HEAT CONDUCTION EQUATION

Prabal TalukdarPrabal TalukdarAssociate Professor

Department of Mechanical EngineeringDepartment of Mechanical EngineeringIIT Delhi

E-mail: prabal@mech.iitd.ac.inp

Heat TransferHeat TransferHeat transfer has direction as well as magnitude, and thus it is a vector quantity

P.Talukdar/Mech-IITD

Coordinate SystemCoordinate System

The various distances and angles involved when describing the location of a point in different coordinate systems

P.Talukdar/Mech-IITD

location of a point in different coordinate systems.

Fourier’s law of heat conductionfor one-dimensional heat conduction:

)Watt(dxdTkAQcond −=&

If n is the normal of the isothermal surface at point P, the rate of heat conduction at that point can be expressed by Fourier’s law as

The heat transfer vector is

)Watt(nTkAQn ∂∂−=&

always normal to an isothermalsurface and can be resolved into its components like any other vectorother vector

kQjQiQQ zyxn

r&

r&

r&

r& ++=

P.Talukdar/Mech-IITD xTkAQ xx ∂∂−=&

yTkAQ yy ∂∂−=&

zTkAQ zz ∂∂−=&

Steady versus Transient Heat Transfer

• The term steady implies no y pchange with time at any point within the medium, while transient implies variationtransient implies variation with time or time dependence. Therefore, the temperature or heat flux remains unchanged with time during steady heat transfer through a medium at gany location, although both quantities may vary from one location to anotherlocation to another

P.Talukdar/Mech-IITD

Multidimensional Heat Transfer

• Heat transfer problems are also classified as being one-p gdimensional, two-dimensional, or three-dimensional, depending on the relative magnitudes of heat transfer rates in different directions and the level of accuracy desireddifferent directions and the level of accuracy desiredEx: 1‐D heat transfer:Heat transfer through the glass of a i d b id d t bwindow can be considered to be one‐

dimensional since heat transfer through the glass will occur predominantly in one direction (the direction normal to the (surface of the glass) and heat transfer in other directions (from one sideedge to the other and from the top edge to the bottom) is negligible

P.Talukdar/Mech-IITD

to the bottom) is negligible

Heat GenerationHeat Generation• A medium through which heat is conducted may involve the

conversion of electrical nuclear or chemical energy into heatconversion of electrical, nuclear, or chemical energy into heat (or thermal) energy. In heat conduction analysis, such conversion 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 generation in a medium is usually specified per unit volumegeneration in a medium is usually specified per unit volume whose unit is W/m3

The rate of heat generation in a medium may vary with time as wellmedium may vary with time as well as position within the medium. When the variation of heat generation with position is known, ∫=

V

dVgG && Watt

P.Talukdar/Mech-IITD

g pthe total rate of heat generation in a medium of volume V can be determined from

V

1-D Heat Conduction Equationq

Assume the density of the wall is ρ, the specific heat is C, and the area of the wall normal to the direction of heat transfer is A.

An energy balance on this thin element during a small time interval t can be expressed asa small time interval t can be expressed as

EΔ

P.Talukdar/Mech-IITDt

EGQQ elementelementxxx Δ

Δ=+− Δ+

&&&

)TT(x.A.C)TT(mCEEE tttttttttelement −Δρ=−=−=Δ Δ+Δ+Δ+

x.A.gVgG elementelement Δ== &&&

EGQQ elementΔ+ &&&

tGQQ element

elementxxx Δ=+− Δ+

)TT(ACAQQ ttt −ΔΔ Δ+&&&

Dividing by

t)(x.A.Cx.A.gQQ ttt

xxx ΔΔρ=Δ+− Δ+

Δ+ &

TTQQ1 −− &&g yAΔx gives

tTTCg

xQQ

A1 tttxxx

Δρ=+

Δ− Δ+Δ+ &

Taking the limit as Δx → 0 and Δt → 0 yields and since from Fourier’s Law:

⎟⎠⎞

⎜⎝⎛

∂∂−

∂∂=

∂∂=

Δ−Δ+

→Δ xTkA

xxQ

xQQ xxx

0xlim

&&&

TT1 ∂⎞⎛ ∂∂

P.Talukdar/Mech-IITD

⎠⎝→Δ 0x

tTCg

xTkA

xA1

∂∂ρ=+⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂

&

Plane wall: A is constant

TT ∂⎞⎛ ∂∂Variable conductivity:tTCg

xTk

x ∂∂ρ=+⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂

&

T1gT2 ∂∂ &Constant conductivity:

where the property k/ρC is the thermal

tT1

kg

xT2 ∂

∂α

=+∂∂

diffusivity

P.Talukdar/Mech-IITD

Heat Conduction Equation in a L C li d

C id thi li d i l h ll l t f

Long CylinderConsider a thin cylindrical shell element of thickness r in a long cylinder

The area of the cylinder normal to theydirection of heat transfer at any location is A = 2πrL where r is the value of the radius at that location. Note that the heat transfer area A depends on r in this case and thus it varies withdepends on r in this case, and thus it varies with location.

E lΔ&&&

P.Talukdar/Mech-IITD

tEGQQ element

elementrrr ΔΔ

=+− Δ+

)TT(r.A.C)TT(mCEEE tttttttttelement −Δρ=−=−=Δ Δ+Δ+Δ+

r.A.gVgG elementelement Δ== &&&

)TT( −&&t

)TT(r.A.Cr.A.gQQ tttrrr Δ

Δρ=Δ+− Δ+Δ+ &&&

TTQQ1 &&dividing by AΔr gives t

TTCgr

QQA1 tttrrr

Δ−

ρ=+Δ−

− Δ+Δ+ &

⎟⎞

⎜⎛ ∂∂∂−Δ+ TkAQQQli

&&&⎟⎠⎞

⎜⎝⎛

∂∂−

∂∂=

∂∂=

ΔΔ+

→Δ rTkA

rrQ

rQQ rrr

0rlim

TCTkA1 ∂+⎟⎞

⎜⎛ ∂∂

&

P.Talukdar/Mech-IITD

tCg

rkA

rA ∂ρ=+⎟

⎠⎞

⎜⎝⎛

∂∂

Different ExpressionsDifferent ExpressionsVariable conductivity: TCgTkr1 ∂ρ+⎟

⎞⎜⎛ ∂∂

&Variable conductivity:t

Cgr

.k.rrr ∂

ρ=+⎟⎠

⎜⎝ ∂∂

T1gT1 ∂⎟⎞

⎜⎛ ∂∂ &

Constant Conductivity:tT1

kg

rTr

rr1

∂∂

α=+⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂

0gdTd1⎟⎞

⎜⎛ &

0kg

rdr

drr=+⎟

⎠⎞

⎜⎝⎛

tT1

rTr

rr1

∂∂

α=⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂

trrr ∂α⎠⎝ ∂∂

0rd

dTrdrd

=⎟⎠⎞

⎜⎝⎛

P.Talukdar/Mech-IITD

Heat Conduction Eq in a SphereHeat Conduction Eq. in a Sphere

A = 4πr2

Variable conductivity:tTCg

rT.k.r

rr1 22 ∂

∂ρ=+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

&

Constant Conductivity: tT1

kg

rTr

rr1 22 ∂

∂α

=+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂ &

P.Talukdar/Mech-IITD

Combined One‐DimensionalHeat Conduction Equation t

TCgrT.k.r

rr1 nn ∂

∂ρ=+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

&

General Heat Conduction E iEquation

E lΔ&&&&&&&

P.Talukdar/Mech-IITD

tEGQQQQQQ element

elementzzyyxxzyx ΔΔ

=+−−−++ Δ+Δ+Δ+

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

z.y.x.gVgG elementelement ΔΔΔ== &&&

tEGQQQQQQ element

elementzzyyxxzyx ΔΔ

=+−−−++ Δ+Δ+Δ+&&&&&&&

tTTz.y.x.Cz.y.x.gQQQQQQ ttt

zzyyxxzyx Δ−

ΔΔΔρ=ΔΔΔ+−−−++ Δ+Δ+Δ+Δ+ &&&&&&&

TTCgQQ1QQ1QQ1 tttzzzyyyxxx −ρ=+

−−− Δ+Δ+Δ+Δ+ &&&&&&&

tCg

zy.xyz.xxz.y Δρ=+

ΔΔΔ−

ΔΔΔ−

ΔΔΔ−

P.Talukdar/Mech-IITD

TTCQQ1QQ1QQ1 tttzzzyyyxxx −−−− Δ+Δ+Δ+Δ+ &&&&&&&

tCg

zQQ

y.xyz.xxQQ

z.ytttzzzyyyxxx

Δρ=+

ΔΔΔ−

ΔΔΔ−

ΔΔΔ− Δ+Δ+Δ+Δ+

⎞⎛ ∂∂⎞⎛ ∂∂∂ TT1Q1QQ1 &&&⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

ΔΔ−∂∂

ΔΔ=

∂∂

ΔΔ=

Δ−

ΔΔΔ+

→Δ xTk

xxTz.y.k

xz.y1

xQ

z.y1

xQQ

z.y1lim xxxx

0x

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

ΔΔ−∂∂

ΔΔ=

∂

∂

ΔΔ=

Δ

−

ΔΔΔ+

→Δ yTk

yyTz.x.k

yzx1

yQ

zx1

yQQ

zx1lim yyyy

0y

&&&

⎠⎝ ∂∂⎠⎝ ∂∂ΔΔ∂ΔΔΔΔΔ→Δ yyyyz.xyz.xyz.x0y

⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

ΔΔ−∂∂

ΔΔ=

∂∂

ΔΔ=

Δ−

ΔΔΔ+

→Δ zTk

zzTy.x.k

zy.x1

zQ

y.x1

zQQ

y.x1lim zzzz

0z

&&&

tTCg

zTk

zyTk

yxTk

x ∂∂ρ=+⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

∂∂+⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂

&

Under what condition?

T1gTTT 222 ∂∂∂∂ &

P.Talukdar/Mech-IITDtT1

kg

zT

yT

xT

222 ∂∂

α=+

∂∂+

∂∂+

∂∂

0kgTTT

2

2

2

2

2

2

=+∂∂+

∂∂+

∂∂ &

kzyx 222 ∂∂∂

tT1

zT

yT

xT

2

2

2

2

2

2

∂∂

α=

∂∂+

∂∂+

∂∂

tzyx ∂α∂∂∂

0zT

yT

xT

2

2

2

2

2

2

=∂∂+

∂∂+

∂∂

y

P.Talukdar/Mech-IITD

Cylindrical and SphericalCylindrical and SphericalTCgTkTrk1Trk1 ∂ρ=+⎟

⎞⎜⎛ ∂∂+⎟⎟

⎞⎜⎜⎛ ∂∂+⎟

⎞⎜⎛ ∂∂

&t

Cgz

.kz

r.krr

r.krr 2 ∂

ρ=+⎟⎠

⎜⎝ ∂∂

+⎟⎟⎠

⎜⎜⎝ φ∂φ∂

+⎟⎠

⎜⎝ ∂∂

TCgTsink1Tk1Trk1 2 ∂ρ=+⎟⎞

⎜⎛ ∂θ∂+⎟⎟

⎞⎜⎜⎛ ∂∂+⎟

⎞⎜⎛ ∂∂

&

P.Talukdar/Mech-IITD

tCgsin.k

sinrk

sinrrr.k

rr 2222 ∂ρ=+⎟

⎠⎜⎝ θ∂

θθ∂θ

+⎟⎟⎠

⎜⎜⎝ φ∂φ∂θ

+⎟⎠

⎜⎝ ∂∂

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 given for each direction of the coordinate system along whichgiven for each direction of the coordinate system along which heat transfer is significant.

Th f d ifTherefore, we need to specify two boundary conditions for one-dimensional problems, four boundary conditions for two dimensional problems and sixtwo-dimensional problems, and six boundary conditions for three-dimensional problems

P.Talukdar/Mech-IITD

• A diti hi h i ll ifi d t ti t 0 i ll d• A condition, which is usually specified at time t = 0, is called the 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 specify an initial conditionp y

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 d i ti t th b d i ll ifi d l f t t

P.Talukdar/Mech-IITD

derivatives at the boundaries as well as specified values of temperature

Specified Temperature Boundary C di iCondition

The temperature of an exposed surface can usually be measured directly and easily. Therefore, one of the easiest ways to yspecify the thermal conditions on a surface is to specify the temperature. For one-dimensional heat transfer through a plane wall of thickness L, for example, the specified temperature boundary conditions can be expressed as

1T)t,0(T =

2T)t,L(T =

P.Talukdar/Mech-IITD

2)(

Specified Heat Flux Boundary C di iCondition

The sign of the specified heat flux isThe sign of the specified heat flux is determined by inspection: positive if the heat flux is in the positive direction of the coordinate a is anddirection of the coordinate axis, and negative if it is in the opposite direction.

Note that it is extremely important to have the correct sign for the specifiedhave the correct sign for the specifiedheat flux since the wrong sign will invert the direction of heat transfer and cause the heat gain to be

P.Talukdar/Mech-IITD

and cause the heat gain to be interpreted as heat loss

For a plate of thickness L subjected to heat flux of 50 W/m2 into theFor a plate of thickness L subjected to heat flux of 50 W/m into the medium from both sides, for example, the specified heat flux boundary conditions can be expressed as

50x

)t,0(Tk =∂

∂− 50x

)t,L(Tk −=∂

∂−and

Special Case: Insulated Boundary

0x

)t,0(Tk =∂

∂ 0x

)t,0(T=

∂∂or

P.Talukdar/Mech-IITD

Another Special CaseAnother Special Case• Thermal SymmetryThermal Symmetry

t,2LT ⎟

⎠⎞

⎜⎝⎛∂

0x2 =∂

⎠⎝

PTalukdar/Mech-IITD

Example Problem

P.Talukdar/Mech-IITD

Comments

P.Talukdar/Mech-IITD

P.Talukdar/Mech-IITD