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Encyclopedia of Physical Science and Technology EN001-13 May 7, 2001 12:29

262 Adsorption (Chemical Engineering)

B. Classification According to Number (2) isothermal, three adsorbable components, no carrier;
of Transitions (3) adiabatic, one adsorbable component plus inert carrier;
(4) adiabatic, two adsorbable components, no carrier.
1. Single-Transition Systems
The only case for which analytical solutions have been
a. One adsorbable component plus inert carrier, obtained is (3) when the equilibrium isotherm is of rectan-
isothermal operation. gular form. For such systems the mass balance equation
i. Trace concentrations. If the concentration of ad- is not coupled to the heat balance, and the solution for
sorbable species is small, variation in flow rate through the concentration profile is the same as for an isothermal
the column may be neglected. Equation (17) reduces to: system. There is thus only one concentration front, the sec-
ond transition being a pure temperature transition with no
2
∂ c v∂c ∂c 1 − ε ∂ ¯ q change in concentration. Solutions for the other cases can

−D L + + + = 0 (19)
∂z 2 ∂z ∂t ε ∂t be obtained numerically, provided that a simple linearized
If the equilibrium is linear, exact analytical solutions for rate expression is used.
the column response can be obtained even when the rate
expression is quite complex. In most of the published so- 3. Multiple-Transition Systems
lutions, axial dispersion is also neglected, but this simplifi-
Only a few full dynamic solutions for systems with more
cation is not essential and a number of solutions including
than two transitions have been derived, and for multicom-
both axial dispersion and more than one diffusional re-
ponent adiabatic systems equilibrium theory offers the
sistance to mass transfer have been obtained. Analytical
only practical approach.
solutions can also be obtained for an irreversible isotherm
with negligible axial dispersion, but the case of an irre-
C. Chromatography
versible isotherm with significant axial dispersion has not
yet been solved analytically. Measurement of the mean retention time and dispersion
For nonlinear systems the solution of the governing of a concentration perturbation passing through a packed
equations must generally be obtained numerically, but adsorption column provides a useful method of determin-
such solutions can be obtained without undue difficulty ing kinetic and equilibrium parameters. The carrier should
for any desired rate expression with or without axial dis- be inert, and the magnitude of the concentration change
persion. The case of a Langmuir system with linear driving must be kept small to ensure linearity of the system.
force rate expression and negligible axial dispersion is a The principle of the method may be illustrated by con-
special case that is amenable to analytical solution by an sidering the response to the injection of a perfect pulse of
elegant nonlinear transformation. sorbateatthecolumninletattimezero.Themeanretention
ii. Nontrace concentration. If the concentration of the time t is given by the first moment of the response peak
adsorbable species is large it is necessary to account for the and is related to the dimensionless Henry constant by:
variations in flow rate through the adsorbent bed. This in- ∝
troduces an additional equation, making the solution more 0 ct dt = L 1 + 1 − ε K (20)
∝ v ε
¯ t ≡
difficult. Numerical solutions can still be obtained, but 0 cdt
few if any analytical solutions have been found for such
where L is column length. Dispersion of the response
systems.
peak, which arises from the combined effects of axial dis-
persion and finite mass transfer resistance, is conveniently
b. Two adsorbable components (no carrier), 2
measured by the second moment σ of the response:
isothermal operation. This is a special case since, in
∝ 2
the absence of a carrier, the rate equations for the two ad- 2 0 c(t − ¯ t) dt
sorbable species are coupled through the continuity equa- σ ≡ ∝ cdt (21)
0
tion so that a single mass transfer zone is still obtained.
The case of tracer exchange is a particularly simple ex- For a dispersed plug flow system with K large (K 1)
ample of this type of system since the adsorption process it can be shown that:
then involves equimolar exchange and the solutions, even σ 2 D L ε 1
v
for a large concentration step, are formally the same as for 2 = + (22)
2¯ t vL 1 − ε L kK
a linear trace component system.
where k is the overall mass transfer coefficient defined
according to Eq. (15).
2. Two-Transition Systems
The relationship with the familiar van Deemter equation
Such systems can be of any of the following types: (1) giving the HETP (height equivalent to a theoretical plate)
isothermal, two adsorbable components plus inert carrier; as a function of gas velocity,