The Heat Conduction Equation

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

An Easy Solution to Industrial Heat Transfer Problems…

The Heat Equation

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

For constant thermal properties and no heat generation.

This is often called the heat equation.

General conduction equation based on Cartesian Coordinates

),(. 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

2

2

tzyxgz

TT

rr

Tk

t

TC p

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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

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Steady-State One-Dimensional Conduction

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

• For one-dimensional steady state conduction with no energy generation, 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-State One-Dimensional Conduction

• For one-dimensional heat conduction in a variable area geometry.

• We can devise a basic description of the process.

• The first law in control volume form (steady flow energy equation) with no shaft work and no mass flow reduces to the statement that Q = 0for all surfaces.

• From Fourier law of conduction, the heat transfer rate in at the left (at x) is:

Taylor’s Theory of Continuum

• For a function converging & well behaving…

......

!3!2

3

3

32

2

2

dx

dx

xQddx

dx

xQddx

dx

xQdxQdxxQ

n

i

n

n

nn

n

dx

dx

xQdxQdxxQ

1 !1

dx

dx

xQdxQdxxQ

x

• For a pure steady state conduction:

0 xQdxxQ

0 xQdxdx

xQdxQ

x

0

xdx

xQd

Substitute Fourier’s law of conduction:

0

x

dxdxdT

kAd

0

dxdxdT

kAd

If k is constant (i.e. if the material is homogeneous and properties of themedium are independent of temperature), this reduces to

02

2

dx

dT

dx

dA

dx

TdA

Pure radial conduction throughA Sphere.

02

2

dr

dT

dr

dA

dr

TdA

Surface area of a sphere at r

rdr

dArA 8 & 4 2

02

12

2

dr

dT

rdr

Td

Heat transfer through a plane slab

2112

2

0 CxCTCdx

dT

dx

TdA

Isothermal Wall Surfaces

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

thR

TQ

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 R = R1 + R2.

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

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

Rconv,1 Rcond Rconv,2

T1 T2