2.40 PROBLEM (Cont.)
q ro, z = 16 W / m K 2 150C / m 1 m 12C /1 m = 4608 W / m r
()
2
2
< <
q r ( ro ) = A r q ro, z r
()
2
where
A r = 2 ro ( 2z o )
q r ( ro ) = 4 1 m 2.5 m 4608 W / m = 144, 765 W
Note that the sign of the heat flux and heat rate in the positive r-direction is negative, and hence the heat flow is into the cylinder. (d) The heat fluxes and the heat rates at end faces, z = + zo and zo, may be calculated using Fouriers law. The direction of the heat rate in or out of the end face is determined by the sign of the heat flux in the positive z-direction.
At the upper end face, z = + zo:
q ( r, + z o ) = k z T z z o
<
= k [ 0 + 0 + 0 + 2dz o ]
q ( r, + z o ) = 16 W / m K 2 300C / m z q z ( + z o ) = A z q ( r, + z o ) z q z ( + z o ) = 1 0.2
2
(
2
(
2
)m
) 2.5 m = +24, 000 W / m where A = (r
2 z 2
<
2 2 o ri
)
< <
2
24, 000 W / m = +72, 382 W
Thus, heat flows out of the cylinder. At the lower end face, z = - zo:
q ( r, z o ) = k z T z
= k [ 0 + 0 + 0 + 2d( z o ) ] zo
2 2
q ( r, z o ) = 16 W / m K 2 ( 300C / m )( 2.5 m ) = 24, 000 W / m z
q z ( z o ) = 72, 382 W
< <
Again, heat flows out of the cylinder. (e) The heat rates from the surfaces and the volumetric heat generation can be related through an overall energy balance the on cylinder as shown in the sketch.
Continued
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PROBLEM 2-40 (Conti.)
E in E out + E gen = 0 where E gen = q = 0
E in = q r ( ro ) = ( 144, 765 W ) = +144, 765 W
< <
E out = + q z ( z o ) q z ( z o ) = [ 72, 382 ( 72, 382 )] W = +144, 764 W
The overall energy balance is satisfied.
COMMENTS: When using Fouriers law, the heat flux q denotes the heat flux in the positive zz direction. At a boundary, the sign of the numerical value will determine whether heat is flowing into or out of the boundary.
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful.