INTRODUCTION TO CONDUCTION

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 a [m2/s] • thermal diffusivity:

3. 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. Fluids k=Cvl/3 → n : number of particles per unit volume Solids k = ke+ kl Pure metal: ke >> kl Alloys: ke ~ kl Non-metallic solids : kl > ke 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 Gases k=Cvl/3 In general, k ↑as T ↑

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

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

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

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

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

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) z f r Steady-state, constant k, no heat generation

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

19. convection boundary condition T∞, h

20. 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 Example 2.2 (at an instant time) • 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.