Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

1. Advanced Transport Phenomena Module 6 Lecture 27 Mass Transport: Two-Phase Flow Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

2. CONVECTIVE MASS TRANSFER IN LAMINAR- AND TURBULENT-FLOW SYSTEMS • Analogies to Momentum Transfer: (High Sc Effects) • Streamwise pressure gradient can break mass/ momentum transfer analogy (St & cf/2) • For laminar or turbulent flows with negligible pressure gradient, Reynolds’- Chilton – Colburn analogy holds:

3. CONVECTIVE MASS TRANSFER IN LAMINAR- AND TURBULENT-FLOW SYSTEMS • Analogies to Momentum Transfer: (High Sc Effects) • For Sc ≈ 1 (e.g., solute gas diffusion through gaseous solvents), Prandtl’s form of extended analogy holds: • In many mass-transfer applications (e.g., aerosols, ions in aqueous solutions), Sc >>1 since D << n • Correlation would underestimate Stm for Sc > 102

4. CONVECTIVE MASS TRANSFER IN LAMINAR- AND TURBULENT-FLOW SYSTEMS • Analogies to Momentum Transfer: (High Sc Effects) • For Sc >> 1: (Shaw and Hanratty, 1977) • Experimental: Stm ~ Sc(-2/3) • Surface roughness effect: when comparable to or greater in height compared to viscous sublayer thickness (dSL ≈ (cf/2)1/2 (5n/U)) increases both cf/2 and St • Effect on St < on friction coeff (hence, pressure drop)

5. CHEMICAL NONEQUILIBRIUM (KINETIC) BOUNDARY CONDITIONS • When dilute species A reacts only at fluid/ solid interface, Stm(Re, Sc) still applies • Mass flux of species A at the wall • This flux appears in BC for species A at fluid/ surface interface

6. CHEMICAL NONEQUILIBRIUM (KINETIC) BOUNDARY CONDITIONS • If species A is being consumed at a local rate given by (irreversible, first-order) chemical reaction: • Surface BC (or jump condition, JC) takes the form:

7. CHEMICAL NONEQUILIBRIUM (KINETIC) BOUNDARY CONDITIONS • JC provides algebraic equation for quasi-steady species A mass fraction, wA,w, at surface, and: and transfer rate as a fraction of maximum (“diffusion-controlled”) rate; C << 1 => fraction is small, rate approaches “chemically controlled” value, kwrwA,∞

8. CHEMICAL NONEQUILIBRIUM (KINETIC) BOUNDARY CONDITIONS • C  surface Damkohler number; “catalytic parameter”; defined by: Resistance additivity approach: adequate for engineering purposes when applied locally along a surface with slowly-varying x-dependences of Tw, kw, wA,w

9. CHEMICAL NONEQUILIBRIUM (KINETIC) BOUNDARY CONDITIONS • If LTCE is achieved at station w due to rapid heterogeneous chemical reactions, then: • wi,w = wi,eq(Tw,….;p) for all species i • Used to estimate chemical vapor deposition (CVD) rates in multicomponent vapor systems with surface equilibrium

10. CHEMICAL NONEQUILIBRIUM (KINETIC) BOUNDARY CONDITIONS • In the presence of homogeneous reactions, similar approach can be used to estimate element fluxes • Effective Fick diffusion flux of each element (k) estimated via: (diffusion coefficients evaluated as weighted sums of Di)

11. COMBINED ENERGY & MASS TRANSPORT: RECOVERY OF MAINSTREAM CHEMICAL & KINETIC ENERGY • If a thermometer is placed in a hot stream with considerable kinetic energy & chemical energy, what temperature will it read? • Neglecting radiation loss, surface temperature will rise to a SS-value at which rate of convective heat loss (Tr gas-dynamic recovery temperature)

12. COMBINED ENERGY & MASS TRANSPORT: RECOVERY OF MAINSTREAM CHEMICAL & KINETIC ENERGY balances rate of energy transport associated with species A mass transport: (Q  energy release per unit mass of A)

13. COMBINED ENERGY & MASS TRANSPORT: RECOVERY OF MAINSTREAM CHEMICAL & KINETIC ENERGY • Adiabatic condition: = 0 (including both contributions) => • In forced-convection systems, (Stm/Sth)  chemical-energy recovery factor, rChE

14. COMBINED ENERGY & MASS TRANSPORT: RECOVERY OF MAINSTREAM CHEMICAL & KINETIC ENERGY • For a laminar BL, rKE ≈ Pr1/2, rChE ≈ Le2/3, and • Tw can be higher or lower than corresponding thermodynamic (“total”) temperature: (depending on Pr, Le)

15. COMBINED ENERGY & MASS TRANSPORT: RECOVERY OF MAINSTREAM CHEMICAL & KINETIC ENERGY • In most gas mixtures, both rKE and rChE ≈ 1 • Probe records temperature near T0, not T∞ • rChE important in measuring temperatures of gas streams that are out of chemical equilibrium • Tw >> T∞ or Trcan be recorded

16. COMBINED ENERGY & MASS TRANSPORT: RECOVERY OF MAINSTREAM CHEMICAL & KINETIC ENERGY • For non-adiabatic surfaces: Tr’ generalized recovery temperature (Tw - Tr’)  “overheat”

17. TWO-PHASE FLOW: MASS TRANSFER EFFECTS OF INERTIAL SLIP & ISOKINETIC SAMPLING • When dynamic coupling between suspended particles (or heavy solute molecules) & carrier fluid is weak  consider particles as distinct phase • Distinction between two-phase flow & flow of ordinary mixtures • Quantified by Stokes’ number, Stk • Above critical value of Stk, 2nd phase can inertially impact on target, even while host fluid is brought to rest

18. TWO-PHASE FLOW: MASS TRANSFER EFFECTS OF INERTIAL SLIP & ISOKINETIC SAMPLING

19. TWO-PHASE FLOW: MASS TRANSFER EFFECTS OF INERTIAL SLIP & ISOKINETIC SAMPLING • Pure inertial impaction at supercritical Stokes’ numbers: Cylinder in cross flow • Particle-laden steady carrier flow of mainstream velocity, U • Suspended particles assumed to be: • Spherical (diameter dp << L) • Negligible mass loading & volume fraction • Large enough to neglect Dp, small enough to neglect gravitational sedimentation • Captured on impact (no rebound)

20. TWO-PHASE FLOW: MASS TRANSFER EFFECTS OF INERTIAL SLIP & ISOKINETIC SAMPLING • Pure inertial impaction at supercritical Stokes’ numbers: Cylinder in cross flow • Each particle moves along trajectory determined by host-fluid velocity field & its drag at prevailing Re (based on local slip velocity) • Capture efficiency function • Calculated from limiting-particle trajectories (upstream locations of particles whose trajectories become tangent to target)

21. TWO-PHASE FLOW: MASS TRANSFER EFFECTS OF INERTIAL SLIP & ISOKINETIC SAMPLING • Pure inertial impaction at supercritical Stokes’ numbers: Cylinder in crossflow • hcapture = 0 for Stk < Stkcrit • Capture occurs only above a critical Stokes’ number (for idealized model of particle capture from a two-phase flow)

22. TWO-PHASE FLOW: MASS TRANSFER EFFECTS OF INERTIAL SLIP & ISOKINETIC SAMPLING • Pure inertial impaction at supercritical Stokes’ numbers: Cylinder in cross flow Particle capture fraction correlation for ideal ( ) flow past a transverse circular cylinder (Israel and Rosner (1983)). Here tflow=(d/2)/U.

23. TWO-PHASE FLOW: MASS TRANSFER EFFECTS OF INERTIAL SLIP & ISOKINETIC SAMPLING • Pure inertial impaction at supercritical Stokes’ numbers: Cylinder in crossflow • In practice, some deposition occurs even at Stk < Stkcrit • Due to non-zero Brownian diffusivity, thermophoresis, etc. • Rates still influenced by Stk since particle fluid is compressible (even while host carrier is subsonic) • Inertial enrichment (pile-up) of particles in forward stagnation region, centrifugal depletion downstream • Net effect: can be a reduction below diffusional deposition rate

24. TWO-PHASE FLOW: MASS TRANSFER EFFECTS OF INERTIAL SLIP & ISOKINETIC SAMPLING • Pure inertial impaction at supercritical Stokes’ numbers: Cylinder in crossflow • Combustion application: sampling of particle-laden (e.g., sooty) combustion gases using a small suction probe • Sampling rate too great => capture efficiency for host gas > that of particles => under-estimation; and vice versa • Sampling rate at which both are equal  isokinetic condition (particle size dependent)

25. TWO-PHASE FLOW: MASS TRANSFER EFFECTS OF INERTIAL SLIP & ISOKINETIC SAMPLING • Pure inertial impaction at supercritical Stokes’ numbers: Cylinder in cross flow Effect of probe sampling rate on capture of particles and their carrier fluid

26. Two-Phase Flow: Mass Transfer Effects of Inertial Slip & Isokinetic Sampling • Effective diffusivity of particles in turbulent flow • Ability to follow local turbulence (despite their inertia) governed by Stokes’ number, Stkt Relevant local flow time = ratio of scale of turbulence, lt, to rms turbulent velocity

27. TWO-PHASE FLOW: MASS TRANSFER EFFECTS OF INERTIAL SLIP & ISOKINETIC SAMPLING • Effective diffusivity of particles in turbulent flow • Alternative form of characteristic turbulent eddy time, where kt turbulent kinetic energy per unit mass, and e  turbulent viscous dissipation rate per unit mass

28. TWO-PHASE FLOW: MASS TRANSFER EFFECTS OF INERTIAL SLIP & ISOKINETIC SAMPLING • Effective diffusivity of particles in turbulent flow • and (for particles in fully turbulent flow, nt >> n) • Data: fct( ) >> 1 for • Alternative approach to turbulent particle dispersion: stochastic particle-tracking (Monte Carlo technique)

29. TWO-PHASE FLOW: MASS TRANSFER EFFECTS OF INERTIAL SLIP & ISOKINETIC SAMPLING • Eddy impaction: • When Stkt is sufficiently large, some eddies project particles through viscous sublayer, significantly increasing the deposition rate • Represented by modified Stokes’ number: • Eddy-impaction augmentation of Stm negligible for Stkt,eff-values < 10-1 • Below this value, turbulent particle-containing BL behaves like single-phase fluid