mass transport: non-ideal flow reactors -...

Dr. R. Nagarajan

Professor

Dept of Chemical Engineering

IIT Madras

Advanced Transport Phenomena

Module 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 ρ&

�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 adistribution 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

PFR

7

( ) ( ) 1

/ exp /−

= = − flow flowE WSR dF dt t t t


( )/ /V m ρ&

�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∞

= ∫

( ) ( )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∞ ∞

≡ − = −

∫ ∫

�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 2

1 2 1 2

flow flow flowt t t

σ σ σ+

+

= +

= +


�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 ρ&

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

f2, f3, C., 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

ω

ω∞

= ∫


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))


�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

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




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, C., N)

18


�Two networks with identical <E(t)> but with different

shot-to-shot variations, as measured by variance:

will perform differently as chemical reactors

19

( )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 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

− =

−


�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 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

26


max<m 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 Vω ω ω ω− =& &



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− = −& &


''' '''/O Fr r f− =−& &


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−

− = − &



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)



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−≈

+&



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

37

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, 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

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

40

/ 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

41

2

1.73 20.216.300

CO 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

mass transport: non-ideal flow reactors -...

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