Adomian decomposition sumudu transform method for solving a solid and porous fin with temperature dependent internal heat generation
- Trushit Patel^{1} and
- Ramakanta Meher^{1}Email author
Received: 21 January 2016
Accepted: 5 April 2016
Published: 19 April 2016
Abstract
In this paper, Adomian decomposition sumudu transform method is introduced and used to solve the temperature distribution in a solid and porous fin with the temperature dependent internal heat generation for a fractional order energy balance equation. In this study, we assume heat generation as a variable of fin temperature for solid and porous fin and the heat transfer through porous media is simulated by using Darcy’s model. The results are presented for the temperature distribution for the range of values of parameters appeared in the mathematical formulation and also compared with numerical solutions in order to verify the accuracy of the proposed method. It is found that the proposed method is in good agreement with direct numerical solution.
Keywords
Adomian decomposition sumudu transform method Fractional order differential equation Caputo fractional derivative Porous fin Temperature dependent thermal conductivity Internal heat generation Thermal analysisBackground
Fins are commonly used to facilitate the dissipation of heat from a heated wall to the surrounding environment. Examples of fin are the radiator in vehicles and heat exchangers in power plants. In electrical devices like motors and transformers, the generated heat can be efficiently transferred. In the study of heat transfer, fin is a surface, made by metallic material which is used to increase the rate of heat transfer to the environment. The rate of heat transfer depends on the surface area of the fin. Fins are extensively used to improve the rate of heat dissipation from a hot surface, especially in thermal engineering applications (Bergman et al.; Nield and Bejan).
In many thermal engineering applications, convective flow through porous media is mandatory for the investigation. Several numerical and analytical revisions has been conducted so far to afford a profound understanding of the transport system of the heat transfer inside the porous medium. Generally, high thermal conductivity porous substrates are employed to improve the rate of forced convection heat transfer in many engineering applications such as reactor heat exchangers, solar collectors and in cooling process (Alkam and Al-Nimr 1999). However, heat transfer in porous fins has attracted a lot of attention of researchers with a wide range of it’s applications, especially in recent years. Kiwan and Al-Nimr (2001) was the first person who introduced the concept of fins made of porous materials by introducing Darcy’s model (Kiwan 2007; Kiwan and Zeitoun 2008).
Now a days, Heat exchanger industries are looking for more compact and cost–effective heat exchanger manufacturing techniques which leads to use porous fins in enhance heat transfer (Kiwan 2007). The heat-transfer enhancement between two parallel-plate channels was investigated by adding porous fin through the channel (Hamdan and Moh’d 2010) and by adding porous insert to one side of the duct walls (Hamdan et al. 2000). Alkam et al. (2002) investigated the thermal analysis of natural convection porous fins. They studied all the geometric flow parameters that influence the temperature distribution in to a single parameter specified S _{ h }. They considered three cases: the infinite fin, a finite fin with an insulated tip and a finite fin with uninsulated tip. Similarly Gorla and Bakier (2011) discussed the thermal analysis of natural convection and radiation in the porous fin and showed that the radiation transfers more heat than a similar model without radiation. Hatami et al. (2013) studied the heat transfer through porous fin with different porous material and compared their results with the Differential Transform Method, Collocation Method and Least Square Method. They Hatami and Ganji (2013) also studied the thermal performance of circular convective–radiative porous fins with different section, shapes and materials. Ghasemi et al. (2014) used the Differential Transform Method for solving the nonlinear temperature distribution in solid and porous fin with temperature dependent internal heat generation. Patel and Meher (2015a, b) studied the fractional solution of longitudinal porous fin for the case of temperature distribution, efficiency and effectiveness and also analysed the variation of temperature distribution for a straight rectangular fin with power-law temperature dependent surface heat flux by using Adomian decomposition sumudu transform method.
It is revealed that, the concept of fractional derivative is more suitable for modeling real world problem than the local derivative. Many researchers have devoted their attention in developing new definition of fractional derivative (Atangana 2016). Baskonus and Bulut (2015) applied the fractional Adams–Bashforth–Moulton Method for obtaining the numerical solutions of some linear and nonlinear fractional ordinary differential equations. Baskonus and Bulut (2015) studied it to obtain some new analytical solutions to the (1 + 1)-dimensional nonlinear Dispersive Modified Benjamin Bona Mahony equation by using modified exp-function method. Roshid et al. (2014) studied solitary wave solutions for vakhnenko-parkes equation via exp-function and Exp \((-\phi (\xi ))\)—expansion method. Also they Roshid et al. (2014) studied traveling wave solutions of nonlinear partial differential equation via new extended \((G^{\prime }/G)\)—expansion method.
In the present study, we fractionalize the energy balance equation in order to understand the anomalous behavior of this system and to find the temperature distribution in solid and porous fin by using Adomian decomposition sumudu transform method.
Preliminaries
Definitions of Caputo fractional derivative
In this part of the paper it would be useful to introduce some definitions and properties of the fractional calculus theory. There are several definitions of fractional derivatives of order \(\alpha >0\) (Miller and Ross; Srivastava et al. 2014). The two most commonly used definitions are Riemann-Liouville and Caputo.
Definition
Definition
Basics of sumudu transform method
The sumudu transform is a new integral transform (Kadem and Baleanu 2012; Atangana and Baleanu 2013; Jarad et al. 2012) which is a little known and not widely used whose defined for the functions of exponential order.
Definition
Definition
Definition
Theorem
- 1.
\(\left( G(1/s)/s\right)\) is a meromorphic function, with singularities having \({\mathrm{Re}} [s] \le \gamma ;\)
- 2.
There exist a circular region \(\gamma\) with radius R and positive constants M and K with \(\left| {G(1/s)/s} \right| < M{R^{ - K}}\), then the function f(t) is given by
Formulation of Adomian decomposition sumudu transform method (ADSTM)
Problem description
Here we considered two cases, namely (1) solid fin and (2) porous fin to study the fin temperature distribution in longitudinal fin with rectangular profile.
Case 1: solid fin with temperature dependent internal heat generation and constant thermal conductivity
Case 2: porous fin with the temperature dependent internal heat generation
Stability analysis via fixed point theorem
Let \(\left( {X,\left\| \cdot \right\| } \right)\) be a Banach space and H a self-map of X. Let \(\theta _{n+1} = f(H,\theta _{n})\) be particular recursive procedure. Suppose that F(H) the fixed point set of H has at lease one element and that \(\theta _{n}\) converges to a point \(p \in F(H)\). Let \({\theta _{n}\subseteq X}\) and define \({e_n} = \left\| {{\theta _{n + 1}} - f(H,{\theta _n})} \right\|\). If \(\mathop {\lim }\limits _{n \rightarrow \infty } {\theta ^n} = p\), then the iteration \(\theta _{n+1}=f(H,\theta _{n})\) is said to be H-stable. Without any loss of generality, we must assume that, our sequence \({\theta _{n}}\) has an upper boundary; otherwise we cannot expect the possibility of convergence. If all these conditions are satisfied for \(\theta _{n+1}=H \theta _{n}\) which is known as Picard’s iteration, then the iteration will be H-Stable. Now we state the following theorem (Atangana 2016).
Theorem 1
Theorem
Proof
Results and discussion
Range of values for physical and thermal parameters
Parameters | M | G | \(I_{G}\) | \(\xi\) |
---|---|---|---|---|
Values | 0.5, 1 and 5 | 0.1–0.9 | 0.1–0.9 | 0.1–0.9 |
Solid fin with temperature dependent internal heat generation and constant thermal conductivity
Temperature distribution for this case (temperature dependent heat generation and constant thermal conductivity) is shown in Figs. 2 and 3 where M = 1 that is common in fin design.
Figure 3 shows the temperature distribution for this state and \(I_{G} =G=0.2, I_{G} =G=0.4\) and \(I_{G} =0.4\), G = 0.6 and for the different fractional order value of \(\alpha\) = 1.75, 1.5 and 1.25. Further, the nature of the graphs depicts that, considered value of \(\alpha\) represents the point of convergence under the given range of interval between 1 and 2.
Porous fin with temperature dependent internal heat generation
Conclusion
In this study, the heat transfer in rectangular solid and porous fin with the temperature dependent internal heat generation is analyzed by using ADSTM and used the concept of T-stable mapping and the fixed point theorem to prove the stability of ADSTM. Here, it is shown that, ADSTM provide a simple, accurate and appropriate technique for simulating the heat transfer in solid and porous fin in a fractional order energy balance equation. The results shows that the temperature distribution strongly depends on different parameter in solid fin as well as on Darcy’s number in porous fin and also on the fractional parameter.
Declarations
Authors' contributions
This work was carried out by the two authors, in collaboration. Both authors read and approved the final manuscript.
Acknowledgements
The authors are thankful to Applied Mathematics and Humanities Department of S. V. National Institute of Technology, Surat for the scholarship, encouragement and facilities.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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