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

Authors: Raseelo J. Moitsheki
Journal: Mathematical Problems in Engineering
 (2008)

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## 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.


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## Body

## 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 interval0<r<a. The unsteady heat conduction problem can be described
by the following governing equation [1, 6]:(2.1)ρcp∂θ∂t=1rm∂∂r(K(θ)rm∂θ∂r)−Q(θ−θ∞)with the initial
condition(2.2)θ(r,t)=θi,att=0,and the boundary
conditions(2.3)∂θ∂r=0,atr=0,K(θ)∂θ∂r=−h(t)(θ−θ∞),atr=a, where θ is the temperature, K(θ)=K0((θ−θ∞)/(θi−θ∞))n is the power law temperature-dependent thermal
conductivity, n is a constant, t is time, θi is the initial temperature of the body, θ∞ is the temperature of the environment, ρ is the density, cp is the specific heat at a constant pressure, Q is the heat loss parameter, and h(t)=h0f(t) is the time-dependent heat transfer
coefficient, with h0 and K0 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: (2.4)r¯=ra,t¯=K0tρcpa2,θ¯=θ−θ∞θi−θ∞,s=Qρcp,Bi=ah0K0.Neglecting the bar symbol for clarity, the
dimensionless boundary value problem (BVP) becomes(2.5)∂θ∂t=1rm∂∂r(θnrm∂θ∂r)−sθsubject to(2.6)∂θ∂r=0,atr=0,θn∂θ∂r=−Bif(t)θ,atr=1, where Bi is the Biot number and s 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; theone-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)r*=r+ϵR(t,r,θ)+O(ϵ2),t*=t+ϵT(t,r,θ)+O(ϵ2),θ*=θ+ϵΘ(t,r,θ)+O(ϵ2), generated by the vector field(3.2)Γ=T(t,r,θ)∂∂t+R(t,r,θ)∂∂r+Θ(t,r,θ)∂∂θ,which
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 Γ is extended to all the derivatives appearing
in the governing equation through second prolongation(3.3)Γ(2)=Γ+Θ[t]∂∂θt+Θ[r]∂∂θr+Θ[rr]∂∂θrr,where(3.4)Θ[t]=Dt(Θ)−θrDt(R)−θtDt(T),Θ[r]=Dr(Θ)−θrDr(R)−θtDr(T),Θ[rr]=Dr(Θ[r])−θrrDr(R)−θrtDr(T),and Dr and Dt 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(3.5)Γ(2)(Eqn(2.5))|Eqn(2.5)=0.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)Γ1=∂∂t,Γ2=enstns(sθ∂∂θ−∂∂t),Γ3=2θn∂∂θ+r∂∂r.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.Table 1
Extra-admitted symmetries.
ConstantsSymmetriesm=3n+4n+2Γ4=r(−2−2n)/(2+n)2n(n+1){(2n+4)θ∂∂θ−(n2+4n+4)r∂∂r}m=1,n=−1Γ4=−2(logr+1)θ∂∂θ+rlogr∂∂rm=0Γ4=∂∂rm=0,n=−43Γ5=−3θr∂∂θ+r∂∂r


## 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 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]:(4.1)Ad(exp(ϵΓi))Γj=∑n=0∞ϵnn!(AdΓi)nΓj=Γj−ϵ[Γi,Γj]+ϵ22[Γi,[Γi,Γj]]−⋯,where [Γi,Γj]=ΓiΓj−ΓjΓi is the commutator of Γi and Γj. To compute the one-dimensional optimal system
of the algebra in (3.6), first a commutator table is constructed and given by Table 2.Table 2
Commutator table.
[Γi,Γj]Γ1Γ2Γ3Γ10nsΓ20Γ2−nsΓ200Γ3000The adjoint representation is constructed using
formula (4.1). We wish to
simplify as much as possible the coefficients a1,a2,
and a3 by carefully applying the adjoint maps
to(4.2)Γ=a1Γ1+a2Γ2+a3Γ3.Starting with a nonzero vector
(4.2) with a1≠0 and rescaling Γ such that a1=1, it follows from Table 3 that acting on Γ by Ad(exp((−a2/ns)Γ2)), one obtains Γ1+αΓ3. No further simplification is possible. For a=0 and assuming a3≠0 (say, a3=1 by rescaling); acting on the remaining vector
(4.2) by Ad(exp(c1Γ2)) yields a2ec1nsΓ2+Γ3. However, depending on the sign a2, the coefficient of Γ2 can be assigned to either +1, −1, or 0.
Finally, for a3=0, we obtain Γ2. Thus the one-dimensional optimal system is
given by (4.3){Γ1+αΓ3;Γ3±Γ2;Γ3;Γ2},where α is an arbitrary constant.Table 3
Adjoint representation for the base vectors
given in (3.6).
AdΓ1Γ2Γ3Γ1Γ1eϵnsΓ2Γ3Γ2Γ1+ϵnsΓ2Γ2Γ3Γ3Γ1Γ2Γ3


## 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 usingΓ2 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 Γ2,
we have to solve the system(5.1)dθsθ=dt−1=dr0.The system (5.1) yields the invariants J1=lnθ+standJ2=r which give rise to the functional
form(5.2)θ=e−stG(r).Table 4
Reductions by elements of the optimal systems.
SymmetryReductionsΓ1+αΓ3ρ=re−αt;θ=e2αt/nG(ρ),
where G satisfies 2αnG−nGn−1(G)2=(mρ+αρ)G′+GnG′′.Γ3+Γ2ρ=rexp(e−nst);θ=exp{(−st−2e−nstn)t}G(ρ), where G satisfies 2sG=nsρG′+nGn−1(G′)2+mρGnG′+GnG′′.Γ3θ=r2/nG(t); with G satisfying G′(t)=(m+2n+1)Gn+1−sG.The time-dependent heat transfer coefficient may be
represented byf(t)=e−nst.
This choice of f(t) renders the boundary condition invariant under Γ2 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(5.3)nG−1(G′)2+mrG′+G′′=0,and the boundary conditions (2.6)
transform to(5.4)dGdr=0,r=0,dGdr=−BiG1−n,r=1.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)
n=−1,m≠1,(5.5)G=c2exp{c1r1−m}.Case (b)
n=−1,m=1,(5.6)G=c2rc1.Case (c)
n≠−1,m=1,(5.7)G=±[c1lnr+c2]−(n+1).Case (d)
n≠−1,m≠1,(5.8)G=±[c1(n+1)r1−m+(m−1)(n+1)c2m−1]1/(n+1).We consider cases (a) and (d) only as examples. For case (a) and in terms of original variables we obtain(5.9)θ=c1(m−1)Biec1exp{−st+c1r1−m}.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 c1<0.
One may rewrite (5.9) as(5.10)θ=c1Biec1exp{−st+c1r−1}.Without loss of generality we letc2=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(5.11)θ=(−Bi)1/n(n+1)1/ne−str1/(n+1),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.Figure 1
Graphical representation of the invariant solution (5.11).
Parameters used n=0.5,s=1, and r=0.55. The temperature profile is given for
varying Biot number.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.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 Table4. For example, the Γ3-invariant is given by(5.12)θ=r(2/n)[mn+2+n+c1nsenstns]−1/n.


## 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 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.



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