# A Study on Convective-Radial Fins with Temperature-dependent Thermal Conductivity and Internal Heat Generation

Trushit Patel and Ramakanta Meher
From the journal Nonlinear Engineering

# Abstract

In this paper, the temperature distribution in a convective radial fins is analyzed through a fractional order energy balance equation with the consideration of internal heat generation and temperature dependent thermal conductivity. Adomian decomposition Sumudu transform method is used to study the influence of temperature distribution and the efficiency of radial fins for different values of thermal conductivity and to determine the role of thermal conductivity, thermo-geometric fin parameter as well as fractional order values in finding the temperature distribution and the fin efficiency of the convective radial fins. Finally, the efficiency of this proposed method has been studied by comparing the obtained results with the classical order results obtained by using numerical method and Variational Iteration Method (Coskun and Atay, 2007).

## 1 Introduction

Extended surfaces, called fins, are used to elevate the heat transfer rate involving a hot body and its surrounding environment and can be modelled in various geometrical shapes such as rectangular, trapezoidal, triangular, and cylindrical. Because of the nature of the material properties, the thermal conductivity is a temperature-dependent parameter which makes the energy balance equation nonlinear.

Several development have been made in solving the nonlinear fin type problems with different parameters by using different semi analytical methods [14, 15, 16, 17]. Coskun and Atay , used variational iteration method and studied the temperature distribution and fin efficiency of convective radial fins with temperature-dependent thermal conductivity with the consideration of classical order energy balance equation. Hatami et. al.  used Differential Quadrature Method (DQM) and Differential Transformation Method (DTM) to study MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates. Dogonchi et al.  applied a Padé approximate which are an analytical solution technique and differential transformation method (DTM) to study the unsteady motion of a vertically falling non-spherical particle in incompressible Newtonian media. Rahimi et. al.  applied collocation method to study boundary layer flow of an Eyring-Powell fluid over a stretching sheet in unbounded domain. Patel and Meher  applied ADSTM to study the temperature distribution in porous fin with uniform magnetic field and compared their result with Runge–Kutta method and Least Square Method (LSM).

Sheikholeslami et al.  investigated the laminar viscous flow in a semi-porous channel in the presence of uniform transverse magnetic field and employed Optimal Homotopy Asymptotic Method (OHAM) to approximate the solution of the governing nonlinear differential equations and studied the influence of the Hartmann number (Ha) and the Reynolds number (Re) on the fluid flow. Sheikholeslami and Ganji  employed OHAM and studied the laminar magneto-hydrodynamic nano fluid flow in a porous channel and compared the obtained results in good agreement with a numerical method (fourth-order Runge-Kutta method). Sheikholeslami et al. [24, 25] employed Homotopy Perturbation Method(HPM) and studied a three-dimensional problem of condensation film on inclined rotating disk and the heat transfer of Cu-water nano fluid flow that is squeezed between parallel plates. Similarly, Sheikholeslami et al.  employed ADM and investigated the unsteady flow of a nano-fluid squeezing between two parallel plates. Sheikholeslami and Ganji [27, 28] used DTM and studied the nano-fluid flow and heat transfer between parallel plates considering Brownian motion obtained the hydrothermal behavior in presence of variable magnetic field and also studied the nano-fluid hydrothermal behavior in existence of Lorentz forces by considering Joule heating effect. Similarly, Sheikholeslami et al.  used HAM and analytically investigated the Lorentz forces effect on nano-fluid Marangoni boundary layer and studied the hydrothermal behavior of nano fluid in presence of magnetic field.

Here in this work, the authors extended the work of Coskun and Atay  to a fractional order energy balance equation from a classical one in order to study the anomalous behavior of the fractional order energy balance equation with the consideration of heat generation as a variable of fin temperature for Convective-Radial fins and used ADSTM to study the behavior of temperature distribution. The ADSTM has a solid application towards non-linearity and is connected here to decide the temperature distribution and fin productivity of a convective-radial fin. For further discussion two cases has been considered. Case 1 discusses the temperature distribution for convective-Radial fins with thermal conductivity and without internal heat generation while case 2 discusses it with temperature dependent thermal conductivity and internal heat generation which is the novelty part of this work.

## 2 Mathematical Formulations of the Problem

Here we considered an one dimensional thin fin having thermal conductivity parameter, k which linearly depends on surface temperature, the emissivity parameter ε and radiates heat with a very low temperature. The surface and the base temperature of the tube surface has been kept constant with negligible effect of convective and radiative exchange between the tubes as shown in Fig 1. Here, the fin tip of length b is considered as the computational domain. ### Fig. 1

Schematics of heat pipe/fin radiating element

The energy balance equation of the radial fin in one dimensional steady state is given by [7, 15]

2wddxk(T)dTdx2εσ(T4TS4)+Q=0(1)

where σ is the Stefan–Boltzmann constant and k(T) is the thermal conductivity of the fin material which is defined by :

k(T)=kb1+λ~(TTb)(2)

where kb characterizes the thermal conductivity at the base temperature and λ͂ characterizes the slant of the thermal conductivity temperature curve.

By using dimensionless variables

θ=TTb,ζ=xb,ψ=εσb2Tb3kbw,β=λ~Tb,θa=TaTb,θS=TSTb,G=b2QTbk0(3)

The dimensionless form of energy balance Eq. (1) can be expressed as

Considering an adiabatic tip, the boundary conditions are

dθdζζ=0=0andθζ=1=1.(5)

Consider a fractional order nonlinear nonhomogeneous differential equation as

Dζαθ(ζ)+Rθ(ζ)+Nθ(ζ)=g(ζ),withn1<αn(6)

with initial condition

θ(0)=K(7)

where Dζα

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

Taking Sumudu Transform (denoted by S) on both sides of Eq. (6), it obtain

S[Dζαθ(ζ)]+S[Rθ(ζ)]+S[Nθ(ζ)]=S[g(ζ)](8)

Using property of Sumudu transform of Caputo derivative , and initial condition, Eq. (8) can be written as

S[θ(ζ)]=K+uαS[g(ζ)]uαS[Rθ(ζ)+Nθ(ζ)](9)

By taking inverse Sumudu Transform on both sides of Eq. (9), it gives

θ(ζ)=G(ζ)S1uαS[Rθ(ζ)+Nθ(ζ)](10)

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

On applying the ADM , the approximate series solution of Eq. (10) can be written as

θ(ζ)=n=0λnθn(ζ),(11)

and the nonlinear term in Eq. (10) can be written as the summation of Adomian polynomials as

Nθ(ζ)=n=0λnAn(θ),(12)

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

An(θ0,θ1,θ2,...,θn)=1n!nλnNi=0λiθiλ=0,forn=0,1,2,...(13)

Substituting Eqs. (11) and (12) in Eq. (10), we get

n=0λnθn(ζ)=G(ζ)λS1uαSRn=0λnθn(ζ)+n=0λnAn(θ)(14)

The resulting Eq. (14) is the coupling of the Sumudu Transform and the Adomian Decomposition Method.

Now by equating the coefficients of like powers of λ, in Eq. (14), the following iterated terms can be obtained as

λ0:θ0(ζ)=G(ζ)λ1:θ1(ζ)=S1uαSRθ0(ζ)+A0(θ)λn:θn(ζ)=S1uαSRθn1(ζ)+An1(θ)(15)

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 Solution by Using ADSTM

Here, the fin temperature distribution is determined by using ADSTM with the consideration of fractional order energy balance equation and to study the anomalous behaviour of the energy balance equation for fractional order 1 < α ≤ 2.

Therefore, Eq. (4) can be expresses as

where 0≤ ζ ≤ 1 and 1 < α ≤ 2 having boundary conditions Eq. (5).

On applying Sumudu transform in Eq. (16), we have

Utilizing the property of Sumudu transform of Caputo fractional , we get

Sdαθdζα=Sθ(ζ)uαθ(ζ=0)uαθ(ζ=0)uα1=Sθ(ζ)uαK,(Kisconstant)(18)

Hence, Eq.(17) becomes

Now, applying inverse Sumudu transform on both sides of Eq. (19), we get

By using, Adomian decomposition method , the recurrence relation of Eq. (20)

n=0λnθn(ζ)=KλS1uαSG+ψθS4βθan=0λnd2θndζ2λS1uαSβn=0λnAn(θ)+βn=0λnBn(θ)n=0λnCn(θ)(21)

where the nonlinear terms An, Bn and Cn are the Adomian’s polynomials and it can be defined as follows

n=0λnAn(ζ)=θd2θdζ2(22)
n=0λnBn(ζ)=dθdζ2(23)
n=0λnCn(ζ)=(θ)4(24)

Some components of the Adomian’s polynomials can be expressed as

A0(θ)=θ0d2θ0dζ2A1(θ)=θ1d2θ0dζ2+θ0d2θ1dζ2A2(θ)=θ2d2θ0dζ2+θ1d2θ1dζ2+θ0d2θ2dζ2(25)
B0(θ)=dθ0dζ2B1(θ)=2dθ0dζdθ1dζB2(θ)=dθ1dζ2+2dθ0dζdθ2dζ(26)
C0(θ)=θ04C1(θ)=4θ1θ03C2(θ)=4θ03θ2+6θ02θ12(27)

Equating the coefficients of similar powers of λ on both sides of Eq. (21), we get

θ0(ζ)=K(28)

Hence, the approximate analytical expression for finding the temperature distribution θ, can be expresses in series form as

θζ=n=0θn(ζ)=θ0(ζ)+θ1(ζ)+θ2(ζ)+...=K+ψK4GψθS4ζαΓ(α+1)+βθaKψK4GψθS4ζ2α2Γ(2α1)+4ψK3ψK4GψθS4ζ2αΓ(2α+1)+...(31)

The value of θ (ζ ) can be calculated with the determination of K at the boundary points θζ =1 =1 in the interval [0, 1].

## 5 Fin Efficiency

By using Newton’s law of cooling, the heat transfer rate from the radial fin can be defined as

E=0bP(TTa)dx(32)

and the fin efflciency can be expressed as

η=EEideal=0bP(TTa)dxPb(TbTa)=ζ=01θ(ζ)dζ(33)

This can be obtained by integrating Eq. (31).

## 6 Results and Discussion

### 6.1 Case 1: Convective-Radial Fins Without Internal Heat Generation

In this case, the dimensionless sink temperatures and the internal heat generation is assumed to be zero i.e. θa = θS = G = 0.

Figure 2, discusses the variation of temperature distribution in a radial fin for β =0.4, α = 2 and for different values of ψ = 1, 10, and 100 keeping thermal conductivity constant. Which shows that the temperature at the fin tip, K, is increasing with the values of thermogeometric fin parameter, ψ. To check consistency and effectiveness of the proposed method, the obtained ADSTM results has been compared with VIM  and with numerical results and it shows that the ADSTM results are very close to the result obtained througn VIM and numerically. ### Fig. 2

Comparison of temperature distributions of convective radial fins for α = 2.0.

Figure 3, discusses the variation of temperature θ in a radial fin with dimensionless coordinate ζ for different fractional order α = 1.75, α = 1.5, and α = 1.25 and for different values of ψ = 1, 10, 100, keeping β fixed. It shows that the temperature distribution is vary with different ψ and with different fractional order and it is maximum for α = 1.75, ψ = 1 and minimum for α = 1.25, ψ = 100. ### Fig. 3

Temperature distributions of convective radial fins for different fractional order α (a) 1.75,(b) 1.5 and (c) 1.25.

Figure 4 and 5, discusses the variation of temperature distribution in a convective radial fins with dimensionless coordinate ζ having variable thermal conductivity for integer order as well as for fractional order having β varying from -0.6, -0.2, 0.2, and 0.6 keeping ψ = 1 constant which shows that fin temperature be maximum for β = 0.6, α = 1.75 and minimum for β = -0.6, α = 1.25. ### Fig. 4

Comparison of temperature distributions of convective radial fins for α = 2.0 ### Fig. 5

Temperature distributions of radial fins of variable thermal conductivity for different fractional order (a) α = 1.75, (b) α = 1.5 and (c) α = 1.25.

Figure 6 discusses the variation of fin tip temperature with the radiation-conduction fin parameter, ψ, for different thermal conductivity parameters, β and for integer order α = 2. It clearly demonstrates that increasing in the values of thermal conductivity parameter results increase in the values of fin tip temperature. The comparison results of ADSTM with VIM  reveal the effectiveness and accuracy of the proposed method. ### Fig. 6

Comparison of the dimensionless fin tip temperature for α = 2.0 and for different thermal conductivity (a) β = –0.6, (b) β = –0.2, (c) β = 0.2 and (d) β = 0.6.

Figures 7 discusses the variation of fin temperature in a convective radial fins with thermo-geometric fin parameter, ψ for different thermal conductivity β = –0.6, –0.2, 0.2, and 0.6 and for different fractional order α = 1.75, 1.5, and 1.25 which shows that the fin tip temperature, K be maximum for β = 0.6 and α = 1.75. ### Fig. 7

Variation of fin tip temperature for various thermal conductivity, β and different fractional order α (a) 1.75, (b) 1.5 and (c) 1.25.

### 6.2 Case 2: Convective-Radial Fin with Internal Heat Generation

The purpose of this work is to observe the simultaneous effects of the governing parameters and the different fractional values α on Convective-Radial fin with internal heat generation. Table (1) depict the physical parametric values used for this case.

### Table 1

Fixed value of dimensionless variable.

 ψ, β, θa, θS, G, Radiation- Thermal Dimensionless Dimensionless Dimensionless -Conduction Conductivity Temperature radiation heat fin parameter Parameter whose K(T) sink generation fin parameter Parameter is constant temperature parameter 1 0.4 0.3 0.2 0.1

Figure 8 discusses the effect of thermo-geometric fin parameter, ψ on temperature distribution in a radial fin keeping β, θa, θS and G fixed with its comparative study to test the accuracy of the said method. Figure 8 shows the effect of radiation conduction parameter, on temperature distribution in the fin. As the radiative transfer becomes stronger, the radiative cooling becomes more effective, which in turn causes the lowering of temperatures in the fin. ### Fig. 8

Comparison of temperature distributions for various, ψ and for α = 2.0.

Figure 9 discusses the effect of thermo-geometric fin parameter, ψ, on temperature distribution in a radial fin and the variation of fin temperature for different fractional order α = 1.75, 1.5, and 1.25 keeping β, θa, θS and G fixed which shows that as the effect of thermo-geometric fin parameter on temperature distribution be increases, the value of fin temperature be decreases implies radiative cooling becomes more effective and it is minimum for α = 1.25 and ψ=10. ### Fig. 9

Variation of temperature distributions for different ψ and for different fractional order α (a)1.75, (b)1.5 and (c) 1.25.

Figure 10 discusses the variation of temperature distribution in a radial fin with thermal conductivity β keeping ψ, θa, θS and G fixed along with its comparative study to test the accuracy of the said method. The curves with positive β correspond to fin materials whose thermal conductivity increases as temperature increases and similarly as the parameter β increases, the average thermal conductivity of the material increases, and as expected, the result is a continuing increase in the local temperature. ### Fig. 10

Comparison of temperature distributions for various, β and for α = 2.0.

Figure 11 discusses the variation of temperature distribution in a radial fin with thermal conductivity, β for different fractional order α = 1.75, 1.5, and 1.25 keeping ψ, θa, θS and G fixed which shows that the fin temperature be increases with thermal conductivity implies the radiation effect becomes more effective and maximum for α = 1.75 and β = 0.6. ### Fig. 11

Variation of temperature distributions for various, β and for different fractional order α (a) 1.75, (b) 1.5 and (c) 1.25.

Figure 12 discusses the effect of θa on temperature distribution in a radial fin keeping ψ, β, θS and G fixed which shows that as θa increases, the temperature within the fin be decreases. This is because of the fact that, as θa becomes lower, thermal conductivity becomes more sensitive to fin temperature. ### Fig. 12

Comparison of temperature distributions for various, θa and for α = 2.0.

Figure 13 discusses the effect of θa on temperature distribution in a radial fin and the variation of fin temperature for different fractional order α = 1.75, 1.5, and 1.25 keeping ψ, β, θS and G fixed which shows that as the effect of θaon temperature distribution be increases, the value of fin temperature decreases which implies that the radiation effect becomes less effective and minimum for α = 1.25 and θa = 0.6. ### Fig. 13

Variation of temperature distributions for various, θa and for different fractional order (a) α = 1.75, (b) α = 1.5 and (c) α = 1.25.

Figure 14 discusses the variation in convection sink temperature θS. The convective heat loss from the fin depends upon the convection sink temperature θS, and as the radiation sink temperature be increases, the convective heat loss from the fin be decreases which results less heat loss from the fin implies the store temperature be more in the fin material having higher sink temperature and the influence of radiation sink parameter be negligible for θS < 0.2. ### Fig. 14

Comparison of temperature distributions for various, θS and for α = 2.0.

Figure 15 discusses the effect of sink temperature θS on temperature distribution and the variation of fin temperature in a radial fin for different fractional order α = 1.75, 1.5, and 1.25 keeping ψ, β, θa and G fixed which shows that as the sink temperature θS be increases, the fin temperature also increases implies less heat be loss from the fin tip and it is minimum for α = 1.25 and θS = 0.1. ### Fig. 15

Variation of temperature distributions for different θS and for different fractional order α (a) 1.75, (b) 1.5 and (c) 1.25.

Figure 16 allows us to evaluate the effect of the heat generation on temperature distribution in Eq. (16), that defined varies from 0.1 to 0.6. Which shows that the temperature inside the fin be increases with G, i.e. with the increase in the effect of heat generation. Figure 17 discusses the effect of heat generation G on temperature distribution in a radial fin and the variation of fin temperature for different fractional order α = 1.75, 1.5, and 1.25 keeping ψ, β, θa and θS fixed which shows that as the heat generation G be increases, the fin temperature also increases within the fin and it is minimum for α = 1.25 and G = 0.1. ### Fig. 16

Comparison of temperature distributions for different G and for α = 2.0. ### Fig. 17

Variation of temperature distributions for different G and for for different fractional order α (a) 1.75, (b) 1.5 and (c) 1.25.

## 7 Conclusion

In this study, the temperature distribution and the fin efficiency for a convective radial fin is alalysed with and without considering the internal heat generation by using ADSTM. The simulation results shows that the thermal conductivity, the thermo-geometric fin parameter and fractional order values has greater role in determining the temperature distribution and the fin efficiency of the convective radial fins. Finally, the efficiency of this proposed method has been evaluated by comparing the obtained numerical results with the classical order results obtained by Range-Kutta method of fourth order and VIM.

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