Page 39 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

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Adsorption (Chemical Engineering) 261

2
∂ c i ∂ ∂c i 1 − ε ∂ ¯ q i
−D L 2 + (vc i ) + + = 0 (17)
∂z ∂z ∂t ε ∂t
(D L is the axial dispersion coefficient, z distance, v the
interstitialfluidvelocity,andε thevoidageoftheadsorbent
bed) together with the adsorption rate expression for each
component, which can be written in the general form:
∂ ¯ q
i = f (¯ q i , q s ,... ; c i , c j ,... ; T ) (18)
∂t
It should be understood that this rate expression may in
factrepresentasetofdiffusionandmasstransferequations
with their associated boundary conditions, rather than a
simple explicit expression. In addition one may write a
differential heat balance for a column element, which has
the same general form as Eq. (17), and a heat balance for
FIGURE 7 Schematic diagram showing (a) approach to heat transfer between particle and fluid. In a nonisothermal
constant-pattern behavior for a system with favorable equilib- system the heat and mass balance equations are therefore
rium and (b) approach to proportionate-pattern limit for a system
coupled through the temperature dependence of the rate
with unfavorable isotherm. Key: c/c 0 , ——; q /q 0 ,–– –; c /c 0 , –·–.
∗
of adsorption and the adsorption equilibrium, as expressed
(Reprinted with permission from Ruthven, D. M. (1984). “Prin-
ciples of Adsorption and Adsorption Processes,” copyright John in Eq. (18).
Wiley & Sons, New York.) Solving this set of equations is a difficult task, and some
simplification is therefore generally needed. Some of the
simplified systems for which more or less rigorous solu-
tions have been obtained are summarized below.
concentration front to broaden due to the effects of mass
For a system with n components (including nonad-
transfer resistance and axial dispersion is exactly balanced
sorbable inert species) there are n − 1 differential mass
by the self-sharpening effect arising from the variation of
balance equations of type (17) and n − 1 rate equations
the characteristic velocity and concentration. Once this
[Eq. (18)]. The solution to this set of equations is a set
state is reached the concentration profile propagates with-
of n − 1 concentration fronts or mass transfer zones sepa-
out further change in shape. This is the basis of the LUB
rated by plateau regions and with each mass transfer zone
(length of unused bed) method of adsorber design, which
propagating through the column at its characteristic veloc-
is considered in greater detail [see Eq. (25)].
ity as determined by the equilibrium relationship. In addi-
In the case of an unfavorable isotherm (or equally for
tion, if the system is nonisothermal, there will be the dif-
desorption with a favorable isotherm) a different type of
ferential column heat balance and the particle heat balance
behavior is observed. The concentration front or mass
equations, which are coupled to the adsorption rate equa-
transfer zone, as it is sometimes called, broadens con-
tion through the temperature dependence of the rate and
tinuously as it progresses through the column, and in a
equilibrium constants. The solution for a nonisothermal
sufficiently long column the spread of the profile becomes
system will therefore contain an additional mass transfer
directly proportional to column length (proportionate pat-
zone traveling with the characteristic velocity of the tem-
tern behavior). The difference between these two limiting
perature front, which is determined by the heat capacities
types of behavior can be understood in terms of the rel-
of adsorbent and fluid and the heat of adsorption. A non-
ative positions of the gas, solid, and equilibrium profiles
isothermal or adiabatic system with n components will
for favorable and unfavorable isotherms (Fig. 7).
therefore have n transitions or mass transfer zones and
as such can be considered formally similar to an (n + 1)-
component isothermal system.
A. Mathematical Modeling
The number of transitions or mass transfer zones pro-
The pattern of flow through a packed adsorbent bed can vides a direct measure of the system complexity and there-
generally be described by the axial dispersed plug flow fore of the ease or difficulty with which the behavior can
model. To predict the dynamic response of the column be modeled mathematically. It is therefore convenient to
therefore requires the simultaneous solution, subject to classifyadsorptionsystemsinthemannerindicatedinSec-
the appropriate initial and boundary conditions, of the tion V.B. It is generally possible to develop full dynamic
differential mass balance equations for an element of the models only for the simpler classes of systems, involving
column, one, two, or at the most three transitions.