Page 126 - Pressure Swing Adsorption

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100 PRESSURE SWING ADSORPTION EQUILIBRIUM THEORY 101

Inserting this into Eq. 4.1 yields: altered form. Since composit10n can be determined aiong any charactenst1c,
ol'r; auPy; knowmg the initial composition, they provide an excellent means for perform-
a("+ /3,a, - 0 (4.4) mg steow1se material balances for PSA cvcles.
The blowdown steo 1s governed by Eas. 4. 7 and 458, while the subsequent
where /3, - {l + [(l - e)Je]f;J-'. which 1s a function of the comoonent
purge step can be analyzed with Eq. 4.7 alone. These steps essentially
partial pressure. Note that when the isotherms of both components are
regenerate the adsorbent. In many convent1onal adsorption systems, regener-
linear, this oarameter is constant, although here tt 1s treated as if it deoends
ation creates a wave that gradually moves through the bed, which 1s com-
on oart1al pressure, Simply adding the appropnate forms of Eo, 4.4 for the
monly referred to as a prooortwnate pattern or disoerstue from (see Section
respective components yields the overall matenal balance, which governs the 2.4). In the remainder of this chapter, it 1s referred to-as a srmpie wave, partly
interstitial flow m the packed bed. Hence, this exoression may be empioyed because it contrasts with the term shock wave which. ts discussed beiow (see
to determine the Jocai veioc1ty in terms of composition, pressure, and
also Section 2.4.1), and to suggest an assoc1at1on with equilibrium behavior
adsorbent-adsorbate mteractions. The result is
(as opposed to kinetic or dispersive effects).
", (!'''' 1 - /3 ·i l + (/3 - l)y, To complete the analysis of even the simplest PSA cycle, 1t 1s necessary to
v, - exp' YA, l + (/3 - l)yA dyA =' l + (/3 - l)y, ( 4,5) account for the uptake step(s). In particular, the feed step mvolves uptake of
the more strongly adsorbed component. As m conventional adsorot,on sys-
where the approximation 1s oniy exact for linear isotherms, and f3 = (3A/f3 , tems, a sham concentration front 1s created by this uptake that 1s sometimes
8
which for nonlinear isotherms may depend on pressure and comoosition, but called a constant pattern or self-sharpening profile. At the extreme of local
for linear isotherms it 1s constant. During pressurization and blowctown, the eauilibrium behavior, the front is a step change, and 1s called a shock wave.
composition, pressure, and mterstitiai velocity vary with time at each axml Since equilibrium effects are emphasized m this chapter, the term shock
position. The details are beyond the· current scope (see Section 4.9), but the wave will be used, even though m reai systems diss1pat1ve effects may
result for linear isotherms ts dimm1sh the sharoness. A shock wave 1s shown m the l)Ottom portion of
-z 1 dP Figure 4.1 m the feed step. It appears as a thick line ,at which charactensucs
(4,6) 1ntersect, and it separates the shaded region (depicting presence of the heavy
" - f3.[1 + (/3 - l)yA] P dt
component) from the plam regmn (depicting the pure light component).
The s1molic1ty of eauilibrrnrn-based theones anses because the couoled Examples of breakthrough data (from expenments rn which methane was
first-order oartiai differential eQuations governing the matenal balances can admitted to a bed of activated carbon previously pressunzect with mtrogcn)
be recast as two ordinary differential eouat1ons that must be simultaneously are shown m Figure 4.2, illustrating the sharpness aua1nable in many PSA
satisfied. The mathematical techmque employed 1s called the method of applicallons,
characteristics. A bnef explanation is given in Appendix A. The resulting It ts the velocity of a shock wave through !he packed bed that governs the
equations are: duration of the feed and nnse steps, Simiiariy, that velocity 1s controlled by
the rate of flow into and out of the system: hence, the material baiance.1s also
( 4,7) affected. The veloc1ty of the shock wave depends On the rnterstitial fluid
velocities at the leading and trailing edges of the wave. These are related by
and equating the shock wave velocities based on the conditions for both compo-
nents A and .B to get
(/3 - 1)(1 - YA)YA
( 4,8)
[l + (/3 - l)yA]P
( 4,9)
Equation 4.7 defines charactenstic trajectories m the z, t oiane. First of all,
Eq. 4.8 shows that when pressure 1s constant, composition ts also constant,
where, m general,
and the characteristics given by Ea. 4.7 are straight lines. Conversely, when
pressure varies with time. the composition varies according to Ea. 4.8, and
the characteristics are curved. Examotes of characteristics are shown m the
bottom portion of Figure 4.1, for example, as lines durmg feed and purge and
as curves during oressunzation and blowdown. It is important to note that l and 2 refer to the leading and trailing edges of the wave, respectively, and
charactenstics do not enct at the end of a step; rather, they continue in an fi; ts given by Eo. 4.2 for component 1 at comoosition- J.

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