4. (a) Let the ends of a copper rod 100 cm long be maintained at 0°C.Suppose that the center of the bar is heated to 75°C by an ex-ternal heat source and that this situation is maintained until asteady-state results. Find this steady-state temperature distri-bution.(b) At a time t0 (after the steady-state of part a has been reached),let the heat source be removed. At the same instant, let the end= 0 be placed in thermal contact with a reservoir at 10°C, whilethe other end remains at 0°C. Find the temperature as a functionof position and time.X ==(c) Plot u versus x for several values of t. Also plot u versus t forseveral values of x.(d) What limiting value does the temperature at the center of the rodapproach after a long time? How much time must elapse beforethe center of the rod cools to within 1°C of its limiting value?

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(a) Let the ends of a copper rod 100 cm long be maintained at 0°C. Suppose that the center of the bar is heated to 75°C by an ex- ternal heat source and that this situation is maintained until a steady-state results. Find this steady-state temperature distri- bution. (b) At a time t = 0 (after the steady-state of part a has been reached), let the heat source be removed. At the same instant, let the end x = 0 be placed in thermal contact with a reservoir at 10°C, while the other end remains at 0°C. Find the temperature as a function of position and time. (c) Plot u versus x for several values of t. Also plot u versus t for several values of x. (d) What limiting value does the temperature at the center of the rod approach after a long time? How much time must elapse before the center of the rod cools to within 1°C of its limiting value?

(a) Let the ends of a copper rod 100 cm long be maintained at 0°C. Suppose that the center of the bar is heated to 75°C by an ex- ternal heat source and that this situation is maintained until a steady-state results. Find this steady-state temperature distri- bution. (b) At a time t = 0 (after the steady-state of part a has been reached), let the heat source be removed. At the same instant, let the end x = 0 be placed in thermal contact with a reservoir at 10°C, while the other end remains at 0°C. Find the temperature as a function of position and time. (c) Plot u versus x for several values of t. Also plot u versus t for several values of x. (d) What limiting value does the temperature at the center of the rod approach after a long time? How much time must elapse before the center of the rod cools to within 1°C of its limiting value?

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Concept explainers

Mechanisms of heat transfer

Heat transfer is an essential discipline in thermal engineering that deals with the transfer, conversion, usability, and creation of heat or thermal energy. Transfer of heat occurs through various mechanisms like conduction, convection, radiation, and change of phases. All these mechanisms have distinct characteristics, but these mechanisms can simultaneously occur in an identical system.

Question

4. (a) Let the ends of a copper rod 100 cm long be maintained at 0°C.
Suppose that the center of the bar is heated to 75°C by an ex-
ternal heat source and that this situation is maintained until a
steady-state results. Find this steady-state temperature distri-
bution.
(b) At a time t
0 (after the steady-state of part a has been reached),
let the heat source be removed. At the same instant, let the end
= 0 be placed in thermal contact with a reservoir at 10°C, while
the other end remains at 0°C. Find the temperature as a function
of position and time.
X =
=
(c) Plot u versus x for several values of t. Also plot u versus t for
several values of x.
(d) What limiting value does the temperature at the center of the rod
approach after a long time? How much time must elapse before
the center of the rod cools to within 1°C of its limiting value?

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