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146 3. Heterogeneous Processes and Reactor Analysis

where
T , T , T w the fluid, solid, and wall temperature, respecti ely (K) v
s
f
c p ,f , c p ,s the specific heat of the fluid and solid, respecti K) v ely (J/kg
f the fluid density (kg/m 3 )
H the heat of reaction (J/mol). Heat of reaction is nee for an v gati
v exothermic reaction and positie for an endothermic reaction.
L the axial ef m K)
fecti
v
vity (J/s
e thermal conducti
R the radial ef (J/s m K)
e thermal conducti
v
vity
fecti
h o the oerall heat transfer coef m v f icient (J/s 2 K)
D the bed diameter (m).
Note that on the right-hand side of eq. (3.301), the accumulation term is a function of the
temperature change of the fluid and solid with time. The heat generated (or consumed)
during the reaction increases (or decreases) the temperature in the solid and at the same
time is transferred to (or is transferred from) the fluid phase.
The last term on the left-hand side of eq. (3.301) corresponds to the heat transfer to the
v
all.
external fed-bed wThe oerall heat transfer resistance is the sum of the internal,
ix
external, and wall resistances. In an adiabatic operation, the oerall heat transfer coef v i- f
cient is zero so the corresponding term in the energy balance expression drops out, while
f in an isothermal operation this coeficient is inf so that inite, T f T s T w .
In general, large industrial fed beds operate under near-adiabatic conditions, whereas ix
small laboratory-scale fed beds may approach isothermal operation (Ruthv 1984). en,
ix
Especially, for most environmental applications, for catalytic, adsorption, and ion-
exchange operations, the species to be remoed are in such low concentrations that the
v
operarion is nearly isothemral. Thus, the heat transfer to the external fed-bed wall is ix
often of minimal importance.
The energy balance (3.301) is applicable for catalysis, adsorption, and ion e xchange.
More specifically, in catalysis, where the steady-state condition e frequently the accu- xists,
mulation term is zero. In contrast, adsorption and ion exchange operate under unsteady-
state condition. The analysis of the energy balance equation for catalytic fed beds is ix
presented in detail in Section 5.3.4.

Pressure drop equation
ix For a fed bed, the Ergun equation for pressure drop in a differential form is

 2 2 
)
)
d P 150 u
(1 1.75 u (1
 s s  (3.302)
d z   2 s d 2 p 3 s d p 3  
where d p is the diameter of spherical particles or the nominal diameter of irre gular -shaped
particles, is the density of the fluid, g is the gravity acceleration constant (9.81 m 2 /cm),
is the dynamic viscosity of the fluid, is the f ix ed-bed v oidage, u is the superf icial fluid
s
velocity, and is the sphericity of the particle. In eq. (3.302), the pressure-dependent
S
icial v parameters are the superfelocity and the gas density, which in turn are related to the