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3.1 Introduction to Heterogeneous Processes 69

Then

( r s ) r g C s S k 2 C ( C g G s ) (3.40)
k
Working as in the first-order reaction case, it can be proved that after the elimination of the
e surface concentration we ha v

k k g 2 k k C 4 gs G
g
C s (3.41)
k 2 s
It is obvious that the reaction rate becomes a complicated expression with the introduction
of the surface concentration.

k g
r ov r ( r ) g kC 2 s G k g k g k kC 4 g s G (3.42)
s
k 2 s
In this case, we cannot work as in the case of the first-order reaction to derie more sim- v
ple expressions. However, the principle of the rate-controlling step is still applicable. If the
rate-controlling step is the diffusion in the gas f the oerall rate ( ilm, v r ov ), is

r ov k C g G (3.43)

If the rate-controlling step is the reaction rate, the oerall rate ( v r ov ), including the ef fec-
tiveness factor, is
r ov s s k C G 2 (3.44)

Three-phase systems
xist, In three-phase systems, twi.e. the gas b ubble–liquid interf ace and the liq- aces e o interf
uid–solid interface and thus, four mass-transfer steps and the corresponding films are
involved in the process (Figure 3.3)

• mass transfer from the bulk gas to the gas bace (gas-bubble f ubble–liquid interf ilm)
• mass transfer from the bubble interface to the bulk liquid (liquid film around the bubble)
• mass transfer to the solid surface (liquid film around the particle)
• mass transfer within the solid phase

All these mechanisms along with any reaction in the solid phase are considered to be
processes in series (Smith, 1981). In three-phase systems, three interface concentrations,
two in the gas–liquid interf ace C G,i and C L,i , and one in the liquid–solid interf ace C ,h v a e
s
to be eliminated. If equilibrium exists at the bubble–liquid interf ace, C G,i and C L,i are
related by Henry’s la w:
C G,i H L,i C (3.45)