Chapter 4: Heat and Mass Transfer in Bulk Materials

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Chapter 4 Heat and Mass Transfer in Bulk
Materials

• What You Will Learn
• Simple Heat Transfer in Fluids
• Heat transfer coefficient
• Numerical Solutions of convective cooling
• Heat Conduction in Metals
• Heat diffusion equation
• Boundary conditions
• Numerical simulation of heat diffusion
• Radiation Heat Transfer in Ceramics
• Radiation boundary conditions
• Numerical solutions radiation heating
• Mass Transfer in Alloys
• Numerical integration of Ficks Laws in 2D
• Application to homogenization

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Consider a cup of coffee
4.1 Simple Heat Transfer in Fluids
T Air
air currents
heat loss surface
T(t)
insulating surface

• How does cooling proceed?
• Air currents (convection) carry away heat from
  the mug
• Assume (for now) that the coffee cup adjusts its
  temperature instantaneously throughout ?more
  rapidly than the cooling rate
• Driving force

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Heat Transfer Coefficient
Heat loss from cup heat gained by Air
Heat transfer coefficient
K
m3
m2
J/Km2s
k/S
J/kgK
kg/m3
Units LHS J/S
RHS J/S
r
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Numerical Solutions Making Time Discrete

• Lets pretend we dont know how to solve equation
• Use numerical methods
• Break time into discrete steps t 0, ?t, 2?t,
  n?t, n 0, 1, 2

• define T(t) ? T(n?t), n 0,1, 2
• for simplicity represent T(n?t) ? Tn
• T1, T2, T3, ?Temperature at discrete time
  increments

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Finite Difference Approximation of Time Derivative
T(t)
One-sided derivative

• Substitute discrete time derivative into cooling
  equation

Discrete form of cooling equation
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Euler Time Marching Scheme

• Re-arranging discrete gives

Euler time stepping method
prediction at previous time
prediction at next time

• Start with

Generate temperature sequence
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Program Algorithm i.e. A Plan of Attack For
Solving the Discrete Cooling Equation Numerically
input value of heat transfer coefficient r
input value of initial coffee temperature
To input value of discrete time step ?t
input value of Air temperature TAir
input
increment discrete time step n from
1?N_max temp change -r ?t (Tn-TAir)
update Tn1 Tn change print (n1)
?t, Tn1 pair to a file Repeat iteration
Repeat for maximum time
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A Working F90 Code for Coffee Cooling
Program coffee_cool !Define variables real 8
r,T0, ?t, T_air, change Integer n,
steps steps 1000 ! Set maximum number of
time steps print , enter heat transfer
coefficient r read , r print, enter
initial cup and air temperatures read,
T0,Tair print, enter discrete time step read,
?t TT0 !initialize
temperature to initial temperature !update
temperature Do n1, steps TT ?t
r(T-Tair) print, n?t ,T and do End
program coffee_cool
Review more detailed code in Directory
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Numerical Simulation of Temperature vs. Time in
a Coffee Cup
T0
T(t)
Tair
Time

• Eventually cup cools to room temperature
• Temperature evolution depends only on r and Tair
  

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Comparison With Exact (Analytical) Solution of ODE

• We are lucky in this case to know T(t) exactly!

Numerical
Exact
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The Effect of Changing Time Step
?t15 ?t21 ?t30.1

• Results stop changing (converge) as ?t ? 0
• Can never make ?t ? 0 exactly! Why?
• Choose ?t small enough to give results
  independent of choice of ?t

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Exact Solution of Discrete Numerical Equation

• In this case, we can also solve the discrete
  equation exactly!
• Original equation

1
call this

• Assume instability of the form

Plug into
1

• simplify (i.e. equate exponential terms and
  constant terms separately)

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Equating Algebraic Expressions

• In general when comparing two algebraic
  expressions, we compare coefficients
  corresponding to like-powers
• are equal
• Similarly when an expression contains exp, sin,
  cos, ln we compare their corresponding terms
  separately
• E.g.
• are equal
  

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Exact Solution of Discrete Numerical Equation

• Equating like terms we obtain

• Use initial condition to find A

Exact numerical solution
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Maximum Bound on Time Step

• Numerical Stability requires that exponential go
  to zero as

Always stable in this range
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4.2 Heat Conduction in Metals

• Consider a solid, cylindrical bar as shown

Insulated surface
insulation
Fixed at 00 C
1000 C
Fixed at 00 C
L
End surfaces Maintained at 0 c

• Aim Determine the temperature evolution in the
  bar as a function of material properties

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Fouriers Law of Heat Conduction
T(x)

• Begin with Fouriers law of heat flow

-dT/dX
-dT/dX
K/m
X
J/m2S
J/mKs Thermal conductivity
heat diffuses down hill
Generalization to 3D
z
y
heat flow in each direction of space
x
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Co-ordinate systems

• Using x, y, z is inconvenient to describe
  cylindrical geometry
• Special co-ordinates known as cylindrical
  co-ordinates were invented specifically to
  describe cylinders!

z
r
r, ? , z (much easier!)
Instead of x, y, z
T
y
x
By symmetry since the bar is insulated
flow along z-directionally!
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Heat Balance Equation

• How does heat flow?

Q(z)
volume of cylinder
Accumulation of heat
Heat change depends on
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Conservation of Heat Equation

• Further simplification gives

in the limit of incrementally tiny volume
and time increments
Fundamental statement of the conservation of heat
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Heat Diffusion Equation (One Dimensional)

• We need a relation between Q and Temperature
• Fouriers Law

Constitutive relation between T Q

• Substitute into heat conservation equation

• If conductivity is constant can define a
  diffusion coefficient

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Boundary Conditions on Temperature Field

• Boundary conditions must be applied in order to
  be able to find a consistent solution of the heat
  equation
• They come in two types
• Dirichlet ?specifying the temperature at the
  surfaces
• Neuman ?specifying the derivatives (fluxes) at
  the surfaces
• Mixed ?Specifying both temperature on (separate)
  parts of the surface

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Numerical Simulation of the Diffusion Equation

• Can use the diffusion equations to compute
  temperature distribution in bar
• Analytical ? Simple geometry and B.Cs
• Complex B.Cs ? Numerical solutions
• Proceeding numerically
• Break up the real world into discrete points
  ?mesh

Temperature snap-shot at discrete points in space
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Discrete Space Time

• Numerical temperature defined as a discrete
  sequence in space
• Numerical temperature sequence also changes with
  time!
• Discrete time
• Thus, we must maintain and store temperature on
  discrete space
• points (mesh) and discrete time intervals

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Initial temp sequence
Temperature Evolution in Discrete Space-Time

• Discrete co-ordinate notation

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Finite Difference Approximation of Time Derivative

• For time gradients, expand temperature in a
  Tailor series about the

Called a forward difference approximation of the
time derivative
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Finite Difference Approximation of First Order
Spatial Derivative

• Now begin with the Tailor series in space, about

This is also a forward difference approximation
of a spatial derivative
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Finite Difference Approximation of Second Order
Derivatives

• For spatial gradient Consider a Tailor series to
  the left and right of x

• Add the two equations

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Discrete First and Second Order Spatial
derivatives

• In discrete co-ordinates first derivatives become

• While the second derivative becomes

• Both become exact in the limit at

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Discrete Form of Heat Diffusion Equation

• Combining all this messy math and numerical
  approximations we obtain

• Explicit time stepping scheme for the diffusion
  equation
• Spatial derivatives referenced with
  respect to old time step n

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Iterative Update of the Temperature Field

• Rearranging last equation gives
• Values of temp at a future time(e.g. )
  depend on values of temp at the previous
  increment of time( )

Called point-wise if each i is updated
separately
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Implementing Fixed Temperature Boundary
Conditions Numerically

• Application of Dirichlet (fixed temperature)
  boundary conditions
• Update Eq. (4) for on nodes
• Use fixed temperatures
  at end nodes

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Implementing Fixed Flux Boundary Conditions
Numerically

• Neuman Boundary conditions
• E.g. assume zero derivative at end points
• Update Eq. (4) for on nodes
• Set to zero the one-sided derivative at
  endpoints

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Initialization of temp
An Algorithm for Simulating Heat Diffusion
Equation
Program Heat_Conduction
Definition of variables
Set boundary temps
Time evolution loop
End Program
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Review of F90 Code For Computing Temperature
Conduction in a Metal Bar
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Simulations of Thermal Conduction in a Metal Bar
Initial temperature
Fixed temperature
Fixed temperature
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Stability of Numerical Solution

• Explicit time integration only stable for certain
  ranges
• For 1-D heat equation can Solve discrete Eq. (4)
  exactly (with fixed boundaries)
• Stability requires

how did we get this??
If you make longer than this on the
computer what happens?