8.2 Heat Balance and Reactor Stability
205
specific heat capacity and the change in temperature as:
˙
q acc = c p
dT
dt
(8.6)
• ˙
q acc : rate of heat accumulation [W/kg]
• c p : specific heat capacity of the reaction mixture [J kg −1 K −1 ]
• T : temperature of the reaction mass [K]
• t: time [s]
• Assumptions: c p is temperature independent, and the heat capacity of the
equipment can be neglected 5
The specific heat capacity of a mixture (c p mix ) can be estimated from the mass
(m) and specific heat capacities (c p ) of the individual components (i) in the mixture
as:
c p mix =
i (m i × c p i )
i m i
(8.7)
8.2.5 Semenov Diagram and the Critical Temperature
Substituting each of the these three definitions for ˙
q acc , ˙
q r , and ˙
q c in Eq. 8.1, the
simplified heat balance for an exothermic batch reaction can be expressed as:
c p
dT
dt
˙
q acc
accumulation
=
1
ρ
(−r A )(− R )
˙
q r
reaction
−
U
ρ
A
V
(T − T c )
˙
q c
external cooling
(8.8)
A helpful way to describe and analyze the heat balance and a reactor’s resulting
stability during normal operating conditions is by plotting the heat balance 6 on a
Semenov diagram, as shown in Fig. 8.2. In this diagram, the heat produced by the
reaction ( ˙
q r ) and the heat removed by the cooling system ( ˙
q c ) are plotted against
reaction temperature (T ). As shown, an increase in reaction temperature, according
to the Arrhenius relationship in Eq. 8.4, is associated with an exponential increase
in the reaction rate and hence in the heat produced. In contrast, heat removal by an
external cooling medium only increases linearly with reaction temperature.
The slope of the heat removal line ( ˙
q c ) is proportional to the overall heat transfer
coefficient (U ) and the ratio of the cooling area (A) to the reaction volume (V ).
5 For stirred tank reactors, such as batch reactors, the heat capacity of the reactor is often negligible
compared to the heat capacity of the reaction mass.
6 This heat balance considers zero-order kinetics, which is a conservative approximation used in
the context of thermal process safety.
205
specific heat capacity and the change in temperature as:
˙
q acc = c p
dT
dt
(8.6)
• ˙
q acc : rate of heat accumulation [W/kg]
• c p : specific heat capacity of the reaction mixture [J kg −1 K −1 ]
• T : temperature of the reaction mass [K]
• t: time [s]
• Assumptions: c p is temperature independent, and the heat capacity of the
equipment can be neglected 5
The specific heat capacity of a mixture (c p mix ) can be estimated from the mass
(m) and specific heat capacities (c p ) of the individual components (i) in the mixture
as:
c p mix =
i (m i × c p i )
i m i
(8.7)
8.2.5 Semenov Diagram and the Critical Temperature
Substituting each of the these three definitions for ˙
q acc , ˙
q r , and ˙
q c in Eq. 8.1, the
simplified heat balance for an exothermic batch reaction can be expressed as:
c p
dT
dt
˙
q acc
accumulation
=
1
ρ
(−r A )(− R )
˙
q r
reaction
−
U
ρ
A
V
(T − T c )
˙
q c
external cooling
(8.8)
A helpful way to describe and analyze the heat balance and a reactor’s resulting
stability during normal operating conditions is by plotting the heat balance 6 on a
Semenov diagram, as shown in Fig. 8.2. In this diagram, the heat produced by the
reaction ( ˙
q r ) and the heat removed by the cooling system ( ˙
q c ) are plotted against
reaction temperature (T ). As shown, an increase in reaction temperature, according
to the Arrhenius relationship in Eq. 8.4, is associated with an exponential increase
in the reaction rate and hence in the heat produced. In contrast, heat removal by an
external cooling medium only increases linearly with reaction temperature.
The slope of the heat removal line ( ˙
q c ) is proportional to the overall heat transfer
coefficient (U ) and the ratio of the cooling area (A) to the reaction volume (V ).
5 For stirred tank reactors, such as batch reactors, the heat capacity of the reactor is often negligible
compared to the heat capacity of the reaction mass.
6 This heat balance considers zero-order kinetics, which is a conservative approximation used in
the context of thermal process safety.