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Chapter 4: Heat and Mass Transfer in Bulk Materials

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Take-home message: Simple Heat Transfer in Fluids Heat transfer coefficient Finite difference method for time derivatives Heat Conduction in Metals – PowerPoint PPT presentation

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Title: Chapter 4: Heat and Mass Transfer in Bulk Materials

1
Chapter 4 Heat and Mass Transfer in Bulk
Materials
• Take-home message
• Simple Heat Transfer in Fluids
• Heat transfer coefficient
• Finite difference method for time derivatives
• Heat Conduction in Metals
• Heat diffusion equation
• Boundary conditions
• Simple numerical solutions of diffusion equation
• Radiation Heat Transfer in Ceramics
• Mass Transfer in Alloys
• Numerical integration of Ficks Laws in 2D
• Application to homogenization

2
4.3 Radiation Heat Transfer in Ceramics
• Consider a green ceramic slab placed into a high
temperature furnace for sintering at 2000K
• Surfaces heated by radiation heat transfer from
walls of furnace

XL
Initial bar at temp
X0
Line of symmetry
• Aim Calculate temperature distribution in

3
At xL heat flux exchange between radiation
heat conduction
Heat radiated to surface of bar
Heat conducted into bar
4
Interior of ceramic bar conducts heat according
to diffusion equation
Conduction and Boundary Conditions in the
Interior of Bar
• At x0, symmetry requires fluxes match

Heat diffusion is a function of position and
temperature
5
Numerical Solution of Temperature Evolution
• Explicit integration method on a 1-D mesh

i0 i1
IN-1 iN
• In the simple case of a constant diffusion
coefficient

boundary conditions

6
Explicit Integration Scheme with Non-Constant
Conductivity
• Finite difference method with non-constant
diffusion
• Where we define, for ease of exposition

7
Explicit Integration Scheme with Non-constant
conductivity
• Point-wise iteration for each node in the
interior nodes

8
Application to Zero flux boundary conditions x0
• Begin with Eq. (1) applied to node
• Symmetry (flux to left-flux to right, think of a
mirror!) requires

9
Application to Radiation Boundary Conditions at
xL
• Begin with Eq. (1) applied to node

10
Material and Numerical Parameters of Problem
11
Numerical Simulation on a Simple 4-Node Mesh
• Noting that for the initial temperature

Initial temp at 4 nodes
1st temperature update at 5 nodes simultaneously
Can iterate this repeatedly!
12
An Algorithm for Computing Temperature
Distribution
Define variables
Initialize temp
Space loop
Time loop
13
Review F90 Code for Radiation Heating
14
Simulation of Temperature Distribution in A
Increasing time
15
4.4- Mass Transfer and Homogenization in Alloys
• Solidification of phase (1st order)
• Grain morphology
• spacing dependent on
• cooling rate

Adapted from Fig. 9.9, Callister 6e
16
Dendrites and Solute Micro-Segregation
solute trace of final alpha phase dendrite
grains
y
x
• Initial a-phase solute concentration distribution
C(x,y,t0) f(x,y) (i.e. some given function,
obtained from micrographs, for example)
• Non-uniformity is know as micro-segregation
• Periodicity on all sides ?mimics situation
whereby alloy repeats in the x and y direction

17
Heat Treatment of Single Phase Alloys
• Find Alloy all in a-phase
• Composition highly inhomogeneous due to
micro-segregation
• Heat treatment used to
• Homogenize its composition
• Grow precipitate particles

18
Modeling Mass Transfer in Homogenization
• Ficks Law
• Boundary conditions
• Initial conditions
• Diffusion constant D(T) drives solute
re-distribution
• high T?diffusion of solute
• Low T?locks in solute segregation

19
Numerical Integration of Ficks Law
• Divide domain into NXN cells
• Finite difference of derivatives

N 2 1 0
0 1 2 . . . . . N
Nodes are labeled by double index(i,j)
Treat interior nodes and boundary nodes separately
20
Updating the Interior Nodes
Point-wise explicit time marching based on
concentrations at previous time step.
21
Updating Boundary nodes Top surface
Drop n for now
By symmetry
22
Updating Boundary nodes Top surface
23
Updating Boundary NodesBottom Surface
By symmetry
24
Update Boundary Nodes Right Surface
By symmetry
25
Update Boundary NodesLeft Surface
By symmetry
26
Corner Boundary Nodes Top/Bottom-Left
i0 ,jN
i0,j0
27
Corner Boundary Nodes Top/Bottom-Right
iN, JN
iN,j0
28
Algorithm for Homogenization Code
integers i,j real8 arrays
DER(1,N-1) C(i,j)f(i,j) for time1,Nmax
increment time by for i1,N-1
for j1,N-1 Apply Interior
Node Update end end
for j1,N-1 Left/Right Surface
Update (i0 N) end for for
i1,N-1 Top/bot Surface Update
(j0,N) end run
Define variables
update 0,0 node update 0,N node
update N,0 node update N,N node print
concentration array at specified times END time
loop
Initialize C
Time loop
29
Review F90 Program to Compute Homogenization
30
Homogenization of Ni-Cr Dendritic Structures
• Try a simple 4X416 node mesh

Initial concentration array Laid out as a single
column
Integrate Ficks Law forward in time
track solute diffusion
31
Time Evolution of Solute in a Simple Phase Alloy
32
Residual Segregation Index
• We track the progress of homogenization via the
residual segregation index
• What we expect to see

Require a minimum value before phase can be
considered homogeneous
time