Everyday Heat Transfer Problems: Sensitivities to Governing Variables

EVERYDAY HEAT TRANSFER PROBLEMS Sensitivities To Governing Variables

by M. Kemal Atesmen

© 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 INFORMATION PUBLISHED IN THIS WORK. NEITHER ASME NOR ITS AUTHORS AND EDITORS SHALL BE RESPONSIBLE FOR ANY ERRORS, OMISSIONS, OR DAMAGES ARISING OUT OF THE USE OF THIS INFORMATION. THE WORK IS PUBLISHED WITH THE UNDERSTANDING THAT ASME AND ITS AUTHORS AND EDITORS ARE SUPPLYING INFORMATION BUT ARE NOT ATTEMPTING TO RENDER ENGINEERING OR OTHER PROFESSIONAL SERVICES. IF SUCH ENGINEERING OR PROFESSIONAL SERVICES ARE REQUIRED, THE ASSISTANCE OF AN APPROPRIATE PROFESSIONAL SHOULD BE SOUGHT. ASME shall not be responsible for statements or opinions advanced in papers or ... printed in its publications (B7.1.3). Statement from the Bylaws. 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. Heat–Transmission–Problems, exercises, etc. 2. Materials–Thermal properties– Problems, exercises, etc. 3. Thermal conductivity–Problems, exercises, etc. 4. Engineering mathematics–Problems, exercises, etc. I. Title. TA418.54.A47 2009 621.402’2–dc22

2008047423

TABLE OF CONTENTS 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 Efficiency 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 a Cryogenic 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

iii

Everyday Heat Transfer Problems 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 Solidification in a Casting Mold ....................................... 213 Chapter 24 Average Temperature Rise in Sliding Surfaces in Contact .............................................................................. 221 References .................................................................................................... 233 Index .............................................................................................................. 235

iv

I NTRODUCTION

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 constant pressure, 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 independent variables. Since these sensitivities are mostly non-linear, one should

1

Everyday Heat Transfer Problems

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 be written 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 the uncertainties of independent variables—for example, experimental measurements of thermophysical properties—the 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 for each problem will be provided, but the focus will be on the sensitivity of the governing dependant variable on the changing independent

2

Introduction

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 listed in 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

3

CHAPTER

HEAT LOSS

1

FROM WALLS IN A TYPICAL HOUSE

H

eat 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 conditions—the temperature inside the house is higher than the temperature outside the house—the 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 these wall layers will occur by conduction, and is presented by the following rate Eqs., (1-2) through (1-4):

5

Everyday Heat Transfer Problems

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) (1-6) + (touter wall/kouter wall) + (1/hout)] The denominator in Eq. (1-6) represents all the thermal resistances between 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

6

Heat Loss From Walls In A Typical House

Tin = 68ºF (inside temperature) Tout = 32ºF (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 material R-value is presented in Figures 1-3 and 1-4. Higher R-value insulation

1.8

Q/A, Btu/hr-ft2

1.79 1.78 1.77 1.76 1.75

5

10

15

20

25

Outside Convection Heat Transfer Coefficient, Btu/hr-ft2-F

Figure 1-1 Wall heat loss versus outside convection heat transfer coefficient

7

Everyday Heat Transfer Problems

Q/A, Btu/hr-ft2

1.8 1.78 1.76 1.74 1.72

1

3

5

7

9

Inside Convection Heat Transfer Coefficient, Btu/hr-ft2-F

Figure 1-2 Wall heat loss versus inside convection heat transfer coefficient

material is definitely the way to go, depending upon the cost and benefit analysis results. The thickness of the insulation material is also very crucial. Thicker insulation material is definitely the best choice, depending upon the cost and benefit analysis results.

Q/A, Btu/hr-ft2

2 1.8 1.6 1.4 1.2

3

3.5

4

Insulation "R" Value,

4.5 hr-ft2-F/Btu-in

Figure 1-3 Wall heat loss versus insulation R-value

8

5

Heat Loss From Walls In A Typical House

Q/A, Btu/hr-ft2

6

4

2

0

0

2

4 6 8 Insulation Thickness, in

10

12

Figure 1-4 Wall heat loss versus insulation thickness

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)/{h2out [(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.

9

Everyday Heat Transfer Problems

∂ (Q/A) / ∂hout, F

0.09

0.06

0.03

0

10 15 5 Outside Convection Heat Transfer Coefficient, Btu/hr-ft2-F

0

20

Figure 1-5 Sensitivity of house wall heat loss per unit area to the outside

convection heat transfer coefficient

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

∂(Q/A) / ∂ Rinsulation, (BTU/hr-ft2)2(in/F)

0 −0.5 −1 −1.5 −2 −2.5

0

1

2

3

4

5

6

Insulation Material R-Value, hr-ft2-F/BTU-in

Figure 1-6 Sensitivity of house wall heat loss per unit area to insulation

material R-value

10

∂ (Q/A) / ∂t insulation, BTU/hr-ft2-in

Heat Loss From Walls In A Typical House

0

−0.5

−1

−1.5

0

2

4 8 6 Insulation Thickness, in

10

12

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 is temperature 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, wall outer board R-value and thickness changes affect heat loss the most, followed by wall inside board R-value and thickness. Wall heat loss is

11

Everyday Heat Transfer Problems

Table 1-1 House wall heat loss change per unit area due to a 10% change in variables nominal values

Variable

Nominal Value

House Wall Heat Loss Change Due To A 10% Decrease In Nominal Value

Tin − Tout

36°F

−10%

+10%

Rinsulation

3.5 hr-ft -F/BTU-in

+7.467%

−6.497%

2

House Wall Heat Loss Change Due To A 10% Increase In Nominal Value

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-ft -F/BTU-in

+0.424%

−0.424%

tinner wall

1 in

+0.424%

−0.424%

2

hin

5 BTU/hr-ft -F

−0.110%

+0.090%

hout

10 BTU/hr-ft -F

−0.055%

+0.045%

2

2

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.

12

CHAPTER

CONDUCTION

HEAT TRANSFER IN A PRINTED CIRCUIT BOARD

2

C

onduction 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-state conditions 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 thermal conductivities ki, Eq. (2-1) can be rewritten as:

13

Everyday Heat Transfer Problems

W Σti 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 = Σ(kiti)/Σti

(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=Σti)/Σti = 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)

kthrough-thickness = Σti/Σ(ti/ki).

(2-8)

where

A printed circuit board is commonly built as layers of conductors separated by layers of insulators. The conductors are mostly alloys of copper, silver or gold, while the insulators are mostly a variety of

14

Conduction Heat Transfer In A Printed Circuit Board

epoxy resins. Therefore the in-plane thermal conductivity Eq. (2-4) for a PCB can be rewritten as: kin-plane = [kconductor Σtconductor + kinsulator (ttotal − Σtconductor)]/ttotal (2-9) Similarly, the through-thickness thermal conductivity Eq. (2-8) for a PCB can be rewritten as: kthrough-thickness = ttotal/[(Σtconductor/kconductor) + (ttotal − Σtconductor)/kinsulator] (2-10) The sensitivities of these two PCB thermal conductivities are analyzed for a 500 µm-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 kconductor and 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-µm. 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-µm. A ten percent variation in variables around the nominal values given above produce the sensitivity results in Table 2-1 for in-plane thermal conductivity. For these nominal values, in-plane thermal

15

Everyday Heat Transfer Problems

kin-plane, W/m-C

400 300 200 100 0

0

20

40 60 80 % Conductor Layers Thickness

100

Figure 2-1 In-plane thermal conductivity versus percent of conductor layers

∂ kin-plane / ∂kconductor & ∂ kin-plane / ∂ kinsulator

thickness to total printed circuit board thickness

1

∂kin-plane / ∂kinsulator

0.8 0.6 0.4 0.2 0

∂kin-plane / ∂kinsulator 0

20 40 60 80 % Conductor Layers Thickness

100

Figure 2-2 Sensitivity of in-plane thermal conductivity to kconductor and

to kinsulator

16

Conduction Heat Transfer In A Printed Circuit Board

Table 2-1 In-plane thermal conductivity change due to a 10% change in variables around nominal values for a 500 micron thick PCB

Nominal Value

In-Plane Thermal Conductivity Change Due To A 10% Decrease In Nominal Value

In-Plane Thermal Conductivity Change Due To A 10% Increase In Nominal Value

kconductor

377 W/m-C

−9.99%

+9.99%

Σtconductor

250 µm

−9.98%

+9.98%

Σtinsulator

250 µm

+9.98%

−9.98%

kinsulator

0.3 W/m-C

−0.01%

+0.01%

Variable

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 increases and becomes significant as the thickness percentage of the insulator layers decreases.

17

Everyday Heat Transfer Problems

kthrough-thickness, W/m-C

20 15 10 5 0

0

20

40 60 80 % Conductor Layers Thickness

100

Figure 2-3 Through-thickness thermal conductivity versus percent

∂kthrough-thickness / ∂ kconductor & ∂ kthrough-thickness / ∂ kinsulator

conductor layer thickness

20

∂ kthrough-thickness / ∂kinsulator

16 12 8 4 0

0

50 % Conductor Layers Thickness

100

∂ kthrough-thickness / ∂ kconductor

Figure 2-4 Sensitivity of through-thickness thermal conductivity to

conductor and insulator thermal conductivities versus percent conductor layer thickness

18

∂kthrough-thickness / ∂ kconductor & ∂ kthrough-thickness / ∂kinsulator, W/m-C-um

Conduction Heat Transfer In A Printed Circuit Board

0.05 ∂ kthrough-thickness / ∂ kconductor

0 −0.05 −0.1 −0.15 −0.2

0

20 40 60 80 % Conductor Layers Thickness

100

∂ kthrough-thickness / ∂ kinsulator

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

conductor and insulator thickness

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-µm 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

19

Everyday Heat Transfer Problems

Table 2-2 Through-thickness thermal conductivity change due 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 Value Nominal Value Nominal Value

Variable Σtinsulator

250 µm

+11.09%

−9.08%

kinsulator

0.3 W/m-C

−9.99%

+9.99%

Σtconductor

250 µm

−9.08%

+11.09%

kconductor

377 W/m-C

−0.01%

+0.01%

kin-plane, W/m-C

400 300 200 100 0 0

20

40 60 80 % Conductor Layers Thickness

100

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

to total thickness

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.

20

Conduction Heat Transfer In A Printed Circuit Board

∂kin-plane / ∂kconductor & ∂ kin-plane / ∂ kinsulator

1

∂ kin-plane / ∂ kconductor

0.8 0.6 0.4 0.2 0

∂ kin-plane / ∂ kinsulator 0

20 40 60 80 % Conductor Layers Thickness

100

Figure 2-7 Sensitivity of in-plane thermal conductivity to kconductor and

to kinsulator

The sensitivity of in-plane thermal conductivity to conductor thickness is a constant at 34.1 W/m-C-µm. 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-µm. 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 of conductor 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 throughthickness thermal conductivity to changes in conductor thermal conductivity is negligible at low percentages of conductor layer thickness. On the other hand, the sensitivity of through-thickness

21

Everyday Heat Transfer Problems

Table 2-3 In-plane thermal conductivity change due to a 10% change in variables around nominal values for a 10-micron thick circuit

Nominal Value

In-Plane Thermal Conductivity Change Due To A 10% Decrease In Nominal Value

In-Plane Thermal Conductivity Change Due To A 10% Increase In Nominal Value

Σtinsulator

8.5 µm

+33.3%

−33.3%

kconductor

377 W/m-C

−6.5%

+6.5%

Σtconductor

1.5 µm

−5.9%

+5.9%

36 W/m-C

−3.5%

+3.5

Variable

kinsulator

kthrough-thickness, W/m-C

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.

400 300 200 100 0

0

20

40

60

80

100

% Conductor Layers Thickness

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

percentage of conductor layer thickness

22

∂ kthrough-thickness / ∂ kconductor & ∂ kthrough-thickness / ∂kinsulator

Conduction Heat Transfer In A Printed Circuit Board

3.5 3

∂kthrough-thickness / ∂ kinsulator

2.5 2 1.5 1 0.5 0

0

20 40 60 80 % Conductor Layers Thickness

100

∂ kthrough-thickness / ∂ kconductor

Figure 2-9 Sensitivity of through-thickness thermal conductivity to

∂ kthrough-thickness / ∂kinsulator & ∂kthrough-thickness / ∂kconductor, W/m-C-µm

conductor and insulator thermal conductivities versus percentage of conductor layer thickness

10

∂ kthrough-thickness / ∂kconductor

0 −10 −20 −30 −40 −50

0

20 60 40 80 % Conductor Layers Thickness

∂ kthrough-thickness / ∂kinsulator 100

Figure 2-10 Sensitivity of through-thickness thermal conductivity to

conductor thickness and to insulator thickness

23

Everyday Heat Transfer Problems

Table 2-4 Through-thickness thermal conductivity change due to a 10% change in variables around nominal values for a 10-micron thick circuit Through-Thickness Through-Thickness Thermal Conductivity Thermal Conductivity Change Due To A 10% Change Due To A 10% Decrease In Increase In Nominal Value Nominal Value

Variable

Nominal Value

kinsulator

36 W/m-C

−9.85%

+9.18%

Σtinsulator

8.5 µm

+9.76%

−8.17%

Σtconductor

1.5 µm

−1.55%

+1.59%

kconductor

377 W/m-C

−0.18%

+0.15%

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.

24

CHAPTER

3 FROM COMBUSTION HEAT TRANSFER

CHAMBER WALLS

C

ooling the walls of a combustion chamber containing gases at high temperatures, i.e., 1000°C, 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) = = εg σ(Tg4 – Twh4)

(3-2)

And so the total heat transfer from the hot gases to a combustion chamber wall is:

25

Everyday Heat Transfer Problems

(Q/A)total = (Q/A)convection + (Q/A)radiation = (hcg + hrg) (Tg – Twh) (3-3) These two heat transfer mechanisms act as parallel thermal resistances, 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 heat transfer coefficient hrg is defined as: hrg = εg σ(Tg4 – Twh4)/(Tg – Twh)

(3-5)

where εg is emissivity of gas and σ 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

26

(3-8)

Heat Transfer From Combustion Chamber Walls

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 = 1000ºC (1273 K) Tc = 100ºC (373 K) kwall = 20 W/m-K L = 0.01m hcg = 100 W/m2-K hc = 50 W/m2-K εg = 0.2 σ = 5.67×10−8 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 3-2 and 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 parameters—thermal conductivity and wall thickness—on 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 wall temperature sensitivity to wall thickness is pretty much a constant, 3.5 C/cm.

27

Everyday Heat Transfer Problems

Iteration Function, K

2000

1000

0

−1000 400

600

800

1000

1200

1400

Twh, K

Figure 3-1 Iteration function versus Twh

Hot Side Wall Temperature, °C

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

2000 1600 1200 800 400 0

0

500

1000 1500 Gas Temperature, °C

Figure 3-2 Hot side wall temperature versus gas temperature

28

2000

Hot Side Wall Temperature, °C

Heat Transfer From Combustion Chamber Walls

1000

900

800

700

0

200

400 600 800 Cold Medium Temperature, °C

1000

Figure 3-3 Hot side wall temperature versus cold medium temperature

Hot Side Wall Temperature, °C

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.

860

840

820

800

0

10

20 30 40 Wall Thermal Conductivity, W/m-K

50

Figure 3-4 Hot side wall temperature versus wall thermal conductivity

29

Hot Side Wall Temperature, °C

Everyday Heat Transfer Problems

840 830 820 810 800

0

0.02

0.04 0.06 Wall Thickness, m

0.08

0.1

Figure 3-5 Hot side wall temperature versus wall thickness

Hot Side Wall Temperature, °C

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 gas emissivity.

1000 900 800 700 600

0

100

200

300

400

Hot Gas Side Convection Heat Transfer Coefficient,

500 W/m2-K

Figure 3-6 Hot side wall temperature versus hot gas side convection heat

transfer coefficient

30

Hot Side Wall Temperature, °C

Heat Transfer From Combustion Chamber Walls

1000 800 600 400 200

0

100

200

300

400

500

Cold Medium Side Convection Heat Transfer Coefficient, W/m2-K

Figure 3-7 Hot side wall temperature versus cold medium side convection

heat transfer coefficient

Hot Side Wall Temperature, °C

When the nominal values of the variables given above are varied ±10%, the results shown in Table 3-1 are obtained. Hot side wall temperature sensitivities to a ±10% change in the governing variables are given in descending order of importance,

950 900 850 800 750 700

0

0.2

0.4

0.6

0.8

1

Hot Gas Emissivity

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

31

Everyday Heat Transfer Problems

Table 3-1 Effects of ±10% change in nominal values of variables to hot side wall temperature

Nominal Value

Change In Hot Side Wall Temperature For A 10% Decrease In Nominal Value

Change In Hot Side Wall Temperature For A 10% Increase 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%

εg

0.2

−0.912%

+0.860%

Tc

373 K

−0.295%

+0.295%

20 W/m-K

+0.056%

−0.046%

0.01 m

−0.051%

+0.051%

Variable

kwall L

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.

32

CHAPTER

HEAT TRANSFER

FROM A HUMAN BODY DURING SOLAR TANNING

4

T

he 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 portions of 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)

33

Everyday Heat Transfer Problems

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, α = ε, 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 30°C. Qconduction to body = (kbody/tbody)(Tskin – Tbody)

(4-5)

where kbody = 0.2 W/m-K, tbody = 0.1 m, and Tbody = 37°C. Qradiation emitted = εσ(T4skin – T4environment)

(4-6)

where emissivity of skin surface ε = 0.8 and σ = 5.67 × 10−8 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

34

Heat Transfer From A Human Body During Solar Tanning

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 = εσ, C2 = h + kbody/tbody, and C3 = Qsolar radiation absorbed + Qatmospheric radiation absorbed – Qperspiration − Qbasal metabolism + εσT4environment + 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 surface—than 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 give equal 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

35

Everyday Heat Transfer Problems

Table 4-1 Effects of ±10% change in nominal values of variables to skin temperature

Variable

Nominal Value

Skin Temperature, Tskin, Change Due To A 10% Decrease In Nominal Value

Tenvironment

30°C

−6.03%

+6.03%

−5.11%

+5.13%

Qsolar radiation absorbed

851 W/m

2

Skin Temperature, Tskin, Change Due To A 10% Increase In Nominal Value

h

28.4 W/m -K

+2.93%

−2.51%

Qperspiration

337.5 W/m

+2.03%

−2.03%

0.8

+0.53

−0.51

85 W/m2

−0.51%

+0.51%

37°C

−0.45%

+0.45%

+0.27%

−0.27%

Emissivity of skin surface, ε Qatmospheric radiation absorbed Tbody

2

2

Qbasal metabolism

45 W/m

Skin-to-body conduction length, tbody

0.1 m

−0.12%

+0.10%

0.2 W/m-K

+0.11%

−0.11%

ktissue

2

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.

36

Heat Transfer From A Human Body During Solar Tanning

Tenvironment=30°C, Tbody=37°C, Qsolar radiation absorbed=851 W/m2 Skin Temperature, °C

70 65 Perspiration=0.5 liters/hr Perspiration=1.0 liter/hr Perspiration=1.5 liter/hr

60 55 50 45 40 35 30

0

100

50

150

200

Convection Heat Transfer Coefficient, W/m2-K

Figure 4-1 Skin temperature versus convection heat transfer coefficient

for Qsolar radiation absorbed = 851 W/m2 and for three different perspiration rates

∂ Tskin / ∂h, m2-K2/W

0 −0.2

Perspiration=0.5 liters/hr Perspiration=1.0 liters/hr Perspiration=1.5 liters/hr

−0.4 −0.6 −0.8 −1

0

50

100

150

200

Convection Heat Transfer Coefficient, W/m2-K

Figure 4-2 Skin temperature sensitivity to convection heat transfer

coefficient versus convection heat transfer coefficient for Qsolar radiation absorbed = 851 W/m2 and for three different perspiration rates

37

Everyday Heat Transfer Problems

Tenvironment=30°C, Tbody=37°C, Qsolar radiation absorbed=425.5 W/m2 Skin Temperature, °C

45 Perspiration=0.5 liters/hr Perspiration=1.0 liter/hr Perspiration=1.5 liter/hr

40 35 30 25

0

50

100

150

200

Convective Heat Transfer Coefficient, W/m2-K

Figure 4-3 Skin temperature versus convection heat transfer coefficient

for Qsolar radiation absorbed = 425.5 W/m2 and for three different perspiration rates

∂ Tskin / ∂h, m2-K2/W

0.2 0.1 0

Perspiration=0.5 liters/hr Perspiration=1.0 liters/hr Perspiration=1.5 liters/hr

−0.1 −0.2 −0.3 −0.4 −0.5

0

50

100

150

200

Convection Heat Transfer Coefficient, W/m2-K

Figure 4-4 Skin temperature sensitivity to convection heat transfer

coefficient versus convection heat transfer coefficient for Qsolar radiation absorbed = 425.5 W/m2 and for three different perspiration rates

38

Heat Transfer From A Human Body During Solar Tanning

60

Tskin, C

55 50 45 40 35 30

20

24

28 32 Tenvironment, C

36

40

Figure 4-5 Skin temperature versus environment temperature

In high perspiration rates, the skin temperature is below the environmental temperature, and it approaches the environmental temperature as the heat transfer coefficient increases.

Tskin, C

50

45

40

35 0.5

0.6 0.7 0.8 0.9 Emissivity Of Human Skin Surface, ε

1

Figure 4-6 Skin temperature versus emissivity of human skin surface

39

Everyday Heat Transfer Problems

∂ Tskin / ∂ε, C

30

28

26

24 0.5

0.6

0.7

0.8

0.9 Emissivity Of Human Skin Surface, ε

1

Figure 4-7 Skin temperature sensitivity to emissivity of human skin surface

versus emissivity of human skin surface

Temperature of the skin behaves as shown in Figure 4-5 with a constant sensitivity to the changes in the temperature of the environment, in the domain of interest. The sensitivity behavior is almost one-to-one; namely, ∂Tskin/∂Tenvironment equals 0.92. Another variable that is analyzed in detail in the region of interest is the emissivity of human skin surface. Skin temperature increases with an increase in emissivity as shown in Figure 4-6. Figure 4-7 provides the skin temperature sensitivity to emissivity of human skin surface. The sensitivity decreases in a linear fashion as the emissivity increases.

40

CHAPTER

EFFICIENCY OF

RECTANGULAR FINS

5

H

eat 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 distribution along 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

41

Everyday Heat Transfer Problems

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 η = 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 fin length, Lc, for the heat lost from the tip of the fin. η = 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

42

(5-7)

Efficiency Of Rectangular Fins

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 of the 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: ∂η/∂h = (0.5/h)[(1/cosh2(mLc)) – (tanh(mLc)/(mLc))]

(5-8)

Fin efficiency as a function of the convection heat transfer coefficient for two different thermal conductivities—aluminum and copper—is 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

1 Fin Efficiency

0.9 kcu=377.2 W/m-C

0.8 0.7

kal=206 W/m-C

0.6 0.5 0.4

0

100

200

300

Convection Heat Transfer Coefficient,

400

L=0.0508 m t=0.002 m

W/m2-C

Figure 5-1 Rectangular fin efficiency versus convection heat transfer

coefficient for two different fin materials with L = 0.0508 m and t = 0.002 m

43

Everyday Heat Transfer Problems

0

∂h/∂ h, m2-C/W

−0.001

kcu=377.2 W/m-C kal=206 W/m-C

−0.002 −0.003

L=0.0508 m t=0.002 m

−0.004 −0.005

0 100 200 300 400 Convection Heat Transfer Coefficient, W/m2-C

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

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 coefficients—natural convection regime and forced convection regime—is given in Figure 5-3. The sensitivity of rectangular fin efficiency with respect to 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 thermal conductivities. The sensitivity diminishes as high fin material thermal conductivities are utilized.

44

Efficiency Of Rectangular Fins

Fin Efficiency, η

1 h=5 W/m2-C

0.8

h=100 W/m2-C

0.6

0.4

0

100 200 300 400 Fin Thermal Conductivity, W/m-C

500

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

∂ η/∂ k, m-C/W

0.0084 0.0063

h=5 W/m2-C

0.0042

h=100 W/m2-C

0.0021 0

L=0.0508 m t=0.002 m 0

100

200

300

400

500

Fin Thermal Conductivity, W/m-C

Figure 5-4 Sensitivity of rectangular fin efficiency to fin material thermal

conductivity versus fin material thermal conductivity for two different convection heat transfer coefficients with L = 0.0508 m and t = 0.002 m

45

Everyday Heat Transfer Problems

Fin Efficiency, η

1 0.9

h=5 W/m2-C

0.8

h=100 W/m2-C

0.7 0.6

t=0.002 m k=377.2 W/m-C 0

0.02

0.04

0.06

0.08

0.1

Fin Length, m

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

Fin efficiency as a function of fin length for two different convection heat transfer coefficients—natural convection regime and forced convection regime—is 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 results can 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 coefficients—natural convection regime and forced convection regime—is given in Figure 5-7. The sensitivity of rectangular fin efficiency with respect to fin thickness is given in Figure 5-8. Figure 5-8 shows sensitivities for combinations of two

46

Efficiency Of Rectangular Fins

8 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

7 ∂ η/∂ L, 1/m

6 5 4 3 2 1 0

t=0.002 m 0

0.05 Fin Length, m

0.1

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

1

Fin Efficiency, η

0.9 h=5 W/m2-C h=100 W/m2-C

0.8 0.7 0.6

k=377.2 W/m-C L=0.0508 m

0.5 0.4

0

0.001

0.002 0.003 Fin Thickness, m

0.004

0.005

Figure 5-7 Rectangular fin efficiency versus fin thickness for two different

convection heat transfer coefficients with k = 377.2 W/m-C and L = 0.0508 m

47

Everyday Heat Transfer Problems

9 8

∂η/∂ t, 1/m

7

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

6 5 4 3 2 1 0

L=0.0508 m 0

0.001

0.002

0.003

0.004

0.005

Fin Thickness, 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

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 of fin 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 convection regime in descending order, from the most sensitive variable to the least. Table 5-2 gives similar results for the forced convection regime.

48

Efficiency Of Rectangular Fins

Table 5-1 Rectangular fin efficiency change due to a 10% change in variables nominal values for the natural convection regime

Variable

Nominal Value

Rectangular Fin Efficiency Change Due To A 10% Decrease 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/m -C

+0.117%

−0.116%

2

Rectangular Fin Efficiency Change Due To A 10% Increase In Nominal Value

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 5-2 Rectangular fin efficiency change due to a 10% change in variables nominal values for the forced convection regime

Variable

Nominal Value

Rectangular Fin Efficiency Change Due To A 10% Decrease In Nominal Value

Rectangular Fin Efficiency Change Due To A 10% Increase 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/m -C

+1.806%

−1.729%

2

49

CHAPTER

HEAT

TRANSFER FROM A HOT DRAWN BAR

6

A

hot drawn bar, assumed to be moving at a constant velocity out of a die at constant temperature, will be treated as a one-dimensional 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 temperature 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 coefficient and radiation heat transfer coefficient, D is the bar diameter, and k is the bar thermal conductivity in W/m-K. Conduction, convection, and radiation heat transfer mechanisms affect the temperature of 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 energy of the element

51

Everyday Heat Transfer Problems

The energy balance on the element can be written as follows: Qconduction at x – Qconduction at x+dx – Qconvection from dx – Qradiation from dx = ρcpAdx(dT/dθ) (6-1) where ρ 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/dθ is the drawn bar velocity in m/s. Assuming that all the bar thermophysical and geometrical properties are constants, the following one-dimensional, second-order and non-linear differential equation is obtained: d2T/dx2 – (hP/kA)(T-Tenvironment) – (σεP/kA)(T4 – T4environment) = (ρcpU/k)dT/dx

(6-2)

where P is the bar perimeter in m, σ is Stefan Boltzmann constant, 5.67×10-8 W/m2-K4, ε is the bar surface emissivity, and U=dx/dθ the speed of the hot drawn bar. The differential equation (6-2) reduces to 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 methods and iteration, in order to determine the temperature at the i'th location along the bar.

52

Heat Transfer From A Hot Drawn Bar

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 = σε(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) = (ρcpU/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/2α) – sqrt((U/2α)2 + m2)]x}

(6-7)

where α = k/ρcp 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 heat transfer 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:

53

Everyday Heat Transfer Problems

D = 0.01 m ρ = 8000 kg/m3 cp = 450 J/kg-K k = 40 W/m-K ε = 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 a drawn 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/νair. 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/νair)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 νair 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 magnitude around the die. As the bar travels away from the die, radiation heat transfer diminishes rapidly.

54

Heat Transfer Coeffieicents W/m2-K

Heat Transfer From A Hot Drawn Bar

50 hconvection

40 30 20 10 0

hradiation 0

2

4 6 8 Distance From Die, m

10

Figure 6-1 Radiation and convection heat transfer coefficients as a function

of distance from die

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 very sensitive 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 heat transfer coefficient is shown in Figure 6-5. Increasing the convection heat transfer coefficient reduces its effects on bar temperature.

55

Everyday Heat Transfer Problems

Temperature, K

1400 1200 1000

Without Radiation With Radiation

800 600 400 200 0

0

2 6 4 8 Distance From Die, m

10

Figure 6-2 Hot drawn bar temperature with and without radiation heat

transfer effects

1400 hconvection=5 W/m2-K hconvection=20 W/m2-K hconvection=40 W/m2-K hconvection=60 W/m2-K

Temperature, K

1200 1000 800 600 400 200

0

2

4

8 6 Distance From Die, m

10

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

fer coefficients with hradiation=0

56

∂T/∂ hconvection, m2-K2/W

Heat Transfer From A Hot Drawn Bar

0 −10 −20 −30 −40 −50

0

10

20

30

40

50

hconvection, W/m2-K

Figure 6-4 Bar temperature sensitivity @ x=10 m to convection heat trans-

Bar Temperature @ x=10 m, K

fer coefficient

1000 800 600 400 200

0

10

20 30 40 hconvection, W/m2-K

50

60

Figure 6-5 Bar temperature @ x=10 m as a function of convection heat

transfer coefficient

57

Bar Temperature @ x=10 m, K

Everyday Heat Transfer Problems

400

380

360

340 7000

7500

8000 Bar Density,

8500

9000

kg/m3

Figure 6-6 Bar temperature @ x=10 m versus bar density

Bar temperature at x=10 m varies close to a linear behavior with bar density, as shown in Figure 6-6. ?T/?ρ 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 weak function 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