Ind. Eng. Chem. Res. 1987,26, 1579-1585
1579
Intensification of Sorption Processes Phillip C. Wankat School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907
Methods t o scale columns which intensify sorption processes are developed. The method utilizes smaller diameter particles with high mass-transfer rates. The column length, diameter, and cycle time are then scaled so that pressure drop, separation, and throughput are the same or better than in the old design. The result is a “pancake” column using significantly less sorbent. Often, the new volume of adsorbent required is l/d,2 times the old volume. Methods for decreasing operating costs are also delineated and show that the designer can easily design to decrease pressure drop or increase column utilization. The Higee distillation process showed that major decreases in the size of distillation and absorption columns are possible (Ramshaw, 1985). Similar intensification of many other unit operations is possible if the appropriate methods are found. There has been a trend in adsorption, chromatography,and ion exchange to shorter columns with more rapid cycles. This paper will codify the reasons for this trend and outline some of the advantages of continuing on this path. Sorption operations such as adsorption, ion exchange, and chromatography can be intensified by simultaneously decreasing the particle size and scaling the column so that the pressure drop remains constant (Wankat, 1986a,b). Theoretically, the column height can be reduced by 1 or 2 orders of magnitude or more. The previous work (Wankat, 1986a,b) was restricted to systems with laminar flow and pore diffusion control. In this paper the scaling procedure will be extended to cases where pore diffusion does not control and to turbulent flow. In addition, new scaling methods which allow a simultaneous decrease in both operating and capital costs will be introduced.
Basic Scaling Concepts In this paper we will assume that an existing design for the sorption column is satisfactory. Then we will search to improve this design by scaling the particle diameter, column length and diameter, fluid velocity, and cycle time. Our goal is to find scaling rules which will give the same or better separation with the same or lower pressure drop in a column using the same or less sorbent. This method is valid for a variety of sorption separations such as gas adsorption, liquid adsorption, ion exchange, and chromatography. It is well-known that decreasing the particle size increases the mass-transfer rates. This decreases the HETP or length of mass-transfer zone, Lm, of the column. This decrease in LMTZ increases the fraction of the bed which can be used. However, as L/LMTZ increases past values in the range from 2 to 3, the increase in fractional bed use becomes quite small (Lukchis, 1973; Wankat, 1986b). Pressure drop also increases as the particle size decreases. Thus, the usual conclusion is that there is a limit to the amount which one wants to reduce the particle size. A detailed example of this conventional thinking is given by Sherwood et al. (1975) for the adsorption of methane using the Thomas solution for design. (See example 10.6.) Wankat (1986a,b) showed that by proper scaling the total pressure drop in the bed can be kept constant while the high mass-transfer rate of small particles is used. This was done for both linear and nonlinear isotherms for laminar flow with pore diffusion control. The linear analysis will be briefly reviewed here. 0888-5885/87/2626-1579$01.50/0
Pressure drop in a packed bed of rigid particles in laminar flow with no channeling and negligible end effects is (Bird et al., 1960) UVL Ap = Kd,2
If we always compare systems with the same velocity u , the pressure drop will remain constant if L/d,2 is kept constant. Velocity will be assumed to be constant during each step of the cycle but can vary from step to step. For linear isotherms, the mass-transfer effects can be included in the Van Deemter equation for the height of an equilibrium stage (Van Deemter et al., 1956) for an isothermal system. B H = A + - + CU (2) u
The A term is due to eddy diffusion
A = Adp
(3)
the B term is due to molecular diffusion
B = 27’D~
(4)
and the C term is due to mass transfer CSMdp2
CMdp2
c = -DM +- DSM
(5)
where A, y, cM, and CSM are constants. The resulting equation has an optimum velocity which minimizes H. However, normal operation is in the region where pore diffusion controls and H a ud;. Since the number of stages in the linear model is
N will be constant if u is constant and L/dp2is constant. This means that LMTZ - = - - NMTZ - constant L N
(7)
and the fractional bed use per cycle is constant. To keep Ap and fractional bed use constant when dp,new
=
udp,old
(8)
b=
(9)
we set LneW= bLold
Obviously, shorter beds will saturate faster. If the cycle time and the different parts of the cycle are scaled so that 0 1987 American Chemical Society
1580 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987
(cycle time),,,
= b(cyc1e time)old= a2(cycle time)old (10)
then the relative movement of the solute wave will be the same. Each cycle will process a fraction, b = a2, of the original amount, but the number of cycles/hour will be increased by a factor of l l b = l/a2. The net result is that the throughput of the column with small particles is the same. The new column does have a volume of packing which is reduced by a factor of b = a2. Thus, the scaling method gives the same throughput, the same Ap, the same fractional bed use, and the same separation, but it does this with a much shorter column which works harder since it is cycled much more frequently. Most desorption methods will also scale. In pressure swing methods for linear systems, the purge step at low pressure scales exactly the same way as the feed step. The blowdown and repressurization steps should scale approximately since the amount of gas to be removed does scale. There may be difficulties with very rapid cycles where pressure gradients can be improtant. In thermal desorption the time for breakthrough of the thermal wave scales, but the length of the thermal transfer zone probably scales as film diffusion (see next section), not pore diffusion control. Thus, there may be a small amount of extra dispersion in the new design. When a desorbent is used for desorption, the desorbent usually moves in approximately plug flow, and this desorption method should also scale. The scaling of the desorption steps is discussed in detail later. This scaling also works for both the adsorption and desorption steps with nonlinear isotherms (Wankat, 1986a,b). The analysis for nonlinear isotherms will be included in the new results reported later in this paper. The scaling method also works when the Thomas method is used for the design calculation even though pore diffusion does not completely control (Wankat, 198613). The columns will be short and fat and will look more like a pancake than a typical column. Practical objections to this scaling are discussed later. Similar methods are being used in commercial designs for ion-exchange separations (EcoTec, 1983, 1984). Thus, at least in this application, the practical problems are not insurmountable.
Scaling When Pore Diffusion Does Not Control Linear Equilibrium. Consider first the situation where flow is laminar so that pressure drop is given by eq 1. For linear equilibrium, an empirical equation to correlate the height equivalent to a theoretical plate H is H = kundpn+l (11) where k and n are constants. When n = 1,this corresponds to pore diffusion control and gives the same results as eq 2 and 5 , while when n = 0.5 this approximates film diffusion control. Equation 11 gives the same results as the usual empirical expressions for mass transfer (see the next section). Equation 11 is also a ljmiting case of the dimensionless form of the Knox equation and reduces to the Snyder equation for constant d , (Grushka et al., 1975). The scaling procedure will be simplest when a simple equation such as eq 11 is used to correlate H, however, the method is valid for any dependence of H on velocity. The number of equilibrium contacts, N, is
In order to do scaling for this case, we will allow the column diameter, D, and the velocity, u, to vary. The velocity, u, can be calculated from
We will assume that the volumetric flow rate, Q, is the same for the old and new designs. Writing eq 1 2 for the old and new conditions, substituting in eq 13, and dividing one equation by the other, we obtain
where RN is the ratio of number of stages and
When RN = 1,we have the same fractional bed use for both beds. The inclusion of RN allows us to change the operating conditions. Writing eq 1 for the old and new conditions, substituting in eq 13, and dividing one equation by the other, we obtain
where R, is the ratio of pressure drops in the two systems. If R, = 1, then the pressure drop for the two systems is the same. Inclusion of R, again allows us to change the operating conditions. Changing the operating conditions is discussed later. Assuming that the particle diameter ratio, a, and the operating ratios, RN and R,, are independent variables, we can solve eq 14 and 16 simultaneously for the required ratios of the column length and diameter
and
The ratio of the volumes of adsorbent in the new and the old systems is
The cycle time can be scaled to keep the relative movement of the solute wave the same. (L/u)new - (cycle time),,, =--(LD2)new -- b C 2 (20) (L/U)old
(cycle time),,d
(LD2),1d
Note that this is the same ratio as the volume ratio in eq 19. If pore diffusion controls, n = 1,these results reduce to
When RN = R, = 1, eq 21 agrees with the previous scaling results (Wankat, 1986a,b). A quick numerical example illustrates the power of this scaling. If n = 1,RN = R, = 1,and a = ' I 2 , we obtain Dnew/Dold= c = 1.0, LnewlLold = b = a2 = l j 4 , and (new adsorber volume/old adsorber volume) = 1 / 4 . Thus, the same separation with the same pressure drop can be obtained with one-fourth as much adsorbent. The effect of changing RN and R, is discussed in detail later. If n = 0.5 which is approximately film diffusion control, the length and diameter ratios become
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1581 Thus, ,,1-n
and
kmapa d,l+" (23)
For a numerical example, consider the case where n = 0.5, RN = R, = 1,and a = l / p Then Dnew/Dold = c = 1.122 and Lnew/Lold= b = 0.315. The ratio (new adsorber volume)/(old adsorber volume) = 0.397. Thus, the volume required is reduced significantly, but not as much as when pore diffusion controls. Nonlinear Equilibrium. Consider a nonlinear system with either a single adsorbate or a binary ion exchange or exchange adsorption with favorable equilibrium. If the column is long enough, a constant pattern forms during the isothermal feed step and a proportional pattern forms during the isothermal desorption step. Unfavorable equilibrium will have a proportional pattern during the feed step, but the scaling methods still work. Mass transfer can usually be modeled in a lumped parameter form
where c* is the equilibrium concentration. To be specific a generalized Langmuir isotherm, a*c* (25) = 1 b*c*
+
will be used, although this does not affect the scaling results. For the constant pattern wave, the length of the mass-transfer zone can be determined from (Wankat, 1986b) -a*
X
where u,h is the shock wave velocity or the velocity of the stoichiometric center of mass of the pattern. U
If we keep concentrations and temperature constant U LMTZ -aL kma& The area/volume ratio up is (Sherwood et al., 1975) 6(1 - t) up = dP while the mass-transfer coefficient kM is proportional to U1-n
k,
a
dP"
When pore diffusion controls, n = 1, while when film diffusion controls, n = 0.415 (Sherwood et al., 1975). Combining eq 28 and 31 and substituting in eq 13 for u , we obtain
L L LMTZ undpl+"
D2"L dpl+"
-cc-a-
(32)
This ratio will control the fractional bed use during the loading step (Lukchis, 1973; Wankat, 1986a,b). Taking the ratio of L/LMm for the old and new conditions, we obtain 1 _ - (L/LMTZ)old = -an+* (33) RN (L/LMTZ)new bCZn The same analysis can be done for pressure drop which is still given by eq 1. This gives eq 16. Since eq 33 has the same form as eq 14, the simultaneous solution of eq 16 and 33 is given by eq 17-20. Thus, the loading step for nonlinear equilibrium scales the same way as for linear equilibrium. If the mass-transfer zone is not fully developed in the shorter column, eq 33 will not be valid; however, this is not a problem since the mass-transfer zone is sharper in the entrance region and separation will be better than predicted by eq 33. Scaling during elution requires a somewhat different analysis (Wankat, 1986a,b). During elution, favorable isotherms such as eq 25 will have proportional pattern waves. Unless mass transfer is very slow, the proportional pattern wave can be fairly accurately predicted from the diffuse wave calculated by local equilibrium theory (Sherwood et al., 1975). Thus, the proportional pattern wave is controlled by equilibrium and its velocity can be estimated from
This solute velocity depends on concentration and temperatures since dq/dc is a function of c and T for nonlinear isotherms. Desorption is often done by passing a wave of hot fluid or a wave of different chemical composition through the bed. The center of the thermal wave will move at a velocity (Wankat, 1986b) U Uth
=
C
11+--
p
PfCpf
~w Cpw ~ ~
+--
4
(35)
CpfPf
The ratio of the time required for the thermal wave to traverse the bed for the new and old systems is (thermal Wave time),,, (thermal wave time),ld
-
(L/Uth)new =--(L/u)new (L/Uth)old (L/U)old
- bC2 (36)
Waves of chemicals will also move at a velocity which is proportional to u. The ratio of times for these desorbent waves is (desorbent wave time),,, =--(L/u)new - bc2 (37) (desorbent wave time),ld (L/u)old The ratios of eq 36 and 37 are the same as the ratio of cycle
1582 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987
L
for the new and old cycles we have
t
( twidth /
cycle) new
-
(twidth/ tcycle)old
L
2
%hLMTZ
LUth
-
Uso1(TH,C=O)
- Usol(cF,Tc)
-
Usol(Chigh,TH)
Figure 1. Times and width for complete cycle for nonlinear system. a
ib
Ads
Des
1
I
I C
Des
Ads
1
Ads
Des
~
Figure 2. Four-column system showing series operation for adsorption and desorption steps.
times given in eq 20. Thus, the fraction of the cycle spent in obtaining breakthrough of the thermal or desorbent wave during the desorption step is unchanged when these scaling methods are used. The width of the proportional pattern wave can be estimated for the desorption step with the aid of Figure 1. Figure 1 is drawn for the case where Uth > U,h(Chigh,TH). twidth = tslow - tfast (38) The slow wave moves at a velocity U,ol(Th,c=o) and starts at z = L if the feed step is run until breakthrough just starts. Thus, (39) The faster wave starts at L - L M T Z and first travels at a velocity usOI(Tc,cF) until it intersects the thermal or desorbent wave. Figure l ignores the spreading of the thermal wave; however, this is linear and will scale in the same way as linear concentration waves. After this intersection, the solute wave has a velocity Usol(TH$high) where Chi& is found from a mass balance around the thermal wave (Wankat, 1986a,b)
' I
where ci are constants. The result after some algebra is
(twidth/ tcycle)new (hdth
1-
-
C3LMTZ/ c3
/ tcycle) old
[ :-
where 2 represents the conditions after the thermal wave has passed. Both the point of the intersection and the time the fast wave exits are easily calculated. After some algebra, the result is
I - - -1LMTZ
UthLMTZ
-
c3
-
c4
1-
I
- c2
2c 5 L
- c2
C3LMTZ/ c3
- c2
-
c4
.])
+ c1 + c3
(46) old
If we set RN = 1 in eq 33, then eq 46 reduces to The total cycle time is tcycle
=
(twidth/ tcycle)new tads
+ tslow + tcool
(42)
where the cooling step may be deleted in some operations. This cycle time can be calculated as
If the column is completely regenerated, we are interested in keeping twidth/ tcycle constant. Forming this ratio
= 1 for R N = 1
(47)
(twidth/ tcycle)old
Thus, the fraction of the cycle required for removal of the proportional pattern wave is constant if RN = 1. Thus, when RN = 1,the scaling laws for the linear and nonlinear systems are the same. Requiring R N = 1makes every step in the old and new cycles scale. This is probably unnecessarily restrictive, and scaling could be done to keep the same total purity.
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1583
If a heel is left in the column, the scaling method still works if RN = 1. Consider the situation where the entire proportional pattern wave is left in the column. The width of the area in the column is Lwidth as shown in Figure 1. This width is = L - tf&Usol(TH,C=O)
LMTZ 1 - c3-(c3 - cz)-l L (Lwidth/ L)new (Lwidth/L)old = [h1 c4
where the ci are from eq 45. If R N = 1, then =1
b = -L- n e ~- a3n+2/n+2RPn/n+2RN 2/n+2
(53)
Lold
and
which gives a volume ratio of (adsorbant volume)new = bC2 = (adsorbant volume),Id
L
(Lwidth/L)old
ac4
Equations 14 and 52 can be solved simultaneously,
(48)
where tw is given by eq 41. The ratio of the proportional pattern bandwidth to the column length for the new and old designs is
(Lwidth/L)new
1
(50)
and the relative amount of the bed containing the heel is constant. Thus, all parts of the cycle will scale. These calculations show that nonlinear systems will scale as long as the mass-transfer coefficient is approximately represented by eq 30, the relative separation is kept constant (RN = l),and separation is based on differences in the equilibrium isotherms. If neither film diffusion nor pore diffusion control, then eq 30 will not be exact. However, the scaling can be done by either approximating the exponent n or by doing numerical calculations (for example, see the Thomas calculation in Wankat (1986b)). In zeolite molecular sieves, mass transfer may be controlled by intracrystalline diffusion. Since the zeolite particle is usually an agglomeration of crystallites held in a binder, the mass transfer rate does not increase as d, is decreased if intracrystalline diffusion controls. The scaling method will not work in this case. In carbon sieves used for separation of N2 and 02,the separation is not an equilibrium separation. Equation 26 is no longer valid, and the conclusions drawn based on this equation may not be valid. More research is required to see if scaling laws exist for the carbon sieves.
Turbulent Flow For turbulent flow with no channeling and negligible end effects, pressure drop can be estimated from (Bird et al., 1960) 3.5 L 1-E A p = - --~iu'( d,
7)
This pressure drop dependence is obviously very different than eq 1. Despite this difference, columns can easily be scaled to use smaller particles. Consider the linear equilibrium system where H is given by eq 11 and N is given by eq 12. The velocity, u, can be calculated from eq 13. The ratio Nold/Nnew is still given by eq 14. Substituting eq 13 into eq 51 and keeping Q constant, we obtain the ratio of pressure drops in two turbulent systems as
Note that these equations differ from eq 17-19 obtained for laminar flow. It is interesting to do a quick numerical calculation. If pore diffusion controls, n = 1, and we want the same separation, RN = 1, and the same pressure drop, R , = 1, the result for a = 'Iz is b = a5I3 = 0.31498 c = all6 = 0.89 which gives a volume ratio (adsorber volume),,/ (adsorber volume)old= bc2 = a2 = l / & Thus, when pore diffusion controls, the same reduction in adsorbent volume is obtained in the laminar and the turbulent cases, but the scaling laws differ. It can be readily shown that nonlinear equilibrium systems will scale in the same way as the linear equilibrium system if R N = 1. If changing the column diameter which changes the velocity takes one from the turbulent into the transition or laminar regions, the scaling laws will have to be adjusted. This is perhaps best done with numerical calculation and will not be demonstrated here. Changing Operating Conditions The ratios RN and R , were introduced so that we could consider changing operating conditions when scaling the column to use smaller diameter packing. In this section we will show that operating costs can be decreased by appropriate scaling. Parameter R , is the ratio of pressure drops of the new and the old designs. Since operating cost depends on the pressure drop, particularly in gas systems, operating costs can be reduced by operating with R , < 1. Consider the case where pore diffusion controls, n = 1. Then eq 21 shows the way to scale the column for laminar flow. If the same fractional bed use is desired, RN = 1, the particle diameter ratio a = lI2, and R , = l I 2 , then b = Lnew/Lold= 1/(4(2'12)) = 0.17678 C =
Dnew/Dold= 2'f4 = 1.1892
and the ratio of adsorbent volumes is (adsorbent volume),,,/(adsorbent volume)old= bc2 = aZN= y4 The same decrease in adsorber volume is achieved as when one scales to keep A p constant, but now the column is even shorter and fatter to reduce Ap. Since pressure drop is less during desorption, less gas can be used for desorption and a more concentrated desorbed gas exists the column. The bed can also be scaled to change R , or RN (in linear systems) without changing the particle diameter. Suppose that pore diffusion controls, n = 1, flow is laminar, the same separation is desired, RN = 1,particle diameter is not
1584 Ind. Eng. Chem. Res., Vol. 26, No. 8 , 1987
1 following the same reasoning used in eq 38-50. Comparing a one-bed system to a beds-in-series system is similar to comparing apples to oranges. It is clear that the beds in series system will have a higher bed utilization which means the feed step is a larger part of the entire cycle. A value of L = LMTZ is adequate for a bed-in-series system, while L = 2-3LMTz for a single bed. Since the bed contains more solute and tslowis the same for the same L , the bed in series must produce a more concentrated product during desorption.
i
Figure 3. Diffuse wave for column which is completely saturated.
changed, a = 1,and we desire to reduce the pressure drop by a factor of 2, Rp = l I 2 . Then from eq 21
new adsorbent '01 = bC2 = a2RN = 1 old adsorbent vol Thus, a shorter, fatter column with the same volume of adsorbent using the same particle diameter can be used to reduce the pressure drop in the system. This ability to reduce A p allows the designer to use column-in-series techniques which might not be economical otherwise because of excessive pressure drop. This should be particularly useful for gas separations. One way of using columns in series for both the adsorption and the desorption steps is shown in Figure 2. The system can be designed with a = ' I 2and R , = ' I z for a single column. The four-column system shown in Figure 2 will have the same pressure drop as a two-column system (one adsorbing and one desorbing) and uses l J 2as much total adsorbent. Because of the use of beds in series, the first bed can be completely saturated during the loading step. During desorption the eluted product will be more concentrated. In addition, more complete desorption can be obtained by using beds in series during desorption. The arrangement shown in Figure 2 may allow skipping the separate cooling step since the hot bed which has just been desorbed receives a cool, purified gas and not feed gas when it is switched to the feed cycle. Many other arrangements of beds in series and parallel can be used. When columns in series are used and the bed is completely saturated at the end of the feed step, the analysis shown in eq 38-50 will change. The diffuse waves are now shown in Figure 3. Equations 38 and 39 are the same, but eq 41 becomes
The total cycle time is still given by eq 42, but since the bed is entirely saturated, eq 43 becomes
where L + LMTZ/2 tFeed
(58)
ush
Now the ratio (twidthltdesorption)new/ (twidthltdesorption)oldis constant for any value of R N since LMTZ does not appear in this ratio. The ratio of (twidth/ tcycle)new/ (twidth/tcycle)old does contain the LMTZ.Beds in series will scale if RN =
Discussion There are many practical considerations which may make the proposed scaling method impractical. To minimize practical difficulties, the ratio of particle diameters, a, was set at 'Izin all of the examples. This relatively small decrease seems to be well within the range of current technology. Use of smaller ratios of particle diameters gives much larger reductions in sorbent volume, but with increased practical difficulty. The practical problems include column design, packing, operation, and user conservatism. With short, fat columns the distributor must be more carefully designed than in a tall, narrow column. With shorter cycles the dead volume or head space in the column must be minimized. At least for modest decreases in d,, these design constraints seem possible. There are also problems induced by the packing. The scaling method developed here assumes that the packing can be reused a large number of times. If this is not the case as may be true in bioseparations, then this scaling method may not be applicable. The rapid cycling used may increase attrition and makes severe poisoning more probable. In bioseparations nonrigid gel packings are often used. This changes the pressure drop laws and will change the scaling rules. New rules for these gels are being developed. The packing needs to be carefully sieved. Keeping the ratio Ad,ld, constant as d is reduced may be difficult. The effect of increased rfispersion in the packing size is not clear and needs to be studied further. Packing small particles uniformly to fully utilize their separation capability is more difficult than packing large particles. However, methods of doing this have been developed in large-scale chromatography and can probably be adapted to other packings. One also needs a commercial source of the smaller packing. Finding such a source may be a chicken and egg problem. Once a market is readily evident, such sources will become available, but until the market is obvious, it may be hard to buy smaller diameter packing at reasonable prices. This prevents the market from developing. In operation suspended solids need to be removed since the columns packed with small diameter particles will be efficient filters. However, these columns are more likely to filter at the surface rather than inside the column; thus, removing the solids may be easier. Accurate timing and rapid valve action is also required for very short cycles. This should not be a problem for modest decreases in size such as a = The user is naturally conservative. If your existing design method is working, why should you take the risk of changing it for an unproven, academic method? Obvious decreases in capital and operating costs have been pointed out in this paper. Another way to look at the question of user conservatism is to turn the world upside down. Suppose that you were currently using short, fat columns with rapid cycles. Under what conditions would you convert to using long, narrow columns with long cycles? Any condition not included in this list is a prime candidate for change.
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1585 The first applications of intensified columns using small diameter packings will probably occur where they have major advantages in addition to those already mentioned. When size and/or weight are very important, these scaling methods should be useful. Examples are off-short platforms, barge and ship mounted plants, and space. Whenever the amount and volume of equipment exposed to a harmful condition such as radiation must be minimized, these methods should be of interest. The obvious example is the nuclear industry. Very expensive packings such as those often employed in bioseparations make these scaling methods attractive. These scaling methods should also be attractive whenever short residence times are required because the compound being purified is laible. There remain many aspects of this scaling technique which need to be studied further. The particle size distribution will affect the results. Velocity can vary due to flow geometry, pressure or temperature changes, and adsorption or desorption. End effects may be important particularly for very short columns. Most industrial gas systems are not isothermal. Multicomponent systems have different behavior then the systems studied here and may scale differently. Packing for biological system is often compressible and has a different Ap dependence. Scaling of different desorption methods needs to be studied in more detail. These studies are either currently in progress or are being planned for the future.
Nomenclature a = ratio of particle diameters in two designs, dp,new/dp,old a, = area/volume ratio of packing, eq 29, m-l a*, b* = constants in Langmuir isotherm, eq 25 A = eddy diffusion term in Van Deemter equation, m A , = cross-sectional area of column, m2 b = ratio of column lengths in two designs, Lnew/Lold B = molecular diffusion term in Van Deemter equation, s-l c = concentration, mol/m3 c* = concentration at equilibrium, mol/m3 c = ratio of column diameters in two designs, Dnew/Dold c I - c ~ = constants in eq 45 cF = feed concentration, mol/m3 cM,CSM = constants in eq 5 C = mass-transfer term in Van Deemter equation, s
C , Cpf, C,, = bulk, fluid, and wall heat capacities, kcal/(kg ,IC)
d = particle diameter, m = column diameter, m DM,DSM = molecular diffusivity in fluid and in stationary phase, m2/s H = height equivalent to a theoretical plate, m
d
k = constant in eq 11 k , = mass-transfer coefficient, m/s K = permeability L = column length, m LMTZ = length of mass-transfer zone, m n = constant in eq 11 and 30 N = number of stages p = pressure q = amount adsorbed, mol/kg q F = q in equilibrium with c mol/kg Q = volumetric flow rate, m?z/’s RN = ratio of number of stages in two designs, Nnew/Nold R, = ratio of pressure drops in two designs, Apnaw/APold t = time, s Udiffuse = velocity of diffuse solute wave, m/s (also uBoJ U,h = velocity of shock wave, m/s Uth = velocity of thermal wave, m/s u = interstitial fluid velocity, m/s W = wall mass per length, kg/m Greek Symbols Ac, Aq = changes in c and q across the shock wave, eq 27 e = porosity p = viscosity X = constant in eq 3 y = tortuosity in eq 4 pB, pf = bulk and fluid densities, kg/m3 Acknowledgment This research was partially supported by NSF Grant CBT-8520700.
Literature Cited Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960; pp 196-200. Eco-Tec, Ltd. “Eco-Tec Ion Exchange Systems”, Technical Report 1983; ”Recoflo-A Breakthrough in Water Deionization Systems”, Technical Report, 1984; Pickering, Ontario, Canada. Grushka, E.; Snyder, L. R.; Knox, J. H. J. Chromatogr. Sci. 1975,13, 25.
Lukchis, G . M. Chem. Eng. 1973,80(13), 111. Ramshaw, C. Chem. Eng. (London) 1985, JulylAug, 30. Sherwood, T. K.; Pigford, R. L.; Wilke, C. R. Mass Transfer; McGraw-Hill: New York, 1975. Van Deemter, J. J.; Zuiderweg, F. J.; Klinkenberg, A. Chem. Eng. Sci. 1956,5, 271. Wankat, P. C. In Ion Exchange: Science and Technology; Rodrigues, A. E., Ed.; Martinus Nijhoff Dordrecht, 1986a; pp 337-368. Wankat, P. C. Large Scale Adsorption and Chromatography; CRC: Boca Raton, FL, 1986b.
Received for reuiew September 22, 1986 Accepted May 5, 1987
