Mass Transport: Non-Ideal Flow Reactors -...
Transcript of Mass Transport: Non-Ideal Flow Reactors -...
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Dr. R. Nagarajan
Professor
Dept of Chemical Engineering
IIT Madras
Advanced Transport Phenomena
Module 6 - Lecture 28
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Mass Transport: Non-Ideal Flow Reactors
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MODELING OF NONIDEAL-FLOW REACTORS
�Simplest approach: apply overall material/ energy/
momentum balances to the reactor
� “black box’ approach, insufficient
�Most rigorous: Divide into small subregions, approximate
each region with PDEs
� Impractical
� Intermediate solution: model as discrete network of small
number of interconnected ideal reactor types (SS PFR &
WSR)
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�RTDF � residence time distribution function (exit-age
DF), E(t)
�E(t) dt � fraction of material at vessel outlet stream that
has been in vessel for times between t and t ± dt
�PFR: E(t) is a Dirac function, centered at residence time
3
MODELING OF NONIDEAL-FLOW REACTORS
( )/ /V m ρ&
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�V � vessel volume
� � feed mass flow rate
�e.g., straight tube through which incompressible fluid
flows with a uniform plug-flow velocity profile
�Partial recycle can alter RTDF
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MODELING OF NONIDEAL-FLOW REACTORS
m&
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MODELING OF NONIDEAL-FLOW REACTORS
Tracer residence-time distribution functions for ideal and real vessels (for e.g., reactors) (adapted from Levenspiel (1972))
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MODELING OF NONIDEAL-FLOW REACTORS
Ideal plug-flow reactor (PFR) with partial “recycle” (recycle introduces adistribution of residence times, and reduces the residence time per pass within the
PFR)
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�WSR:
�Most likely residence time in a WSR is zero!
�Mean residence time =
�Not all fluid parcels have same residence time, unlike
PFR
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( ) ( ) 1
/ exp /−
= = − flow flowE WSR dF dt t t t
MODELING OF NONIDEAL-FLOW REACTORS
( )/ /V m ρ&
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�WSR:
�Dimensionless variance s2 about mean residence time
� indicator of spread of residence times
�Mean residence time related to first moment of E(t), i.e.:
�s2 is related to 2nd moment of E(t):
� = 1 for a WSR, 0 for a PFR
� PFR with infinite recycle behaves like WSR 8
MODELING OF NONIDEAL-FLOW REACTORS
0
. ( )flowt t E t dt∞
= ∫
( ) ( )2
2 2 2
2 2
0 0
1 1. ( )flow flow
flow flow
t t E t dt t E t dt tt t
σ∞ ∞
≡ − = −
∫ ∫
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�RTDF for Composite Systems:
� If RTDF for vessel 1 is E1(t) and for vessel 2 is E2(t),
RTDF for a series combination of the two is:
(convolution formula)
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( ) ( )' ' '
1 2 1 2
0
( ) .t
E t E t E t t dt+ = −∫
MODELING OF NONIDEAL-FLOW REACTORS
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� If vessel 1 is characterized by tflow,1, and s12, and vessel
2 by tflow,2, and s22, then for the series combination,
mean residence times and variances are simply
additive:
10
,1 2 ,1 ,2
2 2 2
1 2 1 2
flow flow flowt t t
σ σ σ+
+
= +
= +
MODELING OF NONIDEAL-FLOW REACTORS
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�RTDF for Composite Systems:
�For a network of n-WSRs of equal volume, for which:
�(tflow � ) for each vessel in series)
11
( )
11
( ) . .exp1 !
−− − = − −
flow
n
flow flow
t t tE n WSRs
n t t
MODELING OF NONIDEAL-FLOW REACTORS
/ ( / )V m ρ&
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�For vessels 1, 2, 3,C., n in parallel, receiving fractions f1,
f2, f3, C., fn of total flow:
�Where , and for each vessel:
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1 1 2 2( ) ( ) ... ( )n nE f E t f E t f E t= + + +
( )0
1 ( 1, 2,..., )
∞
= =∫ iE t dt i n
MODELING OF NONIDEAL-FLOW REACTORS
1iif =∑
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� Real reactors as a network of ideal reactors: Modular
modeling
� Network of ideal reactors can be constructed to
approximate any experimental reactor RTDF:
(where tracer is input as a Dirac impulse function)
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( )exp
0
( )( ) tracer
tracer
reactor exit
tE t
t dt
ω
ω∞
= ∫
MODELING OF NONIDEAL-FLOW REACTORS
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Real reactors as a network of ideal reactors: Modular
modeling
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GT combustor; proposed interconnection of reactors comprising “modular” model (adapted from Swithenbank, et al.(1973))
MODELING OF NONIDEAL-FLOW REACTORS
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�Real reactors as a network of ideal reactors: Modular
modeling
� Info obtained from tracer diagnostics & from
combustor geometry, cold-flow data, etc.
� Important since RTD-data alone cannot discriminate
between alternative networks with identical RTD-
moments
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( ) ( )2
0 0
, , ..., .)
∞ ∞
= ∫ ∫flowt tE t dt t E t dt etc
MODELING OF NONIDEAL-FLOW REACTORS
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� Equivalent vessel network is nonunique
� Each alternative may capture one aspect (e.g.,
combustor efficiency) but not another (e.g., domain
of stable operation)
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MODELING OF NONIDEAL-FLOW REACTORS
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MODELING OF NONIDEAL-FLOW REACTORS
�Real reactors as a network of ideal reactors: Modular
modeling
�Tracer methods can:
� Guide development of “modular” models
� Diagnose operating problems with existing chemical
reactors or physical contactors
� RTD data can show up dead-volumes, flow-
channeling, bypassing (all cause inefficient
operation)
� Geometric or fluid-dynamic changes in design can
correct these flaws
� Perturbation in feed can be used as “tracer”
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�Real reactors as a network of ideal reactors: Modular
modeling
�RTD function, E(t), does not capture role of
concentration fluctuations due to turbulence,
incomplete mixing (at molecular level– “micromixing”)
�When tracer concentration fluctuates at reactor exit,
we only collect data on <E(t)> � arithmetic average of
N tracer shots, each yielding RTD Ej(t) (j = 1, 2, C., N)
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MODELING OF NONIDEAL-FLOW REACTORS
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�Two networks with identical <E(t)> but with different
shot-to-shot variations, as measured by variance:
will perform differently as chemical reactors
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( )2
1 0
1( )lim
∞
→∞ =
− ∑∫N
jN j
E t E t dtN
MODELING OF NONIDEAL-FLOW REACTORS
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�Statistical micro flow (Random Eddy Surface-Renewal)
models of interfacial mass transport in turbulent flow
systems
�Mass/ energy transport visualized to occur during
intervals of contact between turbulent eddies & surface
� “stale” eddies replaced by fresh ones
�Effective transport coefficient calculated by time-
averaging RTDF-weighted instantaneous St(t)
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MODELING OF NONIDEAL-FLOW REACTORS
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�Statistical micro flow (Random Eddy Surface-Renewal)
models of interfacial mass transport in turbulent flow
systems
� If E(t) is defined such that:
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Relative portion of each unit interfacial area
( ) covered by fluid eddies having "ages" between
t and t+dt,
E t dt
≡
MODELING OF NONIDEAL-FLOW REACTORS
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then:
St(t) � calculated from transient micro fluid-dynamical
analysis of individual eddy flow
St � time-averaged transfer coefficient
� Interfacial region being viewed as a thin vessel w.r.t
eddy residence time
22
0
( ). ( )St St t E t dt∞
= ∫
MODELING OF NONIDEAL-FLOW REACTORS
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�Statistical microflow (Random Eddy Surface-Renewal)
models of interfacial mass transport in turbulent flow
systems
�Earliest & simplest model: each eddy considered to
behave like a translating solid body
� Large compared to transient diffusion BL
(penetration) thickness
�Dimensional time-averaged mass-transfer coefficient
given by:
23
MODELING OF NONIDEAL-FLOW REACTORS
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tm � mean eddy contact time (1/(average renewalfrequency))
� Related to prevailing geometry & bulk-flow velocity
� Versatile alternative to Prandtl-Taylor eddy diffusivityapproach
24
( )
[ ]
[ ]
1/2
''
,
1/2, ,
4( ) ( 1935
( ) ( 1951 )
A
mA w
A b A wA
m
Dfor E PFR Higbie
tj
Dfor E WSR Danckwerts
t
π
ρ ω ω
− =
−
MODELING OF NONIDEAL-FLOW REACTORS
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�Extinction, ignition, parametric sensitivity of chemical
reactors:
�Simplest modular model for steady-flow behavior of
combustors: WSR + PFR
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MODELING OF NONIDEAL-FLOW REACTORS
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� �upper limit to total mass flow rate, at each
upstream condition (Tu, pu, mixture ratio Φ) above
which extinction of exoergic reaction (flame-out)
abruptly occurs
�For , two possible SS conditions exist: one
corresponding to high fuel consumption & high
temperature in WSR, the other to negligible fuel
consumption & rise in T
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MODELING OF NONIDEAL-FLOW REACTORS
max<m m& &
maxm& ,m&
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�Extinction, ignition, parametric sensitivity of chemical
reactors:
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MODELING OF NONIDEAL-FLOW REACTORS
Simple, two-ideal reactor “modular” model of gas turbine, ramjet, or rocketengine combustor
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�Extinction, ignition, parametric sensitivity of chemical
reactors:
�Parametric sensitivity: change in reactor performance
for a small change in input or operating parameter
(e.g., Tu)
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MODELING OF NONIDEAL-FLOW REACTORS
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�Example: WSR module with following overall
stoichiometric combustion reaction:
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( )1 O + gm 1 gram P+ cal(heat)gm f F f fQ→ +
MODELING OF NONIDEAL-FLOW REACTORS
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�Extinction, ignition, parametric sensitivity of chemical
reactors:
�Allow a 2nd reactant (oxidant) & associated heat
generation
� Governs WSR operating temperature, T2
�WSR species mass balance:
(i = O, F, P)
30
( ) ( )'''
2 1 2 2 2. , , .i i i O F WSRm r T Vω ω ω ω− =& &
MODELING OF NONIDEAL-FLOW REACTORS
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�Extinction, ignition, parametric sensitivity of chemical
reactors:
�Overall energy balance:
� Source terms for oxidizer & fuel related by:
�So, ωO 2 and ωF 2 can be expressed in terms of T2
31
( ) ( )'''
2 1 2 2. , , .p F O F WSR
mc T T r T QVω ω− = −& &
MODELING OF NONIDEAL-FLOW REACTORS
''' '''/O Fr r f− =−& &
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�Extinction, ignition, parametric sensitivity of chemical
reactors:
�Overall kinetics represented by Arrhenius-type mass-
action rate law:
� LHS � straight line intersecting RHS at 3 distinct T2
values, middle one unstable, upper � ignited WSRSS, lower � extinguished WSR SS (no chemical
reaction)
32
'''
1
1.exp . . . O F
O F
n
v v
F O Fv v
O F
E pMr A
RT M M RTω ω−
− = − &
MODELING OF NONIDEAL-FLOW REACTORS
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�Extinction, ignition, parametric sensitivity of chemical
reactors:
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MODELING OF NONIDEAL-FLOW REACTORS
Influence of feed mass flow rate on WSR operating temperature and space (volumetric) heating rate(SHR);(straight line is the LHS of the energy balance equation)
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MODELING OF NONIDEAL-FLOW REACTORS
�Extinction, ignition, parametric sensitivity of chemical
reactors:
�Maximum volumetric rate of fuel consumption (hence,
maximum chemical heating rate) occurs at WSRtemperature:
� Only slightly > “extinction” temperature (previous
Figure)
�Tb �adiabatic, complete-combustion temperature
�Typical E, n values listed in following Table
34
''' max 1 ( / )
b
rb
TT
n RT E−≈
+&
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MODELING OF NONIDEAL-FLOW REACTORS
�Extinction, ignition, parametric sensitivity of chemical
reactors:
35
aSupplemented, rounded (and selected) values based on Table 4.4 of Kanury (1975)bUnits are: 1014s-1 (g-moles/cm3)-(n-1), where n is the overall reaction order.cunits are: 109 BTU/ft3/hrdStoichiometric mixture, no diluent (“diluent” is N2) unless otherwise specified
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MODELING OF NONIDEAL-FLOW REACTORS
�Extinction, ignition, parametric sensitivity of chemical
reactors:
�Black-box modular-models capture many important
features of real reactors, useful for correlating
performance data on full-scale & small-scale models
�Predictive ability limited compared to more-detailed
pseudo-continuum mathematical models
�All have, as their basis, macroscopic conservation
principles outlined earlier in this course.
36
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PROBLEM1
�The length requirement for a honeycomb-type automotive
exhaust catalytic converter is set by the need to reduce
the CO concentration in the exhaust to about 5% of the
inlet concentration (i.e., 95% conversion). Consider the
basic conditions:
Inlet gas temperature 700K
Inlet gas pressure 1 atm
Inlet gas composition y(N2)=0.93, y(CO)=0.02,
(mole fraction) y(O2)=0.05
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Inlet gas velocity 103 cm/s
Channel cross-section dimensions 1.5mm by 1.5mm (each
channel)
Assumed channel wall temperature 500 K
Assume that the Pt-based catalyst used on the walls of
each channel is active enough to cause the surface-
catalyzed CO oxidation reaction to be diffusion-controlled,
that is, the steady-state value of the CO-mass fraction
established at (1 mean-free-path away from)
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the wall, ωCO,w , is negligible compared to ωCO,b(z) within
each channel. Also assume that the gas-phase kinetics of
CO oxidation under these conditions preclude
appreciable (uncatalyzed) homogeneous CO-
assumption in the available residence times. Answer the
following questions:
a. By what mechanism is CO(g) mass transported to the
channel wall, where chemical consumption (to produce
CO2) occurs? What is the relevant transport coefficient39
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and to what energy-transfer process and transport
properly coefficient is this “analogous”?
b. Are the mass-heat transfer analogy conditions (MAC,
HAC) discussed in this module approximately met in this
application? What is the inlet mass fraction of CO gas?
c. Estimate the Schmidt number mix for CO
Fick diffusion through the prevailing combustion gas
mixture, using the experimental observation that
40
/ CO mixSc v D −≡
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where p is the prevailing pressure (expressed in
atmospheres) and T the mixture temperature (expressed
in kelvins)
d. Under the flow rate, temperature, and pressure
conditions given above and using the mass-transfer
analog, estimate the catalytic duct length
41
2
1.73 20.216.300
CO N
T cmD
p s−
≅
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required to consume 95% of the inlet CO concentration,
and the mixing cup (bulk) stream temperature at this
length.
e. List and defend the principal assumptions made in
arriving at the length estimate (of Part (d))
f. If the catalyst were “poisoned” (e.g., by lead
compounds), what could happen to the CO exit
concentration? Which of the assumptions used in
predicting the required converter length (Part (d)) would
be violated? 42
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g. If the heat of combustion of CO(g) is about 67.8 kcal/g-
mole CO consumed, calculate how much must be
removed to maintain the channel-wall temperature
constant at 500 K?
h. Automatic operating conditions are never strictly steady,
so that in practice the mass-flow rate, temperature, and
gas composition entering the catalytic afterburner will be
time-dependent. Under what circumstances (be
43
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�quantitative) can the design equations you used be
defended if used to predict the conditions exiting the duct
at each instant?(Quasi-steady approximation)
� i. At the design condition, estimate the fractional pressure
drop, , in the honeycomb-type catalytic afterburner.
If, instead of the honeycomb type converter, a packed
bed device were used to achieve the same reduction in
CO-concentration, would you expect to be larger
or smaller than the honeycomb device of your preliminary
design?
44
0/p p−∆
0/p p−∆
PROBLEM1