Everyday Heat Transfer

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)

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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

iv

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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2

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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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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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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10

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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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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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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13

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


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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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%



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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18

0

5

10

15

20

0


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


kthrough-thickne

ss/kconductor&

kthrough-thickness/kinsulator

kthrough-thickness /

kconductor


kinsulator

Figure 2-4 Sensitivity of through-thickness thermal conductivity toconductor and insulator thermal conductivities versus percent

conductor layer thickness


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19

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


kinsulator

kthrough-thickness /kconductor

Figure 2-5 Sensitivity of through-thickness thermal conductivity to

conductor and insulator thickness


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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


Nominal A 10% Decrease In A 10% Increase In


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


kin-plane,W/m-C

Figure 2-6 In-plane thermal conductivity versus percent of conductor layers

to total thickness


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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


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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22

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


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


kthrough-thicknes

s,W/m-C

Figure 2-8 Through-thickness conduction heat transfer coefficient versus

percentage of conductor layer thickness


29/240


23

0

0.5

1

1.5

2

2.5

3

3.5

0 20 40 60 80 100


kthrough-thickness/kconduc

tor&

kthrough-thickness/kinsula

tor


kconductor


kinsulator


conductor and insulator thermal conductivities versus

percentage of conductor layer thickness

50

40

30

20

10

0

10

0 20 60 100


8040kthrough-thickne

ss/kinsulator&

kthrough-thickness/kconductor,

W/m-C-m


kinsulator


kconductor


conductor thickness and to insulator thickness


30/240


24

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%



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%


31/240

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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26

(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)


33/240

Heat Transfer From Combustion Chamber Walls

27

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.


34/240


28

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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29

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.


36/240


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


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.


37/240


31

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


Figure 3-8 Hot side wall temperature versus hot gas emissivity


38/240


32

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



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.


39/240

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


40/240


34

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


41/240

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


42/240


36

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




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%


43/240


37

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


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


44/240


38

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


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


45/240


39

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


46/240


47/240

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


48/240


42

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)


49/240

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


50/240


44

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


51/240


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


52/240


46

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


53/240


47

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


54/240


48

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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49

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




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




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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52

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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Heat Transfer From A Hot Drawn Bar

53

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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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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56

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-K

62

Figure 6-3 Hot drawn bar temperature for different convection heat trans-

fer coefficients with hradiation=0


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57

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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58

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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59

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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60

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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61

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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62

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




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




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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63

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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66

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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Maximum Current In An Open-Air Electrical Wire

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

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