TRANSIENT HEAT CONDUCTION IN A SOLIDIFYING ALLOY
ABSTRACT
Alloys are combinations of two or more metals with the purpose of improving its mechanical
and chemical properties, but majority of the time when combining these metals; a diversion from
the supposed result is experienced. This diversion was majorly seen to occur during the process
of solidification of the alloy, thereby making it paramount to study the solidification process of
alloys since it differs from the solidification process of pure metals. The objective of this project
is to study the temperature distribution in a solidifying alloy over a range of time and to also plot
a graph of temperature against time.
In the course of studying the previous work done on same subjects and related projects, a
discovery of how pertinent the situation is and how there hasn‟t been a major breakthrough in the
subject. A lot of work has been done on the derivation of the heat flux and heat transfer
coefficient of various alloys ranging from AL-SI alloys to MN-CU alloys. The three phases
during solidification has also been studied; the solidus region, the liquidus region and the mushy
region which is the region in-between the solid and the liquid region also fondly called the
problem domain. The mass composition in this region has also been studied
The method adopted for this process is the finite difference approximation using the implicit
method where the time steps that can be used in the calculations are unlimited. This
mathematical modeling procedure involves the discretization of the length into nodal lengths at
varying time steps. The higher the nodes the more accurate the results obtained will be. In this
project, 50 nodes are used.
The result gotten shows a graph depicting the cooling pattern of alloys and suggested points of
phase change which varies from one node to the other. This result will help in guiding
experimental results that would be obtained during the process of carrying out this project
experimentally.
TABLE OF CONTENT
Title page…………………………………………………………………………..i
Certification……………………………………………………………………….ii
Dedication…………………………………………………………………………iii
Acknowledgement…………………………………………………………………iv
Table of content……………………………………………………………………v
Nomenclature……………………………………………………………………..vi
List of figures……………………………………………………………………..vii
List of tables…………………………………………………………………......viii
Abstract…………………………………………………………………………..ix
Chapter 1
Introduction…………………………………………………………………………1
1.1 Problem statement……………………………………………………………… 2
1.2 Objective…………………………………………………………………………3
1.3 Justification………………………………………………………………………3
Chapter 2
2.0 Literature review………………………………………………………………..5
5
2.1 Finite difference approximation………………………………………………10
Chapter 3
Methodology………………………………………………………………………12
Chapter 4
Result and discussion………………………………………………………………51
Chapter 5
Conclusion and recommendation…………………………………………………..57
References…………………………………………………………………………..59
Appendix……………………………………………………………………………60
CHAPTER 1
INTRODUCTION
The process of solidification in alloys has come under great scrutiny due to the consistent flaws
and alteration of the solid alloy created. In majority of the cases, the alloys intended where
different from the one created. In electronic packaging, lead–tin alloy is frequently used to join
electronic components. During usage, the lead–tin alloy usually undergoes a reflow process
which includes spreading, remelting, and then solidification of the alloy. Therefore, the
properties of lead–tin alloy solder joint are altered and in turn the success of electronic packaging
will be significantly affected by the reflow process.
Heat, a form of energy that moves from one body of higher gradient to the other of lower heat
gradient is used as a study parameter and a correctional tool. Heat is an important concept in
engineering and its application in manufacturing is of grave importance such that much study has
been put to understand the concept of heat transfer. Heat is usually added to a system or removed
from the system and in phase transformation the heat added or removed is called latent heat. The
three modes of heat transfer are conduction, convection and radiation.
Solidification which is also a phase transformation process occurs when latent heat is ejected
from the liquid state. During solidification process some factors determine the outcome of the
process which is a basis of constant research to engineers in order to improve the process of
solidification. These factors include the temperature gradient, growth rate, material type and
component. Heat transfer during Solidification in pure metals occur at a steady rate while in
alloys, heat transfer is not constant and varies with time and position. This unsteady state heat
conduction is known as transient heat transfer.
Heat transfer in liquid, gas or solid phase can be easily analyzed with various formulas such as
the Fourier‟s laws of conduction and convection. In contrast to this single phase heat transfer
analysis, multiple phase heat conduction is complex due to presence of solid and liquid phases
separated by solid liquid interface called the mushy phase.
An exact estimation of the heat transfer during the liquid alloy solidification depends on the
determination of the boundary conditions during solidification, property of the alloy and the
temperature distribution in the alloy and for the purpose of accurate solidification modeling, the
correct boundary conditions should be established.
1.1 PROBLEM STATEMENT
A cylindrical mold, shown, is charged with a molten alloy of metals A and B at an initial
temperature of 204 of the liquid alloy. The mold is well insulated except at one end which is
maintained at solidus temperature of 64 by a stream of oil. Obtain the temperature distribution
during solidification.
Many a times in the manufacturing industries, production is based on heat conduction and
solidification. The quality of the products produced are inferior and are short of the expected
qualities intended for such products to have due to the inability to control the rate heat is
conducted during production. This occurrence has made it paramount for such industries to
embark on researches that will stipulate the appropriate quantity of heat to be supplied in order to
yield the best possible product. Numerous researches have been carried out and mathematical
models have been developed such as the Neumann‟s solution to the one dimensional problem of
solidification. These solutions do not take into consideration of the effect of wall conduction,
wall resistance and height of closure. This solution only predicts the solidification rate and
temperature distribution for controlling the speed of fabrication.
In this project, the finite element difference is used to bridge the gap of uncertainty around the
Neumann‟s solution and a more accurate prediction is produced.
1.2 OBJECTIVE
Pure metals solidify at a steady/constant temperature while alloys solidify over a range of
temperature. The objective of the project is to obtain the graph of temperature distribution
against time during the cooling process with the use of finite difference approximation for a one
dimensional multi-phase cooling of a solidifying alloy.
1.3 JUSTIFICATION
Excess heat conduction in an alloy undergoing solidification distorts the molecular composition
of the alloy thereby making the expected result unfeasible. In practice, it has been established
that research work has to be embarked on for a better product. Therefore heat conduction and the
rate of heat conduction into the process of solidification of alloys especially needs to be studied
vigorously due to the anomalous behave or of alloys for the betterment of products.
.