The IEM mixing model in exothermic reactions

Chemical

Engineering Science, Vol. 47, No.

Printed in Great

7, pp. 1727-1731.

ooos-2509p2 s5.00 + 0.00 0 1992 Pergamon Press plc

1992.

Britain.

THE

IEM

MIXING

MODEL N. VATISTAS

Dipartimento

di Ingegneria Chimica, Chin&a

IN EXOTHERMIC

REACTIONS

and P. F. MARCONI

Industriale e Scienza dei Materiali, Universitl di Pisa,

Via Diotisalvi 2. 56100 Piss. Italy

(Received 14 September 1990; accepted for publication 10 September 1991)

Abstract-Incomplete mixing in chemically reactive systems with highly exothennic or endothermic reactions may give rise to segregation,affectingnot only the concentration but also the temperature.Its effecton the former has previouslybeen examined using,among others, the isothermalIEM (interactionby exchange with the mean) model. Here this model has been extended to make it suitable for the study of temperature-related segregation too. The derived non-isothermal IEM model has been employed to study the effect of both concentration- and temperature-related segregation on steady-state multiplicity.

INTRODUCTION

Mixing in chemically reactive systems has been defined using two complementary concepts: (1) determined by the experimentally macromixing, obtained residence time distribution (Danckwerts, 1957); and (2) micromixing, determined by the concept of early mixing (Zwietering, 1959). While macromixing affects linear, non-linear and complex chemically reactive systems, micromixing affects the last two alone. A perfect-macromixing reactor is a reactor with the usual exponential distribution of residence times, while a perfect-micromixing reactor means instantaneous contact between the molecules entering the system and the molecules already in it. The effect of incomplete mixing in chemically reactive systems with highly non-linear or complex reactions is considerable. Experimental research has demonstrated that in complex isothermal systems concentration-related segregation may induce dynamic behaviour not previously accounted for (Luo and Epstein, 1986; Menzinger et al., 1986; Menzinger and Giraudi, 1987; Nagypal and Epstein, 1986). Various theoretical studies have attempted to assess the effect of mixing on the multiplicity of stationary states, for isothermal systems (DudukoviC, 1977; Marconi and Vatistas, 1980; Horsthemke and Harmon, 1984; Kumpinsky and Epstein, 1985; Nicolis and Frisch, 1985; Vatistas and Marconi, 1986; Puhl and Nicolis, 1986, 1987). Incomplete mixing means both concentration- and temperature-related segregation in non-isothermal chemically reactive systems, the second of which is likely to be considerable when the heat given off by the reaction is high. The models used to describe incomplete mixing concern concentration alone, while only a few attempts have been made to take temperature-related (or energy) segregation into account (a) by simulating both macromixing and micromixing by a series of N perfect-mixing stages (Luyben, 1968), and (b) by adapting the generalized recycle model for use only in cases of extremely high or low thermal conductivity (Yang et al., 1974).

The aims of the present paper are: (1) to extend the isothermal IEM (interaction by exchange with the mean) model to cover macro and micro temperature mixing; and (2) to apply the derived non-isothermal IEM model to an exothermic chemically reactive system with multiple stationary states in perfectmacromixing and -micromixing conditions.

THE NON-ISOTT-IRRMAL IRM MODEL

The isothermal IEM model (Harada, 1962; Costa and Trevissoi, 1972; Villermaux and Devillon, 1972) is a single-parameter model. Its parameter is the characteristic micromixing time, relative to the rate at which fluid aggregates (Danckwerts’ “points”) entering the bulk fluid exchange their mass with it. Formal complications have been avoided here by assuming the following conditions: a perfect-macromixing chemically reactive system, and the steady-state condition. The following equation represents the mass balance for a Danckwerts’ “point” entering at input j and concerning the i species: dc,.,(a

+ da) -

de,_,(a)

= [Ei(rx) -

ei.,(or)]+dcr m

+ ai*Cci. ,(cO,

r,(a)1 da (1)

with

where t,,, is the characteristic mass micromixing time, and (1 /t,) da is the fraction of the molecules of the Danckwerts’ “point” removed in the interval da and substituted with other molecules of the environment (Costa and Trevissoi, 1972). Equation (1) becomes dci,, da -

2, -

ci. i + a,r(cl. I, ? ) r,,,

with the following initial conditions: cr.,(O) = Ci.J/’ 1727

(3)

N. VATISTAS and P. F. MARCONI

1728

The isothermal IEM model may be extended to cover non-isothermal chemically reactive systems. The heat transfer between the Danckwerts’ “point” and the environment includes not one, as in the case of mass transfer, but two mechanisms (Brodkey and Hershey, 1988): one due to the fraction of the molecules (l/t,)dcr, subject to migration between the “point” and the environment, and a second mechanism due to the fraction of the molecules ( 1 /t,) da of the “point” that exchange by collision their kinetic energy (heat) with the molecules of the environment, without leaving the “point”. The heat balance has been established on the same Danckwerts’ “point”: q(a

+ da) -

Tj(a) = [F(cz) -

Tj(a)]

-(-AH) rCci,j(a),

?nc1

1+

t

da

u %[Tj(a)PC, v,

(11)

with initial conditions as follows: Tj

(0)= Tjf .

An approximate relationship between t, and t, can be obtained if the microscale 1, of a Danckwerts’ “point” is known. For values of its microscale I, smaller than KolmogorolYs velocity microscale IK,

c1

114

lx=

;

L

(12)

J

only diffusive mass and heat transfer occurs between the “point” and the environment; so, the segregation ratio r, is approximately equal to the Lewis number: (13)

T,3 da

(4)

with Tj ewaIzdu where P is the contact probability between the “point” and the heat transfer wall surface A of the reactor, A, is the mean contact surface between the “point” and heat transfer wall surface A, and VP is the volume of the “point”. The characteristic heat micromixing time t,, has been defined by the equation 1 1 -_=-+-

1

th

tc

tm

f,qP P

is constant. From the above equation we obtain

or fV=PA

A-

-PAP_A -

v

(-AH) PC,

da

dc,, A=

where V is the volume of the reactor. Equation (4), combined with eqs (6) and becomes r(ci, j’ Tj)

da

(lo),

EXOTHERMIC

REACTION

The incomplete micromixing effect is large when combined with unpremixed feed conditions, which may be studied using the IEM model. This model, in the modified form described above, has therefore been employed here in the study of second-order exothermic reactions. The effect of imperfect mass and heat micromixing on the number and respective values of the stationary states of a chemically reactive system has been studied, beginning with a system with multiple stationary states in perfect-macromixing and -micromixing conditions. When applied to the non-isothermal IEM model the equations defining the mass balance relative to a Danckwerts’ “ point” entering the first input become

dc,

and comparing eqs (7) and (9) we have

v,

where the order of magnitude for the Lewis number may be assumed to be Le x 1 for gases and x 100 for liquids. For values of Danckwerts’ “point” microscale 1, larger than Kolmogoroffs velocity microscale I,, the convective transfer becomes effective due to eddies, and the segregation ratio r, approaches 1, then decreasing in the case of liquids, while a rather small variation occurs in the case of gases. The approximate relationship between t, and t, that is obtained shows that the heat segregation introduced with the non-isothermal model is more effective in the case of gases and less effective in the case of liquids.

SECOND-ORDER

where t, is the characteristic collision time. It is assumed here that all the “points” have the same contact probability, mean contact surface and volume; based on these assumptions we get that the fraction

dTj T--j da --th

r,)

Tj(a)lda

PC,

--

[

--$$(I;P

1

-CA-~4.1

_-k

tin c~-c~,l

_-k

L

0

c c

A.lckie

o A.lcB,le

--E,RT,

(14)

-E/XT,

(15)

with the following initial concentration conditions: CA.1(O) = CAY>

CE.i(O) = 0.

Those defining the mass balance relative to a Danckwerts’ “point” entering the second input be-

The IEM

mixing

come

~=%-xtI.2

&

dt?

= CA -

do!

CA.2

_

k

rwl

dc,,

G--B,,_~

da

Gn

A=

c

O

c

0

--E,RTI

A,2cB,2e

- EIRTz

A.2c8.2e

(17)

CA.2(0)

=

0,

cIi3,2m

=

cl?/.

T-T

dT1 -= du

I

FAf.0,

I

0

PC,

th

c

with the following

Y-

d0

dT2 -=-

du

+ (-

dy,

Yl

Y -

_=-

’

-

d0

Y2

following initial temperature conditions:

S[

-1” =

cB=~cBf (-

0

41+q2

+ -c

A*1

q2

41

+

+ q2pCp(T-

Ag2 c72

-cA

~=A.f

- (41 + q2F,,l

Aff)Cqlc,f

92

41

-

=

1

e-“I’

(20) (21)

-1

wC,(T-

T2/) + UA(F

da

-

Tl,) T,).

(22)

The above equations may be rendered dimensionless. Thus, the mass balance relative to a Danckwerts’ “point” entering the first input becomes

dx,.,

2, -

d0

xA.1

%I x

(1

-

B(Y2

+ $1

xA, 1)( 1 -

1

%--B.l

d@

+

%I X(

1 _

+I’)

(27) (1 - x”,~)(

1 - xg.2)eyY2/(1 +y2)

Y,)

(28)

initial temperature y2(0)=

09

=

Da(l

conditions:

&-

c

X”

=

IQ, 0

-

(6 IXA,

XB = 6, -

I

1

1

M(6,

ji= yax,

-

bxA,2)epede

+

(29)

- 2”) 62)

+

1+BY I_ BVYC

+

$

-

1

(30)

62

[

(31)

The aboue equations have been worked out for parameter values such that the system possesses multiple stationary states; a number of results are shown in Figs l-3 [conversion (%;*,) vs dimensionless massmixing time (s,)]. DISCUSSION

OF RESULTS

AND

CONCLUSIONS

isothermal IEM model has been extended to include the temperature-related segregation and a 7’he

eYl)

1 (23)

\

.

. . . ..

..I

.

-

.

.

-

..‘.I

.

-

‘..“p

%==b

---we_mm____

__._..-.---4

dx,

A=

xB, l)eTYl’(i

- XA.1) l

xB*l)eYYII(l

-

Y,(O)

r,(O) = Gf

c

4%

1.

while the values X,, Xs and j are given by the following equations:

while the values CA, Es and T are given by the following equations: EA =

-

+ F

sh

(19)

T,)

q(O) = T,/,

=

the heat balance entering the first

’

P

the

Ep(l

with the following

AH) koca 2cB 2e-EIRT2

-5$AT,

X&2(0)

- B(Yl - Yc)

- EIRTI

PC,

th

0,

conditions:

sh

-

T2

=

Finally, the equations defining relative to a Danckwerts’ “point” and second inputs become

dy, -=-+

(26)

initial concentration

A.lCB.1~

(18) T-

- x,4.2)

x(1 - xB.2)e YYzl(l+Yz) x,t.zw

Finally, the equations defining the heat balance relative to a Danckwerts’ “point” entering the first and second inputs become

+ Dn(l

sm

(16)

with the following initial concentration conditions:

with

1729

model in exothermic reactions

E-=.06

XA.1)

xB,l)eYYll(l+Yl)

(24)

with the following initial concentration conditions: x_4,t(O) = 1,

XII.I(O) = 0.

Those defining the mass balance relative to a Danckwerts’ “point” entering the second input become dx,., d9

X, - x,,~ %n x

(1

-

Da +&I

xB.2)eYY21(l

-

Mass

XA.2)

+Yz)

Fig. 1.

(25)

Conversion non-isothermal

Mixing xA vs mass-mixing

Time time s,

sm obtained

IEM model (low Lewis number).

by

N. VATISTAS and

P. F. MARCONI Acknowledgements-The research described here was carried out with financial assistance from the Italian C.N.R.

B=.05

-

Da=.10

-

H=l.OO

-

M=l.OO

NOTATION

A

A* B

heat transfer area of reactor, mz mean contact surface between the “point” and heat transfer area A, m2 dimensionless adiabatic temperature rise =__ -

AH cA/

PC, c

Mass

Mixing

Time

E

s,

CP

Fig. 2. Conversion x1 vs mass-mixing time s, obtained by non-isothermal IEM model (high Lewis number).

Da

kg m- 3 kg me3 specific heat, J kg- I K- ’ dimensionless DamkGhler number mean

x

E H

TI/ >

concentration,

concentration,

CAfe

-

EIRT,

1

activation energy, J mol- ’ dimensionless ratio of feed -

( = Tk,

,

temperatures

c.f

=2/ > ( frequency factor of reaction rate constant, s- l Kolmogoro~s velocity microscale, m Danckwerts’ “point” microscale, m dimensionless Lewis number ( = ---&) dimensionless

10-=

10-l Mass

Mixing

1 Time

10 sm

Fig. 3. Conversion x,, vs mass-mixing time s, obtained by non-isothermal IEM model (low and high Lewis numbers).

new segregation parameter concerning the temperature has been introduced. The ratio between the two segregation parameters of the model has been approximately estimated. The effects of mass and heat segregation have been studied using the non-isothermal version of the IEM model, and a number of results have been obtained for an exothermic, second-order chemically reactive system with unpremixed inputs (Figs l-3). The results obtained using the non-isothermal IEM model reveal that as the degree of segregation increases (mass-mixing time) the number of stationary states always decreases (at least in the cases studied) from three to one (Figs l-3). For high values of the segregation ratio r, (in the case of liquids) the effect of segregation is larger, which means that imperfect mixing reduces more easily the number of states predicted when assuming complete mixing (Fig. 3). In all the cases studied the diathermic conditions (B > 0) reduce the range of multiplicity in the imperfect-mixing conditions (Figs 1 and 2).

P 4 r

ratio

of feed concentrations

probability, dimensionless volumetric flow rate, m3-s- ’ reaction rate, mol m- ’ s- I

contact

r,

segregation ratio

R

universal gas constant, J mol-’

sh

heat characteristic micromixing time dimensionless mass characteristic

, dimensionless K- ’

micromixing

, time

, dimensionless

V 5

characteristic collision time, s characteristic heat-micromixing time, s characteristic mass-micromixing time, s temperature, K mean temperature, K heat transfer coefficient, W rnPz K-l reactor volume, m3 Danckwerts’ “point” volume, m3

X

conversion ( = y

Y

dimensionless temperature

tc th &?I T T u

), dimensionless =- T =,r 11 >

Greek letters Q age of a molecule, s ai

stoichiometric coefficient, dimensionless

The IEM

dimensionless

B

heat

transfer

mixing model in exothermic reactions

coefficient

(=j$z) Y

dimensionless

6,

proportion

activation

energy

(=&)

of total flow rate entering

through

the jth input local rate of energy dissipation, m* s- 3 thermal conductivity, W m- ’ K-l

&

a P

kinematic viscosity, mz s-l density, kg m - 3

8

dimensionless

V

time

= f (

mean

T

residence

> time in the reactor,

s

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CPS

41:1-s

1731

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