Page 177 - Pressure Swing Adsorption

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152 PRESSURE SWING ADSORPTION EQUILIBRIUM THEORY 153

assumed local eouilibnum and neglected pressure drop, and they solved the consideration to a cycle havmg cornpiete purge .. To: detemune whether a
equat10ns by finite differences. A subsequent moclei that examined pressur- shock wave may form, one must simply examme whether the characteristics
ization and mcluded effects of axial dispersion, but not of mass transfer mtersect (in the soace-!Jme domam bemg considered). The oaths of the
resistance or pressure ctrop, was developed by Rousar and Ditl. 31 Another characteristics are given by Eo. 4.33, and the comno.sition as a function of
local equilibrmm model was proposed by Kumar, 32 and it was one of very few pressure can be determined from EQ. 4.32. When the charactenstics do not
to mcorporate an energy balance. That model was used to analyze adiabatic intersect, regardless of the initial and boundary conditions, that pair of
blowdown behav10r. More detailed modeis are discussed later in this section. equations 1s sufficient to predict the composition nrOfile durmg oressuriza~
This section first examines the simoiest cases of pressurization and blow- , tion. On the other hand, situations that result m the fonnat10n of shock
down,. and suggests that the key features predicted by more sophist1catect waves reQUire a few additional steps to predict the• ultimate composition
models can be obtained analytically. In such cases, much less effort 1s profile. When pressure vanes, the shock wave traJectory can be determined
required, and reasonably accurate estimates of the exoected behavmr can be by emoloymg Ea. A.7 from Appendix A, as follows
obtained. That upproach neglects axial pre5surc drop and mass transfer IJ.1JPyA
resistance, as is the case throughout this chapter. Later m this section, lJsu = OA t::i.PyA (4.69)
however, the impact of pressure drop on pressurization and blowdown 1s
considered. The interstitial velocity is obtained by summmg Eq. 4.4 for components A
It turns out that the coupling of velocity, composition, and pressure 1s and B.
graspable for systems governed by linear isotherms! but when nonlinear
isotherms are invoived the additional complexity makes the set of eouatlons ( 4.70)
unwieldy and to get detailed simulations via the method of charactenstics is
In this equation, the axial dependence of pressure can be neglected, which,
not practical. In addition, as noted in the previous section, the pressunzatmn
given the dependence of pressure on time, leaves a separable ordinary
and blowctown steps also may give rise to significant temperature shifts that
differential eQuation,
affect the validity of predictions based on ·isothermal models.
For oressurizauon a number of possibilities exist, and the two simplest d In P dv dvy;
+
{3
extremes have aiready been covered, viz., pressurization with product (see ~ 8 dz + ({3A - f3a) dz - O (4.71)
Secllon 4.4.1), and with feed (see Sec!Jon 4.4.2). These were assumed to
Integration reouires boundary conditions, and a convenient set 1s: v = u f at
begm with an initial condition m which the bed was purged with the oure
z = L, and u = 0 at z = 0. The result ts the expression that was simolv stated
light component. When the bed has not been completely purged, s1mu1atmg
earlier m this chaoter:
prc·ssurizatwn 1s slightly more complicated (see Sectton 4.4.3). All three cases
-z d In P
represent cyclic steady-state outcomes of operating a PSA cycle at local ( 4.6)
v(P,y,z,t) - f3a[I + ({3- l)y] dt
equilibnum. without dispersive effects. A slightly more comolex s1tua!Jon
occurs durmg startup, when the gas used for purgmg (and possibly pressur-
When Eqs. 4.6 and 4.69 are combined, the shock wave velocity may be found
ization) 1s not pure. Other possible complications anse from mmor vanat10ns from
m operating procedures. For example, a oressure equalization step empioys
the gas evolved from one column {as it ctepressunzes) for pressurizing a -{3z din!' ( 4.72)
parallel column. Such gas may have a slowly or suddenly varymg composition, [1 + (/3 - l)y ](1 + (/3 - J)y ] dt
1
2
due to uneven rates of desorption, poor synchrontzat10n of the valves, or
The dependence of Y1 on P, and that of y on z~ can be expressed vm
2
contamination by residual matenal in the connectmg fittmgs.
Eqs. 4.7 and 4.8, respectively. By combining those with Eq. 4.72, the couoled
Pressurization at startup or with gas having vanable composition can be matenal balances can be solved to get:
regarded as fitting the following possible scenanos: (1) Pressunzing with gas
ay,
that gradually becomes leaner m the heavy component, during which a shock ( 4.73)
Yz= I +(a~ l)y,
wave cannot form, or (2) pressunzmg with gas that gradually becomes richer
m the heavy component, dunng which a shock wave may form if the where a~ y (1 - Y )/lyw(l - y 211 )], and Yw and y are the mitial
10
20
20
composition change is suffic1entiy large, or (3) oressunzmg with gas that is compositions at the leading and trailing edges of the shock wave, respec•
significantly richer in the heavy component than the mitial interstitial gas-, for tively. Thus, a is a sort of select1v1ty, analogous to relative volatility for
which formation of a shocK wave 1s unavoidable. For s1mplic1ty, let us restnct vapor-liouid equilibnum.