5 Two-Step Exothermic Chemical Reaction for Heat Transfer Investigation …
51
Fig. 5.1 The geometry of the problem
Above equations represent a two-step combustion of the methane gas in air, where
the second step helps to eliminate the toxic gas carbon monoxide. Heat loss to the
environment is assumed to be convective and follows Newton’s law of cooling with
the expression −
h
k
[T − T w ], where h is the heat transfer coefficient, k is the thermal
conductivity of the material, T is the absolute temperature of the cylinder, and T w
is the ambient temperature. The geometry of the problem is illustrated in Fig. 5.1
below.
Equation (5.3) below represents the transient heat energy equation for chemical
reaction [10, 12–15],
pc p
∂ T
∂
−
t
=
k
−
r
∂
∂
−
r
−
r
∂ T
∂r
+ Q 1 A 1 C 1
K T
vl
m
e
−E 1 /RT
+ Q 2 A 2 C 2
K T
vl
m
e
−E 2 /RT
−
h
k
(T − T w )
(5.3)
The respective initial and boundary conditions are given in Eq. (5.4):
T
−
r , 0
= T 0 ;
∂ T
∂
−
r
0,
−
t
= 0; T
a,
−
t
= T w
(5.4)
In this case, T 0 is the cylinder initial temperature, Q 1 is the first step heat of
reaction,A 1 is the first step rate constant and C 1 the first step reactant’s concentration.
It follows that Q 2 ,A 2 and C 2 , are respectively, the second step’s heat of reaction, rate
constant, and reactant’s concentration. E 1 and E 2 are the activation energies for the
first and second steps, respectively. m is for the type of kinetics, where m = −2 is for
light induced kinetics, the Arrhenius kinetics is represented by m = 0 and m = 0.5
presents bimolecular type.
The following dimensionless variables were introduced as Eq. (5.5):
θ =
E 1 (T − T b )
RT 2
b
, θ 0 =
E 1 (T 0 − T b )
RT 2
b
, r =
¯
r
a
, ε =
RT b
E 1
, ε =
E 2
E 1
, t =
k ¯
t
pc p a 2 , ϕ =
a 2 h
k
,
ω =
Q 2 A 2 E 2
Q 1 A 1 E 1
e
(E 1 −E 2 )/RT
, α =
K T b
vl
m Q 1 A 1 E 1 a
2 C 1
k RT
2
b
e
−E 1 /RT
.
(5.5)
51
Fig. 5.1 The geometry of the problem
Above equations represent a two-step combustion of the methane gas in air, where
the second step helps to eliminate the toxic gas carbon monoxide. Heat loss to the
environment is assumed to be convective and follows Newton’s law of cooling with
the expression −
h
k
[T − T w ], where h is the heat transfer coefficient, k is the thermal
conductivity of the material, T is the absolute temperature of the cylinder, and T w
is the ambient temperature. The geometry of the problem is illustrated in Fig. 5.1
below.
Equation (5.3) below represents the transient heat energy equation for chemical
reaction [10, 12–15],
pc p
∂ T
∂
−
t
=
k
−
r
∂
∂
−
r
−
r
∂ T
∂r
+ Q 1 A 1 C 1
K T
vl
m
e
−E 1 /RT
+ Q 2 A 2 C 2
K T
vl
m
e
−E 2 /RT
−
h
k
(T − T w )
(5.3)
The respective initial and boundary conditions are given in Eq. (5.4):
T
−
r , 0
= T 0 ;
∂ T
∂
−
r
0,
−
t
= 0; T
a,
−
t
= T w
(5.4)
In this case, T 0 is the cylinder initial temperature, Q 1 is the first step heat of
reaction,A 1 is the first step rate constant and C 1 the first step reactant’s concentration.
It follows that Q 2 ,A 2 and C 2 , are respectively, the second step’s heat of reaction, rate
constant, and reactant’s concentration. E 1 and E 2 are the activation energies for the
first and second steps, respectively. m is for the type of kinetics, where m = −2 is for
light induced kinetics, the Arrhenius kinetics is represented by m = 0 and m = 0.5
presents bimolecular type.
The following dimensionless variables were introduced as Eq. (5.5):
θ =
E 1 (T − T b )
RT 2
b
, θ 0 =
E 1 (T 0 − T b )
RT 2
b
, r =
¯
r
a
, ε =
RT b
E 1
, ε =
E 2
E 1
, t =
k ¯
t
pc p a 2 , ϕ =
a 2 h
k
,
ω =
Q 2 A 2 E 2
Q 1 A 1 E 1
e
(E 1 −E 2 )/RT
, α =
K T b
vl
m Q 1 A 1 E 1 a
2 C 1
k RT
2
b
e
−E 1 /RT
.
(5.5)