Lecture2-The Equations of Heat Conduction

8/3/2019 Lecture2-The Equations of Heat Conduction

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Chapter TwoTHE EQUATIONS OF HEATCONDUCTIONS 2.1 One-Dimensional Heat Conduction Equation

2.2 Boundary Conditions

2-3 Mathematical Formulation of Some Heat

Transfer Problems

Problems


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This chapter is concerned with derivation of

the basic equations and the appropriateboundary conditions that govern the

temperature distribution in solids.

We develop the one-dimensional unsteadyheat conduction equation in rectangular,

cylindrical, and spherical coordinate systems

along with their appropriate boundaryconditions.


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2.1 One-Dimensional Heat Conduction Equation

Consider a solid whose temperature T(x,t)

depends on time and varies in only one direction,

say, along the x coordinate. The x axis in therectangular system refers to the usual x axis, but

in the cylindrical or the spherical coordinate

system it refers to the radial coordinate r. The

heat flow by conduction in the x-direction is given

by the Fourier law

For generality in the analysis, an external

volumetric heat source g(x,t), in W/m3 , generated

within the medium is considered.

x

T

kAQ

(2-1)


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To derive the conduction equation, we consider a

volume element of thickness x and area A normal to

the x axis as illustrated in Fig. 2.1. Applying the

general energy balance equation on this solid element

of constant volume, one obtains:

0

x

Heat flowIn, Qx

x x+x

Heat flow out,Qx+xQx+xEnergy source, (Ax)g


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( ) )12(... =+dt

dEEEE stgoutin

=

+

energyernal

ofincreaseofRate

generation

energyofRate

conductionbygain

heatofrateNet

int

t

TxAC

t

UxgAQQ xxx

= )()(

The above energy equation can now be arranged in the form

t

TACAg

x

QQp

xxx

=

)(

0

xAs

x

Q

x

QQ xxx

+ )(

t

TCgx

Q

Ap

=+

1

(2-2)

(2-3)

(2-4)


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tTCg

xTAk

xAp =+

1

Introducing Q given by Eq.(2.1) into the above equation,

we obtain the one- dimensional conduction equation in

the general form

2-1.1 Rectangular Coordinate

The area A =const., then conduction equation becomes

t

TCxg

x

Tk

xp

=+

)()(

2-1.2 Cylindrical Coordinate

The variable x is replaced by the radial variable r, and A=2rl r.Then the general one-dimensional conduction equation in the

cylindrical coordinate takes the form

t

TCrg

r

Tkr

rrp

=+

)()(

1

(2-5)

(2-6)

(2-7)


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2-1.3 Spherical Coordinate

The variable x is replaced by the radial variable r, and the area

A=4 r2, the one-dimensional conduction equation for the spherical

coordinate becomes

A General Compact One-Dimensional Conduction Equation

t

TCrg

r

Tkr

rrp

=+

)()(

1 22

t

TCrg

r

Tkr

rrp

n

n

=+

)()(

1

=

=

=

coordinatesphericalforn

coordinatelcylindricaforn

xbyreplacedisrandcoordinategularrecforn

2

1

,tan0

(2-8)

(2-9)


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2-1.5 Special Cases

For constant k and Cp ,Eq.(2-9) can be simplified to

= thermal diffusivity of material, m2/s.

Thermal Diffusivity

The physical meaning of thermal diffusivity isassociated with the propagation of heat into themedium during changes of temperature with time.The higher the thermal diffusivity, the faster thepropagation of heat will be through the mediumand consequently the shorter the time is to reachthe steady state condition.

tT

krg

rTr

rr

n

n =

1)()(1

PC

k

(2-10)


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2.2 Boundary Conditions

To solve the one-dimensional, time-dependent heat

conduction equation, two boundary conditions areneeded in addition to the initial condition.

Types of Thermal Boundary Conditions:

a- Prescribed Temperature boundary condition:The mathematical formulation of this type of boundary

condition can be written as:T(x,t) = T1 at x=0, t > 0 (2.14a)

T(x,t) = T2 at x=L, t > 0 (2.14b)

0 L x

T = T1 T = T2


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b- Prescribed Heat Flux Boundary Condition

0L x

0a b

r

Heat Supply

ConductionFlux

ConductionFlux

Heat

Supply

Conduction

Flux

ConductionFlux

HeatSupply

qo= -k

rrrTkq

== )( 11

r

rrTkq

==

)( 22

Lqx

LxTk =

= )(

x

xT

= )0(

0qx

Tk =

Lqx

Tk =

at x=0, t>0

at x=L, t>0


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c- Convection Boundary Conditions

The general thermal boundary condition can be stated as:

0L

x

0r

1

r2 r

Fluid atT1 ,h1

Fluid atT2 ,h2

Qconv. = Qcond.

Qcond. = Qconv.

Qconv. = Qcond.

Qcond. = Qconv.

Fluid atT2 ,h2

Fluid atT1 ,h1

a- Plate b- Hollow cylinder ors here

==

= platetheoxat

surfacethefromConduction

xatsurfacethetoT

atfluidfromConvection

int001

x

TkTTh= )( 11

x

TkTTh

= )( 22

at x=0

at x=L


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Thermal and geometric symmetry

If the solid body has a thermal and geometric symmetry

about a common symmetric axis

Thermal boundary conditions of symmetric slab

Consider a plane wall of thickness 2L initially at a temperature Ti and

for times t>0 is subjected to convection at the boundaries x=0 and x=2L asshown in Fig. (2.6). The environment has a temperature T and a

convection coefficient h at both boundary surfaces. The thermal boundary

conditions at boundary surfaces become

0=

rT at the symmetry axis.

0=

x

T

0)( =+

TThx

Tk

L L

x-x

0

Thermal

&GeometricSymmetry plane

FluidT ,h

FluidT ,h

at x=0

at x=L

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Lecture2-The Equations of Heat Conduction