ROBLEM: IVD-39 BOOK: ENGINEERING THERMOFLUIDS, M. MASSOUD

Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
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PROBLEM: IVD-39

BOOK: ENGINEERING THERMOFLUIDS, M. MASSOUD

39. Show that the steady state temperature distribution in a cylindrical spine is the
solution to:
d²e(x) _ (m? + m? p(x) = 0
dx2
where m. = (Ph JkA)0³, and m, = (Ph,/kA)05. Use the given data to find the tem-
perature distribution at steady state in the above fin. Data: L= 15 cm, D = 1 cm,
TB = 350 C, T; = 27 C, k = 50 W/m-C, h. = 15 W/m².C, ɛ= 0.75. The tip of the fin
is losing energy by both radiation and convection.
[Hint: You may solve this problem analytically or numerically. The analytical
solution follows the method described in Section 8 of Chapter IVa. Since we have
linearized the differential equation by choosing the heat transfer coefficient, as
shown in Problem 38, an iterative solution is required regardless of the solution
method we choose (h,, therefore, m, are unknowns). The initial guess to estimate
h, for thermal radiation is obtained by ignoring thermal radiation (i.e., by setting
m, = 0). Upon obtaining the initial guess for temperature, we then include thermal
radiation but evaluate h, at the temperature calculated in the previous iteration. At
the end of each trial, we find:
%D
%D
Tk - Tk-1
E =
where k is the iteration index. The iteration is terminated when ɛ <ɛ, where &, is
the specified convergence criterion].
Transcribed Image Text:39. Show that the steady state temperature distribution in a cylindrical spine is the solution to: d²e(x) _ (m? + m? p(x) = 0 dx2 where m. = (Ph JkA)0³, and m, = (Ph,/kA)05. Use the given data to find the tem- perature distribution at steady state in the above fin. Data: L= 15 cm, D = 1 cm, TB = 350 C, T; = 27 C, k = 50 W/m-C, h. = 15 W/m².C, ɛ= 0.75. The tip of the fin is losing energy by both radiation and convection. [Hint: You may solve this problem analytically or numerically. The analytical solution follows the method described in Section 8 of Chapter IVa. Since we have linearized the differential equation by choosing the heat transfer coefficient, as shown in Problem 38, an iterative solution is required regardless of the solution method we choose (h,, therefore, m, are unknowns). The initial guess to estimate h, for thermal radiation is obtained by ignoring thermal radiation (i.e., by setting m, = 0). Upon obtaining the initial guess for temperature, we then include thermal radiation but evaluate h, at the temperature calculated in the previous iteration. At the end of each trial, we find: %D %D Tk - Tk-1 E = where k is the iteration index. The iteration is terminated when ɛ <ɛ, where &, is the specified convergence criterion].
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