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Geometrical Variations of Heat Conduction Equation

P M V Subbarao

Professor

Mechanical Engineering Department

IIT Delhi

Same Physics But Shape Matters..

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The Heat Conduction Equation

pC

trgT

t

T

),(

..

For Isotropic Material:

For a homogeneous Isotropic material:

pC

txgT

t

T

),(2

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Mass Diffusion Equation

Using the analogy

),(.. trCDt

C

For a homogeneous & isotropic material:

),(2 trCDtC

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This is a general form of heat conduction equation.

Valid for all geometries.

Selection of geometry depends on nature of application.

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General conduction equation in Cartesian

Coordinate System

xq xxq

yyq

yqzzq

zq

Rate of energy generation

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),(. txgTk

t

TCp

For an isotropic and homogeneous material:

),(2 txgTktTCp

):,,(2

2

2

2

2

2

tzyxgzT

yT

xTk

tTCp

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General conduction equation based on Polar

Cylindrical Coordinates

):,,(12

tzrgzTk

zTk

rrTkr

rtTCp

),( txgTkt

TCp

zrkk ,,

),,(

12 zrgz

T

kz

T

krr

T

krrt

T

Cp

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General conduction equation based on Polar

Spherical Coordinates

):,,(sin

1sin

sin

112

2

222

2

2trg

T

r

T

rr

Tr

rrk

t

TCp

X

Y

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Thermally Heterogeneous Materials

zyxkk ,,

),(. txgTktTCp

),,,( tzyxgz

zTk

y

yTk

x

xTk

t

TCp

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),,,(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

TCp

More service to humankind than heat transfer rate calculations

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Primitive Pottery Kilns

Time

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Thermal Conductivity of Brick Masonry Walls

1,thR

1,2, 2 thth RR

3,thR

3,4, 2 thth RR

1pC

3pC

12 pp CC 34 pp CC

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Geographical Applications of Thermal Mapping

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One Dimensional Heat Conduction problems

P M V Subbarao

Associate Professor

Mechanical Engineering DepartmentIIT Delhi

Simple ideas for complex Problems

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Desert Housing & Composite Walls

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),(. txgTktTCp

General conduction equation in Cartesian

Coordinate System

Assume a homogeneous medium with invariant thermal conductivity ( k

= constant) :

For conduction through a large wall the heat equation reduces to:

),,,(2

2tzyxg

xTk

xT

xk

tTCp

),,,(2

2

tzyxgx

Tk

t

TCp

One dimensional Transient conduction with heat generation.

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General Steady-State One-Dimensional Conduction

Rate of energy generation

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Steady Heat transfer through a plane slab

02

2

dx

TdA

0),,,(2

2

tzyxgx

T

k

No heat generation

211 CxCTC

dx

dT

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Wall Surfaces with Convection

2112

2

0 CxCTCdx

dT

dx

TdA

Boundary conditions:

110

)0(

TThdx

dTk

x

22 )(

TLThdx

dTk

Lx

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Wall with isothermal Surface and Convection Wall

2112

2

0 CxCTCdx

dT

dx

TdA

Boundary conditions:

1)0( TxT

22 )( TLThdxdT

kLx

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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 T

2

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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:

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Composite Walls

The overall thermal resistance is given by

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Desert Housing & Composite Walls