Materials Properties and Characterization
Heat Transfer Thermal Properties of Matter
Dr. Alexandra Teleki
[email protected] phone: 044 632 3952 ML F 18
Particle Technology Laboratory, Department of Mechanical and Process Engineering ETH Zurich, www.ptl.ethz.ch
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Heat transfer (or heat) is energy in transit due to temperature difference Heat is transferred by: Conduction
Convection
Radiation
F. P. Incropera, D. P. DeWitt “Fundamentals of Heat and Mass Transfer” 4th ed, 1996
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Conduction Heat is transferred by random molecular motion in fluids and electron or phonon (lattice vibration) motion in solids.
Fourier’s law
q"x = −k
Thigh
dT dx
qx”: heat flux in x direction per unit area (W/m2) k: thermal conductivity (W/m·K) dT/dx: temperature gradient (K/m)
Analog Fick’s law in mass transfer
j = −D
heat flux
Tlow
dc dx 3
Thermal properties of matter Thermal conductivity: solid > liquid > gas
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Thermal conductivity in solids thermal conduction by: electron movement (ke)
q"x = −k
only in electrically conducting materials, i.e. metals
dT dx
lattice vibrations (kl)
in all materials
k = k e + kl
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Thermal conductivity by lattice vibrations Lattice vibrations can be described as phonons exhibiting particle-like behavior These phonons carry a certain amount of energy in the form of heat By how much does the temperature change by each one of these phonons moving a distance lx ?
As the phonon moves it reduces the temperature by ΔT
ΔT =
T
lx ΔT
dT lx dx x 6
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Thermal conductivity by lattice vibrations The distance a phonon travels (Mean free path) = velocity (vx) x time between collisions (τ)
ΔT =
dT dT lx = vxτ dx dx T
The amount of energy carried by each phonon is:
Energy = cΔT =
lx ΔT
dT cv x τ dx
x 7
Net flux of energy = - (flux of phonons) x (energy/phonon) q"x = −(n < v x > )(cΔT) = −(n < v x > )(v x τc
dT ) dx
dT dx 1 dT 2 = − n < v > cτ 3 dx ≈ −n < v x 2 > cτ
Average velocity in x direction = 1/3 average velocity
with l = vτ and C = nc Heat capacity/unit volume = concentration x heat capacity
1 dT dT =k q"x = − Cvl 3 dx dx
Î
k=
1 Cvl 3 8
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What does the thermal conductivity depend on? kl =
1 Cvl 3
At room temp, where C ~ constant,
phonon velocity = speed of sound in the material (independent of T) −1 the phonon mean free path, l is important (l ∝ T )
What determines the phonon mean free path? 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 However, in the real world, there is a much smaller, finite mean free path for phonons 9
Defects in crystal lattices
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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Thermal conductivity in carbon materials Diamond, Graphite, Carbon nanotubes
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
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Carbon nanotubes c Exhibits high thermal conductivity (k) in a direction and lower k in b and c direction
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 13
Thermal conductivity in suspensions How do particles dispersed in a continuous medium affect it’s thermal conductivity? solid particles
Particles with radius dp, volume fraction Φ, and thermal conductivity kp Continuous phase with thermal conductivity kc
continuous phase (i.e. polymer, liquid)
first approximation: keff = kc(1- Φ) + kp Φ Effective thermal conductivity behaves like two layers
Turner et al., Chem. Eng. Sci. (1976)
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Thermal conductivity in suspensions solid particles
Particles with radius r, volume fraction Φ, and thermal conductivity kp Continuous phase with thermal conductivity kc
continuous phase (i.e. polymer, liquid)
Better is the equation derived by Maxwell for particles that are randomly dispersed: k eff 1 + 2βφ = , β = (α − 1) /(α + 2), α = k p / k c kc 1 − βφ valid for: α < 10, all Φ or α > 10 and Φ < 0.2
Turner et al., Chem. Eng. Sci. (1976)
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Nanofluids 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=
kBT 3πμdp
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Thermal Conductivity of Nanofluids Total thermal conductivity: Energy transport by fluid, particles and brownian motion of particles
k eff = k c (1 − φ) + k p φ + k BMφ Similar as before for phonon heat conduction one can derive equations for the thermal conductivity in the fluid and in the particles kc =
1 lc Cc v c 3
lc: mean free path of fluid Cc: heat capacity of fluid vc: mean velocity of fluid
kp =
1 lpCp v p 3
lp: mean free path of phonon in particles Cp: heat capacity of particle vp: mean phonon velocity in particle 17
Thermal Conductivity of Nanofluids Convection-like effect at nanolevel
k BM = hδT h: Heat transfer coeff. for flow past the particle δT: Thickness of thermal boundary layer
Jang et al., Appl. Phys. Lett., 2004
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Thermal Conductivity of Nanofluids The heat transfer coefficient for flow past nanoparticles (h) can be defined as:
h∼
kc Re2 Pr 2 , dp
Re =
v p dp νc
,
Pr =
ηc Cc kc
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, νc: kinematic viscosity of liquid, ηc: dynamic viscosity of liquid The hydrodynamic boundary layer is ca. 3 times the fluid diameter (dc)
δ ∼ 3dc ,
δT =
δ Pr
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Thermal Conductivity of Nanofluids Combining all equations we get:
k eff = k c (1 − φ) + k p φ + 3
Re =
v p dp μc
,
vp =
dc k c Re2 Pr φ dp
Dpc lc
contribution by brownian motion
,
kBT Dpc = 3πμc dp
Stokes-Einstein
If we include these equations we see that the contribution of Brownian motion to thermal conductivity is proportional to:
k BM ∝ T 2 , dp −1 20
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Thermal Conductivity of Nanofluids Higher temperature → higher conductivity
Smaller particles → higher conductivity
∝ T2
∝ dp −1
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Lecture summary Heat transfer
Conduction, Convection, Radiation
Thermal conductivity
Solids Suspensions Nanofluids
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