This document presents a mathematical model for analyzing heat transfer during phase change processes in a latent heat thermal energy storage (LHTES) unit. The model considers heat transfer between a heat transfer fluid flowing through a circular tube and phase change material (PCM) contained in the cylindrical shell. Dimensionless governing equations are derived to model forced convection in the fluid and natural convection during PCM melting. An analytical solution is obtained for temperature distribution using exponential integral functions. Results show interface position varies with dimensionless time and is affected by parameters like Stefan number, Biot number, and Rayleigh number. Graphs illustrate increased melting rate with natural convection and effect of varying Stefan number and Biot number on interface position.

1. 0.0 0.5 1.0 1.5 2.0 2.5 3.0
1.0
1.5
2.0
2.5
Dimensionless time
InterfacePosition
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.0
0.2
0.4
0.6
0.8
1.0
InterfacePosition
Dimensionless Time
Biot = 10
Biot = 6
Bi = 4
Introduction
The current economic and technological advancement has increased the load of energy sources, many
conventional energy sources are quite limited and have much hazardous effect on the health of living
beings and environment of our earth, besides this emission of Green House Gas is considered the main
cause of environmental changes. In view of these problems a need of an efficient and clean energy
source through the use of advanced technologies and systems are fostering day by day. For this purpose
Different types of solar power plant are now being used to convert solar energy into electricity
production unit.
Phase Change Materials (PCMs) is one of the most basic and important component of TES unit for
storage and retrieval of heat. PCMs store heat by absorbing heat from Heat Transfer fluid of solar plant
by phase transition (Solid-Liquid). Liquid PCM then release heat during discharging process which is being
absorbed by HTF and forwarded to power generation unit to generate electricity. Many researchers used
numerical techniques to study the performance of Thermal energy storage unit (e.g 4, 5) but there is not
much work on the analytical modeling of TES unit to study the forced and natural convection. Recently
Bechiri[1] modeled one tube and shell mode to study the melting and thermal energy storage capacity of
the system using analytical technique. Results obtained showed good agreement with numerical and
experimental data. This goal of this project is to model a system involving Heat transfer fluid as a heat
source and Phase Change Material contained in a cylindrical container to study the effects of convection
on the transition rate of material. Semi Analytical treatment was applied to simulate results and different
dimensionless parameters such as Biot number and Rayleigh number was introduced to study forced and
natural convection.
Dimensionless Mathematical Model
The latent heat Thermal Energy Storage (LHTES) unit as shown in figure 1 consist o several such tubes and
shells for solar thermal power plants. The working Heat Transfer Fluid changes/discharges the PCM stored in
the cylinder. Unsteady heat transfer from circular tube and steady state interface evolution with time is
investigated by using following assumptions.
I. The problem is axissymmetric (no angular variation) and two-dimensional.
II. Thermo-physical properties of HTF and PCM are constant
III. Axial conduction in the fluid and solid material is negligible.
IV. Flow of HTF in circular tube is Laminar with uniform heat flux condition.
V. The molten and solid PCM is homogenous and isotropic.
VI. As pressure in PCM is negligible so the phase change of PCM occur at nearly constant temperature (i.e
KNO3 at 330 0
C)
VII. The effect of natural convection during melting of PCM is taken into account by assuming effective thermal
conductivity [1].
* 7
100 10cRa 
:
0.25 0.25Pr
0.386( ) ( ) ......(1)
0.861 Pr
eff
c
k
Ra
k


*
100cRa 
: 1.....(2)
effk
k

The governing equations in dimensionless form are given as
For thee HTF
1 1
( )
( ( , ) 1) ( ), 1....(3)
2
b
l b
Nu
R R
 
    


  

1(0) 1,0 1b R   
…… (4)
For the PCM
Liquid Phase
2
12
1 1 1
1
1 1
1
1
,1 ( , )....(5)
( , )
( ( , ) 1), 1....(6)
l l l
l
l
R S
R R R
R
Bi R R
R
  
 

 
 
  
   
  

  

1 1( , ) 0, ( , ), 0.....(7)l R R S      
Solid Phase
2
1 22
1 1 1
1
, ( , ) .....(8)s s s
S R R
R R R
  
 

  
   
  
1 1
1
1 2
1 1
( ,0) ( , ), 0............(9)
( , )
0, , 0........(10)
( , ) 0, ( , )............(11)
s
s
s
R F R
R
R R
R
R R S
  
 

   
 

  

 
Solid-Liquid Interface
1 1 1
1
1
1 1 1
( , ) ( , ) 1
, ( , ).........(12)
( , ) ( , ) 0, ( , ), 0............(13)
l s s
l
s l
R k R R
R S
R k R ste
R R R S
   
 

      
  
  
  
   
A Contagion Model of Emergency Airplane Evacuations
Junyuan Lin, Pepperdine University
Advisors: Timothy Lucas and Jesus Rosado
U C L A A Æ Æ L I E D M A T H E M A T I C S Æ EU
Forced and Natural Convection Modeling for Phase Change Problem
Muhammad Zeeshan Khalid, Syed Asad Maqbool, Saqlain Saqib Mukhtar
UNIVERSITY OF ENGINEERING AND TECHNOLOGY TAXILA
1 1 1
1
( , , ) ( , , ) 1l s s
l
R k R R
R k R ste
     

  
 
  
1 1( , ,0) ( , )s R F R  
( , )S  
1
1
1
( , , )
( ( , , ) )
2
l
f l
f
R Nu
R
R K
  
   

  

Initial Condition
(Known)
Inlet Heat Transfer
Fluid
Forced Convection
Solution of the Problem
For the HTF
The solution for equation 3 for the system of HTF is given as
1
0
( ) ( ( ( , ) 1) ....(14)
2
b l
Nu
exp R d

      
11 1( ) ( ( , ) 1) ......(15)
2
l R
Nu
R      
Equation 14 becomes:
0
( ) ( ( ) )....(16)b exp d

     
Solution for Liquid Phase
Exponential Integral function method as described by Oziki [2] was used to get the temperature distribution for liquid
phase and final expression is given as:
22
1
1 2
( ( ) ( )
4 4( , ) ....(17)
1 1
2exp( ) ( ( ) ( ))
4 4 4
l
RS
Bi Ei Ei
R
S
Bi Ei Ei
  
  
   

     
Bulk Temperature Relation
22
1
1 2
( ( ) ( )
4 4( , , ) ( )....(18)
1 1
2exp( ) ( ( ) ( ))
4 4 4
l b
RS
Bi Ei Ei
R
S
Bi Ei Ei
    
  
   

     
Solution for Solid Phase
The temperature distribution for solid phase of equations (8-11) are solved by using the eigen function expansion
method technique as discussed in chapter 3. Recalling solutions of these equations from chapter 3, transcendental
equation and temperature distribution in solid is written as
Transcendental equation:
0 1 2 1 2 0( ) ( ) ( ) ( ) 0.....(19)n n n nJ S Y R J R Y S    
Temperature distribution in solid phase:
2
( )
1 0 1 0 0 0 1
1
( , ) ( ( ) ( ) ( ) ( )).............(20)n
s n n n n n
n
R C e J R Y S J S Y R 
     



 
Coefficient:
2
2
1 0 1 0 0 0 1 1 1
( )
2
0 1 0 0 0 1 1
( )
( , )( ( ( ) ( ) ( ) ( ) )
......(21)
( ( ) ( ) ( ) ( ))
R
n n n n
S
n R
n n n n
S
F R J R Y S J S Y R R dR
C
J R Y S J S Y R RdR


    
   





Bulk Temperature Relation
2
( )
1 0 1 0 0 0 1
1
( , , ) ( ) ( ( ) ( ) ( ) ( )).............(22)n
s b n n n n n
n
R C e J R Y S J S Y R 
        



 
Interface Equation:
2
2
( )2
1 02
11
0 1
2 ( ) exp( / 4 ) ( )
( ( ( ( )) ( ( ))
2 exp( 1/ 4 ) ( ( / 4 ) ( 1/ 4 ))
( ( )) ( ( )))
1 ( )
....(23)
nb b
n n n
n
n n n
Bi S k
C e J S Y S
S Bi Ei S Ei k
J S t Y S
dS
Ste d
     
   
  
   






 
    



Results and Discussions
In order to find the interface location of the melted PCM, equation 23 is solved using Mathematica.
Equation was solved by varying transition parameter i.e Stefan number and convective parameters
such as Biot number and Rayleigh Number. Results were plotted as shown in figures below.
0 1 2 3 4 5 6
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Dimensionless time
InterfacePosition
With Natural Convection
Without Natural Convection
Figure 2: Effect of natural convection on the melting rate ( 0.433  , Ste=0.15, ξ=1)
Ste= 3
Ste=2.0
Ste=0.5
Figure 3: Effect of Stefan Number on the melting rate ( 0.433  , ξ=1)
Figure 4: Effect of Biot Number on the melting rate ( 0.433  , Ste=0.15, ξ=1)
Figure 5: 3D representation of temperature distribution of axial and radial position at (a) t=0 (b) t=1 (c) t=2.5 for PCM
( 0.433  , Ste=1)
References
[1] Bechiri, Mohammed, and Kacem Mansouri. "Analytical solution of heat transfer in a shell-and-
tube latent thermal energy storage system." Renewable Energy. 2015, 74, 825-838
[2] Ozisik, M. Necati. Heat Conduction. John Wiley & Sons, 1993.
[3] Wolfram Research, Inc., Mathematica, Version 5.2, Champaign, IL, 2005.
[4] T. Saitoh, Numerical method for multi-dimensional freezing problems in arbitrary domains,
ASMEJ. Heat Transfer 100, 1978. 294–299.
[5]Ismail, Kamal AR, and E. Maria das Gracas. "Numerical solution of the phase change problem
around a horizontal cylinder in the presence of natural convection in the melt region." International
journal of heat and mass transfer. 2003, 46(10), 1791-1799.
ξ
ξ
ξ
Φ