INTRODUCTION TO CONDUCTION Thermal Transport Properties Heat Diffusion (Conduction) Equation...
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Transcript of INTRODUCTION TO CONDUCTION Thermal Transport Properties Heat Diffusion (Conduction) Equation...
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