An Example Heat Conduction in a Brass
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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