Chapter 2HEAT CONDUCTION EQUATION

Heat Transfer: A Practical ApproachSecond Edition

Yunus A. CengelMcGraw-Hill, 2002

Universitry of Technology Materials Engineering DepartmentMaE216: Heat Transfer and Fluid

bjectivesnderstand multidimensionality and time dependence of heat transfer, nd the conditions under which a heat transfer problem can be pproximated as being one-dimensional.

btain the differential equation of heat conduction in various oordinate systems, and simplify it for steady one-dimensional case.

entify the thermal conditions on surfaces, and express them athematically as boundary and initial conditions.

olve one-dimensional heat conduction problems and obtain the mperature distributions within a medium and the heat flux.

nalyze one-dimensional heat conduction in solids that involve heat eneration.

valuate heat conduction in solids with temperature-dependent ermal conductivity.

NTRODUCTIONAlthough heat transfer and temperature are closely related, they are of a

ifferent nature.emperature has only magnitude. It is a scalar quantity.eat transfer has direction as well as magnitude. It is a vector quantity.

We work with a coordinate system and indicate direction with plus or minus igns.

he driving force for any form of heat transfer is the temperature difference.

he larger the temperature difference, the larger the rate of heat ransfer.hree prime coordinate systems: rectangular T(x, y, z, t) cylindrical T(r, , z, t) spherical T(r, , , t).

Steady implies no change with time at any point within he medium

Transient implies variation with time or time dependencen the special case of ariation with time but not

with position, theemperature of the medium hanges uniformly with me. Such heat transferystems are called lumped ystems.

eady versus Transient Heat Transfer

ultidimensional Heat TransferHeat transfer problems are also classified as being: one-dimensional two dimensional three-dimensionaln the most general case, heat transfer through a medium is three-dimensional. However, some problems can be classified as two- or one-dimensional depending on the relative magnitudes of heat ransfer rates in different directions and the level of accuracy desired.One-dimensional if the temperature in the medium varies in onedirection only and thus heat is transferred in one direction, and thevariation of temperature and thus heat transfer in other directions arenegligible or zero.Two-dimensional if the temperature in a medium, in some cases,varies mainly in two primary directions, and the variation ofemperature in the third direction (and thus heat transfer in thatdirection) is negligible.

he rate of heat conduction through a medium in a specified directionay, in the x-direction) is expressed by Fourier’s law of heat

onduction for one-dimensional heat conduction as:

eat is conducted in the direction decreasing temperature, and us the temperature gradient is

egative when heat is conducted the positive x -direction.

The heat flux vector at a point P on he surface of the figure must be erpendicular to the surface, and it

must point in the direction of ecreasing temperature

f n is the normal of the isothermal urface at point P, the rate of heat onduction at that point can be xpressed by Fourier’s law as

Heat Generation

xamples: electrical energy being converted to heat at a rate of I2R, fuel elements of nuclear reactors, exothermic chemical reactions.eat generation is a volumetric phenomenon.he rate of heat generation units : W/m3 or Btu/h·ft3.he rate of heat generation in a medium may vary with time as well as osition within the medium.

NE-DIMENSIONAL HEAT CONDUCTION QUATION

sider heat conduction through a large plane wall such as the wall of ae, the glass of a single pane window, the metal plate at the bottom of ssing iron, a cast-iron steam pipe, a cylindrical nuclear fuel element, ectrical resistance wire, the wall of a spherical container, or a rical metal ball that is being quenched or tempered.

conduction in these and many other geometries can be oximated as being one-dimensional since heat conduction through e geometries is dominant in one direction and negligible in other tions.

we develop the onedimensional heat conduction equation in ngular, cylindrical, and spherical coordinates.

(2-6)

Heat Conduction Equation in a Large Plane Wall

Heat Conduction Equation in a Long Cylinder

t Conduction Equation Sphere

mbined One-Dimensional Heat Conduction ation

examination of the one-dimensional transient heat conduction ations for the plane wall, cylinder, and sphere reveals that all e equations can be expressed in a compact form as

n = 0 for a plane walln = 1 for a cylinder n = 2 for a sphere

he case of a plane wall, it is customary to replace the variable y x. s equation can be simplified for steady-state or no heat

ti d ib d b f

NERAL HEAT CONDUCTION EQUATIONe last section we considered one-dimensional heat conduction assumed heat conduction in other directions to be negligible.

t heat transfer problems encountered in practice can be roximated as being one-dimensional, and we mostly deal with h problems in this text.

wever, this is not always the case, and sometimes we need to sider heat transfer in other directions as well.

uch cases heat conduction is said to be multidimensional, and is section we develop the governing differential equation in

h systems in rectangular, cylindrical, and spherical coordinate ems.

tangular Coordinates

ylindrical Coordinatesations between the coordinates of a point in rectangular

d cylindrical coordinate systems:

pherical Coordinatesations between the coordinates of a point in rectangular

d spherical coordinate systems:

OUNDARY AND INITIAL CONDITIONSescription of a heat transfer problem in a medium is not complete without a full ption of the thermal conditions at the bounding surfaces of the medium.

ndary conditions: The mathematical expressions of the thermal conditions at the aries.

mperature at any n the wall at a ed time depends condition of the try at theng of the heat

ction process. condition, which lly specified at

= 0, is called the condition, which athematicalsion for the ature distribution

Specified Temperature Boundary Condition

Specified Heat Flux Boundary Condition

Convection Boundary Condition

Radiation Boundary Condition

Interface Boundary Conditions

Generalized Boundary Conditions

Boundary Conditions

pecified Temperature Boundary Condition

temperature of an exposed surface usually be measured directly andy. efore, one of the easiest ways to ify the thermal conditions on a surface specify the temperature.

one-dimensional heat transfer through ne wall of thickness L, for example, pecified temperature boundary itions can be expressed as

re T1 and T2 are the specified peratures at surfaces at x = 0 and L, respectively.

ifi d t t b

ecified Heat Flux Boundary Condition

plate of thickness L subjected to heat f 50 W/m2 into the medium from both , for example, the specified heat flux dary conditions can be expressed as

eat flux in the positive x-direction anywhere in the m, including the boundaries, can be expressed by

ecial Case: Insulated Boundary

ell-insulated surface can be modeled a surface with a specified heat flux of o. Then the boundary condition on a ectly insulated surface (at x = 0, for mple) can be expressed as

an insulated surface, the first ivative of temperature with respecthe space variable (the temperature dient) in the direction normal to the ulated surface is zero.

ther Special Case: Thermal Symmetryheat transfer problems possess thermal etry as a result of the symmetry in imposed al conditions. xample, the two surfaces of a large hot plate kness L suspended vertically in air is cted to the same thermal conditions, and thus mperature distribution in one half of the plate same as that in the other half. s, the heat transfer problem in this plate sses thermal symmetry about the center at x = L/2. fore, the center plane can be viewed as an ted surface, and the thermal condition at this of symmetry can be expressed as

onvection Boundary Conditione-dimensional heat transfer in the x-direction ate of thickness L, the convection boundary ons on both surfaces:

adiation Boundary Condition

ne-dimensional heat transfer in the ction in a plate of thickness L, thetion boundary conditions on both ces can be expressed as

ation boundary condition on a surface:

terface Boundary Conditionsboundary conditions at an interface ased on the requirements that

wo bodies in contact must have the e temperature at the area of contact

n interface (which is a surface) ot store any energy, and thus the flux on the two sides of an interface be the same.

boundary conditions at the interface o bodies A and B in perfect contact at

can be expressed as

Generalized Boundary Conditions

eneral, however, a surface may involve convection, ation, and specified heat flux simultaneously.

e boundary condition in such cases is again obtainedm a surface energy balance, expressed as

LUTION OF STEADY ONE-DIMENSIONALAT CONDUCTION PROBLEMSsection we will solve a wide range of heat ction problems in rectangular, cylindrical,

pherical geometries. ll limit our attention to problems that result inary differential equations such as the y one-dimensional heat conduction ms. We will also assume constant thermal

uctivity.olution procedure for solving heat uction problems can be summarized asrmulate the problem by obtaining the able differential equation in its simplest nd specifying the boundary conditions, tain the general solution of the differential on, and

AT GENERATION IN A SOLIDpractical heat transfer applications e the conversion of some form of energy

hermal energy in the medium. mediums are said to involve internal heat ation, which manifests itself as a rise in rature throughout the medium. examples of heat generation are

stance heating in wires, hermic chemical reactions in a solid, and ear reactions in nuclear fuel rods electrical, chemical, and nuclear es are converted to heat, respectively.

generation in an electrical wire of outer ro and length L can be expressed as

antities of major interest in a medium with eneration are the surface temperature Tse maximum temperature Tmax that occurs medium in steady operation.

ARIABLE THERMAL CONDUCTIVITY, k(T)When the variation of thermal conductivity with temperature in a specified temperature interval is large, it may be necessary to account for thisvariation to minimize the error.When the variation of thermal conductivity with temperature k(T) is known, the average value of the thermal conductivity in the temperature range between T1 and T2 can be determined from

mperature coefficient ermal conductivity.

average value of thermal conductivity e temperature range T1 to T2 in thiscan be determined from

average thermal conductivity in this is equal to the thermal conductivity

e at the average temperature.

variation in thermal conductivity of a material with erature in the temperature range of interest can often be oximated as a linear function and expressed as

ummaryIntroduction Steady versus Transient Heat Transfer Multidimensional Heat Transfer Heat Generation

One-Dimensional Heat Conduction Equation Heat Conduction Equation in a Large Plane Wall Heat Conduction Equation in a Long Cylinder Heat Conduction Equation in a Sphere Combined One-Dimensional Heat Conduction Equation

General Heat Conduction Equation Rectangular Coordinates Cylindrical Coordinates Spherical Coordinates

Boundary and Initial Conditions Solution of Steady One-Dimensional Heat Conduction Problems

G S