AME Int. Heat Trans. D. B. GoSlide 1 Non-Continuum Energy Transfer: Overview

AME Int. Heat Trans. D. B. GoSlide 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  macroscopic

AME Int. Heat Trans. D. B. GoSlide 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  non- continuum –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!)

AME Int. Heat Trans. D. B. GoSlide 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

AME Int. Heat Trans. D. B. GoSlide 5 Four Energy Carriers Phonons – bond vibrations between adjacent atoms/molecules in a solid –not a true “particle”  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”  radiation –no charge/no mass Atoms/Molecules –freely (random) moving energy carriers in a gas/liquid

AME Int. Heat Trans. D. B. GoSlide 6 Appreciating Length Scales Consider length in meters: “nano” “micro” “milli” “kilo” 10 6 “mega” 10 9 “giga” simple molecule (caffeine) You Are Here

AME Int. Heat Trans. D. B. GoSlide 7 The Scale of Things

AME Int. Heat Trans. D. B. GoSlide 8 The Importance of Non-Continuum Technology Perspective –scaling down of devices is possible due to advances in technology  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

AME Int. Heat Trans. D. B. GoSlide 9 mems.sandia.gov

AME Int. Heat Trans. D. B. GoSlide 10 mems.sandia.gov

AME Int. Heat Trans. D. B. GoSlide 11 Kinetic Description of Thermal Conductivity Conduction is how thermal energy is transported through a medium  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

AME Int. Heat Trans. D. B. GoSlide 12 Kinetic Description of Thermal Conductivity Consider a box of particles G. Chen Consider the small distance: 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. The ½ assumes only half of the particles in the distance v x τ move in the positive direction

AME Int. Heat Trans. D. B. GoSlide 13 Kinetic Description of Thermal Conductivity We can Taylor expand this relationship just as we did in the derivation of the heat equation: If the speed in the x-direction is 1/3 of the total speed & we use the chain rule Specific heat defined as how much the temperature increases for a given amount of heat transfer

AME Int. Heat Trans. D. B. GoSlide 14 Kinetic Description of Thermal Conductivity compare to Fourier’s Law To determine thermal conductivity we need to understand how heat is stored and how energy carries collide