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

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 onedimensional optimal systems for the admitted Lie algebras. Following symmetry reductions, we construct invariant solutions.


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 2 Mathematical Problems in Engineering 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.

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 0 < r < a.The unsteady heat conduction problem can be described by the following governing equation 1, 6 : with the initial condition θ r, t θ i , at t 0, 2.2 and the boundary conditions where is the power law temperaturedependent thermal conductivity, n is a constant, t is time, θ i is the initial temperature of the body, θ ∞ is the temperature of the environment, ρ is the density, c p is the specific heat at a constant pressure, Q is the heat loss parameter, and h t h 0 f t is the time-dependent heat transfer coefficient, with h 0 and K 0 being constants.The geometry of the body is specified by m 0, 1, 2 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 becomes subject to ∂θ ∂r 0, at r 0, where Bi is the Biot number and s is the heat loss parameter.

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 8-13 .In essence, determining symmetries for the governing equation 2.5 implies seeking transformations of the form 3.1 generated by the vector field Constants Symmetries where and D r and D t are the operators of total differentiation with respect to r and t, respectively.The operator Γ is a point symmetry of the governing equation 2.5 , if Γ 2 Eqn 2.5 | Eqn 2.5 0.

3.5
Since the coefficients of Γ do not involve derivatives, we can separate 3.5 with respect to the derivatives of θ 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 3.6 Extra symmetries may be obtained for the cases a m 3n 4 / n 2 ; b m 0, n 0; c n 0, m 2; d m 1, n −1; and e m 0, n −4/3.One may note that except for case a , all these cases are realistic.n 0 renders the governing equation 2.5 linear and we herein omit this case.Extra symmetries, for which n / 0, are listed in Table 1.Physically the parameters m and n are not related since m represents the geometry and n is exponent of the thermal conductivity .However, it is interesting from symmetry analysis point of view to note that the given relationship between m and n leads to extra symmetries being admitted.

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 Table 3: Adjoint representation for the base vectors given in 3.6 .
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 l dimensional subalgebras such that every l dimensional subalgebra is equivalent to a unique element of the set under some element of the adjoint representation 11 : where To compute the one-dimensional optimal system of the algebra in 3.6 , first a commutator table is constructed and given by Table 2.
The adjoint representation is constructed using formula 4.1 .We wish to simplify as much as possible the coefficients a 1 , a 2 , and a 3 by carefully applying the adjoint maps to Starting with a nonzero vector 4.2 with a 1 / 0 and rescaling Γ such that a 1 1, it follows from Table 3 that acting on Γ by Ad exp −a 2 /ns Γ 2 , one obtains Γ 1 αΓ 3 .No further simplification is possible.For a 0 and assuming a 3 / 0 say, a 3 1 by rescaling ; acting on the remaining vector 4.2 by Ad exp c 1 Γ 2 yields a 2 e c 1 ns Γ 2 Γ 3 .However, depending on the sign a 2 , the coefficient of Γ 2 can be assigned to either 1, −1, or 0. Finally, for a 3 0, we obtain Γ 2 .Thus the one-dimensional optimal system is given by where α is an arbitrary constant.

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 Γ 2 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 Γ 2 , we have to solve the system Symmetry Reductions The system 5.1 yields the invariants J 1 ln θ st and J 2 r which give rise to the functional form θ e −st G r .

5.2
The time-dependent heat transfer coefficient may be represented by f t e −nst .This choice of f t renders the boundary condition invariant under Γ 2 and it is also realistic note that the form of f t is obtained by substituting 5.2 into 2.6 see also 6,16 .Substituting this expression for f t and the functional form 5.2 into the governing equation 2.5 , one obtains and the boundary conditions 2.6 transform to dG dr 0, r 0,

5.4
Note that the trivial solution to 5.3 is given by a constant.Four cases arise for the nontrivial solution of

5.8
We consider cases a and d only as examples.For case a and in terms of original variables we obtain 5.9 The invariant solution 5.9 satisfies the prescribed boundary condition 2.6 at r 0 only if m 2 spherical geometry and for any constant c 1 < 0. One may rewrite 5.9 as

5.10
Without loss of generality we let c 2 0 in case d and in terms of original variables, the invariant solution satisfying the boundary conditions 2.6 for m 0 rectangular geometry is given by where values of n must be chosen such that the singularity at r 0 is avoided.Furthermore, we obtain real solutions for −1 < n < 0 and 0 < n < 1. Invariant solution 5.11 is depicted in Figures 1 and 2.
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 Γ 3 -invariant is given by θ r 2/n mn 2 n c 1 nse nst ns

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 r 0, 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.

Figure 2 :
Figure 2: Graphical representation of the invariant solution 5.11 .Parameters used n −0.33, s 1, and t 1. Temperature is plotted against spatial variable.

Table 4 :
Reductions by elements of the optimal systems.