by
Jessica Lyn Michalski
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
August, 2009 i

LIST OF SYMBOLS ......................................................................................................... iii
LIST OF TABLES .............................................................................................................. iv
LIST OF FIGURES ............................................................................................................. v
ACKNOWLEDGMENT .................................................................................................... vi
ABSTRACT ........................................................................................................................ 1
1. Introduction ...................................................................................................................... 2
2. Theory .............................................................................................................................. 4
2.1 Effective Heat Capacity Method .......................................................................... 4
2.2 General Heat Equation ......................................................................................... 4
2.3 Moving Boundary Equations ............................................................................... 5
2.4
Newmann’s Solution for the Moving Boundary Problem ................................... 5
2.5 Description of Finite Element Analysis in COMSOL Multiphysics ................... 6
3. Results and Discussion .................................................................................................... 8
3.1 One-Dimensional Analysis Results...................................................................... 8
3.1.1 Solutions from Analytical Methods ......................................................... 8
3.1.2 Finite Difference Method for One-Dimensional Analysis ....................... 9
3.1.3 Solutions from COMSOL Multiphysics Methods ................................. 11
3.1.4 Discussion of One-Dimensional Results ................................................ 18
3.2 Analysis of a Two-Dimensional System ............................................................ 20
3.2.1 Definition of the Two – Dimensional System ....................................... 20
3.2.2 Solutions from COMSOL Multiphysics Methods ................................. 21
3.2.3 Discussion of Two-Dimensional Results ............................................... 23
4. Conclusions.................................................................................................................... 24
5. References ...................................................................................................................... 25
ii

K1 – Thermal Conductivity of Solid [W/mK]
K2 – Thermal Conductivity of Liquid [W/mK]
ρ1 – Density of Solid [kg/m 3
]
ρ2 – Density of Liquid [kg/m 3
]
C p1
– Specific Heat of Solid [J/kgK]
C p2
– Specific Heat of Liquid [J/kgK] v – Temperature of material [K] v1 – Temperature of the Solid [K] v2 – Temperature of the Liquid [K]
T1– Temperature of Solid – Liquid Interface [K]
V – Initial Temperature [K]
T w
– Temperature of Cooling Boundary [K]
H f
– Heat of Fusion [J/kg] t – Time [s]
X – Location of Solid / Liquid Interface [m]
κ – Thermal Diffusivity [m 2
/s]
κ1 – Thermal Diffusivity of Solid [m 2
/s]
κ2 – Thermal Diffusivity of Liquid [m 2
/s]
λ – Solidification Constant x – Location in System [m] x
COMSOL
– Location of Solid – Liquid Interface in COMSOL System [m] x
Exact
– Location of Solid – Liquid Interface in Exact System [m] iii

Table 1: Physical Properties Used in Analysis .................................................................. 3
Table 2: Physical properties used in baseline COMSOL analysis .................................. 12
Table 3: Thermal Conductivity Definition of One-Dimensional Analysis ..................... 15
Table 4: Heat Capacity Definition in One-Dimensional Analysis .................................. 15
Table 5: Transient Analysis Time Step Options .............................................................. 16
Table 6: Average percent difference in solid – liquid interface location compared to exact solution ................................................................................................................... 18
Table 7: Percent Difference between 2-D Strict and Exact 1-D Solid – Liquid Interface
Location ........................................................................................................................... 23 iv

Figure 1: Depiction of Problem Definition ........................................................................ 2
Figure 2: Plot of temperature distribution within the material .......................................... 8
Figure 3: Solid – Liquid Interface location as function of time ........................................ 9
Figure 4: Depiction of a segmented 1-D finite difference system ................................... 10
Figure 5: Comparison of finite difference and exact solid – liquid interface locations .. 11
Figure 6: Depiction of 1D system used in COMSOL analysis ........................................ 12
Figure 7: Typical one-Dimensional COMSOL plot of temperature within material ...... 13
Figure 8: Optimized one-Dimensional COMSOL plot of temperature within material .. 14
Figure 9: Comparison of COMSOL solutions to exact solution of solid – liquid interface location ............................................................................................................................ 17
Figure 10: Comparison of FDM, COMSOL and Exact Solid – Liquid Interface Locations
......................................................................................................................................... 19
Figure 11: Depiction of 2D system being analyzed......................................................... 20
Figure 12: Depiction of strict time step solution to 2-D analysis .................................... 21
Figure 13: Solid – liquid interface location comparison ................................................. 22 v

I would like to thank Ernesto Gutierrez – Miravete for his help in completing this project. I would also like to thank my heterosexual life mate for her understanding when
I continually blew her off to work on this project. vi

The goal of this study was to determine the possible ability to alter the parameters of the
COMSOL Multiphysics program in order to obtain a high accuracy solution that approximates the solidification of a copper system. Analytical and finite difference methods were employed in creating a comparable solution to the problem attempting to be replicated.
COMSOL Multiphysics was used to solve the one – dimensional heat conduction with change of phase problem and parameters were altered and compared to determine values that produced the optimal solution. This solution was later compared to that of a similar two – dimensional system in an attempt to determine the scope of the usefulness of this process.
1

The phenomenon of phase change within a material is often seen during the course of manufacture of materials, particularly metals. For a metal, in the very least, two phase changes must occur during the creation of the base material. During the production of an ingot the initial material must be smelted from the ore that has been removed from the earth in order to be processed to obtain a more pure element and a second phase change is required to obtain a solid ingot from the molten metal that has been obtained. It is frequently required that additional phase changes be used in the creation of a more final product, in the case of forged metals, or even in the creation of final components, as is true in the formation of castings.
The usefulness of a finite element analysis tool that can accurately describe a phase change phenomenon should not be undervalued. When solidification or melting process is fully understood, such as melting of materials and solidification times of cast structures, the process can be optimized to ensure that the highest quality, lowest cost methods of manufacture can be utilized.
The particular problem being analyzed in this project is that of the time dependent heat conduction with solidification in copper. The conditions of this system are that initially all of the copper has been heated to a temperature above the copper melting point of 1356 K. The liquid copper is assumed to be in an insulated, semi-infinite, system with only one side free to allow for the extraction of energy from the system, as
shown in Figure 1 below.
Figure 1: Depiction of Problem Definition
2

In order to create a significant comparison between the analytical, finite difference analysis and COMSOL Multiphysics finite element analysis being conducted it, was important to create a complete definition of physical properties. In addition to the physical properties used to define the problem, it was also important to make consistent assumptions about temperatures in order to create a relevant relationship between the
analyses being performed. Table 1 shows the complete set of input that was used.
Table 1: Physical Properties Used in Analysis
Property Name
Thermal Conductivity of Solid
Thermal Conductivity of Liquid
Density of Solid
Density of Liquid
Specific Heat of Solid
Specific Heat of Liquid
Temperature of Solid / Liquid Interface
Physical Property Symbol
K1
K2
ρ1
Value
365 W/m·K
174 W/m·K
8920 kg/m 3
ρ2
Cp1
Cp2
T1
8920 kg/m 3
435 J/kg·K
494.5 J/kg·K
1356 K
Temperature Initial
Heat of Fusion
Temperature Cooling Boundary
V
Hf
Tw
1400 K
205000 J/kg
400 K
In previous studies, the use of the enthalpy method has been used in phase change systems to determine the thermal performance of latent heat storage [1]. This same principle can be utilized in the analysis of the solidification of copper. The ability for the enthalpy method to properly account for the latent heat lost during the solidification process makes it well suited to this scenario. Also, the use of an effective heat capacity method has been previously found to have a high accuracy in other studies on phase change phenomena. [1]
3

The effective heat capacity method of solving heat conduction problems with change of phase is based on the use of an effective heat capacity over the melting range, directly proportional to the latent heat. The effective heat capacity uses the concept of three temperature ranges of thermal capacitance within the system; above melting, during melting/solidification and below solidification. The region of melting must be assumed to be greater than just a single temperature point, even for a pure material, and a range where melting and solidification occurs is defined. This range will have an effective heat capacity that is governed by (1) where, H f
is the latent heat of fusion and Δ T f
is the relatively small range of temperatures it is assumed that melting occurs over [1].
C
P eff
=
H f
ΔT f
(1)
The range of temperature is required to be chosen due to the fact that a pure metal, such as copper, melts at a single point, not over a range of temperatures as required for the effective heat capacity equation. The other heat capacities regions can be defined by using the average of known values for the heat capacity from experimental data, which is readily available for most materials, in those regions. These heat capacities can then be used in heat equations and a piecewise curve can be created that incorporates the latent heat of fusion into the effects on temperature and time to solidification or melting. The use of these heat transfer ranges is required due to the non-linear parabolic partial differential equation that governs the heat conduction process.
The heat equation (2) is a parabolic partial differential equation which describes the distribution of heat within a system, or region of a system, as a function of time.
 v
 t
    2  v (2)
In phase change systems, two such equations are required, one for the solid phase and one for the liquid phase. The heat equation is such that the system will have a maximum
4

temperature within itself that is either defined as occurring at the initial time or occurring at one of the boundary locations. This property of the heat equation is important due to the fact that heat cannot be generated within the system without some kind of external heat source being added [2].
Another important characteristic of the heat equation is its ability to create smooth transitions within the material where initially there could be large discontinuities. For instance in a system of length x = 5, when the initial condition at location x = 0 is T = 0
K and at x = 5 is T = 1600 K the system will very quickly create a smooth distribution of temperature between the two discontinuous endpoints in the attempt of the system to find an equilibrium condition.
In order to obtain an exact analytical solution to a moving boundary problem that can be used to describe the problem being analyzed, it is required to write the heat equation for each phase and the following conditions at the solid – liquid interface. v
1
= v
2
= T
1
, when x = X (t) (3)

1 v x

2 v
K = x f
  dt
(4)
These conditions at the moving boundary, called Stefan conditions, define the situation where the separation of liquid and solid material lies and account for the latent heat in the system. That means that for x > X the material is a liquid at temperature v
2
(x,t) and for x < X the material is a solid of temperature v
1
(x,t) and takes into consideration the latent heat of the system that must be lost or gain in order for a phase change to occur and for the interface to move[3].
One specific solution that is particularly helpful in defining the temperature distribution found for the one dimensional situation to be analyzed is the Newmann solution [3]. In this exact solution it is assumed that the region being analyzed is from x > 0 where the initial temperature is V and the surface x = 0 is maintained at T w
for all times t > 0. In this solution set of equations, the boundary conditions are redefined to be:
5

v
2

V
, as x
  v
1
= T w
, when x = 0
(5)
(6)
By solving the heat equation for the two phases together with the Stefan conditions yield the following transcendental equation: e
  2 erf (

)

K
1
 
1 /
2
K
2
2

 
1
1 /
( T
1
2


( V
T w
)


T
1
)
 e
 
1
  2 erfc (
 
(

1
/

2
/

2
)
1 / 2
)
=
 c
1

H

( T
1 f
 
1 / 2

T w
)
(7) for the solidification constant λ. The position of the solid – liquid interface is simply given by:
X

(

1
 1 2 t ) (8) and it is possible to obtain a curve that defines the exact solution for the initial Newmann problem. This solution can then be compared with various outputs from finite element analysis programs, such as COMSOL Multiphysics.
In addition to obtaining an equation defining the location of the solid – liquid interface it is also possible to obtain the expressions that define the temperatures of the solid phase with respect to location and time: v
1

T w

T
1 erf

(
T w

)
 erf (
2

(

1 x
 t
1 / 2
)
)
and the temperature of the liquid phase with respect to location and time. v
2

V

( V erfc (
 

(

T
1

1
2
)
)
1 / 2
)
 erfc (
2

(

2 x
 t )
1 / 2
)
(9)
(10)
The COMSOL Multiphysics program was created as an implementation of the method of finite element analysis in order to, among many other applications, determine solutions to heat transfer problems. The program has been designed to solve problems which are undergoing conduction, convection and radiation methods of heat transfer. In particular the use of COMSOL Multiphysics has been used in applications where heat generation, or heat flow are of interest.
In COMSOL Multiphysics, a domain geometry is created and one is able to assign a number of Subdomain Setting parameters including thermal conductivity,
6

density, heat capacity, and initial temperatures, which are used in the following study.
Also able to be defined, are the initial and boundary conditions. The geometry, once fully defined physically, is then meshed and a solution can be obtained. This solution can either be steady-state or transient with the transient solution having another series of parameters that can be altered in order to obtain a more accurate solution depending on the time intervals that are defined. These solutions can be used in order to obtain a reasonably accurate depiction of the heat transfer within many systems.
In the case of phase change phenomenon, these parameters are not able to fully define all of the physical changes that are occurring within the system. In addition to the basic definitions of material properties is it required that additional functions be created in order to accurately describe the phase change phenomenon itself. For this it is possible to use the Function option that has been created within COMSOL Multiphysics to assign values to specific parameters, such as specific heat and thermal conductivity, to adjust to the changes in heat transfer seen in specific regions.
7

3.1.1
Solutions from Analytical Methods
Using the Newmann's method described in Section 2.4 above, MAPLE was used in
order to determine an exact solution to the moving boundary problem described, the complete one – dimensional MAPLE code is shown in Appendix A. Using this
information, together with the exact solution and the input data in Table 1 the following
plot has been created to show the temperature distribution as a function of distance into
the material after a time of 30 seconds has elapsed, in Figure 2.
Figure 2: Plot of temperature distribution within the material
From the MAPLE calculation one is able to show easily the discontinuity in the slope of the temperature distribution that occurs due to the loss of latent heat in the solidification process, which in this case is shown approximately between .08 and .15 meters into the material after 30 seconds of time elapsing. This feature is what needs to be replicated within the COMSOL Multiphysiscs analysis in order to obtain an accurate analysis of
8

0.08
0.07
0.06
0.05
0.04
0.03
0.02
the solidification process. For this purpose, the location of the solid – liquid interface within the material will be tracked as a function of time. In the exact solution, the definition of the solid – liquid interface location is given by Equation 8 [3] as the material cools and solidifies.
The results of using this as the defining feature of the solidification process is
shown in Figure 3, where the location of the solidus is found to move steadily away from
the origin as the material cools and solidification can begin to occur. This plot was created using the exact equation for the solid – liquid interface as determined in the
Maple code, found in Appendix A, and the values were determined in Excel over a range of 30 seconds.
Solid Liquid Interface Location VS Time
0.09
0.01
0
0 5 10 15
Time (s)
20 25 30 35
Figure 3: Solid – Liquid Interface location as function of time
3.1.2
Finite Difference Method for One-Dimensional Analysis
The use of a finite difference method can be utilized to determine the temperature of the system over its length for given times as well as for determining the location of the
9

solidus within the system. Here the finite difference method is based on an implementation of the enthalpy method, as shown in Appendix B [7].
Using the enthalpy method, a system of nodes can be created where for the given material properties and initial conditions of the system, a set of algebraic equations
(finite difference equations) can be solved and an approximate solution achieved. The initial step to achieving an approximate solution to a finite difference problem is to create a discrete series of uniformly sized segments bounded by nodes to define the
system, as shown in Figure 4 below.
Figure 4: Depiction of a segmented 1-D finite difference system
Using the explicit method to solve the finite difference equation, it is possible to approximately solve for an unknown value by using the known values that neighbor that node. Using a known solution code for the finite difference explicit method for the 1-D heat equation with freezing – enthalpy method, Appendix B, a solution set was obtained that solves for the solidus location as a function of time. This result was compared to that
of the exact analytical solution, Figure 5.
10

Figure 5: Comparison of finite difference and exact solid – liquid interface locations
This shows that with an approximately 3.2% accuracy the solid – liquid interface location of the system can be approximated using the explicit solution to the finite difference analysis.
3.1.3
Solutions from COMSOL Multiphysics Methods
Initially in the COMSOL program no modifications were made to the equations used to define an analysis in order to obtain a baseline COMSOL solution to which the modified solutions can be compared, in addition to comparing to the analytical solutions. The
initial conditions describe in Table 2 were used in this analysis.
11

Table 2: Physical properties used in baseline COMSOL analysis
Property Name
Thermal Conductivity
Physical Property Symbol
K
Density
Specific Heat
Melting Temperature
Temperature Initial
Temperature External
ρ
Cp
T1
V
T w
Value
269.5 W/m·K
8920 kg/m 3
464.5 J/kg·K
1356 K
1400 K
400 K
These property values are identical to those used in the Table 1 with the exception of the
physical properties that define the heat transfer with phase change within the system.
The values for thermal conductivity and specific heat were averaged over the temperature range being analyzed and the actual melting temperature was used.
The physical system that was created to obtain all one – dimensional COMSOL
Multiphysics solutions is shown below in Figure 6.
Figure 6: Depiction of 1D system used in COMSOL analysis
The resulting solution that was obtained for COMSOL Multiphysics using the
parameters defined after 30 seconds is shown in Figure 7.
12

Figure 7: Typical one-Dimensional COMSOL plot of temperature within material
The smooth curve that the COMSOL Multiphysics program creates does not properly depict the phase change region as it is known to occur though testing and analytic solutions that are available. If an inexperienced person were to attempt to complete an analysis using the output from COMSOL without further research, it is possible that the results could be inaccurate, especially if the occurrence of a phase change was not expected within the system.
The result of the initial analysis when using the COMSOL supplied equations with no modifications is such that there is no discernable loss of energy when the material solidifies at 1356 K. This is not the case in the analytical solution, where a distinct
“knee” is formed in the temperature versus distance plot at the region where
solidification occurs, as shown in Figure 2. With modification of the parameters within
the COMSOL Multiphysics program it is possible to create an accurate and meaningful
13

representation of the heat transfer through the material over time including phase change regions.
When the equations are modified as discussed in Section 3.1.1 above, it is possible
to notice the three distinct region that are formed when accounting for the loss of latent heat within the system. As the thermal conductivity and heat capacity of liquid and solid copper are not equal and since removal of heat is required for the interface to advance, it is noted that the point above the melting temperature should have a certain slope, while
the point below the melting temperature should have a different slope, as seen in Figure
2. When the one-dimensional analysis is performed using the physical properties defined
in Table 1, after 30 seconds of time has elapsed the result is Figure 8.
Figure 8: Optimized one-Dimensional COMSOL plot of temperature within material
14

In creating this analysis no properties were altered from the previous case with the
exception of including the varied thermal conductivity, as described in Table 3, and the
varied heat capacity which includes an equivalent heat capacity during solidification as
described in Table 4.
Table 3: Thermal Conductivity Definition of One-Dimensional Analysis
Range Start (K) Range End (K) Thermal Capacity Value (W/m·K)
289
1350
1360
1350
1360
1500
365
1/2*(365+174)
174
Table 4: Heat Capacity Definition in One-Dimensional Analysis
Temperature (Kelvin) Heat Capacity (J/kg·K)
289
1340
435
435
1350
1360
10250
10250
1370
1500
494
494
By using the new ranges of thermal conductivity implicitly and by interpolating the values of heat capacity inputted into the system the results were much improved over the initial analysis.
A series of six improved solutions were created with identical material properties, and only altering the time step solution methods. The time step for the solution is the way in which the COMSOL Multiphysics program chooses to divide the transient analysis. This parameter can be modified in order to obtain a solution that can
more accurately describe the transient phenomenon. Table 5 below shows the six
variations chosen for this analysis. It is important to be aware that while decreasing the time step for the solution can possibly improve the results of the analysis this also
15

greatly increases the resources required to the analysis, both in computing power and time to complete the analysis.
Table 5: Transient Analysis Time Step Options
Option Time Step Name Initial Step Max Step
3
4
1
2
5
6
Free
Intermediate
Strict
Manual
Manual
Manual
-
-
-
.001
.0001
.00001
-
-
-
.01
.001
.0001
While this depiction of the temperature over the region being analyzed is a helpful indication that the modifications being made are allowing for a more realistic solution, as previously stated it is necessary to compare the exact solution solid – liquid interface location to that of the COMSOL Multiphysics solution. In order to do this, values for the solid – liquid interface were found at intervals over the transient solution
up to 30 seconds. The resulting solutions are shown in Figure 9. This plot of solutions
was created by exporting the raw solution data from COMSOL Multiphysics for each time step option into Excel and overlaying the results.
16

COMSOL Options VS Exact Solution for Solidus Location
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0 5 10 15 20 25 30
Option 1
Option 2
Option 3
Option 4
Option 5
Option 6
Exact
Time (s)
Figure 9: Comparison of COMSOL solutions to exact solution of solid – liquid interface location
In addition to the visual comparison in Figure 9 above, an analytical comparison of the
percent difference between the individual points from the COMSOL Multiphysics solutions and the corresponding exact solution points was determined using equation 11.
% Difference
 x
COMSOL
( x
COMSOL
2

 x
Exact x
Exact )

100 (11)
These individual percent differences were then averaged to obtain the average percent
difference for each of the options was created, the results of which are shown in Table 6
below.
17

Table 6: Average percent difference in solid – liquid interface location compared to exact solution
Option Average Percent Difference
1
2
6.07 %
12.28 %
3
4
6.03 %
12.28 %
5 6.87 %
6 5.92 %
Similar to what was shown in Figure 9 the overall percent difference between the
exact solution and the COMSOL solution for Option 6 gives the highest accuracy solution for solid – liquid interface location. The solution that defines the solid – liquid interface location is comparable to those seen in the finite difference method.
3.1.4
Discussion of One-Dimensional Results
The one – dimensional solution that is obtained by incorporating the effective heat capacity method to the COMSOL Multiphysics program is such that the solution is of a significantly higher accuracy than would be initially given, with only minor changes to the standard COMSOL program by introducing two functions.
The resulting solution from the COMSOL Multiphysics solution is able to obtain a similarly accurate solution as the use of finite difference analysis utilizing the explicit
method for enthalpy with the Dirichlet condition. As shown in Figure 10 below, the
finite difference analysis creates a solution that averages within 3.2% difference at determining the location of the solid – liquid interface with respect to the exact analytical solution for the initial 30 seconds of the solidification process.
18

Figure 10: Comparison of FDM, COMSOL and Exact Solid – Liquid Interface Locations
In addition to the alteration of the defining material properties, the use of an appropriate time step interval is required to complete the transient analysis within
COMSOL Multiphysics. It has been shown that the resulting solution from COMSOL
Multiphysics program with a manual time step with an initial time step of .00001 seconds and a maximum time step of .0001 seconds was able to accurately describe the location of the solidus to within 6% overall during the initial 30 seconds of the solidification process.
It is important to note that the improvement in the solidus location over time between the six solution methods cannot be determined to be significant. Additional work should be completed in the form of testing of other systems and actual temperature results compared to COMSOL solutions in order to obtain more significant results proving the necessity of utilizing a time step analysis such as Option 6, which takes a large period of time to complete, even in such a simplified system.
19

3.2.1
Definition of the Two – Dimensional System
While an exact solution can be obtained analytical for the one – dimensional system identified, no such exact solution exists for any two – dimensional system. In order to obtain a data point to compare the COMSOL Multiphysics solution it is required to choice a system that can be related back to the only exact solution that exists. For this
reason the system shown in Figure 11 has been chosen.
Figure 11: Depiction of 2D system being analyzed
Using this initial condition it is possible to create a correlation between the one - dimensional analysis that has been validated and the two - dimensional analysis that is being conducted.
By creating the correlation back to the one - dimensional analysis it can be assumed that the parameters being used are accurate for multiple scenarios and appropriate for most freezing analysis in a copper system. It is the goal that the two - dimensional analysis accurately describe the freezing within the two - dimensional system along the wall far from the corner that will initially freeze and will closely match
20

the solution that was obtained in the one – dimensional system. While the results from the one – dimensional analysis determined the need to use a larger analysis to obtain an accurate solution, two time step intervals were chosen to be used in the two – dimensional analysis, the strict time step and the manual .00001 initial and .0001 max time step.
3.2.2
Solutions from COMSOL Multiphysics Methods
Using an initial model representing the one shown in Figure 11 and the physical
properties defined in Table 1, Table 3 and Table 4 the COMSOL Multiphysics program
was loaded. Using the Strict time step method as defined by COMSOL Multiphysics the resulting definition of temperature within the system after 30 seconds elapsed is shown
in Figure 12.
Figure 12: Depiction of strict time step solution to 2-D analysis
This plot is in keeping with the expected results of heat conducting through the system into the surrounding cooler areas.
21

Using this output it was again required to create a system that could adequately define the solidus location within the system as a function of time for both the x and y direction within the system and compare these results to those of the exact solution and
the optimized one – dimensional solution, as shown in Figure 13.
Solidus Location VS Time
0.05
0.04
0.03
0.02
0.01
0
0
0.1
0.09
0.08
0.07
0.06
Exact
1-D Strict
2-D (x-direction) Strict
2-D (y-direction) Strict
5 10 15
Time (s)
20 25 30
Figure 13: Solid – liquid interface location comparison
It can be seen from these results that the x-direction results of the two – dimensional analysis has significantly more error than that of the similar one – dimensional analysis and the two – dimensional analysis in the y-direction. The percent differences between
the strict two – dimensional analysis and the exact solution are shown in Table 7.
22

Table 7: Percent Difference between 2-D Strict and Exact 1-D Solid – Liquid Interface Location
Direction Percent Difference x y
10.30 %
4.00 %
In attempts to implement the manual time step solution for the two – dimensional analysis for a .0001 time step, the computational power required by the system to solve this problem was not available in the computers available to perform this analysis. Due to this fact no results were able to be obtained for this solution set and all figures and calculations above have been created using the less accurate strict time step interval.
3.2.3
Discussion of Two-Dimensional Results
Due to the limitations caused by the computing power available to complete the analysis, no conclusive results were able to be obtained for the two – dimensional system being analyzed. For the results that were obtained using the less accurate strict time step interval, it can be shown that it is possible to obtain results that are realistic to the system being analyzed and the possibility that a system can be created where accurate results can be achieved using this method may be possible.
23

The result of this study into the use of the COMSOL Multiphysics in creating an accurate representation of the problem of heat conduction with phase change in a copper system is promising. While the range of this study was not enough to definitively show the high accuracy results that are required in many real world applications, the results show the potential for a use of this concept in creating high accuracy models.
The initial results of this study were able to show that by the introduction of two simple functions to define the thermal conductivity and most importantly the specific heat of copper, one was able to greatly increase the ability for COMSOL Multiphysics to predict the thermal behavior of copper as it solidifies. The introduction of these functions made possible to create a system where the latent heat of fusion was able to be accounted for as a function of time.
With additional modification of parameters and the availability of high power computers, work can be continued to optimize this concept and create a fully developed high accuracy model of the solidification of copper within COMSOL Multiphysics. In addition to more computational work, the use of testing to create a series of sample sets where the actual inputs and outputs of the system are known would greatly increase the validity of this method.
It has been shown in this study that the concept of altering the COMSOL
Multiphysics program moderately, can have a significant positive impact on the solution approximation that is created. It can be deduced that in further time it will be possible to create a heat transfer program that is able to properly define any phase change phenomenon within a system with the addition of a few new parameter inputs.
24

[1]
Lamberg, Piia, R. Lehtiniemi. “Numerical and Experimental Investigation of
Melting and Freezing Processes in Phase Change Material Storage.”
International Journal of Thermal Sciences . 200, Volume 43, pp. 277 - 287.
[2] Askeland, Donald. The Science and Engineering of Materials [ISBN
0534916570] . Boston: PWS-KENT, 1989.
[3] Carslaw, H.S. and J.C. Jaeger. Conduction of Heat in Solids [OCLC 535528] .
London: Clarendon Press, 1959.
[4] Butts, Allison. Copper: The Science and Technology of the Metal, Its Alloys and
Compounds [OCLC 2139600].
New York: Reinhold Publishing Corporation,
1954.
[5] Lewis, R.W., K. Morgan, and O.C. Zienkiewicz. Numerical Methods in Heat
Transfer [ISBN 0471906328] . New York: Wiley, 1981.
[6] Seetharamu, K.N., R. Paragasam, G. Quadir, Z. Zainal, B. Prasad, and T.
Sundarajan. “Finite Element Modeling of Solidification Phenomena.” Sadhana .
Volume 26, Parts 1&2, Febuary – April 2001, pp. 103 – 120. India: Springer,
2001.
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C Finite Difference Explicit Method for the 1D Heat Equation
C with Freezing - Enthalpy Method
C Material: Copper
C
C Find u(x,t) such that
C
C dH/dt = d( k du/dx)/dx
C x in [0,L] ; t > 0
C
C IC at t=0, x in [0,L]
C u = ui
C BC at x =0, t>0
C du/dx = dH/dx = 0
C BC at x=L, t>0
C Dirichlet: u = uw OR
C Newman: -k du/dx = qs OR
C Robin: -k du/dx = hconv*(u-uw)
C
parameter(nx=101) ! Use nx = odd
C
dimension u(nx),uo(nx),H(nx),Ho(nx)
real L,ks,kl,k(nx),kp,km,Ny
C
C System size, maximum time and initial temperature
L = 0.25
tmax = 30.0
ui = 1400.0
ur = 298.0
uw = 400.0
hconv = 3e3
C
C Thermophysical properties
ks = 365.0
kl = 174.0
rhos = 8920.0
rhol = 8920.0
Cps = 435.0
Cpl = 494.5
Hf = 205000.0*rhos
ufs = 1350.0
ufl = 1360.0
ufsl = 0.5*(ufs+ufl)
PI = 4.0*ATAN(1.0)
C
C Preliminaries
dt = 0.1 ! Trial; program checks fulfillment of CFL condition
dx = L/float(nx-1)
alphas = ks/(rhos*Cps)
alphal = kl/(rhol*Cpl)
alpha = AMAX1(alphas,alphal)
dtmax = (dx**2)/(4.0*alpha)
if(dt.gt.dtmax) dt=dtmax
nt = int(tmax/dt) + 1
31

C
Hs = rhos*Cps*(ufs-ur)
Hl = Hs + Hf
C
ipick1 = 1
ipick2 = nx/2+1
ipick3 = nx
C
C Exact Solution
C
DALAMB=0.00001
ALAMB=0.0
DO I=1,100000
ALAMB=ALAMB+DALAMB
FLHS=EXP(-ALAMB**2)/ERF1(ALAMB)
COEF1=kl*SQRT(alphas)/(ks*SQRT(alphal))
COEF2=(ui-ufsl)/(ufsl-uw)
COEF3=EXP(-alphas*(ALAMB**2)/alphal)
COEF4=1.0-ERF1(ALAMB*SQRT(alphas/alphal))
SLHS=-COEF1*COEF2*COEF3/COEF4
RHS=ALAMB*(Hf/rhos)*SQRT(PI)/(Cps*(ufsl-uw))
DIF=FLHS+SLHS-RHS
IF(DIF.LE.0.0001) GO TO 2 c write(6,*) FLHS,SLHS,RHS,DIF,ALAMB
ENDDO
2 CONTINUE
C
write(6,*) 'alamb =',ALAMB
C
C Initial Condition
C
x = 0
do i=1,nx
uo(i) = ui
if(uo(i).gt.ufl) then
k(i)=kl
Ho(i) = Hl + rhol*Cpl*(uo(i)-ufl)
endif
if(uo(i).le.ufl.and.uo(i).gt.ufs) then
k(i)=0.5*(kl+ks)
Ho(i) = Hs + Hf*(uo(i)-ufs)/(ufl-ufs)
endif
if(uo(i).le.ufs) then
k(i)=ks
Ho(i) = rhos*Cps*(uo(i)-ur)
endif
x = x + dx
enddo
C
C March forwards in time
C
t = 0.0
write(71,*) t,uo(ipick1)
write(72,*) t,uo(ipick2)
write(73,*) t,uo(ipick3)
C
do j=1,nt
32

t = t+dt
do i=2,nx-1
kp = 0.5*(k(i+1)+k(i))
km = 0.5*(k(i)+k(i-1))
dqdxp = -kp*(uo(i+1)- uo(i))/dx
dqdxm = -km*(uo(i)- uo(i-1))/dx
H(i) = Ho(i) - dt*(dqdxp-dqdxm)/dx
enddo
C
C BC at x = 0
H(1) = H(2)
C BC at x = L (Choose one; comment the other two)
C Dirichlet Condition
uo(nx) = uw
if(uo(nx).gt.ufl) then
H(nx) = Hl + rhol*Cpl*(uo(nx)-ufl)
endif
if(uo(nx).lt.ufs) then
Ho(nx) = rhos*Cps*(uo(nx)-ur)
endif
if(uo(nx).le.ufl.and.uo(nx).ge.ufs) then
Ho(nx) = Hs + Hf*(uo(nx)-ufs)/(ufl-ufs)
endif
C Newman condition c qout = qs(t) c qin = - k(nx)*(uo(nx)-uo(nx-1))/dx c H(nx) = Ho(nx)-(2.0*dt/dx)*(qout - qin)
C Robin Condition c qout = hconv*(uo(nx) - uw) c qin = - k(nx)*(uo(nx)-uo(nx-1))/dx c H(nx) = Ho(nx)-(2.0*dt/dx)*(qout - qin)
C
C Compute new temperature
C
do i=1,nx
if(H(i).gt.Hl) then
u(i) = ufl + (H(i)-Hl)/(rhol*Cpl)
k(i) = kl
endif
if(H(i).le.Hl.and.H(i).gt.Hs) then
u(i) = ufs+(ufl-ufs)*((H(i)-Hs)/Hf)
k(i) = 0.5*(kl+ks)
endif
if(H(i).le.Hs) then
u(i) = ur + Ho(i)/(rhos*Cps)
k(i) = ks
endif
enddo
C
if(uo(ipick2).gt.ufl.and.u(ipick2).le.ufl) then
write(6,*) 'Liquidus Crossed at x=L/2 at time =', t
endif
if(uo(ipick2).gt.ufs.and.u(ipick2).le.ufs) then
write(6,*) 'Solidus Crossed at x=L/2 at time =', t
endif
C
do i=1,nx-1
33

dudx=(u(i+1)-u(i))/dx
dudt=(u(i)-uo(i))/dt
Ny=dudx/sqrt(dudt)
write(77,*) t,float(i)*dx/2,Ny
enddo
C
do i=1,nx
write(76,*) t,float(i-1)*dx,u(i)
uo(i) = u(i)
Ho(i) = H(i)
enddo
C
C Track Interface
C
do i=nx,2,-1
if(i.eq.nx.and.u(nx).gt.ufl) then
write(90,*) t,0.0
endif
if(i.lt.nx.and.u(i).le.ufl.and.u(i-1).gt.ufl) then
xliq = L-0.5*(float(i)+float(i-1))*dx
write(90,*) t,xliq
endif c
if(i.eq.nx.and.u(nx).gt.ufsl) then
write(91,*) t,0.0
endif
if(i.lt.nx.and.u(i).le.ufsl.and.u(i-1).gt.ufsl) then
xsl = L-0.5*(float(i)+float(i-1))*dx
write(91,*) t,xsl
endif c
if(i.eq.nx.and.u(nx).gt.ufs) then
write(92,*) t,0.0
endif
if(i.lt.nx.and.u(i).le.ufs.and.u(i-1).gt.ufs) then
xsol = L-0.5*(float(i)+float(i-1))*dx
write(92,*) t,xsol
endif c
enddo
if(u(1).lt.ufl.and.u(2).lt.ufl) write(90,*) t,L
if(u(1).lt.ufsl.and.u(2).lt.ufsl) write(91,*) t,L
if(u(1).lt.ufs.and.u(2).lt.ufs) write(92,*) t,L
C
xslex = 2.0*ALAMB*sqrt(alphas*t)
write(80,*) t,xslex
C
write(71,*) t,u(ipick1)
write(72,*) t,u(ipick2)
write(73,*) t,u(ipick3)
if(u(nx).lt.ur.or.u(nx).gt.ui) go to 100
enddo
100 continue
C
write(6,*) 'Temperatures at x=0, x=L/2 and x=L at t=',tmax
write(6,*) u(ipick1),u(ipick2),u(ipick3)
C
34

write(6,*) 'Interface Location (exact and FDM) at t=',tmax
write(6,*) xslex,xsl
C
stop
end
FUNCTION qs(t)
if(t.ge.0.0) qs = 3e6
RETURN
END
FUNCTION ERF1(ARG)
A1=0.0705230784
A2=0.0422820123
A3=0.0092705272
A4=0.0001520143
A5=0.0002765672
A6=0.0000430638
C
ATX=A1*ARG
ATX=ATX+A2*ARG**2
ATX=ATX+A3*ARG**3
ATX=ATX+A4*ARG**4
ATX=ATX+A5*ARG**5
ATX=ATX+A6*ARG**6
C
ERF1=1.0-1.0/(1.0+ATX)**16
C
RETURN
END
35