2-Heat Conduction Equation

7/25/2019 2-Heat Conduction Equation

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Heat transfer has direction as well as

magnitude, and thus it is a vector quantity


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The various distances and angles involved when describing the

.


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Fouriers law of heat conduction

for one-dimensional heat conduction:

)Watt(dx

dTkAQcond =&

If n is the normal of the isothermal surface atpoint P, the rate of heat conduction at that point

can be expressed by Fouriers law as

The heat transfer vector is

)Watt(n

TkAQn

=&

always normal to an isothermal

surface and can be resolved

into its components like any

kQjQiQQ zyxn

r&

r&

r&

r& ++=

P.Talukdar/Mech-IITD x

T

kA

Q xx

=&

y

T

kAQ yy

=&

z

T

kAQ zz

=&


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Steady versus Transient Heat

Transfer

The term stead im lies no

change with time at any pointwithin the medium, while

with time or time dependence.

Therefore, the temperature orheat flux remains unchanged

with time during steady heat

transfer throu h a medium at

any location, although both

quantities may vary from one


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Multidimensional Heat

Transfer

Heat transfer roblems are also classified as bein one-

dimensional, two-dimensional, or three-dimensional,depending on the relative magnitudes of heat transfer rates in

Ex:1Dheattransfer:

Heattransfer

through

the

glass

of

a

w n owcan econs ere to eone

dimensionalsinceheattransferthrough

theglasswilloccurpredominantlyinone

direction(thedirectionnormaltothe

surfaceof

the

glass)

and

heat

transfer

in

otherdirections(fromoneside

edgetotheotherandfromthetopedge


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A medium through which heat is conducted may involve the

, ,

(or thermal) energy. In heat conduction analysis, suchconversion processes are characterized as heat generation.

Heat generation is a volumetric phenomenon. That is, it occurs

throughout the body of a medium. Therefore, the rate of heat

whose unit is W/m3

The rate of heat generation in a

as position within the medium.

When the variation of heat

generation with position is known,= dVgG && Watt

P.Talukdar/Mech-IITD

g p

the total rate of heat generation in

a medium of volume V can bedetermined from


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1 D Heat Conduction Equation

Assume the density of the wall is , the specificheat is C, and the area of the wall normal to thedirection of heat transfer is A.

An energy balance on this thin element during

t

GQQ elementelementxxx

=+ + &&&


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)TT(x.A.C)TT(mCEEE tttttttttelement === +++

x.A.gVgG elementelement == &&&

E&&&

t

elementxxx

=+ +

TT &&

Dividin b

tx..x..gxxx

=+ +

&&Ax gives

tCg

xA

tttxxx

=+

++ &

Taking the limit as x0 and t0 yields and since from Fouriers Law:

=

=+

x

TkA

xx

Q

x

QQ xxxlim

&&&

t

Cg

x

kA

xA

=+

&


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Plane wall: A is constant

Variable conductivity:t

Cgx

kx

=+

&

2&

Constant conductivity:

where the property k/C is the thermal

tk

g

x2 =+

diffusivity


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Heat Conduction Equation in a

L C li dong Cylinder

ons er a n cy n r ca s e e emen o

thickness r in a long cylinder

The area of the c linder normal to the

direction of heat transfer at any location is A =

2rL where r is the value of the radius at that

location. Note that the heat transfer area A,

location.

&&&

tGQQ e ementelementrrr

=+ +


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)TT(r.A.C)TT(mCEEE tttttttttelement === +++

r.A.gVgG elementelement == &&&

t

r.A.Cr.A.gQQ tttrrr =+ ++ &

&&dividing by Argives t

CgrA

tttrrr

=+

++ &

T&&&

== rrrrrrr

0rm

TT1 &

tg

rrA =


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TT1

&

tr

..

rr

=

&Constant Conductivity:

tkrr

rr =+

dTd1 &

krdr

drr=+

t

T1

r

Tr

rr

1

=

0rd

dTr

dr

d=


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.

A = 4r2

Variable conductivity:

t

TCg

r

T.k.r

rr

1 22

=+

&

Constant Conductivity:t

T1

k

g

r

Tr

rr

1 22

=+

&

P.Talukdar/Mech-IITD

CombinedOneDimensional

Heat

Conduction

Equation t

TCg

r

T.k.r

rr

1 nn

=+

&


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General Heat Conduction

Equation

&&&&&&&

tGQQQQQQ e ementelementzzyyxxzyx

=+++ +++


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)TT.(z.y.x.C)TT(mCEEE tttttttttelement === +++

z.y.x.gVgG elementelement == &&&

t

EGQQQQQQ elementelementzzyyxxzyx

=+++ +++ &&&&&&&

t

TT

z.y.x.Cz.y.x.gQQQQQQttt

zzyyxxzyx

=+++ +

+++ &

&&&&&&

TTQQ1QQ1QQ1 tttzzzyyyxxx ++++ &&&&&&&

tzy.xyz.xxz.y


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TT1QQ11 &&&&&&&

tg

zy.xyz.xxz.y =+

&&&

=

=

=

+

x

k

xx

z.y.k

xz.yxz.yxz.y

lim xxxx

0x

=

=

=+ T

kT

z.x.k1Q1QQ1

limyyyy

&&&

...

=

=

=

+

z

Tk

zz

Ty.x.k

zy.x

1

z

Q

y.x

1

z

QQ

y.x

1lim zzzz

0z

&&&

t

TCg

z

Tk

zy

Tk

yx

Tk

x

=+

+

+

&

Under what condition?

222&

tkzyx222

=+

+

+


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0gTTT 222=+

+

+

&

zyx

T1TTT2

2

2

2

2

2 =

+

+

0z

TT

x

T2

2

2

2

2

2

=

+

+


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TTT1T1

&tz

.z

.rr

.rr 2

TT1T1T1 2 &

t.

sinrsinrr.

rr 2222


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Boundary and Initial Conditions

The temperature distribution in a medium depends on the

conditions at the boundaries of the medium as well as the heat

transfer mechanism inside the medium. To describe a heat

transfer problem completely, two boundary conditions must be

heat transfer is significant.

T ere ore, we nee to spec y two

boundary conditions for one-dimensional

problems, four boundary conditions for

- ,

boundary conditions for three-dimensional

problems


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con on, w c s usua y spec e a me = , s ca ethe initial condition, which is a mathematical expression for

the temperature distribution of the medium initially.

)z,y,x(f)0,z,y,x(T =

Note that under steady conditions, the heat conduction

equation does not involve any time derivatives, and thus we do

not need to s ecif an initial condition

The heat conduction equation is first order in time, and thus the initial

condition cannot involve any derivatives (it is limited to a specified

temperature).However, the heat conduction equation is second order in space

coordinates, and thus a boundary condition may involve first

er vat ves at t e oun ar es as we as spec e va ues o temperature


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Specified Temperature Boundary

C di iondition

The temperature of an exposed surface can

usually be measured directly and easily.

Therefore, one of the easiest ways to

specify the thermal conditions on a surface

is to specify the temperature. For one-

dimensional heat transfer through a planewall of thickness L, for example, the

specified temperature boundary conditions

can be expressed as

1T)t,0(T =Tt,LT =


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Specified Heat Flux Boundary

ondition

determined by inspection: positive ifthe heat flux is in the positive

rec on o e coor na e ax s, an

negative if it is in the opposite

direction.

Note that it is extremely important to

have the correct si n for the s ecified

heat flux since the wrong sign will

invert the direction of heat transfer

interpreted as heat loss


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2medium from both sides, for example, the specified heat flux boundary

conditions can be expressed as

50x

)t,0(Tk =50

x)t,L(Tk =

and

Special Case: Insulated Boundary

0x

)t,0(Tk =

0x

)t,0(T=

or


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t,L

T

0x

=

2-Heat Conduction Equation

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