Heat Transfer Solutions - James Van Sent

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Conduction heat transfer solutions

James H. VanSant

u

March 1980

.:'•• •:•.•! i s UHUMITEE

CONTENTS

Preface

v

Nomenclature Introduction

vii .

1

Steady-State Solutions 1.

Plane Surface - Steady State 1.1 1.2

Solids Bounded by Plane Surfaces

1-1

Solids Bounded by Plane Surfaces —With Internal Heating

2.

Cylindrical Surface - Steady State 2.1 2.2

3.

1-27

Solids Bounded by Cylindrical Surfaces — N o Internal Heating Solids Bounded by Cylindrical Surfaces —With Internal Heating

2-1 2-33

Spherical Surface - Steady State 3.1

Solids Bounded by Spherical Surfaces — N o Internal Heating

3.2

Solids Bounded by Spherical Surfaces

3-1

—With Internal Heating 4.

4.1 5.

3-10

Traveling Heat Sources Traveling Heat Sources

4-1

Extended Surface - Steady State 5.1

Extended Surfaces—No Internal Heating

5.2

Extended Surfaces—With Internal Heating

. •

5-1 5-31

Transient Solutions 6.

7.

8.

Infinite Solids - Transient 6.1

Infinite Solids—No Internal Heating

6-1

6.2

Infinite Solids—With Internal Heating

6-22

Semi-Infinite Solids - Transient 7.1

Semi-Infinite Solids—No Internal Heating

7.2

Semi-Infinite Solids—With Internal Heating

7-1 . . . .

7-22

Plane Surface - Transient 8.1 8.2

Solids Bounded by Plane Surfaces — N o Internal Heating Solids Bounded by Plane Surfaces —With Internal Heating

iii

8-1 8-52

9.

Cylindrical Surface - Transient 9.1 9.2

10.

10.2

12.

9-1

S o l i d s Bounded by C y l i n d r i c a l Surfaces — W i t h Internal Heating

9-24

Spherical Surface - Transient 10.1

11.

S o l i d s Bounded by Cylindrical Surfaces —No Internal Heating

Solids Bounded by Spherical Surfaces —No Internal Heating S o l i d s Bounded by Spherical Surfaces —With Internal Heating

10-1 10-19

Change of Phase 11.1

Change of Phase—Plane Interface

.

11.2

Change of Phase—Nonplanar Interface

.

, ' .

.

.

.

.

n-i 11-13

Traveling Boundaries 12.1

Traveling Boundaries

12-1

Figures and Tables for Solutions

F-i

Miscellaneous Data 13.

Mathematical Functions

13-1

14.

Roots of Some Characteristic Equations

14-1

15.

Constants and Conversion Factors

15-1

16.

Convection Coefficients

16-1

17.

Contact C o e f f i c i e n t s

17-1

18.

Thermal Properties

.

16-1

References

R-l

iv

PREFACE

This text is a collection of solutions to a variety of heat conduction problems found in numerous publications, such as textbooks, handbooks, journals, reports, etc.

Its purpose is to assemble these solutions into one

source that can facilitate the search for a particular problem solution. Generally, it is intended to be a handbook on the subject of heat conduction. Engineers, scientists, technologists, and designers of all disciplines should find this material useful, particularly those who design thermal sys­ tems or estimate temperatures and heat transfer rates in structures.

More

than 500 problem solutions and relevant data are tabulated for easy retrieval. Having this kind of material available can save time and effort in reaching design decisions. There are twelve sections of solutions which correspond with the class of problems found in each. Geometry, state, boundary conditions, and other cate­ gories are used to classify the problems. A case number is assigned to aach problem for cross-referencing, and also for future reference. Each problem is concisely described by geometry and condition statements, and many times a descriptive sketch is also included.

At least one source reference is given

so that the user can review the methods used to derive the solutions.

Problem

solutions are given in the form of equations, graphs, and tables of data, all of which are also identified by problem case numbers and source references. The introduction presents a synopsis on the theory, differential equa­ tions, and boundary conditions for conduction heat transfer. is given on the use and interpretation of solutions. lem solutions are included.

Some discussion

Also, some example prob­

This material may give the user a review, or ever

some insight, on the phenomenology of heat conduction and its applicability tc specific problems. Supplementary data such as mathematical functions, convection correla­ tions, and thermal properties are included for aiding the user in computing numerical values from the solutions.

Property data were taken from some of

the latest publications relating to the particular properties listed. the international system of units (SI) is used.

v

Only

Consistency in nomenclature and terminology is used throughout, making this text more readable than a collection of different references.

Also, dimension-

less parameters are frequently used to generalize the applicability of the solutions and to permit easier evaluation of the effects of problem conditions. Even though some of the equational solutions are lengthy and include several different mathematical functions, this should not pose a formidable task for most users.

Modern computers can make complicated calculations easy

to perform.

Even many electronic calculators can be used to compute complex

functions.

If, however, these tools are not available, one can resort to hand

computing methods.

The table of mathematical functions and constants would be

useful in this case. Heat conduction has been studied extensively, and the number of published solutions is large. in this text.

In fact, there are many solutions that are not included

For example, some solutions are found by a specific computa­

tional process that cannot be described briefly. Moreover, new solutions are constantly appearing in technical journals and reports. Nevertheless, this collection contains most of the published solutions. The differential equations and boundary-condition equations for heat flow are identical in form to those for other phenomena such as electrical fields, fluid flow, and mass diffusion.

This similarity gives additional utility to

the heat conduction solutions. The user needs only identify equivalence of conditions and terms when selecting a proper solution.

This practice is pre­

scribed in many texts on applied mathematics, electrical theory, heat transfer, and mass transfer. A search for particular solutions has frequently been a tedious and dif­ ficult task. Too often, countless hours have been spent in searching for a problem solution.

Locating and obtaining a proper reference can require con­

siderable effort.

Also, it is frequently necessary to study a theoretical

development in order to find the applicable solution.

In so doing, there are

sometimes misinterpretations which lead to erroneous results.

This text

should help alleviate some of these problems. Science gives us information for reaching new frontiers in technology. It is, thus, appropriate to give something back.

I hope this text is at least

a small contribution. James H. VanSant

VI

NOMENCLATURE A

=

2 A r e a , ro

b

=

Time constant, s

c

=

S p e c i f i c heat, J/kg* C

C = Circumference, m d, D =

Diameter, depth, m

h

=

Heat transfer c o e f f i c i e n t , W/m • C

k

=

Thermal conductivity, W/m* C

m =

1

"V hC/kA, m""

d, L = q

=

Length, m

2 Heat flux r a t e , W/m

%,' %.>

For isotropic homogeneous media this becomes 2

2

2

3t _ k_ 3 t , 3 t ^ 3 t l + ^ — 3 T " pc 3x 3y 3z J pc 2

2

a

= art +

.

(16)

2

pc

When q'" = 0 , Eq. (16) becomes Fourier's equation. In steady-state conditions, 3t/3x = 0 and Eg. (16) becomes the Poisson equation. When q ' " = 3t/3x = 0, Eg. (16) reduces to the Laplace equation. Nonisotropic materials, such as laminates, can have directionally sensitive properties. For such materials the conduction differential equation in two dimensions is expressed in the following form: 2 pc || = ( k cos B + k sin B ) ^-| + l k sin B + 1^ cos B ) 2

?

2

n

2

2

?

2 ~

2 + (k -k )(sin B)y - + q'" , 2

c

7

T 1

y

(17)

•-X

FIG. 4. Coordinate system for a nonisotropic medium.

where k_ and k are directional thermal conductivities, and 3 is the angle of laminations as indicated in Fig. 4. When the geometrical axes of the nonisotropic material are oriented with the principal axes of the thermal conductivities, then Eg. (17) simplifies to the form of Eq. (14)

n

3

p c

2.2

+

t

X

a?

4 * *4+*-' 3x 3y 2

(18)

2

Cylindrical Coordinate System Rectangular coordinates can be transformed into cylindrical coordinates

by the relations x = r cos 9, y = r sin 9, and z = z. The partial differential equations (15) and (16) transformed to cylindrical coordinates are thus

P c

=

37 7 37 (

t _ » (& T

U

2

+

3?) 7 3 9 \

rk

+

i & r 8 r

k

39 ) 3l ( 37) +

k

i_ i*t . aft\ a

r ae

2

+

3^1

q

(19)

(20)

p c

a.V

For nonisotropic materials with the conductivity and geometry axes aligned as in Eq. (18) the differential equation is

c

P 3?"r

r

3rl 3r)

+ r

2

3

e

2

+

k Z

8

3

z

2

+

*

(21)

2.3

Spherical Coordinate System A transformation from rectangular t o spherical coordinates can be accom­

plished by s u b s t i t u t i n g the r e l a t i o n s x = r s i n i|i cos ij>, y = r s i n ty s i n § and z = cos \|i into Eqs. (15) and (16), which y i e l d the partial d i f f e r e n t i a l equations for i s o t r o p i c heterogeneous and homogeneous m a t e r i a l s , r e s p e c t i v e l y .

3t 1 3 / 2 . 3t\ 1 3 /. 3 t \ P° 37 - ~2 3T( 37) * 2 . 2 ^ 36 ( 39 j A

r

k

k

s

n

+

JL

—— IV ( r

2

at *

m

/a t

2 at E

" "W

3

2

1

t

r

2

sin

3 t 2

sin * 36

r

2

+ q

^ If) '"

x

ty

(22)

'

1

2

k sin

3t\ 8

tan * * )