Page 322 - Pressure Swing Adsorption

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298 PRESSURE SWING ADSORPTION
MEMBRANE PROCESSES 299
however, reqmres that the active membrane be on the mtenor of the hollow
for assessing the effect of flow pattern on performance. but m generai one
fiber tubes. It 1s much easier to apply a uniform membrane film to the
would try to avoid this condition in an operatmg svstem.
extenor surface of the tubes, but to take advantage of such an arrangement
the feed must be applied to the shell side, on which some deviation from
ideal plug flow 1s mev1table. Such deviations can> howc:;;ver, be mtmmized ?Y
8.3 Calculation of Recovery - Purity Profiles
good design; so this arrangement 1s in fact widely used in commercial
systems. 8.3.1 Mixed Flow
The fractional recovery (R) 1s defined simply as the fraction of the less
permeable species that emerges m tile raffinate product stream:
8.2.1 Effect of Flow Pattern
L (1-x )
2
2
R--.''-;-;--~
Since the effective separation factor 1s reduced by back pressure (Eq. 8.10), L,(I - x,) (8.12)
the flow oattern has a pronounced effect on the performance of a membrane
system. This may be clearly shown by caiculating the punty-recovery orofiles For a well-nuxed system the mole fractions x 2 and y 2 m the raffinate and
for different flow schemes. As in any mass transfer process, countercurrent permeate streams are related through Eq. 8.2. The separation factor o/ is
constant throughout the system and 1s given by Eq. 8.10 with x = x •
flow maximizes the average ctnving force and therefore provides the most 2
Calculation of the recovery-ounty orofile 1s therefore straightforward, re-
efficient arrangement. It is relatively easy to achieve _a reasonable a~prox1ma-
Qmrtng only the cornbinat10n of an overall mass balance for the Jess perme-
tion to plug flow on the high:.pressure side, but this is much more_ drfficult on able species:
the Iow-oressure side because of the wide variation m the gas velocity (from
close to zero at the closed end to a significant value at the permeate exit): If L 1(1 - x 1 ) = L (1 - x ) + ( L, - L )(1 - y ) (8.13)
2
2
2
2
the pressure ratio is large, deviations from plug flow on the l?w-oressure side
with Eas. 8.2, 8. 10, and 8.12.
hav~ a relatively minor effect on performance, provided that plug flow ts
mamtamed on the high-pressure side. The operatton of many membrane
modules, part1cularly those of the hollow fiber type, is therefore w~ll reore- 8.3.2 Cross-Flow
sented by the "cross-flow" model, which assumes plug flow on the high-pres-
The calculation 1s slightly more complex for the cross-flow case, smcc it 1s
sure sidC with perfect m1xmg on the low-pressure side [Figure 8.?(b)].
necessary to account for the variation of partial pressure with positron on the
The worst case from the point of view of process efficiency 1s oerfect
high-pressure side. For the ideal cross-flow system sketched m Figure 8.Mh),
mixmg on both sides of the membrane. This provides a useful limiting case
a differential mass balance for the more rapidly diffusing species gives:
ydL = d(Lx) = Ldt + xdL
(al (8.14)
, l · Low p J • rermeale where L is the (local) molar flow rate on the high,-pressure side. The locai
concentrations x and y on the high- and low-oressure sides of the membrane
are related by Eo. 8.2. Substitution m Ea. 8.14 and :rearranging yields:
Raffina1e7:.:: H~ p ::::- lb•~--Feed
dL dx dx
-L = ~( ,,-. ---!~) (~1---x~)-x + 70 1--=--x') (8.15)
(
(bi Permeate (L,-L,l. Y which may be mtegrated from the mlet ( x = x ) to: any arbitrary exist mole
1
fraction (x ):
2
LowP v dl
_t_. t ' t in('~')= 1n(!....=_:_o_) + r"' dx (8.16)
Ratfinate . L-dl •1 .'--• i-x 2 1 ,,(a'-1)(1-x)x
L .x 2 Hi h P x .. dx L,x Combining Eas. 8.12 and 8.16, we obtam:
2
x, dx
Figure 8.7 (a) Countercurrent and (b) cross-flow membrane elements showing defi- lnR =
nition of vanables used in EQs. 8.14-8.27. f x, [a'(x) - lj(l -x)x (8.17)