Fundamentals of Heat and Mass Transfer
7th Edition
ISBN: 9780470917855
Author: Bergman, Theodore L./
Publisher: John Wiley & Sons Inc
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Chapter 2, Problem 2.51P
A spherical shell of inner and outer radii
Are conditions steady-state or transient? How do the heat flux and heat rate vary with radius?
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In this question, we are concerned with the evolution of the temperature u(x, t) in a
homogeneous thin heat conducting rod of length L = 1. We can consider that the rod
is laterally insulated as to have a one-dimensional problem. The evolution of the temperature
is governed by the one-dimensional heat equation
ди
0 0
= K
Ət
Əx2'
Assume that this equation is subject to the following initial conditions
u(x,0) = f(x)
and boundary conditions
(0, t) = 0 and
ди
(1,t) + и(1,t) — 0
(i) Discuss briefly the physical meaning of the boundary conditions.
0
k(T) = k₂(1+B7)
Plane
wall
L X
Example:-Consider a plane wall of
thickness L whose thermal conductivity
varies linearly in a specified temperature
range as
K(T) =k₁ (1+BT)
where k, and B are constants.
The wall surface at x=0 is
maintained at a constant temperature
1
of T₁ while the surface at x =L is
maintained at T2, as shown in Figure .
Assuming steady one-dimensional
heat transfer, obtain a relation for:-
(a) the heat transfer rate through the wall..
(b) the temperature distribution T(x) in the
wall.
Consider a solid sphere of radius R with a fixed surface temperature, TR. Heat is generated within
the solid at a rate per unit volume given by q = ₁ + ₂r; where ₁ and ₂ are constants.
(a) Assuming constant thermal conductivity, use the conduction equation to derive an expression
for the steady-state temperature profile, T(r), in the sphere.
(b) Calculate the temperature at the center of the sphere for the following parameter values:
R=3 m 1₁-20 W/m³ TR-20 °C k-0.5 W/(m K) ₂-10 W/m³
Chapter 2 Solutions
Fundamentals of Heat and Mass Transfer
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