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 è macroscopic
AME 60614 Int. Heat Trans.
D. B. Go Slide 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 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.
D. B. Go Slide 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 60614 Int. Heat Trans.
D. B. Go Slide 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
AME 60614 Int. Heat Trans.
D. B. Go Slide 6
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.
D. B. Go Slide 7
The Scale of Things
AME 60614 Int. Heat Trans.
D. B. Go Slide 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
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.
D. B. Go Slide 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
Thot Tcold
AME 60614 Int. Heat Trans.
D. B. Go Slide 12
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.
D. B. Go Slide 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
€
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.
D. B. Go Slide 14
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