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

Dr. R. Nagarajan

Professor

Dept of Chemical Engineering

IIT Madras

Advanced Transport PhenomenaModule 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 & cf/2)

For laminar or turbulent flows with negligible pressure

gradient, Reynolds’- Chilton – Colburn analogy holds:

2

2/3.( )2f

m

cSt Sc

CONVECTIVE MASS TRANSFER IN LAMINAR- AND TURBULENT-FLOW SYSTEMS


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 Stm for Sc > 102

3

1 1/22

12

1 5( ) Pr 1

f

hf

CSt

C



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 (SL ≈ (cf/2)1/2

(5/U)) increases both cf/2 and St

Effect on St < on friction coeff (hence, pressure drop) 4

1/20.7040.08. .( )

2f

m

cSt Sc


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

5

'', , , ,Re, .A w m A A A A wj USt Sc

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

''.A w A w

kinetic rate of consumptionr k

of A at surface

'' '',A w Aj r


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 << 1 => fraction is small, rate approaches

“chemically controlled” value, kwA,∞7

,

,

Local reactant "starvation"11

A w

A C

'', , ,.

1A w m A ACj UStC


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, kwA,w

8

,

w w m

m A A

k kC

USt D


If LTCE is achieved at station w due to rapid

heterogeneous chemical reactions, then:

i,w = i,eq(Tw,….;p) for all species i

Used to estimate chemical vapor deposition (CVD)

rates in multicomponent vapor systems with surface

equilibrium

9


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)

10

''( ) ( ) 1,2,...,k k elemk mixD k N j grad


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)11

'' . Re,Pr . .w h p w rconvq U St c T T

balances rate of energy transport associated with species A

mass transport:

(Q energy release per unit mass of A)

12

'',. Re, . w m A Adiff

q U St Sc Q


KINETIC ENERGY

Adiabatic condition: = 0 (including both contributions)

=>

In forced-convection systems, (Stm/Sth) chemical-

energy recovery factor, rChE

13

2

Pr,Re .2r KE

p

UT T rc

2,Re,

Pr,Re . .2 Re,Pr

Am Aw KE

p h p

QSt ScUT T rc St c


KINETIC ENERGY

''wq

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)14

2

1/2 2/3 ,Pr .2

Aw

p p

QUT T Lec c

2,

0 2A

p p

QUT Tc c


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 Tr can be recorded

15


KINETIC ENERGY

For non-adiabatic surfaces:

Tr’ generalized recovery temperature

(Tw - Tr’) “overheat”

16

'' '. Re,Pr .w h p w rq U St c T T


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, 2nd phase can inertially

impact on target, even while host fluid is brought to rest17

18


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) 19


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

,Re, ,capture Stk shape orientation


Pure inertial impaction at supercritical Stokes’ numbers:

Cylinder in crossflow

capture = 0 for Stk < Stkcrit

Capture occurs only above a critical Stokes’ number

(for idealized model of particle capture from a two-

phase flow)

21



22

Particle capture fraction correlation for ideal ( ) flow past a transversecircular cylinder (Israel and Rosner (1983)). Here tflow=(d/2)/U.

Re


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

23


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



25


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, Stkt

Relevant local flow time = ratio of scale of turbulence, lt, to rms turbulent velocity

26

1/2/ '. ' p

t

t

tStk

l

Two-Phase Flow: Mass Transfer Effects of Inertial Slip & Isokinetic Sampling


Effective diffusivity of particles in turbulent flowAlternative form of characteristic turbulent eddy time,

where kt turbulent kinetic energy per unit mass, and turbulent viscous dissipation rate per unit

mass

27

/eddy tt k :eddyt

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

1, / .p eff t t tD v Sc fct Stk


/ /t p tStk t k

tStk 110tStk


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 29

, / ( / )p

t eff pw

tStk t

v

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