Page 153 - Pressure Swing Adsorption

LI

128 PRESSURE SWING ADSORPTION EQUILIBRIUM THEORY 129

completely ourged as a result of expansion of the oure light component from 1s diminishing concentration of the heavy component (which continues until
the product-end dead volume. Similar additional possibilities arise m other the adsorbent is purged). The column outlet concentration durmg the first
steps of the cycle, leading to many permutattons and contmgencies. Hence, phase 1s given by:
the overall problem is difficult to generalize.
To ctetermme the effluent composition durmg the countercurrent blow- ( 4.56)
down step, the material baiance for. the feed-end dead volume 1s combined
with the foregomg balances of the adsorbent bed and product-end dead
volume. The former yields y (P(t )) at z = L via Eo. 4.50. The overall
80
balance equat10n is:
The second phase 1s charactenzed by gradual cleansmg of the feed-end
dead voiume, which deoends on the relative volume at the product end, as
( 4.52)
follows:
By appiymg the cham rule for differentiation and simplifying, this reduces to: e°'C9o- ll) l
-
dy°'" = e/{3 8 - Ap (. Yeo - Yo"' ) (4.53) j
dt Ac (1 - /3)(1 - Y 80 )Yeo ( 4.57)
where Ai::-= V 0 vF'/Vc. This exoression may be integrated either analytically 112 112
where a= 2e/,\ 0 {3A, 0 = (t* /tL) = [I + ({3 - 1).v'](l - X') . and
0
or via a Runge-Kutta routme .. The resuit of the former approach 1s given by
X* is given by Ea. 4.51. When the product-end dead volume ts small, the
Ec1. 4.54, although in terms of time and effort, the numerical approach may ultimate composition leaving the column durmg purge 1s:
well be easier.
(1
_ . - Yeo Ye _ Ye - 1 )
Ysoout - YF. 1 - YP y ) "( 1 1J 00 L ( 4.54)
BO •=O (4.58)
f1(1J -j)
,-o
00 Yao - 1
+ 1JYeo L
!=tl
f1(1J -j) The pressurizatmn step demands excess rnatenai to pressurize the dead
,-o volume. The· precise amount, according to the ideal gas· law, 1s mdeoendent
where 1J = (e/{3 8 - Ao)/,\e· of composition. Nevertheless, at the feed end, since matenal flows mto the
The dead volume at the feed end is assumed to be well mixed, and thus dead volume as oressure mcreases, the composition changes. The moles
cannot be completely purged. For practicai ourooses, it 1s sufficient to purge reqmred for pressurizatton become:
the adsorbent completely. In fact, m view of the earlier discusSion of I
tncompiete purge, even this 1s often unnecessary. The auanttty of light (4.59)
component essential for comoiete purging of the adsorbent bed is:
!
' The composition m the feed-end dead volume at the end of oressurizatlon
'
( 4.55) ' becomes:
This 1s iess than the amount reqmred for a column having V nvr = 0, by the y PU011t11
fraction of the bed that hact been purged by residual product ctunng the YPRrma1 = ~ ( 4.60)
blowdown steo.
To continue the analysis, 1t 1s necessary to find the final composition in the Finally, tl1e feed step IS ootenl!ally somewhat involved ctue to the presence
feed-end dead volume durmg the purge step. Two ohases are reievant: of the heavy component m the feed-end dead volume. The feed immediately
during the first, the heavy component concentratmn mcreases (reflecting begins to shift that comoosition, but it displaces ma:tenal into the bed
residual matenai m the feed-end dead volume), and durmg the second there s1mu1taneousjy. The comoosition m the feed-end dead. volume vanes with