Numerical Investigation into CO 2 Emission , O 2 Depletion , and Thermal Decomposition in a Reacting Slab

The emission of carbon dioxide CO2 is closely associated with oxygen O2 depletion, and thermal decomposition in a reacting stockpile of combustible materials like fossil fuels e.g., coal, oil, and natural gas . Moreover, it is understood that proper assessment of the emission levels provides a crucial reference point for other assessment tools like climate change indicators and mitigation strategies. In this paper, a nonlinear mathematical model for estimating the CO2 emission, O2 depletion, and thermal stability of a reacting slab is presented and tackled numerically using a semi-implicit finite-difference scheme. It is assumed that the slab surface is subjected to a symmetrical convective heat and mass exchange with the ambient. Both numerical and graphical results are presented and discussed quantitatively with respect to various parameters embedded in the problem.


Introduction
Studies relating to transient heating of combustible materials due to exothermic oxidation chemical reactions are extremely important and have a wide range of applications in industry, engineering, and environmental science 1 .For instance, fossil fuels coal, oil, and natural gas account for 85% of world's primary energy supply, 70% of worlds electricity and heat generation and over 94% of energy for transportation 2 .The production and use of these combustible materials contribute up to 80% of CO 2 emission.Given expected increases in global population, economic growth, and energy demand, a continuous rise in emissions is expected unless fundamental technology changes occur in global energy systems which  are currently dominated by fossil fuels.The CO 2 pollution is the principal human cause of global warming and climate change 3 .Meanwhile, for proper assessment of the CO 2 emission and O 2 depletion levels together with their impact on both the environment and life on Earth, knowledge of the mathematical models of these complex chemical systems is essential.These provide crucial reference points for other assessment tools like thermal stability of the materials, climate change indicators, and mitigation strategies.It may also help in developing medium to long-term action plans for climate change research and reliable design of the systems 4 .
An extensive review of detailed chemical kinetic models for the heating-up of combustible materials is given by Simmie 5 .His review considered post-1994 work and focuses on the modeling of hydrocarbon fuel oxidation in the gas phase by detailed chemical kinetics and those experiments which validate them.Moreover, thermal combustion analysis has received much attention in the literature 6-8 .Several studies have been directed towards obtaining critical conditions for thermal ignition to occur in the form of a critical value for the Frank-Kamenetskii parameter 9 .Usually, chemical processes include many, up to a several hundred, intermediate elementary reactions 10 .For example, in combustion science, it is very common to use complex multistep reaction mechanisms to predict the oxidation of hydrocarbons 11 .However, the use of one-step decomposition kinetics clearly simplifies the complicated chemistry involved in the problem but is both practical and necessary without additional information about the individual decomposition reaction steps 12, 13 .Meanwhile, analytical solutions of the partial differential equations governing transient heating of the combustible material undergoing oxidation reactions are usually impossible or extremely difficult to obtain.The exothermic nature of such reactions leads to complex nonlinear transient interaction of heat conduction, mass diffusion, and chemical reactions, resulting in steep concentration and temperature gradients 14 .In such circumstances, a better understanding of the system behavior can only be accomplished by conducting numerical simulations to capture the frontal behavior of the processes.
The basic objective of this study is to provide a numerical estimate for the thermal stability together with the rate of CO 2 emission and O 2 depletion in transient heating of a slab of combustible material in the presence of convective heat and mass exchange with the ambient at the slab surface.The mathematical formulation of the problem is established in section two.In section three, the semi-implicit finite difference technique is implemented to tackle the problem.Both numerical and graphical results are presented and discussed quantitatively with respect to various parameters embedded in the system in Section 4.

Mathematical Model
We consider a stockpile of combustible material in a rectangular slab.It is assumed that the slab is undergoing an nth-order oxidation chemical reaction, and its surface is subjected to a symmetrical convective heat and mass exchange with the ambient see Figure 1 .The complicated chemistry involved in this problem can be simplified by assuming a one-step finite-rate irreversible reaction between the combustible material hydrocarbon and the oxygen of the air; that is, Following 1, 5, 6, 12, 14 , the nonlinear partial differential equations describing temperature, oxygen, and carbon dioxide concentration in the combustible material can be written as with initial and boundary conditions as follows:

−D
where T is the absolute temperature, C is the depletion oxygen concentration, P is the carbon dioxide emission concentration, T a is the ambient temperature, C a is the oxygen concentration in the surrounding air, P a is the carbon dioxide concentration in the surrounding air, t is the time, T 0 is the slab initial temperature, the initial depletion of oxygen in the slab is zero, ρ is the density, c p specific heat at constant pressure, k is the thermal conductivity of the reacting slab, D is the diffusivity of oxygen in the slab, γ is the diffusivity of carbon dioxide in the slab, Q is the exothermicity, A is the rate constant, E is the activation energy, R is the universal gas constant, l is the Planck number, K is the Boltzmann constant, ν is the vibration frequency, a is the slab half width, y is the distance measured transverse direction, h 1 is the coefficient of heat transfer between the slab and its surroundings, h 2 is the coefficient of oxygen transfer between the slab and its surroundings, h 3 is the coefficient of carbon dioxide transfer between the slab and its surroundings, n is the order of exothermic chemical reaction, and m is the numerical exponent such that m ∈ {−2, 0, 0.5}.The three values taken by the parameter m represent the numerical exponent for sensitized, Arrhenius and Bimolecular kinetics, respectively, see 1, 6 .We introduce the following dimensionless variables into 2.2 -2.3 : and we obtain the dimensionless governing equations as

2.5
The corresponding initial and boundary conditions then become θ y, 0 0, Φ y, 0 0.5, Ψ y, 0 0, ∂Φ ∂y where λ, ε, β 1 , β 2 , α, σ, Bi 1 , Bi 2 , and Bi 3 represent the Frank-Kamenetskii parameter, activation energy parameter, oxygen consumption rate parameter, carbon dioxide emission rate parameter, oxygen diffusivity parameter, carbon dioxide diffusivity parameter, the thermal Biot number, oxygen Biot number, and carbon dioxide Biot number, respectively.A body of material releasing heat to its surroundings may achieve a safe steady state where the temperature of the body reaches some moderate value and stabilizes.However, when the rate of heat generation in the material exceeds the rate of heat loss to the surroundings, then ignition can occur.In the following sections, 2.5 -2 are solved numerically using a semi-implicit finite difference scheme.

Numerical Solution
Our numerical algorithm is based on the semi-implicit finite difference scheme 15-18 .The implicit terms are taken at the intermediate time level N ξ where 0 ≤ ξ ≤ 1.The algorithm employed in 18 uses ξ 1/2, we will, however, follow the formulation in 15-17 and take ξ 1 in this paper so that we can use larger time steps.In fact being nearly fully implicit, our numerical algorithm presented in this paper is conjectured to work for any value of the time step!The discretization of the governing equations is based on a linear Cartesian mesh and uniform grid on which finite differences are taken.We approximate both the second and first spatial derivatives with second-order central differences.The equations corresponding to the first and last grid points are modified to incorporate the boundary conditions.The semiimplicit schemes for the temperature, O 2 concentration, and CO 2 concentration, respectively, read The equations for θ N 1 , Φ N 1 , and Ψ N 1 , thus, become −rξθ −rξσΨ where r Δt/Δy 2 .The solution procedures reduce to inversion of tridiagonal matrices.The schemes 3.2 were checked for consistency.For ξ 1, these are first-order accurate in time but second order in space.The schemes in 18 have ξ 1/2 which improves the accuracy in time to second order.We use ξ 1 here so that we are free to choose larger time steps and still converge to the steady solutions.As already conjectured, our algorithm works for any value of the time step!The code was checked for both spatial and temporal convergence.In particular, solutions calculated from, say Δt 1, using 200 time steps are exactly the same as those after 40 time steps with Δt 5 or those after 20,000 time steps with Δt 0.01.Similarly solutions using Δy 0.02 converge to the same results as those say for Δt 0.025 or Δt 0.05, and so forth.Our code, thus, runs extremely fast, and; hence, we can easily obtain and, thus, present all our results at steady state using nearly insignificant computational times.

Results and Discussion
Unless otherwise stated, we employ the parameter values: These will be the default values in this work, and; hence, in any graph where any of these parameters is not explicitly mentioned, it will be understood that such parameters take on the default values.

Transient and Steady Flow Profiles
We display the transient solutions in Figures 2, 3, and 4. At the given parameter values, Figure 2 shows a transient increase in temperature until a steady-state is reached.A similar scenario obtains in Figure 4 in which a transient increase of carbon dioxide emission is observed until a steady state is reached.An opposite situation is noticed in Figure 3 where a decrease in oxygen concentration is observed with increasing time until a steady-state concentration is attained.These results are consistent with intuition regarding exothermic oxidation reactions.

Parameter Dependance of Solutions in Steady State
It is understood from Figures 2-4 that, at the default parameter values, solutions have reached steady state at, say, times t ≥ 40.All the solutions at t 200 given below will, thus, be understood to be steady solutions.The dependence of solutions on parameter n is illustrated in Figures 5, 6, and 7.As n increases, the results show a decrease in both the temperature and the CO 2 emission.The reduced oxidation reactions mean lower oxygen consumption, hence, lead to a corresponding increase in O 2 concentration.
The dependence of solutions on parameter m is illustrated in Figures 8, 9, and 10.As m increases, the results show an increase in both the temperature and the CO 2 emission.The increased oxygen consumption from higher oxidation reactions correspondingly decreases O 2 concentration.
The dependence of solutions on parameter Bi 1 is illustrated in Figures 11, 12, and 13.As Bi 1 increases, the results show a decrease in both the temperature and the CO 2 emission.The reduced oxidation reactions mean lower oxygen consumption, hence, lead to a corresponding increase in O 2 concentration.
The dependence of solutions on parameter Bi 2 is illustrated in Figures 14, 15, and 16.An increase in Bi 2 directly corresponds to an increase in O 2 concentration.This correspondingly increases the oxidation reactions and, hence, increases the temperature and, hence, the CO 2 emission.The dependence of CO 2 emission on Bi 3 is shown in Figure 17.An increase in Bi 3 decreases the CO 2 emission.No noticeable effects are observed in the temperature and the O 2 concentration.
The dependence of solutions on the reaction parameter λ is illustrated in Figures 18,  19, and 20.An increase in λ directly corresponds to an increase in the exothermic oxidation reactions and, hence, increases the temperature and, hence, the CO 2 emission.The increased oxidation reactions increase oxygen consumption and thus decrease O 2 concentration.
The dependence of solutions on parameter β 1 is shown in Figures 21, 22, and 23.An increase in β 1 directly corresponds to a decrease in O 2 concentration.This correspondingly decreases the oxidation reactions and, hence, decreases the temperature and also CO 2 emission.
The dependence of CO 2 emission on β 2 is shown in Figure 24.An increase in β 2 increases the CO 2 emission.No noticeable effects are observed in the temperature and the O 2 concentration.
The dependence of solutions on the parameter ε is illustrated in Figures 25, 26, and 27.An increase in ε decreases the temperature and the CO 2 emission.The reduced oxygen consumption leads to an increase in O 2 concentration.The dependence of solutions on the parameter α is shown in Figures 28, 29, and 30.An increase in α directly corresponds to an increase in O 2 concentration.This correspondingly increases the oxidation reactions and hence increases the temperature and also CO 2 emission.
The dependence of CO 2 emission on σ is shown in Figure 31.An increase in σ decreases the CO 2 emission.No noticeable effects are observed in the temperature and O 2 concentration.

Thermal Stability and Blowup
In Figures 32, 33, 34, and 35, we plot θ y 1, t against λ for varying values of n, m, Bi 1 , and Bi 2 , respectively.The solutions are given up to the values of λ at which the onset of blowup in temperature is observed.We notice that parameters that increase the temperature θ correspondingly increase θ y 1, t .The results also show that early onset of blowup and, hence, thermal instability can be delayed by using higher values of n, lower values of m, higher values of Bi 1 , lower values of Bi 2 , and so forth.

Conclusion
We develop an unconditionally stable works for any time step size and convergent semiimplicit finite-difference scheme and utilize it to computationally investigate the transient dynamics of CO 2 emission, O 2 depletion, and thermal decomposition in a reacting slab.The solutions show that those processes that increase the oxidation reactions lead to enhanced oxygen depletion as well as increased carbon dioxide emission.The results also show that enhanced thermal stability and reduced carbon dioxide emission are best achieved by cutting down on those processes that would otherwise increase the oxidation reactions.

Figure 1 :
Figure 1: Geometry of the problem.

Figure 5 :
Figure 5: Effects of n on temperature.

Figure 6 :
Figure 6: Effects of n on O 2 .

5 Figure 32 : 5 Figure 33 :
Figure 32: Effects of variation of n on the thermal criticality or blowup in the system.

Figure 34 : 1 Bi 2 = 2 Figure 35 :
Figure 34: Effects of Bi 1 variation on the thermal criticality or blowup in the system.