macquarie university 20041 the heat equation and diffusion phys220 2004 by lesa moore department of...
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Macquarie University 2004 1
The Heat Equation and Diffusion
PHYS220 2004by Lesa Moore
DEPARTMENT OF PHYSICS
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Diffusion of Heat
The diffusion of heat through a material such as solid metal is governed by the heat equation.
We will not try to derive this equation.
We will compare results from the heat equation with our studies of the random walk.
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Initial Temperature Distribution
-3 -2 -1 0 1 2 3 x
Consider diffusion in 1D (let a thin copper wire represent a one-dimensional lattice).
Let u(t,x) be the heat at point x at time t, with x and t integers, u(t=0,x=0)=1 and u(t=0,x)=0 if x is not zero.
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The Partial Differential Equation The heat equation is a partial
differential equation (PDE):
k is the diffusion coefficient. Assume the initial distribution is a spike
at x=0 and is zero elsewhere.
2
2
x
uk
t
u
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Partial Derivatives For functions of more than one
variable, the partial derivative is the rate of change with respect to one variable with the other variable(s) fixed.
:
:
:
t
xtuxttuxt
t
ut
),(),(lim),(
0
x
xtuxxtuxt
x
ux
),(),(lim),(
0
x
xtxuxxtxuxt
x
ux
),)(/(),)(/(lim),(
02
2
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The PDE in full
:
:
:
x
xxxtuxtu
xxtuxxtu
kt
xtuxttuxt
),(),(),(),(
lim),(),(
lim00
x
xtxuxxtxuk
t
xtuxttuxt
),)(/(),)(/(lim
),(),(lim
00
2
2
x
uk
t
u
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Converting to a Difference Equation Don’t take the limits as intervals
approach zero. Take finite time steps (t=1) and finite
positions steps (x=1).
x
xxxtuxtu
xxtuxxtu
kt
xtuxttuxt
),(),(),(),(
lim),(),(
lim00
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Simplifying …
x
xxxtuxtu
xxtuxxtu
kt
xtuxttuxt
),(),(),(),(
lim),(),(
lim00
11
1
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Rearranging …
)1,(),(),()1,(),(),1( xtuxtuxtuxtukxtuxtu
),()1,(),(2)1,(),1( xtuxtuxtuxtukxtu
Want all t+1 terms on l.h.s. and everything else on r.h.s.
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Modelling in Excel Columns are x-values. Rows are t-values.
The difference equation relates each cell to three cells in the row above.
),()1,(),(2)1,(),1( xtuxtuxtuxtukxtu
1234
Y Z AA AB AC-2 -1 0 1 2
u(t,x-1) u(t,x) u(t,x+1)u(t+1,x)
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The Excel Spreadsheet The first row (t=0) is all zeros except
for the initial spike: u(t=0,x=0) = 1. The same formula is entered in every
cell from row 2 down: A1 holds the value of k (k = 0.1) AA3=$A$1*(Z2-2*AA2+AB2)+AA2
12345
X Y Z AA AB AC AD-3 -2 -1 0 1 2 3
0 0 0 1 0 0 00 0 0.1 0.8 0.1 0 00 0.01 0.16 0.66 0.16 0.01 0
0.001 0.024 0.195 0.56 0.195 0.024 0.001
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Filling the Spreadsheet
In Excel, it is easiest to insert the formula in the top left cell of the range, select the range and use Ctrl+R, Ctrl+D to fill the range: -20 ≤ x ≤ 20; 0 ≤ t ≤ 60.
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Boundary Conditions What happens at the boundaries? Setting columns at x=±21 equal to
zero stops the spatial evolution of the model – is this a problem? Provided that values in neighbouring
columns (x=±20) are still small at the end of the simulation, the choice of boundary conditions is not so important.
u=0 is equivalent to an absorbing boundary.
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SnapshotsSpread of Heat in 1D: t = 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-20 -15 -10 -5 0 5 10 15 20
Space (x) units
Mea
sure
of
hea
t
Spread of Heat in 1D: t = 11
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-20 -15 -10 -5 0 5 10 15 20
Space (x) units
Mea
sure
of
hea
t
Spread of Heat in 1D: t = 51
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-20 -15 -10 -5 0 5 10 15 20
Space (x) units
Mea
sure
of
hea
t
Spread of Heat in 1D: t = 3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-20 -15 -10 -5 0 5 10 15 20
Space (x) units
Mea
sure
of
hea
t
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Plotting the Heat SpreadSpread of Heat in 1D
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-20 -15 -10 -5 0 5 10 15 20
Space (x) units
Mea
sure
of
hea
t
t=1
t=11
t=21
t=31
t=51
t=81
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Spreadsheet Results Conservation of heat can be
demonstrated by adding the values in a row (a row is a time step). Values in a row should add to 1.
Checking the sum in a row is good test of numerical accuracy.
Heat diffusion looks like a Gaussian distribution.
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The Distribution
The simulation satisfies conservation of energy (total heat along a row = 1).
Does the Gaussian distribution satisfy this condition too (area under curve = 1)? The initial spike can be thought of as a
very sharp, very narrow Gaussian. For t>0, need to integrate the Gaussian. “Normalised” if integral yields unity.
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Normalisation of the Gaussian Formula for Gaussian with = 0.
Use a trick for the integral:
2)(
22 2/xexf
dxdyyfxfdxxf )()()(
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The integral becomes
dxdyeedx
e yxx
2222
22
2/2/2/
2
1
2
dxdye yx 222 2/)(
2
1
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But using and
222 ryx
0
2
0
rdrddxdy
2
0 0
2/ 22
2
1)( drreddxxf r
0
2/ 22
22
1drre r
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Cancelling
0
2/ 22
22
1)( drredxxf r
0
2/ 221drre r
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Then use the substitution:
dvrdr
drrdv
rv
)/(
)/(
2/
2
2
22
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And finally:
1
1
0
0
0
2
v
v
v
e
dve
dvr
re
dx
e x
2
22 2/
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The integral proves that the Gaussian is normalised to unity – the area under the curve is one.
But the heat equation is a function of x and t, and uses a constant k.
k and t must be included in the term of the Gaussian if we are to say our model satisfies this distribution.
2)(
22 2/xexf
2
2
x
uk
t
u
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What is ? From the Random Walk, we learned
that √t. Try a guess: The Gaussian becomes:
kt2
kt
etxf
ktx
4),(
4/2
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Derivatives of the Gaussian Space derivatives:
The time derivative is left as an exercise …
fkt
x
ktx
f
fkt
x
x
f
122
1
22
2
2
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The Gaussian satisfies the Heat Equation It can be shown
that the heat equation is satisfied by our guess.
2
2
x
uk
t
u
The distribution integrates to unity (conservation of energy).
The spread of heat is given by of the Gaussian (normal) distribution.
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Diffusion and the Random Walk
The initial temperature spike grows into a Gaussian distribution according to the 1D heat equation.
The width grows in proportion to the square root of elapsed time.
Heat and diffusion can be understood in terms of the “random walk”.
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Other Conditions
The initial condition may not be a spike, but could be some initial distribution: u(x,0)=g(x).
The boundary conditions may not be absorbing, but could be continuous.
The thermal diffusivity constant k may not be constant, but may vary with x or t.
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Summary The heat equation is a PDE. By separating space and time variables,
we see that a Gaussian that spreads as √t is a solution.
We can model the differential equation as a difference equation in Excel and see the same effect.
The spread of heat is a physical example of a random walk.
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Acknowledgements
This presentation was based on lecture material for PHYS220 presented by Prof. Barry Sanders, 2000-2003.
Additional Reference: Folland, Fourier Analysis and its
Applications, 1992.