Adsorption - Principles of Chemical Separations with Environmental Applications - page 200

if the system can be assumed to be isothermal. Since most of the heat is generated in the
MTZ where adsorption is occurring, the rate at which the heat can be transferred out of
this zone compared to the movement of the MTZ is the basis for one method [7]. This
comparison is shown by the “crossover ratio” R :

C p f ( X i − X res )

R = (7.7)

C p s ( Y i − Y o )

where Y is the molar ratio of sorbate to the carrier fluid (i denotes inlet, o denotes outlet),
and the fluid and sorbent heat capacities, C p f and C p s , include the effect of the sorbate.
X is the sorbent loading (wt sorbate/wt sorbent); X res is the residual loading in the bed
prior to the adsorption step. When R

1, the heat is easily removed from the MTZ and
adsorption can be assumed to be isothermal. As R approaches a value of 1, more and
more heat will be retained in the MTZ. An increase in the temperature of the “leading”
or breakthrough end of the MTZ will lower the equilibrium loading from the isothermal
value based on the inlet temperature and cause the curve to become less favorable relative
to the operating line, until ultimately the MTZ has no stable limit but continues to expand
as it moves through the bed. When R 1, the heat front is moving through the bed at the

same velocity as that of the MTZ, and essentially all the heat of adsorption is found in the
MTZ. For cases where R

=

1, the heat front will lag the adsorption front and heat will
be stored in the equilibrium section. Here the temperature rise will cause the equilibrium
loading to decrease. Thus, the crossover ratio is an indication of non-isothermal operation,
the extent of the harmful effects of the temperature rise due to adsorption, and the location
of the temperature change.
A second method computes the temperature rise under equilibrium conditions [10]

<

q H / C p g

T = T max − T f = (7.8)

( q / Y ) f − C p s / C p g

= /

where q solute adsorbed mass of sorbent

H = heat of sorption

Y = mass solute in fluid phase / mass of carrier gas

heat capacity,
and the subscripts are:
s

C p =

= sorbent

g = gas

f = feed.

For many operating conditions C p s / C p g ( q / Y ) f , so the above equation reduces to

T = Y f H / C p g . (7.9)