An Example Heat Conduction in a Brass

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Consider an insulated 10-cm brass rod, initially at a temperature of 0°C. One end of the rod is heated to 100°C. Equation 12-20 describes the heat flow in the rod as a function of time. (For simplicity, we assume that there is no heat loss through the sides of the rod.) For brass, the coefficient of thermal conductivity k is 0.26 cal s~! cm-1 deg~', the heat capacity c is 0.094 cal g_1 deg-1 and the density p is 8.4 g cm-3. From these values, the coefficient k in equation 12-22 is 3.04 s cm"2. Figure 12-5 shows part of the spreadsheet used to calculate the temperature along the rod, in 1-second and 1-cm intervals. The table extends to t = 100 seconds (row 113).

A

B C

D

E ! F

G

H

I

J

K

L

M

N

1 J

Time-dependent Temperature Distribution in a Brass Rod

2

{Temperature values in bold are constant)

3

length, cm

I L ,.L...J

j

10

4

heat capacity of brass, cal/g/deg

________L A J

0.1

(heap)

5

thermal conductivity of brass, oal/secfcrctfdeg

0.3

00 (rho)

6

density of brass, g/cm3

j

8,4

7

Coefficient e in general PDE, =k.'(hcap':rho)

0,3

Ce)

8

Ùx

F

1

(Dx)

9

a

1

(Dt)

10

f=e*DtiCDxA2)

0.3

(0

Distance x (cm)

12

0 1 2 3 4 5 6 7

8 9 10

13

0

160 0

0 0

o

0

0 0

0 I 0

o

14

Î 3

100 32.9

0.0 10.8

0.Ü

0.0

0.0

0.0

0,0

OX) I 0.0

0

15"

100

44.2

00

0.0

0.0

0.0

0.0

D.O 0.0

0

16

100

51.6

16 2 3.6

0.0

0.0

0.0 : 0.0

0.0

0.0

0

_______

Ï7

£ 4

100

56.5

244 7.2

1.2

0.0

0.0

0.0

0.0

0.0

0

18

,§ 6

100

50.3

29.3 10 9

2.8

0.4

0.0

0.0

0.0

0,0

0

19

100

63.2

33.4

14.3

4,7

1.0

0.1

0.0

0.0

G.O

0

TO"

7

100

65.5

36.9

17.4

6.6

1.9

0.4

0.0

0.0

0.0

0

21

8 9

100

67.5

39.9 20.3

8.6

3.0

0.8

0.1

Oil

0.0

0

22

100

69.1

42.5 22.9

10.6

4.1

1.3 I 0.3

0,1

0,0

a

23

10

100

70,5

44.0:25.3

12 5

5.3

1.9 0.5

0.1

0.0

«

Figure 12-5. Calculation of heat flow in a brass rod. The text in cells M4:M10 are the names assigned to the cells L4:L10. (folder 'Chapter 12 (PDE) Examples, workbook 'Parabolic PDE', sheet 'Temp distribution')

Figure 12-5. Calculation of heat flow in a brass rod. The text in cells M4:M10 are the names assigned to the cells L4:L10. (folder 'Chapter 12 (PDE) Examples, workbook 'Parabolic PDE', sheet 'Temp distribution')

Cells K3:K9 contain constants used in the calculations; these cells were assigned the names shown in parentheses in column M. The formulas in cells K6, K7, K8 and K9 are, respectively

[In the spreadsheet, the range name f was used for the parameter r in equation 12-26, since r can't be used as a name in Excel.]

The values in cells on the edges of the table of temperatures (column C and column M) are the constant temperature values at the ends of the rod; the values in row 13 are the initial temperature of the interior of the rod. The formula in the remaining cells in the body of the temperature table (D14:L113) is based on equation 12-22. For example, the formula in cell D14 is

Experience has shown that the factor/must be less than 1/2 in order to avoid instability in the calculations. For a given problem, this requires adjustment of both Ax and At.

=k/(hcap*rho) (coefficient k in general PDE, equation 12-22)

Figure 12-6. Temperature vs. time in a brass rod. (folder 'Chapter 12 (PDE) Examples, workbook 'Parabolic PDE', sheet 'Temp distribution')

0 20 40 60 80 100 Time, seconds

Figure 12-6. Temperature vs. time in a brass rod. (folder 'Chapter 12 (PDE) Examples, workbook 'Parabolic PDE', sheet 'Temp distribution')

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