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Course SD 2225

Heat transfer by conductionin a 2D metallic plate

Pau Mallol, Georgios Spanopoulos, Alan Vargas KTH, April 2008

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Physical Background• Heat transfer: thermal energy in transit due to a

spatial temperature difference within/between media.

• Modes of heat transfer:

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Differential Equation• The equation that governs the process is:

Heat sources

Convection with air

Radiation

• Assumptions: - no heat sources in plate - no convection - no radiation - constant conduction thermal conductivity k

02

2

2

2

y

T

x

TT

Poisson’s Equation

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Boundary Conditions• One side is thermally insulated, whereas the

rest kept at a certain constant temperature

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Meshing• 3 meshes with both COMSOL and MATLAB

a) 19 x 13 b) 49 x 31 c) 124 x 76

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COMSOL: Resolution & Results

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100LENGTHWISE TEMPERATURE Y=0.1 COMSOL

X [m]

T [º

C]

Coarse 13x19

Medium 31x49Fine 76x124

COMSOL: Resolution & Results

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COARSE19 X 13

MEDIUM49 X 31

FINE124 X 76

Number of elements

247 1519 9424

ComputingTime [s]

0.094 0.157 0.939

BenchmarkTemperature [oC]Point (0.2,0.6)m

67.347493 67.346702 67.346667

COMSOL: Resolution & Results

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MATLAB: Discretization• DG: using same stepsize h in both directions

• DD: 2nd order Finite Difference Method

11

h

Nh

M6.0

2

11

2

2

2

11

2

2

2),(

2),(

h

TTTyx

y

T

h

TTTyx

x

T

ijijijii

jiijjiii

02

2

2

2

y

T

x

TT

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• DD (cont.): discretized DE

04 1111 ijijjijiij TTTTT

• DB: 1st and 2nd order Finite Difference Method

0),( 1

h

TTyx

y

T ijijii

02

),( 11

h

TTyx

y

T ijijii

1st order

2nd order

ijij TT 1

11 ijij TT

MATLAB: Discretization

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MATLAB: Linear Sytems of Eq.

• Analytical 2D problem results to be 1D problem after discretization.

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• Elliptic DE has been reduced to a linear system of MxN EQUATIONS to be solved.

• There are MxN UNKNOWNS, the discretized temperatures in all points of the grid.

• STIFFNESS & STABILITY ?

MATLAB: Linear Sytems of Eq.

• System is of very SPARSE nature -> treat it this way to save computational effort.

ACOARSE19 X 13

MEDIUM49 X 31

FINE124 X 76

λMAX -0.0381 -0.0064 -0.0016

λMIN -7.9215 -7.9861 -7.9964

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MATLAB: Resolution & ResultsTEMPERATURE DISTRIBUTION

20 40 60 80 100 120

10

20

30

40

50

60

70

0

10

20

30

40

50

60

70

80

90

100

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MATLAB: Resolution & Results

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100LENGTHWISE TEMPERATURE Y=0.1 MATLAB

X [m]

T [

ºC]

Coarse 13x19

Medium 31x49Fine 76x124

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COMSOL & MATLAB: comparison

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100LENGTHWISE TEMPERATURE Y=0.1 COARSE

X [m]

T [

ºC]

COMSOL

MATLAB

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100LENGTHWISE TEMPERATURE Y=0.1 MEDIUM

X [m]

T [

ºC]

COMSOL

MATLAB

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100LENGTHWISE TEMPERATURE Y=0.1 FINE

X [m]

T [

ºC]

COMSOL

MATLAB • COMSOL insensitive to mesh fineness.

• MATLAB depends strongly upon mesh fineness -> ACCURACY

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COMSOL & MATLAB: comparison

• COMSOL is more efficient with big systems.

COARSE19 X 13

MEDIUM49 X 31

FINE124 X 76

Number of elements

247 1519 9424

COMSOLTime [s] 0.094 0.157 0.939

MATLABTime [s]

0.041 0.082 1.826

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Conclusions

• Max/Min temperatures not consistent in COMSOL (depend on mesh); MATLAB is OK.

• COMSOL: easier, faster, more accurate and efficient than MATLAB.

• But COMSOL is particular use and MATLAB offers infinite possibilities (general).

• STABILITY: numerical systems to these PDE’s are always stable, no matter what h.

• ACCURACY: in COMSOL does not depend on h, in MATLAB strongly depends on h -> limitation: backward slash operator A\b size of A limited to about 10000.