AME  60614    Int.  Heat  Trans.  

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Non-Continuum Energy Transfer: Overview

AME  60614    Int.  Heat  Trans.  

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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

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 è 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” 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

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 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

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 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

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: vxτvx ≡ x-component of velocityτ ≡ 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.

€

qx =12(NEvx )x+vxτ

−12(NEvx )x−vxτ

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

€

qx = −vxτdNEvxdx

= −vx2τdNEdx

= −vx2τdUdx

qx = −13v2τ dU

dTdTdx

CV =∂U∂T"

#$

%

&'V

Specific heat defined as how much the temperature increases for a given amount of heat transfer

C = ΔQΔT

dU = δQ− pdV∂U∂T#

$%

&

'(V

=∂Q∂T#

$%

&

'(V

AME  60614    Int.  Heat  Trans.  

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Kinetic Description of Thermal Conductivity

qx = −13v2τ dU

dTdTdx

compare to Fourier’s Law

k = 13vg2τC

qx = −kdTdx

To determine thermal conductivity we need to understand how heat is stored and how energy carries collide

qx = −13v2τCv

dTdx