Numerical investigation of CO 2 emission and thermal stability of a convective and radiative stockpile of reactive material in a cylindrical pipe

Abstract

In this article, we investigate the combined effects of emission of CO₂ and O₂ depletion on thermal stability in a long cylindrical pipe of combustible reactive material. The cylindrical pipe loses heat by convection and radiation at the surface, and the nonlinear differential equations governing the heat and mass transfer problem are tackled numerically using Runge–Kutta–Fehlberg method coupled with shooting technique. The effects of various thermo-physical parameters on the temperature, CO₂ and O₂ fields, and thermal stability are presented graphically and discussed quantitatively.

Introduction

The study of CO₂ emission and thermal stability due to exothermic chemical reaction in a stockpile of combustible reactive materials is of importance in the study of the environment and understanding of heat transfer in engineering processes. Moreover, spontaneous ignition in stockpiles of combustible materials such as industrial waste fuel, coal, hays, and wool wastes is usually experienced leading to loss of life and properties. ¹ For instance, Figure 1 illustrates hazard that a self-ignited stockpile of cylindrical hays can cause to properties and the ambient environment.

Figure 1.

Ignited cylindrical hays.

Theoretical study of thermal stability in a stockpile of combustible materials due to exothermic chemical reactions has a wide range of industrial applications. These applications include processes such as heavy oil recovery, incineration of waste material, storage of cellulose material, combustion of solids, pyrolysis of biomass and coal, and also in design of internal combustion engines and automobile exhaust systems.^2–4 The heat generated as a result of exothermic chemical reaction in a stockpile of combustible material may exceed the heat loss to the surrounding environment, leading to the phenomenon called thermal explosion.^5–8 Analysis of thermal explosion criticality was explored by Frank-Kamenetskii ⁹ who also developed the steady-state theory of exothermic reactive materials. This theory is very helpful in the determination of the critical values, which are limit values not to be exceeded in order to avoid self-ignition from taking place, so that explosions can be controlled in exothermic chemical reactions.

Mathematical modeling has provided useful insight into this phenomenon in order to achieve reliable thermal system design that is of importance in respect of safety and hazard assessment.^9–11 Previous research study has shown that at most 80% of CO₂ emission that contributes to the Greenhouse effect may be due to exothermic chemical reactions in stockpiles of combustible reactive materials. ¹² There is need, therefore, for more knowledge that can help to control CO₂ emission for the preservation of O₂ that is essential for life. The complicated chemistry involved in combustion of reactive material in a stockpile is tackled in Williams, ¹³ Makinde, ¹⁴ and Legodi and Makinde ¹⁵ using one-step decomposition kinetics. The combustion reaction due to exothermic chemical reaction is very complicated and includes many radicals, especially when large hydrocarbons are involved. ¹⁶ The process involves highly nonlinear interaction of reacting species with products’ diffusion and heat conduction which ultimately bring about reactants, products, and temperature gradients that are steep in nature. ¹⁷ Detailed reviews of models that involve chemical kinetics for reactions of hydrocarbons with oxygen are outlined in Simmie. ¹⁸ Due to the highly nonlinear nature of the differential equations describing these complicated systems, it is practically impossible to obtain an exact solution to the problem; hence, it is therefore necessary to apply numerical or semi-numerical schemes in order to obtain an approximate solution. One of such methods is the application of a special type of Hermite–Pade approximation technique^1,2,19,20 coupled with perturbation method to solve nonlinear differential equations modeling thermal stability in exothermic reactive materials in various geometrical configurations.

The purpose of this article is to examine the complex interaction of heat and mass transfer in an exothermic chemically reactive and thermally radiating material in cylindrical geometry with respect to oxygen (O₂) depletion and carbon dioxide (CO₂) emission. The thermal stability analysis is performed on the model nonlinear differential equations using shooting method coupled with the Runge–Kutta–Fehlberg integration scheme. Pertinent results are displayed graphically and discussed quantitatively.

Mathematical model

An exothermic chemical reaction in a stockpile of combustible material is studied in a long cylindrical pipe model. The cylindrical pipe of combustible reactive material is considered to be of constant thermal conductivity k with surface emissivity $\epsilon$ . It is assumed that the reactive cylinder is undergoing an nth order oxidation chemical reaction and that it is radiating with convection. A one-step finite rate irreversible chemical kinetics mechanism between the material and the oxygen of the air is assumed, and it is represented as follows ¹¹

$${\mathrm{C}}_i{\mathrm{H}}_j+(i+\frac{j}{4}){\mathrm{O}}_2\to i\mathrm{C}{\mathrm{O}}_2+\left(\frac{j}{2}\right){\mathrm{H}}_2\mathrm{O}+\mathrm{heat}$$

Heat loss as a result of radiation at the cylindrical pipe surface to the surrounding environment is expressed as $q=\epsilon \sigma (T^4-T_0^4)$ , following Stefan–Boltzmann’s law. Convective heat loss at the surface of the cylinder obeys Newton’s law of cooling. Figure 2 illustrates the geometry of the problem.

Figure 2.

Geometry of the problem.

The steady-state nonlinear ordinary differential equations governing the heat and mass transfer problem in the presence of convection, thermal radiation, CO₂ emission, and O₂ depletion are given by^{1,2,12,15,19,20}

$$\frac{k}{\overline{r}}\frac{d}{d\overline{r}}\left(\overline{r}\frac{\mathrm{dT}}{d\overline{r}}\right)+\mathrm{QA}{\left(\frac{\mathrm{KT}}{\mathrm{vl}}\right)}^m{C}^n\exp \left(\frac{-E}{\mathrm{RT}}\right)-\epsilon \sigma (T^4-T_{\infty }^4)=0$$

(1)

$$\frac{D}{\overline{r}}\frac{d}{d\overline{r}}\left(\overline{r}\frac{\mathrm{dC}}{d\overline{r}}\right)-\mathrm{QA}{\left(\frac{\mathrm{KT}}{\mathrm{vl}}\right)}^m{C}^n\exp \left(\frac{-E}{\mathrm{RT}}\right)=0$$

(2)

$$\frac{\gamma }{\overline{r}}\frac{d}{d\overline{r}}\left(\overline{r}\frac{\mathrm{dP}}{d\overline{r}}\right)+\mathrm{QA}{\left(\frac{\mathrm{KT}}{\mathrm{vl}}\right)}^m{C}^n\exp \left(\frac{-E}{\mathrm{RT}}\right)=0$$

(3)

The appropriate boundary conditions on the reactive solid cylinder are, respectively

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\mathrm{dT}}{d\overline{r}}=\frac{\mathrm{dC}}{d\overline{r}}=\frac{\mathrm{dP}}{d\overline{r}}=0,\mathrm{on}}\\ \multicolumn{1}{c}{\overline{r}=0(\mathrm{axial}-\mathrm{symmetric}\mathrm{condition})}\end{array}$$

(4)

$$\begin{array}{c}\multicolumn{1}{c}{-k\frac{\mathrm{dT}}{d\overline{r}}=h_1(T-T_{\infty }),-\gamma \frac{\mathrm{dP}}{d\overline{r}}=h_2(P-P_{\infty }),}\\ \multicolumn{1}{c}{C=C_w,\mathrm{on}\overline{r}=a}\end{array}$$

(5)

where $T$ is the reactive solid cylinder’s absolute temperature, $C$ is the O₂ concentration within the cylinder, $P$ is the CO₂ concentration emitted, $T_{\infty }$ is the ambient temperature, and $C_w$ is the O₂ concentration at the cylinder surface. The CO₂ concentration at the cylinder surface is denoted by $P_w$ , $k$ is the thermal conductivity of the reacting cylinder, and $\epsilon$ is the solid cylinder emissivity ( $0<\epsilon <1$ ). The notation $\sigma$ represents Stefan–Boltzmann constant ( $5.6703\times {10}^{-8}\mathrm{W}/{\mathrm{m}}^2{\mathrm{K}}^4$ ), while $D$ is the diffusivity of O₂ in the solid cylinder and $\gamma$ is the diffusivity of CO₂ at the cylinder surface. The symbol $Q$ is the heat of reaction, $A$ is the rate constant, $E$ is the activation energy, $R$ is the universal gas constant, and $l$ is the Planck number. Vibration frequency and Boltzmann constant are, respectively, indicated by $v$ and $K$ , and $\overline{r}$ represents radial distance. $h_1$ and $h_2$ are, respectively, the heat transfer coefficient at the surface of the cylinder and the CO₂ transfer coefficient at the surface of the cylinder, while $n$ is the order of exothermic chemical reaction and $m$ the numerical exponent such that $m\in \{-2,0,0.5\}$ . The three values taken by the parameter $m$ represent the numerical exponent for sensitized, Arrhenius, and bimolecular kinetics, respectively.^1,2,12,15 The boundary conditions (4) and (5) describe the temperature and CO₂ conditions at the surface of the reactive cylinder. The following dimensionless parameters are introduced in equations (1)–(5)

$$\begin{array}{c}\hfill \theta =\frac{E(T-T_{\infty })}{{\mathrm{RT}}_{\infty }^2},\Phi =\frac{C}{C_w},\Psi =\frac{P}{P_w},\mathrm{B}{\mathrm{i}}_1=\frac{a{h}_1}{k}\hfill \\ \multicolumn{1}{c}{\mathrm{B}{\mathrm{i}}_2=\frac{a{h}_2}{\gamma },{\beta }_1=\frac{{\mathrm{kRT}}_{\infty }^2}{\mathrm{QED}{C}_w},{\beta }_2=\frac{{\mathrm{kRT}}_{\infty }^2}{\mathrm{QE}\gamma P_w}}\\ \multicolumn{1}{c}{\lambda ={\left(\frac{K{T}_{\infty }}{\mathrm{vl}}\right)}^m\frac{\mathrm{QAE}{a}^2{(C_w)}^n}{{\mathrm{kRT}}_{\infty }^2}\exp (-\frac{E}{R{T}_{\infty }})}\\ \multicolumn{1}{c}{r=\frac{\overline{r}}{a},\mu =\frac{R{T}_{\infty }}{E},\mathrm{Rd}=\frac{\epsilon \sigma E{a}^2{T}_{\infty }^2}{\mathrm{kR}}}\end{array}\}$$

(6)

and we obtain

$$\begin{array}{c}\multicolumn{1}{c}{\frac{1}{r}\frac{d}{\mathrm{dr}}\left(r\frac{d\theta }{\mathrm{dr}}\right)+\lambda {(1+\mu \theta )}^m{\mathrm{\Phi }}^n\exp \left(\frac{\mathrm{\theta }}{1+\mu \theta }\right)}\\ \multicolumn{1}{c}{-\mathrm{Rd}[{(\mu \theta +1)}^4-1]=0}\end{array}$$

(7)

$$\frac{1}{r}\frac{d}{\mathrm{dr}}\left(r\frac{d\Phi }{\mathrm{dr}}\right)-\lambda {\beta }_1{(1+\mu \theta )}^m{\mathrm{\Phi }}^n\exp \left(\frac{\mathrm{\theta }}{1+\mu \theta }\right)=0$$

(8)

$$\frac{1}{r}\frac{d}{\mathrm{dr}}\left(r\frac{d\Psi }{\mathrm{dr}}\right)+\lambda {\beta }_2{(1+\mu \theta )}^m{\mathrm{\Phi }}^n\exp \left(\frac{\mathrm{\theta }}{1+\mu \theta }\right)=0$$

(9)

with

$$\frac{d\theta }{\mathrm{dr}}=\frac{d\Phi }{\mathrm{dr}}=\frac{d\Psi }{\mathrm{dr}}=0,\mathrm{at}r=0$$

(10)

$$\frac{d\theta }{\mathrm{dr}}=-B{i}_1\theta ,\frac{d\Psi }{\mathrm{dr}}=-B{i}_2\mathrm{\Psi },\mathrm{\Phi }=1,\mathrm{at}r=1$$

(11)

where $\lambda$ is the Frank-Kamenetskii parameter, $\mu$ is the activation energy parameter, ${\beta }_1$ is the O₂ consumption rate parameter, ${\beta }_2$ is the CO₂ emission rate parameter, Bi₁ is the thermal Biot number, Bi₂ is the CO₂ Biot number, Rd is the radiation parameter, and $r$ is the dimensionless radial distance. The dimensionless heat and mass transfer rates at the cylinder surface are expressed in terms of local Nusselt number and Sherwood numbers as follows

$$\mathrm{Nu}=-\frac{d\theta }{\mathrm{dy}}(1),S{h}_1=-\frac{d\Phi }{\mathrm{dy}}(1),S{h}_2=-\frac{d\Psi }{\mathrm{dy}}(1)$$

(12)

Equations (7)–(9) and the boundary conditions (10) and (11) are solved numerically using the Runge–Kutta–Fehlberg method coupled with shooting technique. ¹⁵ From the process of numerical computation, the local Nusselt number and the local Sherwood numbers in equation (12) are calculated and their numerical values presented in a table.

Results and discussion

In this section, the computational results obtained for the modeled nonlinear ordinary differential equations are presented graphically and discussed for various values of thermo-physical parameters embedded in the system. It is important to note that combustible reactive materials may include cylindrical storage of petroleum materials, industrial waste, sugarcane bagasse, coal, hays, wool waste, and so on. Moreover, our results below reveal the effect of storage geometry on the exothermic reaction and CO₂ emission in reactive materials which invariably affect the environment through the production of ozone layer leading to climate change and global warming.

Effects of thermo-physical parameters on temperature profiles

The graphical results in Figures 3–9 illustrate the effects of different parameters on the steady-state temperature profiles in the exothermic reactive cylindrical material. In Figure 3, we see that increasing the value of $\lambda$ increases the temperature profiles. This is because as the rate of reaction accelerates, the exothermic chemical reaction increases leading to an increase in internal generated heat. We see the same scenario in Figure 5 where the increase in the numerical exponent m also corresponds to a temperature increase, with bimolecular reactions (m = 0.5) producing highest temperature profile. A different scenario is observed in Figures 4 and 6–9. The increase in the parameters n, Rd, $\mu$ , ${\beta }_1$ , and $\mathrm{B}{\mathrm{i}}_1$ results in a decrease in temperature profiles. We may say that these parameters retard the exothermic chemical reaction to some extent, and as a result less heat production is observed with increasing parameter values.

Figure 3.

Temperature profile: $n=1,m=0.5,\mathrm{Rd}=1,\mu =0.1,{\beta }_1=0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_1=1,\mathrm{B}{\mathrm{i}}_2=1$ .

Figure 4.

Temperature profile: $\lambda =0.1,m=0.5,\mathrm{Rd}=1,\mu =0.1,{\beta }_1=0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_1=1,\mathrm{B}{\mathrm{i}}_2=1$ .

Figure 5.

Temperature profile: $\lambda =0.1,n=1,\mathrm{Rd}=1,\mu =0.1,{\beta }_1=0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_1=1,\mathrm{B}{\mathrm{i}}_2=1$ .

Figure 6.

Temperature profile: $\lambda =0.1,n=1,\mu =0.1,{\beta }_1=0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_1=1,\mathrm{B}{\mathrm{i}}_2=1$ .

Figure 7.

Temperature profile: $\lambda =0.1,n=1,\mathrm{Rd}=1,{\beta }_1=0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_1=1,\mathrm{B}{\mathrm{i}}_2=1$ .

Figure 8.

Temperature profile: $\lambda =0.1,n=1,\mathrm{Rd}=1,\mu =0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_1=1,\mathrm{B}{\mathrm{i}}_2=1$ .

Figure 9.

Temperature profile: $\lambda =0.1,n=1,\mathrm{Rd}=1,\mu =0.1,{\beta }_1=0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_2=1$ .

Effects of thermo-physical parameters on O₂ profiles

Figures 10–16 illustrate effects of different parameters on O₂ profiles. We observe from Figures 10, 12, and 15 that the following parameters $\lambda$ , m, and ${\beta }_1$ lower O₂ profiles as their numerical values increase. These parameters reduce the O₂ concentration in the reactive cylinder by accelerating the exothermic reaction that uses more oxygen. From Figures 11, 13, 14, and 16, we observe that the parameters n, Rd, $\mu$ , and $\mathrm{B}{\mathrm{i}}_1$ favor the conservation of O₂ concentration in a reactive cylinder. An increase in each of the parameters mentioned shows a slight increase in the O₂ profiles. In other words, these parameters slow down the exothermic chemical reaction, and in turn, less O₂ is consumed when the parameters are considered.

Figure 10.

O₂ profile: $n=1,m=0.5,\mathrm{Rd}=1,\mu =0.1,{\beta }_1=0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_1=1,\mathrm{B}{\mathrm{i}}_2=1$ .

Figure 11.

O₂ profile: $\lambda =0.1,m=0.5,\mathrm{Rd}=1,\mu =0.1,{\beta }_1=0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_1=1,\mathrm{B}{\mathrm{i}}_2=1$ .

Figure 12.

O₂ profile: $\lambda =0.1,n=1,\mathrm{Rd}=1,\mu =0.1,{\beta }_1=0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_1=1,\mathrm{B}{\mathrm{i}}_2=1$ .

Figure 13.

O₂ profile: $\lambda =0.1,n=1,m=0.5,\mu =0.1,{\beta }_1=0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_1=1,\mathrm{B}{\mathrm{i}}_2=1$ .

Figure 14.

O₂ profile: $\lambda =0.1,n=1,m=0.5,\mathrm{Rd}=1,{\beta }_1=0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_1=1,\mathrm{B}{\mathrm{i}}_2=1$ .

Figure 15.

O₂ profile: $\lambda =0.1,n=1,m=0.5,\mathrm{Rd}=1,\mu =0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_1=1,\mathrm{B}{\mathrm{i}}_2=1$ .

Figure 16.

O₂ profile: $\lambda =0.1,n=1,m=0.5,\mathrm{Rd}=1,\mu =0.1,{\beta }_1=0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_2=1$ .

Effects of thermo-physical parameters on CO₂ profiles

We also study how various parameters play a role in the CO₂ emission due to exothermic chemical reaction taking place within a reactive cylinder of combustible material. The observations are illustrated in Figures 17–25. As in the case of temperature profiles, we see that the parameters $\lambda$ and m increase the CO₂ profiles as their numerical values are increased. This is due to accelerated exothermic reaction that produces more CO₂. The illustrations are given in Figures 17 and 19. We observe the same scenario from Figure 23 where an increase in ${\beta }_2$ shows a corresponding increase in the CO₂ profiles. From Figures 18, 20–22, 24, and 25, we see that increasing the parameters n, Rd, $\mu$ , ${\beta }_1$ , $\mathrm{B}{\mathrm{i}}_1$ , and $\mathrm{B}{\mathrm{i}}_2$ reduces the CO₂ profiles. The last parameter shows a huge decline in the CO₂ emission profiles. This is good because the emission of CO₂ is lessened when these parameters are considered. It is important to note that in all processes, that is, temperature, O₂ depletion, and CO₂ emission, the parameter ${\beta }_1$ reduces the profiles as it increases.

Figure 17.

CO₂ profile: $n=1,m=0.5,\mathrm{Rd}=1,\mu =0.1,{\beta }_1=0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_1=1,\mathrm{B}{\mathrm{i}}_2=1$ .

Figure 18.

CO₂ profile: $\lambda =0.1,m=0.5,\mathrm{Rd}=1,\mu =0.1,{\beta }_1=0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_1=1,\mathrm{B}{\mathrm{i}}_2=1$ .

Figure 19.

CO₂ profile: $\lambda =0.1,n=1,\mathrm{Rd}=1,\mu =0.1,{\beta }_1=0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_1=1,\mathrm{B}{\mathrm{i}}_2=1$ .

Figure 20.

CO₂ profile: $\lambda =0.1,n=1,m=0.5,\mu =0.1,{\beta }_1=0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_1=1,\mathrm{B}{\mathrm{i}}_2=1$ .

Figure 21.

CO₂ profile: $\lambda =0.1,n=1,m=0.5,\mathrm{Rd}=1,{\beta }_1=0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_1=1,\mathrm{B}{\mathrm{i}}_2=1$ .

Figure 22.

CO₂ profile: $\lambda =0.1,n=1,m=0.5,\mathrm{Rd}=1,\mu =0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_1=1,\mathrm{B}{\mathrm{i}}_2=1$ .

Figure 23.

CO₂ profile: $\lambda =0.1,n=1,m=0.5,\mathrm{Rd}=1,\mu =0.1,{\beta }_1=0.1,\mathrm{B}{\mathrm{i}}_1=1,\mathrm{B}{\mathrm{i}}_2=1$ .

Figure 24.

CO₂ profile: $\lambda =0.1,n=1,m=0.5,\mathrm{Rd}=1,\mu =0.1,{\beta }_1=0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_2=1$ .

Figure 25.

CO₂ profile: $\lambda =0.1,n=1,m=0.5,\mathrm{Rd}=1,\mu =0.1,{\beta }_1=0.1,{\beta }_2=0.1,\mathrm{B}{\mathrm{i}}_1=1$ .

Effects of parameter variation on thermal criticality values or blowups

In this case, we discuss the variation in the rate of reaction, Frank-Kamenetskii parameter $\lambda$ , with the dimensionless heat transfer rate at the reactive cylinder surface, and Nusselt number Nu, for various values of some parameters as shown by the figures plotted. It is noteworthy at blowup point that both Nu and Sh have the same critical value λ _c. The thermal critical values λ _c obtained are presented in Table 1.

Table 1.

Computations showing the effects of various thermo-physical parameters on thermal criticality values, where we have that β₁ = β₂ = 0.1.

Rd	n	µ	$\mathrm{B}{\mathrm{i}}_1$	$\mathrm{B}{\mathrm{i}}_2$	m	Nu	λ c
1	1	0.1	1	1	0.5	1.01214717	0.8268
5	1	0.1	1	1	0.5	1.25413389	1.6961
10	1	0.1	1	1	0.5	1.70814487	2.9771
1	5	0.1	1	1	0.5	1.34025335	0.9874
1	10	0.1	1	1	0.5	2.06022198	1.2658
1	1	0.2	1	1	0.5	3.11461714	1.6181
1	1	0.3	1	1	0.5	1.82112818	2.4538
1	1	0.1	5	1	0.5	2.35466641	1.8537
1	1	0.1	10	1	0.5	2.70427097	2.2171
1	1	0.1	1	5	0.5	1.01214717	0.8268
1	1	0.1	1	10	0.5	1.01214717	0.8268
1	1	0.1	1	1	0	1.09860431	0.8821
1	1	0.1	1	1	-2	1.89070258	1.2338

The numerical values for the Nusselt number, Nu, for each parameter indicated versus the reaction rate or Frank-Kamenetskii parameter, $\lambda$ , presented in the table above are presented graphically in Figures 26–30. From Figures 26–29, we observe that thermal stability is attained as the parameters Rd, n, µ, and ${\mathrm{Bi}}_1$ are increased. The higher values of the parameters result in higher values of both the Nusselt number and rate of reaction, with an exception of µ, where the highest value gives the highest value of only $\lambda$ . An increase in the convective and radiative heat loss to the ambient will enhance heat transfer in the reacting system and maintain thermal stability. A different scenario is observed in Figure 30 where Nu is lowest at 0, and it is observed that thermal stability is attained by keeping m very low.

Figure 26.

Effect of increasing Rd on cylinder thermal criticality values.

Figure 27.

Effect of increasing n on cylinder thermal criticality values.

Figure 28.

Effect of increasing µ on cylinder thermal criticality values.

Figure 29.

Effect of increasing $\mathrm{B}{\mathrm{i}}_1$ on cylinder thermal criticality values.

Figure 30.

Effect of increasing m on cylinder thermal criticality values.

Conclusion

This article considered the analysis of thermal stability and CO₂ emission that involves O₂ depletion in a convective and radiating cylindrical combustible reactive material. It was observed from the results that reaction processes that increase both the temperature and carbon dioxide emission profiles also increase the depletion of oxygen. The parameters that favor the slowing down of CO₂ emission in an exothermic chemical reaction of combustible material were found to be n, Ra, $\mu$ , ${\beta }_1$ , $\mathrm{B}{\mathrm{i}}_1$ , and $\mathrm{B}{\mathrm{i}}_2$ . It was also observed that increasing numerical values of these parameters also helps to attain thermal stability of exothermic chemical reactions. As mentioned earlier, the significance of the study of reactive materials in cylindrical geometry may include cylindrical storage of carbon-containing materials such as industrial waste, sugarcane bagasse, coal, hays, and wool wastes, to mention a few. The discussion of results obtained reveals the effect of storage geometry due to exothermic chemical reaction. The emission of CO₂ in reactive materials invariably affects the environment through the production of ozone layer. It is well known that increased ozone layer has led to climate change and global warming.

Finally, the adoption of mathematical model to simulate this complex phenomenon helps to provide deep insight into this intricate problem in a cheaper and safer way since it may involve thermal runaways through exothermic chemical reaction.