This document presents a novel method called the Eigenfunction Expansion Method (EFEM) for analytically solving transient heat conduction problems with phase change in cylindrical coordinates. The method involves formulating the governing equations and associated boundary conditions, introducing coefficients, solving the eigenvalue problems, and representing the solution as a series expansion of the eigenfunctions. Dimensionless parameters are introduced to simplify the problem. The EFEM is then applied to solve a one-dimensional phase change problem. Results show that increasing the number of terms in the series expansion decreases the truncation error and that the Stefan number affects the melting fraction evolution over time.

1. A Novel Method for Analytical Solution of
Transient Heat Conduction and Stefan
Problem in Cylindrical Coordinate
Presented by:
Muhammad Zeeshan Khalid
Authors:
Muhammad Zeeshan Khalid a
, Dr. Muhammad Zubair b
, Dr. Majid Ali c
a
Department of Basic Sciences, University of Engineering and
Technology, 47050-Taxila, Pakistan
b
Nuclear Engineering Department, University of Sharjah, 27272
Sharjah, United Arab Emirates
c
Centre for Advanced Studies in Energy, NUST, H-12 Islamabad,
Pakistan

2. 2
Outline
 Introduction
 Problem Formulation
 Methodology
 Solution
 Application of EFEM for Phase Change Problem
 Assumptions and Boundary Conditions
 Results and Discussions
 Conclusions

3. 14-Jan-2016 3
Introduction
 Need of Sustainable energy
 Latent Heat based Thermal Energy
Storage Systems
 Phase Change Problem
 Solution Methods
 Eigen function expansion Method

4. 14-Jan-2016 4
One can write respective Governing equation and their corresponding
boundary conditions for cylinder with i (i=1, 2 ,…..n) layers as;
2 2
2 2
0
( , , )1 1
....(1)
( , , ), ,1 ,
0 , 0
i i i i i
i i
i i n in i
T T T g r z t T
r r r z k t
T T r z t r r r i n r r r
z L t
α
∂ ∂ ∂ ∂
+ + + =
∂ ∂ ∂ ∂
= ≤ ≤ ≤ ≤ ≤ ≤
≤ ≤ ≥
Problem Formulation

5. The associated boundary conditions are given as:
i. Inner surface of first layer
1 0
1 0
( , , )
( , , ) ....(2)in in in
T r z t
A B T r z t C
r
∂
+ =
∂
ii. Outer surface for n layers (i=1,2…..n)
( , , )
( , , ) ....(3)n n
out out n n out
T r z t
A B T r z t C
r
∂
+ =
∂
iii. Surface Initial Condition z=0 (i=1,2…n)
Ti
(r, 0, t) = 0….. (4)
( ,0, )
0iT r t
r
∂
=
∂
iv. Boundary Surface z=L (i=1,2….n)
Ti(r, L, t) = 0 . ……. (6)
( , , )
0iT r L t
r
∂
=
∂
, i=1, 2,….n …. (7)
v. Inner Interface of ith Layer
1 1 1
1 1 1
1
( , , ) ( , , )......(8)
( , , ) ( , , )
......(9)
i i i i
i i i i
i i
T r z t T r z t
T r z t T r z t
k k
r r
− − −
− − −
−
=
∂ ∂
=
∂ ∂
1
1
1
( , , ) ( , , ).....(10)
( , , ) ( , , )
.....(11)
i i i i
i i i
i i
T r z t T r z t
T r z t T r z t
k k
r r
+
+
+
=
∂ ∂
=
∂ ∂
vi. Outer Surface of the ith Layer

6. 14-Jan-2016 6
EFEM
Introduction of
Coefficients
Separation of
Variable
Determination of
Eigen Values
Transcendental
Equation
Solution in Series
Expansion
Determination of
Coefficients
Orthogonality
Property
Stefan
Condition
For Phase Change
Problem
Results

7. 14-Jan-2016 7
Introduction of Coefficients
2 2
2 2
( , , )1 1
......(13)i i i i i
i i
T T T g r z t T
r r r z k tα
∂ ∂ ∂ ∂
+ + + =
∂ ∂ ∂ ∂
( , , ) ( ) ( ) ( )....(14)i i iT r z t R r Z z t= Γ
2
''( ) ( ) ( ) '( ) ( ) ( ) ( ) ''( ) ( )1
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) '( )
....(15)
( ) ( ) ( )
''( ) '( ) ''( )1
....(16)
( ) ( ) ( )
''( ) '1
( )
i i i i i i
i i i i i i
i i
i i
i i i
i i i
i i
i
R r Z z t R r Z z t R r Z z t
R r Z z t r R r Z z t R r Z z t
R r Z z t
R r Z z t
R r R r Z z
R r r R r Z z
R r R
R r r
β
Γ Γ Γ
+ +
Γ Γ Γ
Γ
=
Γ
+ + = −
+ 2 2( ) ''( )
....(17)
( ) ( )
i
i i
r Z z
R r Z z
β µ= − − = −
All spatial directions are first found by associated eignvalue problem which is
solved by separation of variable technique. Then the dependent variable
Temperature is written as the series expansion of the obtained function.
We introduce the coefficient to find the solution by separation of variable
Multiplying and dividing equ (14) to equa (13) we get

8. 14-Jan-2016 8
Determination of Eigen Values
2 2
2 2 2
''( ) ( ) 0.....(18)
( )....(19)
i iZ z Zβ µ
λ β µ
+ − =
= −
1 2( ) sin cos ...(19)il il ilZ z C z C zλ λ= +
Eigen Value Problem I
Now putting equation (19) into (18) and solving
Boundary Conditions (iii) and (iv) are the corresponding conditions to find the
eignvalue of the problem.
Eigen Value Problem II
2'' '1
...........(20)i i
i i
R R
R r R
µ+ = −
2 2 2 2
2 2 2 2
3 4
'' ' ( ) ..........(22)
'' ' ( ) 0..........(23)
( ) ( ) ( )......(24)
i i i
i i i
im vm im vm im
r R rR r R v
r R rR r v R
R r C J r C Y r
µ
µ
µ µ
+ + =
+ + − =
= +
1 1 /im m iµ µ α α=

9. 14-Jan-2016 9
Solution in Series Expansion
Substituting Eigen value equations into Coefficients and representing into double-
series expansion.
0 1
( , , ) ( ) ( ) ( ).....(26)i ilm im il
l m
T r z t t R r Z z
∞ ∞
= =
= Γ∑∑
( )ilm tΓ is constant which is determined by initial condition given by:
( , ,0) ( , ).....(12)
1
i iT r z f r z
i n
=
≤ ≤
Similarly we can represent heat source term as;
0 1
( , , ) ( ) ( ) ( ).....(27)i ilm im il
l m
g r z t g t R r Z z
∞ ∞
= =
= ∑∑
( )ilmg tWhere can be found by using orthogonality properties of Eigen function and
found
1
1
0
2
0
( , , ) ( ) ( )
( ) ......(34)
( ) ( ) ( )
i
i
i
i
rL
i im il
r
ilm rL
im il il
r
g r z t rR r Z z drdz
g t
rR r Z z Z z drdz
−
−
=
∫ ∫
∫ ∫

10. 14-Jan-2016 10
APPLICATION OF EFEM FOR PHASE CHANGE
PROBLEM
Figure 3: Schematic of the Phase change problem
Axial conduction is negligible so z=L is equal to zero.
It is a One Phase Problem.
Phase Change Material Stored is in slid form at t=0.
Melting of PCM occurs at t>0.
Material is melting homogenously.

11. 14-Jan-2016 11
GOVERNING EQUATIONS
One Dimensional Equation for Phase Change Material in Cylindrical
Geometry
2
22
1 1
....(40), ( )i i i
i
T T T
t r r
r r r t
δ
α
∂ ∂ ∂
+ = ≤ ≤
∂ ∂ ∂
The equation has following boundary conditions:
1 1
2
1
( , )
0, ....(40)
( , ) , 0..........(41)i
T r t
r r
r
T r t T t
∂
= =
∂
= =
 Interface boundary conditions are given as ;
1 1
1
1
( , ) ( , ), ( ), 0.............(42)
( , ) ( )
, ( ), 0...........(43)
mT r t T r t r t t
T r t d t
k L r t t
r dt
δ
δ
ρ δ
= = >
∂
= = >
∂

12. 14-Jan-2016 12
DIMENSIONLESS PARAMETERSDIMENSIONLESS PARAMETERS
In order to make this problem simple, less time consuming and
general, we introduce dimensionless parameters. There are the
following benefits of introducing dimensionless parameters[8].
It reduces the number of variables in equation.
Helps to analyze the system behavior without any consideration
of units used to measure variables.
 It helps to differentiate between relevant and irrelevant
variable and how it may affects the system.
It rescales variables and parameters so that we can have all the
computed quantities of the same order.
It reduces the computation time during numerical and analytical
simulation.
Prevent round off errors.

13. 14-Jan-2016 13
DIMENSIONLESS FORM OF EQUATIONDIMENSIONLESS FORM OF EQUATION
One Dimensional Equation for Phase Change Material in Cylindrical
Geometry
2
1 1 1
22
1
....(45), ( )S t R
θ θ θ
ε
ε ε ε τ
∂ ∂ ∂
+ = ≤ ≤
∂ ∂ ∂
Dimensionless boundary conditions:
1
2
1
( , )
0, .........(46)
( , ) 0, ( )...........(47)
R
S t
θ ε τ
ε
ε
θ ε τ ε
∂
= =
∂
= =
1( , ) ( , )
, ( , ), 0.....(48)
d
Ste S
d
θ ε τ δ η τ
ε η τ τ
ε τ
∂
= = >
∂
 Interface boundary conditions are given as ;
( )
, m
i m
T Tc T T
Ste
L T T
θ∞ −−
= =
−
1
r
r
ε =
2
1
t
r
α
τ = 2
2
1 1
(, )
( ) ,
rt
S t R
r r
δ
= =

14. 14-Jan-2016 14
SOLUTION USING EFEMSOLUTION USING EFEM
 Coefficients for this problem are assumed as;
( , ) ( ) ( )....(49)Rφ ε τ ε τ= Γ
 Solution of Eigenvalue Problem I
general solution: 1 0 2 0( ) ( ) ( )......(50)n nR C J C Yε β ε β ε= +
Transcendental equation: 1 2 0 0 1 2( ) ( ( )) ( ( )) ( ) 0.....(52)n n n nJ R Y S t J S t Y Rβ β β β− + =
5 1 0 1 5 2 0 2 5 3 0
 0 . 1 0
 0 . 0 5
0 . 0 5
0 . 1 0
0 . 1 5
Figure 3: Plot of bassel equation eignvalue against.[8-9]

15. 14-Jan-2016 15
Solution in Series Expansion
Substituting General Solution into Coefficients and representing into
double-series expansion.
2
( )
0 0 0 1
1
( , ) ( ( ) ( ( )) ( ( )) ( ))....(54)n n n n n
n
C e J Y S t J S t Yβ τ
φ ε τ β ε β β β ε
∞
−
=
= −∑
Where Cn can be found by using orthogonality properties of Eigen function and found
2 2
2 2 2
0 0 0 0
( ) ( )
2 2 2 2
0 0 0 0 0 0 0 0
( ) ( ) ( )
( ( )) (( ) ( ( )) (( )
...(55)
( ( )) (( ) ( ( )) (( ) 2 ( ( )) ( ( )) (( ) ( ( ))
R R
n n n n
S S
n R R R
n n n n n n n n
S S S
Y S J d J S Y d
C
Y S J d J S Y d J S Y S J Y S d
τ τ
τ τ τ
β τ ε β ε ε β τ ε β ε ε
β τ ε β ε ε β τ ε β ε ε β τ β τ ε β ε β τ ε
−
=
+ −
∫ ∫
∫ ∫ ∫
Melted fraction evolution with time can be calculated by putting (54) into Interface
Equation resulting in form:
2
( )
1 1 0 0 1
1
1 ( )
( ( ( ( )) ( ( )) ( ( )) ( ( ))) ....(56)n n n n n n n
n
dS
k C e J S Y S J S t Y S
Ste d
β τ τ
β β τ β τ β β β τ
τ
∞
−
=
− + =∑

16. 14-Jan-2016 16
RESULTS AND DISCUSSIONS
The equations were solved using Mathematica notebook. Time dependent equation
54 is truncated at n=N leading to [8];
2
( )
0 0 0 1
1
( , ) ( ( ) ( ( )) ( ( )) ( )) ( , , )....(57)
N
n n n n n
n
C e J Y S t J S t Y Nβ τ
φ ε τ β ε β β β ε ζ ε τ−
=
= − +∑
Where is truncation error.( , , )Nζ ε τ
0 0 0 1
1
( , , ) ( ( ) ( ( )) ( ( )) ( ))....(58)
N
n n n n n
n
N C J Y S t J S t Yζ ε τ β ε β β β ε
=
= −∑
Since bn is directly related to n so by increasing n value of bn also increases, hence
the maximum truncation errors at t=0. Since =1 so we have;( , )φ ε τ
By increasing the N truncation error decreases.

17. 14-Jan-2016 17
EFFECT OF STEFAN NUMBER
0 .0 0 .5 1 .0 1 .5 2 .0
1 .0 0
1 .0 5
1 .1 0
1 .1 5
1 .2 0
1 .2 5
1 .3 0
1 .3 5
D im e n s io n le s s tim e
Temperature
1 .0 1 .2 1 .4 1 .6 1 .8 2 .0
0 .0
0 .2
0 .4
0 .6
0 .8
R
R,t
0 .0 0 .5 1 .0 1 .5 2 .0
1 .0
1 .1
1 .2
1 .3
1 .4
1 .5
1 .6
1 .7
D im e n s io n le s s tim e
InterfacePosition
Ste=2
Ste=1
Ste=0.5
Ste= 2
Ste= 1
Ste=1.5
Ste=1.2
Ste=1.0

18. 14-Jan-2016 18
ADVANTAGES AND LIMITATION OF THIS METHODADVANTAGES AND LIMITATION OF THIS METHOD
On the theoretical level eigen function expansion method is simple,
powerful, straightforward and takes less computational time.
. This method has been found accurate and rapid in convergence in
literature [11,12].
We can calculate temperature distribution or interface position at any
point after the calculations of eigenvalues, eigen functions and
coefficients of eignefunctions.
The speed of calculation by this method is many times faster than
fourier transform method [13].
One of the limitation or disadvantage of this method is it sometime
gives imaginery eigenvalues due to the explicit dependence of
transverse eigenvalue on the other direction of cylinder.
We obtained real eigenvalues and dependence of eigenvalues on the
other direction of cylinder is not explicit.

19. 14-Jan-2016 19
FUTURE WORKFUTURE WORK
We can further extend the applicability of this method by considering
the geometry of tube for Heat Transfer Fluid flow and coupling it with
cylinder filled with PCM.
Analytical solution using this method can also be induced by
considering the fins geometry for heat transfer enhancement.
Multidimensional formulation and exact analytical solutions
Modeling the whole Thermal Energy Storage system and calculate the
efficiency of the system and validation of analytical results with numerical
and experimental data.
Consideration of heat transfer enhancement geometries.

20. 14-Jan-2016
20
CONCLUSIONCONCLUSION
Eigenfunction expansion method for multiple layers of cylinder was
introduced for transient heat conduction problem.
The heat equation for cylindrical coordinated was solved by introducing
the coefficient to solve the eigenvalue by separation of variables.
 Coefficients are then expanded by using the orthogonality properties.
 Roots found by eigenvalue are used to find heat conduction solutions
and are dependent upon system components and geometry.
EFEM was applied for Phase Change Problem.
Dimensionless parameters were introduced in order to make problem
general and simple, and to study the effect of Stefan number on the phase
transition of PCM.
 The results are discussed by varying the Stefan value and illustrated the
results in figures showing the low phase transition time at high Stefan
numbers.

21. References
1. Christian Grossmann; Hans-G. Roos; Martin Stynes. Numerical Treatment of Partial Differential Equations.
Springer Science & Business Media. p. 23. ISBN 978-3-540-71584-9. 2007
2. Cao Y, Faghri A. A study of thermal energy storage systems with conjugate turbulent forced convection. J Heat
Transf 1992.
3. Bellecci, C., and M. Conti. "Transient behaviour analysis of a latent heat thermal storage module." International
journal of heat and mass transfer 36.15, 3851-3857, 1993
4. Lacroix, Marcel. "Study of the heat transfer behavior of a latent heat thermal energy storage unit with a finned
tube." International Journal of Heat and Mass Transfer 36.8, 2083-2092, 1993.
5. Bechiri, Mohammed, and Kacem Mansouri. "Analytical solution of heat transfer in a shell-and-tube latent thermal
energy storage system." Renewable Energy74, 825-838,2015 Dalir, Nemat, and S. Salman Nourazar. "Analytical
solution of the problem on the three-dimensional transient heat conduction in a multilayer cylinder."Journal of
Engineering Physics and Thermophysics 87.1, 89-97,2014
6. Dalir, Nemat, and S. Salman Nourazar. "Analytical solution of the problem on the three-dimensional transient heat
conduction in a multilayer cylinder."Journal of Engineering Physics and Thermophysics 87.1, 89-97,2014
7. B.G Higgens, “Transient Heat Conduction in a Finite Length”, unpublished.
8. Jain, Prashant K., and Suneet Singh. An exact analytical solution for two-dimensional unsteady multilayer heat
conduction in spherical coordinates. International Journal of Heat and Mass Transfer, 53(9),2133-2142,2013
9. Wolfram Research, Inc., Mathematica, Version 5.2, Champaign, IL, 2005
10. Ozisik, M. Necati. Heat Conduction. John Wiley & Sons, 1993.
11. R. Blackmore and B. Shizgal, Phys. Rev. A31, 1855 ~1985!
12. B. Shizgal, J. Comput. Phys.41, 309 ~1981!.
13. Boyarchenko, Nina, and Sergei Levendorskiǐ. "The eigenfunction expansion method in multifactor quadratic term
structure models." Mathematical finance vol.17.4,pp.503-539, 2007.
21

22. 22