Materials Properties and Characterization
Dr. Frank O. Ernst ernst@ptl.mavt.ethz.ch
phone: 044 632 6273
ML F 22
Particle Technology Laboratory,
Department of Mechanical and Process Engineering
ETH Zurich, www.ptl.ethz.ch
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Heat is transferred by:
Conduction Convection Radiation
F. P. Incropera, D. P. DeWitt “Fundamentals of Heat and Mass Transfer” 4th ed, 1996 2
1

Heat is transferred by random molecular motion in fluids and electron or phonon (lattice vibration) motion in solids.
T high
Fourier’s law q " x
= − k dT dx q x
” : heat flux in x direction per unit area (W/m 2 ) k: thermal conductivity (W/m·K) dT/dx: temperature gradient (K/m) heat flux T low
Analog Fick’s law in mass transfer j = − D dc dx
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Thermal conductivity: solid > liquid > gas
4
2

thermal conduction by:
ƒ electron movement (k
ƒ e
) only in electrically conducting materials, i.e. metals
ƒ lattice vibrations (k l
)
ƒ in all materials k k e
+ k l q " x
= − k dT dx
Lattice vibrations can be described as phonons exhibiting particle-like behavior
ƒ These phonons carry a certain amount of energy in the form of heat
ƒ By how much does the temperature change by each one of these phonons moving a distance l x
?
ƒ As the phonon moves it reduces the temperature by Δ T
T l x
Δ T dT dx l x x
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5
3

The distance a phonon travels (Mean free path) velocity (v x
=
) x time between collisions ( τ ) dT dx l x
= dT dx v x
τ
The amount of energy carried by each phonon is:
T l x
Δ T dT dx cv x
τ x
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Net flux of energy = - (flux of phonons) x (energy/phonon) q " x
(n v x
)(c T)
= − < x
> τ dT dx
) x
2
= −
1
3 n v 2 c dT dx dT dx
Average velocity in x direction
= 1/3 average velocity with l = v τ and C = nc
Heat capacity/unit volume = concentration x heat capacity q " x
= −
1
3
Cvl dT dx
= k dT dx
Î k =
1
Cvl
3
8
4

k l
=
1
3
Cvl
ƒ At room temp, where C ~ constant,
ƒ
ƒ phonon velocity = speed of sound in the material (independent of T) l ∝ T
ƒ What determines the phonon mean free path?
ƒ For a material with purely harmonic interactions, and a perfect lattice
(i.e. no defects such as dislocations), there would be nothing to stop the phonons: mean free path ~ size of crystal
ƒ However, in the real world, there is a much smaller, finite mean free path for phonons
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a) Interstitial impurity atom, b) Edge dislocation, c) Self interstitial atom, d)
Vacancy, e) Precipitate of impurity atoms, f) Vacancy type dislocation loop, g) Interstitial type dislocation loop, h) Substitutional impurity atom
The mean free path of phonons is also limited by collisions with other phonons
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5

diamond graphite carbon nanotube
These materials exhibit extremely high thermal conductivity: k = 2000 W/m·K k = 400 W/m·K k = 3000 W/m·K
Pierson, “Handbook of Carbon, Graphite, Diamond and Fullerenes”, 1993
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Graphite c
Exhibits high thermal conductivity
(k) in ab direction and lower k in c direction b a
• k (ab): 400 W/m·K (can be up to 4180 W/m·K)
• k (c): 2.2 W/m·K
• waves are very little scattered in ab direction (basal planes)
Î good heat conductor in ab direction, insulator in c direction
12
6

Carbon nanotubes
Exhibits high thermal conductivity
(k) in a direction and lower k in b and c direction c b a
• k (a): up to 3000 W/m·K (difficult to measure)
• waves are very little scattered in a direction (along the tube)
Î good heat conductor in a direction, insulator in b and c direction
1 dimensional heat transfer
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How do particles dispersed in a continuous medium affect it’s thermal conductivity?
solid particles
Particles with radius d p volume fraction Φ ,
, and thermal conductivity k p
Continuous phase with thermal conductivity k c continuous phase
(i.e. polymer, liquid) first approximation: k eff
= k c
(1Φ ) + k p
Φ
Turner et al., Chem. Eng. Sci. (1976)
Effective thermal conductivity behaves like two layers
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7

solid particles
Particles with radius r, volume fraction Φ , and thermal conductivity k p
Continuous phase with thermal conductivity k c continuous phase
(i.e. polymer, liquid)
Better is the equation derived by Maxwell for particles that are randomly dispersed: k eff k c
=
1 − βφ
, β = α − 1) /( α + 2), α = c valid for: α < 10, all Φ or α > 10 and Φ < 0.2
Turner et al., Chem. Eng. Sci. (1976) 15
Nanofluids are suspensions of very small particles
(nanoparticles < 50 nm) dispersed in a liquid.
They exhibit much increased thermal conductivity compared to the pure fluid even at very low particle concentrations:
• Important for increasing the heat conductivity of cooling liquids
• Suspension of nanoparticles exhibit low sedimentation, no clogging and less corrosion compared to larger particles
Maxwell’s relation fails when describing thermal conductivity in nanofluids
Small particles exhibit much higher brownian motion (Stokes-Einstein)
Î Heat transfer by brownian motion
Jang et al., Appl. Phys. Lett., (2004); Das et al., Heat Trans. Eng. (2006)
D =
3 k T
πμ d p 16
8

Total thermal conductivity:
Energy transport by fluid, particles and brownian motion of particles k eff
= k (1 − φ + φ + k
BM
φ
Similar as before for phonon heat conduction one can derive equations for the thermal conductivity in the fluid and in the particles k c
=
1
3 l C v c l c
: mean free path of fluid
C c v c
: heat capacity of fluid
: mean velocity of fluid k p
=
1
3 l C v p l p
: mean free path of phonon in particles
C p v p
: heat capacity of particle
: mean phonon velocity in particle
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Thermal Conductivity of Nanofluids
Convection-like effect at nanolevel k
BM h
T h: Heat transfer coeff. for flow past the particle
δ
T
: Thickness of thermal boundary layer
Jang et al., Appl. Phys. Lett., 2004
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9

Thermal Conductivity of Nanofluids
The heat transfer coefficient for flow past nanoparticles (h) can be defined as: h
k c d p
Re Pr , Re =
ν c
, Pr =
η c
C c k c
Re: Reynolds # for particles, relates inertial forces to viscous forces
Pr: Prandtl # relates the momentum boundary layer to the thermal boundary layer or the viscous diffusion rate to the thermal diffusion rate,
ν c
: kinematic viscosity of liquid, η c
: dynamic viscosity of liquid
The hydrodynamic boundary layer is ca. 3 times the fluid diameter (d c
)
δ ∼
δ
Pr
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Thermal Conductivity of Nanofluids
Combining all equations we get: k eff
= k (1 − φ + φ + 3 d c k Re Pr d p
φ contribution by brownian motion
Re =
μ c
,
Stokes-Einstein v p
=
D pc , l c
D pc
=
3 πμ d
If we include these equations we see that the contribution of
Brownian motion to thermal conductivity is proportional to: k
BM
p
− 1
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Thermal Conductivity of Nanofluids
Higher temperature
→ higher conductivity
Smaller particles
→ higher conductivity
T 2
ƒ Heat transfer
ƒ
Conduction, Convection, Radiation
ƒ Thermal conductivity
ƒ
ƒ
ƒ
Solids
Suspensions
Nanofluids
d p
− 1
21
22
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