Non-Continuum Energy Transfer: Overview

Transcription

1 Non-Continuum Energy Transfer: Overview D. B. Go Slide 1

2 Topics Covered To Date Conduction - transport of thermal energy through a medium (solid/ liquid/gas) due to the random motion of the energy carriers Fourier s law, circuit analogy (1-D), lumped capacitance (unsteady), separation of variables (2-D steady, 1-D unsteady) Convection transport of thermal energy at the interface of a fluid and a solid due to the random interactions at the surface (conduction) and bulk motion of the fluid (advection) Netwon s law, heat transfer coefficient, energy balance, similarity solutions, integral methods, direct integration Radiation transport of thermal energy to/from a solid due to the emission/absorption of electromagnetic waves (photons) We studied these topics by considering the phenomena at the continuum-scale è macroscopic D. B. Go Slide 2

3 Continuum Scale The continuum-scale is a length/time scale where the medium of interest is treated as continuous individual or discrete effects are not considered Properties can be defined as continuous and averaged over all the energy carriers thermal conductivity viscosity density When the characteristic dimension of the system is comparable to the mechanistic length of the energy carrier, the energy carriers behave discretely and cannot be treated continuously è noncontinuum the mechanistic length is the mean length of transport or mean free path of the energy carrier between collisions even at large length scale this is possible (gas dynamics in a vacuum!) D. B. Go Slide 3

4 Continuum Scale At the continuum-scale, local thermodynamic equilibrium is assumed temperature is only defined at local thermodynamic equilibrium Ultrafast processes may induce non-equilibrium during the timescale of interest (e.g., laser processing) At the non-continuum scale (both time and length) we treat energy carriers statistically D. B. Go Slide 4

5 Four Energy Carriers Phonons bond vibrations between adjacent atoms/molecules in a solid not a true particle è can often be treated as a particle can be likened to mass-spring-mass primary energy carrier in insulating and semi-conducting solids Electrons fundamental particle in matter carries charge (electricity) and thermal energy primary energy carrier in metals Photons electromagnetic waves or light particles è radiation no charge/no mass Atoms/Molecules freely (random) moving energy carriers in a gas/liquid D. B. Go Slide 5

6 Appreciating Length Scales Consider length in meters: 10-9 nano 10-6 micro 10-3 milli kilo 10 6 mega 10 9 giga You Are Here simple molecule (caffeine) D. B. Go Slide 6

7 The Scale of Things D. B. Go Slide 7

8 The Importance of Non-Continuum Technology Perspective scaling down of devices is possible due to advances in technology è take advantage of non-continuum physics potential for high impact in essential fields (healthcare, information, energy) in order to control the transport at these small scales we must understand the nature of the transport Scientific/Academic Perspective study non-continuum phenomena helps us understand the physical nature of the principles we ve come to accept we can define, from first principles, entropy, specific heat, thermal conductivity, ideal gas law, viscosity by understanding non-continuum physics we can better appreciate our world D. B. Go Slide 8

9 mems.sandia.gov D. B. Go Slide 9

10 mems.sandia.gov D. B. Go Slide 10

11 Kinetic Description of Thermal Conductivity Conduction is how thermal energy is transported through a medium è solids: phonons/electrons; fluids: atoms/molecules We will use the kinetic theory approach to arrive at a relationship for thermal conductivity valid for any energy carrier that behaves and be described like a particle T hot T cold D. B. Go Slide 11

12 Kinetic Description of Thermal Conductivity Consider a box of particles Consider the small distance: v x τ v x x-component of velocity G. Chen τ avg time between collisions (relaxation time) If each particle carries with it thermal energy, the total heat flux across the face is the difference between particles moving in the forward direction and those moving in the reverse direction. q x = 1 2 (NEv x ) x +v x τ 1 2 (NEv x ) x v x τ The ½ assumes only half of the particles in the distance v x τ move in the positive direction D. B. Go Slide 12

13 Kinetic Description of Thermal Conductivity We can Taylor expand this relationship just as we did in the derivation of the heat equation: q x = v x τ dnev x dx = v x 2 τ dne dx = v 2 xτ du dx If the speed in the x-direction is 1/3 of the total speed & we use the chain rule q x = 1 3 v2 τ du dt D. B. Go Slide 13 dt dx Specific heat defined as how much the temperature increases for a given amount of heat transfer C = ΔQ ΔT du = δq pdv # U & % ( $ T ' V # = Q & % ( $ T ' V " C V = U % $ ' # T & V

14 Kinetic Description of Thermal Conductivity q x = 1 3 v2 τ du dt dt dx compare to Fourier s Law q x = 1 3 v2 τc v dt dx q x = k dt dx k = 1 3 v 2 τc g To determine thermal conductivity we need to understand how heat is stored and how energy carries collide D. B. Go Slide 14