Dynamic profiles in chemical reactors with recycle

DYNAMIC

Polish

Academy

PROFILES

of Scicncer, Institute

IN CHEMICAL RECYCLE

MAREK BEREZOWSKI of Chemical Engineerin&

REACTORS

WITH

ul. Baltycka 5. M-100 Gliwice.

Poland

(Received 9 Novawtber 1992: accepted2 February 1993) Abstract-Timcand por;ition&pcndcnt algebraic equations arc derived which dcr;crik the dynamics of prooesr as occurring in an adiabatic homogcnaous tubular reactor with recycle. The equations are valid in the neighbourhood of a steady state and may be useful in oontrohng the reactor opxation.

INTRODUcriON

The present paper is a sequel to the study on the phenomena occurring in a chemical reactor with recycle. The articles published so far have dealt with (a) multiplicity of .steady states (Berezowski and Burghardt. 1989; Beremwskj, 1WCI). and (b) local and global stability of steady states (Eerezowski, t991a, b). Jn each case, to eliminate any additional feedback the simple5t type OF readtor was discussed, namely, an adiabatic homogeneous tubular reactor with recycle and plug flow. The only feedback in the above system was that restllting from the recycle. It Was demonstrated that such a feedback is sufficient to give rise to multiplicity of the steady states of the reactor, not all 01 which are stable. Berezowski and Burgharbt (1989) presented a method for determining the regions in which multiple states appear. Elcrezowski (1990) derived an analytical sufficient condition for the existcuti of single-steady states. Tn subsequent papers @erezowski. 1!@1a. b) methods for analysing local and global stability of steady states have been derived. Unfortunately, none of these analyses enables us to reproduce the quantitative nature of the dynamic phenomena, since they fail to provide transient trajectories Ibr the state v;rriablea of a reactor [i.e. concentration and temperature). In the present paper time- and positiondependent algebraic equations are derived which dcscritK the dynamic behaviour of the state variables or a reactor, Owing to the strong non-linear character of the kinetic equations of the model, the analyti& solutions cannot be sought outside the neighbourhwd of a steady state ol the reactor. Consequently, such an assumption leads to two time- and position-dependent recurrence solutions.‘The first concerns a dynamic relation between the gas temperature and the ccncxmtration under adiabatic conditions, whereas the other provides an analytical picture dthe temporalchanges in the concentration profile. An interesting conclusion has been reached: similarly as in steady states the mutuaI iulhrenoe of concentration and temperature does not depend, under adiabatic conditions, on the rate of a reaction. lt is also shown that in transient states the relationship between the two parameters is only a function of the initial conditions and the time.

In an adiabatic tubular reactor, in the steady state, a rigorously defined relation is valid between the state variables. i.e. between the composition and the temperature of the reacting mixture. This relation can be written generally as y{ol, T) = 0. The validity of this relation during the dynamic (non-stationary) changes in both temperature and concentration profiles in the reactor depends on an initial state of the dynamic process discussed. If, in the initial state, the relation ~(a, T) = 0 is valid, it will remain so throughout the whole dynamic course of the process. If, on the other hand, the initial state has been chosen such that this 4ation is not identically equal to zero, i.e. @(K, T) = E*(Z),then it will differ from ZCFOduring the whole period of reaching the equilibrium. Whereas the analytical solution describing the concentration and temperature profiles is impostible in this case due to the non-linearity of the expression determining the reaction rate, the changes in the relation ~(01, T) with time and position may be found analytically in a rigorous manner. Thus it only remains necessary to calculate approximately the concentration proMe as a function of time and temperature, making use of the function g(cr, T) = E(Z, t). The method develod may be useful in both designing and controlling a recycle reactor. This appreach, as can be seen from the assumptions, can be employed only in the vicinity of a steady state. This does not necessarily have to restrict the use of the method since, in most cases. the control does not mean transferring the state of the system into another distant steady state. Usually, the objective is rather to maintain the state variables in the closest pGssible neighhourhood of the operating point of the apparafns. Thus, the dynamic equations derived in this work should be usefuf for practical purposes, THE MOTIEr. Consider a usual model of an adiabatic tubular reactor with recycle (Fig. 1) (cf. Berezowski and Burghardl. 1989; Berezowski, IWla, b), with the mass

and heat balance equations

2799

given as fohows:

MAREK BEREZOWSKI

aT(z, 0

-+ppwc,~=(-AH~)(-r~). psp 43

(2)

If we assume that no reaction and no heat losses occur in the recycle loop, the inlet conditions may k determined by formulating the mass and energy balance equations for the mixing point (Douglas, 1972; Perlmutter, 1972; Barezowski, 1991% b). Then equations are written in a more ,general form than those presented elsewhere (Bcrezowski, 199 1a):

we may write the mass and energy balances

(7)

(8) The boundary conditions and rkulting from following form:

tions

u(0, r) =JDI(i, r - r,)l(r +su;(tJ T(0, t) =fT(tR. +fT$(r,

- 010,

- t) + (I -S)UAa

as

valid for the above quaeqs (3) and (4) have the

- rr) +fiEl;(rJ

- r)l(rf

(3)

- r) (9)

t - tJ)l(t - t,) - t)l(t,

- f) + (1 -f)r.

where the initial conditions and the delay time in the recycle loop are defined as follows:

T$(r, - r):= T,[w,(r,

- t), 01;

+StiI(sr

(41

v, 1 -s tr:= KS(5)

The term ws(l - t) in the description of u; and T; is a measure of location itl the recycle lwp in an initial state. This simply means that at a given instant t -z tI a portion of matter will appear at the inlet to the reactor, which at an initA moment t = 0 was at such a particular position within the recycle loop that the time necessary to cover the distance from this position to the inlet is t. For Instance, at e = tr, at the inlet to the reactor we deal with a reacting mixture which left the reactor exactly at the moment t = 0. Our objective is to find an analytical solution to the eqs (l)- (4) in the neighbourhood of a steady state. Introducing the following dimensionless variables and numbers:

- r)l(tf

- r).

00)

It should be stressed that the above model is valid only if constant flow in the reactor or in the recycle loop is mumed. since the varktion in the inlet flow rate affects the whole system. leading to cyclic changes of the flow in the reactor. Therefore, the balance equations should be supplemented with two differential quations describing the flow dynamics. For the reactor:

For the recycle loop:

It this is not done, al1 dynamic changes will be valid subject,to the Following assumption:

Of

ANALYTICAL

Using the following

BASIS

OF

THE MEXHOD

increments x:=a--a,

(II)

y:= K - K=

(12)

mnamic

profiles in chemical reactors

the balance eqs (7) and (8) be written as

with recwk

2801

where

fi + ax + 3 = Ip~,+z,
H(s) =_Lw_&o,(~J - t)l(r,

(13)

- dl.

&=

1 iA< (14)

stn,0)

I + TV. (27)

Consequently

1IW

1

and the corresponding

boundary

conditions

G(1, s) =

are

0

x(0, .c) ==JX(I,T - TJ)l(Z - r,) +sxT(zJ

- r)l(rf

vto. i) =_fv(l, T - r,)lCt + fi;crs Equations

- z)

(IS)

- .c)1 (TI - 7).

s

G(0, s) =

(16)

(17)

I?((, s) =

Since

B

+ H,(s)

e -s

; Ate- s(l+CJ-*fdq + H,(s)

s

ar (?e -+-+x=

JK,

s)e-“f

.

~at1+r,h

1 -fe

1291

Thus, eq. (25) takes the form

t?K.

&

gBRFdq CfG(1,

and

- ZJ)

(13) and (14) yield

ay

15

aa.

Cgje-“((-r) s0

d?j

0

aE+Y,z= , we have

We shall now find a solution transfomed to

g+$+&(g+$>=ck Upon introduction

of the definition

(it is obvious that for the stationary is equal to zero) we obtain

state this function

ti(O*r)=fg(I,z

ANALYTICAL

ts)+~~,(t~--)lIz,-T).

DLTERMINATION

OF

PUNCTl0Fi

7

THE

ADIABATIC

r- l-rs -

0 -c r

<

1.

g$e” dq. + G(O, s)

1

s)e--’

+ H,(s)

+ 2-l

(24

s(L r) =1;4(& r -

[H,(s)e-“
(33)

soIution

1 - rs)l(T - 1 - 7,)

t- &IT - f)CI(Z)g+.

1

which leads to the following

lb - E)l +fg;(T

t 1

+ t, - @Cl(T - e - tf) - I(t - 1 - T,,)] 125)

The unknown function G(0, s) is determined using the boundary condition which, by virtue of eq. (22). is G(O,s) =jG(l,

S(u) du

(23)

of eq. (23) is given by

[S 0

&Wdu

8

c G(L s) =

1 - r,,)

l-c-ff

(32)

s -II

A general solution

form of eq. (31) is

r - 1 - Tr)l(T -

SC 7) -MC,

(31)

(22)

We shall now prove that the temperature T is closely related to the wonversion degree a via a socalled adiabatic equation g which, for unsteady states, may Aso be non-stationary. This will be shown by solving eq. (21). Thus we have

&“R:= a(LO).

t H,(s)e-““.

=Z si S(u) du + f s 7-r s r-l-7,

condition -

+I‘ <’ff$Pi+C+r’-q’dq s The time-dependent

czl) and the boundary

tci eq. (30). First, it will be

(261

+ iv- ‘[H&)e-“q_

(34)

The last term of eq. (34) is discussed. The right-hand side of eq. (271 can be transformed to H,(s) =s2{g;(TJ

- T)P(r)-

l(t -

7J)ll

MAR~K

2MI2

r

-f#f(fs

- 7)

J

r - r,

S(u) drr

(35)

k&(s) -f

‘I &(~)e-““~-‘ld~. s0

BEREZOWSKI

It may be geen that the larger the recycle parameter, f, the slower the convergence of the process. We now return to the original problem, i.e. to determining the relationship between the temperatue and the conversion degrw in an adiabatic process undo unsteady conditions. It follows from eq. (20) that the following relations are valid:

(36)

B

K=,q--u

Therefore, H&e

- *c = W{$i(~l

T-

f r - TI

x P(z - 0 - 10 - 4 - rj)] ).

(37)

It is clearly seen that g is a pseudo-periodic function of the time variable, t. with a wudo-period (1 + r,-). The term “pseudo-periodic” refers here to a function whose profik, while remaining the same for the suhsequent periods, changes its overall level. Therefore, z = k(l + ?I) + 0,

(46)

VA

0 & 0 & 1 f t, k = 0, 1,2,3,

...

(y + l)T, -B

PA

Toa.

(47)

Using the above formulae, the set of eqs (13) and (14) can be written as a single equation (43) Upon linearization of eq. (48) around x = 0,1 = 0, g = 0 we obtain

the steady state

(38)

The function, 8, given by eqs (3-S) and (37), may be written as ece, k(t + r,I + @I =J%K

- I(@ - t11 +P+‘Qgc

- @)ClW

ANALYTICAL

- @ -t TJ)[W

- I@ - 5 - Tdl +Sk+‘&(C

- 0

An operator

- @I

x [I(@ - r - TX)- I(0 - 1 - ts)]

UETERMIKATION

TRAJECTORY

form

OF THE

of eq.

OF

A

VMIIABLE

DYNAMIC x

(49) is

I391

since

+ BIOG(T, ~1. 9%

+ 1 + ?J) = #W.

(W

Consequently, as gk + , -c gk, the solution to eq. (21) is always stable and g tends to the stationary values, i.e. g = 0. Therefore, no matter whether during the dynamic process the system tends to the same or to another steady state, the function Q always tends to Zero. Let us try to evaluate the rate of convergence of the geometric series eq. (39). The sum of this series can be wriHkn as

The rate at which this sum is convergent sum is

to the infinite

The general solution of eq. (50) is a given by

since

and

S,,=L.

ev

1-f

Let us define an error of deviation

from a steady state

: s

A,,+$.

0

6

(53)

The unknown quantity X(0, S) can be calculated from eq. (51) making use of the boundary condition eq. (15) which, in a symbolic form, can bz written ~5

Hence, In.?, tie==--. l%f

(50)

where

2803

Dynamic proi%a in chemkai rcactow with recycle

Therefore,

Introduction of eq, (56) into (51) and a simple transformation yields

To solve the above equation- transformations will be carried out identical to those employed in determining the function g [eqs (31)-(37)]. As a result, we obtain a mlution analogous to eq. (34):

XK 4 = f-

9*Q.(u x(C,‘F - 1 - t,)l@ -

mm

1-

t,)

X

s ‘B

-[1(@.,~-~(0,,-~lp)lI(~-~++

c 9.

+tl-

1-

af)dq.

(58)

Again, it is clearly seen that x is a pseudo-periodic function of the time variable T, with a pseudo-period (1 + tr). Therefore+ T = k(l + tr) + B,

0 G 8 < 1 + TJ, k = 0, 1.2.3..

The function

x, givm

._

(59)

by eq. (58), can be written as

Q:-@-~+v-~-T,

*9Sll

xi(t - 0 + rjtr)

Wb)

+5--rp<0<1+rf @.1:= e-If+?/

k+,

forl+r, forO<8cI+~~+~-,.

Equation (&I) is not as 00mpkx as it might seem. Most of the problems arc encountered~~ in determining the values of,the terma containing the integral. How-

MAREK

2804

BEREZOWSK I

ever, the value of this integral in periodical and contjequently has to be calculated only once (either numerically or analytically depending on the degree of complexity of the knction B/cp,). Thus, eq. (60) leads dir&ly to a condition for the local asymptotic stability of a reactor. This condition is given as

lim x[t, k(l + T,) f k-m

e] = 0

9

The error of deviation

from a steady

1% < 1 (63) a

whence

cp*Kb>fPm.

(64)

The same result was obtained elsewhere by using Lapunov’s first method (Berezowski, 19913. Let us examine the eonvergenm of the stabilization process for 9 = 0 i.e. the convergence of the geometric sequence eq+ (60) under the assumption given by eq. (64). The sum of the series eq. (60) is

It ia clearly seen t&t, for I+ 0, the stabilization txcups instantaneously. For f1 the time of stabilization, characterized by the numkr of cycles, n,, is definite (tix 4 w).

NUMERICALEXAMPLE A numeiical example is presented to illustrate the analytical conclusions obtained in the present paper. For clarity, the example will deal with one of the simpler cases concerning an initial state of the reactor. Assume that #= 0, which, by virtue of eqs (45)-(47) yieIds

Fig. 2. Bihrcation diagrams roara nxycle reactor.

-__r’~______________----I

a-oo5 o.oao -

state is

fL.t 0

7

4

6

Fig. 3. Representative dynamic profile at the outlet or B reactor.

n

Dynamic protilts in &em&l Assume

additionally

that r, 7 0 and x(i.0)

# 0.

(70)

Our goal is to transfer the system from a given steady state to some other neighbquring .stmdy state by changing and shifting the whole bifurcation diagiam (Fig. 2). Let us assumb that the observation was carried out at the outlet from the reactor (5 = 1). Equation (60) then reduces to

x(1. k f e) =

.[

f-

1

Y%(l) t rpI (0)

JBR(1

rp,(O %(I

I:;-

(71)

Therefore, it follows from the fact that the initial state is stationary that at each moment 0 < 8 -z 1 the state at the outlet will remain unchanged. Conscqucntly, by virtue ofeq. (71) we have

41)

x(1 - e)

= c = con&,

0 < 8 < 1. (72)

rp.(l - W

The transient profile of the variable x has therefore the form of the diagram presented in Fig 3. The numerical data were taken from the bifurcation diagrams illustrated in Fig. 2. tt is assumed that the initial state is given by d(l) = O-915, Da = 0.0645 whereas the steady state is defind by a.(l) = 0.88, Da = 0.06. Assuming the following kinetics of the action:

r=KC*A and a process of the type (Berezowski, 1991a) rp=(l Therefore,

Substituting we have

-fW(l the ratio

(73)

A --* 3, the function

-wexP(Y&). cp,(l)/~dO)

the numerical

Q is

(74)

is given by

values

taken

from

Fig. 3

Finally, we can estimate the num&r of the cycles, fir. necessary to attain a stationary state for an assumed error E,. Taking E, as equal to 0.01 we tid from cq. (68) nX = 6.14. Therefore, .it can be concluded that the steady state will be reached after the reacting mixture has been rccyclcd sjx times through the reaclor. SUMMARY

paper presents an analytical solution of the dynamic model of an adiabatic tubular homogeneous reactor with recycle. The aoiution is valid in the immediate neighbourhood of a steady state of the reactar. Based on mathematical analyses, two basic rwurThis

reanors with recycle

2805

fence equations were darived. The first describes the transient profile of the adiabatic function d<, T) [cq. (39)], while the other determines the dyuarnic behaviour of the’ state variable x(5. r) [cq. .(Bo)]. The adiabatic function relates the temperature in the reactor, T, with the conversion degm a. This function becomes equal to zero for a steady state, while for a transient state it may assume different values depending on the initial conditions of the reactor. It has been demonstrated that the transient changes in the State variable, ~(6, r), are analogous to those occuris due to the fact ring in the function g. The differ&e that, for the variable X, the raw ot reaction has an important effect on the profile of this vatiable. In contrast. the function y is not am&c&d by the rate of reaction. Finally. a numerical example was present4 which shows the control operation transferring the system from one into another, closely situated, steady state. Stepwise asymptotic dynamic changes in the sfate of the reactor were obtained {Fig. 3).

NOTATION

heaat capacity, kJ/(kg K) A, kmol/m’ coricentration of component DamkChIer number activation energy, kJ/kmol (= fif/rir) recycle parameter volumetric Aow rate, m3/s adiabatic function heat of reaction, kJ/kmoi length, m mass flux, kg/5 molecular mass. kg/km01 gas constant, kJ/(kmol K) operator variable, l/s time, s temperature, K mass fraction volume, m3 velmity, m/s deviation variable for conversion degree deviation variable for dimensionIess temperature position coordinate along reactor, m Greek letters conversion degree dimensionless numbeT related to adiabatic ; increase of temperature dimensionless number related to activation Y enfli!Y c = WR,~)l 6 Dirac delta K dimwsionless temperature stoichiometric coefficient dimensionless position co-ordinate along ; the reactor density, kg@2 P dimensionless time r,@

chemic4 reactors with recycle. Chem Engng Sci, 44, 2927-2933. Beraowski. M.. 1990, A sufficient condition for fhe existence of single steady state in chemical reactors with recycle. Chem. Bngng &‘ci.4s$ 1325-1329. Berezawski. M, 1991& Method for analysing lo4 stability of aseudohomoaencous chemical rcact~rs with recvde. < ~~~

c&m. Engng SC:4&

REFERD4CeS

Beremwski. M. ad

lytical

method

Burghardt, A+, 1989. A genedid anafor determining multiplicity features in

557-562.

Berezowski, M., 1991b, Method for analyysingglobd stability of pseudohomogcnwus chemi;cal reactors with reqcle. GBem. Engng Sci. 46, 17%1-1785. VW&S, J. hi.. 1972, Process Dymmks ad Control, VoL 2, Conr~l System Syulppsfs, pp. 128-130. PreatimHall. EngIewood CliRq NJ. P&mutter, I). IL, 1972, Srnbiliry of Chemiral Rearrow. pp. 249-276. Prcntioe-Hall. En&wood Cliffs, NJ.