Dr. R. Nagarajan

Professor

Dept of Chemical Engineering

IIT Madras

Advanced Transport PhenomenaModule 6 Lecture 24

1

Mass Transport: Ideal Reactors &

Transport Mechanisms

MASS TRANSPORT

2

Transport-controlled situations:

In applications where reagents are initially separated,

mass-transfer determines reactor behavior

e.g., in combustors, fuel & oxidant must “find each

other”

Observed reaction rate is “transport controlled”

RELEVANCE

3

Kinetically-limited situations:

Kinetics play important role, but local reactant

concentrations also do

e.g., premixed fuel/ oxidizer systems

“law of mass action” => mass transport still plays

significant role

Especially true for “nonideal” reactors

Gradients in concentration & temperature

RELEVANCE

4

Overall reactor volume required to convert reactants to

products depends on:

Apparent chemical kinetics, and

Reagent-product contacting pattern in reactor

Two limiting ideal cases:

Plug-flow reactor (PFR)

Well-stirred reactor (WSR/ CSTR)

IDEAL STEADY-FLOW CHEMICAL REACTORS

5

''' Ar

6Basic chemical reactor types

IDEAL STEADY-FLOW CHEMICAL REACTORS

PFR

Reacting fluid mixture moves through vessel (e.g.,

long tube) in one predominant () direction

Negligible recirculation, backmixing, streamwise

diffusion

Governing equations: Quasi 1D

Species A mass balance: (ODE)

'' ''',. .AA w A

Pdmj r

A d A

7

constantzm v A

A cross-section area-averaged reactant mass

fraction

j”A,w wall diffusion flux of species A

P() local wetted perimeter

A() local flow area

Lumping heterogeneous term into an effective (pseudo-)

homogeneous term:

PFR

''' ''' '', , .A eff A A w

Pr r j

A

8

Increment in reactor volume

Then:

If can be uniquely related to local reagent mass

fraction A , then vessel volume, VPFR, required to reduce

reactant composition from feed (A1) to reactor exit (A2):

PFR

''',

AA eff

dm r

dV

1

'''2,

.( )

A

A

APFR

A eff A

dV m

r

9

Ad dV

''', A effr

PFR

10

Reciprocal reaction rate vs reagent composition plot to determine ideal reactorvolume required (per unit mass flow rate of feed)

WSR

Intense backmixing even in steady flow

All intra-vessel composition nonuniformities rendered

negligible

Reaction takes place at single composition, nearly

equal to exit value

Mass balance equations simplify to:

, and

'''1 2 , 2 .A A A eff A WSRm r V

11

1 2 constm m m

Hence:

From previous Figure, when rate increases monotonically

with reagent concentration, VWSR > VPFR

WSR

1 2

''', 2

. A AWSR

A eff A

V mr

12

Physical appearance can be deceiving:

At low gas pressures, a short straight tube with axial

flow behaves like a WSR (due to molecular backmixing)

At high pressures, turbulent stirring action produced by

reactant jet injection can make a tubular reactor behave

like a WSR (e.g., aircraft gas turbine combustor)

Radial-flow, thick annualar bed can perform like a PFR

WSR

13

Real reactor can deviate considerably from both PFR &

WSR

Momentum, energy & mss transport laws must be

applied together for design & scale-up

Reaction rate laws can involve transport factors

(especially for multiphase reactors)

Can often be represented as a network of

interconnected ideal reactors

WSR

14

MASS-TRANSPORT MECHANISMS & ASSOCIATED TRANSPORT PROPERTIES

Three mechanisms of mass transport:

Convection

Diffusion

Free-molecular flight

Analogous to energy transport

First two collaborate in continuum (Kn << 1) regime

15

Two types:

Due to motion of host fluid

Due to solute drift through host fluid (as a result of net

forces applied directly to solute)

CONVECTION

16

Host-fluid convection:

For fluid mixture in Eulerian CV, local convective mass

flux

Local chemical species convective mass flux

CONVECTION

17

'' m v

'' ''i i iconv

m m v

Solute convection:

“phoresis”

In response to local applied force– gravitational,

electrostatic, thermal, etc.

Quasi-steady drift velocity, ci

Relative to local mixture velocity v

Contributes mass flux vector

CONVECTION

18

''i i idriftm c

CONCENTRATION DIFFUSION

Random-walk contributes net drift of species:

Fick diffusion fluxFor suspended particles: Brownian flux In local turbulent flow, time-averaged mass flux: eddy

diffusion flux Actually, result of time-averaging species convective

flux

Total mass flux of species

19

'' '',i i i eff idiffusion

D m j grad

'' '',i i i i i eff iD m m c grad

FREE-MOLECULAR FLIGHT

Travel in the absence of collisions with other molecules

Net flux: algebraic sum of fluxes in different directions

and at different speeds

Mechanism analogous to energy transport by photons

20

Di,eff effective mass diffusivity of species I in prevailing

medium

Not always a scalar, but frequently treated as such

Can be a tensor defined by 9 (6 independent) local

numbers

When diffusion is easier in some directions, e.g.:

Anisotropic solids (single crystals)

Anisotropic fluids (turbulent shear flow)

Diffusion not always “down the concentration

gradient”, but skewed

SOLUTE DIFFUSIVITIES

21

gi force acting on unit mass of species i

mi particle mass

ci quasi-steady drift speed, given by:

where

fi friction coefficient (inverse of mobility)

Relates drag force to local slip (drift) velocity

Example: Stokes coeff.= 3i,eff for solutes much

larger than local solvent mean free path

DRIFT VELOCITIES DUE TO SOLUTE - APPLIED FORCES

22

ii

i

m

figc =

Sedimentation:

gi,eff gravitational body force

c = ci,s = settling or sedimentation velocity

DRIFT VELOCITIES DUE TO SOLUTE - APPLIED FORCES

23

DRIFT VELOCITIES DUE TO SOLUTE-APPLIED FORCES

Electrophoresis:

gi,eff due to presence of electrostatic field, and

either a net charge on species i, or an induced

dipole on a neutral species

c = ci,e = electrophoretic velocity

Determines motion of ions & charged particles in

fields (e.g., fly-ash removal in electrostatic

precipitators)24

Thermophoresis:

gi,eff due to temperature gradient, proportional to –

grad(ln T)

c = ci,T = thermophoretic velocity

ici,T = thermophoretic flux

Important in some gaseous systems (e.g., high MW

disparity, steep temperature gradients), and in

Most aerosol systems (e.g., soot and ash transport in

combustion gases, deposition on cooled surfaces)

DRIFT VELOCITIES DUE TO SOLUTE - APPLIED FORCES

25

Single-phase flow:

Only if particles or heavy molecules follow host fluid closely

Criterion: stopping time, tp

Compared to characteristic flow time, tflow (L/U)

tp time required for velocity to drop by factor e-1 in prevailing

viscous fluid

where

p particle mass density

dp diameter

26

2

18p p p

p

m dt

f Stokes

PARTICLE “SLIP”, INERTIAL SEPARATION

PARTICLE “SLIP”, INERTIAL SEPARATION

Stk Stokes’ number, given by

Inverse Damkohler number governing dynamical

nonequilibrium

<< 1 => particles follow host fluid closely

> 10-1 => separate momentum equations required for

each coexisting phase

27

p

flow

tStk

t

CONCENTRATION FIELDS, SURFACE MASS- TRANSFER RATES & COEFFICIENTS

mass fraction field for chemical species i (or

particle class i)

Measurable: local fluxes, , at important boundary

surfaces

e.g., local rate of naphthalene sublimation into a gas

stream

or, average flux for entire surface,

28

,xi t

'',i wm

'' /i i wm m A

ji,w” diffusional contribution to mass flux

Reference values:

Dimensionless Nusselt number for mass transport

Widely used for quiescent systems,and for forced & natural convection systems

29

, ,'',

, ,

/ ,i i w i

i ref

i w i

D L orm

U

,,

, ,

/

/i w w

m i

i i w i

j ANu

D L

CONCENTRATION FIELDS, SURFACE MASS- TRANSFER RATES & COEFFICIENTS

Dimensionless Stanton number for mass transport

Used only for forced convection systems

area-weighted average of normal component of

diffusion flux, i.e.,

Rarely measured in this manner

30

,

, ,

/i w wmi

i w i

j ASt

U

'', , .i w i eff i wj D grad n

CONCENTRATION FIELDS, SURFACE MASS- TRANSFER RATES & COEFFICIENTS

'',i wj

Dimensional coefficients:

Mass flux per unit driving force

May be based on , or on , or on (for gases)

System of units becomes important in the definition

31

CONCENTRATION FIELDS, SURFACE MASS- TRANSFER RATES & COEFFICIENTS

i in ip

Capture Fraction, :

When species i is contained in mainstream feed

Ratio of actual collection rate ( ) to rate of flow of species

through projected area of target, i.e.:

When i,w << i,∞ , , and:

Generally < (1/2) cap since Aw,proj ≤ (1/2) Aw

32

,

, ,

i wcap

i w proj

m

U A

, .w projm cap

w

ASt

A

CONCENTRATION FIELDS, SURFACE MASS- TRANSFER RATES & COEFFICIENTS

cap

, i wm

, ,i w i wm j