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 28

1

Mass Transport: Non-Ideal Flow Reactors

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)

2

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


/ /V m

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

4


m

5


Tracer residence-time distribution functions for ideal and real vessels (for e.g., reactors) (adapted from Levenspiel (1972))

6


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)

WSR:

Most likely residence time in a WSR is zero!

Mean residence time =

Not all fluid parcels have same residence time, unlike

PFR7

1/ exp /

flow flowE WSR dF dt t t t


/ /V m

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


0

. ( )flowt t E t dt

22 2 22 2

0 0

1 1. ( )flow flow

flow flow

t t E t dt t E t dt tt t

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)

9

' ' '1 2 1 2

0

( ) .t

E t E t E t t dt


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 21 2 1 2

flow flow flowt t t


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


/ ( / )V m

For vessels 1, 2, 3,…., n in parallel, receiving fractions f1,

f2, f3, …., fn of total flow:

Where , and for each vessel:

12

1 1 2 2( ) ( ) ... ( )n nE f E t f E t f E t

0

1 ( 1,2,..., )

iE t dt i n


1iif

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)

13

exp

0

( )( ) tracer

tracer

reactor exit

tE t

t dt


Real reactors as a network of ideal reactors: Modular modeling

14

GT combustor; proposed interconnection of reactors comprising “modular” model (adapted from Swithenbank, et al.(1973))



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

15

2

0 0

, , ..., .)

flowt tE t dt t E t dt etc


Equivalent vessel network is nonunique

Each alternative may capture one aspect (e.g.,

combustor efficiency) but not another (e.g., domain

of stable operation)

16



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”

17


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, …., N)

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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


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)

20


Statistical micro flow (Random Eddy Surface-Renewal)


systems

If E(t) is defined such that:

21

Relative portion of each unit interfacial area

( ) covered by fluid eddies having "ages" between

t and t+dt,

E t dt


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


Statistical microflow (Random Eddy Surface-Renewal)


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


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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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


Extinction, ignition, parametric sensitivity of chemical

reactors:

Simplest modular model for steady-flow behavior of

combustors: WSR + PFR

25


upper limit to total mass flow rate, at each

upstream condition (Tu, pu, mixture ratio ) 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

26


maxm m

maxm ,m


reactors:

27


Simple, two-ideal reactor “modular” model of gas turbine, ramjet, or rocketengine combustor


reactors:

Parametric sensitivity: change in reactor performance

for a small change in input or operating parameter

(e.g., Tu)

28


Example: WSR module with following overall

stoichiometric combustion reaction:

29

1 O + gm 1 gram P+ cal(heat)gm f F f fQ



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 V


Extinction, ignition, parametric sensitivity of chemical reactors:Overall energy balance:

Source terms for oxidizer & fuel related by:

So, O 2 and F 2 can be expressed in terms of T2

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'''2 1 2 2. , , .p F O F WSRmc T T r T QV


''' ''' /O Fr r f

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 WSR SS, lower extinguished WSR SS (no chemical reaction)

32

'''1

1.exp . . . O F

O F

nv v

F O Fv vO F

E pMr A

RT M M RT


Extinction, ignition, parametric sensitivity of chemical reactors:

33


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)


Extinction, ignition, parametric sensitivity of chemical reactors:Maximum volumetric rate of fuel consumption (hence,

maximum chemical heating rate) occurs at WSR temperature:

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



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



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 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)

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 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 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 mix for CO Fick

diffusion through the prevailing combustion gas mixture,

using the experimental observation that

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/ CO mixSc v D

PROBLEM1

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

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2

1.73 20.216.300CO N

T cmD

p s

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, , 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

Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

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