Chapter_2_Introduction to Conduction

8/17/2019 Chapter_2_Introduction to Conduction

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Chapter 2 Perpindahan Panas/Heat Transfer 1

Chapter 2

Conduction


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Conduction Heat Transfer

• Conduction refers to the transport of energy in a medium (solid, liquid or

gas) due to a temperature gradient.• The physical mechanism is random atomic or molecular actiity

• !oerned "y #ourier$s la%

• &n this chapter %e %ill learn The definition of important transport properties and %hat goerns

thermal conductiity in solids, liquids and gases

The general formulation of #ourier$s la%, applica"le to any geometry

and multiple dimensions

Ho% to o"tain temperature distri"utions "y using the heat diffusionequation.

Ho% to apply "oundary and initial conditions


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Thermal Properties of Matter

• 'ecall from Chapter , equation for heat conduction

L

T

k L

T T

k q x∆

=−

=21"

The proportionality constant is a transport property , *no%n as

thermal conductivity k (units +/m.)

• -sually assumed to "e isotropic (independent of the direction oftransfer) k x =k y =k z =k

Is thermal conductivity different between gases, liquids and solids?


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Thermal Conductivity: Solids

• olid comprised of free electrons and atoms "ound in lattice

• Thermal energy transported through

0igration of free electrons, *e

1attice i"rational %aes, *l

l e k k k +=)(y,resistivitelectrical

1

eρ

≈ek %here

? What is the relative magnitude in pure metals, alloys and non-metallic

solids?

See Figure 2.5, Appendix Tables A.1, A.2 and A.3 text


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Thermal Conductivity: Fluids

• &ntermolecular spacing is much larger

• 0olecular motion is random• Thermal energy transport less effectie than in solids thermal

conductiity is lo%er

inetic theory of gases

λ∝ cnk

%here n the num"er of particles per unit olume, the mean

molecular speed and λ the mean free path (aerage distance traelled"efore a collision)

? What are the effects of temperature, molecular weight and pressure?

c


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Thermal Conductivity: Fluids

• Physical mechanisms controlling thermal conductiity not %ell

understood in the liquid state

• !enerally * decreases %ith increasing temperature (e3ceptions

glycerine and %ater)

• * decreases %ith increasing molecular %eight.

• 4alues ta"ulated as function of temperature. ee Ta"les 5.6 and 5.7,

te3t.


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Thermal Conductivity: Insulators

Can disperse solid material throughout an air space fi"er, po%der andfla*e type insulations

Cellular insulation #oamed systems

eeral modes of heat transfer inoled (conduction, conection, radiation)

8ffectie thermal conductiity depends on the thermal conductivity and

radiative properties of solid material, volumetric fraction of the air space,

structure/morphology (open s. closed pores, pore olume, pore si9e

etc.) bulk density (solid mass/total olume) depends strongly on the

manner in %hich the solid material is interconnected. ee Ta"le 5.:.

? How can we design a solid material with low thermal

conductivity?


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The Conduction Rate !uation

'ecall from Chapter

dx

dT kAq x −=

• Heat rate in the

3=direction

• Heat flu3 in the

3=direction dx

dT k

A

qq x −==

"

3

We assumed that T varies only

in the x-direction, T=T(x)

Direction of heat flux is normal

to cross sectional area A, where

A is isothermal surface (plane

normal to x-direction)

T(high)

T2 (lo%)

q3”

3 32

A


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The Conduction Rate !uation

&n reality %e must account for heat transfer in three dimensions

• Temperature is a scalar field T(3,y,9)• Heat flu3 is a ector quantity. &n Cartesian coordinates

for isotropic medium z T k q

yT k q

xT k q z y x ∂

∂−=∂∂−=

∂∂−= """ ,,

T k

z

T

y

T

x

T k ∇=

∂

∂+

∂

∂+

∂

∂−=∴ k i!

+here three dimensional del operator in cartesian coordinates

z y x ∂∂

+∂∂

+∂∂

=∇ k i

"""" z y x qqq k i! ++=


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The Heat Diffusion !uation

• >"?ectie to determine the temperature field, ie. temperature

distri"ution %ithin the medium.• @ased on *no%ledge of temperature distri"ution %e can

compute the conduction heat flu3.

'eminder from fluid mechanics ;ifferential control olume.

We will apply the enery

c!nservati!n e#ati!n t!

the $i%%erential c!ntr!lv!l#&e'

8lement of olumed3 dy d9

(*,y,+)


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i&ensi $i%%erential c!ntr!l v!l#&e dx, dy $an dz -


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Heat Diffusion !uation

.nery '!nservati!n .#ati!n st

st

ut g in !

dt

d! ! ! !

==−+

dz z dy ydx xut

z y xin

qqq !

qqq !

+++ ++=

++=

%here from #ourier$s la%

z

T dxdyk

z

T kAq

y

T dxdz k y

T kAq

x

T dydz k

x

T kAq

z z

y y

x x

∂∂

−=∂∂

−=

∂∂−=

∂∂−=

∂∂

−=∂∂

−=

)(

)(

)(

(2-1)

+

y

*

*

y

+


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• Thermal ener%y %eneration due to an ener%y source

0anifestation of ener%y conversion process ("et%eenthermal energy and chemical&electrical&nuclear energy)

Positie (source) if thermal energy is generated

Aegatie (sin*) if thermal energy is consumed

)(

$

dz dydxq

q ! g

=

=

• ner%y stora%e term

'epresents the rate of change of thermal energy

stored in the matter in the a'sence of phase chan%e.

)( dz dydxt

T c ! p st

∂∂

ρ=

t T c p ∂∂ρ / is the time rate of change of the sensi'le (thermal)energy of the medium per unit olume (+/m:)

qis the rate at %hich ener%y is %enerated perunit olume of the medium (+/m:)


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u"stituting into 8q. (2.)

t

T cq

z

T k

y y

T k

y x

T k

x p ∂

∂ρ=+

∂∂

∂∂

+

∂∂

∂∂

+

∂∂

∂∂

"eat#!uation

Aet conduction of heat into the C4rate of

energy

generation

per unit

olume

time rate of

change of

thermal

energy per

unit olume

5t any point in the medium the rate of energy transfer "y conduction into

a unit olume plus the olumetric rate of thermal energy generation must

equal the rate of change of thermal energy stored %ithin the olume

(2-2)


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Heat Diffusion !uation( )ther forms

• &f *Bconstant

t

T

k

q

z

T

y

T

x

T

∂∂

α=+

∂∂+

∂∂+

∂∂ 1

2

2

2

2

2

2 is the thermal diffusiity

pc

k

ρ=α

• #or steady state conditions

0=+

∂∂

∂∂

+

∂∂

∂∂

+

∂∂

∂∂

q z

T k

y y

T k

y x

T k

x

• #or steady state conditions, one=dimensional transfer in 3=direction

and no energy generation

0!r0

"

==

dx

dq

dx

dT k

dx

d x Heat flu3 is constant inthe direction of transfer

(2-3)

(2-4)


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Cylindrical Coordinate



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Spherical Coordinates

t

T cq

T k

r

T k

r r

T kr

r r p ∂

∂ρ=+

θ∂∂

θθ∂∂

θ+

φ∂

∂φ∂∂

θ+

∂∂

∂∂

sinsin

1

sin

11222

2

2


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*ample

The steady=state temperature distri"ution in a one=dimensional %all of

thermal conductiity 6 +/m. and thic*ness 6 mm is o"sered to

"e T(DC)BaE"32, %here aB2DC, "B=2DC/m2, and 3 is in meters.

a) +hat is the heat generation rate in the %allF

") ;etermine the heat flu3es at the t%o %all faces. &n %hat manner are

these heat flu3es related to the heat generation rateF


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+oundary and Initial Conditions

• Heat equation is a differential equation

econd order in spatial coordinates Aeed 2 "oundary conditions

#irst order in time Aeed initial condition

@oundary Conditions

Example a surface is in contact with a melting solid or a boiling liquid ) @.C. of first *ind (;irichlet condition)

*

(*,t)

s


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Example What happens when an electric heater is attached to a

surface? What if the surface is perfectly insulated?

2) @.C. of second *ind (Aeumann condition) Constant heat flu3 at thesurface

*(*,t)

*

(*,t)

q3”


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:)@.C. of third *ind +hen convective heat transfer occurs at the surface

(*,t)

(0,t)

*

"T ,∞


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*ample 2,-

5 long copper "ar of rectangular cross section, %hose %idth % is much greater

than its thic*ness 1, is maintained in contact %ith a heat sin* (for e3ample anice "ath) at a uniform initial temperature To. uddenly an electric current is

passed through the "ar, and an airstream of temperature is passed oer

the top surface, %hile the "ottom surface is maintained at To. >"tain the

differential equation, the "oundary and initial conditions that can "e used to

determine the temperature as a function of position and time inside the "ar.

∞T


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*ample

Passage of an electric current through a long conducting rod of radius

r i and thermal conductiity *r results in uniform olumetric heating at arate of . The conducting rod is %rapped in an electrically

nonconducting cladding material of outer radius r o and thermal

conductiity *c, and conection cooling is proided "y an ad?oining

fluid.

#or steady=state conditions, %rite appropriate forms of the heatequations for the rod and cladding. 83press appropriate "oundary

conditions for the solution of these equations.

q


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Matur than. you / your attention,


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Chapter_2_Introduction to Conduction

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