# An Analytical and a Numerical Method for Nonlinear Convection-Radiation Problems in Porous Fins

Authors: E. D. Correa; J. M. Quirino; R. L. Sobral; J. F. Corrêa; R. M. S. Gama
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

The present work shows an analytical and a numerical method for heat transfer nonlinear problems in porous fins using the Darcy model. Numerical simulations are carried out with the aid of a sequence of linear problems, each of them possessing an equivalent minimum principle, that has as its limit the solution of the original problem. The nonlinear convection-radiation heat transfer process is considered and simulated by means of a finite difference scheme. Results showed the relevance of the radiation for realistic thermal mapping in porous media with percentage errors of up to 40% for the last nodes.


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## Body

## 1. Introduction

In general, the first choice for the enhancement of rate of heat transfer from/to a body consists of using fins in order to provide an increase of the effective heat transfer surface. Since the need of optimizing heat transfer processes is encountered in several practical situations, the study of the thermal behavior of fins becomes a fundamental issue in the heat transfer area.Nowadays, high-performance heat transfer components are related to the need to achieve high heat transfer rates with low cost and, above all, limited space and weight. Heat transfer on extended surfaces is, currently, one of the main focuses of high-performance heat transfer studies, accounting for the most varied parameters and respective behavior in the thermal mapping [1–3].The extended surface is widely used in several engineering systems that incorporate from well-known heat exchangers to heat pipes as shown in recent studies [4–6]. Effective sizing of fins and heat sinks requires accurate knowledge of heat transfer on extended surfaces. Usually, convection and conduction heat transfers are the main considered mechanisms, but radiation heat transfer has significant effect in specific applications mainly involving high temperatures and/or rarefied atmospheres [7, 8].Due the larger effective surface area, porous fins have better thermal performance when compared to conventional solid ones with equivalent size. In the literature, porous constituents of high thermal conductivity have been used to improve the thermal performance of different thermal systems. Bioengineering, electronic technologies, and especially the most recent nanotechnology have been concerned with knowing the effects of heat transfer in porous media.Some studies leveraged the application the porous fins. Pop et al. [9] conducted analyses of steady-state conjugate free convection about a vertical fin embedded in a porous medium.Huang and Vafai [10] presented an investigation of forced convection enhancement in a channel using multiple emplaced porous blocks.Kim et al. [11] experimentally investigated the impact of porous fins on the pressure drop and heat transfer characteristics in plate-fin heat exchangers.Kiwan [12] introduced a simple method of analysis to study the performance of porous fins in a natural convection environment.El-Hakiem and Rashad (2007) [13] examined the effects of both radiation and the nonlinear Forchheimer terms on free convection from a vertical cylinder embedded in a fluid-saturated porous medium with the fluid viscosity varying as an inverse linear function of temperature. Rashad [14] employed magnetohydrodynamics and thermal radiation effects in heat and mass transfer of a vertical flat plate embedded in a fluid saturated porous media.Kundu and Bhanja [15] developed an analytical model for determination of the performance and optimum dimensions of porous fins accounting for different models of predictions. Gorla and Bakier [16] considered radiation and convection effects in porous media on rectangular profile fin.Rashad et al. [17] investigated theoretically the effects of thermal radiation and the nonlinear Forchheimer terms on boundary-layer flow and heat transfer by non-Darcy natural convection from a vertical cylinder embedded in a porous medium saturated with nanofluids. Rashad et al. [18] studied the combined effects of thermal radiation and thermophoresis on heat and mass transfer by mixed convection over a vertical rotating cone in a fluid saturated porous medium.Darvishi et al. [19] conducted a numerical study of steady-state heat transfer in porous rectangular fin under the influence of natural convection and radiation using homotopy analysis method. Darvishi et al. [20] numerically investigated the transient thermal effect on porous fin performance and the comparative study between the fin with or without radiation. Darvishi et al. [21] solved energy equation with spectral collocation method on wet longitudinal fin under natural convection and radiation.Sobamowo [22] analyzed the heat transfer in porous fin with temperature-dependent thermal conductivity. Results show that increases in convective and porosity parameter improved the efficiency of the fin. Jooma and Harley [23] numerically verified heat transfer in a porous radial fin with the differential transformation method.Sowmya et al. [24] numerically studied the thermal behavior of a porous longitudinal fin under convection-radiation effect. The nonlinear partial differential equation was nondimensionalized and solved numerically with the help of Maple software by the finite difference method.This work presents an analytical and numerical method for heat transfer nonlinear problems in porous fins. The Darcy model is utilized to simulate the flow through the porous media. Conduction-radiation-convection heat transfer process is an inherently nonlinear phenomenon in which the coupling on the boundary of the body is mathematically represented by a nonlinear relationship between the absolute temperature and its normal derivative, in which the unknown is the temperature distribution. The solution to the problem is given by the limit of a sequence whose elements are obtained, each one, from the minimization of a quadratic functional.


## 2. Mathematical Model

The modeling of heat transfer that occurs in a porous radial fin requires several feasibility considerations in its analysis. This study comprises such an analysis considering the material that constitutes the fin as being isotropic and homogeneous, in addition to the fin being analyzed as a strict heat sink. Regarding the fin geometry, this work considers a radial fin, being analyzed in cylindrical coordinates, which is connected to a tubular body, as exemplified in Figure1.Figure 1
Radial porous fin.Analyzing the fin as a porous medium leads to infiltration of the fluid in which the fin is inserted, thus allowing a greater exchange of heat between them. The general heat exchange equation used in this work arises from(1)qx−qx+Δx=ṁcpTx−T∞+hPΔxTx−T∞+PΔxσεT4x−αfεT∞4,where the fluid mass flow rate is
(2)ṁ=ρvwΔxW,in which the interactions between the porous medium and the fluid are described by the Darcy law and Boussinesq approximation, which provides the model
(3)vwx=gKβfνTx−T∞.The Fourier law of thermal conduction states that(4)q=−keffAcdTdx.Combining the equations, evaluating atΔx⟶0, and assuming that αf=ε, we can rewrite Equation (1), adapting it to cylindrical coordinates, such as
(5)1rddrrdTdr−ρfgβfKνfkefftT−Ta−−2εσFf−akefftT3T=0.The termT3T is used to ensure coerciveness, since the function for the temperature T is strictly increasing, in all applied temperature values, ensuring the correct physical description of the phenomenon. [25]By rearranging the equation so that its algebraic manipulation is convenient, you can rewrite Equation (6) as
(6)d2Tdr2+1rdTdr−AT−T∞T−T∞−BT3T−T∞4=0,where the constant A gathers all flow and geometric parameters that affect the solution of the problem and B characterizes the radiation parameter, which carries the effects of the porous body’s emissivity.For a more comprehensive mathematical analysis, it is convenient that the temperature values and the geometric position values will be dimensionless through the following relationships:(7)θ=TrTb,ξ=rR.This equation is considered by being subject to the following boundary conditions:(8)ξ=1⟶θ=Tb=1base,ξ=R⟶dθdξ=0tip.In order to seek possible approaches to the problem, it is assumed that the solutionθ of the original problem is given by
(9)θ=limi⟶∞φi.That is, the solution of the problem analyzed by this work is given by the limit of the sequenceφ1,φ2,φ3,⋯, whose elements are obtained from
(10)1ξddξξdφi+1dξ−αφi+1+βi=0,βi=αφi−Aφi−T∞φi−T∞−Bφi3φi−T∞4.If we consider thatφi is known, the problem described by Equation (11) is a classical linear boundary value problem, which can be easily solved. Therefore, to ensure that the solution is applicable to the problem, evidence of some aspects of the problem must be presented. The constant α will be discussed in more depth later.


### 2.1. Nondecreasing Proof

In order to prove that the sequence ofφi is nondescending, the difference between two consecutive terms of the sequence is considered.
(11)1ξddξξddξφi+1−φi−αφi+1−φi+αφi−φi−1−Aφi−T∞φi−T∞−Bφi3φi−T∞4++Aφi−1−T∞φi−1−T∞+Bφi−13φi−1−T∞4=0.Since(12)φ∧i+11−φ∧i1=0,ddξφ∧i+1−φ∧iξ=R=0,defining the function φ1as the solution of
(13)1ξddξξddξφ1−φ0−αφ1−φ0=Aφ0−T∞φ0−T∞−Bφ03φ0−T∞4≥0with
(14)φ∧11=1,dφ∧1dξξ=Rext=0,we have, since φ0≡0, that
(15)φ1−φ0≥0⇒φ1≥φ0≡0.So, we can writei=1(16)1ξddξξddξφ2−φ1−αφ2−φ1+αφ1−φ0−Aφ1−T∞φ1−T∞−Bφ13φ1−T∞4+Aφ0−T∞φ0−T∞+Bφ03φ0−T∞4=0with
(17)φ∧21−φ∧11=0,ddξφ∧2−φ∧1ξ=Rext=0.So, choosingα large enough for ensuring that
(18)αφ1−φ0≥Aφ1−T∞φ1−T∞+Bφ13φ1−T∞4−Aφ0−T∞φ0−T∞−Bφ03φ0−T∞4,the following inequality holds:
(19)φ2≥φ1≥φ1≡0.Repeating the above procedure, we have(20)⋯≥φi+1≥φi≥φi−1≥⋯≥φ1≥φ0≡0provided the constant α is large enough to ensure that
(21)α≥Aφi+1−T∞φi+1−T∞φi+1−φi+Bφi+13φi+1−T∞4φi+1−φi−Aφi−T∞φi−T∞−Bφi3φi−T∞4φi+1−φi.It is sufficient to chooseα in such a way that
(22)α≥ddθAθ−T∞θ−T∞+Bθ3θ−T∞4θ=1=2Aθ−T∞+4Bθ3θ=1=2AT∞+4B.


### 2.2. Upper Bound

Combining the above problems, we have(23)1ξddξξddξθ−φi+1+αφi+1−φi−−Aθ−T∞θ−T∞+Bθ3θ−T∞4−Aφi−T∞φi−T∞−Bφi3φi−T∞4=0,with the boundary conditions
(24)θ−φi+1=0atξ=1,ddξθ−φi+1=0atξ=R.Now, let us assume thatθ−φi+1 is negative at some point. Since
(25)1ξddξξddξθ−φi+1−αθ−φi+1−−Aθ−T∞θ−T∞+Bθ3θ−T∞4−Aφi−T∞φi−T∞−Bφi3φi−θ∞4=0,we must have
(26)αθ−φi+1−Aθ−T∞θ−T∞+B∣θ∣θ3−T∞4−Aφi−T∞φi−T∞−Bφi3φi−θ∞4≤0.Sinceθ≥0, φ0≡0, and α is large enough, we conclude that the above inequality never holds and, so, θ−φ1 is nonnegative everywhere. Repeating this procedure, we can conclude that θ≥φi, for any i. In other words, θ is an upper bound for the elements φi. This ensures the existence of the limit θ=limi⟶∞φi.


### 2.3. Exact Solution

The maximum ofθ−φi+1 occurs when its second derivative is not positive. So, this maximum occurs when
(27)αφi+1−φi≥Aθ−T∞θ−T∞+Bθ3θ−T∞4−Aφi−T∞φi−T∞−Bφi3φi−T∞4.Since the sequenceφ0,φ1,φ2,φ3 has an upper bound and Aθ−T∞θ−T∞+Bθ4−T∞4 is a strictly increasing function, we can conclude that the limit of the sequence is the solution of the original problem.


## 2.1. Nondecreasing Proof

In order to prove that the sequence ofφi is nondescending, the difference between two consecutive terms of the sequence is considered.
(11)1ξddξξddξφi+1−φi−αφi+1−φi+αφi−φi−1−Aφi−T∞φi−T∞−Bφi3φi−T∞4++Aφi−1−T∞φi−1−T∞+Bφi−13φi−1−T∞4=0.Since(12)φ∧i+11−φ∧i1=0,ddξφ∧i+1−φ∧iξ=R=0,defining the function φ1as the solution of
(13)1ξddξξddξφ1−φ0−αφ1−φ0=Aφ0−T∞φ0−T∞−Bφ03φ0−T∞4≥0with
(14)φ∧11=1,dφ∧1dξξ=Rext=0,we have, since φ0≡0, that
(15)φ1−φ0≥0⇒φ1≥φ0≡0.So, we can writei=1(16)1ξddξξddξφ2−φ1−αφ2−φ1+αφ1−φ0−Aφ1−T∞φ1−T∞−Bφ13φ1−T∞4+Aφ0−T∞φ0−T∞+Bφ03φ0−T∞4=0with
(17)φ∧21−φ∧11=0,ddξφ∧2−φ∧1ξ=Rext=0.So, choosingα large enough for ensuring that
(18)αφ1−φ0≥Aφ1−T∞φ1−T∞+Bφ13φ1−T∞4−Aφ0−T∞φ0−T∞−Bφ03φ0−T∞4,the following inequality holds:
(19)φ2≥φ1≥φ1≡0.Repeating the above procedure, we have(20)⋯≥φi+1≥φi≥φi−1≥⋯≥φ1≥φ0≡0provided the constant α is large enough to ensure that
(21)α≥Aφi+1−T∞φi+1−T∞φi+1−φi+Bφi+13φi+1−T∞4φi+1−φi−Aφi−T∞φi−T∞−Bφi3φi−T∞4φi+1−φi.It is sufficient to chooseα in such a way that
(22)α≥ddθAθ−T∞θ−T∞+Bθ3θ−T∞4θ=1=2Aθ−T∞+4Bθ3θ=1=2AT∞+4B.


## 2.2. Upper Bound

Combining the above problems, we have(23)1ξddξξddξθ−φi+1+αφi+1−φi−−Aθ−T∞θ−T∞+Bθ3θ−T∞4−Aφi−T∞φi−T∞−Bφi3φi−T∞4=0,with the boundary conditions
(24)θ−φi+1=0atξ=1,ddξθ−φi+1=0atξ=R.Now, let us assume thatθ−φi+1 is negative at some point. Since
(25)1ξddξξddξθ−φi+1−αθ−φi+1−−Aθ−T∞θ−T∞+Bθ3θ−T∞4−Aφi−T∞φi−T∞−Bφi3φi−θ∞4=0,we must have
(26)αθ−φi+1−Aθ−T∞θ−T∞+B∣θ∣θ3−T∞4−Aφi−T∞φi−T∞−Bφi3φi−θ∞4≤0.Sinceθ≥0, φ0≡0, and α is large enough, we conclude that the above inequality never holds and, so, θ−φ1 is nonnegative everywhere. Repeating this procedure, we can conclude that θ≥φi, for any i. In other words, θ is an upper bound for the elements φi. This ensures the existence of the limit θ=limi⟶∞φi.


## 2.3. Exact Solution

The maximum ofθ−φi+1 occurs when its second derivative is not positive. So, this maximum occurs when
(27)αφi+1−φi≥Aθ−T∞θ−T∞+Bθ3θ−T∞4−Aφi−T∞φi−T∞−Bφi3φi−T∞4.Since the sequenceφ0,φ1,φ2,φ3 has an upper bound and Aθ−T∞θ−T∞+Bθ4−T∞4 is a strictly increasing function, we can conclude that the limit of the sequence is the solution of the original problem.


## 3. Numerical Analysis

The problem is modeled as(28)1ξddξξdφi+1dξ−αφi+1+βi=0.with
(29)βi=αφi−Aφi−T∞φi−T∞−Bφi3φi−T∞4that has, as its solution the limit of the sequence φ0,φ1,φ2,φ3⋯, where we chose to use the finite difference method to solve the ODE.Thus, the numerical model adopted becomes(30)φni+1=l2+αl2φn+1i+1+φn−1i+1l+φn+1i+1−φn−1i+12ξn+βnil,where
(31)βni=αφni−A∣φni−T∞∣φni−T∞−Bφni3φni−T∞4.Therefore, it can be understood thati is the index that represents the numerical iteration comprising each element of the sequence whose limit is the solution to the problem.Meanwhile,n is the index that represents the numerical iterations of each point in space (geometric) along the radial fin.A more in-depth study ofα demonstrates that the larger its value, the larger the number of iterations required to ensure sequence convergence, as can be seen in Figure 2.Figure 2
Alpha vs. required iterations.Figure2 also shows that such increase in the number of iterations is linear to the increase in the value of α; however this linearity is valid in the region where α is large.For values ofα close to zero, it can be noticed that the number of iterations varies, seeking to approach linearity as it moves away from values close to zero, as can be seen in Figure 3.Figure 3
Alpha vs. iterations, for small alpha.Disregarding the variation in computational effort for eachα due to the increase in the number of iterations, the thermal profile of the fin is identical for any α and is given by Figure 4.Figure 4
Thermal profile.The convergence of the numerical sequence can be noticed, according to the adopted parameters, through the visualization of a well-behaved curve, where after a certain point, there is no longer a considerable difference between previous values.It is interesting to analyze the behavior of the numerical convergence of specific points on the fin, since, for comparative purposes, the variation in the convergence speed is noted. For this, the points were analyzed: 0.25, 0.5, 0.75, and 1.0, as illustrated in Figures5–8.Figure 5
Sequence behavior at point 0.25.Figure 6
Sequence behavior at point 0.5.Figure 7
Sequence behavior at point 0.75.Figure 8
Sequence behavior at point 1.0.It is possible to better visualize how the parameters of temperature, sequence elements, and fin nodal points vary in a combined way through Figure9.Figure 9
Result surface.Thus, as expected, it is noted that convergence occurs more quickly at points closer to the base.In order to further analyze the influence of radiation on the thermal profile that was modeled in this work, it was studied how the variation of the parameterB affects the temperature distribution along a porous radial fin. For this, the parameter B was taken with different values, where such values were compared with the situation where B=0. The consideration of B=0 implies a situation where thermal radiation is not considered; that is, the nonlinear temperature term due to radiation is removed from the modeling. It is noteworthy that disregarding the effects of thermal radiation greatly alters the problem analyzed, since the radiation emitted by the fin through its pores considerably affects how the temperature is dissipated in the body.Figure10 shows six superimposed temperature profiles, in which the value of the parameter B varies from B=0 (a situation that disregards the effects of thermal radiation), increasing the value of B, passing by B=20, B=40, B=60, B=80, up to B=100.Figure 10
Thermal radiation influence analysis.For a better analysis, Table1 shows the dimensionless values of the temperature in some selected nodes, comparing such values for the situations where B=0 with B=20.Table 1
Comparison of radiation valuesB=20.
NodeB=0B=20B20−B0%11.00001.00000.00000.00%20.92280.78620.136614.80%50.73810.49960.238532.31%100.53340.32190.211539.65%200.31570.18890.126840.16%300.20870.13080.077937.33%400.14860.09790.050734.12%500.11210.07730.034831.07%600.08880.06370.025228.32%700.07380.05460.019125.94%800.06440.04890.015524.07%900.05930.04570.013622.88%1000.05780.04480.013022.49%Increasing the value ofB means increasing the influence of the effects of thermal radiation. It is interesting to take this increment gradually, so that it is possible to analyze the thermal behavior of the fin as the radiation becomes considerable. Thus, analyzing the thermal profile for the value of B=40, we have Table 2.Table 2
Comparison of radiation valuesB=40.
NodeB=0B=40B40−B0%11.00001.00000.00000.00%20.92280.73430.188520.43%50.73810.43480.303341.09%100.53340.27310.260348.80%200.31570.16030.155549.24%300.20870.11220.096546.24%400.14860.08510.063642.76%500.11210.06800.044139.38%600.08880.05660.032236.27%700.07380.04900.024733.54%800.06440.04420.020231.38%900.05930.04150.017829.99%1000.05780.04070.017129.54%It can be noticed, analyzing Tables1 and 2, that the greatest discrepancy between temperature values occurs between nodes 10 and 20. As the analysis progresses, the temperature values for B=60 are taken, as seen in Table 3.Table 3
Comparison of radiation valuesB=60.
NodeB=0B=60B60−B0%11.00001.00000.00000.00%20.92280.70060.222224.08%50.73810.39750.340646.15%100.53340.24620.287353.85%200.31570.14450.171354.24%300.20870.10180.106951.22%400.14860.07780.070947.69%500.11210.06250.049644.21%600.08880.05240.036440.97%700.07380.04570.028138.11%800.06440.04130.023135.84%900.05930.03890.020434.36%1000.05780.03820.019633.89%Going forward in the analysis of the influence of thermal radiation, it can already be noted that, from here on, large increments in the value ofB change the profile little, as can be seen in Table 4.Table 4
Comparison of radiation valuesB=80.
NodeB=0B=80B80−B0%11.00001.00000.00000.00%20.92280.67560.247226.79%50.73810.37170.366449.64%100.53340.22810.305457.24%200.31570.13390.181957.60%300.20870.09470.114054.61%400.14860.07270.075951.08%500.11210.05880.053347.57%600.08880.04950.039344.28%700.07380.04330.030541.36%800.06440.03930.025139.03%900.05930.03710.022237.51%1000.05780.03640.021437.02%The analysis ends with the temperature values forB=100 as shown in Table 5, where it is noticed that the thermal profile varies little in relation to values of B a little smaller. The greatest discrepancy between temperature values for B=100 and B=80 is less than 6%, bringing the hypothesis of negligible variation in values much larger in B.Table 5
Comparison of radiation valuesB=100.
NodeB=0B=100B100−B0%11.00001.00000.00000.00%20.92280.65570.267128.95%50.73810.35230.385852.27%100.53340.21460.318859.76%200.31570.12600.189760.09%300.20870.08950.119257.14%400.14860.06890.079753.63%500.11210.05590.056250.12%600.08880.04720.041646.81%700.07380.04140.032443.87%800.06440.03770.026741.50%900.05930.03560.023739.97%1000.05780.03500.022839.47%It can be seen, through this analysis, that the parameter that carries the effect of thermal radiation has a great influence on the thermal profile, since the discrepancy in temperature values between profiles with higher values ofB reaches more than 60% (comparison between the cases B=0 and B=100).


## 4. Conclusions

This work provides a comprehensive approach to heat dissipation in extended surfaces, where several parameters were considered, among which the application of thermal radiation effects and the porosity of the fin material stand out. The developed model seeks an adequate solution to the problem, applying a cylindrical geometry, widely used in engineering, mainly in heat dissipation in industrial pipes.The mathematical description of the solution proves that the model used in this work has an exact solution, which was presented, provided that a specific criterion of nondecrease of the thermal profile is met, which was also mathematically proven.The simple methodology employed considers realistic modeling of heat transfer with specific boundary conditions in different porous media. Applications of this procedure seeks to extrapolate finned array analysis of heat sinks and the relevance of your parameters on final heat flux.



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