Page 67 - Pressure Swing Adsorption

42 PRESSURE SWING ADSORPTION FUNDAMENTALS OF ADSORPTION 43

such c1rcumstances the sorption/desorpt1on rate depends on both the resis- Uotake:
tance to mass transfer and the time dependence of the local gas-phase
-3k ; \
ii
concentration. The modeling of such systems ts considered in Section 2.4. -= ! -exp -r) (2.37b)
f
However, in order to understand their behavior, 1t is helpful first to consider \ e
the simpler problem .of sorot10n m a smgte adsorbent particle subjected to a
step change m surface concentration. To do this 1t 1s necessary to consider m
sequence the various oossible mass transfer resistances that may control the 2.3.6 Micropore Diffusion
sorotion rate. Of course in practice more than one of these resistances may
We assume mstantaneous eouilibrallon at the external surface with the
be significant, but in order to avoid undue complex1ty we assume here approach to equilibrmm m the mtenor of the sphencal particle controlled bv
spherical adsorbent oart1cies and a smgle rate-controlling process. We as• Fickian diffusion with the diffus1v1ty defined on the basis of the gradient of
sume a general expression for the eauilibnum isotherm {q* = f(c)} and in all
the adsorbed ohase concentratmn. Local sorot1on rate:
cases given here the assumed initial and boundary conditions are:
, < 0, q = c = 0; t > 0, c = c , alR, = Kc 0 (2.35) (2.38a)
0

Uptake:
2.3.4 External Fluid Film Resistance
Sorpt10n rate: (2.38b j
dii 3k, •
di = R [ Co - c ], c* =f(li) (2.36a)
p At short times this expression 1s approximated by:
Uptake: q, _ 6 JD,; Dt
3
q,,,, -,;v1r - 7i (2.39)
ii = 1 _ exp( ~3k,i ), (2.36b) '
qu . RP . This expression is accurate to within 1 % for m,/m"" < 0.85 (or Dct/r'; < 0.4).
The mass transfer coefficient (k,) depends m general on the hydrodynamic The first term alone provides an adequate approximation for the initial
condihons but m the special case of a stagnant gas (Sh = 2.0)k, = Dm/R,. In region (m,lm~ < 0.15 or D,t /r; < 0.002). Confdrm1tv with these exores-
practice the external fluid film resistance 1s normally smaller than the sions is illustrated m Figure 2.17. The difference between the fonns of the
mternal (intraparticle or mtracrystalline) diffusional resistances; so this pro- uptake curve denvect from the diffusion model and the surface resistance
cess 1s seldom if ever rate controlling, although in many systems it makes models (Eq. 2.37 or 2.38) 1s illustrated m Figure 2.20, while the temperature
some contribut10n to the overall resistance. dependence of D 1s shown m Figure 2.18.
0
The situation 1s more complicated m binarv or muit1component svstems,
smce It is then necessary to take account of the effect of component B on the
2.3.5 Solid Surface Resistance
chem1cal ootent1al of component A. As the simtilest realistic example we
If mass transfer resistance 1s much higher at the surface than in the intenor consider an idealized system m which the cross terms m the flux equation can
of the adsorbent particle, for example, as a result of partial closure of the be neglected and m which the mobility 1s mdependent of composttmn. The
pore mouths, the concentration profile will show a steplike form with a sharp detailed analysis has been given by Round, Newton, and Habgood 48 and bv
change m concentratt0n at the surface and an essentially constant concentra- Karger and Biilow. 49 We have for the fluxes:
tion through the mtenor regton. In this situation the expression for the
uotake rate ts similar to the case of external film resistance but with the mass -D (dlnpA)ilqA (2.40)
OA dlnqA i!z
transfer coeffident ks representmg the diffusional resistance at the solid
surface. Sorotion rate:
N = -D /dlnp.)aq.
8 08
da 3k, _ \dlnqn oz
dt = R(qo - q), (2.37a)
p If the eauilibnum isotherm 1s of binary Langmuir torm (Ea. (2.13), the