Page 79 - Pressure Swing Adsorption

54 PRESSURE SWING ADSORPTION FUNDAMENTALS OF ADSORPTION 55

cally unrealistic overhanging profile s!wtched m the ·figure. In fact this does
Simple Wove (o) not occur; wJ1en cauilibnum theory predicts an overhanging profile tl1e
contrnuous soiut1on JS m fact reoiaced by the corresponding shock, which
0 travels with a velocity ( w') dictated by a mass balance over the trans1t1on:
~
u
uf w; - v/[ 1 + (Y) ~:• ] ( 2.52)
If the isotherm has an inflexmn oomt (e.g., a type II isotherm), It may be
C/c.
regarded as a combination of "favorable." and ".unfavorable·· segments.
, Equilibrium theory then predicts that the asymptotic form of the concentra-
Shock , I (bl
qiq. , , tmn profile will be a compos1ie wave cons1stmg of a shock front with a
, , prooort1onate pattern wave or a proportionate pattern wave followed by a
•
u , 't3 shock lsee Figure 2.23(c)l.
u , , Another s1tuat1on m which a shock solution 1s ·obtained ansc . .., in hulk
r,:f I seoarat1ons, where the change m flow rate due to adsorotmn 1s rclat1vc1v
;
large. For a bulk separation we have in place of Eq,, 2.48:
(' l -
1c ') iJij
ac
iJ o
v- + c- + - ilc + -- -:-- = 0 (2.53)
az dz <Jt e ilt
where, for an isobaric system with an adsorbable component m an Inert
earner:
V 1 - Yo
Vo T=v (2.54)
Expressed m terms of the mole fract10n of the adsorbable (or more ad~
sorbable) component, Eq. 2.53 becomes. for a linear equilibnum system:
{vo(l - Y 0 )/(I -y)'f 1 + ( ~; ")K]} ~~ + ~; = n (2.55)
Figure 2.23 Dcveloomem of the conccntralion profile m an adsorpi1on column with
negligible mass transfer resistance. (a) For an "unfavorable" equilibrmm relationship which evidently represents a traveling wave with the wave veiocitv given bv:
the profile spreads as 1t propagates, approaching proponionate pattern behavior. (b)
For a "favorable" equilibnum relat10nship an mitially dispersed profile 1s sharpened :'.".. = {(1 -yn)/(1 -y) [1 + (~ )K]}
2
as 1t propagates, approaching a shock wave. (c) For a BET-type 1soltlerm the (2.56)
asymptotic form 1s a combinat1on of a shock and a proportmnate pattern wave. Vo _ L £ ·'
Clearly w mcreases with mcreasmg y, Just as m the case of a trace sy5item
with favorable equilibrium, so that, according to eouilibnum theory, there
soreads as it prooagates fFigure 2.23(a)]. Since the profile spreads 111 direct will be a shock transition.
orooortwn to the distance traveled, this is referred to as "orooortionate
pattern·· behavior.
2.4.2 Asymptotic Behav10r: Effect of Mass Transfer ReSistance
The case of a favorable cauilibnum isotherm is slightly more comoiex.
and Axial Dispersion
tla* /de decreases with concentratmn; so. according to Eo. 2.49; w will
mcrease with concentration. This leads to what 1s commonly referred to as When the isotherm is of unfavorable form, mass transfer resistance and axial
"self-sharoemng" behavior. An m1tially dispersed profile will become les_s dispersion have only a relatively minor effect on the asvmotot1c form of the
and less dispersed as 1t propagates [Figure 2.23(b)], eventually approaching a concentration profile, This may be understood from Figure 2.24, which shows
shock transition. Eauation 2.50 predicts that the sharpening of the. profile the qualitative form of the concentration orofiles m a column followmg a step
would continue, even beyond the rectangular sI1ock form, to give the ohys1- change m concentrations at the mlet. Because the isotherm 1s of unfavorable