Heat Transfer Solutions - James Van Sent

/f^

Conduction heat transfer solutions

James H. VanSant

u

March 1980

Preface v Nomenclature vii Introduction . 1 Steady-State Solutions

1. Plane Surface - Steady State

1.1 Solids Bounded by Plane Surfaces 1-1 1.2 Solids Bounded by Plane Surfaces

—With Internal Heating 1-27 2. Cylindrical Surface - Steady State

2.1 Solids Bounded by Cylindrical Surfaces

— N o Internal Heating 2-1 2.2 Solids Bounded by Cylindrical Surfaces

—With Internal Heating 2-33 3. Spherical Surface - Steady State

3.1 Solids Bounded by Spherical Surfaces

— N o Internal Heating 3-1 3.2 Solids Bounded by Spherical Surfaces

—With Internal Heating 3-10 4. Traveling Heat Sources

4.1 Traveling Heat Sources 4-1 5. Extended Surface - Steady State

5.1 Extended Surfaces—No Internal Heating . • 5-1 5.2 Extended Surfaces—With Internal Heating 5-31 Transient Solutions

6. Infinite Solids - Transient

6.1 Infinite Solids—No Internal Heating 6-1 6.2 Infinite Solids—With Internal Heating 6-22 7. Semi-Infinite Solids - Transient

7.1 Semi-Infinite Solids—No Internal Heating 7-1 7.2 Semi-Infinite Solids—With Internal Heating . . . . 7-22

8. Plane Surface - Transient

8.1 Solids Bounded by Plane Surfaces

— N o Internal Heating 8-1 8.2 Solids Bounded by Plane Surfaces

—With Internal Heating 8-52

9 . C y l i n d r i c a l Surface - Transient

9.1 S o l i d s Bounded by C y l i n d r i c a l Surfaces

—No Internal Heating 9-1 9.2 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 10. Spherical Surface - Transient

10.1 Solids Bounded by Spherical Surfaces

—No Internal Heating 10-1 10.2 S o l i d s Bounded by Spherical Surfaces

—With Internal Heating 10-19 11. Change of Phase

1 1 . 1 Change of Phase—Plane Interface . . , ' . . . n - i

11.2 Change of Phase—Nonplanar Interface 11-13 12. Traveling Boundaries

12.1 Traveling Boundaries 12-1 Figures and Tables for Solutions F-i Miscellaneous Data

13. Mathematical Functions 1 3 - 1 14. Roots of Some C h a r a c t e r i s t i c Equations 14-1

15. Constants and Conversion Factors 15-1

16. Convection C o e f f i c i e n t s 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

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. Some discussion is given on the use and interpretation of solutions. Also, some example prob­ lem solutions are included. 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. Only the international system of units (SI) is used.

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 fact, there are many solutions that are not included in this text. 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

2 A = 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 c o n d u c t i v i t y , W/m* C m = "V hC/kA, m""1 d, L = Length, m 2 q = Heat flux r a t e , W/m 2 %,' %.> <3_ = Heat flux i n x , y , z d i r e c t i o n s , W/m q "1 = Volumetric heating r a t e , W/m Q = Heat transfer r a t e , W r, R = Radius, m t , T = Temperature, C, K Y = V e l o c i t y , m/s w = Width, m x, y , z , = Cartesian c o o r d i n a t e s , m 2 a = Thermal d i f f u s i v i t y , k / p c , m / s B = Temperature c o e f f i c i e n t , C Y = Heat of evaporation, J/kg Y = Latent heat of f u s i o n , J/kg A = Difference

e = E m i s s i v i t y for thermal radiation

„. ... ,_. ( a c t u a l heat transferred) n = Fin e f f e c t i v e n e s s , )heat transferred without f i n s )

. _. „ . ( a c t u a l heat transferred <p = f i n e r r e c t i v e n e s s , ( h e a f c tTaRStsct^ from i n f init e c o n d u c t i v i t y f i n s ) p = D e n s i t y , kg/m a = 2 "V a x , m - 2 - 4 a = Stefan-Boltzmann c o n s t a n t , W/m *K T = Time, s

Jf = Radiation configuration—emissivity factor

DIMENSIONLESS GROUPS

Bi

v

B

h'

P o

r

2 Fo = Fourier modulus = at/X Fo

*v

*

, Fo = Modified Fourier modulus = l/(: Fo

*v

Gr = Grashof number = gSAtH /v

Ki

v

N ud, Nu = Nusselt number = hd/k

*\, Pd = Predvoditelev modulus = bS,2 /a

*°v

R = Radius ratio = r/r

R ed, Re = Reynolds number = vd/V X = Length ratio = x/SL

Y = Width ratio = y/w Z = Length ratio = z/5.

MATHEMATICAL FUNCTIONS

exp = Exponential function Ei = Exponential integral erf = Error function

erfc = Complementary error function

inerfc = Complementary error function integral I = Modified Bessel function of the first kind

n

J = Bessel function of the first kind n

K = Modified Bessel function of the second kind n

&n = Natural log

Y = Bessel function of the second kind n

P = Legendre polynomial of the first kind r = Gamma function

1 . HEAT CONDUCTION

E n e r g y i n t h e form o f h e a t h a s been u s e d by man e v e r s i n c e he b e g a n

walking on t h i s earth. Moreover, the transfer of heat i s e s s e n t i a l t o our very e x i s t e n c e . Not only do our own p h y s i o l o g i c a l functions require some form of heat t r a n s f e r , but so do most l i f e - s u s t a i n i n g p r o c e s s e s of nature and many man-c o n t r o l l e d a man-c t i v i t i e s . The importanman-ce of the thermal sman-cienman-ces i n the t o t a l sphere of s c i e n c e can, t h u s , hardly be disputed.

Conduction i s one of the three principal heat transfer modes, the others being convection and r a d i a t i o n . I t i s customarily distinguished as being an energy d i f f u s i o n process i n m a t e r i a l s which do not contain molecular convection. Kinetic energy i s exchanged between molecules r e s u l t i n g in a net t r a n s f e r between regions of d i f f e r e n t energy l e v e l s , t h e s e energy l e v e l s are commonly c a l l e d temperature. P a r t i c u l a r l y , heat conduction i n metals i s mainly a t t r i ­ buted t o the motion of f r e e e l e c t r o n s and i n s o l i d e l e c t r i c a l i n s u l a t o r s t o the l o n g i t u d i n a l o s c i l l a t i o n s of atoms. In f l u i d s , the e l a s t i c impact of molecules i s considered as the heat conduction process.

The process of heat t r a n s f e r i n materials has been studied for many c e n t u r i e s . Even early Greek philosophers, such as Lucretius ( c . 98-55 B . C . ) , meditated on the subject and recorded t h e i r c o n c l u s i o n s . Much l a t e r , the famous mathematical p h y s i c i s t , Joseph B. J. Fourier (1768-1830), developed a mathematical expression t h a t became the b a s i s of p r a c t i c a l l y a l l heat conduc­ t i o n s o l u t i o n s . He p o s t u l a t e d that a l o c a l heat f l u x rate in a m a t e r i a l i s proportional t o the l o c a l temperature gradient i n the direction of heat flow:

where g^ i s the heat flow i n the x - d i r e c t i o n per u n i t area as i l l u s t r a t e d i n F i g . l a . Material p r o p e r t i e s are accounted f o r by including a p r o p o r t i o n a l i t y constant:

* X = - K ! X - , (2,

•

I

t

/{///ft

FIG. 1. I l l u s t r a t i o n of heat flow and temperature gradient.

there the constant k i s c a l l e d thermal c o n d u c t i v i t y . (The minus s i g n must be .ncluded to s a t i s f y the second law of thermodynamics.) This equation i s a i l e d Fourier's law for heterogeneous i s o t r o p i c continua.

A simpler form of F o u r i e r ' s law i s for homogeneous i s o t r o p i c continua. For example, consider a p l a t e of t h i s of type material having isothermal surfaces md insulated edges as shown in F i g . 1. The p l a t e has a width Ax, surface area A, and thermal conductivity k. The heat flow in the p l a t e i s expressed as

Qx = -= -kA ]

( t

2 - V

Ax (3)

Chis expression becomes Eg. (2) when Ax diminishes to zero.

fx lim At . 3t

A

%.

~

Ax+O Ax

"

3x

The heat f l u x q i s presumed t o have both magnitude and d i r e c t i o n . Thus, Lt can be given as a v e c t o r , g which i s normal t o an isothermal s u r f a c e . For example, i n Cartesian coordinates

q = qxi +

j

+

* z

Since Eq. (4) defines q^ = -k3t/3x, and similarly qy = -k3t/3y, q

= -k3t/3z,

we can state

q = -k (i3t/3x + j3t/3y + k3t/3z) (6)

i = -kVt . (7)

In anisotropic continua the direction of the heat flux vector i s not

necessarily normal to an isothermal surface. Example materials are crystals,

laminates, and oriented fiber composites. In such materials we may assume

each component of the heat flux vector to be linearly dependent on a l l com­

ponents of the temperature gradient at a point. The vector form of Fourier's

law for heterogeneous anisotropic continua becomes

q = -K • Vt , (8)

where K i s the conductivity tensor; the components of this tensor are called

the conductivity coefficients.

In Cartesian form, Eq. (8) i s

*x -

-("ll

£

12

£

*13

H)

S ' -(

21 fe

*22 ft

23 l l ) <

>

/ 3t 3t 3t\

%

' ^ 3 1 3x

32 3y

33 Szj *

To compute heat flow by Fourier's law, a thermal conductivity value i s

needed. I t can be estimated fran theoretical predictions for some ideal

materials, but mostly, i t i s determined by measurement and Fourier's law,

Eq. (2* -. As illustrated in Fig. 2, thermal conductivity can have a large

range, which depends on materials and temperature. For example, copper at

20 K has a thermal conductivity of approximately 1000 W/m*K and diatomaceous

earth at 200 K has a conductivity of 0.05 W/m'K. Consequently, heat flow in

materials can have a very large range, depending on a combined effect of

temperature gradient and material property.

Thermal properties of some selected materials are given in Section IS.

1000

_{1 r}

1

_L

200

400

FIG. 2. Thermal conductivity of some selected solids.

Solutions to heat conduction problems are usually found by some mathe­ matical technique which begins with a differential equation of the temperature field. The^appropriate equation should include all energy sources and sinks pertinent to a particular problem. Also, the equation should be expressed in terms of a convenient coordinate system such as rectangular, cylindrical, or spherical. Then analytical or differencing methods can be used to solve for temperature or heat flow.

A common method for deriving the generalized differential equation for heat conduction is to apply the first law of thermodynamics (conservation of energy) to a volume element in a selected coordinate system. By accounting for all the thermal energy transferred through the element faces, the change of internal energy and thermal sources or sinks in the element, and by letting the element dimensions approach zero, the differential equation can be derived. This procedure is typified by the following heat energy accounting of the rectangular solid element shown in Fig. 3. The net heat flow through six faces is

"net x+Ax *y ~ ^ + A y z ~ °z+Az ' (10)

FIG. 3. Coordinate systems for heat conduction equations: (a) rectangular, (b) cylindrical, (c) spherical.

where t y p i c a l l y

Q = -AyAz

Q = -AxAz

y

Ml'

Q = -AxAy

and k , k , and k are d i r e c t i o n a l c o n d u c t i v i t i e s . x ' y z

An i n c r e a s e in i n t e r n a l energy of the element i s represented by

A I = AxAyAzpc | ^ , (11)

where t is the mean temperature of the element, p is the material density,

and c is its specific heat.

Internal energy sources can be expressed as

QUI

t i i

A y A

(12)

where

q" '

is the unit volume source rate. Examples of internal heating in

materials are joule, nuclear, or radiation heating. Summing these energies in

accordance with the energy conservation law yields

For the limits Ax, Ay, Az -»• 0, we obtain

3t BCL.

3qy 3 q

pc

3T

3x 3y 3z »

. 3t

q

x

x 3x '

qy " " y 3y '

q = - k -5— . ^z z oz

Using Eg. (14) as a general differential equation, we can derive the

following specific equations.

2.1 Rectangular Coordinate System

For isotropic heterogeneous media

D c

3 t

3_/ 3t\ 3_/ i t \ ^ i _ / 3t\

pc

3x

\

3

/

3y \

Syj 3z \

3

/

"

For isotropic homogeneous media t h i s becomes

3t _ k_

" pc

3

t , 3

t ^ 3

t l

3x

3y

3 z

J

+ ^ — = art +

. (16)

pc 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 2

pc | | = ( k

cos

B + k

s i n

B ) ^-| + l k

sin

B + 1^ cos

B )

~

2

+ ( k

- k

) ( s i n

B ) y

- + q ' " , (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 3t

p c

a ?

4

+

* 4+*-'

(18)

2.2 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

Pc

37

7 37 (

3?

)

7 3 9

\

39

)

3l (

37

)

(19)

t _ » (&

i & i_ i*t . aft\ 3^1

T

U

r

a e

a.V

(20)

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

P

3 ? " r 3 r l

3 r )

2

2

*

(21)

A transformation from rectangular t o s p h e r i c a l coordinates can be accom­ p l i s h e d 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 i n t o Eqs. (15) and (16), which y i e l d the p a r t i a l 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 . 3 t \

1 3 /. 3 t \

P° 37 - ~2 3T(

37) * 2

.

2 ^ 36 (

39 j

—

— IV (

^ I f )

' "

r s i n ty

'

a t

/ a

t 2 a t 1 3

t 1 3 t \ <?•"

The d i f f e r e n t i a l equation for nonisotropic materials with a l i g n e d conductivity and geometric axes i s

n

„ 3t .

r 3

t

d) 3

t , *Sl» 3 . ,., 3t

PC j r = — — 7 + - 5 — * - = 5- + - ^ JT. s i n \p •55- + q' . (24) 3 T r

3 r

r

s i n

i|» 3<|)

r

s i n *

* **

3 . SPECIAL DIFFERENTIAL EQUATIONS

Some defining equations can have implied assumptions and boundary

c o n d i t i o n s . They are u s u a l l y employed i n s p e c i a l cases t o s i m p l i f y the method of s o l u t i o n . However, F o u r i e r ' s law i s the b a s i s for deriving t h e s e s p e c i a l e q u a t i o n s .

3 . 1 Combined Conduction-Convection

Thin materials having r e l a t i v e l y high thermal conductivity have very small l a t e r a l temperature g r a d i e n t s . I f t h e surfaces are c o n v e c t i v e l y heated or c o o l e d , the convection condition becomes part of a heat accounting on a d i f f e r e n t i a l element. Referring t o F i g . 3 , l e t Az be the t h i c k n e s s b of a t h i n s o l i d . On the surface z and z + Az, t h e s o l i d has a convection boundary described by q = h (t - t ) , where t . i s the convection f l u i d temperature.

Applying the same principles used to develop the general equations for rectangular coordinate systems should result in

/

£

S<*-v-«

fc(

lf)

fc(

ij)-

If the geometry is a thin rod of circumference C, the appropriate equation is

p°{f

r < * -

f > - «

f c (

i i ) - <

>

3.2 Moving heat sources

The general heat conduction Eq. (15) can also be used for moving heat sources, but a simpler quasi-steady-state equation can be derived by

coordinate transformation. If the coordinates are relative to the traveling source, the temperature distributions appear to be stationary. For example, if a point source of strength Q is moving at a velocity U parallel to the x-axis, the transformation would be x = x' + U T , where x' is the x-direction distance from the source. By substitution in Eq. (16) we can obtain

I F ('fc-Vfc(»|*)*fc ( * £ ) • » . & •«•••-. •

^ I X T V ^ I X T ^ " ^

! i t - v

• <

>

The moving source strength is accounted for in the boundary conditions for a particular problem solution.

4. BOUNDARY CONDITIONS

Solutions to heat conduction problems require statements of conditions. For general solutions there must be given at least a definition of the solu­

tion region, such as infinite, semi-infinite, quarter-infinite, finite, etc. Additionally, limits can be specified for any of these regions.

defined. They could i n c l u d e , for example, i n i t i a l , i n t e r n a l , and surface c o n d i t i o n s . Other conditions might include property d e f i n i t i o n s . Whether deriving a s o l u t i o n or searching for e x i s t i n g s o l u t i o n s , one must decide which c o n d i t i o n s are applicable t o the problem and how they can be s u i t a b l y

expressed.

4 . 1 I n i t i a l Condition

Unsteady-state problems must have an initial condition defined. Mostly, this implies a temperature distribution at x = 0 but, also, internal or

surface conditions could have initial values. A problem solution for x = 0 depends, of course, on whatever is specified at x = 0.

4.2 Surface Conditions

The most commonly employed surface c o n d i t i o n s i n heat conduction problems are prescribed surface c o n v e c t i o n , temperature, heat f l u x , or r a d i a t i o n . I t i s even acceptable t o prescribe two of *.hes3 for the same surface, such a s com­ bined r a d i a t i o n and convection. Other surface conditions could include phase change, a b l a t i o n , chemical r e a c t i o n s , or mass transfer fror a porous s o l i d .

4 . 2 . 1 Convection boundary

Conduction and convection heat transfer r a t e s on a surface are equated t o s a t i s f y c o n t i n u i t y of heat f l u x according t o Newton's law:

a t( xh T )

- k 3 x' = h [ t ( xb, T) - tf] . (29)

where t_ is the temperature of the convection fluid, and x. is the boundary location. The convection coefficient h must be determined from suitable sources that give predicted values satisfying the conditions of the fluid.

(Some correlations of convection coefficients are given in Section 16.) The method for defining h can vary depending on the type of convection or the methods prescribed by those researchers who have supplied values for the coefficient. However, h is usually defined as

TABLE 1. Sample convection coefficient values.

F l u i d Condition h, W/nT-K

Air Free convection on v e r t i c a l p l a t e s 10

Air Forced convection on p l a t e s 100

Air Forced flow i n tubes 200

Steam Forced flow in tubes 300

Oil Forced flow in tubes 500

Water Forced flow in tubes 2,000

Water Nucleate b o i l i n g 5,000

Liquid helium Nucleate b o i l i n g 8,000

Steam Film condensation 10,000

Liquid metal Forced flow i n tubes 20,000

Steam Dropwise condensation 50,000

Water Forced convection b o i l i n g 100,000

h

• t(K

?

) - t

'

>

where t can be given as

t

=

[ t <x

, x, - t j + t

^ (31)

and where 3 £ 1, and t^ is the temperature outside the thermal boundary layer of the fluid. In this respect, one must take care to use the proper fluid

temperature and convection coefficient.

Some typical order-of-magnitude values for the convection coefficient h are given in Table 1.

4.2.2 Surface temperature

Of all boundary conditions, this is probably the simplest in a

mathematical sense. It can be variable or constant with respect to position and time. In the real sense, it is very difficult to achieve a prescribed surface temperature, but it can be closely approached by imposing a relatively high convection rate.

Fourier's law defines the flux on a boundary by

3 t ( xb, T )

* = "k ~ 3 x • <3 2 )

An adiabatic surface can be defined by either setting q in Eq. (32) or h in Eq. (29) to zero. Inversely, if the solid's temperature di.stribution has been solved, then Eq. (32) can be used to determine the surface heat flux.

Understandably, heat flux can be, in particular cases, time and position dependent.

4.2.4 Thermal radiation

Heat transfer from an opaque surface by radiation can be expressed as

T 4 41

V

a*\v

(xb,

- T*J = -k —

where a is the Stefan-Boltzmann radiation constant, J*" is the combined configuration-emissivity factor for inuiltiple-surface radiation exchange, and T is the sink or source temperature for radiation. Because Eq. (33) is a nonlinear expression, it is frequently difficult to find exact solutions to problems with this condition.

A common method for dealing with radiation problems is to treat the radiation boundary as a convection boundary. According to Eq. (29) we can write

3t(x , T )

- k — ^ hr[ t ( xb, x) - t j , (34)

where

hr = o y [ T ( xb, x) + T ] [ T2( xb, T) + T* J .

Using this method means that T(x. , T) must first be estimated in order to compute a value of h . After a value for T(x. , T) has been computed from the problem solution, then the estimated value for h can be improved. This, of course, becomes an iterative process.

4 . 3 Interface Conditions 4 . 3 . 1 Contact

Two contacting s o l i d s , e i t h e r s i m i l a r of d i s s i m i l a r , w i l l almost always have some i n t e r f a c e thermal r e s i s t a n c e t o heat flow between them. The magnitude of t h i s r e s i s t a n c e can depend g r e a t l y on the condition of the two contacting s u r f a c e s . Properties t h a t can e f f e c t the surface condition include c l e a n l i n e s s , roughness, waviness, y i e l d s t r e n g t h , contact p r e s s u r e , and the thermal c o n d u c t i v i t i e s of the s o l i d s and i n t e r s t i t u a i f l u i d . Since there are s o many influences on the contact thermal r e s i s t a n c e , i t i s d i f f i c u l t t o t h e o r e t i c a l l y p r e d i c t i t s value. Consequently, experimental r e s u l t s are frequently used. Some representative values of the inverse thermal contact r e s i s t a n c e , commonly referred to as the thermal contact c o e f f i c i e n t , are given i n Section 17.

The generally accepted d e f i n i t i o n of the contact c o e f f i c i e n t i s

ho - A ^ ' ( 3 5 )

where q i s the steady heat flux corresponding t o a f i c t i t i o u s i n t e r f a c e temperature drop of At. defined by extrapolating the v i r t u a l l i n e a r

temperature gradient i n each s o l i d to the contact c e n t e r l i n e . This temperature drop, which i s i l l u s t r a t e d in Fig. 5, would diminish t o zero i f the i n t e r f a c e s were in perfect c o n t a c t .

4 . 3 . 2 Phase change

Other i n t e r f a c e conditions include t h o s e caused by endothermic reactions such as m e l t i n g , s o l i d i f i c a t i o n , sublimation, vaporization, and chemical d i s s o c i a t i o n .

A statement of an i n t e r f a c e reaction condition defines the difference of heat flux across the i n t e r f a c e . If the i n t e r f a c e which s e p a r a t e s two phases o f a material i s l o c a t e d a t x - x . , the heat balance for a phase change r e a c t i o n i s given i n t h e form

3 t (x ) a t (x ) dx

h -hr~ - h ~^r- =

* r • <

>

FIG. 5. Illustration of interface contact between solids.

whece y is the latent heat or chemical heat capacity, and subscripts 1 and 2 refer to the two phases.

5.0 Solutions

5.1 Extending solutions

A solution can be retrieved after identifying a problem by boundary conditions, geometry, and other pertinent data. Usually, a temperature solution is given, but heat flow can be derived from the temperature

distribution by using Fourier's law, i.e. Eq. (2). If cumulative heat flow is required, a time and surface integration of local heat flux is necessary.

where n is the direction normal to the surface s.

Steady-state solutions can be considered as the infinite-time condition for unsteady-state solutions. That is, problems which have a time-asymptotic

solution exhibit steady-state solutions for x •*• <*>. Thus, steady-state solutions can be derived from transient solutions.

A steady surface temperature condition can be implied from a convection boundary condition. For h •* °°, the surface temperature approaches the fluid

temperature. Therefore, a solution which includes a convection boundary can be transformed into a constant temperature boundary solution by solving for the implied limiting case.

5.2 Dimen&'ionless parameters

Grouping particular variables yields dimensionless numbers that can be useful. Symbolically, they can shorten an eguational expression. But, they can also give insight to the behavior of heat transfer in a particular problem.

One very useful parameter is the Biot number, Bi = ht/k. which results from convection boundary conditions. This parameter is proportional to the ratio of the conduction resistance to the convection resistance. Thus, we could say that for

Bi > 1, conduction is highest resistance to heat transfer, Bi < 1, convection is highest resistance to heat transfer, Bi « 1, the solid behaves like k = ».

2 Another dimensionless parameter is the Fourier number, Fo = crr/S. , which is found in transient solutions. This number is a dimensionless time value, but it is also considered an indicator of the degree of thermal penetration

2

into a solid. Since crt/S. = (kTAt/£)/(pc8.Ai.;. it is proportional to the ratio of conduction heat transferred to thermal capacity. Thus, an increasing Fo value implies approaching thermal equilibrium.

occurs in transient problems having a convection boundary. This is also a dimensionless time parameter, but it is based on convection heat transfer instead of conduction as in the Fourier number.

Solutions to problems having an internal heat source q ' " usually have a dimensionless heating parameter called the Pomerantsev modulus

po = q"'Jl2/kAt. This number is a ratio of internal heating to heat

conduction rates. Large values of Po imply large temperature differences will occur in the solid.

The parameter Fo = 1/2S&Z is a form of the reciprocal of the Fourier number and occurs in many solutions for transient temperatures in

semi-infinite solids.

When time dependent boundary conditions have a time constant, the solution will frequently include a dimensionless group called the

Predvoditelev modulus, Pd = bfl, /d, where b is the inverse time constant. Small values of Pd imply a slow changing condition. It signifies: the ratio of the change rate of the boundary condition to the change rate of the solid temperature.

5.3 Example Problems

5.3.1 Steady heat-transfer in a pipe wall

Hot water flows at 0.5 m/s in a 2.5 cm i.d., 2.66 cm o.d. smooth copper pipe. The pipe is horizontal in still air and covered with a 1-cm layer of polystyrene foam insulation. For a 65°C water temperature and 20°C air

temperature, estimate the heat loss rate per unit length. The solution given in case 2.1.2 is 2ir ( tx - t2) q = . 1 . r2 ^ 1 . r3 __ 1 A 1 .— In — + ;— Jin — + — — + — r -h rl k2 r2 rlhl r3h3 17

From the problem d e s c r i p t i o n tx = 65°C t . = 20°C 4 r. = 1.25 cm r = 1.33 cm r , = 2.33 cm kx = 400 W/m-°C (from Table 18.1) k2 = 0.038 W/m«°C (from Table 18.2) hd/k = 0.0155 P r ° '5R e0 , 9 (from S e c t . 16.1) hl = ^ w a t e r /2 1^(0 J 1 5 5 P r ° '5R e0-9) kwater = ° '6 5 9 w/m* °c la t 6 5°c> Pr = 2.73 Re = 2pvr]/n p = 980 kg/mJ v = 0.5 m/s p = 4.3 x 10"4 kg/m's Re = 2 ( 9 8 0 ) ( 0 . 5 ) ( 0 . 0 1 2 5 ) = M 488 4.3 x 10 hx = ( 0 . 6 5 9 / 0 . 0 2 5 ) ( 0 . 0 1 5 5 ) ( 2 . 7 3 ) ° "5( 2 8 4 8 8 )0 , 9 = 6895 W/(m2-°C) h3 = ( ka i r/ 2 r3) C l^a*1)1* (from S e c t . 16.8) ka i r = ° -0 2 5 w/m*0c Pr = 0.71 Gr = g B ( t4 - t3) ( 2 r3) W g = 9.8 m/s B = 1 / T4 = 1/293 K- 1 V = 1 6 . 5 5 x 1 0 "6 m2/ s G . (9.8) ( 0 . 0 4 6 6 )3 (0.71) = ^ _ = ^ a (293) (16.55 x l O ' V C = 1.14, m = 1/7 (from Table 16.3) h3 = (0.025/0.0466) (1.14) ( 8 7 7 4 )1 / / 7 = 2.24 W/m2«°C q = _{1 „ / 1 . 3 3 \ . 1 » / 2 . 3 3 V} 2n(65-20) 400£n (1^5 )+ 0^38 la(l^2) (0.0125) (6895) (0.233)(2.24) 2ir(45) 1.55 x 1 0 "4 + 14.76 + 0.012 + 19.16 = 8.33 W/m

18

^-u

Using this new estimate of (t. - t.), we can recalculate h,. h3 = 2.24 (25.4)1 / 7 = 3.56 W/m2*°C,

q = 10.54 W/m.

Additional iterations on h, would little improve this result.

Note that the copper tube and water film have a small effect on the results because they present little resistance to heat transfer by comparison to the insulation and air film.

5.3.2 Transient heat conduction in a slab

A billet of 304 stainless steel measuring 2 x 2 x 0.1 m thick and having a uniform temperature of 30 C is heated by sudden immersion into a 450°C molten salt bath. The mean convection coefficient is 350 W/m • C. Determine the time required for the center temperature of the billet to reach 400°C. The solution is found in the solution table (case 8.1.8 and Fig 8.4a):

t - t. r_ 400 - 450 30 - 450 = 0.119 From Table 18.1 k = 21 W/m«°C# -6 2 , a = 7 x 10 m /s k 21 hi (350)(0.05) = 1.2 . — h . tf

From Fig. 8.4a O T A2 = 3 . 4 ,

_ = 3 . 4 &2 _ ( 3 . 4 H 0 . 0 5 )2 =

a 7 x 1 0 "6

5.3.3 Transient heat conduction in a semi-infinite plate

For the conditions given in 5.3.2, find the temperature at 0.05 m from

the end and sides of the billet.

The solution is found in case 7.1.21 and Fig. 9.4a for a semi-infinite

plate.

hVax

35oV(7x 10~

) (1214)

^

l

V

I

~

t

t = (0.066>(30 - 450) + 450 = 423°C

5.3.4 Extended surface steady-state heat transfer

A 160 °C uniform-temperature copper plate has a long rectangular rib

brazed to it. All surfaces are convectively cooled by 30 C air having a

convection coefficient of 53 W/(m • C ) . The rib is yellow brass extending

4 cm from the flat surface and 2 cm wide. Estimate the additional heat loss

from the flat surface caused by the rib.

The solution for temperature distribution in the rib is given in

Case 1.1.17.

\ l

*^\

T

I

: t (

20

1 " tt ^ cos (X ) ( B i2 + X2 + Bi) fx cosh (X B) + Bi sinh (X B)"l

n=l n n L n n n J

t

X tan (X ) = Bi (characteristic equation) B = b/a = 4/1 = 4

X = x/a, Y = y/a

B i =

I r

i 3 o *

°

-

From Table 14.1

Xx = 0.0632, X2 = 3.1429, X3 = 6.2838, X4 = 9.4252, Xg = 12.5667, Heat loss without the rib attached would be

q = hA (tx - tf) = 53 (0.02)(160 - 30) = 138 W/m . The additional heat loss is thus

Aq = 666 - 138 = 528 W/m .

5.3.5 Rectangular fin heat transfer

Use the straight rectangular fin solution to estimate heat loss from the rib described in 5.3.4.

The solution is found in Case 5.1.4 and Fig. 5.2.

= \ k a " V(130M0.01> = 6'385 m l = 0.04 + 0.01 = 0.05 m c mfc = 0.3193 c

tanh {mSLj_

.

_

^

5.3.6 Semi-infinite plate heat transfer

Find an equation for the heat transfer rate through the edge of the

semi-infinite plate described in case 1.1.5 with f(x) = t .

Using the given temperature solution and Eqs. (2) and (37) we can find

the heat transfer in the following manner:

. 3t k .. . . 3T .

"

l

S

37

" I

2 " V 3? '

t~^t

9Y|Y = Q

~

Z

^ " ^ C

~

G

n=l

Y | Y = o

/ ^XIY = o ^

2 " V i It,

^ ^ C

-

t ™ ) ] ^

= 2k(t

- t ) 2 , El - costfflT)]

= J k ( t

- t

) 2 , J '

- 1» 3 '

'

n=l an n=l

Case No. References Description 1.1.1 19, Convectively heated and

p. 3-103 cooled plate.

w

i

i - V

- h

W

o

- V

Mi'S.

]£ (w

/k

+ 1/IK) + l/h

J a

/k

+ l/h

) + (l/h

)

i=l

Section 1.1. Solids Bounded by Plane Surfaces—No Internal Heating.

Case No. References Description Solution

1.1.3 1, p. 138 Plate with temperature dependent conductivity, k = k + B (t - tx) .

< w w ^

\—*~~\

- h - i H

v

- v f -

-

o

= expT-

(£

x)

v

2 -

l " " » ~ f ? "

]

M*

V ^ = | ^ [1 - «*p«

«>] , 0 < X < 5

Case No. References Description Solution 1.1.5 2, p. 122 Semi-infinite plate. 9, p. 164 t = t:, x a 0, H, y > 0. t = f(x), 0 > x > A, y = 0. V I UJ t - tx = 2 2 _ exp(-rarY) s i n (mrX) n = l 1 x f ff (X) - ti"| s i n (mrX)dX F o r £ ( x ) = t2s -t = f(x> t - t

n=l

t = t

, x = 0.

"

V - ^

~

t = t_ on other surfaces.

^ - 2

2

Section 1.1. Solids Bounded by Plane Surfaces—No Internal Beating.

Case Ho. References Description S o l u t i o n

1 . 1 . 7 2, p . 130 Rectangulac i n f i n i t e rod. t « P j U ) , 0 < i < I , y = 0. i t a t

i

u* 'in

4v

i

- 2

[fo

-

')]

A

i

i

Case No. References Description Solution 1 . 1 . 8 2 , p . 147 Thin rectangular p l a t e , t = tQ, x = 0, 0 < y < x. 0. w. in t = tQ, 0 < x < l, y t - t0 = 2 t = tQ, 0 < x < «,, y t = G ( y ) , x = Si, 0 < y < w. 6 . 0. hrt0 at z h2, tQ a t • V sinh [(Bi + n2i r2L2) \ ] . . „„. y &— 2 2 2 1.1 s i n ( n i r Y )

~L sinh [(Bi + n V l . V J

x

j [G(Y) - t

]

Z.I-' j

/7777&777777777777777777777777

JL

Section 1.1. Solids Bounded by Plane Surfaces—No Internal Heating.

Case No. References Description Solution

1.1.10 4, p. 41 Infinitely long thin plate Vertical plate: in a semi-infinite solid. ... . . k < t

l " V d

° ~

..

, ,0.24

-

°'

< w < "

1

i^k)

1

K

ii

J V

w /

1.1.11 4, p. 43 Thin rectangular plate on the kwir(t - t ) surface of a semi-infinite Q = / /M \ —

Case Ho. References Description _Solution

1.1.12 4, p. 44 Thin rectangular plate in an 2irwk(t1 - t )

i n f i n i t e s o l i d . Q =- — —

in

m

T 1.1.13 5, p. 54 Rectangular parallelpiped

"J with Wall thickness of &.

Qt = P £ ( d w + db + wb) + 2.16(d + w + b) + 1.2<5J(t2 - tj_ = total heat flow through six walls

i Ll _', _ i

T

a

1

Section 1.1. Solids Bounded by Plane Surfaces—No Internal Heating.

Case No. References Description Solution

1.1.14 4, p. 37 Infinite hollow square rod.

„ h2, t2 2irk(t2 - tx) 9 = It o l.OBw Tlk hlr0 2 r0 2 h2W 1 . 1 . 1 5 9, p . 166 Rectangular i n f i n i t e rod. t = f ( x ) , 0 < x < a, y = 0. T t » t , 0 < x < a, y = b. t = t , , x • 0 , 0 < y < b. t = t , , x = a , 0 < y < b .

t - t = Y An s i n (mix) sinh j (1 - Y) ( ^ Y l coaech 0&\

n=l 1 A » 2 I [f (X) - O s i n (nirX)dX , L = l/v w—»

V

Case No. References Description Solution 1.1.16 9, p. 167 Rectangular infinite rod.

t = f (x), 0 < x < I, y = 0. a » 0, 0 < x < I, y = w. a = 0, x = 0, 0 < y < w. i V w

<

/

_^

1

!

^ ( B i2 + X2 ) cos (X X) cosh [*(1 - Y)WX ]

nTl [ ( B i2 + X * )+B i ] c o s h <XnW,

/

Section 1.1. Solids Bounded by Plane Surfaces—No Internal Heating.

Case No. References Description Solution

1.1.17 9, p. 168 Case 1.1.16 with q = 0, y = w , Q - « x < J ! . i s replaced by convection boundary h,tf.

t - tf = 2

~ ( B i2 + X*J cos (AnX){Xn cosh [Xn(W - Y)] + Bi sinh [\n(W - Y}]}

^ [(Bi2 + X2) + Bil{Xn cosh (XRW) + Bi sinh <\,W)}

X

i

t - tf ^ . Bi cos (x n x) Un cosh [Xn(W - Y)] + Bi sinh [Xn<W - Y)]} fcl " fcf ~^ cos (Xn>r^Bi2 + XM + Bi"J[Xn cosh (XnW) + Bi sinh (^nW)]

Case No. References Description S o l u t i o n 1 . 1 . 1 8 9 , p . 166 Case 1.1.16 with t = t£,

y a w, 0 < x < £ .

t - t . « 2

" i

B

( B i2 + X2 ) cos (X X) sinh fX (W - Y)"| r1

* *" — "*• x I [ f (X) - tf] c o s (XnX)dX

"

* » :

j + Bij sinh (X

W)

P

\ r(

• x

)

t

- t

.. ^ cos (XX) {sinh f X W - Jf)"| - sinh (X Y) ( t - t )/<t_ - t . ) }

•— = 2 B i > — = ^

-n=l [ ^ B i2 + X2) + B i l cos (Xn) sinh <XnW)

X tan (X ) = Bi , W «= w/X, n n

Section 1.1. Solids Bounded by Plane Surfaces—No Internal Heating.

Case No. References

Description

Solution

1.1.20 9, p. 178 Rectangular parallelpiped.

t » t, , x = 0 , 0 <"y < w ,

"

0 1 6 V V fsinh (L - LX) + T sinh (LX)] sin (mtY) sin (nmZ)

1

t, - t„

_2Z Z <nm) sinh (L) '

0 < z < d .

t = t

, x = & , 0 < y < w ,

0 < z < d .

Remaining surfaces at t

.

"0 TT

_{n=l m=l}

n = 1, 3, 5,

m = 1, 3/ 5,

L

= (nirVw)

+ (Jim£/d)

, z = z/d

T = (t

- t

) / (

- t

)

Case No. References Description Solution 1.1.21 9, p. 179 Rectangular parallelpiped. t = tt , x = 0 , -w < y < +w f -d < z < +d . t = t2 , x = i. , -w < y < +w , -d < z < +d . Remaining s u r f a c e s convec­ t i o n boundary with h , t , .

t - t . Jl> ~ Tsinh (L - X) + T sinh (LX)3 c o s (X Y) cos (B Z)

t _ . _, z _ x x ^ n TO

fcl " ** ~ _ , cos (X ) c o s <B ) f x2 + B i2 + Bil^B2 + B i2 D2 + Bi D^sinh (h)

-—, r-i i sinn ^L, - a.) v T sinn tidw J COS JA XJ cos IP zj

x x ^ n TO

~ „ , cos (X ) c o s <B ) T x2 + B i2 + BilZe2 + B i2 D2 + Bi D^sinh

n=l m=i n m |_ n J\ m / X tan (X ) = B i , B tan (B ) = Bi D, Bi = hw/k

n n m m

L2 = X2&2/ w2 + B / V d2, D = d/w, X = x/i., Y = y/w

Section 1.1. Solids Bounded by Plane Surfaces—Mo Internal Heating.

Case No. References Description Solution 1.1.22 9, p. 180 Case 1.1.21 except t = t ,

x = 0 , -w < y < +w , -c < z < +c .

Remaining faces are correction boundaries with h,t .

oa oo

fc ~ fcf . _ . 2

[A Bi sinh (L - LX) + L cosh (L - LX)] cos (X Y) c o s (B Z) _ _ L * « sin

2* 2, [A Bi

t, - t. 2 . Z . [A Bi sinh (L) + L cosh (L)] NM cos (X ) cos < B )

1 * n=l jn=l

A = £/w , N = X2 + B i2 + Bi , M = &2 + B i2 C2 + BiC. n in X , 3 , L , D , X , Y , Z, and Bi are defined in

n m case 1.1.21 •

1.1.23 2, p. 175 Infinite plate with Case 2.2.10 cylindrical heat source.

Case No. References Description Solution

m

1 . 1 . 2 4 9 , p. 428 I n f i n i t e s t r i p with stepped temp boundary, t - tf l . y - 0 . - » < x < +°» . t = t 1 , y = w , x > 0 . t = t 2 , y = w , x < 0 . k = k , , 0 < y < w , x < 0 • k = k , 0 < y < v f x > 0 •

h

„ *, .

"

o

i

2 {

{

2

-

l

2 [

V

l> - ^ f (-1)"

Section 1.1. Solids Bounded by Plane Surfaces—No Internal Heating.

Case No. References

Description

Solution

1.1.25 9, p. 452 Heated planes on a

semi-infinite medium.

t = t

, x < -I , y = 0

t = t

, x > . + J , , y - 0

a

=> 0 ,

-I

< x < +8, ,

y = 0 .

Q

J

-l

(^

_

, _

_{< x <}

_-a

+£

1.1.26 9, p. 453 Heated parallel planes in Heat flow from bottom side of semi-infinite plane:

an infinite medium.

t = t , , x > . O

y = s .

Heat flow from top side of semi-infinite plane:

t l

Q = £ ^[(irxj/s) + l] (^ - t

) , 0 < x <

.

C = k [(Xj/s) + (1/TI)J (tj^ - t

) , 0 < x <

x

.

t = t

, -» < x + «

Yt

i 1 r

I — x J

1

Case No. References Description Solution 1.1.27 9, p. 454 Infinite right-angle cocnec.

t » t l f x > 0 , y » 0 . t = t l f x = 0 , y > 0 . t » t j . , x > Wj^ , y = w2 . t » t j , i • ^ , y > »2 . 2 . 2\ w2 W ; L nw2 ^ w J it y4 W lw2 | — t a n "1 ( — ) ( t , - t_) , 0 < x < x, , 0 < y < y, wl \w 2 / 1 X

I

I *

-K

-K

1.1.28 9, p. 462 A wedge with stepped surface temp, t = tx , 0 < r < rQ ,

e =

.

Section 1.1. Solids Bounded by Plane Surfaces—No Internal Heating.

Case No. References Description Solution

1.1.29

15

convection boundary. x = 0 y = 0 fcf-fcl \ exp(-miY) s i n (mix) C* 2 2 . ' n=l mr + (n IT / B I ) 1. 3 , 5, Convection boundary at

y = 0 . Heat transfer into strip at y = 0:

-s^.fi—»—...i...

k(t

h, tf * = w 1.1.30 88

27

Periodic strip heated plate q ^ W j y ) = 0, on 2b wide strips.

t(w,y) • t , on (2a - b) wide strips spaced 2a on centers.

-y = 2a- b

See Fig. 1.6a and b for values of maximum differences on y = 0 surface, i.e. Ct(0,a) - t(0,0)J = At . and heat transfer.

tf, h —

£-y

= b

Case No. References Description Solution

1.1.31 28 Case 1.1.30 except

Q = surface heat flux on (2a - b) wide strips spaced 2a on centers.

t(x, y) - tf k .

£ L - §j. (1 + BiX) + 2j

Qa

°° r Bi i

^ sin (rotB) cos (raiY) cosh (nirX) + — sinh (fflrX)J /. 2~T id T

n=l n sinh (nnw) + — • cosh (miW)

']

B = b/a , Bi = ha/k » W « w/a , X = x/a , Y = y/a .

See Pig. 1.7 for values of T = t(0 , a) - t(0 , 0),

1.1.32 29

i 10

Spot insulated infinite plate with constant temp on one face and convection boundary with an insulating spot on the other.

h,tf

T777?77m \

w

? r w

See Fig. 1.9 for values of 0(r/w,z/w) =

Section 1.1. Solids Bounded by Plane Surfaces—Ho Internal Heating.

Case No. References Description Solution

1.1.33 32 Infinite plate containing

an insulating strip. t = t1, -00< x < < * > , y = w . t = tQ , - » < x < » , y = -w q t » 0 , 0 < x < A , y = 0 . i t<x , y) - t, t ' *' U0 _ 1 . i _ n„ - l f VI F 1 f y •

1

T TH

1 + cosh (TTL)

G = F o r L •*• <° :

F = 2 cos (2irY)exp(-2TTX) - 1 , 6 = 2 sin <2TTY)exp(-2TrX)-l , X = x/w , Y = y/w , L = V w

1.1.34 19, Infinite thin plate with p. 3-110 heated circular hole.

t ~ «*. K0(Br/6) t = tx , r = rx . fc

3 "

» " V

V

if

Case No. References

Description

Solution

1.1.35 19,

p. 3-110

Case 1.1.34 with t.

replaced by a heat

source of strength g.

k5(t - t j K

(Br/5)

q

2v{Bi

/S)

K^Brj/S) '

l

B and t_ defined in case 1.1.34 ,

1.1.36 19, Case 1.1.34 with h„ = 0.

p. 3-111

t -

t

i

(Br/6)

I

(Br,/6) '

1

B given in case 1.1.34 .

1.1.37 60 Infinite plate with wall

19, cuts as shown. Heat flow

p. 3-123 normal to cuts,

q

See Table 1.3 for conductance data K/K

K = ka/fi,

uncut

Q = K t

- t