1. INTRODUCTION TO CONDUCTION
Thermal Transport Properties
Heat Diffusion (Conduction) Equation
Boundary and Initial Conditions

2. Thermal Properties of Matter
transport properties: physical structure of
matter, atomic and molecular
thermal conductivity:
isotropic medium: kx = ky = kz = k
thermal diffusivity:
a [m2/s]

3. Fourier law of heat conduction
Kinetic theory
C : volumetric specific
heat [J/m3K]
l : mean free path [m]
v : characteristic velocity
of particles [m/s]
Fourier law of heat conduction
First term : lattice (phonon) contribution
Second term : electron contribution

4. Solids
k = ke + kl
Pure metal: ke >> kl
Alloys: ke ~ kl
Non-metallic solids : kl > ke
Fluids
k = Cvl/3 →
n : number of particles per unit
volume
weak dependence on pressure
As p goes up,
n increases
but l becomes shorten → k = k (T)

5. Range of thermal conductivity for various of states of matter at normal temperature and pressure

6. Temperature dependence of thermal conductivity of soilds
Solids
k = ke + kl
Pure metal: ke >> kl
Alloys: ke ~ kl
Non-metal : kl > ke
In general, k ↓as T ↑

7. Temperature dependence of thermal conductivity of gases
k = Cvl/3
In general, k ↑as T ↑

8. Temperature dependence of thermal conductivity of nonmetallic liquids under saturated conditions

9. Phonon-Boundary Scattering
Nanoscale Structure
d >> l
d ~ l or less
Phonon-Boundary Scattering d
phonon l
= 300 nm

10. Thermal Conductivity of Silicon
100
120
140
160
102
103
Silicon layer thickness (nm) 40 60 80
Thermal conductivity (W/mK)
Phonon boundary scattering predictions
Asheghi et al. (1998)
Ju & Goodsen (1999)
Escobar & Amon (2004)
T = 300 K
Bulk silicon
104

11. Heat Diffusion (Conduction) Equation
density r
net out-going heat flux
specific heat cp
differential volume dV
differential area dA
control volume V
surface area A
: internal heat generation rate per unit volume [W/m3]

12. Divergence theorem:
Thus,
Since V can be chosen arbitrary, the integrand should be zero. or

13. Fourier law :
Without radiation,
For constant k :
Steady-state :
Constant k :
No heat generation :

14. y
Cartesian coordinates (x, y, z) x z
Steady-state, constant k, no heat generation

15. Cylindrical coordinates (r, f, z)
Steady-state, constant k, no heat generation

16. Spherical coordinates (r, f, q)
Steady-state, constant k, no heat generation

17. Boundary and Initial Conditions
Equations: Elliptic, Parabolic, Hyperbolic eq.
elliptic in space and parabolic in time
1. Initial condition
condition in the medium at t = 0 :
or constant

18. 2. Boundary conditions 1) Dirichlet (first kind)
variable value specified at the boundaries
or constant
2) Neumann (second kind)
gradient specified at the boundaries
or constant
when b(t) = 0 : adiabatic condition
3) Cauchy (third kind)
mixed boundary condition
or constant

19. convection boundary condition
T∞, h

20. Example 2.2 Find Heat rates entering, , and leaving, , the wall
(at an instant time)
Find
Heat rates entering, , and leaving, , the wall
Rate of change of energy storage in the wall,
Time rate of temperature change at x = 0, 0.25, and 0.5 m
Assumptions :
1-D conduction in the x-direction
Homogeneous medium with constant properties
Uniform internal heat generation

21. (at an instant time) 1)

22. (at an instant time) 2) 3)

23. Example 2.3 initially at T0 (uniform) copper bar Air Air Heat sink
Find :
Differential equation and B.C. and I.C. needed to determine T
as a function of position and time within the bar
Assumptions :
Since w >> L, side effects are negligible and heat transfer within the bar is 1-D in the x-direction
Uniform volumetric heat generation
Constant properties

24. Air
governing equation
1-D unsteady with heat generation
B.C. at the bottom surface :
B.C. at the top surface : convection boundary condition
I.C.