 X 10-2/s
(50)
Taking molecular diffusion to be D N 1.8 X cm2/s, eq 48 gives an estimate for optimum velocity as (Aw),,,
N
1.5 X
cm/s
(51)
which is small indeed, quite outside practical operating conditions. Nevertheless, eq 48 should prove very useful for experimental design. It will be pertinent later in the
discussion to note that the above computation shows one expects normally k,a, > w . The result given by eq 48 appears to be the first time a simple, theoretically based expression has appeared in the literature which allows a first estimate of optimum velocities in parapumps. Experimental Studies A dual column or “back to back” parapumping system was used in the present experiments as illustrated in Figure 1. Two 2-cm i.d. glass columns each with 77 cm of packed height were used to separate 0.05 M aqueous NaCl solutions. The sorbent was a carefully washed ion retardation resin (Bio Rad AG11A8) of approximately 3 mm diameter (4C-80 mesh). Operating temperature was cycled between 10 and 50 “C. Fluid displacement was accomplished using an infusion-withdrawal pump manufactured by Harvard Apparatus Co. (Model 940 with 2 x 50 mL syringes). Cycle times ranging from 15 min to 63 h were obtainable. Composition was monitored using two flow-through conductance cells with cell constants of approximately unity and 0.02 for the rich and lean streams, respectively. Additional details of the experimental unit can be found elsewhere (Gough, 1977). Sorption and hold-up characteristics of the washed resin have been published previously (Rice and Foo, 1979). A compilation of the relevant physical properties and sorption parameters is given in Table I. Computations giving Figures 2-4 are based on parameters given in this table. Temperature response characteristics were determined by measurements using a thermistor inserted to a distance of a quarter of the inside packed diameter, midway along the column length. The system thermal time constant and time delay were determined from response curves and found to be 85 s ( T ~ and ) 4 s ( T ~ ) re, spectively. These values were considered to be insignificant relative to the large cycle times used; hence thermal effects were neglected in the theoretical analysis (Le., ATb AT y 0). The measurement of packed column voidage is discussed elsewhere (Gough, 1977) and average results are given in Table I. A constant fluid displacement was used in all experiments with measured interstitial fluid displacement equal to eight-tenths of packed length, giving an amplitude ratio A I L = 0.4. Cycle time was varied from 15 to 301.9 min giving separation factors of 9 to 3179, respectively; the time to reach steady state varied from 3.5 to 10 days for the respective cycle times of 15 and 301.9 min. The largest cycle time ( - 5 h) yielded reservoir compositions of 0.15 M. Saturated solutions M and approximately 8 X
-
-
122
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
--..---_ 9 5 X Confidence bound
1 10’
I
, 81/11#1
,
io4
,,,,,,,I
(P~J,,,
, ,,,,,,,I
, ,,,,
106
105 Pe L
Figure 3. Comparison of complete separation theory (eq 40) with experiments for fast thermal response (ATb = A? y = 0).
3t
Y
1-
0
I
# # # # 1 1 1
I
I m111111
I
I
i i i i i t l
I
4
to give 2.39 x 10-2/s, which is of the same order as the result using the Gueckauf-Coates expression. In Figure 3, the complete separation theory (eq 40) is presented using the parameters determined from the approximate theory (eq 39 shown in Figure 2). It is noted in Figure 2 that the slope is a 1/ShL while the intercept is a ql. Figure 3 illustrates the extremes taken by the complete theory when parameters take values a t the limits of the confidence interval determined from parameter estimation using the linear theory. Thus, minimum slope-maximum intercept gives ShL = 0.857 X lo7 and q1 = 18.5, while maximum slope-minimum intercept gives ShL = 0.722 X lo7 and q1 = 14.5. It is seen that peak height is quite sensitive to the two parameters giving a variation in separation factor order 10 a t the confidence boundaries. If experiments can be made around the maximum in separation factor, eq 48 may also be used to estimate the transfer coefficient. The experimentally determined dispersion precoefficient (vl) was approximately four times higher than expected. However, one expects larger mixing in parapumps than in steady-flow isothermal systems because: (i) shrinking and swelling of resin gives a mini-mixer effect; (ii) oscillating temperatures give rise to induced mixing by natural convection; and (iii) oscillating velocities enhance mixing. The present work seems to show that axial dispersion is a key dissipative effect in operating parapumps and should not be ignored in the modeling exercise. Finally, because of the excessive time required, separations on or about the optimum frequency were not undertaken. Estimations from Figure 3 show that a cycle time of around 20 h is required (for the present system) to obtain the expected maximum separation of around lo4. This would require an estimated 40 days to reach steady-state using the present experimental system.
Comments and Conclusions A simple lumped model to describe the particle phase transfer resistance was used to predict limiting separations under quasi-steady conditions in a batch, direct-thermal-mode parapump. The theory shows that maximum separations occur when fluid velocity is of order (kpaJl)i/z. An asymptotic form of theory useful for parameter estimation showed that the inverse of the natural logarithm of separation factor varies linearly with velocity when the mass transfer coefficient is independent of velocity (particle diffusion controlled); if film resistance controls, this variation is expected to be linear with the square root of velocity (i.e., taking kpap0: u1lz). This asymptotic result is tied to the condition k,ap >> w , which is a reasonable approximation for liquid systems but is probably incorrect for gaseous systems. Experiments closely approaching the region of the predicted maximum separation produced separation factors exceeding 3000. The comparison of the complete theory with experiments (Figure 3) was quite good; however, it is yet to be proven that the predicted maxima in separation factor exists. Axial mixing in the parapump separator was determined to be significantly larger than that expected in a steady-flow, isothermal situation. The quantitative nature of this increased mixing can be quickly estimated using the asymptotic theory herein described. The technique presented here has also been used recently (Rice and Foo, 1977) to predict and correlate preformance parameters in open systems with continuous product withdrawal. Finally, we make special note of our forecast (Foo and Rice, 1975) several years ago that separation factors exceeding 3000 should be obtainable in an apparatus similar to the present experimental setup. These predictions
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
based on first principles have now been experimentally verified. Acknowledgments Support by the Australian Research Grants Committee is gratefully acknowledged. Special thanks are given to Mr. John Muller (University of Queensland) for uncovering a missing term in the numerical calculations. Nomenclature A = fluid amplitude a p = particle mass transfer area per unit volume interstitial fluid a ( T I = isotherm intercept (defined by eq 3) C = bulk fluid composition C* = equilibrium fluid composition d, = particle diameter d, = column diameter D = molecular diffusion coefficient D , = particle (solid) diffusion coefficient E = axial dispersion coefficient J = axial flux k , = mass transfer coefficient k ( T ) = isotherm slope (defined by eq 3) ko, k l = adsorption isotherm parameters (defined by eq 15) L = column length PeL = Peclet number, L 2 w / D q p = particle phase concentration R e = Reynolds number, pdptbU/p s = Laplace transform variable ShL = Sherwood number, L 2 k p a p / D T = bed temperature AT = amplitude of applied temperature oscillation ATb = amplitude of bed temperature oscillation (defined by eq 12) Tj = jacket temperature
u =
123
dissipation function (defined by eq 25)
u = interstitial fluid velocity x = axial coordinate
Greek Letters a , = separation factor (CB/CT) p = solids holdup, p (1 - t b ) / t b y = phase angle (dehned by eq 13) t b = bulk bed voidage tp = particle voidage vl = dispersion pre-coefficient (defined by eq 32 and 33) p = fluid viscosity T~ = first-order lag time constant for bed temperature response TD = time-delay for bed temperature response C#Jl, C#J2 = defined by eq 22 and 23
frequency Literature Cited w =
Foo, S. C., Rice, R. G., AIChE J.. 21, 1149 (1975). Foo, S. C., Rice, R. G., AIChE J . , 23, 120 (1977). Foo, S. C., Ph.D. Thesis, University of Queensland, 1977. Foo, S. C., Rice, R. G., Ind. Eng. Chem. Fundam., 18, 68 (1979). Gough, G. J., B. E. Thesis, University of Queensiand, 1977. Gueckauf, E., Coates. J. I., J . Chem. SOC.,1315 (1947). Gunn, D. J., Chem. Eng., No. 210, C.E. 153 (1968). Harris, H. G., Goren, S. L., Chem. Eng. Sci., 22, 1571 (1967). Klinkenberg, A,, Sjentzer, F., Chem. Eng. Sci., 5, 256 (1956). Rice, R. G., Eagleton, L. C., Can. J . Chem. Eng., 48. 46 (1970). Rice, R. G., Mackenzie, M., Ind. Eng. Chem. Fundam.. 12, 486 (1973). Rice, R. G., Ind. Eng. Chem. Fundam.. 14, 202 (1975). Rice, R. G., "Separation and Purification Methods", Vol. 5, pp 139-188, Marcel-Dekker, New York, N.Y., 1976. Rice, R. G.. Foo. S.C.. "Proceedings, 5th Australian Conference on Chemical Engineering", pp 179-83, Canberra, 1977. Rice, R. G., Foo, S. C.. Ind. Eng. Chem. Fundam., 18, 63 (1979). Sweed, N. H.,Gregory, R. A., AIChE J., 17, 171 (1971). Wihelm, R. H., Rice, A. W.. Roke. R. W., Sweed, N. H.,I d . E%. Chem. Fmdam., 7, 337 (1968).
Received for review January 9, 1978 Accepted December 7, 1978
Vertical Pneumatic Conveying Using Four Hydrodynamic Models Hamld Arastoopour Institute of Gas Technology, Chicago, Illinois 606 16
Dlmltrl Gidaspow Deparlment of Chemical Engineering, Illinois Institute of Technology, Chicago, Illinois 606 16
Pressure drop in vertical transport of solids was calculated using four hydrodynamic models previously proposed. The calculations were done using literature correlations of friction factors. Three of the models predict a minimum in pressure drop vs. superficialgas velocity. The three models-a relative velocity model, a model with pressure drop in the gas and solid phases, and a model with a pressure drop in the gas phase only-predict a choking behavior associated with flow reversal of particles. The relative velocity model compares well with Zenz's experimental data with a fixed inlet volume fraction.
Background of Hydrodynamic Models A hydrodynamic approach to fluidization was started by Davidson in 1961. He analyzed single bubble motion in an infinite fluid bed using two continuity equations and an expression for relative velocities in terms of Darcy's law for flow in porous media. Davidson assumed that the 0019-7874/79/1018-0123$01 .OO/O
solids flow around a bubble was irrotational. This assumption (Gidaspow and Solbrig, 1976) can be justified based on the mixture momentum equation. It can also be shown (Gidaspow and Solbrig, 1976) that the use of Darcy's law and the mixture momentum equation is in the limit equivalent to the use of two separate phase momentum
0 1979 American Chemical
Society
