Transcript Mass Transport: Non-Ideal Flow Reactors

Mass Transport: Non-Ideal Flow Reactors
Advanced Transport Phenomena
Module 6 - Lecture 28
Dr. R. Nagarajan
Professor
Dept of Chemical Engineering
IIT Madras
1
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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MODELING OF NONIDEAL-FLOW REACTORS
 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
V / m /  
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MODELING OF NONIDEAL-FLOW REACTORS
 V  vessel volume

m  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
4
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 a
distribution of residence times, and reduces the residence time per pass within the
PFR)
6
MODELING OF NONIDEAL-FLOW REACTORS
 WSR:
E WSR   dF / dt   t flow  exp  t / t flow 
1
 Most likely residence time in a WSR is zero!
 Mean residence time =
V / m /  
 Not all fluid parcels have same residence time, unlike
PFR
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MODELING OF NONIDEAL-FLOW REACTORS
 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.:

t flow   t.E (t ) dt
s2
is related to
 
1
2
t
2nd
moment of E(t):

 t  t 
2
flow 0
0
flow
2



1
2
2
.E (t )dt  2   t E  t  dt  t flow 
t flow  0

 = 1 for a WSR, 0 for a PFR
 PFR with infinite recycle behaves like WSR
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MODELING OF NONIDEAL-FLOW REACTORS
 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:
t
E12 (t )   E1  t '  .E2  t  t '  dt '
0
(convolution formula)
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MODELING OF NONIDEAL-FLOW REACTORS
 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:
t flow,1 2  t flow,1  t flow,2
 12 2   12   22
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MODELING OF NONIDEAL-FLOW REACTORS
 RTDF for Composite Systems:
 For a network of n-WSRs of equal volume, for which:
1
 t 
E (n  WSRs) 
.


 n  1!  t flow 
t flow
(tflow  V
n 1
 t 
.exp 

 t flow 
/ (m /  ) ) for each vessel in series)
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MODELING OF NONIDEAL-FLOW REACTORS
For vessels 1, 2, 3,…., n in parallel, receiving fractions f1,
f2, f3, …., fn of total flow:
E  f1E1 (t )  f2 E2 (t )  ...  f n En (t )
Where

i
fi  1 , and for each vessel:

 E t  dt  1
i
(i  1, 2,..., n)
0
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MODELING OF NONIDEAL-FLOW REACTORS
 Real reactors as a network of ideal reactors: Modular
modeling

Network of ideal reactors can be constructed to
approximate any experimental reactor RTDF:




tracer (t ) 

Eexp (t )  


  tracer  t  dt 
0
 reactor exit
(where tracer is input as a Dirac impulse function)
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MODELING OF NONIDEAL-FLOW REACTORS
Real reactors as a network of ideal reactors: Modular
modeling
GT combustor; proposed interconnection of reactors comprising “modular”
model (adapted from Swithenbank, et al.(1973))
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MODELING OF NONIDEAL-FLOW REACTORS
 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 RTDmoments

t flow   tE  t  dt ,
0

 t E t  dt, ...,
2
etc.)
0
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MODELING OF NONIDEAL-FLOW REACTORS
 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
 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, flowchanneling, 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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MODELING OF NONIDEAL-FLOW REACTORS
 Real reactors as a network of ideal reactors: Modular
modeling
 RTD
function, E(t), does
concentration
fluctuations
not capture role of
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, …., N)
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MODELING OF NONIDEAL-FLOW REACTORS
 Two networks with identical <E(t)> but with different
shot-to-shot variations, as measured by variance:

2
1 N
 E j  t   E (t )  dt

lim

N  N j 1 0
will perform differently as chemical reactors
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MODELING OF NONIDEAL-FLOW REACTORS
 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
 Statistical micro flow (Random Eddy Surface-Renewal)
models of interfacial mass transport in turbulent flow
systems
 If E(t) is defined such that:
Relative portion of each unit interfacial area

E (t )dt  covered by fluid eddies having "ages" between
 t and t+dt,

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MODELING OF NONIDEAL-FLOW REACTORS
then:

St   St (t ).E(t )dt
0
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
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MODELING OF NONIDEAL-FLOW REACTORS
 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:
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MODELING OF NONIDEAL-FLOW REACTORS

 j A'' , w
  A ,b   A , w

 4 D 1/2
A

 for E ( PFR) ( Higbie 1935
  tm 

1/2
 DA 
for E (WSR) ( Danckwerts 1951)


 tm 
tm  mean eddy contact time (1/(average renewal
frequency))
 Related to prevailing geometry & bulk-flow velocity
 Versatile alternative to Prandtl-Taylor eddy diffusivity
approach
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MODELING OF NONIDEAL-FLOW REACTORS
 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

mmaxupper limit to total mass flow rate, m, at each
upstream condition (Tu, pu, mixture ratio F) above
which extinction of exoergic reaction (flame-out)
abruptly occurs
 For
m  mmax , 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
 Extinction, ignition, parametric sensitivity of chemical
reactors:
Simple, two-ideal reactor “modular” model of gas turbine, ramjet, or rocket
engine combustor
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MODELING OF NONIDEAL-FLOW REACTORS
 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
 Example: WSR module with following overall
stoichiometric combustion reaction:
1 gm O + f gm F  1  f  gram P+fQcal(heat)
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MODELING OF NONIDEAL-FLOW REACTORS
 Extinction, ignition, parametric sensitivity of chemical
reactors:
 Allow a 2nd reactant (oxidant) & associated heat
generation

Governs WSR operating temperature, T2
 WSR species mass balance:
m. i 2  i
1
'''

r
 i O
2
, F
2
, T 2  .VWSR
(i = O, F, P)
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MODELING OF NONIDEAL-FLOW REACTORS
 Extinction, ignition, parametric sensitivity of chemical
reactors:
 Overall energy balance:
mc p . T 2  T
'''


r
1
F  O 2 ,  F
2
, T  QV
. WSR
 Source terms for oxidizer & fuel related by:
r  r / f
'''
O
 So, O
2
'''
F
and F 2 can be expressed in terms of T2
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MODELING OF NONIDEAL-FLOW REACTORS
 Extinction, ignition, parametric sensitivity of chemical
reactors:
 Overall kinetics represented by Arrhenius-type massaction rate law:

1
 E 
 pM  vO vF
r   A.exp  
  . vO vF 1 . 
 .O F
 RT   M O M F
 RT 

n
'''
F
 LHS  straight line intersecting RHS at 3 distinct T2
values, middle one unstable, upper  ignited WSR
SS, lower  extinguished WSR SS (no chemical
reaction)
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MODELING OF NONIDEAL-FLOW REACTORS
 Extinction, ignition, parametric sensitivity of chemical
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)
33
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 WSR
temperature:
Tr '''  max
Tb

1  n( RTb / E )
 Only slightly > “extinction” temperature (previous
Figure)
Tb adiabatic, complete-combustion temperature
 Typical E, n values listed in following Table
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MODELING OF NONIDEAL-FLOW REACTORS
 Extinction, ignition, parametric sensitivity of chemical
reactors:
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/hr
dStoichiometric mixture, no diluent (“diluent” is N ) unless otherwise specified
2
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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
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
Inlet gas pressure
Inlet gas composition
(mole fraction)
700K
1 atm
y(N2)=0.93, y(CO)=0.02,
y(O2)=0.05
37
PROBLEM1
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 surfacecatalyzed 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)
38
PROBLEM1
the wall, CO,w , is negligible compared to CO,b(z) within
each channel. Also assume that the gas-phase kinetics of
CO
oxidation
appreciable
under
these
(uncatalyzed)
conditions
preclude
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 coefficient
39
PROBLEM1
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 Sc  v / DCOmix mix for CO
Fick diffusion through the prevailing combustion gas
mixture, using the experimental observation that
40
PROBLEM1
1.73
DCO N2
0.216  T 

.

p  300 
cm2
s
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
PROBLEM1
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
PROBLEM1
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
PROBLEM1
 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, p / p0 , 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 p / p0 to be larger
or smaller than the honeycomb device of your preliminary
design?
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