## 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')

0 0