REACTORS: IDEALIZED CHEMICAL REACTORS - Biomass as a Sustainable Energy Source for the Future - page 171

The enthalpy introduced by the flow to a slice of the reactor is
X

+ X

φ n , i

φ n , i H i T re ðÞ c p , i T ð − T ref Þ ð Eq : 6 : 31 Þ

The enthalpy that leaves the slice is
X

+ X

φ n , i

φ n , i H i T re ðÞ c p , i T+dT ð − T ref Þ ð Eq : 6 : 32 Þ

The enthalpy generated in the slice due to the reaction of component A is

dV ð − Δ r H Þ − ð R A X A ,T ð Þ Þ ð Eq : 6 : 33 Þ

If the reactor operates in steady state, there is no accumulation of heat in the slice.
The energy balance then reads

Rate of accumulation = rate of supply − rate of release + rate of production

+ heating = cooling )

0= X + X

X

φ n , i

φ n , i H i T re ðÞ c p , i T ð − T ref Þ− φ n , i H i T re ðÞ

+dV

+ X

+d Q φ n , i c p , i T+dT ð − T ref Þ − Δ r H Þ − ð R A X A ,T ð Þ Þ ð Eq : 6 : 34 Þ

This balance neglects any shaft work introduced to the system, for instance, the work
done by the turbine that pumps the flow. Simplifying the balance, we obtain

X

+d Q

φ n , i c p , i dT = dV ð − Δ r H Þ − R A X A ,T ð ð Þ Þ ð Eq : 6 : 35 Þ

Dividing by dV and rearranging gives

+ d Q
dV

dV = − Δ r H

ð Þ − ð R A X A ,T ð Þ Þ

dT

X

ð Eq : 6 : 36 Þ

φ n , i c p , i

Q = hA T f −

The heat removed from the system by the coolant, ð Þ T , can be expressed as

d Q = h T f −

= : :

ð Þ T d A = h 4 ðÞ d ð Þ T f − T dV ð Eq 6 37 Þ

where d is the reactor diameter, h is the heat transfer coefficient, and T f is the temper-
ature of the coolant. Substituting the expression for d Q in Equation (6.36), we obtain
the final expression for the temperature gradient inside the reactor:

dT
dV =

ð − Δ r H Þ − ð R A X A ,T ð Þ Þ + h 4 ðÞ = d ð Þ T f − T

X

ð Eq : 6 : 38 Þ

φ n , i c p , i