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174 3. Heterogeneous Processes and Reactor Analysis

Finally, the same equations can be used after the appropriate modifications for (Hopper
et al ., 2001)
• three-phase fluidized beds
• concurrent upflow three-phase fed bed (packed bubble bed reactor) ix
• slurry bubble column reactors for the continuous flow of gas and slurry

ver all r ate—F Material balances using an oogler’s approach
The above analysis, using the material balances for both phases for the gas-phase reactant A,
has been derived by Smith (1981). From eq. (3.367), it is obvious that the individual rates of
v mass transfer are not equal, and thus it is not possible to derie an overall rate based on the
gas-phase concentration of A. However, Fogler (1999) followed the overall rate approach and
proposed the use of a single material balance for the gas reactant A. Obviously, this can be
applied if in the material balance for A in the liquid phase (eq. (3.367)),
C d AL
u sL 0 (3.374)
z d

and axial dispersion is negligible. Condition (3.374) means that the concentration of A
remains practically constant along the reactor length due to very low liquid flow rates or
because of very low solubility of A. The last relation is also valid if the liquid phase is sat-
urated with A, thus resulting in a constant concentration of A in the liquid phase through-
out the reactor.
As in the fluidized beds analysis (Section 3.8.3), a similar simplif ication has been made
in Kunii–Levenspiel model for the material balances in the emulsion phase, where again the
corresponding derivatives have been omitted (eqs. (3.529) and (3.530)). As in the case of
liquid flow in trickle beds, the flow of the gas in the emulsion phase is considered too small
and so the superficial velocities can be neglected. Thus, in trickle beds, from eq. (3.367),

C  
( Ka )  AG C  ( k a ) C C 0 (3.375)
LG A AL f u A AL AS
L
 H A 
or
C  
( k a
( Ka LG A )  AG C AL  f u A ) C AL C AS (3.376)
L
 H A 
From the definition of the effectiveness factor and for a first-order reaction with respect to A

( ka ) C C k C (3.377)
fu A AL AS b m AS
Now, w wing the method used for slurry bubble columns orking these equations and follo
(see Section 3.4.6), C AS can be eliminated and an oerall reaction rate can be written in v
terms of the gas-phase concentration:

( R ) (3.378)
C
K
A AG