A quasi-analytical solution of homogeneous extended surfaces heat diffusion equation
- Ernest Léontin Lemoubou^{1} and
- Hervé Thierry Tagne Kamdem^{1}Email author
https://doi.org/10.1186/s40712-017-0084-4
© The Author(s). 2017
Received: 1 July 2017
Accepted: 13 July 2017
Published: 16 August 2017
Abstract
Background
In this study, a quasi-analytical solution for longitudinal fin and pin heat conduction problems is investigated.
Methods
The differential transform method, which is based on the Taylor series expansion, is adapted for the development of the solution. The proposed differential transform solution uses a set of mathematical operations to transform the heat conduction equation together with the fin profile in order to yield a closeform series of homogeneous extended surface heat diffusion equation.
Results and conclusions
The application of the proposed differential transform method solution to longitudinal fins of rectangular and triangular profiles and pins of cylindrical and conical profiles heat conduction problems showed an excellent agreement on both fin temperature and efficiencies when compared to exact results. Therefore, the proposed differential transform method can be useful for optimal design of practical extended surfaces with suitable profile for temperature response.
Keywords
Analytical solution Differential transform method Heat conduction Longitudinal fin PinsBackground
Extended surfaces in the forms of longitudinal or radial fins or spines with various cross sections are widely used in industrial applications such as air-cooled craft engines, cooling of computer processors and electrical components, air-conditioning, refrigeration, heat exchangers, and solar collectors. These devices, which provide a considerable increase in the surface area for heat transfer between a heated source and a cooler ambient fluid, are most effective to enhance heat transfer between a surface and an adjacent fluid (Kraus et al. 2001). In designing extended surfaces, the first step consists of assuming that the heat transfer is governed by a one-dimensional homogeneous steady conduction along an extended surface and a uniform convection at the surface area (Arauzo et al. 2005; Kang 2009; Brestovic et al. 2015). Although an exact solution for this problem can be found in the literature, it may be either difficult or not possible to obtain when designing practical extended surfaces with profile matching suitable geometry for temperature response. Moreover, the exact solution of some problems requires difficult manipulation of special functions such as Bessel functions.
Several efficient analytical or quasi-analytical methods have been developed over the past two decades to solve linear and nonlinear problems in science and engineering including the power series, the Adomian decomposition, the homotopy, and the differential transformation methods (Diez et al. 2009; Aziz and Bouaziz 2011; Torabi and Zhang 2013; Hayat et al. 2016). In general, exact solutions are mostly based on the tedious computation of special functions (Kraus et al. 2001; Diez et al. 2009). The power series method has been used by Arauzo et al. (2005) to approximate the solution of the one-dimensional steady heat conduction equation that governs the temperature variation in annular fins of hyperbolic profile. This method is a standard technique for solving linear ordinary differential equations with variable coefficients. The Adomian decomposition method has been used by Arslanturk (2005) and Bhowmik et al. (2013) to compute a closed-form solution for a straight convecting rectangular and hyperbolic profile annular fin with temperature-dependent thermal conductivity. This method represents the solution by an infinite series of the so-called Adomian polynomials and uses an iterative method such as the Newton–Raphson for the evaluation of the undetermined temperature at the fin tip. The homotopy method is a quasi-analytical solution for solving nonlinear boundary value problem (Domairry and Fazeli 2009; Hayat et al. 2017a). This method does not require the calculation of Adomian polynomials as required for the Adomian decomposition method, but it requires an initial approximation (Roy and Mallick 2016; Hayat et al. 2017b; Waqas et al. 2016). Inc (2008) used the homotopy analysis method to evaluate the efficiency of straight fins with temperature-dependent thermal conductivity and determined the temperature distribution within the fin. The results show that the homotopy analysis method presents faster convergence and higher accuracy than the Adomian decomposition method and the homotopy perturbation method for nonlinear problems in science and engineering. The differential transform method (DTM) is based on the Taylor series expansion and constructs an analytical solution in the form of a polynomial. In general, the solution necessitates an iterative method such as the Newton–Raphson methods for the determination of the initial temperature transform function value. Joneidi et al. (2009) applied the differential transform method to predict temperature and efficiency of convective straight fins with temperature-dependent thermal conductivity. Their obtained results compared to exact and numerical results reveal that the differential transform method is an effective and accurate method for analyzing extended surfaces’ nonlinear heat transfer problems. Kundu and Lee (2012) determined the performance of different fin geometries by analyzing heat transfer in rectangular, triangular, convex, and exponential geometric longitudinal fins using the differential transform method. These authors demonstrated that the differential transform is precise and cost efficient for analyzing nonlinear heat and mass transfer effects in extended surfaces.
The literature presents several quasi-analytical methods for analyzing heat transfer problems in extended surfaces. The different quasi-analytical methods, which used an iterative method such as the Newton–Raphson for the evaluation of an undetermined parameter, are applied mostly for nonlinear problems. On the other hand, the solution of linear problems is generally obtained as a particular case of nonlinear problems. The contribution of the present work is to present close-form series solution of the homogeneous extended surface heat diffusion equation using the differential transform method. This can be a useful strategy in developing an analytical solution when designing practical extended surfaces with suitable geometry for temperature response. The proposed differential transform solution uses a set of mathematical operations to transform the heat conduction equation together with the fin profile in order to yield a close-form series of homogeneous extended surface heat diffusion equation which avoid using an iterative method. The homogeneous extended surfaces in the forms of longitudinal fins of rectangular and triangular profiles and pins of cylindrical and conical profiles are attached to a primary surface at constant temperature heat losses by convection to the surrounding medium and where the heat loss from the tip of the extended surfaces is assumed to be negligible. The temperature distribution and efficiency within extended surfaces are analyzed and compared against exact results.
Methods
Problem description
Differential transform operations
One-dimensional DTM fundamental operations
Functions | Transform functions |
---|---|
f(x)= | F(k) |
af _{1}(x) + bf _{2}(x) | F(k) = aF _{1}(k) + bF _{2}(k) |
\( \frac{df}{dx} \) | F(k) = (k + 1)F(k + 1) |
\( \frac{d^2f}{dx^2} \) | F(k) = (k + 1)(k + 2)F(k + 2) |
x ^{ n } | \( F(k)=\delta \left(k-n\right)=\left\{\begin{array}{c}1\kern0.75em si\kern0.75em k=n\\ {}0\kern0.75em si\kern0.5em k\ne n\end{array}\right. \) |
exp(λx) | \( F(k)=\frac{\lambda^k}{k!} \) |
f _{1}(x)f _{2}(x) | \( F(k)=\sum_{l=0}^k{F}_1(l){F}_2\left(k-l\right) \) |
f _{1}(x)f _{2}(x)f _{3}(x) | \( F(k)=\sum_{h=0}^k\sum_{l=0}^h{F}_1(l){F}_2\left(h-l\right){F}_3\left(k-h\right) \) |
The transform condition (13) is applied only for rectangular and cylindrical fins whereas (14) is adapted for all the geometries.
Results and discussion
For stationary heat conduction through fins and pins with a dry surface and constant thermal conductivity cooled by air of constant heat transfer coefficient, exact analytical solution using a standard method of ordinary differential equations can be found in the literature (Kraus, et al., 2001; Kundu and Lee 2012). In the present work, the DTM is considered as alternative for obtaining analytical solution and the predictions are compared to the results from the standard method of ordinary differential equations.
Longitudinal fin with rectangular profile
Comparison of the DTM and exact results for different approximations order
mb = 0.6 | mb = 1.4 | |||||||
---|---|---|---|---|---|---|---|---|
Exact | DTM | Exact | DTM | |||||
ζ | n = 1 | 2 | n = 1 | 2 | 3 | 4 | ||
0.0 | 0.8436 | 0.0039 | 0 | 0.4649 | 0.0401 | 0.0024 | 0.0001 | 0 |
0.1 | 0.8451 | 0.0039 | 0 | 0.4695 | 0.0405 | 0.0024 | 0.0001 | 0 |
0.2 | 0.8496 | 0.0039 | 0 | 0.4833 | 0.0416 | 0.0024 | 0.0001 | 0 |
0.3 | 0.8573 | 0.0039 | 0 | 0.5065 | 0.0431 | 0.0026 | 0.0001 | 0 |
0.4 | 0.8680 | 0.0039 | 0 | 0.5397 | 0.0445 | 0.0027 | 0.0001 | 0 |
0.5 | 0.8818 | 0.0038 | 0 | 0.5836 | 0.0452 | 0.0029 | 0.0001 | 0 |
0.6 | 0.8988 | 0.0036 | 0 | 0.6388 | 0.0444 | 0.0030 | 0.0001 | 0 |
0.7 | 0.9191 | 0.0032 | 0 | 0.7066 | 0.0409 | 0.0030 | 0.0001 | 0 |
0.8 | 0.9426 | 0.0025 | 0 | 0.7883 | 0.0335 | 0.0027 | 0.0001 | 0 |
0.9 | 0.9696 | 0.0015 | 0 | 0.8855 | 0.0205 | 0.0018 | 0.0001 | 0 |
1.0 | 1.0000 | 0.0000 | 0 | 1.0000 | 0.0000 | 0.0000 | 0.0000 | 0 |
Longitudinal fin of triangular profile
Pin of cylindrical profile
Pin of conical profile
Conclusions
A differential transform method to analyze stationary heat conduction through homogeneous extended surfaces with negligible heat loss from the tip has been presented. The major conclusion of this work is that for stationary heat conduction through extended surfaces, the DTM solution can be obtained in a closed series solution which does not necessitate an iterative method such as the Newton–Raphson methods for the determination of the initial value of temperature transform function. The proposed method is shown to converge with a few Taylor series for both low and high values of the thermal length characteristic parameter. Application of the present DTM solution to longitudinal fins of rectangular and triangular profiles and pins of cylindrical and conical profiles show an excellent agreement with exact results. For all cases studied, the magnitude of temperature decreases with increasing thermal length characteristic parameter. This indicates that the loss of heat from extended surfaces is more significant for low convection coefficient of the cooling fluid than that for extended surfaces with high thickness and conduction coefficient.
Nomenclature
A fin section area, m^{2}
b height of the fin, m
h heat convectivity coefficient W ∙ m ^{−2} ∙ K ^{−1}
k _{ c } thermal conductivity heat coefficient, W ∙ m ^{−1} ∙ K ^{−1}
q heat flux, W/m^{2}
T temperature, K
x coordinate, m
p profile function
P profile transform function
W thickness of the fin, m
Greek symbols
β dimensionless temperature
θ dimensionless temperature
Θ dimensionless temperature transform function
η fin efficiency
ζ dimensionless coordinate
Subscripts
b fin base
i ideal parameter
∞ ambient environment
Declarations
Authors’ contributions
ELL implemented the quasi-analytic differential transform method and HTTK defined the research problem. All authors contributed to the problem formulation, drafted the manuscript, and read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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