1. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Three Modes of Heat Transfer 1 “conduction” “radiation” “convection”

2. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Conduction 2

3. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Convection = Cooling by mass motion (diffusion + advection) in a fluid 3

4. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Radiation Note: Usually nothing is a perfect “black body” and parts of the emissive spectrum may be missing (ex: photonic band gap crystals). 4 Linearize (when and why?): For black body ( Ɛ =1) at 300 K:

5. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips All problems have boundaries! Heat diffusion equation needs boundary conditions Dirichlet (fixed T): Neumann (fixed flux ~ dT/dx): When is it OK to “lump” a body as a single R or C? Biot number: Boundaries and Lumped Elements 5

6. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Transient Cooling of Lumped Body 6 Source: Lienhard book, http://web.mit.edu/lienhard/www/ahtt.html (2008)http://web.mit.edu/lienhard/www/ahtt.html

7. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips What if Biot Number is Large Bi = hL/k b << 1 implies T b (x) ~ T surf (lumped OK) Bi = hL/k b >> 1 implies significant internal T b (x) gradient 7

8. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Lumped Body Examples (Steady State) 8 Boundary conditions: T L = 400 o C, T R = 100 o C 1)Assume NO internal heat generation (how does the temperature slope dT/dx scale qualitatively within each layer?) 2) Assume UNIFORM internal heat generation

9. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Contact Resistance R C = 1/h C A BUT, also remember the fundamental solid-solid contact resistance given by density of states, acoustic/diffuse phonon mismatch ~Cv/4! (prof. Cahill’s lecture) 9

10. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Notes on Finite-Element Heat Diffusion 10 RCRC T0T0 T0T0 TNTN TiTi T i-1 T i+1 ΔxΔx L T1T1 Boundary conditions: (heat flux conservation) = MTb T1T1 TNTN Matlab: T = M\b b1b1 bNbN …… M 11 M 12 0 … M NN … … … M 21 M 22 M 23 0 …

11. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips More Comments on “Fin Equation” Same as Poisson equation with various BC’s BC’s can be given flux (dT/dx) or given temperature (T 0 ) Very useful to know: –Thermal healing length L H (Poisson: screening length) –General, qualitative shape or solution 11 general solution sinh, cosh, tanh … etc.

12. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Fin Efficiency (how long is too long?) Fin efficiency η = actual heat loss by fin / heat loss if entire fin was at base temperature TB Actual heat loss: Here 12 sinh cosh tanh T=T B dT/dx ≈ 0 L W d Not worth making cooling fins much >> L H ! exp

13. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Poisson Equation Analogy Thermal fin is ~ mathematically same problem as 1-D transistor electrostatics, e.g. nanowire or SOI transistor L < λ  short fin, or “short channel” FET L >> λ  long fin (too long?!), or “long channel” FET 13 with solution and electrostatic screening length Liu (1993) Knoch (2006)