REACTORS: IDEALIZED CHEMICAL REACTORS - Biomass as a Sustainable Energy Source for the Future - page 166

=

Rate of accumulation X m i c p , i dT
dt

Rate of supply = 0 ð − rate of release = 0 ðÞ +

Þ Q Rate of production V ð − Δ r H Þ − ð R A X A ,T ð Þ Þ Þ + heating or cooling ð

ð Eq : 6 : 14 Þ

The term Q represents any addition or removal of heat from the reactor. The energy
balance states that the variation of temperature in the reactor (dT/dt) depends on four
key factors, namely, the rate of heat generation V(

R A ( X A , T)), the heat added
to or removed from the system Q , the mass of the species m i , and the specific heat
capacity of the species involved c p ,i , which in this case is considered constant.
A strongly exothermic reaction such as combustion (

− Δ r H )( −

Δ r H has a large negative value)
will increase (dT/dt) unless the removal of heat Q counteracts the production of heat.
The term Q can be manipulated to get the desired (dT/dt) in the reactor by heating the
reactor or cooling it.
The solution of the energy balance leads to

X m i c p , i dT

+ Q

dt =V

ð − Δ r H Þ − ð R A X A ,T ð Þ Þ ð Eq : 6 : 15 Þ

For adiabatic systems, Q = 0. If the reactor is heated/cooled by interchanging
heat with an external fluid, e.g., through a heat exchange coil,

Q will have the form

Q = hA T f −

.T f is the temperature of the heating/cooling fluid, T is the temperature
of the reactor, h is the heat transfer coefficient, and A is the heat exchange area.
Substituting Equation (6.12) in Equation (6.15), we obtain

ð Þ T

X m i c p , i dT

n A0 d X A
dt

= Q

dt −− Δ r H

ð Þ

if = 0 ! adiabatic ð Eq : 6 : 16 Þ

Thus,

= ð

t

X m i c p , i T

Q dt

Tð Þ−− Δ r H − ð Þ n A0 X A − ð X A 0 Þ

0

= Q t

ð Eq : 6 : 17 Þ

Q = constant

ð if Þ

if = 0 ! adiabatic

Although in some cases it is possible to obtain an analytical solution, most of the cases
require numerical methods to solve the combined mass and energy balances.