# Thermal Analysis of porous fin with uniform magnetic field using Adomian decomposition Sumudu transform method

Trushit Patel and Ramakanta Meher
From the journal Nonlinear Engineering

# Abstract

In this paper, we consider a Roseland approximation to radiate heat transfer, Darcy’s model to simulate the flow in porous media and finite-length fin with insulated tip to study the thermal performance and to predict the temperature distribution in a vertical isothermal surface. The energy balance equations of the porous fin with several temperature dependent properties are solved using the Adomian Decomposition Sumudu Transform Method (ADSTM). The effects of various thermophysical parameters, such as the convection-conduction parameter, Surface-ambient radiation parameter, Rayleigh numbers and Hartman number are determined. The results obtained from the ADSTM are further compared with the fourth-fifth order Runge-Kutta-Fehlberg method and Least Square Method(LSM) (Hoshyar et al. 2016 [1])) to determine the accuracy of the solution.

## 1 Introduction

The rate of heat transfer is mostly depends upon the temperature variation involving the surface and the surrounding fluids, an existing surface area and heat transfer coefficient. However, this necessity is frequently justified through the high cost of the high thermal conductivity metals. Magnetohydrodynamics (MHD) is the study of the magnetic properties of electrically conducting fluids in different porous geometries that is of considerable attention due to its frequent occurrence in geothermal, industrial and technological applications. The theoretical MHD’s study has been a subject of large interest owing to its widespread applications, such as petroleum industries, plasma studies, the boundary layer control in aerodynamics, MHD power generators, crystal growth, and cooling of nuclear reactors.

Now a days nonlinear fractional order types of problems and phenomenon plays an essential role in engineering, physics, applied mathematics, and new branches of science specially heat transfer problems. Several approximate analytical techniques such as Variational iteration method [20], Homotopy perturbation method [20, 21], Least Square Method [1, 2224], Differential Transform Method [2527], Adomian decomposition method [28], have been used to solve such type of problems.

In this paper, the energy balance equation is modeled through a nonlinear fractional order differential equation and ADSTM is applied to find the series solution for temperature held of a rectangular porous fin with multiple nonlinearities.

## 2 Mathematical Formulation

Here, a rectangular porous fin is considered (Fig. 1) having constant cross sectional area, width W, length L and thickness t with the following assumptions

### Fig. 1

The geometry of a rectangular fin profile.

1. The fin is made of porous material that allows the flow of infiltrate through it.

2. The porous medium is isotropic, homogeneous and saturated with a single-phase fluid.

3. A uniform magnetic field is applied in the direction of y- axis having the temperature inside the fin is a function of x only.

4. The Darcy’s model is used to study the flow velocity in a porous medium with negligible effect of the imposed and induced magnetic field, and induced electrical field due to polarization effect.

The one-dimensional energy balance equation at steady state condition to the slice segment of the fin thickness ΔX is given by [1],

q(x)q(x+Δx)=m¯cpT(x)T()+hPΔx(1ε)T(x)T()+Jc×Jcσ+PΔxσstε¯T(x)4γε¯T()4(1)

where Jc is conduction current intensity, and it is defined by

Jc=σV×B+E(2)

and, J is total current intensity defined by

J=Jc+ρε¯V

The mass flow rate of the fluid passing through the porous material which is stated as

m¯=ρwv¯wΔx(3)

The passage velocity w be supposed to estimated from the consideration of the flow in porous medium, then Darcy’s model yields,

v¯w=kβg(T(x)T)v(4)

The relation between conduction and radiation at the base of fin can be defined as

using Fourier’s law of conduction and the radiation heat flux term, the Rosseland diffusion approximation can be defined as

Substituting Eq. (2)(6) in Eq. (1), it gives

ddxdTdx+4σ3βrkeffdT4dx=ρcpgkβbvkeffT(x)T()2+hP(1ε)keffT(x)T()+Jc×JcσkeffAb+Pσstε¯keffAbT(x)4T()4(7)

With the neglect of magnetic field and the induced current, the electromagnetic force in Eq. (1) takes the form Jc×Jcσ=σB02u2.

Using the dimensionless parameters

θ=(T(x)T)(TbT),ζ=xL,θb=TbT,Ra=gkβb(TbT)γvkr,Nc=PbhkeffA,Nr=4σstLT3keffH=σB02u2k0A,Rd=4σstT33βrkeff(8)
Eq. (7) gives
d2θdζ2Ra(1+4Rd)θ2(Nc(1ε)+Nr+H)(1+4Rd)θ=0(9)

with the boundary conditions

dθdζζ=1=0andθζ=0=1.(10)

where Ra, is a modified Rayleigh number, Nc, is a convection-conduction parameter, Nr, is a Surface-ambient radiation parameter, H, is Hartman parameters, Rd, is Radiation-conduction parameter and and ɛ, is porosity.

In this paper, we considered and studied finite-length fin with insulated tip, so that there won’t be any heat transfer at the insulated tip.

## 3 Mathematical Formulation of ADSTM

Consider a fractional order nonlinear nonhomogeneous differential equation as

Dζαθ(ζ)+Rθ(ζ)+Nθ(ζ)=g(ζ),withn1<αn(11)

with initial condition

θ(0)=K(12)

where Dζα is the Caputo fractional derivative of the function θ(ζ), R is reminder term or linear differential operator, N represents the general nonlinear differential operator, and g(ζ) is the source term.

Taking Sumudu Transform (denoted by S) on both sides of Eq. (11), it obtain

S[Dζαθ(ζ)]+S[Rθ(ζ)]+S[Nθ(ζ)]=S[g(ζ)](13)

By using Sumudu transform of Caputo fractional derivative [33, 34], Eq. (11) can be written as

S[θ(ζ)]=θ(0)+uαS[g(ζ)]uαS[Rθ(ζ)+Nθ(ζ)](14)

By taking inverse Sumudu Transform on both sides of Eq. (14), it gives

θ(ζ)=G(ζ)S1uαS[Rθ(ζ)+Nθ(ζ)](15)

where G(ζ) = S-1 [θ(0) + uαS [g(ζ)]] represents the term arising from the source term and the prescribed initial conditions.

Now by applying the ADM [28], the approximate series solution of Eq. (15) can be written as

θ(ζ)=n=0λnθn(ζ),(16)

and the nonlinear term in Eq. (15) can be written as the summation of Adomian polynomials as

Nθ(ζ)=n=0λnAn(θ),(17)

where the Adomian polynomials An(θ) arising in differential equation can be defined as

An(θ0,θ1,θ2,...,θn)=1n!nλnNi=0λiθiλ=0,(18)

forn = 0, 1, 2, ...

Substituting Eqs. (16) and (17) in Eq. (15) and applying the inverse Sumudu Transform, we find

n=0λnθn(ζ)=G(ζ)λS1uαSRn=0λnθn(ζ)+n=0λnAn(θ)(19)

The resulting Eq. (19) is the coupling of the Sumudu Transform and the Adomian Decomposition Method.

Now by equating the coefficients of like powers of λ, in Eq. (19), the following iterated terms can be obtained as

λ0:θ0(ζ)=G(ζ)λ1:θ1(ζ)=S1uαSRθ0(ζ)+A0(θ)λ2:θ2(ζ)=S1uαSRθ1(ζ)+A1(θ)λ3:θ3(ζ)=S1uαSRθ2(ζ)+A2(θ)(20)

By considering this above procedure, θn (ζ) can be completely obtained implies the series solution can be determined subsequently and hence the resulted approximate analytical solution n=0Mθn(ζ) converges to the exact solution θ(ζ) as M → ∞.

## 4 Application of ADSTM for finding solution for porous fin having uniform magnetic field

In the last two decades, fractional calculus starts to intervene significantly in engineering, physics, economics, etc [29]. It is due to new possibilities in which fractional calculus brings into the modelling of various problems [30, 31]. Therefore, In this work, to understand the anomalous behavior of this system, the energy balance Eq. (9) fractionalize into fractional order (α > 0), and applied Adomian decomposition Sumudu transform method in order to find the fin temperature distribution in a rectangular porous fin as,

dαθdζαRa(1+4Rd)θ2(Nc(1ε)+Nr+H)(1+4Rd)θ=01<α2and0ζ1(21)

and the boundary conditions given in Eq. (10).

Now, by applying Sumudu transform (denoted by S)[32] on both sides of Eq. (21), it obtains

Sdαθdζα=SRa(1+4Rd)θ2+S(Nc(1ε)+Nr+H)(1+4Rd)θ(22)

By using Sumudu transform of Caputo fractional derivative [33, 34], we can write

Sθ(ζ)uαθ(0)uαθ(0)uα1=SRa(1+4Rd)θ2+S(Nc(1ε)+Nr+H)(1+4Rd)θ(23)

Using initial and boundary conditions, and Inverse Sumudu transform, we get

θ(ζ)=1+Kζ+S1uαSRa(1+4Rd)θ2+uαS(Nc(1ε)+Nr+H)(1+4Rd)θ(24)

By using the Adomian decomposition method in Eq. (24), it gives

n=0λnθn(ζ)=1+Kζ+S1uαSRa(1+4Rd)n=0λnAn(θ)+S1uαS(Nc(1ε)+Nr+H)(1+4Rd)n=0λnθn(ζ)(25)

where An(θ)’s are the Adomian’s polynomial that represents nonlinear terms to be determined. The first few Adomian polynomials are obtained by using Eq. (18),

A0=(θ0)2A1=2θ0θ1A2=2θ0θ2+(θ1)2A3=2θ0θ3+2θ1θ2

On comparing the coefficients of like power of λ in Eq. (25), the iterated components can be expressed as,

θ0(ζ)=1+Kζθn+1(ζ)=S1uαSRa(1+4Rd)An(θ)+(Nc(1ε)+Nr+H)(1+4Rd)θn(ζ),n>1.(26)

and it gives,

θ0(ζ)=1+Kζθ1(ζ)=Ra1+4RdζαΓα+1+2Kζα+1Γα+2+Γ(3)K2ζα+2Γα+2+Nc1E+Nr+HK1+4RdζαΓα+1+Kζα+1Γα+2

Hence, the temperature held for the rectangular porous profile fin is obtained in expressions of finite series,

θ=k=0mθk(ζ)=θ0(ζ)+θ1(ζ)+θ2(ζ)+(27)

and the value of θ can be evaluated if the temperature at the fin tip K is known, and it must lie in the interval [0, 1]. The constant, K can be determined by applying Newton-Raphson method.

## 5 Fin Efficiency

Fin performance can also be characterized by fin efficiency and it can be defined as

η=qfqmax=(1+4Rd)dθdζζ=1(Nc(1ε)+Nr)+H+Ra(28)

## 6 Results and Discussion

Here, we discusses the temperature distribution in a rectangular porous fin with the consideration of uniform magnetic held. The effects of different dimensionless parameters such as modified Rayleigh number, Ra, convection-conduction parameter, Nc, Surface-ambient radiation parameter, Nr, Hartman parameters, H and Radiation-conduction parameter, Rd, on temperature distribution are investigated during heat transfer in a rectangular porous fin.

The parameter Ra measures the ratio of thermal convection to diffusion, when there is a balance between buoyancy and Lorentz forces, and it is the determining parameter for the flow. The parameter Nc indicates that the strength of convection versus conduction and the parameter Nr, measures the strength of surface radiation versus conduction. The values of Nc and Nr range between 0 (ideal fin of infinite thermal conductivity) and 2 for most of the fins. The parameter H measures the ratio of electromagnetic force to the viscous force.

Table 1 discusses the numerical results obtained for temperature distribution by using ADSTM and the obtained results has been compared with the standard numerical results obtained by Runge-Kutta method and Least Square Method (LSM)[1]. The absolute error has been discussed to test the accuracy of the present method and the obtained results shows that the ADSTM results is very close to the numerical results obtained by Runge-Kutta method and LSM [1].

### Table 1

Comparison of ADSTM solution, LSM [1] solution and the numerical method for dimensionless temperature 6(() and for dimensionless parameters Nr = 0.3, Nc = 0.4, H = 0.9, Rd = 0.5 and Ra = 0.1.

XADSTMLSM [1]NMAbsolute error between ADSTM and NMAbsolute error between LSM [1] and NM
0.01.0000000001.0000000001.0000000000.0000000000.000000000
0.10.9569879430.9569876650.9569880200.0000000770.000000355
0.20.9191324420.9191320600.9191325130.0000000710.000000453
0.30.8862178870.8862178280.8862179640.0000000770.000000136
0.40.8580575900.8580579700.8580576900.0000001000.000000280
0.50.8344924020.8344929690.8344925090.0000001070.000000460
0.60.8153895500.8153899060.8153896680.0000001180.000000238
0.70.8006416760.8006415890.8006418050.0000001290.000000216
0.80.7901660630.7901656680.7901661870.0000001240.000000519
0.90.7839040490.7839037600.7839041540.0000001050.000000394
1.00.7818205940.7818205690.7818207290.0000001350.000000160

Fig. 2 discusses the variation of temperature distribution for ɛ = 0.4, Rd = 0.7, Ra = 0.5, Nc = 0.2, H = 0.4 and α = 2 with different values of Nr. It shows that the fin temperature decreases with Nr results, strong cooling implies lesser the radiant temperature distribution within the fin. Similarly, Fig. 3 discusses the variation of temperature distribution with different Nr and for different fractional values α = 1.75, 1.5, 1.25 results the behavior of the solution curve is closer to the integer order solution implies the solution curve is valid for the fractional order energy balance equation.

### Fig. 2

Effect of surface-ambient radiation parameter, Nr and comparison between the ADSTM with the numerical solution for classical order α = 2.0.

### Fig. 3

The ADSTM solutions for a different value of Nr and for (a) α = 1.75 (b) α = 1.5 (c) α = 1.25.

To study the effect of varying the convection-conduction parameter Nc on the performance of the fins, Fig. 4 and 6 was plotted. Fig. 4 discusses the variation of temperature distribution for ɛ = 0.4, Rd = 0.7, Ra = 0.5, Nr = 0.3, H = 0.4 and for classical α = 2 with different values of Nc. It shows that the fin temperature decreases with Nc increases results, a decline in fin temperature causes a stronger decrease in local temperature of insulated tip fin. Similarly, Fig. 6 discusses the variation of temperature distribution with different Nc and for different fractional values α = 1.75, 1.5, 1.25 results the behavior of the solution curve is closer to the integer order solution implies the solution curve is valid for the fractional order energy balance equation.

### Fig. 4

Effect of convection-conduction parameter, Nc and comparison between the ADSTM with the numerical solution for classical order α = 2.0.

### Fig. 5

Effect of Hartman parameters, H and comparison between the ADSTM with the numerical solution for classical order α = 2.0.

### Fig. 6

The ADSTM solutions for a different value of Nc and for (a) α = 1.75 (b) α = 1.5 (c) α = 1.25.

Figs. 5 and 7 depicts the results of dimensionless temperature variation for parameter H and for classical order solution α = 2. The values of H are varied from 0.1, 0.3, 0.6 and 0.9, while remaining parameters are kept at, ɛ = 0.4, Rd = 0.7, Ra = 0.5, Nc = 0.3 and Nr = 0.3. It shows that the fin temperature be decreases with the increase of the parametric values of H and for fractional order α = 1.75, 1.5, 1.25 respectively.

### Fig. 7

The ADSTM solutions for a different value of H and for (a) α = 1.75 (b) α = 1.5 (c) α = 1.25.

Figs. 8 and 9 depicts the results of dimensionless temperature variation for different parameters Ra and for classical order solution α = 2. The values of Ra are varied from 0.1, 0.3, 0.6 and 0.9, while remaining parameters are kept at, ɛ = 0.4, Rd = 0.7, H = 0.4, Nc = 0.3 and Nr = 0.3. It shows that the fin temperature be decreases with the increase of the parametric values of Ra results stronger thermal convection implies decline in fin temperature and for fractional order α = 1.75, 1.5, 1.25 respectively. Accordingly, Fig. 8 clearly demonstrates that the radiation-conduction parameter has a minimum effect on the fins surface temperature for the rectangular porous fin.

### Fig. 8

Effect of Radiation-conduction parameter, Ra and comparison between the ADSTM with the numerical solution for classical order α = 2.0.

### Fig. 9

The ADSTM solutions for a different value of Ra and for (a) α = 1.75 (b) α = 1.5 (c) α = 1.25.

Figs. 10(a) illustrate the variation of fin efficiency with Ra with the effect of different parameters Rd, H, Nr and Nc by keeping its parametric values fixed which shows that the fin efficiency be decreases as Ra increases and porosity decreases. In Fig. 10(b), we have plotted the fin tip temperature as a function of the radiation-conduction parameter Rd for fixed parametric values Ra, H, Nr and Nc and for the different porosity parameter ϵ. When radiation is stronger, it results in lower fin tip temperatures. Figs. 10(c) illustrate the variation of fin efficiency with Nr with the effect of different parameters Ra, RdH and Nc by keeping its parametric values fixed which shows that the fin efficiency be decreases as Nr increases and porosity decreases. Figs. 10(d) illustrate the variation of fin efficiency with Nc with the effect of different parameters Ra, RdH and Nr by keeping its parametric values fixed which shows that the fin efficiency be decreases as Nc increases and porosity decreases.

### Fig. 10

The variation of efficiency with variation of (a)Ra (b)Rd (c)Nr (d)Nc for different porosity .

## 7 Conclusions

In this study, the solution of a rectangular porous fin with a uniform magnetic field in a vertical isothermal surface obtained by using the ADSTM. A dimensionless expression for the temperature distribution and fin efficiency has been derived and discussed the effects of different parameters on porous fin. Analytical and numerical results for the temperature distribution are presented through the graphs and the table in various values of the parameter. Finally, the ADSTM results has been compared with the fourth-fifth order Runge-Kutta-Fehlberg method and Least Square Method(LSM).

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