Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras
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Transcript of Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras
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Dr. R. Nagarajan
Professor
Dept of Chemical Engineering
IIT Madras
Advanced Transport PhenomenaModule 6 Lecture 24
1
Mass Transport: Ideal Reactors &
Transport Mechanisms
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MASS TRANSPORT
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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
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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
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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
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''' Ar
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6Basic chemical reactor types
IDEAL STEADY-FLOW CHEMICAL REACTORS
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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
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constantzm v A
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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
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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
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Ad dV
''', A effr
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PFR
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Reciprocal reaction rate vs reagent composition plot to determine ideal reactorvolume required (per unit mass flow rate of feed)
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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
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1 2 constm m m
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Hence:
From previous Figure, when rate increases monotonically
with reagent concentration, VWSR > VPFR
WSR
1 2
''', 2
. A AWSR
A eff A
V mr
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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
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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
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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
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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
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Host-fluid convection:
For fluid mixture in Eulerian CV, local convective mass
flux
Local chemical species convective mass flux
CONVECTION
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'' m v
'' ''i i iconv
m m v
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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
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''i i idriftm c
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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
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'' '',i i i eff idiffusion
D m j grad
'' '',i i i i i eff iD m m c grad
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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
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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
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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
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ii
i
m
figc =
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Sedimentation:
gi,eff gravitational body force
c = ci,s = settling or sedimentation velocity
DRIFT VELOCITIES DUE TO SOLUTE - APPLIED FORCES
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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
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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
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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
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2
18p p p
p
m dt
f Stokes
PARTICLE “SLIP”, INERTIAL SEPARATION
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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
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p
flow
tStk
t
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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,
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,xi t
'',i wm
'' /i i wm m A
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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
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, ,'',
, ,
/ ,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
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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
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,
, ,
/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
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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
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CONCENTRATION FIELDS, SURFACE MASS- TRANSFER RATES & COEFFICIENTS
i in ip
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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
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,
, ,
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