ch2-heatconductionequation

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Chapter 2: Heat Conduction Equation( )Y.C. Shih S rin 2009

Chapter 2 Heat Conduction Equation

2-1 Introduction

2-2 Fouriers Law2-3 Derivation of Heat Conduction Equation

2-4 Boundary Conditions


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Although heat transfer and temperature are closely

related, they are of a different nature.Temperature has only magnitude

it is a scalar quantity.Heat transfer has direction as well as magnitude

it is a vector quantity.

We work with a coordinate system and indicate directionwith plus or minus signs.

2-1 Introdution (1)


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The driving force for any form of heat transfer is thetemperature difference.

The larger the temperature difference, the larger therate of heat transfer.Three prime coordinate systems:

rectangular (T(x, y, z, t)) ,cylindrical (T(r, f, z, t)),spherical (T(r, f, q, t)).

2-1 Introdution (2)


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Classification of conduction heat transferproblems:steady versus transient heat transfer,multidimensional heat transfer,heat generation.

2-1 Introdution (3)


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2-1 Introdution (5)

Multidimensional Heat TransferHeat transfer problems are also classified as

being:one-dimensional,two dimensional,

three-dimensional.In the most general case, heat transfer through amedium is three-dimensional . However, some

problems can be classified as two- or one-dimensional depending on the relativemagnitudes of heat transfer rates in differentdirections and the level of accuracy desired.


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Heat Generation

Examples:electrical energy being converted to heat at a rate of I2R ,fuel elements of nuclear reactors,exothermic chemical reactions.

Heat generation is a volumetric phenomenon.The rate of heat generation units : W/m 3 or Btu/h ft 3.The rate of heat generation in a medium may vary with time as

well as position within the medium.The total rate of heat generation in a medium of volume V canbe determined from

2-1 Introdution (6)


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2-2 Fouriers Law (1)

The rate of heat conduction through a medium in a specifieddirection (say, in the x-direction) is expressed by Fouriers

law of heat conduction for one-dimensional heat conductionas:

Heat is conducted in the directionof decreasing temperature, and thus

the temperature gradient is negativewhen heat is conducted in the positive x-direction.

(W)cond dT

Q kAdx

= &


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The heat flux vector at a point P on the surface of the figure must be perpendicular to the surface, andit must point in the direction of decreasingtemperatureIf n is the normal of theisothermal surface at point P ,the rate of heat conduction at

that point can be expressed byFouriers law as

(W)ndT

Q kAdn

= &


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In rectangular coordinates, the heat conduction vectorcan be expressed in terms of its components as

which can be determined from Fouriers law asn x y zQ Q i Q j Q k = + +

rrr r& & & &

x x

y y

z z

T Q kA xT

Q kA

yT

Q kA z

=

= =

&

&

&


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2-3 Derivation of Heat ConductionEquation (1)


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Heat Conduction Equation or

Heat Diffusion Equation

(1) k = constant

(2) Steady stateThermal diffusivity m

2

/s


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2-4 Boundary Conditions

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