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

 Analogies to Momentum Transfer: (High Sc Effects)  Streamwise pressure gradient can break mass/ momentum transfer analogy (St & c f /2)  For laminar or turbulent flows with negligible pressure gradient, Reynolds’- Chilton – Colburn analogy holds: 2 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 <<  Correlation would underestimate St m for Sc > CONVECTIVE MASS TRANSFER IN LAMINAR- AND TURBULENT-FLOW SYSTEMS

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

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

 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: 6 CHEMICAL NONEQUILIBRIUM (KINETIC) BOUNDARY CONDITIONS

 JC provides algebraic equation for quasi-steady species A mass fraction,  A,w, at surface, and: and transfer rate as a fraction of maximum (“diffusion- controlled”) rate; C fraction is small, rate approaches “chemically controlled” value, k w  A,∞ 7 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 T w, k w  A,w 8 CHEMICAL NONEQUILIBRIUM (KINETIC) BOUNDARY CONDITIONS

 If LTCE is achieved at station w due to rapid heterogeneous chemical reactions, then:   i,w =  i,eq (T w,….;p) for all species i  Used to estimate chemical vapor deposition (CVD) rates in multicomponent vapor systems with surface equilibrium 9 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 D i ) 10 CHEMICAL NONEQUILIBRIUM (KINETIC) BOUNDARY CONDITIONS

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 (T r  gas-dynamic recovery temperature ) 11

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

 Adiabatic condition: = 0 (including both contributions) =>  In forced-convection systems, (St m /St h )  chemical- energy recovery factor, r ChE 13 COMBINED ENERGY & MASS TRANSPORT: RECOVERY OF MAINSTREAM CHEMICAL & KINETIC ENERGY

 For a laminar BL, r KE ≈ Pr 1/2, r ChE ≈ Le 2/3, and  T w can be higher or lower than corresponding thermodynamic (“total”) temperature: (depending on Pr, Le) 14 COMBINED ENERGY & MASS TRANSPORT: RECOVERY OF MAINSTREAM CHEMICAL & KINETIC ENERGY

 In most gas mixtures, both r KE and r ChE ≈ 1  Probe records temperature near T 0, not T ∞  r ChE important in measuring temperatures of gas streams that are out of chemical equilibrium  T w >> T ∞ or T r can be recorded 15 COMBINED ENERGY & MASS TRANSPORT: RECOVERY OF MAINSTREAM CHEMICAL & KINETIC ENERGY

 For non-adiabatic surfaces: T r ’  generalized recovery temperature (T w - T r ’)  “overheat” 16 COMBINED ENERGY & MASS TRANSPORT: RECOVERY OF MAINSTREAM CHEMICAL & KINETIC ENERGY

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, 2 nd phase can inertially impact on target, even while host fluid is brought to rest 17

18 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 d p << L)  Negligible mass loading & volume fraction  Large enough to neglect D p, small enough to neglect gravitational sedimentation  Captured on impact (no rebound) 19 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) 20 TWO-PHASE FLOW: MASS TRANSFER EFFECTS OF INERTIAL SLIP & ISOKINETIC SAMPLING

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

 Pure inertial impaction at supercritical Stokes’ numbers: Cylinder in cross flow 22 Particle capture fraction correlation for ideal ( ) flow past a transverse circular cylinder (Israel and Rosner (1983)). Here t flow =(d/2)/U. 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 < Stk crit  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 23 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) 24 TWO-PHASE FLOW: MASS TRANSFER EFFECTS OF INERTIAL SLIP & ISOKINETIC SAMPLING

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

 Effective diffusivity of particles in turbulent flow  Ability to follow local turbulence (despite their inertia) governed by Stokes’ number, Stk t Relevant local flow time = ratio of scale of turbulence, l t, to rms turbulent velocity 26 Two-Phase Flow: Mass Transfer Effects of Inertial Slip & Isokinetic Sampling

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 k t  turbulent kinetic energy per unit mass, and   turbulent viscous dissipation rate per unit mass 27

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

 Eddy impaction:  When Stk t is sufficiently large, some eddies project particles through viscous sublayer, significantly increasing the deposition rate  Represented by modified Stokes’ number:  Eddy-impaction augmentation of St m negligible for Stk t,eff -values <  Below this value, turbulent particle-containing BL behaves like single-phase fluid 29