Non-Continuum Energy Transfer: Overview

1. Non-Continuum Energy Transfer: Overview

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

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 lengthof the energy carrier, the energy carriers behave discretely and cannot be treated continuously non-continuum • the mechanistic length is the mean length of transport ormean free path of the energy carrier between collisions • even at large length scale this is possible (gas dynamics in a vacuum!)

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

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

6. Appreciating Length Scales 10-9 “nano” 10-6 “micro” 10-3 “milli” 100 103 “kilo” 106 “mega” 109 “giga” You Are Here simple molecule (caffeine) Consider length in meters:

7. The Scale of Things

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

9. mems.sandia.gov

10. mems.sandia.gov

11. Kinetic Description of Thermal Conductivity Tcold Thot • 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

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 vxτmove in the positive direction

13. Kinetic Description of Thermal Conductivity Specific heat defined as how much the temperature increases for a given amount of heat transfer 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

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