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 25

1

Mass Transport: Composite Planar Slab

Conservation Equations:Concentration fields are coupled by the facts that: Homogeneous reaction rates involve many local

species All local mass fractions must sum to unity (only N-1

equations are truly independent)Species i mass conservation condition may be written

as:

local mass rate of production of species i2

'' '''( ) ( 1,2,..., ),mii idiv r i N

t

'''ir

CONCENTRATION FIELDS, SURFACE MASS- TRANSFER RATES & COEFFICIENTS

Conservation Equations:Since i = i, by virtue of total mass conservation:

and the species balance becomes:

LHS proportional to (Lagrangian) rate of change of ifollowing a fluid parcel

RHS is “forced” diffusion flux3

''( ) 0midivt

'' ''',.v grad j ci

i i diff i i idiv rt


i ic

Conservation Equations:

PDEs for coupled to each other, and to PDEs governing linear momentum density v(x,t) & temperature field, T(x,t)

All must be solved simultaneously, to ensure self-consistency

Simplest PDE governing is Laplace equation:

4

,xi t

,xA t

0grad Adiv



Laplace eq. holds when there are no: Transients Flow effects Variations in fluid properties Homogeneous chemical reactions involving species A Forced diffusion (phoresis) effects

In Cartesian coordinates:

5

2 2 2

2 2 2 0A A A

x y z



Boundary conditions on are of two types:

i along each boundary surface, or

Some interrelation between flux (e.g., via heterogeneous chemical kinetics)

Solution methods: Numerical or analytical Exact or approximate Solutions could be carried over from corresponding

momentum or energy transfer problems 6

,xi t

'', ,&i w i wm


ANALOGIES & ANALOGY-BREAKERS

Heat-Transfer Analogy Condition (HAC) applies when:

Fluid composition is spatially uniform

Boundary conditions are simple

Fluid properties are nearly constant

All volumetric heat sources, including viscous

dissipation and chemical reaction, are negligible

known as functions of Re, Pr, Rah, etc.

7

,h hSt Nu

Corresponding temperature field:

Under HAC, the rescaled chemical species concentration

And corresponding coefficients, , will be

identical functions of resp. arguments.8

* * *, ;Re,Pr,...xw

w

T T T tT T

, * * *

, ,

, ;Re, ,...xA w

A w A

t Sc


m mSt and Nu

A


MACs:

Species A concentration is dilute ( ) specified constant along surface

Negligible forced diffusion (phoresis)

No homogeneous chemical reaction

Analogy holds even when:

Fluid flow is caused by transfer process itself (e.g., natural convection in a body force field)

Analogy to linear momentum transfer breaks down due to streamwise pressure gradients

9

1A

,A w

MAC:Schmidt number plays role that Pr does for

heat transferMass-transfer analog of Rah is:

where defines dependence of local fluid density on A:

10

/ ASc D

, ,

1 1

A Ap T p T


3, ,2 . .A w A

m mA

g L vRa Gr Scv D

Correction Factors for Analogy-Breaking Phenomena:

Two analogy breaking mechanisms:

Phoresis

Homogeneous chemical reaction

Have no counterpart in energy equation T(x,t)

BL situation: Substance A being transported from

mainstream to wall A,w << A,∞

11



Phoresis toward the wall:

Distorts concentration profile

Affects wall diffusional flux,

where Num,0 mass transfer coefficient without phoretic

enhancement; analogous to Nuh

F(suction) augmentation factor

12

,0( ).m mNu F suction Nu


'',A wj


F(suction) is a simple function of a Peclet number based

on drift speed, -c, boundary-layer thickness, m,o, and

diffusion coefficient, DA:

or

In most cases

13

,0

,0

msuction

mA A

c c LPe

D DNu

,0

1.suctionm

cPe

U St

( )1 exp( )

suction

suction

PeF suctionPe



Homogeneous reaction within BL:

A profile distorted, diffusional flux affected

or

14

,0.m mNu F reaction Nu

,0.m mSt F reaction St



F(reaction) depends on Damkohler (Hatta) number:

where k”’ relevant first-order rate constant; time-1

can be rewritten using:

15

2 ''' 2,0 ,0

'''

/

1/mdiff m

chem A

Dt kDam

t Dk

,0,0 ,0

mm m

L DNu USt


Correction Factors for Analogy-Breaking Phenomena:For an irreversible reaction with A,w << A,∞

(F(reaction) 1 when Dam 0)

Reaction augmentation factors Hatta factors

F(Reaction) = m,o /m

If only heterogeneous reactions occur, analogy is intact: Num = Num,0 16

1/2 1/2

1/2 1/2

/ sin( ) ( )

/ sinh( ) (sink)

Dam Dam sourceF reaction

Dam Dam


QUIESCENT MEDIA

Above conditions not sufficient in nondilute systemsMass transfer itself gives rise to convection normal to

surface, Stefan flow

vw fluid velocity @ interfaceAdditional condition for neglect of convective transport

in mass transfer systems:

Inevitably met in dilute systems17

, ,

,1A w Aw m

mA w

vBD

, ,

,

11A w Aw m

A w

vD

Stefan flow becomes very important when A,w ≠ A,∞,

and A,w 1

e.g., at surface temperatures near boiling point of

liquid fuel

18

QUIESCENT MEDIA

COMPOSITE PLANAR SLAB

T continuous going from layer to layer, but not A

Only chemical potential is continuous

Two unknown SS concentrations at each interface

Linear diffusion laws to be reformulated using a

continuous concentration variable

Applies to nonplanar composite geometries as well

19

20


x

Mass transfer of substance A across a composite barrier: effect of piecewisediscontinuous concentration (e.g., mass fraction A(x))

Continuous composition variable = -phase mass fraction of solute A

Corresponding mass flux through a composite solid (or liquid membrane)

,l concentration-independent dimensionless equilibrium solute A partition coefficients, (A

()/A(l))LTCE, between phase and

phase l (= …)m,eff stagnant film (external) thickness (resistance)

21

'',

,

..

A overall

A x A A overall

llA l

j KL

D


Dilute solute SS diffusional transfer between two

contacting but immiscible fluid phases in separation/ extraction devices

Modeled as through two equivalent stagnant films of

thicknesses m,eff () and m,eff ()

In series, negligible interfacial resistance between them

“Two-film” theory (Lewis and Whitman, 1924)

22


KA() overall interface mass transfer coefficient

(conductance)Satisfies “additive-resistance” equation

(symmetrical replacement of and yields KA())

23

( ) ( )

. ,Overall

α-phase β-phasemass transferresistance resistanceresistance

1 1

/ /A A m eff A m effK D D


Gas absorption/ stripping:One phase (say ) vapor phaseK relevant partition coefficient; inversely

proportional to Henry constant, H:

where M solvent molecular weight pA partial pressure of species A in vapor

phase

24

.( ) . /A A solvent A Asolventp H x H M M


H dimensional inverse partition (distribution) coefficient

(if -phase (vapor mixture) obeys perfect gas law)

25

( )

1.RTM H


Addition of reagents to solvent phase :

Reduces m,eff ()

Simultaneous homogeneous chemical reaction increases

liquid-phase mtc’s, accelerates rate of uptake of sparingly-

soluble (large H) gases

Additive (B) in sufficient excess => pseudo-first-order reaction

( linearly proportional to A, with rate constant k”’)

26


'''Ar

where

and

27

( ), ;0( )

, ( )m eff

m eff F reaction

1/21/2tanh( )

DamF reaction

Dam

2 ( ), ;0

'''

/

1/m eff AD

Damk


When reaction is so rapid that the two reagents meet in stoichiometric ratio at a thin reaction zone (sheet):Distance between reaction zone & phase boundary

plays role of

B,b concentration of additive B in bulk of solvent

i concentration of transferred solute A at solvent interfaceb gms of B are consumed per gram of A

28

( )

,

,

11 . . B bB

A A i

DF reactionb D


( ),m eff

dr. r. nagarajan professor dept of chemical engineering iit madras

Documents

concentration fields

mass transport

transfer process

transfer analogy condition

linear momentum transfer

mass conservation condition

energy transfer problems

analogybreaking phenomena