AME 60614 Int. Heat Trans. D. B. GoSlide 1 Non-Continuum Energy Transfer: Overview.

AME 60614 Int. Heat Trans.

D. B. Go Slide 1

Non-Continuum Energy Transfer: Overview

AME 60614 Int. Heat Trans.

D. B. Go Slide 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 60614 Int. Heat Trans.

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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 60614 Int. Heat Trans.

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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 60614 Int. Heat Trans.

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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 60614 Int. Heat Trans.

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Appreciating Length Scales

Consider length in meters:

10-9

“nano”10-6

“micro”10-3

“milli”100 103

“kilo”106

“mega”109

“giga”

simple molecule(caffeine)

You Are Here

AME 60614 Int. Heat Trans.

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The Scale of Things

AME 60614 Int. Heat Trans.

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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 60614 Int. Heat Trans.

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

AME 60614 Int. Heat Trans.

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

AME 60614 Int. Heat Trans.

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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

Thot Tcold

AME 60614 Int. Heat Trans.

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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 vxτ move in the positive direction

AME 60614 Int. Heat Trans.

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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 60614 Int. Heat Trans.

D. B. Go Slide 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