Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2008 (2008), Article ID 347568, 9 pages
http://dx.doi.org/10.1155/2008/347568
Research Article

Transient Heat Diffusion with Temperature-Dependent Conductivity and Time-Dependent Heat Transfer Coefficient

School of Computational and Applied Mathematics, University of the Witwatersrand, Private bag 3, Wits 2050, South Africa

Received 8 April 2008; Revised 5 June 2008; Accepted 18 July 2008

Academic Editor: Yuri V. Mikhlin

Copyright © 2008 Raseelo J. Moitsheki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Lie point symmetry analysis is performed for an unsteady nonlinear heat diffusion problem modeling thermal energy storage in a medium with a temperature-dependent power law thermal conductivity and subjected to a convective heat transfer to the surrounding environment at the boundary through a variable heat transfer coefficient. Large symmetry groups are admitted even for special choices of the constants appearing in the governing equation. We construct one-dimensional optimal systems for the admitted Lie algebras. Following symmetry reductions, we construct invariant solutions.

1. Introduction

For many years, considerable attention has been paid to the collection, storage, and use of thermal energy to meet various energy demands. The use of solar energy to meet the thermal demands of industries, electronics devices, and residential establishments, and so forth, is fast growing in many countries of the world [1]. Solar energy is provided by the light energy that comes from the sun. An important component of thermal systems designed for such purposes is a thermal energy storage unit. Solar collectors transform short wavelengths into long wavelengths and trap this energy in the form of heat which is transferred and transported into a heat storage vault. The medium in which the energy is stored may be fluid or solid [2]. For instance, in middle- and low-temperature solar energy systems, water and stones are the best and cheapest storing energy medium [3]. The heat energy collected by solar energy collectors increases the temperature of the medium, so the heat energy is stored in the medium. When needed, the heat energy is desorbed for use. The effectiveness of a liquid thermal storage system is determined by how the temperature of the system decays as a result of heat losses to the environment [2, 4]. The thermal energy storage problem in a medium with temperature-dependent thermal conductivity constitutes an unsteady nonlinear heat diffusion problem and the solutions in space and time may reveal the appearance of thermal decay in the system. In order to predict the occurrence of such phenomena, it is necessary to analyze a simplified mathematical model from which insight might be gleaned into an inherently complex physical mechanism.

Meanwhile, the solution of unsteady nonlinear heat diffusion equations in rectangular, cylindrical, and spherical coordinates remains a very important problem of practical relevance in the engineering sciences [5]. Recently, the ideas of hybrid analytical-numerical schemes for solving nonlinear differential equations have experienced a revival (see, e.g., [6]). One such trend is related to the combination of group theoretic approach and Adomian decomposition method [7]. This hybrid analytical-numerical approach is also extremely useful in the validation of purely numerical schemes.

In the present work, we study an unsteady nonlinear heat diffusion problem modeling thermal energy storage in a medium with power law temperature-dependent thermal conductivity and subjected to a convective heat transfer to the surrounding environment at the boundary. The mathematical formulation of the problem is established in Section two. In Section three, we introduce and apply some rudiments of Lie group techniques. In Section four, we construct the one-dimensional optimal systems, and perform reductions by one variable and construct invariant solutions in Section five. Some discussions and conclusions are presented in Section six.

2. Governing Equations

Consider an unsteady thermal storage problem in a body whose surface is subjected to heat transfer by convection to an external environment having a heat transfer coefficient that varies with respect to time. The energy equation in a rectangular, cylindrical, or spherical coordinate system can be used to find the temperature distribution through a region defined in an interval The unsteady heat conduction problem can be described by the following governing equation [1, 6]:with the initial conditionand the boundary conditions where is the temperature, is the power law temperature-dependent thermal conductivity, is a constant, is time, is the initial temperature of the body, is the temperature of the environment, is the density, is the specific heat at a constant pressure, is the heat loss parameter, and is the time-dependent heat transfer coefficient, with and being constants. The geometry of the body is specified by representing rectangular, cylindrical, and spherical coordinates, respectively. Equations (2.1)–(2.3) are made dimensionless by introducing the following quantities:

Neglecting the bar symbol for clarity, the dimensionless boundary value problem (BVP) becomessubject to where is the Biot number and is the heat loss parameter.

3. Classical Lie Point Symmetry Analysis

In brief, a symmetry of a differential equation is an invertible transformation of the dependent and independent variables that does not change the original differential equation. Symmetries depend continuously on a parameter and form a group; the one-parameter group of transformations. This group can be determined algorithmically. The theory and applications of Lie groups may be obtained in excellent texts such as those of [813]. In essence, determining symmetries for the governing equation (2.5) implies seeking transformations of the form generated by the vector fieldwhich leaves the governing equation invariant. Note that we seek symmetries that leave a single equation (2.5) invariant rather than the entire boundary value problem, and apply the boundary condition to the obtained invariant solutions. It is well known that the dimension of symmetry algebra admitted by the governing equation may be reduced if one seeks invariance of the entire BVP (see, e.g., [9]). The action of is extended to all the derivatives appearing in the governing equation through second prolongationwhereand and are the operators of total differentiation with respect to and , respectively. The operator is a point symmetry of the governing equation (2.5), ifSince the coefficients of do not involve derivatives, we can separate (3.5) with respect to the derivatives of and solve the resulting overdetermined system of linear homogeneous partial differential equations known as the determining equations. Further calculations are omitted at this stage as they were facilitated by a freely available package Dimsym [14], a subprogram of Reduce [15].

The admitted Lie algebra is three-dimensional and spanned by the base vectors

Extra symmetries may be obtained for the cases and . One may note that except for case (a), all these cases are realistic. renders the governing equation (2.5) linear and we herein omit this case. Extra symmetries, for which are listed in Table 1. Physically the parameters and are not related (since represents the geometry and is exponent of the thermal conductivity). However, it is interesting from symmetry analysis point of view to note that the given relationship between and leads to extra symmetries being admitted.

tab1
Table 1: Extra-admitted symmetries.

4. One-dimensional Optimal Systems of Subalgebras

In this section, we determine nonequivalent subalgebras of the symmetry algebra admitted by (2.5) that is, we construct the one-dimensional optimal system of the symmetry algebra given in (3.6). Reduction of independent variables by one is possible using any linear combination of the admitted base vectors. In order to ensure that a minimal complete set of reductions is obtained from the admitted Lie algebra, an optimal system is constructed (see, e.g., [11, 12]). An optimal system of a Lie algebra is a set of dimensional subalgebras such that every dimensional subalgebra is equivalent to a unique element of the set under some element of the adjoint representation [11]:where is the commutator of and To compute the one-dimensional optimal system of the algebra in (3.6), first a commutator table is constructed and given by Table 2.

tab2
Table 2: Commutator table.

The adjoint representation is constructed using formula (4.1). We wish to simplify as much as possible the coefficients , and by carefully applying the adjoint maps toStarting with a nonzero vector (4.2) with and rescaling such that it follows from Table 3 that acting on by one obtains No further simplification is possible. For and assuming (say, by rescaling); acting on the remaining vector (4.2) by yields However, depending on the sign , the coefficient of can be assigned to either +1, −1, or 0. Finally, for we obtain Thus the one-dimensional optimal system is given by where is an arbitrary constant.

tab3
Table 3: Adjoint representation for the base vectors given in (3.6).

5. Symmetry Reductions and Invariant Solutions

In this subsection, we reduce the variables of the governing BVP by one. We provide the invariant solution constructed using in (3.6) which satisfies the prescribed boundary condition. Reductions by members of the one-dimensional optimal system are listed in Table 4. In order to find invariants of , we have to solve the systemThe system (5.1) yields the invariants which give rise to the functional form

tab4
Table 4: Reductions by elements of the optimal systems.

The time-dependent heat transfer coefficient may be represented by . This choice of renders the boundary condition invariant under and it is also realistic (note that the form of is obtained by substituting (5.2) into (2.6)) (see also [6, 16]). Substituting this expression for and the functional form (5.2) into the governing equation (2.5), one obtainsand the boundary conditions (2.6) transform to

Note that the trivial solution to (5.3) is given by a constant. Four cases arise for the nontrivial solution of (5.3) subject to different choices of m and n (see also [17, page 365]). Case (a)Case (b)Case (c)Case (d)

We consider cases (a) and (d) only as examples. For case (a) and in terms of original variables we obtain

The invariant solution (5.9) satisfies the prescribed boundary condition (2.6) at only if (spherical geometry) and for any constant . One may rewrite (5.9) as

Without loss of generality we let in case (d) and in terms of original variables, the invariant solution satisfying the boundary conditions (2.6) for (rectangular geometry) is given bywhere values of must be chosen such that the singularity at is avoided. Furthermore, we obtain real solutions for and Invariant solution (5.11) is depicted in Figures 1 and 2.

347568.fig.001
Figure 1: Graphical representation of the invariant solution (5.11). Parameters used , and . The temperature profile is given for varying Biot number.
347568.fig.002
Figure 2: Graphical representation of the invariant solution (5.11). Parameters used and . Temperature is plotted against spatial variable.

Symmetry analysis may lead to extra solutions, if we use the linear combinations of the admitted symmetries or elements of the optimal systems as listed in Table 4. For example, the -invariant is given by

6. Some Discussions and Conclusions

We have determined some examples of group invariant solutions which satisfy the realistic boundary conditions (it is a well-known fact that more often symmetries do not lead to solutions which satisfy the boundary conditions). In this manuscript, Lie group analysis resulted in some exotic admitted point symmetries. Furthermore, reduction by one variable of the governing equation has been performed using members of the optimal system.

The invariant solution (5.11) shows thermal decay due to heat losses by convection to the surrounding environment. Figure 1, depicts an increase in the Biot number due to the resistance of the medium surface heat losses which leads to an increase in the medium temperature, and hence enhancing its energy storage capabilities. The medium temperature decreases with time. In Figure 2, temperature is much lower at the device surface than at and this is due to heat loss to the surrounding.

We have given some exact (invariant) solutions to nonlinear heat diffusion equations with temperature-dependent conductivity and time-dependent heat transfer coefficient.

Acknowledgments

Raseelo J. Moitsheki wishes to thank the National Research Foundation of South Africa under Thuthuka program, for the generous financial support. The author is also grateful to the anonymous reviewers for their useful comments.

References

  1. J. A. Duffie and W. A. Becham, Solar Engineering of Thermal Processes, John Wiley & Sons, New York, NY, USA, 1980.
  2. Y. Jaluria and S. K. Gupta, “Decay of thermal stratification in a water body for solar energy storage,” Solar Energy, vol. 28, no. 2, pp. 137–143, 1982. View at Publisher · View at Google Scholar
  3. M. N. A. Hawlader and B. J. Brinkworth, “An analysis of the non-convective solar pond,” Solar Energy, vol. 27, no. 3, pp. 195–204, 1981. View at Publisher · View at Google Scholar
  4. T. W. Davies, “Transient conduction in a plate with counteracting convection and thermal radiation at the boundaries,” Applied Mathematical Modelling, vol. 9, no. 5, pp. 337–340, 1985. View at Publisher · View at Google Scholar
  5. O. D. Makinde, “Solving microwave heating model in a slab using Hermite-Padé approximation technique,” Applied Thermal Engineering, vol. 27, no. 2-3, pp. 599–603, 2007. View at Publisher · View at Google Scholar
  6. M. B. Abd-el-Malek and M. M. Helal, “Group method solution for solving nonlinear heat diffusion problems,” Applied Mathematical Modelling, vol. 30, no. 9, pp. 930–940, 2006. View at Publisher · View at Google Scholar
  7. O. D. Makinde and R. J. Moitsheki, “Group method solution of nonlinear heat diffusion for thermal energy storage,” submitted.
  8. G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations, vol. 154 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2002. View at Zentralblatt MATH · View at MathSciNet
  9. G. W. Bluman and S. Kumei, Symmetries and Differential Equations, vol. 81 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989. View at Zentralblatt MATH · View at MathSciNet
  10. N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, vol. 4 of Wiley Series in Mathematical Methods in Practice, John Wiley & Sons, Chichester, UK, 1999. View at Zentralblatt MATH · View at MathSciNet
  11. P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1986. View at Zentralblatt MATH · View at MathSciNet
  12. L. V. Ovisianikov, Group Analysis of Differential Equations, Academic Press, New York, NY, USA, 1982. View at Zentralblatt MATH · View at MathSciNet
  13. H. Stephani, Differential Equations: Their Solution Using Symmetries, Cambridge University Press, Cambridge, UK, 1989. View at Zentralblatt MATH · View at MathSciNet
  14. J. Sherring, “Dimsym: symmetry determination and linear differential equation package,” Latrobe University, Mathematics Department, Melbourne, Australia, 1993.
  15. A. C. Hearn, “Reduce user's manual version 3.4,” Rand Publication CP78, The Rand Corporation, Santa Monica, Calif, USA, 1985. View at Google Scholar
  16. M. M. Helal, M. B. Abd-el-Malek, and S. M. A. El-Mansi, “Group method solution for internal flow through conducting thin duct subjected to a variable heat transfer coefficient,” International Journal of Applied Mathematics and Mechanics, vol. 1, no. 3, pp. 52–66, 2005. View at Google Scholar
  17. A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press, Boca Raton, Fla, USA, 1995. View at Zentralblatt MATH · View at MathSciNet