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8/7/2019 Everyday Heat Transfer
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EVERYDAY HEATTRANSFER
PROBLEMSSensitivities To
Governing Variables
by M. Kemal Atesmen
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2009 by ASME, Three Park Avenue, New York, NY 10016, USA (www.asme.org)
All rights reserved. Printed in the United States of America. Except as permitted
under the United States Copyright Act of 1976, no part of this publication may be
reproduced or distributed in any form or by any means, or stored in a database or
retrieval system, without the prior written permission of the publisher.
INFORMATION CONTAINED IN THIS WORK HAS BEEN OBTAINED BY THE
AMERICAN SOCIETY OF MECHANICAL ENGINEERS FROM SOURCES BELIEVED
TO BE RELIABLE. HOWEVER, NEITHER ASME NOR ITS AUTHORS OR EDITORS
GUARANTEE THE ACCURACY OR COMPLETENESS OF ANY INFORMATIONPUBLISHED IN THIS WORK. NEITHER ASME NOR ITS AUTHORS AND EDITORS
SHALL BE RESPONSIBLE FOR ANY ERRORS, OMISSIONS, OR DAMAGES ARISING
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THE UNDERSTANDING THAT ASME AND ITS AUTHORS AND EDITORS ARE
SUPPLYING INFORMATION BUT ARE NOT ATTEMPTING TO RENDER
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ASME shall not be responsible for statements or opinions advanced in papers
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For authorization to photocopy material for internal or personal use under those
circumstances not falling within the fair use provisions of the Copyright Act, contact
the Copyright Clearance Center (CCC), 222 Rosewood Drive, Danvers, MA 01923,
tel: 978-750-8400, www.copyright.com.
Library of Congress Cataloging-in-Publication Data
Atesmen, M. Kemal.
Everyday heat transfer problems : sensitivities to governing variables / by M. Kemal
Atesmen.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-7918-0283-0
1. HeatTransmissionProblems, exercises, etc. 2. MaterialsThermal properties
Problems, exercises, etc. 3. Thermal conductivityProblems, exercises, etc.
4. Engineering mathematicsProblems, exercises, etc. I. Title.
TA418.54.A47 2009
621.4022dc22 2008047423
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iii
Introduction ................................................................................................... 1
Chapter 1 Heat Loss from Walls in a Typical House ............................. 5
Chapter 2 Conduction Heat Transfer in a Printed Circuit Board.... 13
Chapter 3 Heat Transfer from Combustion Chamber Walls.............. 25
Chapter 4 Heat Transfer from a Human Body During
Solar Tanning ............................................................................ 33
Chapter 5 Effi ciency of Rectangular Fins.............................................. 41
Chapter 6 Heat Transfer from a Hot Drawn Bar.................................. 51
Chapter 7 Maximum Current in an Open-Air Electrical Wire .......... 65
Chapter 8 Evaporation of Liquid Nitrogen in aCryogenic Bottle ...................................................................... 77
Chapter 9 Thermal Stress in a Pipe ........................................................ 85
Chapter 10 Heat Transfer in a Pipe with Uniform Heat
Generation in its Walls ......................................................... 93
Chapter 11 Heat Transfer in an Active Infrared Sensor .................. 103
Chapter 12 Cooling of a Chip ................................................................. 113
ABLE OF CONTENTST
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Everyday Heat Transfer Problems
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Chapter 13 Cooling of a Chip Utilizing a Heat Sink
with Rectangular Fins......................................................... 121
Chapter 14 Heat Transfer Analysis for Cooking in a Pot ................ 131
Chapter 15 Insulating a Water Pipe from Freezing........................... 139
Chapter 16 Quenching of Steel Balls in Air Flow .............................. 147
Chapter 17 Quenching of Steel Balls in Oil......................................... 155
Chapter 18 Cooking Time for Turkey in an Oven .............................. 161
Chapter 19 Heat Generated in Pipe Flows due to Friction............. 169
Chapter 20 Sizing an Active Solar Collector for a Pool................... 179
Chapter 21 Heat Transfer in a Heat Exchanger ................................. 195
Chapter 22 Ice Formation on a Lake .................................................... 203
Chapter 23 Solidifi cation in a Casting Mold ....................................... 213
Chapter 24 Average Temperature Rise in Sliding Surfaces
in Contact .............................................................................. 221
References .................................................................................................... 233
Index.............................................................................................................. 235
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1
Everyday engineering problems in heat transfer can be very
complicated and may require solutions using finite element or finite
difference techniques in transient mode and in multiple dimensions.
These engineering problems might cover conduction, convection and
radiation energy transfer mechanisms. The thermophysical properties
that govern a particular heat transfer problem can be challenging to
discover, to say the least.
Some of the standard thermophysical properties needed to
solve a heat transfer problem are density, specific heat at constantpressure, thermal conductivity, viscosity, volumetric thermal
expansion coefficient, heat of vaporization, surface tension, emissivity,
absorptivity, and transmissivity. These thermophysical properties
can be strong functions of temperature, pressure, surface roughness,
wavelength and other properties. in the region of interest.
Once a heat transfer problem's assumptions are made, equations
set up and boundary conditions determined, one should investigate
the sensitivities of desired outputs to all the governing independentvariables. Since these sensitivities are mostly non-linear, one should
NTRODUCTIONI
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Everyday Heat Transfer Problems
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analyze them in the region of interest. The results of such sensitivity
analyses will provide important information as to which independent
variables should be researched thoroughly, determined accurately,and focused on. The sensitivity analysis will also provide insight into
uncertainty analysis for the dependent variable, (Reference S. J. Kline
and F. A. McClintock [9]). If the dependent variable y is defined as a
function of independent variables x1, x2, x3, xn as follows:
y = f(x1, x2, x3, xn)
then the uncertainty U for the dependent variable can bewritten as:
U = [(y/x1 u1)2 + (y/x2 u2)2 + (y/x3 u3)2 + + (y/xn un)2]0.5
where y/x1, y/x2, y/x3, , y/xn are the sensitivities of the
dependent variable to each independent variable and u1, u2, u3, ,
un are the uncertainties in each independent variable for a desired
confidence limit.
In this book, I will provide sensitivity analyses to well-known
everyday heat transfer problems, determining y/x1, y/x2,
y/x3, , y/xn for each case. The analysis for each problem will
narrow the field of independent variables that should be focused
on during the design process. Since most heat transfer problems
are non-linear, the results presented here would be applicable only
in the region of values assumed for independent variables. For theuncertainties of independent variablesfor example, experimental
measurements of thermophysical propertiesthe reader can find the
appropriate uncertainty value for a desired confidence limit within
existing literature on the topic.
Each chapter will analyze a different one-dimensional heat transfer
problem. These problems will vary from determining the maximum
allowable current in an open-air electrical wire to cooking a turkey
in a convection oven. The equations and boundary conditions foreach problem will be provided, but the focus will be on the sensitivity
of the governing dependant variable on the changing independent
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Introduction
3
variables. For the derivation of the fundamental heat transfer
equations and for insight into the appropriate boundary conditions,
the reader should refer to the heat transfer fundamentals books listedin the references.
Problems in Chapters 1 through 6 deal with steady-state and
one-dimensional heat transfer mechanisms in rectangular coordinates.
Chapters 7 through 10 deal with steady-state and one-dimensional
heat transfer mechanisms in cylindrical coordinates. Unsteady-state
problems in one-dimensional rectangular coordinates will be tackled
in Chapters 11 through 14, cylindrical coordinates in Chapter 15, and
spherical coordinates in Chapters 16 through 18.The following six chapters are allocated to special heat transfer
problems. Chapters 19 and 20 deal with momentum, mass and heat
transfer analogies used to solve the problems. Chapter 21 analyzes
a counterflow heat exchanger using the log mean temperature
difference method. Chapters 22 and 23 solve heat transfer problems
of ice formation and solidification with moving boundary conditions.
Chapter 24 analyzes the problem of frictional heating of materials in
contact with moving sources of heat.
I would like to thank my engineering colleagues G. W. Hodge,
A. Z. Basbuyuk, E. O. Atesmen, and S. S. Tukel for reviewing some of
the chapters. I would also like to dedicate this book to my excellent
teachers and mentors in heat transfer at several universities and
organizations. Some of the names at the top of a long list are
Prof. W. M. Kays, Prof. A. L. London, Prof. R. D. Haberstroh,
Prof. L. V. Baldwin, and Prof. T. N. Veziroglu.
M. Kemal Atesmen
Ph. D. Mechanical Engineering
Santa Barbara, California
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5
C H A P T E R
Heat loss from the vertical walls of a house is analyzed under
steady-state conditions. Walls are assumed to be large and
built in a planar fashion, so that one-dimensional heat transfer
rate equations in rectangular coordinates may be used, and only
conduction and convection heat transfer mechanisms are considered.
In this analysis, radiation heat transfer effects are neglected. No air
leakage through the wall was assumed. Also, the wall material thermal
conductivities are assumed to be independent of temperature in the
region of operation.Assuming winter conditionsthe temperature inside the house is
higher than the temperature outside the housethe convection heat
transferred from the inside of the house to the inner surface of the
inner wall is:
Q/A = hin (Tin Tinner wall inside surface) (1-1)
Most walls are constructed from three types of materials: inner wall
board, insulation and outer wall board. The heat transfer from thesewall layers will occur by conduction, and is presented by the following
rate Eqs., (1-2) through (1-4):
1
1EAT LOSS
FROM WALLS
IN A TYPICAL HOUSE
H
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Q/A = (kinner wall/tinner wall) (Tinner wall inside surface Tinner wall outside surface) (1-2)
Q/A = (kinsulation/tinsulation) (Tinner wall outside surface Touter wall inside surface) (1-3)
Q/A = (kouter wall/touter wall) (Touter wall inside surface Touter wall outside surface) (1-4)
The heat transfer from the outer surface of the outer wall to the
atmosphere is by convection and can be expressed by the following
rate Eq. (1-5):
Q/A = hout (Tout Touter wall outer surface) (1-5)
Eliminating all the wall temperatures from Eqs. (1-1) through (1-5),
the heat loss from a house wall can be rewritten as:
Q/A = (Tin Tout)/[(1/hin) + (tinner wall/kinner wall) + (tinsulation/kinsulation)
+ (touter wall/kouter wall) + (1/hout)] (1-6)
The denominator in Eq. (1-6) represents all the thermal resistancesbetween the inside of the house and the atmosphere, and they are
in series.
In the construction industry, wall materials are rated with their
R-value, namely the thermal conduction resistance of one-inch
material. R-value dimensions are given as (hr-ft2-F/BTU)(1/in). The
sensitivity analysis will be done in the English system of units rather
than the International System (SI units). The governing Eq. (1-6)
for heat loss from a house wall can be rewritten in terms of R-values
as follows:
Q/A = (Tin Tout)/[(1/hin) + Rinner wall tinner wall + Rinsulation tinsulation
+ Router wall touter wall + (1/hout)] (1-7)
where the definitions of the variables with their assumed nominal
values for the present sensitivity analysis are given as:
Q/A = heat loss through the wall due to convection and conduction
in Btu/hr-ft2
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Heat Loss From Walls In A Typical House
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Tin = 68F (inside temperature)
Tout = 32F (outside temperature)
hin = 5 BTU/hr-ft2
-F (inside convection heat transfer coefficient)Rinner wall = 0.85 hr-ft2-F/BTU-in (wall inside board R-value)
tinner wall = 1 in (wall inside board thickness)
Rinsulation = 3.5 hr-ft2-F/BTU-in (insulation layer R-value)
tinsulation = 4 in (insulation layer thickness)
Router wall = 5 hr-ft2-F/BTU-in (wall outside board R-value)
touter wall = 1 in (wall outside board thickness)
hout = 10 BTU/hr-ft2-F (outside convection heat transfer coefficient).
The heat loss through a wall due to changes in convection heat
transfer is presented in Figures 1-1 and 1-2. Changes in the convection
heat transfer coefficient affect the heat loss mainly in the natural
convection regime. As the convection heat transfer coefficient increases
into the forced convection regime, heat loss value asymptotes.
Resistances from both inside and outside convection heat transfer are
too small to cause any change in heat loss through the wall.
The heat loss through a wall due to changes in insulation materialR-value is presented in Figures 1-3 and 1-4. Higher R-value insulation
1.75
1.76
1.77
1.78
1.79
1.8
5 10 15 20 25
Outside Convection Heat Transfer Coefficient, Btu/hr-ft2-F
Q/A,Btu/hr-
ft2
Figure 1-1 Wall heat loss versus outside convection heat transfer coefficient
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Everyday Heat Transfer Problems
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material is definitely the way to go, depending upon the cost and
benefit analysis results. The thickness of the insulation material is alsovery crucial. Thicker insulation material is definitely the best choice,
depending upon the cost and benefit analysis results.
1.72
1.74
1.76
1.78
1.8
1 5 9
Inside Convection Heat Transfer Coefficient, Btu/hr-ft2-F
Q/A,Btu/hr-ft2
3 7
Figure 1-2 Wall heat loss versus inside convection heat transfer coefficient
1.2
1.4
1.6
1.8
2
3 3.5 4 4.5 5
Insulation "R" Value, hr-ft
2
-F/Btu-in
Q/A,Btu/hr-ft2
Figure 1-3 Wall heat loss versus insulation R-value
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Heat Loss From Walls In A Typical House
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The effects on heat loss of inner and outer wall board R-values
and thicknesses are similar to the effects of insulation R-value and
thickness, but to a lesser extent. Sensitivities of heat loss to all the
governing variables around the nominal values given above will be
analyzed later.
Sensitivity of heat loss to the outside convection heat transfer
coefficient can be determined in a closed form by differentiating the
heat loss Eq. (1-7) with respect to hout:
(Q/A)/
hout = (Tin Tout)/{h
2
out [(1/hin) + Rinner wall tinner wall+ Rinsulation tinsulation + Router wall touter wall + (1/hout)]2} (1-8)
Sensitivity of heat loss to the outside convection heat transfer
coefficient is given in Figure 1-5. Similar sensitivity is experienced
for the inside convection heat transfer coefficient. The sensitivity
of heat loss to the convection heat transfer coefficient is high in
the natural convection regime, and it diminishes in the forced
convection regime.Sensitivities of heat loss to insulation material R-value and
insulation thickness are given in Figures 1-6 and 1-7 respectively.
0
2
4
6
0 2 4 6 8 10 12
Insulation Thickness, in
Q/A,Btu/hr-ft2
Figure 1-4 Wall heat loss versus insulation thickness
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0
0.03
0.06
0.09
0 5 10 15 20
Outside Convection Heat Transfer Coefficient,Btu/hr-ft2-F
(Q/A)/hout,F
2.5
2
1.5
1
0.5
0
0 2 5 6
Insulation Material R-Value, hr-ft2-F/BTU-in
(Q/A)/Rinsulation,
(BTU/hr-ft2)2(in/F)
3 41
Figure 1-5 Sensitivity of house wall heat loss per unit area to the outside
convection heat transfer coefficient
Figure 1-6 Sensitivity of house wall heat loss per unit area to insulation
material R-value
These two sensitivities are similar, as can be expected, since the
linear product of insulation material R-value and insulation thickness
affects the heat loss, as shown in the governing heat loss Eq. (1-7).
Absolute sensitivity values are high at the low values of insulation
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Heat Loss From Walls In A Typical House
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1.5
1
0.5
0
0 10 12Insulation Thickness, in
(Q/A)/tinsulation,BTU/hr
-ft2-in
2 4 68
Figure 1-7 Sensitivity of house wall heat loss per unit area to insulation
thickness
material R-value and insulation thickness. Sensitivities approach
zero as insulation material R-value and insulation thickness values
increase.
A ten-percent variation in independent variables around the
nominal values given above produces the sensitivity results given in
Table 1-1 The sensitivity results are given in a descending order and
they are applicable only in the region of assigned nominal values,
due to their non-linear effect to heat loss. The one exception istemperature potential, (Tin Tout), which will always be 10% due
to its linear effect on heat loss. Material R-value and its thickness
change have the same sensitivity, since their linear product affects the
governing heat loss equation.
Heat loss through the wall is most sensitive to the temperature
potential between the inside and outside of the house. Changes in
wall insulation R-value and thickness affect heat loss as much as
the temperature potential. Continuing in order of sensitivity, wallouter board R-value and thickness changes affect heat loss the most,
followed by wall inside board R-value and thickness. Wall heat loss is
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Everyday Heat Transfer Problems
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least sensitive to both the inside and outside heat transfer coefficient
changes. Wall heat loss sensitivity to both the inside and outside
heat transfer coefficient changes is an order of magnitude less than
sensitivity to temperature potential changes.
Table 1-1 House wall heat loss change per unit area due to a10% change in variables nominal values
House Wall House Wall
Heat Loss Heat Loss
Change Due To Change Due To
A 10% A 10%
Nominal Decrease In Increase In
Variable Value Nominal Value Nominal Value
Tin Tout 36F 10% +10%
Rinsulation 3.5 hr-ft2
-F/BTU-in +7.467% 6.497%tinsulation 4 in +7.467% 6.497%
Router wall 5 hr-ft2-F/BTU-in +2.545% 2.545%
touter wall 1 in +2.545% 2.545%
Rinner wall 0.85 hr-ft2-F/BTU-in +0.424% 0.424%
tinner wall 1 in +0.424% 0.424%
hin 5 BTU/hr-ft2-F 0.110% +0.090%
hout 10 BTU/hr-ft2-F 0.055% +0.045%
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C H A P T E R
Conduction heat transfer in printed circuit boards (PCBs) has been
studied extensively in literature i.e., B. Guenin [4]. The layered
structure of a printed circuit board is treated using two different
thermal conductivities; one is in-plane thermal conductivity and the
other is through-thickness thermal conductivity. One-dimensional
conduction heat transfers in in-plane direction and through-thickness
direction are treated independently. Since the significant portion of
the conduction heat transfer in a PCB occurs in the in-plane direction
in the conductor layers, this is a valid assumption. Under steady-stateconditions and with constant thermophysical properties, the in-plane
(i-p) conduction heat transfer equation for a PCB can be written as:
Qin-plane = Q1i-p + Q2i-p + + Qni-p (2-1)
where the subscript refers to the layers of the PCB. Using the
conduction rate equation in rectangular coordinates for a PCB with
a width of W, a length of L, layer thicknesses ti and layer thermalconductivities ki, Eq. (2-1) can be rewritten as:
2
2ONDUCTION
HEAT TRANSFER
IN A PRINTED
CIRCUIT BOARD
C
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W ti kin-plane (TL=0 TL=L)/L = W t1 k1 (TL=0 TL=L)/L
+ W t2 k2 (TL=0 TL=L)/L
+ W tn kn (TL=0 TL=L)/L (2-2)
In-plane conduction heat transfer in a PCB represents a parallel
thermal resistance circuit which can be written as:
(1/Rin-plane) = (1/R1) + (1/R2) + + (1/Rn) where Ri = L/(kitiW) (2-3)
where
kin-plane = (kiti)/ti (2-4)
Through-thickness (t-t) conduction heat transfer in a PCB represents
a series thermal resistance circuit, and the through-thickness conduction
heat transfer equation for a PCB can be written as:
Qthrough-thickness = Q1t-t = Q2t-t = = Qnt-t (2-5)
which can be expanded into following equations:
W L kthrough-thickness (Tt=0 Tt=ti)/ti = W L k1 (Tt=0 Tt=t1)/t1
= W L k2 (Tt=t1 Tt=t2)/t2 = = W L kn (Tt=tn-1 Tt=tn)/tn (2-6)
Inter-layer temperatures can be eliminated from Eqs. (2-6), and a
series thermal resistance equation extracted as follows:
Rthrough-thickness = R1 + R2 + + Rn where Ri = ti/ki (2-7)
where
kthrough-thickness = ti/(ti/ki). (2-8)
A printed circuit board is commonly built as layers of conductorsseparated by layers of insulators. The conductors are mostly alloys
of copper, silver or gold, while the insulators are mostly a variety of
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Conduction Heat Transfer In A Printed Circuit Board
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epoxy resins. Therefore the in-plane thermal conductivity Eq. (2-4)
for a PCB can be rewritten as:
kin-plane = [kconductortconductor + kinsulator (ttotal tconductor)]/ttotal (2-9)
Similarly, the through-thickness thermal conductivity Eq. (2-8) for
a PCB can be rewritten as:
kthrough-thickness
= ttotal/[(tconductor/kconductor) + (ttotal tconductor)/kinsulator] (2-10)
The sensitivities of these two PCB thermal conductivities are
analyzed for a 500 m-thick printed circuit board, with the assumed
nominal values for thermal conductivity of the conductor and
insulator layers given below:
kconductor = 377 W/m-C for copper conductor layers and
kinsulator = 0.3 W/m-C for glass reinforced polymer layers.
In-plane thermal conductivity versus percent of conductor layers
to total printed circuit board thickness is given in Figure 2-1. In-plane
thermal conductivity starts at the all-insulator thermal conductivity
value of 0.3 W/m-C, and increases linearly to conductor thermal
conductivity at no insulator layers.
Sensitivities of in-plane thermal conductivity to changes in kconductorand kinsulator are represented in Figure 2-2. As you can see, the two
sensitivities are opposite.
The sensitivity of in-plane thermal conductivity to conductor
thickness is a constant, 0.75 W/m-C-m. The sensitivity of in-plane
thermal conductivity to insulator thickness is also a constant,
and it is the opposite of sensitivity to conductor thickness, namely
0.75 W/m-C-m.
A ten percent variation in variables around the nominal valuesgiven above produce the sensitivity results in Table 2-1 for in-plane
thermal conductivity. For these nominal values, in-plane thermal
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0
100
200
300
400
0 20 40 60 80 100
% Conductor Layers Thickness
kin-plane,W/m-C
Figure 2-1 In-plane thermal conductivity versus percent of conductor layers
thickness to total printed circuit board thickness
0
0.2
0.4
0.6
0.8
1
0 20 100
% Conductor Layers Thickness
kin-plane/kconductor&
kin-plane/
kinsulator kin-plane /kinsulator
kin-plane /kinsulator
40 60 80
Figure 2-2 Sensitivity of in-plane thermal conductivity to kconductor and
to kinsulator
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Conduction Heat Transfer In A Printed Circuit Board
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conductivity is 188.65 W/m-C. Insulator thermal conductivity is
the least effective independent variable in this case due to its low
value. Variations in the sum of conductor thicknesses and the sum of
insulator thicknesses affect in-plane thermal conductivity in opposing
directions, but with the same magnitude.
Through-thickness thermal conductivity versus percent of
conductor thickness has a non-linear behavior, and it is given
for a 500-micron PCB in Figure 2-3. Through-thickness thermal
conductivity is similar to insulator layer thermal conductivity for up
to 80% conductor layer thickness of the total printed circuit board,and therefore is not a good conduction heat transfer path for printed
circuit boards.
Sensitivities of through-thickness thermal conductivity to
kconductor and to kinsulator are given in Figure 2-4. The sensitivity of
through-thickness thermal conductivity to changes in conductor
thermal conductivity is negligible throughout the percent conductor
layer thickness. The sensitivity of through-thickness thermal
conductivity to changes in insulator thermal conductivity increasesand becomes significant as the thickness percentage of the insulator
layers decreases.
Table 2-1 In-plane thermal conductivity change due to a 10%change in variables around nominal values for a
500 micron thick PCB
In-Plane Thermal In-Plane Thermal
Conductivity Change Conductivity Change
Due To A 10% Due To A 10%
Nominal Decrease In Increase In
Variable Value Nominal Value Nominal Value
kconductor 377 W/m-C 9.99% +9.99%
tconductor 250 m 9.98% +9.98%tinsulator 250 m +9.98% 9.98%
kinsulator 0.3 W/m-C 0.01% +0.01%
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0
5
10
15
20
0
% Conductor Layers Thickness
kthrough-thickness,W/m-C
20 40 60 80 100
Figure 2-3 Through-thickness thermal conductivity versus percent
conductor layer thickness
0
4
8
12
16
20
0 50 100
% Conductor Layers Thickness
kthrough-thickne
ss/kconductor&
kthrough-thickness/kinsulator
kthrough-thickness /
kconductor
kthrough-thickness /
kinsulator
Figure 2-4 Sensitivity of through-thickness thermal conductivity toconductor and insulator thermal conductivities versus percent
conductor layer thickness
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Conduction Heat Transfer In A Printed Circuit Board
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Sensitivities of through-thickness thermal conductivity to
conductor and insulator thickness are given in Figure 2-5. The
sensitivities become significant as conductor thickness
approaches 100%.
A ten-percent variation in independent variables around the
nominal values given above produce the sensitivity results given in
Table 2-2 for through-thickness thermal conductivity, which has a
nominal value of 0.6 W/m-C. In this case, through-thickness thermal
conductivity is very resistant to conductor thermal conductivity
variations in the region of interest.A second and a similar analysis can be performed for a plated or
sputtered thinner circuit. A 10-m thick circuit is considered with the
assumed nominal thermal conductivities below:
kconductor = 377 W/m-C for copper conductor layers and
kinsulator = 36 W/m-C for aluminum oxide insulating layers.
In-plane thermal conductivity versus percent of conductor layers
to total thickness is given in Figure 2-6. In-plane thermal conductivity
0.2
0.15
0.1
0.05
0
0.05
0 20 40 60 80 100% Conductor Layers Thickness
kthrough-thickness/kcondu
ctor&
kthrough-thickness/kinsulator,
W/m-C-um
kthrough-thickness /
kinsulator
kthrough-thickness /kconductor
Figure 2-5 Sensitivity of through-thickness thermal conductivity to
conductor and insulator thickness
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Everyday Heat Transfer Problems
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starts at the all insulator thermal conductivity value of 36 W/m-C and
increases linearly to a conductor thermal conductivity of 377 W/m-C.Sensitivites of in-plane thermal conductivity to kconductor and kinsulator are
given in Figure 2-7. As you can see, the two sensitivities are opposite.
Table 2-2 Through-thickness thermal conductivity changedue to a 10% change in variables around nominal
values for a 500 micron thick PCB
Through-Thickness Through-Thickness
Thermal Conductivity Thermal Conductivity
Change Due To Change Due To
Nominal A 10% Decrease In A 10% Increase In
Variable Value Nominal Value Nominal Value
tinsulator 250 m +11.09% 9.08%
kinsulator 0.3 W/m-C 9.99% +9.99%tconductor 250 m 9.08% +11.09%
kconductor 377 W/m-C 0.01% +0.01%
0
100
200
300
400
0 20 40 60 80 100
% Conductor Layers Thickness
kin-plane,W/m-C
Figure 2-6 In-plane thermal conductivity versus percent of conductor layers
to total thickness
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Conduction Heat Transfer In A Printed Circuit Board
21
The sensitivity of in-plane thermal conductivity to conductor thickness
is a constant at 34.1 W/m-C-m. The sensitivity of in-plane thermal
conductivity to insulator thickness is a constant, and it is the opposite of
sensitivity to in-plane thermal conductivity, namely 34.1 W/m-C-m.
A ten percent variation in variables around the nominal values given
above produce the sensitivity results in Table 2-3 for in-plane thermal
conductivity, which has a nominal value of 87.15 W/m-C. Insulator
thickness is the dominant independent variable in this region of interest.
Through-thickness thermal conductivity versus percent ofconductor thickness has a non-linear behavior, and it is given in
Figure 2-8. The percentage of the conductor layers thickness to total
circuit thickness affects the through-thickness thermal conductivity
at all conductor layer thicknesses. Through-thickness conduction heat
transfer is much more prominent in thin-plated or sputtered circuits.
Sensitivities of through-thickness thermal conductivity to kconductor
and to kinsulator are given in Figure 2-9. The sensitivity of through-
thickness thermal conductivity to changes in conductor thermalconductivity is negligible at low percentages of conductor layer
thickness. On the other hand, the sensitivity of through-thickness
0
0.2
0.4
0.6
0.8
1
0 20 40 60 100
% Conductor Layers Thickness
kin-plane/kconductor&
kin-plane/kinsulator
kin-plane /kinsulator
kin-plane /kconductor
80
Figure 2-7 Sensitivity of in-plane thermal conductivity to kconductor and
to kinsulator
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thermal conductivity to changes in insulator thermal conductivity
is significant; it increases to a maximum at around 90% conductor
layer thickness, and finally decreases sharply as conductor thermal
conductivity starts to dominate.
Table 2-3 In-plane thermal conductivity change due to a 10%change in variables around nominal values for a
10-micron thick circuit
In-Plane Thermal In-Plane Thermal
Conductivity Conductivity
Change Due To Change Due To
Nominal A 10% Decrease A 10% Increase
Variable Value In Nominal Value In Nominal Value
tinsulator 8.5 m +33.3% 33.3%
kconductor 377 W/m-C 6.5% +6.5%tconductor 1.5 m 5.9% +5.9%
kinsulator 36 W/m-C 3.5% +3.5
0
100
200
300
400
0 20 40 60 80 100
% Conductor Layers Thickness
kthrough-thicknes
s,W/m-C
Figure 2-8 Through-thickness conduction heat transfer coefficient versus
percentage of conductor layer thickness
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Conduction Heat Transfer In A Printed Circuit Board
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0
0.5
1
1.5
2
2.5
3
3.5
0 20 40 60 80 100
% Conductor Layers Thickness
kthrough-thickness/kconduc
tor&
kthrough-thickness/kinsula
tor
kthrough-thickness /
kconductor
kthrough-thickness /
kinsulator
Figure 2-9 Sensitivity of through-thickness thermal conductivity to
conductor and insulator thermal conductivities versus
percentage of conductor layer thickness
50
40
30
20
10
0
10
0 20 60 100
% Conductor Layers Thickness
8040kthrough-thickne
ss/kinsulator&
kthrough-thickness/kconductor,
W/m-C-m
kthrough-thickness /
kinsulator
kthrough-thickness /
kconductor
Figure 2-10 Sensitivity of through-thickness thermal conductivity to
conductor thickness and to insulator thickness
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Table 2-4 Through-thickness thermal conductivity changedue to a 10% change in variables around nominal
values for a 10-micron thick circuit
Sensitivities of the through-thickness heat transfer coefficient
to conductor and insulator thickness are given in Figure 2-10. The
sensitivity to insulator thickness becomes significant as the percent of
conductor thickness approaches 100%.
A ten percent variation in independent variables around the
nominal values given above produce the sensitivity results in
Table 2-4 for through-thickness thermal conductivity, which has a
nominal value of 41.65 W/m-C. In this case, through-thickness thermal
conductivity is most sensitive to insulator thermal conductivity and
insulator thickness variations.
Through-Thickness Through-Thickness
Thermal Conductivity Thermal Conductivity
Change Due To A 10% Change Due To A 10%
Nominal Decrease In Increase In
Variable Value Nominal Value Nominal Value
kinsulator 36 W/m-C 9.85% +9.18%
tinsulator 8.5 m +9.76% 8.17%
tconductor 1.5 m 1.55% +1.59%
kconductor 377 W/m-C 0.18% +0.15%
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25
C H A P T E R
Cooling the walls of a combustion chamber containing gases at
high temperatures, i.e., 1000C, results in parallel modes of heat
transfer, convection and radiation. This problem can be approached
by assuming one-dimensional steady-state heat transfer in rectangular
coordinates and with constant thermophysical properties.
Convection heat transfer per unit area, from hot gases to the hot
side of a wall that separates the cold medium and the hot gases, can
be written as:
(Q/A)convection = hcg (Tg Twh) (3-1)
Radiation heat transfer per unit area from hot gases which are
assumed to behave as gray bodies to the hot side of a wall can be
written as:
(Q/A)radiation = hrg (Tg Twh) = = g(Tg4 Twh
4) (3-2)
And so the total heat transfer from the hot gases to a combustion
chamber wall is:
3
3EAT TRANSFER
FROM COMBUSTION
CHAMBER WALLS
H
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(Q/A)total = (Q/A)convection + (Q/A)radiation = (hcg + hrg) (Tg Twh) (3-3)
These two heat transfer mechanisms act as parallel thermalresistances, namely:
(1/Rtotal) = (1/Rconvection) + (1/Rradiation) (3-4)
where hcg is the convection heat transfer coefficient between gas
and the hot side of a wall, hrg is the radiation heat transfer coefficient
between gas and the hot side of a wall, Tg is average gas temperature
and Twh is average hot side wall temperature. The radiation heattransfer coefficient hrg is defined as:
hrg = g(Tg4 Twh
4)/(Tg Twh) (3-5)
where g is emissivity of gas and is the Stefan Boltzmann constant.
Heat transfer occurs through the wall by conduction and is
defined as:
(Q/A)total = (kwall/L)(Twh Twc) (3-6)
where kwall is wall material thermal conductivity, L is thickness of the
wall and Twc is the wall temperature at the cold medium side of the wall.
Heat transfer between the cold medium side of the wall and the
cold medium occurs by convection and is defined as:
(Q/A)total = hc (Twc Tc) (3-7)
where hc is the convection heat transfer coefficient between the cold
medium side of the wall and the cold medium, and Tc is the average
temperature of cold medium.
In this example, Twh is going to be the dependent variable, and it
will be solved by iterating a function using a combination of above
Eqs. (3-3), (3-5), (3-6) and (3-7), as follows:
(Tg Tc)/{[1/(hcg + hrg)] + (L/kwall) + (1/hc)}
(hcg + hrg)(Tg Twh) = 0 (3-8)
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The above governing equation can be rewritten as an iteration
function K as follows:
K = (Tg Tc) + {[1/(hcg + hrg)]
+ (L/kwall) + (1/hc)}(hcg + hrg)(Tg Twh) (3-9)
During iteration to determine Twh, all temperatures should be in
degrees Kelvin because of the fourth power behavior of radiation
heat transfer. Also, all thermophysical properties are assumed to
be constants. Nominal values of these variables for the sensitivity
analysis are assumed to be as follows:
Tg = 1000C (1273 K)
Tc = 100C (373 K)
kwall = 20 W/m-K
L = 0.01m
hcg = 100 W/m2-K
hc = 50 W/m2-K
g = 0.2 = 5.67108 W/m2-K4
For these nominal variables, the iteration function crosses zero at
1075.93 K as shown in Figure 3-1, and 42.5% of the total heat transfer
from hot gases to the hot side of the wall comes from radiation mode;
the rest, 57.5%, comes from convection mode.
The effects of hot gas temperature and cold medium temperature
to hot side wall temperature are shown respectively in Figures 3-2and 3-3. Hot gas temperature affects hot side wall temperature almost
one-to-one, namely a slope of 0.925. However, cold side medium
temperature affects hot side wall temperature almost five-to-one,
namely a slope of 0.221.
The effects of wall parametersthermal conductivity and wall
thicknesson hot side wall temperature are shown in Figures 3-4
and 3-5. Changes in wall thermal conductivities below 10 W/m-K
are more effective on hot side wall temperature. Hot side walltemperature sensitivity to wall thickness is pretty much a constant,
3.5 C/cm.
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1000
0
1000
2000
400 600 800 1000 1200 1400Twh, K
IterationFunction,K
Figure 3-1 Iteration function versus Twh
0
400
800
1200
1600
2000
0 500 1000 1500 2000
Gas Temperature, C
HotSideWallTemperature,C
Figure 3-2 Hot side wall temperature versus gas temperature
The convection heat transfer coefficients on both sides
of the wall have opposite effects on hot side wall temperature,
as shown in Figures 3-6 and 3-7. As the hot gas side convection
heat transfer coefficient increases, hot side wall temperature
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Heat Transfer From Combustion Chamber Walls
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700
800
900
1000
0 200 400 600 800 1000
Cold Medium Temperature, CHotSideWallTemperature,C
Figure 3-3 Hot side wall temperature versus cold medium temperature
800
820
840
860
0 10
Wall Thermal Conductivity, W/m-K
HotSideWallTempe
rature,C
20 30 40 50
Figure 3-4 Hot side wall temperature versus wall thermal conductivity
increases as well. Sensitivity of hot side wall temperature to
variations in the hot gas side convection heat transfer coefficient
is more prominent at lower convection heat transfer coefficient
values.
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30
800
810
820
830
840
0 0.02 0.04 0.06 0.08 0.1Wall Thickness, m
H
otSideWallTemperature,C
Figure 3-5 Hot side wall temperature versus wall thickness
600
700
800
900
1000
0 100 200 300 400 500
Hot Gas Side Convection Heat Transfer Coefficient, W/m2-K
HotSideWallTemperature,C
Figure 3-6 Hot side wall temperature versus hot gas side convection heat
transfer coefficient
The variation of hot side wall temperature at different hot
gas emissivities is given in Figure 3-8. Hot side wall temperature
is more sensitive to changes in lower values of hot gasemissivity.
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Heat Transfer From Combustion Chamber Walls
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200
400
600
800
1000
0 100 200 300 400 500
Cold Medium Side Convection Heat Transfer Coefficient,
W/m2-K
HotSideWallTemperatu
re,C
Figure 3-7 Hot side wall temperature versus cold medium side convection
heat transfer coefficient
When the nominal values of the variables given above are
varied 10%, the results shown in Table 3-1 are obtained. Hotside wall temperature sensitivities to a 10% change in the
governing variables are given in descending order of importance,
700
750
800
850
900
950
0 0.2 0.4 0.6 0.8 1
Hot Gas Emissivity
HotSideWallTemperature,C
Figure 3-8 Hot side wall temperature versus hot gas emissivity
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Table 3-1 Effects of 10% change in nominal values ofvariables to hot side wall temperature
Change In Hot Change In Hot
Side Wall Side Wall
Temperature For Temperature For
Nominal A 10% Decrease A 10% Increase
Variable Value In Nominal Value In Nominal Value
Tg 1273 K 11.493% +11.981%
hc 50 W/m2-K +2.070% 1.991%
hcg 100 W/m2-K 1.262% +1.136%
g 0.2 0.912% +0.860%
Tc 373 K 0.295% +0.295%
kwall 20 W/m-K +0.056% 0.046%
L 0.01 m 0.051% +0.051%
and they are applicable around the nominal values assumed for
this study.
Hot side wall temperature is most sensitive to variations in hot
gas temperature. Next in order of sensitivity are the convection heat
transfer coefficients on both sides of the wall. Changes to emissivity
of hot gases affect the dependent variable at the same level as the
convection heat transfer coefficients. Next in order of sensitivity is
the cold medium temperature. Hot side wall temperature is least
sensitive to variations in wall thermal conductivity and wall thickness.
This variable order of sensitivity is applicable around the nominal
values assumed for this case, due to the nonlinear relationship
between the dependent variable and the independent variables.
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33
C H A P T E R
The solar tanning of a human body was analyzed under
steady-state conditions with one-dimensional rate equations
in rectangular coordinates, and using temperature-independent
thermophysical properties. Human skin that is exposed to direct solar
radiation is considered to be in an energy balance. Energy goes into
the skin from both direct solar radiation and solar radiation scattered
throughout the atmosphere. Energy leaves the skin through a variety
of means and routes: by convection heat transfer and radiation heat
transfer (into the atmosphere), by conduction (to the inner portionsof the body), by perspiration, and by body basal metabolism. Other
energy gains and losses, such as those due to terrestrial radiation,
breathing and urination, are negligible.
Energy balance at the human skin gives the following heat transfer
equation:
Qsolar radiation absorbed + Qatmospheric radiation absorbed Qconvection
Qconduction to body Qradiation emitted Qperspiration Qbasal metabolism = 0 (4-1)
4
4EAT TRANSFER
FROM A HUMAN
BODY DURING
SOLAR TANNING
H
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For the present sensitivity analysis, the heat transfer rate equations
that will be used, and the nominal values that will be assumed for
energies outlined in Eq. (4-1), are given below.
Qsolar radiation absorbed = 851 W/m2 (4-2)
which assumes a gray body skin with an absorptivity, = , of 0.8, on
a clear summer day at noon, with full sun exposure
Qatmospheric radiation absorbed = 85 W/m2 (4-3)
which is assumed to be about 10% of Qsolar radiation absorbed.
Qconvection = h(Tskin Tenvironment) (4-4)
where h is the heat transfer coefficient between the skin surface that
is being tanned and the environment. In the present analysis, h is
assumed to be 28.4 W/m2-K and Tenvironment is 30C.
Qconduction to body = (kbody/tbody)(Tskin Tbody) (4-5)
where kbody = 0.2 W/m-K, tbody = 0.1 m, and Tbody = 37C.
Qradiation emitted = (T4
skin T4
environment) (4-6)
where emissivity of skin surface = 0.8 and = 5.67 108 W/m2-K4.
Qperspiration = 337.5 W/m2 (4-7)
which corresponds to a 1 liter/hr perspiration rate for a human body
with a perspiration area of 2 m2.
Qbasal metabolism = 45 W/m2 (4-8)
which represents a 30-year-old male at rest.There are ten independent variables that govern the dependent
variable Tskin in this heat transfer problem. Sensitivities to these ten
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Heat Transfer From A Human Body During Solar Tanning
35
variables are analyzed in the region of the nominal values given above.
The governing Eq. (4-1) takes the following form, and can be solved
for Tskin by trial and error.
C1T4skin + C2Tskin = C3 (4-9)
where C1 = , C2 = h + kbody/tbody, and
C3 = Qsolar radiation absorbed + Qatmospheric radiation absorbed Qperspiration
Qbasal metabolism + T4environment + hTenvironment + kbody/tbodyTbody
All the calculations are performed in degrees Kelvin for
temperature, since the governing equation is non-linear in
temperature. These sensitivities are presented in Table 4-1 below
in the order of their significance.
The most effective variable on the skin temperature is
Tenvironment and the least effective is the thermal conductivity of
human tissue. The skin temperature is an order of magnitude less
sensitive to changes in the thermal conductivity of human tissue
skin-to-body conduction heat transfer length, heat transfer due to
basal metabolism, body temperature, atmospheric radiation absorbed,
and emissivity of skin surfacethan changes in the temperature of
the environment, solar radiation absorbed, convection heat transfer
coefficient, and heat lost due to perspiration. Changes in some
variables, such as the environmental temperature and heat transfer
due to perspiration, behave linearly in the region of interest, and giveequal percentage changes to the dependent variable on both sides of
the variable's nominal value.
It is important to remind the reader that the order shown in
Table 4-1 is only useful in this region of the application due to
non-linear behavior of the sensitivities. The non-linear affects of
variables such as the convection heat transfer coefficient are given
in Figure 4-1 for Qsolar radiation absorbed = 851 W/m2 and for three different
perspiration rates.The sensitivity of skin temperature to the convection heat transfer
coefficient is significant up to 50 W/m2-K. The heat transfer coefficient
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from the skin surface to the environment can be determined from
appropriate empirical relationships found in References [6] and [10].The heat transfer coefficient in the natural convection regime is
around 5 W/m2-K. If the wind picks up to, say, 8.9 m/s (20 mph),
then the heat transfer coefficient is in the turbulent flow regime, and
it increases to 20 W/m2-K. As the perspiration rate goes down, this
sensitivity increases. The sensitivity curves are given in Figure 4-2.
Similar results are obtained for an afternoon solar radiation
by assuming half the noon solar radiation, i.e., Qsolar radiation absorbed =
425.5 W/m2, and they are given in Figures 3-3 and 3-4.As the solar radiation goes down, the skin temperature and its
sensitivity to the convection heat transfer coefficient decreases.
Table 4-1 Effects of 10% change in nominal values ofvariables to skin temperature
Skin Skin
Temperature, Temperature,
Tskin, Change Tskin, Change
Due To A 10% Due To A 10%
Nominal Decrease In Increase In
Variable Value Nominal Value Nominal Value
Tenvironment 30C 6.03% +6.03%
Qsolar radiation absorbed 851 W/m2
5.11% +5.13%h 28.4 W/m2-K +2.93% 2.51%
Qperspiration 337.5 W/m2 +2.03% 2.03%
Emissivity of
skin surface, 0.8 +0.53 0.51
Qatmospheric radiation absorbed 85 W/m2 0.51% +0.51%
Tbody 37C 0.45% +0.45%
Qbasal metabolism 45 W/m2
+0.27% 0.27%Skin-to-body
conduction 0.1 m 0.12% +0.10%
length, tbody
ktissue 0.2 W/m-K +0.11% 0.11%
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Heat Transfer From A Human Body During Solar Tanning
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Tenvironment=30C, Tbody=37C, Qsolar radiation absorbed=851 W/m2
30
35
40
45
50
55
60
65
70
0 50 100 150 200
Convection Heat Transfer Coefficient, W/m2-K
SkinTemperature,C
Perspiration=0.5
liters/hr
Perspiration=1.0
liter/hr
Perspiration=1.5
liter/hr
1
0.8
0.6
0.4
0.2
0
0 50 100 150 200
Convection Heat Transfer Coefficient, W/m2-K
Tskin/h,m2-K2/W
Perspiration=0.5
liters/hrPerspiration=1.0
liters/hr
Perspiration=1.5liters/hr
Figure 4-1 Skin temperature versus convection heat transfer coefficient
for Qsolar radiation absorbed = 851 W/m2 and for three different
perspiration rates
Figure 4-2 Skin temperature sensitivity to convection heat transfer
coefficient versus convection heat transfer coefficient forQsolar radiation absorbed = 851 W/m2 and for three different perspiration
rates
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Tenvironment=30C, Tbody=37C, Qsolar radiation absorbed=425.5 W/m2
25
30
35
40
45
0 50 100 150 200
Convective Heat Transfer Coefficient, W/m2-K
SkinTemperature,C
Perspiration=0.5
liters/hr
Perspiration=1.0
liter/hrPerspiration=1.5
liter/hr
Figure 4-3 Skin temperature versus convection heat transfer coefficient
for Qsolar radiation absorbed = 425.5 W/m2 and for three different
perspiration rates
0.5
0.4
0.3
0.2
0.1
0
0.1
0.2
0 50 100 150 200
Convection Heat Transfer Coefficient, W/m2-K
Tskin/h,m2-K2/W
Perspiration=0.5
liters/hr
Perspiration=1.0
liters/hr
Perspiration=1.5
liters/hr
Figure 4-4 Skin temperature sensitivity to convection heat transfer
coefficient versus convection heat transfer coefficient forQsolar radiation absorbed = 425.5 W/m2 and for three different
perspiration rates
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Heat Transfer From A Human Body During Solar Tanning
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In high perspiration rates, the skin temperature is below
the environmental temperature, and it approaches theenvironmental temperature as the heat transfer coefficient
increases.
30
35
40
45
50
55
60
20 24 28 32 36 40
Tenvironment, C
Tskin,C
Figure 4-5 Skin temperature versus environment temperature
35
40
45
50
0.5 0.6 0.7 0.8 0.9 1
Emissivity Of Human Skin Surface,
Tskin,C
Figure 4-6 Skin temperature versus emissivity of human skin surface
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41
C H A P T E R
Heat transfer from a surface can be enhanced by using fins.
Heat transfer from surfaces with different types of fins has been
studied extensively, as seen in References by Incropera, F. P. and
D. P. DeWitt [6] and by F. Kreith [10].
The present sensitivity analysis represents rectangular fins under
steady-state, one-dimensional, constant thermophysical property
conditions without radiation heat transfer. Energy balance to a
cross-sectional element of a rectangular fin gives the following second
order and linear differential equation for the temperature distributionalong the length of the fin.
d2T/dx2 (hP/kA) (T Tenvironment) = 0 (5-1)
where h is the convection heat transfer coefficient between the
surface of the fin and the environment in W/m2-C, k is the thermal
conductivity of the fin material in W/m-C, P is the fin cross-sectional
perimeter in meters, and A is the fin cross-sectional area in m2.There can be different solutions to Eq. (5-1) depending upon the
boundary condition that is used at the tip of the fin. If the heat loss
FFICIENCY OFRECTANGULAR
FINS
5
5E
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to environment from the tip of the fin is neglected, the following
boundary conditions can be used:
T = Tbase at x = 0 and (dT/dx) = 0 at x = L (5-2)
The solution to Eqs. (5-1) and (5-2) can be written as
(T Tenvironment) = (Tbase Tenvironment) [cosh m(L-x)/cosh (mL)] (5-3)
where L is the length of the rectangular fin in meters and m =
(hP/kA)0.5 in 1/m.The heat transfer from the rectangular fin can be determined from
Eq. (5-3) by finding the temperature slope at the base of the fin,
namely
Qfin = kA(dT/dx) at x = 0 or (5-4)
Qfin = (Tbase Tenvironment) sqrt(hkPA) tanh(mL) (5-5)
Here the sensitivities of variables that affect the efficiency of a
rectangular fin will be analyzed. Fin efficiency is generally defined by
comparing the fin heat transfer to the environment with a maximum
heat transfer case to the environment, where the whole fin is at the
fin base temperature, namely = Qfin/Qmax where Qmax = hAfin(Tbase
Tenvironment). For a rectangular fin, the fin heat transfer efficiency
is approximated by using Eq. (5-5), and by adding a corrected finlength, Lc, for the heat lost from the tip of the fin.
= tanh(mLc)/(mLc) (5-6)
where m = [h 2(w + t)/k wt ]0.5 and Lc = L + 0.5t.
For cases where the fin width, w, is much greater than its
thickness, t, m becomes
m = (2h/kt)1/2 (5-7)
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Efficiency Of Rectangular Fins
43
There are four independent variables that affect the rectangular
fin heat transfer efficiency. These are the convection heat transfer
coefficient, h; the thermal conductivity of the fin material, k; length ofthe fin, l; and thickness of the fin, t.
The sensitivity of efficiency to these four independent variables can
be obtained in closed forms by differentiating the efficiency equation
with respect to the desired independent variable. For example:
/h = (0.5/h)[(1/cosh2(mLc)) (tanh(mLc)/(mLc))] (5-8)
Fin efficiency as a function of the convection heat transfercoefficient for two different thermal conductivitiesaluminum and
copperis given in Figure 5-1. The sensitivity of rectangular fin
efficiency with respect to the convection heat transfer coefficient is
given in Figure 5-2.
Fin efficiency is good in the natural convection regime and
degrades as high forced convection regimes are used. Sensitivity of
fin efficiency to the convection heat transfer coefficient is high in the
0.4
0.5
0.6
0.7
0.8
0.9
1
0 100 200 300 400
Convection Heat Transfer Coefficient, W/m2-C
FinEfficiency kcu=377.2
W/m-C
kal=206
W/m-C
L=0.0508 m
t=0.002 m
Figure 5-1 Rectangular fin efficiency versus convection heat transfercoefficient for two different fin materials with L = 0.0508 m and
t = 0.002 m
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natural convection regime and decreases as the forced convection
heat transfer coefficient increases.
Fin efficiency as a function of fin material thermal conductivity
for two different convection heat transfer coefficientsnatural
convection regime and forced convection regimeis given in
Figure 5-3. The sensitivity of rectangular fin efficiency with respectto fin material thermal conductivity is given in Figure 5-4.
Fin material thermal conductivity does not affect fin efficiency in
the natural convection regime except in the region of low thermal
conductivity materials. However, in the forced convection regime
the behavior is quite different. Fin material thermal conductivity
affects fin efficiency, and high thermal conductivity materials have to
be used in order to achieve high fin efficiency. The sensitivity of fin
efficiency to fin material thermal conductivity is high for low thermalconductivities. The sensitivity diminishes as high fin material thermal
conductivities are utilized.
0.005
0.004
0.003
0.002
0.001
0
0 100 200 300 400
Convection Heat Transfer Coefficient, W/m2-C
h/h,m2-C/W
kcu=377.2
W/m-C
kal=206
W/m-C
L=0.0508 m
t=0.002 m
Figure 5-2 Sensitivity of rectangular fin efficiency to convection heat
transfer coefficient versus convection heat transfer coefficient
for two different fin materials with L = 0.0508 m and t = 0.002 m
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Efficiency Of Rectangular Fins
45
0.4
0.6
0.8
1
0 100 200 300 400 500
Fin Thermal Conductivity, W/m-C
FinEfficiency,
h=5
W/m2-C
h=100
W/m2-C
L=0.0508 m
t=0.002 m
Figure 5-3 Rectangular fin efficiency versus fin material thermal
conductivity for two different convection heat transfer
coefficients with L = 0.0508 m and t = 0.002 m
0
0.0021
0.0042
0.0063
0.0084
0 100 200 300 400 500
Fin Thermal Conductivity, W/m-C
/k,m-C/W
h=5
W/m2-C
h=100
W/m2-C
L=0.0508 m
t=0.002 m
Figure 5-4 Sensitivity of rectangular fin efficiency to fin material thermal
conductivity versus fin material thermal conductivity for twodifferent convection heat transfer coefficients with L = 0.0508 m
and t = 0.002 m
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Fin efficiency as a function of fin length for two different convection
heat transfer coefficientsnatural convection regime and forced
convection regimeis given in Figure 5-5. The sensitivity of rectangular
fin efficiency with respect to fin length is given in Figure 5-6. Figure 5-6
shows sensitivities for combinations of two different convection heat
transfer coefficients and two different thermal conductivities.
Figure 5-5 shows that fin efficiency is a weak function fin length in
the natural convection regime, but this weakness becomes a strong
function of fin length in the forced convection regime. These resultscan also be seen in Figure 5-6. In the natural convection regime,
sensitivity of fin efficiency to fin length is low, but increases as the fin
length increases. In the forced convection regime, sensitivity of fin
efficiency to fin length starts low, goes through a maximum as the fin
length increases, and decreases as the fin length increases further.
Fin efficiency as a function of fin thickness for two different
convection heat transfer coefficientsnatural convection regime
and forced convection regimeis given in Figure 5-7. The sensitivityof rectangular fin efficiency with respect to fin thickness is given in
Figure 5-8. Figure 5-8 shows sensitivities for combinations of two
0.6
0.7
0.8
0.9
1
0 0.02 0.04 0.06 0.08 0.1
Fin Length, m
FinEfficiency,
h=5
W/m2-C
h=100
W/m2-C
t=0.002 m
k=377.2 W/m-C
Figure 5-5 Rectangular fin efficiency versus fin length for two different
convection heat transfer coefficients with t = 0.002 m and
k = 377.2 W/m-C
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0
1
2
3
4
5
6
7
8
0 0.05 0.1
Fin Length, m
/L,1/m
h=5 W/m2
-C & @k=377.2 W/m-C
h=100 W/m2-C & @
k=377.2 W/m-C
h=5 W/m2-C & @
k=206 W/m-C
h=100 W/m2-C & @
k=206 W/m-C
t=0.002 m
Figure 5-6 Sensitivity of rectangular fin efficiency to fin length versus fin
length for combinations of two different convection heat transfer
coefficients and two different thermal conductivities with t = 0.002 m
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.001 0.002 0.003 0.004 0.005
Fin Thickness, m
FinEfficiency,
h=5
W/m2-C
h=100
W/m2-C
k=377.2 W/m-C
L=0.0508 m
Figure 5-7 Rectangular fin efficiency versus fin thickness for two differentconvection heat transfer coefficients with k = 377.2 W/m-C and
L = 0.0508 m
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different convection heat transfer coefficients and two different
thermal conductivities.
Figure 5-7 shows that fin efficiency is a weak function of fin
thickness in the natural convection regime, but this weakness
becomes a strong function of fin thickness in the forced convection
regime. In Figure 5-8, in the natural convection regime, sensitivity offin efficiency to fin thickness starts high at low fin thickness values,
but decreases as the fin thickness increases. In the forced convection
regime, sensitivity of fin efficiency to fin thickness starts low, goes
through a maximum as the fin thickness increases, and decreases as
the fin thickness increases further.
A ten percent variation in independent variables around the
nominal values produces the following sensitivity results (Table 5-1)
for fin efficiency. The results are given for the natural convectionregime in descending order, from the most sensitive variable to the
least. Table 5-2 gives similar results for the forced convection regime.
0
1
2
3
4
5
6
7
8
9
0 0.001 0.002 0.003 0.004 0.005
Fin Thickness, m
/t,1/m
h=5 W/m2-C & @
k=377.2 W/m-C
h=100 W/m2-C & @
k=377.2 W/m-C
h=5 W/m2-C &
k=206 W/m-C
h=100 W/m2-C & @
k=206 W/m-C
L=0.0508 m
Figure 5-8 Sensitivity of rectangular fin efficiency to fin thickness versus
fin thickness for combinations of two different convection heat
transfer coefficients and two different thermal conductivities
with L = 0.0508 m
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Efficiency Of Rectangular Fins
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The order of significance in fin efficiency change follows the same
pattern with respect to variables in both the natural convection and
forced convection regimes. However, in the forced convection regime,fin efficiency changes are an order of magnitude higher than the
natural convection regime.
Table 5-1 Rectangular fin efficiency change due to a 10%change in variables nominal values for the natural
convection regime
Rectangular Rectangular
Fin Efficiency Fin Efficiency
Change Due To Change Due To
Nominal A 10% Decrease A 10% Increase
Variable Value In Nominal Value In Nominal Value
L 0.0508 m +0.218% 0.239%
k 377.2 W/m-C 0.129% +0.106%
t 0.002 m 0.124% +0.102%
h 5 W/m2-C +0.117% 0.116%
Table 5-2 Rectangular fin efficiency change due to a10% change in variables nominal values for the
forced convection regime
Rectangular Rectangular
Fin Efficiency Fin Efficiency
Change Due To Change Due To
Nominal A 10% Decrease A 10% Increase
Variable Value In Nominal Value In Nominal Value
L 0.0508 m +3.436% 3.478%
k 377.2 W/m-C 1.916% +1.639%
t 0.002 m 1.844% +1.574%
h 100 W/m2-C +1.806% 1.729%
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51
C H A P T E R
Ahot drawn bar, assumed to be moving at a constant velocity out of
a die at constant temperature, will be treated as a one-dimension-
al heat transfer problem. The Biot numberfor the bar, htD/2k, will be
assumed to be less than 0.1, to assure no radial variation of tempera-
ture in the bar. Here ht is the total heat transfer coefficient from the
bar surface in W/m2-K, the sum of the convection heat transfer coef-
ficient and radiation heat transfer coefficient, D is the bar diameter,
and k is the bar thermal conductivity in W/m-K. Conduction, convec-
tion, and radiation heat transfer mechanisms affect the temperatureof the drawn bar.
Energy balance can be applied to a small element of the bar with a
width of dx:
Conduction heat transfer into the element Conduction heat
transfer out of the element Convection heat transfer out of the
element to the environment Radiation heat transfer out of the
element to the environment = Rate of change of internal energyof the element
6
6EAT
TRANSFER FROM
A HOT DRAWN BAR
H
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The energy balance on the element can be written as follows:
Qconduction at x Qconduction at x+dx Qconvection from dx Qradiation from dx= cpAdx(dT/d) (6-1)
where is the density of the bar in kg/m3, cp is the specific heat of the
bar at constant pressure in J-kg/K, A is the bar cross-sectional area
in m2, T is the temperature of the bar element in K, and dx/d is the
drawn bar velocity in m/s.
Assuming that all the bar thermophysical and geometrical
properties are constants, the following one-dimensional, second-orderand non-linear differential equation is obtained:
d2T/dx2 (hP/kA)(T-Tenvironment) (P/kA)(T4 T4environment)
= (cpU/k)dT/dx (6-2)
where P is the bar perimeter in m, is Stefan Boltzmann constant,
5.6710-8 W/m2-K4, is the bar surface emissivity, and U=dx/d the
speed of the hot drawn bar. The differential equation (6-2) reducesto steady-state heat transfer from fins with a uniform cross-sectional
area; if the radiation heat transfer and the rate of change of internal
energy are neglected, see References by F. Kreith [10] and by
Incropera, F. P. and D. P. DeWitt [6].
The boundary condition for this heat transfer problem can be
specified as follows:
T = Tx=0 (temperature of the drawn bar at the die location)at x = 0 (6-3)
and
T = Tenvironment as x goes to 4 (6-4)
The governing differential equation (6-2), along with boundary
conditions (6-3) and (6-4), can be solved by finite difference methodsand iteration, in order to determine the temperature at the i'th
location along the bar.
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Another method to solve this non-linear heat transfer problem is to
define a radiation heat transfer coefficient utilizing the temperature of
the previous bar element, i-1, as follows:
hradiation = (T4i-1 T4environment)/(Ti-1 Tenvironment) (6-5)
and write the differential equation for location i along the bar as
d2Ti/dx2 [(hconvection + hradiation) P/kA](Ti -Tenvironment)
= (cpU/k)dTi/dx (6-6)
The solution reached by linear differential equation (6-6) for
location i along the bar is valid for small x increments along the bar,
i.e. < 0.01 m, since the radiation heat transfer coefficient is calculated
using the temperature of the previous element i-1.
The solution to the above second order linear differential equation
(6-6) which satisfies both boundary conditions, (6-3) and (6-4), is:
(Ti Tenvironment)/(Tx=0 Tenvironment)
= exp{[(U/2) sqrt((U/2)2 + m2)]x} (6-7)
where = k/cp is the thermal diffusivity of the bar in m2/s and
m2 = (hconvection + hradiation)P/kA in 1/m2. (6-8)
This temperature distribution solution reduces to steady-state heattransfer from rectangular fins with a uniform cross-sectional area
and the above applied boundary conditions, (6-3) and (6-4), if the
radiation heat transfer and the hot drawn bar velocity are neglected,
i.e., hradiation=0 and U=0 (see References by F. Kreith [10] and by
Incropera, F. P. and D. P. DeWitt [6]).
Sensitivity to governing variables is analyzed by fixing the drawn bar
temperature at the die, i.e., Tx=0=1273 K. There are eight independent
variables that affect the temperature distribution of the hot drawn bar.The sensitivities of bar temperature to these variables are analyzed by
assuming the following nominal values for a special steel bar:
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54
D = 0.01 m
= 8000 kg/m3
cp = 450 J/kg-Kk = 40 W/m-K
= 0.5
U = 0.02 m/s
Tenvironment = 298 K
The last independent variable is the convention heat transfer
coefficient between the surface of the bar and the environment.
The convection heat transfer coefficient can be determined from adrawn bar temperature requirement at a distance from the die. In
the present analysis, Tx=10 is specified to be 373 K. At approximately
x=10 m the radiation heat transfer contribution almost diminishes.
The convection heat transfer coefficient that meets the Tx=10 = 373 K
requirement is determined from the above solution to be:
hconvection = 46.17 W/m2-K
which is in the turbulent region of forced cooling air over the
cylindrical bar, i.e., ReD = 1083 where ReD= VairD/air. The empirical
relationship for the convection heat transfer coefficient for air flowing
over cylinders is given in Reference [10] by F. Kreith as:
hconvectionD/kair = 0.615 (VairD/air)0.466 for 40 < ReD < 4000 (6-9)
where Vair is mean air speed over the cylinder (2.18 m/s in this case),
kair is air thermal conductivity, and air is air kinematic viscosity.
Air thermophysical properties are calculated at film temperature,
namely the average of bar surface temperature and environmental
temperature.
A comparison of convection and radiation heat transfer coefficients
as a function of distance from the die is given in Figure 6-1.
Heat transfer due to radiation is at the same order of magnitudearound the die. As the bar travels away from the die, radiation heat
transfer diminishes rapidly.
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Heat Transfer From A Hot Drawn Bar
55
Hot drawn bar temperature distributions, both with and without
radiation heat transfer, are shown in Figure 6-2. Radiation heat
transfer effects on bar temperature cannot be neglected below x=4
meters from the die.
In the initial sensitivity analysis, radiation heat transfer effects
will be neglected, namely hradiation=0. Hot drawn bar temperatures
as a function of distance from the die for different convection heat
transfer coefficients are given in Figure 6-3. Temperatures are verysensitive to low convection heat transfer coefficients.
The sensitivity of bar temperature at x=10 m to the convection heat
transfer coefficient is given in Figure 6-4. Bar temperature sensitivity
is high at natural convection and at low forced-convection heat
transfer regions. As the forced-convection heat transfer coefficient
increases, bar temperature sensitivity to the convection heat transfer
coefficient decreases.
Bar temperature at x=10 m as a function of the convection heattransfer coefficient is shown in Figure 6-5. Increasing the convection
heat transfer coefficient reduces its effects on bar temperature.
0
10
20
30
40
50
0 2 4 6 8 10Distance From Die, m
HeatTransferCoeffieice
nts
W/m2-K
hradiation
hconvection
Figure 6-1 Radiation and convection heat transfer coefficients as a function
of distance from die
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0
200
400
600
800
1000
1200
1400
0 4 8 10
Distance From Die, m
Temperature,K
Without Radiation
With Radiation
2 6
Figure 6-2 Hot drawn bar temperature with and without radiation heat
transfer effects
200
400
600
800
1000
1200
1400
0 4 8 10
Distance From Die, m
Tempera
ture,K
hconvection=5
W/m2-Khconvection=20
W/m2-Khconvection=40
W/m2-Khconvection=60
W/m2-K
62
Figure 6-3 Hot drawn bar temperature for different convection heat trans-
fer coefficients with hradiation=0
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Heat Transfer From A Hot Drawn Bar
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50
40
30
20
10
0
0 10 20 30 40 50
hconvection, W/m2-K
T/hconvection,m2-K2/W
Figure 6-4 Bar temperature sensitivity @ x=10 m to convection heat trans-
fer coefficient
200
400
600
800
1000
0 10 20 30 40 50 60
hconvection, W/m2-K
BarTemperature@x=10m,K
Figure 6-5 Bar temperature @ x=10 m as a function of convection heat
transfer coefficient
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Bar temperature at x=10 m varies close to a linear behavior with
bar density, as shown in Figure 6-6. ?T/? is 0.0239 K-m3/kg in the
7000 to 8000 kg/m3 bar density range.
Bar temperature at x=10 m also varies close to a linear behavior
with bar-specific heat at constant pressure, as shown in Figure 6-7.
?T/?cp is 0.4254 K2-kg/J in the 400 to 500 J/kg-K bar specific heat
range.
Bar temperature at x=10 m varies linearly with bar thermal
conductivity, as shown in Figure 6-8. Bar temperature is a weakfunction of bar thermal conductivity in this problem. ?T/?k is 0.0007
K2-m/W in the 20 to 60 W/m-K bar thermal conductivity range.
Bar temperature at x=10 meters versus bar velocity is given in
Figure 6-9.
Hot drawn bar velocity does not affect bar temperature at low
velocities, i.e., U
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340
360
380
400
400 420 440 460 480 500Bar Specific Heat, J/kg-K
BarTemperature@x=10
m,K
Figure 6-7 Bar temperature @ x=10 m versus bar specific heat at constant
pressure
373
373.01
373.02
373.03
373.04
20 30 40 50 60
Bar Thermal Conductivity, W/m-K
Bar
Temperature@x=10m,K
Figure 6-8 Bar temperature @ x=10 m versus bar thermal conductivity
Bar temperature at x=10 meters versus bar diameter is given
in Figure 6-10. Bar temperature at x=10 meters is not sensitive tobar diameter changes in small diameter values, i.e., D
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maximum of about 19,500 K/m at around D=0.02 meters, and starts to
decrease as the bar diameter increases.Bar temperature at x=10 meters versus environmental temperature
is given in Figure 6-11. The relationship is linear as expected.
200
300
400
500
600
700
0 0.01 0.02 0.03 0.04 0.05Bar Velocity, m/s
BarTemperature@x=10
m,K
Figure 6-9 Bar temperature @ x=10 m versus bar velocity
200
400
600
800
1000
0 0.01 0.02 0.03 0.04 0.05
Bar Diameter, m
Ba
rTemperature@x=10m,K
Figure 6-10 Bar temperature @ x=10 m versus bar diameter
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The slope of the curve is 0.9231 K/C under the assumed nominal
conditions.
When the nominal values of the independent variables given above
are varied +-10%, the results shown in Table 6-1 are obtained. The
sensitivity analysis is conducted by neglecting radiation heat transfer
at x=10 meters.
The convection heat transfer coefficient, bar diameter, bar velocity,
bar density and bar-specific heat at constant pressure have the same
order of magnitude sensitivity on bar temperature at x=10 m. Changes
in the temperature of the environment affect bar temperature at x=10m, at an order of magnitude less. Changes to the thermal conductivity
of the bar have the least effect on bar temperature at x=10 m. The
sensitivity magnitudes and order that are shown in Table 6-1 are only
valid around the nominal values that are assumed for the independent
variables for this analysis. Bar velocity, bar density and bar-specific
heat at constant pressure have the same sensitivity effects on the
temperature of the bar as can be seen in Eq. (6-7).
Another interesting sensitivity analysis can be performed aroundx=0.5 m, where both the convection and the radiation heat transfers
are in effect.
360
365
370
375
380
15 20 25 30Environment Temperature, C
BarTemperature@x=10m,K
Figure 6-11 Bar temperature @ x=10 m versus environment temperature
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Table 6-2 Effects of 10% change in nominal values ofvariables to bar temperature @ x=0.5 m, including
radiation heat transfer
Bar Temperature Bar Temperature
@ x=0.5 m Change @ x=0.5 m Change
Due To A 10% Due To A 10%
Nominal Decrease In Increase In
Variable Value Nominal Value Nominal Value
D 0.01 m 1.386% +1.183%
U 0.02 m/s 1.386% +1.183%
8000 kg/m3 1.386% +1.183%
cp 450 J/kg-K 1.386% +1.183%
hconvection 46.17 W/m2-K +0.788% 0.783%
0.5 +0.500% 0.482%
Tenvironment 25C (298 K) 0.046% +0.046%
k 40 W/m-K 0.0003% +0.0003%
Table 6-1 Effects of 10% change in nominal values of vari-ables to bar temperature @ x=10 m
Bar Temperature Bar Temperature
@ x=10 m Change @ x=10 m Change
Due To A 10% Due To A 10%
Nominal Decrease In Increase In
Variable Value Nominal Value Nominal Value
hconvection 46.17 W/m2-K +5.879% 4.549%
D 0.01 m 4.986% +5.280%
U 0.02 m/s 4.986% +5.279%
8000 kg/m3 4.986% +5.279%
cp 450 J/kg-K 4.986% +5.279%
Tenvironment 25C (298 K) 0.619% +0.619%
k 40 W/m-K 0.0007% +0.0007%
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Sensitivities to a +-10% change in independent variables are given
in descending order of importance in Table 6-2.
All the sensitivities to governing independent variables at x=0.5meters are at the same order of magnitude, except Tenvironment and k.
Variations in radiation and convection heat transfer losses from the
bar have similar effects on bar temperature close to the die.
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65
C H A P T E R
Maximum current in an open-air single electrical wire (not in a
bundle), will be analyzed under steady-state and one-dimensional
cylindrical coordinates, with constant property conditions. The wire is
assumed to be a cylindrical conductor with a certain diameter and with
certain material characteristics; i.e., resistivity. The wire conductor is
insulated with concentric layers of insulation material that can stand
up to a certain wire conductor temperature, Tc, which will be the
temperature rating of the wire. Heat generated in the wire conductor
is I2
R and the temperature within the wire conductor is assumed to beuniformthe conductor temperature does not vary radially from the
center to the outer radius of the wire conductor.
Heat is transferred by conduction through the wire insulator and by
convection from the surface of the wire insulator to the environment.
Radiation heat transfer from the surface of the wire insulator is
neglected. The conduction heat transfer from the conductor to the
outer radius of the wire insulator can be written by the rate equation
in cylindrical coordinates:
Q = 2Lkins (Tc Tinsulation outer radius)/ln(rw/rc) (7-1)
7
7AXIMUM
CURRENT IN
AN OPEN-AIR
ELECTRICAL WIRE
M
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The convection heat transfer from the outer surface of the wire
insulator to the environment can be written by the rate equation in
cylindrical coordinates:
Q = 2rwLh (Tinsulation outer radius Tenv) (7-2)
The heat transfer mechanisms in Eqs. (7-1) and (7-2) are in a series
thermal resistance path. The energy balance for this heat transfer
problem, heat generated by the conductor equals heat lost to the
environment, can be written as follows, by eliminating Tinsulation outer radius
from Eqs. (7-1) and (7-2):
I2R = (Tc Tenv)/{[ln(rw/rc)/(2Lkins)] + (1/2rwLh)} (7-3)
where the first term in the denominator is the conduction heat
transfer resistance in the wire insulation, and the second term is
the convection heat transfer resistance at the insulated wire's outer
surface.
Definitions of variables:
I = Current through the conductor in amps
R = Resistance of the wire can also be written as L/rc2 where
= Resistivity of the conductor material in -m
L = Length of the conductor in meters
rc = Conductor radius in meters
Tc = Temperature rating of the wire in C
Tenv = Temperature of the environment in C
rw = Radius of insulated wire in meters
kins = Thermal conductivity of wire insulation in W/m-C
h = Convection heat transfer coefficient in W/m2-C
The maximum current that a wire can stand can be written from
Eq. (7-3) by replacing resistance of the wire with resistivity of the
conductor material:
Imax = rc {(2/)(Tc Tenv)/[ln(rw/rc)/kins + (1/rwh)]}1/2 (7-4)
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67
There are six independent variables that govern the maximum
current allowed in the wire. These are the conductor radius, rc, the
conductor resistivity, , the temperature potential (the temperaturerating of the wire minus the temperature of the environment), Tc
Tenv, the radius of the