2d Heat Conduction

Anoop Pinjala 11110016 TWO DIEMENSIONAL HEAT CONDUCTION (ELECTRICAL ANALOG)

AIM: To measure voltages at different nodes of two dimensional electrical resistance grid and compare it with the temperature predicted by 2D heat conduction method.

APPARARUS: 1) 2-D Electrical resistance Grid 2) Stable D.C power supply (9 Volts) 3) Digital Multimeter of 0.01 volts accuracy for measuring electrical voltage.

THEORY: In 1-D steady state heat conduction the controlling equation was found to be:d2T/dx2 =0 In an isotropic material the same is true in the direction at right angles to the x-direction, so we can deduce that the controlling equation for 2-D steady state conduction is:d2T/dx2 + d2T/dy2 =0 The equation for 2-D steady conditions can be solved using finite difference techniques. Consider the area to be divided up by a rectangular grid where the grid lines are spaced 'a' apart

1

a 4

0

2

3

a

The temperature gradient between any two nodes is given by T0-T4/a between 4 and 0 T2-T0/a between 2 and 0 Therefore at 0 d2T/dx2 = [( T2-T0/a) - (T0-T4/a)]/a => d2T/dx2 = ( T2+T4- 2T0)/a2 Also d2T/dy2= ( T1+T3- 2T0)/a2 therefore d2T/dx2 + d2T/dy2 = ( T1+T2- +T3+T4- 4T0)/a2 = 0  T0 = (T1+T2+T3+T4)/4 Thus, initially some temperature profile will be considered and then iterations will be carried out until temperature profile becomes constant. In the grid if a constant voltage of 9V is given across inner (i) and outer (o) surfaces then for the iteration at the middle  Tm = (Tm-1+Tm+1+Ti+To)/4 And at the corner it will be

 Tc = 2(Tc-1+To)/4

OBSERVATIONS: Voltage supplied across the resistance grid

= 10.1 V

OBSERVATION TABLE: Point

10

11

12

13

14

15

16

17

7.62

7.75

8.58

7.65

7.40

7.12

6.53

4.83

7.62

7.37

7.13

7.28

7.29

7.12

6.51

4.83

0

-4.92

-19.44

-4.83

-1.45

0.00

-0.31

0

number Voltage measured Voltage predicted %error

Point

20

21

22

23

24

25

26

27

28

4.86

4.62

3.77

4.59

4.67

4.51

3.99

3.09

2.05

4.86

4.90

4.95

4.88

4.82

4.53

4.24

3.15

2.05

%error

0

6.12

31.18

6.39

3.24

-0.83

6.34

1.83

0

Point number

30

number Voltage measured Voltage predicted

31

32

33

34

35

36

37

38

39

Voltage 2.36 measured

2.29

2.08

2.22

2.26

2.17

1.92

1.51

1.03

0.51

Voltage predicted

2.36

2.26

2.07

2.23

2.27

2.19

1.92

1.51

1.03

0.57

%error

0

-1.24

-0.68

0.56

0.60

1.00

0.17

0.10

0.28

0

Voltage vs nodal point for interpolating the voltage values at odd nodes.

RESULT:



As temperature is an analogy to the voltage , we can say that the temperature at the nodes (10,11,12,13,14,15) and (20,21,22,23,24,25) and (30,31,32,33,34,35) are almost constant and the temperature at the nodes (16,17) and (26,27,28) and (36,37,38,39) decreases.

DISCUSSION:

The temperature is almost constant in the nodes (10,11,12,13,14,15) and (20,21,22,23,24,25) and (30,31,32,33,34,35) because the molecular arrangement is linear in that grid resistance up to those nodes and at the nodes (16,17) and (26,27,28) and (36,37,38,39) the molecular arrangement is not linear and hence the rate of heat transfer decreases and hence the temperature at the corner of the grid decreases. We can conclude that heat conduction depends on the shape. Moreover there might a chances of experimental in case when you do not make a right contact on the node with voltage measuring device. CONCLUSION: The variation of temperature along the X and Y axis is analogous to the variation of voltage.