Basic Equations - Heat Diff-2

8/10/2019 Basic Equations - Heat Diff-2

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Introduction A. Oztekin 2005 1

INTEGRAL FORMS OF BASIC CONSERVATION LAWS

WHEN CONTROL VOLUME IS AT REST RELATIVE TO INERTIAL AXES

VB

n

n

n

Typical Control Volume

n = unit outward normal to the surface

Conservation Laws


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The density of the material is defined bythe condition that mass of material in V + B

dV mV

The mater ial, or bulk, veloci ty v is defined bythe condition thatLinear Momentum of material in V + B

dV V

vp

Mass and linear momentum


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Conservation of Mass

.acrossleavingismasswhichatrate

inmassof changeof rate inmassof changeof rate

B

V B V

dAdV t dt

dm BV

)( nv


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Newton's Second Law

.outsidefrominmaterialonexertedforcetotal

inmaterialof momentumlinearof changeof rate

BV

BV

.)(

)(

dAdV dAvdV t dt

d

BV

BV

f gnv

vp

)or viscous(elasticareaunit perforcesurface

gravity)(massunit perforce body

f

g


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First Law of Thermodynamics (Energy Equation)

Usually,

V V

dV iedV E vv21

where

massunitpersystemtheof energye

systemtheof energytotalE

materialtheof massunitperenergyinternali

vv21

ie


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Conservation of Energy - Continued

.boundarytheacrossconductedisheatwhichatrate

ingeneratedisenergywhichatrateatwork doforcessurfacewhichatrate

inwork doforcesbodywhichatrate

changesinmaterialof ,energy,whichatrate

B

V

B

V

B V E


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Conservation of Energy - Continued

dAdV q

dAdV

dAedV t e

dt dE

BV

BV

BV

)(

)()(

)(

nq

f vgv

nv

massunitpergeneratedisenergywhichatrate

areaunitperfluxheatq

q


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HEAT CONDUCTION IN A RIGID MATERIAL

constant.p ,npf ,0g,0v,constant

Energy Equation Becomes

.0)( dV qdAdV t i V BV nq

Assumptions:


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HEAT CONDUCTION IN A RIGID MATERIAL -Continued

If zyx q,q,qq

DIVERGENCE THEOREM implies that

dV z q

yq

xqdA z y x

V B)()( nq

so that

0)( dV q z

q

y

q

x

q

t

i z y x V


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Consequently, at all points of the region

.q z

q

y

q

x

q

t

i z y x

Assumptions:Internal energy is a function only of temperature and

Fourier's law holds so

.)(,)(,)(or

)(q),(

z

T T k q

y

T T k q

x

T T k q

T T k T i i

z y x


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Equation for :)t,z,y,x(T

.

cheat,specifictheof Definition

))(())(())((

)(

p

dT di c

q z

T T k

z y

T T k

y x

T T k

x

t

T T c

p

p


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q z x

T k k

z y

T k k

y x

T k k

z T k

yT k

xT k

t T T c

zx xz zy yz yx xy

zz yy xx p

222

2

2

2

2

2

2

)(

Non-isotropic Material

When k is constant and second order tensor

k will be different in different directions


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Isotropic Material

y diffusivit thermal c

k

q c

T t

T

q c z

T

y

T

x

T

c

k

t

T

p

p

p p

:

1

1

2

2

2

2

2

2

2

Coefficients are constant and isotropic material


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Suggestion:

Use of Telegraph Equation Damped Wave EquationHyperbolic Equation the disturbances propagate with afinite velocity

T t

T c Usually

t ime relaxation c where

T

t

T

t

T

c

2102

00

2

22

2

2

110

11


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T z y x

T

2

2

2

2

2

22Cartesian Coordinate System

Cylindrical Coordinate System

Spherical Coordinate System

T

z r r

r

r r

T

z z and rSin yr xwith

2

2

2

2

22 11

,cos

T r r

r r r

T

r z and r yr xwith

2

2

2222

22

sin1

sinsin11

cossinsin,cossin


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Thermal Conductivity k (W/(m K) for heat diffusion

Metallic Solid (W/mK)Silver 418Cu 387

Stainless Steel 16

Nonmetallic SolidsPericlas, MgO 42Quartz 19Quartz fused 2Pyrex 1

Liquids W/(m K)Hg 8Water 5

Freon 0.07

Gases H 2 0.175He 0.141Air 0.0243


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Thermal Conductivity k

k is affected by

(a) impurities (most values are given for pure substances)

(b) radiation damage(c) alloying

k metals > k nonmetals

k solid > k liquid > k gas

the effect of density solid > liquid > gas


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Thermal Conductivity k

For metals,Electrical conductivity and thermal conductivity are related byk = L 0 Twhere L 0 : Lorentz constant, : electrical conductivity

For gases,

k cv for monatomic gases (rigid sphere molecules) k = 5/2 c v

In general,k depends on temperature Tk may also depend on pressure (p) and composition (c)

k usually is decreasing function of T for solids and liquidsand is increasing function of T for gases


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Boundary Conditions at a Surface

1. Specify the temperature

).,,(),,,0( 0 t z yT t z yT

2. Specify the heat flux

).,,(0 t z yq xT

k

Special case when the surface is insulated

.0 xT

:0x

x = 0

Solid

T(x,y,z,t)

T = T 0(x,y,z)

Another Material


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Boundary Conditions at a Surface - Continued 3. Surface convection

).),,((10

T t z y T h x

T k

4. Surface radiation

).),,(()),,(( 414

010 T t z y T T t z y T h x T

k

In general coefficients k, h, and vary from point to point on the surface

:0x

x = 0

Solid

T(x,y,z,t)

T = T 0(x,y,z)

Another Material

T 1

).),,(( 414

0 T t z y T x

T k

: Stefan - Boltzman constant: emissivity (0 1)

5. Surface convection and radiation

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Basic Equations - Heat Diff-2