5 Two-Step Exothermic Chemical Reaction for Heat Transfer Investigation …
55
5.3.1 Effects of Thermo-physical Parameters on Temperature
(Unsteady State)
Figures 5.2, 5.3, 5.4, 5.5 and 5.6 show the effects of α, m, ε, ω and ϕ on the temperature behavior during combustion. From Figs. 5.2, 5.3, 5.4 and 5.5, it can be observed
that an increase in α (reaction rate parameter), m (chemical kinetics type), ε (activation energy) and ω (two-step exothermic chemical reaction) shows that the profiles
of temperature increase correspondingly. Therefore, a combusting stockpile will
increase its heat release rate under the effects of these parameters and fire may ultimately ignite if no control measures are done. In other words, these parameters
enhance spontaneous reaction of oxygen trapped within the stockpile with its carbon
or hydrocarbon substances to sustain the combustion process, which results with heat
release. A high temperature profile increase in Fig. 5.2 shows that α’s effect on the
exothermic chemical reaction is so intense to fast-track the combustion process for
emission of more heat within the stockpile. Figure 5.3 indicates that the chemical
reaction that is exothermic is very quick in the kinetic type of bimolecular and that it
slows down in the light induced one. In Fig. 5.6 a different scene is illustrated which
shows that an increase in ϕ (heat loss parameter) reduces the levels of temperature.
The parameter in the latter figure enhances the loss of more heat from the cylinder’s
surface to reduce the accumulation of heat inside the stockpile, which is the reason
for the decline in the levels of temperature.
5.3.2 Steady State Heat Transfer Analysis
The temperature independent state is represented by the following ordinary differential equation (Eq. (5.10)):
∂
2
θ
∂r 2 +
1
r
∂θ
∂r
+ α(1 + εθ)
m e
[θ/(1+εθ)]
+ αω(1 + εθ)
m e
[θ/(1+εθ)]
− ϕθ = 0,
(5.10)
with the boundary conditions as shown in Eq. (5.11)
∂θ
∂r
(0) = 0; θ (1) = 0.
(5.11)
The numerical solution in this case is obtained by using the Runge–Kutta Fehlberg
(RKF45) method coupled with Shooting technique. Equation (5.10) is reduced to 1
st
order ordinary differential equation by letting θ = s 1 , θ
= s 2 . We express Eqs.
(5.10)–(5.11) as Eqs. (5.12), (5.13) and (5.14):
s
1 = s 2
(5.12)
55
5.3.1 Effects of Thermo-physical Parameters on Temperature
(Unsteady State)
Figures 5.2, 5.3, 5.4, 5.5 and 5.6 show the effects of α, m, ε, ω and ϕ on the temperature behavior during combustion. From Figs. 5.2, 5.3, 5.4 and 5.5, it can be observed
that an increase in α (reaction rate parameter), m (chemical kinetics type), ε (activation energy) and ω (two-step exothermic chemical reaction) shows that the profiles
of temperature increase correspondingly. Therefore, a combusting stockpile will
increase its heat release rate under the effects of these parameters and fire may ultimately ignite if no control measures are done. In other words, these parameters
enhance spontaneous reaction of oxygen trapped within the stockpile with its carbon
or hydrocarbon substances to sustain the combustion process, which results with heat
release. A high temperature profile increase in Fig. 5.2 shows that α’s effect on the
exothermic chemical reaction is so intense to fast-track the combustion process for
emission of more heat within the stockpile. Figure 5.3 indicates that the chemical
reaction that is exothermic is very quick in the kinetic type of bimolecular and that it
slows down in the light induced one. In Fig. 5.6 a different scene is illustrated which
shows that an increase in ϕ (heat loss parameter) reduces the levels of temperature.
The parameter in the latter figure enhances the loss of more heat from the cylinder’s
surface to reduce the accumulation of heat inside the stockpile, which is the reason
for the decline in the levels of temperature.
5.3.2 Steady State Heat Transfer Analysis
The temperature independent state is represented by the following ordinary differential equation (Eq. (5.10)):
∂
2
θ
∂r 2 +
1
r
∂θ
∂r
+ α(1 + εθ)
m e
[θ/(1+εθ)]
+ αω(1 + εθ)
m e
[θ/(1+εθ)]
− ϕθ = 0,
(5.10)
with the boundary conditions as shown in Eq. (5.11)
∂θ
∂r
(0) = 0; θ (1) = 0.
(5.11)
The numerical solution in this case is obtained by using the Runge–Kutta Fehlberg
(RKF45) method coupled with Shooting technique. Equation (5.10) is reduced to 1
st
order ordinary differential equation by letting θ = s 1 , θ
= s 2 . We express Eqs.
(5.10)–(5.11) as Eqs. (5.12), (5.13) and (5.14):
s
1 = s 2
(5.12)