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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.