Abstract

Heat and mass transfer in porous media have critical applications in engineering, groundwater management, and oil recovery. Reactive flows, involving chemical reactions within the fluid phase, generate heat, presenting challenges in thermal systems. This study gives the theoretical and practical analysis of entropy generation in the unsteady flow of a reactive viscous fluid through a porous circular pipe under Arrhenius kinetics. The flow is modeled as an unsteady incompressible viscous fluid subjected to a pressure gradient, with constant wall temperature. Key parameters such as permeability, Prandtl number, and viscous heating are analyzed for their impact on entropy generation. Results indicate that the entropy generation peaks near the pipe wall, with maximum values observed between r = 0.8 and r = 0.9 from the centre. Entropy increases with high permeability until a critical value of 4.2887, beyond which it diverges. Similarly, increases in the Prandtl number and viscous heating parameter enhance entropy generation, particularly midway toward the wall. These findings provide insights into minimizing entropy in reactive flow systems.

Keywords

Entropy Generation, Reactive Viscous Flow, Porous Pipe, Unsteady Flow, Thermodynamics, Isothermal Wall

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1. Introduction

Entropy is a fundamental thermodynamic concept that quantifies the degree of disorder within a system and is pivotal for understanding process irreversibility [1]. According to the second law of thermodynamics, systems naturally progress towards higher entropy unless external energy is applied to maintain order [2]. In reactive flow systems, factors such as heat transfer, mass diffusion, and chemical reactions contribute to entropy production, which impacts energy efficiency by increasing energy loss and degrading system performance. The principles of thermodynamics, particularly the first and second laws, form the basis for analyzing thermofluid systems. Entropy production, indicative of irreversibility in complex processes, is crucial for optimizing industrial designs including solar energy collectors, electronic cooling systems, and combustion systems. Eliminating energy waste, often linked with increased disorder, remains a significant challenge for Scientists and Engineers. Consequently, studying entropy generation offers valuable insights for enhancing flow and heat transfer systems. Research by Reddy, Kumar, and Bég [3] on viscosity variations in viscoelastic flows, Goyal and Srinivas [4] on converting wasted energy into usable forms, and Aiyesimi et al. [5] on the effects of chemical reactions, magnetic fields, and Hall effects in magnetohydrodynamic flows demonstrates that a deeper understanding of entropy can improve heat transfer efficiency, reduce energy consumption, and minimize environmental impact. Additionally, entropy analysis aids in predicting fluid behaviour and refining modeling accuracy. Further exploration can be found in [6]-[10].

Unsteady reactive viscous flow in porous cylindrical pipes is critical for various engineering applications, necessitating a thorough understanding of fluid dynamics and heat transfer [11]. In chemical reactors, porous cylindrical pipes enhance the interaction between reactants and catalysts, thereby improving system performance . Farayola showed that adjusting thermal conductivity and viscous heating parameters in a porous cylindrical pipe with a reactive variable viscous fluid can effectively manage fluid temperature and boost system efficiency. The unsteady flow conditions and reactive processes within these reactors are crucial for optimizing reaction rates and product yields. Paul [13] explored the impact of a magnetic field on the unsteady natural convective one-dimensional laminar flow of an incompressible, slightly electrically conducting fluid over a vertical cylinder, finding that increased porosity enhances fluid velocity. Additionally, viscous dissipation is a major contributor to entropy generation by converting mechanical energy into thermal energy, which increases system disorder and affects entropy management in chemical processes [14]. Shehata et al. [15] numerically analyzed entropy generation due to viscous dissipation around a well’s turbine blade and concluded that understanding viscous dissipation enables Engineers to optimize energy inputs, improve reaction efficiency, and develop more sustainable chemical processes.

Porous cylindrical pipes are also vital in filtration systems due to their ability to facilitate unsteady flow through the porous medium, which improves particle and contaminant separation. The effectiveness of these pipes depends on factors such as fluid dynamics, viscosity, and interactions within the filter material. Understanding entropy generation is crucial for designing filtration systems that maximize separation performance while minimizing energy costs. In thermal management applications, these pipes play a significant role in heat dissipation, essential for the safe operation of electronic devices and automotive engines [16]. Studying unsteady reactive viscous flow in these pipes allows engineers to enhance thermal conductivity and heat dissipation efficiency, resulting in better cooling performance and reduced thermal resistance [17]. The study of entropy in porous cylindrical pipes addresses several key aspects and challenges [18]. The porous structure and cylindrical geometry significantly impact fluid flow and heat transfer. While the porous medium enhances heat and mass transfer by increasing the interaction surface area, it also introduces frictional resistance and flow maldistribution, leading to additional entropy generation [19]. Accurate modeling requires considering the complex interactions between fluid dynamics, thermal effects, and reactive processes. Although isothermal wall conditions simplify thermal analysis, they may not always reflect real-world variations [20]. Additionally, experimental measurement of entropy generation and validation of theoretical models pose challenges due to the need for precise instrumentation and data interpretation. Addressing these aspects is essential for optimizing the design and performance of systems involving porous cylindrical pipes, such as heat exchangers and chemical reactors.

Despite the critical importance of analyzing entropy production for designing heat exchangers that minimize energy losses and enhance thermal performance which is key for applications requiring precise temperature control and high thermal efficiency in power plants and industrial heating systems, limited research specifically addresses entropy generation in unsteady reactive viscous flow within a porous cylindrical pipe with an isothermal wall. The objective of this paper, therefore, is to study the unsteady flow of a reactive variable viscous fluid in a porous cylindrical pipe with an isothermal wall under Arrhenius kinetics and to report the effects of porosity and heating parameters, and Prandtl number on the entropy generation of the flow.

2. Mathematical Formulation of the Problem

Unsteady flow of a laminar, incompressible, viscous heat generation/absorption fluid in a porous cylindrical pipe in the presence of a pressure gradient has been considered. The temperature of the wall of the cylindrical pipe is assumed to be constant throughout the flow. The radius of the pipe is $a$ unit. The governing equations of the problem with the corresponding initial and boundary conditions are stated as follows:

Energy equation:

$\rho C_p\left(\frac{\partial T}{\partial \overline{t}}+\overline{V}\frac{\partial T}{\partial \overline{r}}\right)=\frac{k}{\overline{r}}\frac{\partial }{\partial \overline{r}}\left(\overline{r}\frac{\partial T}{\partial \overline{r}}\right)+Q{C}_{0}A{\text{e}}^{-\frac{E}{RT}}+\mu {\left(\frac{\partial \overline{u}}{\partial \overline{r}}\right)}^2$ (1)

Momentum equation:

$\rho \left(\frac{\partial \overline{u}}{\partial \overline{t}}+\overline{V}\frac{\partial \overline{u}}{\partial \overline{r}}\right)=\frac{1}{\overline{r}}\frac{\partial }{\partial \overline{r}}\left(\mu \overline{r}\frac{\partial \overline{u}}{\partial \overline{r}}\right)+\frac{\mu }{k_1}\overline{u}+G$ (2)

Initial and boundary conditions:

$\begin{array}{l}\overline{u}\left(\overline{r},t\right)=0,\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}T\left(\overline{r},t\right)=T_0\text{}\text{}\text{}\text{}\text{}\text{}\text{on}\overline{r}=a\text{};\\ \overline{u}\left(\overline{r},0\right)=0,T\left(\overline{r},0\right)=0\\ \frac{\partial T\left(\overline{r},t\right)}{\partial \overline{r}}=\frac{\partial \overline{u}\left(\overline{r},t\right)}{\partial \overline{r}}=0\text{on}\overline{r}=0\end{array}$(3)

where $\overline{V}$ is the suction velocity, and $\rho ,C_P,k,k_1,\mu ,Q,C_0,A,T,\overline{u},G$ are as defined in the nomenclature. The source term $Q$ , represents the heat generation when $Q>0$ and represents the heat absorption term when $Q<0$ .

Figure 1 shows the schematic diagram of problem. The flow is in the z-direction and the velocity varies along $\overline{r}$ , i.e. velocity varies from the centre to the wall of the cylindrical pipe. There is constant concentration of the chemical species, that is, the materials are not consumed, the fluid is reactive and incompressible, and it is an Initial Boundary Value forced convention problem.

Figure 1. Schematic diagram of unsteady flow in a porous cylindrical pipe.

Introducing the dimensionless variables in Equation (4) into Equations (1)-(3),

$\begin{array}{l}\theta =\frac{E}{R{T}_0^2}\left(T-T_0\right)\Rightarrow T=T_0+\frac{R{T}_0^2}{E}\theta \text{and}\partial T=\frac{R{T}_0^2}{E}\partial \theta \\ \epsilon =\frac{R{T}_0^2}{E},\overline{u}=u_{0}u\text{}\text{}\text{}\text{}\Rightarrow \partial \overline{u}=u_0\partial u\text{}\text{}\text{},\overline{n}=\frac{V_0^{2}n}{\nu }\\ \overline{t}=\frac{\nu t}{V_0^2}\Rightarrow \frac{\partial }{\partial \overline{t}}=\frac{1}{\nu /V_0^2}\frac{\partial }{\partial t}\text{and}\frac{\partial }{\partial \overline{t}}=\frac{V_0^2}{\nu }\frac{\partial }{\partial t}\\ \overline{r}=\frac{\nu r}{V_0}\Rightarrow \partial \overline{r}=\frac{\nu }{V_0}\partial r\text{and}\frac{\partial }{\partial \overline{r}}=\frac{V_0}{\nu }\frac{\partial }{\partial r}\end{array}$ (4)

we have

$\frac{\partial \theta }{\partial t}-\left(1+\epsilon A_{*}{\text{e}}^{nt}\right)\frac{\partial \theta }{\partial r}=\frac{1}{Pr}\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial \theta }{\partial r}\right)+\frac{\lambda }{Pr}{\text{e}}^{\frac{\theta }{1+\epsilon \theta }}+\lambda He\cdot {\text{e}}^{\frac{-\theta }{1+\epsilon \theta }}{\left(\frac{\partial u}{\partial r}\right)}^2$ (5)

$\frac{\partial u}{\partial t}-\left(1+\epsilon A_{*}{\text{e}}^{nt}\right)\frac{\partial u}{\partial r}=\frac{\delta }{r}\frac{\partial }{\partial r}\left({\text{e}}^{\frac{-\theta }{1+\epsilon \theta }}r\frac{\partial u}{\partial r}\right)+{\delta }_1{\text{e}}^{\frac{-\theta }{1+\epsilon \theta }}u+1$ (6)

with dimensionless initial and boundary conditions

$\begin{array}{l}\theta \left(1,t\right)=u\left(1,t\right)=0,\theta \left(r,0\right)={\theta }_c\left(r\right)\\ u\left(r,0\right)=h\left(r\right),\frac{\partial \theta }{\partial r}\left(0,t\right)=\frac{\partial u}{\partial r}\left(0,t\right)=0\end{array}$ (7)

where

$\lambda =\frac{QEA{a}^2{C}_0{\text{e}}^{-\frac{E}{R{T}_0}}}{kR{T}_0^2}=\frac{QEA{v}^2{C}_0{\text{e}}^{-\frac{E}{R{T}_0}}}{kR{T}_0^2{V}_0^2}$ (the Frank-Kamenetskii parameter), $He=\frac{k{\mu }_0{u}_0^2{V}_0^2{\text{e}}^{\frac{2E}{R{T}_0}}}{\rho C_P{v}^{3}QA{C}_0}$ (the viscous heating parameter), $a^2=\frac{v^2}{V_0^2}$ , $P_r=\frac{\rho C_{p}v}{k}$ (the Prandtl number), $\delta =\frac{V_0}{v}{\text{e}}^{\frac{E}{R{T}_0}}>0$ , $V_0\ll 1$ (the suction parameter), $G=\frac{\rho u_0{V}_0^2}{\nu }$ (the constant axial pressure gradient), and ${\delta }_1=\frac{v{\mu }_0{\text{e}}^{\frac{E}{R{T}_0}}}{\rho V_0^2{k}_1}=\frac{v^2{\text{e}}^{\frac{E}{R{T}_0}}}{V_0^2{k}_1}$ (the porosity (permeability) parameter).

The suction velocity takes the exponential form

$\overline{V}=-V_0\left(1+\epsilon A_{*}{\text{e}}^{\overline{n}\overline{t}}\right)$(8)

where $A_{*}$ is a real positive constant, $\epsilon$ and $\epsilon A_{*}$ are small less than unity, and $V_0$ is a scale suction velocity which has non-zero positive constant [21]-[24], and in non-dimensional form, $\overline{V}$ can be written as

$\overline{V}=-V_0\left(1+\epsilon A_{*}{\text{e}}^{nt}\right)$(9)

For all fuels of interest, the parameter $\epsilon$ is assumed to be small. By using the method of activation energy asymptotics and for $\epsilon \ll 1$ , it gives [25]-[27]

$\frac{\partial \theta }{\partial t}-\left(1+\epsilon A_{*}{\text{e}}^{nt}\right)\frac{\partial \theta }{\partial r}=\frac{1}{Pr}\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial \theta }{\partial r}\right)+\lambda \left(\frac{1}{Pr}{\text{e}}^{\theta }+He\cdot {\text{e}}^{-\theta }{\left(\frac{\partial u}{\partial r}\right)}^2\right).$ (10)

and

$\frac{\partial u}{\partial t}-\left(1+\epsilon A_{*}{\text{e}}^{nt}\right)\frac{\partial u}{\partial r}=\frac{\delta }{r}\frac{\partial }{\partial r}\left({\text{e}}^{-\theta }r\frac{\partial u}{\partial r}\right)+{\delta }_1{\text{e}}^{-\theta }u+1$ (11)

with the dimensionless initial and boundary conditions

$\begin{array}{l}\theta \left(1,t\right)=u\left(1,t\right)=0,\theta \left(r,0\right)={\theta }_c\left(r\right)\\ u\left(r,0\right)=h\left(r\right),\frac{\partial \theta }{\partial r}\left(0,t\right)=\frac{\partial u}{\partial r}\left(0,t\right)=0\end{array}$ (12)

The coupled nonlinear partial differential equations with the initial and boundary conditions are difficult to solve to obtain exact solutions. And since the cylindrical coordinate has singularity at $r=0$ , singular perturbation technique was employed as it was obtained in [8] [28] and the solutions are assumed to be of the form:

$u\left(r,t\right)=u_0\left(r\right)+\epsilon {\text{e}}^{nt}{u}_1\left(r\right)+0\left({\epsilon }^2\right)$ (13)

$\theta \left(r,t\right)={\theta }_0\left(r\right)+\epsilon {\text{e}}^{nt}{\theta }_1\left(r\right)+0\left({\epsilon }^2\right)$ (14)

Now,

${\text{e}}^{\theta \left(r,t\right)}={\text{e}}^{{\theta }_0\left(r\right)}\left(1+\epsilon {\text{e}}^{nt}{\theta }_1\left(r\right)\right)$ neglecting $0\left({\epsilon }^2\right)$ (15)

and

${\text{e}}^{-\theta \left(r,t\right)}={\text{e}}^{-{\theta }_0\left(r\right)}\left(1-\epsilon {\text{e}}^{nt}{\theta }_1\left(r\right)\right)$ neglecting $0\left({\epsilon }^2\right)$ (16)

Substituting Equations (13), (14), (15) and (16) in Equations (10) and (11), and setting $\lambda =\epsilon {\text{e}}^{nt}\Lambda$ , in Equation (10) gives

$\begin{array}{l}n\epsilon {\text{e}}^{nt}{\theta }_1\left(r\right)-\left(1+\epsilon A_{*}{\text{e}}^{nt}\right)\frac{\partial }{\partial r}\left[{\theta }_0\left(r\right)+\epsilon {\text{e}}^{nt}{\theta }_1\left(r\right)\right]\\ =\frac{1}{Pr}\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial }{\partial r}\left({\theta }_0\left(r\right)+\epsilon {\text{e}}^{nt}{\theta }_1\left(r\right)\right)\right)+\epsilon {\text{e}}^{nt}\Lambda [\frac{1}{Pr}{\text{e}}^{{\theta }_0\left(r\right)}\left(1+\epsilon {\text{e}}^{nt}{\theta }_1\left(r\right)\right)\\ +He\cdot {\text{e}}^{-{\theta }_0\left(r\right)}\left(1-\epsilon {\text{e}}^{nt}{\theta }_1\left(r\right)\right){\left[\frac{\partial }{\partial r}\left(u_0\left(r\right)+\epsilon {\text{e}}^{nt}{u}_1\left(r\right)\right)\right]}^2]\end{array}$ (17)

and

$\begin{array}{l}n\epsilon {\text{e}}^{nt}{u}_1\left(r\right)-\left(1+\epsilon A_{*}{\text{e}}^{nt}\right)\frac{\partial }{\partial r}\left(u_0\left(r\right)+\epsilon {\text{e}}^{nt}{u}_1\left(r\right)\right)\\ =\frac{\delta }{r}\frac{\partial }{\partial r}\left[{\text{e}}^{-{\theta }_0\left(r\right)}\left(1-\epsilon {\text{e}}^{nt}{\theta }_1\left(r\right)\right)\frac{\partial }{\partial r}\left(u_0\left(r\right)+\epsilon {\text{e}}^{nt}{u}_1\left(r\right)\right)\right]\\ +{\delta }_1{\text{e}}^{-{\theta }_0\left(r\right)}\left(1-\epsilon {\text{e}}^{nt}{\theta }_1\left(r\right)\right)\left(u_0\left(r\right)+\epsilon {\text{e}}^{nt}{u}_1\left(r\right)\right)+1\end{array}$ (18)

Equating powers of $\epsilon$ and simplifying Equations (17) & (18), and applying the dimensionless initial and boundary conditions in Equation (12), we have

$\theta \left(r,t\right)=\epsilon {\text{e}}^{nt}\left(a_4+a_{10}{r}^2+a_9{r}^3+a_8{r}^4+a_7{r}^5+a_6{r}^6+a_5{r}^7\right)$ (19)

$u\left(r,t\right)=a_1+a_2{r}^3+a_3{r}^5+\epsilon {\text{e}}^{nt}\left(a_{11}+a_{12}{r}^3+a_{13}{r}^5\right)$ (20)

where

$a_1=\frac{20\delta -3}{120{\delta }^2-20\delta {\delta }_1+3{\delta }_1}$ (21)

$a_2=-\frac{1}{6\delta }\left[1+\frac{{\delta }_1\left(20\delta -3\right)}{120{\delta }^2-20\delta {\delta }_1+3{\delta }_1}\right]$ (22)

$a_3=\frac{1}{40{\delta }^2}\left[1+\frac{{\delta }_1\left(20\delta -3\right)}{120{\delta }^2-20\delta {\delta }_1+3{\delta }_1}\right]$ (23)

$a_4=\frac{\begin{array}{l}\Lambda (-6350400+1411200Pr-264600P{r}^2-396900nPr+42336P{r}^3\\ +119952nP{r}^2-6350400He\cdot a_2^{2}Pr-24010nP{r}^3-11025n^{2}P{r}^2\\ -5880P{r}^4+3804nP{r}^4+3798n^{2}P{r}^3+777600He{a}_2^{2}P{r}^2+270P{r}^5)\end{array}}{\begin{array}{l}-25401600-6350400nPr+1411200nP{r}^2-264600nP{r}^3-396900n^{2}Pr\\ +42336nP{r}^4+119952n^{2}P{r}^3-5880nP{r}^5-24010n^{2}P{r}^4-11025n^{3}P{r}^3\\ +720nP{r}^6+3798n^{3}P{r}^4+3804n^{2}P{r}^5\end{array}}$ (24)

$\begin{array}{c}{a}_5=-\frac{1}{4233600}{a}_4\left(120nP{r}^2+633n^3+634n^{2}Pr\right)P{r}^4\\ +\frac{1}{1225}\left(-\frac{1}{4}\left(-\frac{1}{6}\Lambda P{r}^2-\frac{1}{4}\Lambda nPr\right)Pr+\frac{1}{18}\Lambda P{r}^{2}n\right)nPr\\ -\frac{1}{294}(-9\Lambda He\cdot a_2^{2}Pr+\frac{1}{16}\left(-\frac{1}{6}\Lambda P{r}^2-\frac{1}{4}\Lambda nPr\right)nPr\\ -\frac{1}{5}\left(-\frac{1}{4}\left(-\frac{1}{16}\Lambda P{r}^2-\frac{1}{4}\Lambda nPr\right)Pr+\frac{1}{18}\Lambda P{r}^{2}n\right)Pr)Pr\end{array}$ (25)

$\begin{array}{c}{a}_6=\frac{a_4}{103680}\left(24nP{r}^2+98n^{2}Pr+45n^3\right)P{r}^3\\ -\frac{1}{4}\Lambda He\cdot a_2^{2}Pr+\frac{1}{576}\left(-\frac{1}{6}\Lambda P{r}^2-\frac{1}{4}\Lambda nPr\right)nPr\\ -\frac{1}{180}\left(-\frac{1}{4}\left(-\frac{1}{6}\Lambda r^2-\frac{1}{4}\Lambda nPr\right)Pr+\frac{1}{18}\Lambda P{r}^{2}n\right)Pr\end{array}$ (26)

$a_7=-\frac{1}{100}\left(-\frac{1}{6}\Lambda P{r}^2-\frac{1}{4}\Lambda nPr\right)Pr+\frac{1}{450}\Lambda P{r}^{2}n-\frac{1}{3600}{a}_4\left(6nPr+17n^2\right)P{r}^3$ (27)

$a_8=-\frac{1}{96}\Lambda P{r}^2-\frac{1}{64}\Lambda nPr+\frac{1}{192}{a}_4\left(2nPr+3n^2\right)P{r}^2$ (28)

$a_9=-\frac{1}{18}{a}_{4}nP{r}^2+\frac{1}{18}\Lambda Pr$ (29)

$a_{10}=-\frac{1}{4}{a}_{4}nPr+\frac{1}{4}\Lambda$ (30)

$\begin{array}{c}{a}_{11}=-\frac{1}{120{\delta }^2+20\delta n-20{\delta }_1\delta -3n+3{\delta }_1}(-18A{a}_2\delta +6{\delta }_1{a}_{10}{a}_1\delta +20\delta {\delta }_1{a}_4{a}_1\\ +120{\delta }^2{a}_4{a}_2+120{\delta }^2{a}_4{a}_3+72{\delta }^2{a}_{10}{a}_2-3{\delta }_1{a}_4{a}_1-18\delta a_4{a}_2)\end{array}$ (31)

$a_{12}=\frac{1}{6\delta }\left(n{a}_{11}+{\delta }_1{a}_4{a}_1+6\delta a_4{a}_2-{\delta }_1{a}_{11}\right)$ (32)

$\begin{array}{c}{a}_{13}=-\frac{1}{40{\delta }^2}(-2{\delta }_1{a}_{10}{a}_1\delta -40{\delta }^2{a}_4{a}_3-24{\delta }^2{a}_{10}{a}_2+6A{a}_2\delta \\ +n{a}_{11}+{\delta }_1{a}_4{a}_1+6\delta a_4{a}_2-{\delta }_1{a}_{11})\end{array}$ (33)

3. Dimensionless Entropy Generation Rate

Okedoye, Lamidi and Ayeni [29] defined the two-dimensionless entropy generation rate as:

${\Gamma }_n={\left(\frac{\partial \theta }{\partial y}\right)}^2+{\lambda }_1{\left(\frac{\partial u}{\partial y}\right)}^2+{\lambda }_2{\left(\frac{\partial c}{\partial y}\right)}^2+{\lambda }_3\left(\frac{\partial \theta }{\partial y}\right)\left(\frac{\partial c}{\partial y}\right)$ (34)

The following can now be defined:

${\Gamma }_{n,h}={\left(\frac{\partial \theta }{\partial y}\right)}^2$ , ${\Gamma }_{n,f}={\lambda }_1{\left(\frac{\partial u}{\partial y}\right)}^2$ , ${\Gamma }_n^c{}_{,d}={\lambda }_2{\left(\frac{\partial c}{\partial y}\right)}^2$ , and ${\Gamma }_{n,d}^{c,T}={\lambda }_3\left(\frac{\partial \theta }{\partial y}\right)\left(\frac{\partial c}{\partial y}\right)$ ,

where ${\Gamma }_{n,h}$ and ${\Gamma }_{n,f}$ are thermal and viscous irreversibility respectively, while ${\Gamma }_n^c{}_{,d}+{\Gamma }_{n,d}^{c,T}$ is the diffusive irreversibility. Dimensionless terms denoted by ${\lambda }_i\left(1\le i\le 3\right)$ , and called irreversibility distribution ratios, are given by:

${\lambda }_1=\frac{\mu T_0}{k}{\left(\frac{a}{L\left(\Delta T\right)}\right)}^2,{\lambda }_2=\frac{RD{T}_0}{k{c}_0}{\left(\frac{\Delta c}{\Delta T}\right)}^2,{\lambda }_3=\frac{RD}{k}\left(\frac{\Delta c}{\Delta T}\right)$

where $C_0$ and $T_0$ are respectively the reference concentration and temperature, which are in our case, the bulk concentration and the bulk temperature.

Now for the discussion, obtained solutions in Equations (19) and (20) are made use of, which are respectively

$\theta \left(r,t\right)=\epsilon {\text{e}}^{nt}\left(a_4+a_{10}{r}^2+a_9{r}^3+a_8{r}^4+a_7{r}^5+a_6{r}^6+a_5{r}^7\right)$

$u\left(r,t\right)=a_1+a_2{r}^3+a_3{r}^5+\epsilon {\text{e}}^{nt}\left(a_{11}+a_{12}{r}^3+a_{13}{r}^5\right)$

representing both energy and velocity distributions. Since the model does not involve chemical species, we have that ${\lambda }_3=0$ . Therefore Equation (34) becomes

${\Gamma }_n={\left(\frac{\partial \theta }{\partial r}\right)}^2+{\lambda }_1{\left(\frac{\partial u}{\partial r}\right)}^2$

i.e.

$\begin{array}{l}{\Gamma }_n={\left(3a_2{r}^2+5a_3{r}^4+\epsilon {\text{e}}^{nt}\left(3a_{12}{r}^2+5a_{13}{r}^4\right)\right)}^2\\ +{\lambda }_1{\left(\epsilon {\text{e}}^{nt}\left(2a_{10}r+3a_9{r}^2+4a_8{r}^3+5a_7{r}^4+6a_6{r}^5+7a_5{r}^7\right)\right)}^2\end{array}$ (35)

where

$a_1$ , $a_2$ , $a_3$ , $a_4$ , $a_5$ , $a_6$ , $a_7$ , $a_8$ , $a_9$ , $a_{10}$ , $a_{11}$ , $a_{12}$ , $a_{13}$ are as defined in Equations (21), (22), (23), (24), (25), (26), (27), (28), (29), (30), (31), (32), (33) respectively.

4. Results and Discussion

The graph of Entropy generation rate against $r$ for various flow parameters are shown in Figure 2 and Figure 3. Figure 2 shows the permeability parameter (${\delta }_1$ ) with the entropy generation rate. Entropy against $r$ with different values of Prandtl number (Pr) are shown in Figure 3 while the graph of entropy generation rate with different values of viscous heating parameter (He) are displayed in Figure 4. Figure 5 shows the 3D graph of Entropy against $r$ and permeability parameter.

In Figure 2, the entropy generation rate against $r$ for various values of permeability parameter is displayed. It is observed that the degree of disorderliness in the fluid is high towards the wall of the cylindrical pipe and maximum entropy is observed to be between $r=0.8$ and $r=0.9$ away from the centre of the cylindrical pipe. As permeability parameter increases, the degree of disorderliness increases until when permeability parameter approaches 4.2887, a point at which entropy run away i.e. blow up. This is depicted in Figure 5 where the value of the entropy generation rate approaches infinity.

Figure 3 shows the graph of entropy generation rate against $r$ for different values of Prandtl number. It is observed that increase in Prandtl number causes increase in entropy with the effect felt halfway towards the wall of the cylindrical pipe. This is in line with the results of [7] [9] [14], where it was observed that increasing the Prandtl number decreases the entropy generation firstly around the pipe centreline, then it enhances entropy rapidly towards the pipe wall, which supports the findings of [18] [21] [22]. The combined effect of Prandtl number on velocity and entropy generation rate is that as the Prandtl number increases, the velocity decreases towards the wall of the cylindrical pipe while the entropy generation rate increases towards the wall of the pipe.

Figure 2. Profile of Entropy against r for different values of Permeability parameter (δ₁) when Pr = 0.71, He = 1.0, ϵ = 0.01, n = 0.1, Λ = 45.24, δ = 1.25, β = 0.5, t = 1.0.

Figure 3. Plots of Entropy against r for different values of Prandtl number (Pr) when He = 1.0, δ₁ = 1.0, ϵ = 0.01, n = 0.1, Λ = 45.24, δ = 1.25, β = 0.5, t = 1.0.

Figure 4. Plots of Entropy against r for different values of Heating parameter (He) when Pr = 1.0, δ₁ = 1.0, ϵ = 0.01, n = 0.1, Λ = 45.24, δ = 1.25, β = 0.5, t = 1.0.

Figure 5. Plots of Entropy against r and δ₁ when He = 0.5, Pr = 1.0, ϵ = 0.01, n = 0.1, Λ = 45.24, δ = 1.25, β = 0.5, t = 1.0.

Similar trend is observed in Figure 4 where increase in viscous heating parameter causes increase in entropy generation [26] [28]. This is obvious as increase in heat energy usually agitates the molecules of the fluid and hence increase the rate of disorderliness in the fluid.

5. Conclusions

This study provided an in-depth analysis of entropy generation in an unsteady reactive viscous flow within a porous cylindrical pipe with an isothermal wall. By investigating key flow parameters namely: permeability, viscous heating, and the Prandtl number, the following key findings were established regarding the behaviour of entropy within the system:

1) Entropy distribution in the flow field: The degree of disorderliness within the fluid was found to be significantly high near the wall of the cylindrical pipe. This is attributed to the interaction between thermal and viscous forces, which leads to enhanced irreversible energy dissipation in the boundary layer region.

2) Radial Entropy Peak: The analysis revealed that maximum entropy generation occurs within the radial range r = 0.8 to r = 0.9, away from the centre of the pipe. This suggests that entropy generation is not uniformly distributed but is concentrated in regions where thermal gradients and fluid velocity interactions are pronounced.

3) Effect of Permeability: Increasing the permeability parameter generally leads to a rise in entropy generation, indicating intensified fluid interaction within the porous medium. However, when the permeability parameter reaches 4.2887, the system undergoes a critical transition, causing entropy generation to diverge. This suggests that beyond this threshold, the porous structure significantly alters the flow dynamics, potentially introducing instability.

4) Influence of Prandtl Number on the entropy: A higher Prandtl number corresponds to increased entropy generation, with the effect becoming more pronounced midway toward the pipe’s wall. Since the Prandtl number governs the ratio of momentum diffusivity to thermal diffusivity, its influence on entropy highlights the strong interplay between heat conduction and viscous dissipation within the system.

Overall, the study offers important insights into the thermodynamic behavior of porous pipe flows, with implications for the design and optimization of thermal systems such as heat exchangers and energy-efficient industrial processes.

6. Applications

Entropy generation is a crucial concept in assessing the efficiency and performance of diverse engineering systems, particularly those involving fluid flow, heat transfer, and reactive processes, where optimizing energy utilization and minimizing losses are paramount. The findings of the study provide actionable insights for several applications, with quantifiable benefits in efficiency, energy savings, and system reliability.

The findings on entropy generation can guide the design of heat exchangers with improved thermal efficiency. And by optimizing permeability and viscous heating parameters, engineers can achieve between 5% and 15% reduction in thermal losses, resulting in improved heat transfer efficiency [7] [16] . The entropy trends observed in this study can help determine the ideal material selection and flow control strategies to minimize energy dissipation in heat exchangers. Chemical reactors rely on efficient mixing and heat transfer to maximize reaction rates while minimizing energy losses. Adjusting permeability parameters based on entropy generation analysis can enhance mixing efficiency, potentially increasing reaction yield between 10% and 20% in industrial chemical processes . The study’s insights can be used to design reactors with optimized flow conditions, reducing entropy-related inefficiencies in large-scale production.

In porous media, structures are used in filtration systems, petroleum extraction, and biomedical fluid transport. Engineers can use entropy-based optimization to design porous structures that increase fluid transport efficiency between 8% and 12% while minimizing entropy-dependent losses [31]. The findings can inform the development of advanced filtration membranes and enhanced oil recovery techniques. Entropy generation directly affects energy consumption in industrial systems, including turbines, compressors, and fluid transport networks. By implementing entropy-based design modifications, industries can achieve between 3% and 7% reductions in overall energy consumption, leading to significant cost savings [32]. The study’s results can be integrated into computational models for predictive maintenance and system optimization.

Acknowledgements

I appreciate the Tertiary Education Trust Fund (TETFund), Abuja, Nigeria for sponsoring me, on behalf of other authors of this paper, to attend the 2025 Joint Mathematics Meeting (2025 JMM) organized by the American Mathematical Society held at Seattle, WA, USA between 8 and 11 January 2025, and the Management of Emmanuel Alayande University of Education, Oyo, Nigeria for releasing me to attend this 2025 JMM for the first time.

Nomenclature

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References