KINETIC THEORY AND MICRO/NANOFLUDICS

1. KINETIC THEORY AND MICRO/NANOFLUDICS • Kinetic Description of Dilute Gases • Transport Equations and Properties of Ideal Gases • The Boltzmann Transport Equation • Micro/Nanofludics and Heat Transfer

2. Kinetic Description of Dilute Gases simple kinetic theory of ideal molecular gases limited to local equilibrium based on the mean-free-path approximation Hypotheses and Assumptions • molecular hypothesis ▪ matter: composition of small discrete particles • ▪ a large number of particles in any macroscopic • volume (27×106 molecules in 1-mm3 at 25ºC and 1 atm) • statistic hypothesis ▪ long time laps: longer than mean-free time or relaxation time • ▪ time average

3. kinetic hypothesis ▪ laws of classical mechanics: Newton’s law of motion • molecular chaos ▪ velocity and position of a particle: uncorrelated (phase space) • ▪ velocity of any two particles: uncorrelated • ideal gas assumptions ▪ molecules: widely separated rigid spheres • ▪ elastic collision: energy and momentum conserved • ▪ negligible intermolecular forces except during collisions • ▪ duration of collision (collision time) << mean free time • ▪ no collision with more than two particles

4. ▪ number of particles in a volume element of the phase space in • Distribution Function : particle number density in the phase space at any time ▪ number of particles per unit volume (integration over the velocity space)

5. density: ▪ total number of particles in the volume V In a thermodynamic equilibrium state, the distribution function does not vary with time and space.

6. : additive property of a single molecule such as kinetic energy and momentum • Local Average and Flux ▪ local average or simply average (average over the velocity space) ▪ ensemble average (average over the phase space)

7. number of particles with velocities between and that passes through the area dA in the time interval dt q dA vdt flux of y within ▪ flux of y : transfer of y across an area element dA per unit time dt per unit area dt is so small that particle collisions can be neglected. total flux of y :

8. ▪particle flux: In an equilibrium state For an ideal gas: Maxwell’s velocity distribution

9. ▪ average speed For an ideal gas: Maxwell’s velocity distribution ▪ mass flux

10. ▪ kinetic energy flux ▪ momentum flux

11. d 2d d • The Mean Free Path • Mean Free Path : average distance between two subsequent collisions for a gas molecule. m0 m1 m1 m2 Mean Free Path 11

12. d 2d d • ndV particles will collide with the moving particle. • number of collisions per unit time : (frequency) 12

13. relative movement of particles • magnitude of the relative velocity : Since and are random and uncorrelated,

14. relative movement of particles - Ideal gas : based on the Maxwell velocity distribution 14

15. : probability that a molecule travels at least x between collisions Probability for the particle to collide within an element distance dx : probability not to collide within dx probability to travel at least x + dx between collision probability not to collide within x + dx 15

16. Probability density function (PDF) : 16

17. is the mean free-path PDF. 17

18. : probability for molecules to have a free path less than x Free-path distribution functions 18

19. Transport Eqs and Properties of Ideal Gases • Average Collision Distance Molecular gas at steady state (Local equilibrium) Average collision distance dAcosq : projected area x: coordinate along gradient Lcosq : average projected length

20. Shear Force and Viscosity Momentum exchange between upper layer and lower layer Average momentum of particles Momentum flux across y0 plane Velocity in y direction Flow direction, x

21. Net momentum flux : Shear force Dynamic viscosity : Order-of-magnitude estimate weak dependence on pressure Dynamic viscosity from more detailed calculation and experiments Simple ideal gas model → Rigid-elastic-sphere model

22. Heat Diffusion Thermal energy transfer Molecular random motion → Net energy flux across x0 plane Temperature,T xdirection

23. Heat flux Thermal conductivity T dependence

24. Thermal conductivity versus Dynamic viscosity < Tabulated values for real gases Same Laof momentum transport & energy transfer Eucken’s formula: Gas T(K) Pr (Eq.) Pr (Exp.) Air 273.2 0.74 0.73 ≈ Tabulated values for real gases Monatomic gas Diatomic gas

25. Mass Diffusion Fick’s law Gas A nA = n nB = 0 Gas B nA = 0 nB = n nA(x) nB(x) x direction

26. Uniform P Net molecular flux Diffusion coefficient

27. Intermolecular Forces Rigid-elastic-sphere model → Not actual collision process Attractive force (Van der Waals force) Fluctuating dipoles in two molecules Repulsive force Overlap of electronic orbits in atoms Intermolecular potential Empirical expression (Lennard-Jones) Repulsive Attractive Intermolecular potential, φ

28. Force between molecules Newton’s law of motion for each molecule Computer simulation of the trajectory of each molecule Molecular dynamic is a powerful tool for dense phases, phase change → Not good for dilute gas → Direct Simulation Monte Carlo (DSMC)

29. and • Thermal Conductivity L : mean free path [m] u : energy density of particles [J/m3] : characteristic velocity of particles [m/s] z +Lz L  z z - Lz heat flux in the z-direction Taylor series expansion

30. Averaging over the whole hemisphere of solid angle 2

31. Assuming local thermodynamic equilibrium: u is a function of temperature Fourier law of heat conduction First term :latticecontribution Second term :electroncontribution

32. The Boltzmann Transport Equation Volume element in phase space Without collision, same number of particles in

33. Liouville equation In the absence of collision and body force

34. : number of particles that join the group in as a result of collisions : number of particles lost to the group as a result of collisions : scattering probability the fraction of particles with a velocity that will change their velocity to per unit time due to collision With collisions, Boltzmann transport equation

35. Relaxation time approximation under conditions not too far from the equilibrium f0 : equilibrium distribution t : relaxation time

36. Hydrodynamic Equations The continuity, momentum and energy equations can be derived from the BTE The first term local average The second term

37. Since velocity components are independent variables in the phase space,

38. The third term Integrating by parts

39. When : bulk velocity, : random velocity • Continuity equation

40. When • Momentum equation : shear stress

41. combination of all terms

42. applying the mass balance equation

43. : energy flux vector • Energy equation : only random motion contributes to the internal energy u: mass specific internal energy

44. using the continuity equation

45. Fourier’s Law and Thermal conductivity BTE under RTA Assume that the temperature gradient is in the only x-direction, medium is stationary local average velocity is zero, distribution function with x only at a steady state If not very far away from equilibrium

46. : 1-D Fourier’s law : 3-D Fourier’s law heat flux in the x direction Under local-equilibrium assumption and applying the RTA

47. Micro/Nanofluidics and Heat Transfer Microdevices involving fluid flow : microsensors, actuators, valves, heat pipes and microducts used in heat engines and heat exchangers Biomedical diagnosis (Lab-on-a-chip), drug delivery, MEMS/NEMS sensors, actuators, micropump for ink-jetprinting, microchannel heat sinks for electronic cooling Fluid flow inside nanostructures, such as nanotubes and nanojet

48. The Knudsen Number and Flow Regimes • Knudsen Number ratio of the mean free path to the characteristic length • Knudsen number relation with Mach number and Reynolds number g : ratio of specific heat

49. Knudsen Number Rarefaction or Continuum Flow Regimes based on the Knudsen Number

50. centerline 3 1 1 3 2 2 Velocity profiles Temperature profiles • Flow regimes Continuum flow (Kn < 0.001) The Navier-Stokes eqs. are applicable. The velocity of flow at the boundary is the same as that of the wall The temperature of flow near the wall is the same as the surface temperature. Conventionally, the flow can be assumed compressibility. If Ma< 0.3, the flow can be assumed incompressible. Consider compressibility : pressure change, density change