Chemical Engineering CH EN 3453 Heat Transfer from University of Utah, 2014 Fall Homework HW 02 Solutions

2. PROBLEM 2.23
KNOWN: Temperature distribution in a one-dimensional wall with prescribed thickness and thermal
conductivity.
FIND: (a) The heat generation rate, q, in the wall, (b) Heat fluxes at the wall faces and relation to q.
SCHEMATIC:
ASSUMPTIONS: (1) Steady-state conditions, (2) One-dimensional heat flow, (3) Constant
properties.
ANALYSIS: (a) The appropriate form of the heat equation for steady-state, one-dimensional
conditions with constant properties is Eq. 2.19 re-written as
q=-k d dT
⎡ ⎤
⎢⎣ ⎥⎦
dx dx
Substituting the prescribed temperature distribution,
q=-k d d (a+bx2 ) k d [2bx] 2bk
⎡ ⎤ = − = − ⎢⎣ ⎥⎦
dx dx dx
q=-2(-2000DC/m2 )×50 W/m⋅K=2.0×105 W/m3.
(b) The heat fluxes at the wall faces can be evaluated from Fourier’s law,
q x k dT .
′′ x ( )
= − ⎤⎥⎦
x
dx
Using the temperature distribution T(x) to evaluate the gradient, find
q x k d a+bx 2kbx.
( ) 2
x
′′ = − ⎡ ⎤ = − ⎢⎣ ⎥⎦
dx
The fluxes at x = 0 and x = L are then
q′x′ (0) = 0
( ) ( 2 )
q′x′ L = −2kbL=-2×50W/m⋅K -2000 C/m ×0.050m D
( ) 2
q′x′ L =10,000 W/m .
COMMENTS: From an overall energy balance on the wall, it follows that, for a unit area,
( ) ( )
− + = ′′ − ′′ +
′′ − ′′
−
E in E out E g 0 q x 0 q x
L qL=0
q x ( L ) q ( 0 )
q= x = 10,000 W/m 2 0 = 2.0 ×
10 5 W/m 3
.
L 0.050m
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