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  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


                                    CHAPTER 1


   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.


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.


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.


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. 

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