Solutions of the Conduction Equation

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

An Idea Generates More Mathematics….Mathematics Generate Mode Ideas…..

The Conduction Equation

),(''. trgqt

H

),(.. trgTkt

TC p

Incorporation of the constitutive equation into the energy equation above yields:

Dividing both sides by Cp and introducing the thermal diffusivity of the material given by

s

mm

s

m

C

k

p

2

Thermal Diffusivity

• Thermal diffusivity includes the effects of properties like mass density, thermal conductivity and specific heat capacity.

• Thermal diffusivity, which is involved in all unsteady heat-conduction problems, is a property of the solid object.

• The time rate of change of temperature depends on its numerical value.

• The physical significance of thermal diffusivity is associated with the diffusion of heat into the medium during changes of temperature with time.

• The higher thermal diffusivity coefficient signifies the faster penetration of the heat into the medium and the less time required to remove the heat from the solid.

pp C

trgT

C

k

t

T

),(

..

This is often called the heat equation.

pC

trgT

t

T

),(

..

For a homogeneous material:

pC

txgT

t

T

),(2

This is a general form of heat conduction equation.

Valid for all geometries.

Selection of geometry depends on nature of application.

General conduction equation based on Cartesian Coordinates

xqxxq

yyq

yqzzq

zq

),(. txgTkt

TC p

For an isotropic and homogeneous material:

),(2 txgTkt

TC p

):,,(2

2

2

2

2

2

tzyxgz

T

y

T

x

Tk

t

TC p

General conduction equation based on Polar

Cylindrical Coordinates

):,,(1

2

2

2

2

2tzrg

z

TT

rr

Tr

rk

t

TC p

General conduction equation based on Polar Spherical Coordinates

):,,(sin

1sin

sin

112

2

2222

2trg

T

r

T

rr

Tr

rrk

t

TC p

X

Y

Thermal Conductivity of Brick Masonry Walls

Thermally Heterogeneous Materials

zyxkk ,,

),(. txgTkt

TC p

),,,( tzyxgz

zT

k

y

yT

k

xxT

k

t

TC p

),,,(2

2

2

2

2

2

tzyxgz

Tk

z

T

z

k

y

Tk

y

T

y

k

x

Tk

x

T

x

k

t

TC p

More service to humankind than heat transfer rate calculations

Satellite Imaging : Remote Sensing

Thermal Imaging of Brain

One Dimensional Heat Conduction problems

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

Simple ideas for complex Problems…

Desert Housing & Composite Walls

Steady-State One-Dimensional Conduction

• Assume a homogeneous medium with invariant thermal conductivity ( k = constant) :

• For conduction through a large wall the heat equation reduces

to:

),,,(2

2

tzyxgx

Tk

x

T

x

k

t

TC p

),,,(2

2

tzyxgx

Tk

t

TC p

One dimensional Transient conduction with heat generation.

Steady Heat transfer through a plane slab

02

2

dx

TdA

0),,,(2

2

tzyxgx

Tk

No heat generation

211 CxCTCdx

dT

Isothermal Wall Surfaces

Apply boundary conditions to solve for constants: T(0)=Ts1 ; T(L)=Ts2

211 CxCTCdx

dT

The resulting temperature distribution is:

and varies linearly with x.

Applying Fourier’s law:

heat transfer rate:

heat flux:

Therefore, both the heat transfer rate and heat flux are independent of x.

Wall Surfaces with Convection

2112

2

0 CxCTCdx

dT

dx

TdA

Boundary conditions:

110

)0(

TThdx

dTk

x

22 )(

TLThdx

dTk

Lx

Wall with isothermal Surface and Convection Wall

2112

2

0 CxCTCdx

dT

dx

TdA

Boundary conditions:

1)0( TxT

22 )(

TLThdx

dTk

Lx

Electrical Circuit Theory of Heat Transfer

• Thermal Resistance• A resistance can be defined as the ratio of a

driving potential to a corresponding transfer rate.

i

VR

Analogy:

Electrical resistance is to conduction of electricity as thermal resistance is to conduction of heat.

The analog of Q is current, and the analog of the temperature difference, T1 - T2, is voltage difference.

From this perspective the slab is a pure resistance to heat transfer and we can define

q

TR

R

Tq th

th

WKmW

Kmm

kA

L

L

TTkA

TT

q

TR

ss

ss

condth /

1.2

12

21

WKmW

Km

hATThA

TT

q

TR

s

s

convth /

1.12

2

WKmW

Km

AhTTAh

TT

q

TR

rsurrsr

surrs

radth /

1.12

2

The composite Wall

• The concept of a thermal resistance circuit allows ready analysis of problems such as a composite slab (composite planar heat transfer surface).

• In the composite slab, the heat flux is constant with x.

• The resistances are in series and sum to Rth = Rth1 + Rth2.

• If TL is the temperature at the left, and TR is the temperature at the right, the heat transfer rate is given by

21 thth

RL

th RR

TT

R

Tq

Wall Surfaces with Convection

2112

2

0 CxCTCdx

dT

dx

TdA

Boundary conditions:

110

)0(

TThdx

dTk

x

22 )(

TLThdx

dTk

Lx

Rconv,1 Rcond Rconv,2

T1 T2

Heat transfer for a wall with dissimilar materials

• For this situation, the total heat flux Q is made up of the heat flux in the two parallel paths:

• Q = Q1+ Q2

with the total resistance given by:

Composite Walls

• The overall thermal resistance is given by

Desert Housing & Composite Walls