INTRODUCTION TO CONDUCTION

• Thermal Transport Properties

• Heat Diffusion (Conduction) Equation

• Boundary and Initial Conditions

Thermal Properties of Matter

/x

x

qk

T x

transport properties: physical

structure of matter, atomic and molecular• thermal conductivity:

isotropic medium: kx = ky = kz = k

• thermal diffusivity:

p

k

c

ability to conduct thermal energyability to store thermal energy

[m2/s]

Kinetic theory

Fourier law of heat conduction

First term : lattice (phonon) contributionSecond term : electron contribution

1

3x

dTq Cv

dx

3

Cvk

x

dTq k

dx

1

3l ek Cv Cv

C : volumetric specific heat [J/m3K] : mean free path [m]v : characteristic velocity of particles [m/s]

Fluids

k = Cv/3 → n : number of particles per unit volume

k nv

Solids

k = ke + kl

Pure metal: ke >> kl

Alloys: ke ~ kl

Non-metallic solids : kl > ke

weak dependence on pressureAs p goes up,

n increasesbut becomes shorten → k = k

(T)

Range of thermal conductivity for various of states of matter at normal temperature

and pressure

Solids

k = ke + kl

Pure metal: ke >> kl

Alloys: ke ~ kl

Non-metal : kl > ke

Temperature dependence of thermal conductivity of soilds

In general, k ↓as T ↑

Gases

k = Cv/3

Temperature dependence of thermal conductivity of gases

In general, k ↑as T ↑

Temperature dependence of thermal conductivity of nonmetallic liquids under saturated conditions

Phonon-Boundary Scattering

Si@300K = 300 nm

d phonon

phonon

d >> d ~ or less

Nanoscale Structure

100

120

140

160

102 103

Silicon layer thickness (nm)

40

60

80

Ther

mal

con

duct

ivity

(W/m

K)

40

Phonon boundary scattering predictions

Asheghi et al. (1998)Ju & Goodsen (1999)Escobar & Amon (2004)

T = 300 KBulk silicon

104

Thermal Conductivity of Silicon

Heat Diffusion (Conduction) Equation

control volume Vsurface area A

differential volume dV

differential area dA

n

: internal heat generation rate per unit volume [W/m3]

q

ˆ ,Aq ndA

q

VqdV

in g out stE E E E

in outE E gE

stE pV

TdVc

t

q net out-

going heat flux

ˆq n

p

TdVc

t

density

pV

Tc dV

t

specific heat cp

dA

Divergence theorem:

Since V can be chosen arbitrary, the integrand should be zero.

ˆpV A V

Tc dV q ndA qdV

t

Vq dV

Thus, 0pV

Tc q q dV

t

0p

Tc q q

t

or p

Tc q q

t

ˆAq ndA

cond radq q q , cond radp

Tc q q q

t

Steady-state :

Constant k :

No heat generation :

p

Tc k T q

t

0k T q

2 0q

Tk

2 0T

For constant k : 2

p

T qT

t c

cond radp

Tc q q q

t

Fourier law : condq k T

Without radiation,

Cartesian coordinates (x, y, z)

T T Tq k i j k

x y zˆ ˆ ˆ

p

T T T Tc k k k q

t x x y y z z

Steady-state, constant k, no heat generation2 2 2

2 2 20

T T T

x y z

x

y

zj i

k

Cylindrical coordinates (r, , z)

p

T Tc kr

t r r r

1

r z

T T Tq k e e e

r r z

1ˆ ˆ ˆ

Steady-state, constant k, no heat generation2 2

2 2 2

1 10

T T Tr

r r r r z

T Tk k q

r z z2

1

r

z

ree

ze

Spherical coordinates (r, ,)

p

T Tc kr

t r r r2

2

1

Steady-state, constant k, no heat generation2

2

2 2 2 2 2

1 1 1sin 0

sin sin

T T Tr

r r r r r

r

T T Tq k e e e

r r r

1 1ˆ ˆ ˆ

sin

T Tk k q

r r2 2 2

1 1sin

sin sin

r

reee

Equations: Elliptic, Parabolic, Hyperbolic eq.

elliptic in space and parabolic in time

Boundary and Initial Conditions

condition in the medium at t = 0 :

1. Initial condition

p

Tc k T q

t

or constant

( , )T r t

rbr

( ,0) ( )T r f r

1) Dirichlet (first kind) variable value specified at the boundaries

3) Cauchy (third kind) mixed boundary condition

2. Boundary conditions

( , ) ( )bT r t t or constant

or constant( , ) ( )b

Tr t t

n

when (t) = 0 : adiabatic condition

( , ) ( , ) ( )b b

Ta r t bT r t t

n

or constant

( , )T r t

rbr

2) Neumann (second kind) gradient specified at the boundaries

T∞, h

convection boundary condition

( , ) ( , )b b

Tk r t h T r t T

n

( , )bT r t

( , )b

Tk r t

n

( , )bh T r t T

( , )T r t

inq

x

outqstE

Example 2.2

210 mA2( )T x a bx cx

3

3

1000 W/m

40 W/m K

kg/m

4 kJ/kg Kp

q

k

c

1mL

2

900 C

300 C/m

50 C/m

a

b

c

(at an instant time)

Find 1) Heat rates entering, , and leaving, , the wall2) Rate of change of energy storage in the wall,3) Time rate of temperature change at x = 0, 0.25, and 0.5 m

in ( 0)q x out ( 1)q x

stET

t

Assumptions : 1) 1-D conduction in the x-direction2) Homogeneous medium with constant properties3) Uniform internal heat generation

1) in (0)xqq 0

,x

TkA

x

out ( )xx L

Tq L kAq

x

2

Tb cx

x

inq kAb 240 W/m K 10 m 300 C/m 120 kW

out 2k cq A b L

2 240 W/m K 10 m 300 C/m 2 50 C/m 1 m 160 kW

2( )T x a bx cx 2 2900 C 300 C/m 50 C/mx x

2 21173 K 300 K/m 50 K/mx x

inq

x

outqstE

210 mA2( )T x a bx cx

3

3

1000 W/m

40 W/m K

kg/m

4 kJ/kg Kp

q

k

c

1mL

2

900 C

300 C/m

50 C/m

a

b

c

(at an instant time)

2)

3)

n gst i outE EE E in outq qAL q

30 kW

2

2p p

k T q

C xt C

T

44.69 10 C

2

22

Tc

x

3 2120 kW 1000 W/m 10 m 1 m 160 kW

inq

x

outqstE

210 mA2( )T x a bx cx

3

3

1000 W/m

40 W/m K

kg/m

4 kJ/kg Kp

q

k

c

1mL

2

900 C

300 C/m

50 C/m

a

b

c

(at an instant time)

Air,T h

wL

z

I

xy

Heat sink

0T

copper bar

( ,k ( , , , ) ( , )T x y z t T x t

Find : Differential equation and B.C. and I.C. needed to

determine T as a function of position and time within the bar

Example 2.3

Assumptions : 1) Since w >> L, side effects are negligible and heat

transfer within the bar is 1-D in the x-direction2) Uniform volumetric heat generation3) Constant properties

q Lx

Air,T h

initially at T0

(uniform)

0 (0, )T T t

B.C. at the bottom surface :

B.C. at the top surface : convection boundary condition

I.C.

q Lx

Air,T h

0 (0, )T T t

( , )T L t

condq

convq

cond convq q ( , )x L

Tk h T L t T

x

0( ,0)T x T

0(0, )T t T

2

2p

T Tc k q

t x

governing equation

1-D unsteady with heat generation