Heat Transfer Thermal Properties of Matter

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1 Materials Properties and Characterization Heat Transfer Thermal Properties of Matter Dr. Aleandra Telei phone: ML F 8 Particle Technology Laboratory, Department of Mechanical and Process Engineering ETH Zurich, 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, 996 2

2 Conduction Heat is transferred by random molecular motion in fluids and electron or phonon (lattice vibration) motion in solids. T high Fourier s law q " dt = d q : heat flu in direction per unit area (W/m 2 ) : thermal conductivity (W/m K) dt/d: temperature gradient (K/m) heat flu T low Analog Fic s law in mass transfer j= D dc d Thermal properties of matter Thermal conductivity: solid > liquid > gas 4 2

3 Thermal conductivity in solids thermal conduction by: electron movement ( e ) only in electrically conducting materials, i.e. metals lattice vibrations ( l ) in all materials = + e l q " dt = d 5 Thermal conductivity by lattice vibrations Lattice vibrations can be described as phonons ehibiting particle-lie 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 l? As the phonon moves it reduces the temperature by ΔT dt Δ T = l d T l ΔT 6

4 Thermal conductivity by lattice vibrations The distance a phonon travels (Mean free path) = velocity (v ) time between collisions (τ) dt dt Δ T = l = vτ d d The amount of energy carried by each phonon is: T l ΔT dt Energy = cδ T = cv τ d 7 Net flu of energy = - (flu of phonons) (energy/phonon) " q = (n< v > )(cδt) dt = (n < v > )(v τc ) d 2 dt n< v > cτ d 2 dt = n< v > cτ d Average velocity in direction = / average velocity with l = vτ and C = nc Heat capacity/unit volume = concentration heat capacity " dt dt q = Cvl = d d = Cvl 8 4

5 What does the thermal conductivity depend on? l = Cvl At room temp, where C ~ constant, phonon velocity = speed of sound in the material (independent of T) 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 0 5

6 Thermal conductivity in carbon materials Diamond, Graphite, Carbon nanotubes diamond graphite carbon nanotube These materials ehibit etremely high thermal conductivity: = 2000 W/m K = 400 W/m K = 000 W/m K Pierson, Handboo of Carbon, Graphite, Diamond and Fullerenes, 99 Graphite Ehibits high thermal conductivity () in ab direction and lower in c direction (ab): 400 W/m K (can be up to 480 W/m K) (c): 2.2 W/m K c b a waves are very little scattered in ab direction (basal planes) good heat conductor in ab direction, insulator in c direction 2 6

7 Carbon nanotubes Ehibits high thermal conductivity () in a direction and lower in b and c direction c b a (a): up to 000 W/m K (difficult to measure) waves are very little scattered in a direction (along the tube) good heat conductor in a direction, insulator in b and c direction dimensional heat transfer Thermal conductivity in suspensions 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 p Continuous phase with thermal conductivity c continuous phase (i.e. polymer, liquid) Turner et al., Chem. Eng. Sci. (976) first approimation: eff = c (- Φ) + p Φ Effective thermal conductivity behaves lie two layers 4 7

8 Thermal conductivity in suspensions solid particles Particles with radius r, volume fraction Φ, and thermal conductivity p Continuous phase with thermal conductivity c continuous phase (i.e. polymer, liquid) Better is the equation derived by Mawell for particles that are randomly dispersed: eff + 2βφ =, β= ( α )/( α+ 2), α = p /c βφ c valid for: α < 0, all Φ or α > 0 and Φ < 0.2 Turner et al., Chem. Eng. Sci. (976) 5 Nanofluids Nanofluids are suspensions of very small particles (nanoparticles < 50 nm) dispersed in a liquid. They ehibit 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 ehibit low sedimentation, no clogging and less corrosion compared to larger particles Mawell s relation fails when describing thermal conductivity in nanofluids Small particles ehibit much higher brownian motion (Stoes-Einstein) Heat transfer by brownian motion Jang et al., Appl. Phys. Lett., (2004); Das et al., Heat Trans. Eng. (2006) T B D = πμd p 6 8

9 Thermal Conductivity of Nanofluids Total thermal conductivity: Energy transport by fluid, particles and brownian motion of particles eff = c( φ ) + pφ+ BMφ Similar as before for phonon heat conduction one can derive equations for the thermal conductivity in the fluid and in the particles = l C v c c c c = l C v p p p p l c : mean free path of fluid C c : heat capacity of fluid v c : mean velocity of fluid l p : mean free path of phonon in particles C p : heat capacity of particle v p : mean phonon velocity in particle 7 Thermal Conductivity of Nanofluids Convection-lie effect at nanolevel BM = hδ T h: Heat transfer coeff. for flow past the particle δ T : Thicness of thermal boundary layer Jang et al., Appl. Phys. Lett.,

10 Thermal Conductivity of Nanofluids The heat transfer coefficient for flow past nanoparticles (h) can be defined as: vd 2 2 p p η C h Re Pr, Re =, Pr = d ν c c c p c 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 : inematic viscosity of liquid, η c : dynamic viscosity of liquid The hydrodynamic boundary layer is ca. times the fluid diameter (d c ) δ d c, δ T = δ Pr 9 Thermal Conductivity of Nanofluids Combining all equations we get: dc 2 eff = c( φ ) + pφ+ c Re Prφ d p contribution by brownian motion vd p p Dpc T B Re =, v p =, Dpc = μ l πμ d c c c p Stoes-Einstein If we include these equations we see that the contribution of Brownian motion to thermal conductivity is proportional to: 2 BM T, dp 20 0

11 Thermal Conductivity of Nanofluids Higher temperature higher conductivity Smaller particles higher conductivity T 2 d p 2 Lecture summary Heat transfer Conduction, Convection, Radiation Thermal conductivity Solids Suspensions Nanofluids 22

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