30079_43a.pdf

CHAPTER 43

HEAT TRANSFER FUNDAMENTALS

G. P. "Bud" Peterson

Executive Associate Dean and Associate Vice Chancellor of Engineering

Texas A&M University

College Station, Texas

43.1 SYMBOLS AND UNITS 13 67

43.3.4 The Log Mean

Temperature Difference 1400

43.2 CONDUCTION HEAT

TRANSFER 13 69

43.2.1 Thermal Conductivity 13 70

43.2.2 One-Dimensional Steady-

State Heat Conduction 1375

43.2.3 Two-Dimensional Steady-

State Heat Conduction 1377

43.2.4 Heat Conduction with

Convection Heat Transfer

on the Boundaries 1381

43.2.5 Transient Heat Conduction 1383

43.4 RADIATION HEAT

TRANSFER 1400

43.4.1 Black-Body Radiation 1400

43.4.2 Radiation Properties 1404

43.4.3 Configuration Factor 1407

43 A A Radiative Exchange

among Diffuse-Gray

Surfaces in

an Enclosure

1410

43.4.5 Thermal Radiation

Properties of Gases 1415

43.3 CONVECTION HEAT

TRANSFER

13 85

43.5 BOILING AND

CONDENSATION HEAT

TRANSFER 1417

43.5.1 Boiling 1420

43.5.2 Condensation 1423

43.5.3 Heat Pipes

43.3.1 Forced Convection —

Internal Flow

1385

43.3.2 Forced Convection —

External Flow 13 93

43.3.3 Free Convection 13 97

1424

43.1 SYMBOLS AND UNITS

A area of heat transfer

Bi Biot number, hL/k, dimensionless

C circumference, m, constant defined in text

Cp specific heat under constant pressure, J/kg • K

D diameter, m

e emissive power, W/m2

/ drag coefficient, dimensionless

F cross flow correction factor, dimensionless

Ff_j configuration factor from surface i to surface j, dimensionless

Fo Fourier number, atA2/V2, dimensionless

FO-\T radiation function, dimensionless

G irradiation, W/m2; mass velocity, kg/m2 • sec

g local gravitational acceleration, 9.8 m/sec2

gc proportionality constant, 1 kg • m/N • sec2

Gr Grashof number, gL3/3Ar/f2, dimensionless

h convection heat transfer coefficient, equals q/AAT, W/m2 • K

hfg heat of vaporization, J/kg

J radiocity, W/m2

k thermal conductivity, W/m • K

Mechanical Engineers' Handbook, 2nd ed., Edited by Myer Kutz.

K wick permeability, m2

L length, m

Ma Mach number, dimensionless

N screen mesh number, m"1

Nu Nusselt number, NuL = hL/k, NuD = hDlk, dimensionless

Nu Nusselt number averaged over length, dimensionless

P pressure, N/m2, perimeter, m

Pe Peclet number, RePr, dimensionless

Pr Prandtl number, Cpjjilk, dimensionless

q rate of heat transfer, W

cf' rate of heat transfer per unit area, W/m2

R distance, m; thermal resistance, K/W

r radial coordinate, m; recovery factor, dimensionless

Ra Rayleigh number, GrPr; RaL = GrLPr, dimensionless

Re Reynolds number, ReL = pVLI /n, Re^, = pVDI /a, dimensionless

S conduction shape factor, m

T temperature, K or °C

t time, sec

Tas adiabatic surface temperature, K

Tsat saturation temperature, K

Tb fluid bulk temperature or base temperature of fins, K

Te excessive temperature, Ts — Tsan K or °C

Tf film temperature, (Tx + Ts)/2, K

T. initial temperature; at t = 0, K

T0 stagnation temperature, K

Ts surface temperature, K

^ free stream fluid temperature, K

U overall heat transfer coefficient, W/m2 • K

V fluid velocity, m/sec; volume, m3

w groove width, m; or wire spacing, m

We Weber number, dimensionless

x one of the axes of Cartesian reference frame, m

Greek Symbols

a thermal diffusivity, kl pCp, m2/sec; absorptivity, dimensionless

(3 coefficient of volume expansion, 1/K

r mass flow rate of condensate per unit width, kg/m • sec

y specific heat ratio, dimensionless

A7 temperature difference, K

8 thickness of cavity space, groove depth, m

e emissivity, dimensionless

e wick porosity, dimensionless

A wavelength, /nm

T]f fin efficiency, dimensionless

jji viscosity, kg/m • sec

v kinematic viscosity, m2/sec

p reflectivity, dimensionless; density, kg/m3

or surface tension, N/m; Stefan-Boltzmann constant, 5.729 X 10~8 W/m2 • K4

T transmissivity, dimensionless, shear stress, N/m2

M* angle of inclination, degrees or radians

Subscripts

a adiabatic section, air

b boiling, black body

c convection, capillary, capillary limitation, condenser

e entrainment, evaporator section

eff effective

/ fin

/ inner

/ liquid

m mean, maximum

n nucleation

o outer

0 stagnation condition

P PiPe

r radiation

s surface, sonic or sphere

w wire spacing, wick

v vapor

A spectral

oo free stream

— axial hydrostatic pressure

+ normal hydrostatic pressure

The science or study of heat transfer is that subset of the larger field of transport phenomena that

focuses on the energy transfer occurring as a result of a temperature gradient. This energy transfer

can manifest itself in several forms, including conduction, which focuses on the transfer of energy

through the direct impact of molecules; convection, which results from the energy transferred through

the motion of a fluid; and radiation, which focuses on the transmission of energy through electro-

magnetic waves. In the following review, as is the case with most texts on heat transfer, phase change

heat transfer, that is, boiling and condensation, will be treated as a subset of convection heat transfer.

43.2 CONDUCTION HEAT TRANSFER

The exchange of energy or heat resulting from the kinetic energy transferred through the direct impact

of molecules is referred to as conduction and takes place from a region of high energy (or temper-

ature) to a region of lower energy (or temperature). The fundamental relationship that governs this

form of heat transfer is Fourier's law of heat conduction, which states that in a one-dimensional

system with no fluid motion, the rate of heat flow in a given direction is proportional to the product

of the temperature gradient in that direction and the area normal to the direction of heat flow. For

conduction heat transfer in the ^-direction this expression takes the form

,A dT

qx = -kA —

where qx is the heat transfer in the ^-direction, A is the area normal to the heat flow, dT/dx is the

temperature gradient, and k is the thermal conductivity of the substance.

Writing an energy balance for a three-dimensional body, and utilizing Fourier's law of heat con-

duction, yields an expression for the transient diffusion occurring within a body or substance.

d / dT\ d / dT\ d / df\

d dT

—Ik —} + —[k — \ +—\k — } + q = pcv

dx\ dx/ dy\ dy/ dz\ dz/

p dx dt

This expression, usually referred to as the heat diffusion equation or heat equation, provides a basis

for most types of heat conduction analysis. Specialized cases of this equation, such as the case where

the thermal conductivity is a constant

tfT <PT #T q = pCpdT

dx2 + dy2 + dz2 + k ~ k dt

steady-state with heat generation

d (, dT\ d /, dT\ d f, dT\

—Ik — + —Ik — + —\k — + q = 0

dx\ dx/ dy\ dy/ dz\ dz/

steady-state, one-dimensional heat transfer with heat transfer to a heat sink (i.e., a fin)

-iprW'-o

dx\dx/ k

or one-dimensional heat transfer with no internal heat generation

Ji(?I\ = №p?L

dx\dx) k dt

can be utilized to solve many steady-state or transient problems. In the following sections, this

equation will be utilized for several specific cases. However, in general, for a three-dimensional body

of constant thermal properties without heat generation under steady-state heat conduction, the tem-

perature field satisfies the expression

v2r= o

43.2.1 Thermal Conductivity

The ability of a substance to transfer heat through conduction can be represented by the constant of

proportionality k, referred to as the thermal conductivity. Figures 43.la, b, and c illustrate the char-

acteristics of the thermal conductivity as a function of temperature for several solids, liquids and

gases, respectively. As shown, the thermal conductivity of solids is higher than liquids, and liquids

higher than gases. Metals typically have higher thermal conductivities than nonmetals, with pure

metals having thermal conductivities that decrease with increasing temperature, while the thermal

conductivities of nonmetallic solids generally increase with increasing temperature and density. The

addition of other metals to create alloys, or the presence of impurities, usually decreases the thermal

conductivity of a pure metal.

In general, the thermal conductivities of liquids decrease with increasing temperature. Alterna-

tively, the thermal conductivities of gases and vapors, while lower, increase with increasing temper-

ature and decrease with increasing molecular weight. The thermal conductivities of a number of

commonly used metals and nonmetals are tabulated in Tables 43.1 and 43.2, respectively. Insulating

materials, which are used to prevent or reduce the transfer of heat between two substances or a

substance and the surroundings are listed in Tables 43.3 and 43.4, along with the thermal properties.

The thermal conductivities for liquids, molten metals, and gases are given in Tables 43.5, 43.6 and

43.7, respectively.

Fig. 43.1 a Temperature dependence of the thermal conductivity of selected solids.

Fig. 43.1 b Selected nonmetallic liquids under saturated conditions.

Fig. 43.1 c Selected gases at normal pressures.1

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