Parametric sensitivity in tubular reactors with co-current external cooling

Chemical Engineering ScLnce, Prmted in Great Britain.

Vol. 45. No. 5, pp. 1301-1307,

PARAMETRIC

SENSITIVITY IN TUBULAR REACTORS CO-CURRENT EXTERNAL COOLFNG EVAN

Department

BAUMAN

and ARVIND

WITH

VARMAt

of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, U.S.A. JEAN

Dipartimento

ooo9~2509/90 $3.00 + 0.00 0 1990 Pergamon Press plc

1990.

LORUSSO

and

MARIO

DENTE

di Ingegneria Chimica e Chimica Industriale, Politecnico di Milano, 20133 Milan, Italy and

MASSIMO MORBIDELLI Dipartimento di Ingegneria Chimica e Materiali, Universit$ di Cagliari, 09123 Cagliari, Italy (Received 24 August 1988; accepted 21 August 1989) Abstract-The parametric sensitivity behavior of a tubular reactor with co-current external cooling is investigated using a generalized parametric sensitivity criterion. This criterion is based on the occurrence of a maximum value for the normalized sensitivity of the hot-spot with respect to any of the input parameters of the reactor. Comparisons are made with the Adler and Enig criterion, which is based on a geometric feature of the temperature profile; this criterion is shown to give qualitatively erroneous results in some cases. The regions of parametric sensitivity in the reactor parameter space are also reported for various values of the involved dimensionless parameters.

INTRODUCTION

Reactor parametric sensitivity or runaway indicates a particular condition of reactor operation where, for any given small variation in the reactor input parameters, the steady-state reactor behavior undergoes large changes. This is best represented by the temperature profile along the length of a tubular reactor, and particularly by its maximum value (or hot-spot), which is usually present in the case of exothermic reactions. Obviously, large temperature excursions are deleterious in applications, and so the parametric sensitivity region of reactor operation should be avoided. For this reason several a priori parametric sensitivity criteria have been developed in the literature, aimed at defining the boundary between sensitive and non-sensitive behavior in a rigorous fashion. In most cases, the parametric sensitivity criteria first originated in the context of thermal explosion theory, and were later applied to chemical reactor theory. In general, these criteria are geometric in nature, and based on some physical intuition about the shape of the temperature vs reactor length or conversion profile, as discussed in more detail in a recent review (Morbidelli et al., 1987). A new criterion has recently been proposed (Morbidelli and Varma, 1988), again with reference to thermal explosions, which is based on a rigorous definition of normalized sensitivity; it does not require any geometric consideration of the temperature profile [cf. Chemburkar et al. (1986)]. The purpose of this work is to apply this criterion to a pseudohomogeneous plug-flow reactor, with special emphasis on

‘Author

to whom correspondence

should be addressed.

the influence of co-current cooling on reactor sensitivity behavior. Several papers have appeared recently in the literature dealing with this particular aspect, and have adopted various types of geometric sensitivity criteria [cf. Soria Lopez et al. (198 l), Henning and Perez (1986) and Hosten and Froment (1986)]. The present work can therefore be used to gain some further insight towards the definition of the most reliable criterion for parametric sensitivity.

MODEL

AND

SENSITIVITY

EQUATIONS

The mass and energy balances for a pseudohomogeneous tubular reactor with co-current external cooling are represented by the following equations: dC _=

__

dz

dT

-= dz

(1)

(-AH)r

dT,_ dz -

with initial conditions z=O:

r

u

2PR’cnU(T wcC,

_ T,)

(3)

(ICs)

C=C,,

T=

TO,

TC=TCo

(4)

and r = k,exp(

-

E/RT)C”

(5)

for an nth-order irreversible reaction. Not by observation of eqs (l)-(4) that an algebraic relation between the variables C, T and T, can be developed readily. These equations can also be made dimensionless to

1302

EVAN

give the compact

BAUMAN

form:

dv

as described in detail elsewhere (Morbidelli and Varma, 1988), it is elegant and computationally more efficient to evaluate local sensitivity for any 4, by calculating the adjoint sensitivity, 9, first. The latter is obtained by solving the ODE

B(u - r&o)

dx=a-

(1 - xYexp[y(l

et al.

-:)I

@r(v - U. - ax)

-

(1 -xYexp[r(l

di

-;J=s(v,@)

dF --A au

dx=

(6)

(13)

with the TC with the IC

x=0: x=0:

(7)

v = vo

;1=1

(14)

and by definition

and uc = ye0 + asx + 7(uo -

u)

where the dimensionless groups a, fl, y and r are defined in the Notation. The variable ZJrefers to dimensionless temperature, and x represents the fractional reactant conversion. The reactor behavior is then determined completely by the following seven dimensionless parameters: heat of reaction (a) and heat transfer (fl) parameters, activation energy (‘y), reaction order (n), the heat capacity ratio between the reacting and the cooling fluid streams (t), and the inlet reactant (uo) and coolant (v,,) temperatures. As r -+ 0, eq. (6) approaches the well-known case where the external coolant is at a constant temperature, which is identical to the equation for thermal explosion theory previously examined in detail (Morbidelli and Varma, 1988). Let us now consider the value of the hot-spot along the reactor length, a*; this value obviously depends upon each of the seven parameters noted above. Thus, indicating any of these parameters as 4, we can define the local sensitivity of u to the generic parameter, 4, as

and so for the local sensitivity

of the hot-spot

1..*

(10) Equation (6) can be differentiated with respect to 4, to yield an equation which describes s: ds aF c?F dx=z&+avs

(11)

along with the IC

dv

s = s(0)

Since g

(12)

x=o is 0 for all choices of 4, except u.

for which it equals 1. The expressions can be readily tion of F.

The procedure

obtained

analytically

for g

and g a+ from the defini-

depends on the specific choice of 4. eq. (11)

would need to be solved repeatedly for every #, for which the hot-spot sensitivity, s*, is desired. However,

(15)

to obtain s* is then as follows:

(1) Equations (6) and (13) are integrated simultaneously, using ICs (7) and (14), to the location x = x*, where the reactor exhibits a hot-spot, lF.

(2) With v(x) and 2(x) thus determined:

1 (1’5)

where s (0) = 1/2(x*)

(17)

so that the sensitivity, s*, of the hot-spot to any parameter, 4, can be readily calculated. Now, although the use of the adjoint variable S is efficient, the technique for computing s* noted above has in it the implicit assumption that a hot-spot is present. However, a co-currently cooled tubular reactor may operate in a manner such that the reactor temperature increases continuously with reactor length, with no hot-spot present. This type of operation was termed pseudo-adiabatic by Soria Lopez et al. (1981). The method of adjoints cannot be used to calculate the sensitivity of a reactor operating in a pseudo-adiabatic state. In such cases, the sensitivity of the hot-spot temperature can be calculated by simultaneous direct integration of the model eq. (6) and the sensitivity eq. (1 l), for each specific choice of 4, over the entire reactor. In these cases, the hot-spot is located at x*, where x* is the conversion at the reactor exit. RESULTS

x = 0: where s(O) = G

A(x) = g.

(8)

AND

DKKXJSSION

The procedure described above yields the local sensitivity of the hot-spot temperature, v*, with respect to the parameter, 4. However, it is more desirable to examine the normalized sensitivity, which may be defined as s(v*,

4)

=

9

v*

av* A,*, aqb v* &

(18)

In Fig. 1, the values of normalized sensitivities are shown as a function of the heat capacity ratio parameter, 7, for various selections of the parameter, 4. It can be observed that S(v*, #) exhibits a maximum at

Parametric sensitivity in tubular reactors with co-current external cooling -3 8 *> v) G 2 .Z :Z! g C% ‘D 1 R Z E b 2

0

8 L_ >

1303

200

cn p :z 100 5 cn $ ,N g

0.2

0.1

0.3

Heat Capacity

Ratio

0.4

0.5

Parameter.

=

0.6

0

0.035

0.040 Heat Capacity

T

Ratio

0.045 Parameter,

z

Fig. 1. Normalized sensitivity S(u*, I#J)as a function of r for various choices of the reactor input parameter, 4 (1 = y and a, 2 = vO, 3 = p, 4 = n, 5 = 7).

Fig. 2. Normalized sensitivity S(v*, &) as a function of T for various choices of the reactor input parameter, r$ (1 = CC, 2 =vO, 3 - y, 4 = p, 5 = 7, 6 = n).

a specific value of z, which is substantially independent of the specific parameter, C#J,adopted in the sensitivity definition. This observation suggests defining this specific value of z as the critical value, t,, separating sensitive (to the right of the maximum) from non-sensitive (to the left of the maximum) reactor behavior. This constitutes the so-called generalized criterion for parametric sensitivity, which accounts simultaneously for the effects of all input parameters, 4, and does not rely on the shape of the temperature profile. Figure 2 shows similar results as Fig. 1, but now with increased values of the activation energy parameter, y, and the heat transfer parameter, B_ Since, in this case, the reaction rate is more sensitive to temperature changes, the values of the hot-spot exhibit a larger sensitivity to all the parameters, 4. However, the general picture, i. e. the existence of a sensitivity peak at the same T location, independent of the choice of the input parameter, C#J,remains unchanged. These features of the generalized sensitivity criterion have been discussed in detail previously (Morbidelli and Varma, 1988), and do not need to be repeated here since they are not affected by the variable external coolant temperature along the reactor length. However, it is worth reiterating that the generalized criterion holds for sufficiently large (although still quite moderate) values of the activation energy, y, and heat of reaction parameter, CC.For lower values of these parameters, the location of the sensitivity peak becomes a function of the choice of the input parameter, d, so that the generalized criterion does not hold any longer. However, in such cases, the reacting system is generally non-sensitive anyway, so that runaway is not a relevant issue (Morbidelli and Varma, 1988). As noted earlier, the final aim of developing u priori

z values (i.e. decreasing heat capacity for the external coolant), reactor runaway becomes more and more likely. It should however be noted that, while for increasing z values the runaway region enlarges, the magnitude of the normalized sensitivity at the hotspot decreases, as may be seen in Fig. 4. These values represent the sensitivity maximum as a function of the critical t value, i.e. S(u*, I+,) at 7 = T,, for the three situations examined in Fig. 3(aHc). It should also be noted that in Fig. 3(a), (b) and (d), the sensitivity criterion is no longer generalized for sufficiently high values of 7. In such cases, the choice of the parameter d will affect the critical value of the heat capacity ratio parameter, rC. It is safest to choose the value of C$which yields the smallest critical value of Z, (i.e. corresponding to the highest coolant flow rates). In Fig. 3(a), (b) and (d), this choice of C#J is the coolant parameter, z, itself. The loss of generalized behavior for large T can be explained by recalling from the definition that, as 7 + co, the reactor approaches adiabatic operating conditions. When this occurs, the reactor exhibits nearly 100% conversion, and the value of the sensitivity is relatively small. In Fig. 3(aHd), the critical curves computed by the Adler and Enig (1964) criterion (i.e. occurrence of a region with a positive second-order derivative before the hot-spot in the temperature-conversion plane) are also shown. In addition, the pseudoadiabatic region of operation (hatched region) as discussed earlier is also shown. As may be seen, the Adler-Enig criterion occasionally predicts runaway within the region of pseudo-adiabatic operation. However, in this region the reactor temperature continuously increases with reactor length, indicating a non-sensitive reactor behavior by the absence of a hot-spot. Therefore, the Adler-Enig criterion appears not to apply. This point can be further demonstrated by the trajectories in the temperatureconversion phase plane, shown in Fig. 5 for z = 0.0525, and various p values. These refer to the case considered in Fig. 3(c), from which we may note that the critical values predicted by the generalized and the Adler-Enig criteria are 114 and 140, respectively, while for /? larger than 127 the reactor enters

sensitivity

criteria

is to identify

the runaway

region

in

the reactor parameter space, so that it can be avoided during reactor design and operation. Examples of such plots are shown in Fig. 3(a)--(d), in terms of the heat transfer parameter, 8, vs the heat capacity ratio parameter, 2. The solid curves in Fig. 3(aHd) represent the critical T values computed according to the sensitivity criterion described above. For larger

1304

EVAN

BALJMAN

etal. (b)

80

v

$=a,vo

I

n = 1,

“0

a=l,

y=ZO

1 , , , , , a,;:;:+“, 0.2

a

0.4

Heat Capacity

0.6 Ratio

0.8

0’0

I

Parameter,

’

’

’

0.2

Heat Capacity

7

’

0.4

’

’

0.6

Ratio

’

’

0.8

Cd)

a=2.

0

Fig.

3. Comparison

the Adler-Enig

Ratio

Parameter,

a consequence

of

I

/

y=15

I

I

5

Heat Capacity

T

Ratio

Parameter,

T

and of regions of parametric sensitivity predicted by the generalized sensitivity (-) (- - -) criteria. Below and to right of curves is sensitive. Hatched zone = region of pseudo-adiabatic operation.

Since, at a pseudo-adiabatic operating region. /3= 110, the reactor exhibits a sharp rise in the temperature-conversion trajectory, but at fl = 120 it does not, it is evident from Fig. 5 that p = 114 is indeed a quite reasonable estimate of the boundary between sensitive and non-sensitive reactor behavior. The failure of the Adler-Enig criterion is a quite surprising result, since this criterion has previously provided the same answer as the generalized criterion in a large number of cases, including heterogeneous reactors, although always with constant-temperature external cooling, i.e. t = 0 (Morbidelli and Varma, 1986, 1988). This is indeed true also in Fig. 3(a)--(d), where it may be seen that the answers of the two criteria become identical as z -+ 0. This seems to indicate that the Adler-Enig criterion fails because of the variable-temperature external cooling, most likely as shape

z

= 1

y=40 = "0

Heat Capacity

’

Parameter.

600

“0

1

“co= “0

I

o

=

the

of the temperature

increased profiles

complexity

(cf. Fig. 5).

of

the

A final observation concerns the effect of the activation energy, y, and the heat of reaction parameter, CI, on reactor sensitivity behavior. From Fig. 3(a)-(c), it appears that, as y increases, the sensitivity region enlarges asymptotically up to a finite size as y + m. On the other hand, u has a continuous enhancing effect on the sensitivity, as can be seen by comparing the results shown in Fig. 3(a) and (d).

CONCLUDING

REMARKS

The generalized sensitivity criterion previously proposed by Morbidelli and Varma (1988) in the context of thermal explosion theory and heterogeneous fixedbed reactors (Morbidelli and Varma, 1986), has been applied to pseudohomogeneous tubular reactors with co-current external cooling. It is shown that this criterion provides very reasonable predictions of the

Parametric sensitivity in tubular reactors with co-current external cooling

1305

n=l.vo=l a=l.+=vo p = pa

I

0.1

i

I

0.2

0.3

Heat Capacity Fig. 4. Values of the maximum

Ratio

I

0.4

Parameter.

5

normalized sensitivity S(v*, uO) as a function of the critical value, Z,

1.7 -

1.6

-

n=1.

“0 = 1

a=l, y=40 ‘I: = 0.0525 pc-= 114.0

($I=+)

ppo = 127.4 P Adler-EniQ = 14”’

Conversion

Fig. 5. Temperature-conversion

profiles for various p values and r = 0.0525.

boundary between sensitive and non-sensitive reactor operation. It can be used conveniently to identify the runaway region in the reactor parameter space, as summarized in Fig. 6(a)-(d). In Fig. 6(a)-(d), the boundary curves separating runaway from non-runaway behavior are shown in the 8-r parameter space, for various values of a third parameter. In particular, the effects of the heat of reaction parameter, u, the activation energy, y, the inlet temperature, vor and the reaction order, n, have been investigated. As can be expected, all of these except the last one have an enhancing effect on reactor runaway. Thus the possibility of runaway increases as the heat of reaction, the

activation energy, or the inlet temperature increases; however, runaway is more likely for reactions of lower order. These effects are the same as observed before for pseudohomogeneous and heterogeneous reactors, with constant-temperature external cooling (Morbidelli and Varma, 1982, 1986, 1987, 1988). An important result of the analysis is the failure of the Adler-Enig criterion under certain conditions. This criterion was found to give results consistent with the generalized criterion in all cases examined previously (Morbidelli and Vat-ma, 1986, 1988) where constant-temperature external cooling was considered. However, in the case where the external cool-

1306

EVAN BAUMAN et al. 600

0 Heal Capacity Ratio Parameter.

150-

“0 = 1

n=l.

T

(b)

I

I

0.1

0.2

I

0.3

I

a

0.4

Heat Capacity Ratio Parameter,

n=1,

300 - vO= 1,@=vg a= 1, y=20 vcO= “0

T

(d) /

CcL s

100 -

g

$ d

200

!-?! b a & E E ; 100

.S kz m Iiii 2

8

0

0

I 0.1

1 0.2

I 0.3

# 0.4

Heat Capacity Ratio Parameter,

I 0.5

0

z

Fig. 6. Parametric sensitivity regions in the /S-T reactor parameter reaction parameter, a; (b) activation energy, y; (c) inlet temperature, right of curves is sensitive.

ant varies in temperature with reactor length, the Adler-Enig criterion leads to the inconsistent result of predicting runaway within the pseudo-adiabatic region of reactor operation. This points of the intrinsic weakness of criteria based on some geometric feature of the temperature profile, and stresses the reliability of the generalized criterion which is firmly based on the observation of the quantity of direct interest for establishing reactor parametric sensitivity, i.e. the hotspot sensitivity towards all the input parameters of the reactor. NOTATION

C CP

reactant concentration, mol/cm3 specific heat, cal/mol K

0.1

0.2

0.3

0.4

Heat Capacity Ratio Parameter,

E F

ko k(T) n I

R

Rt S

s^ WV*, 4) T L

U

u

0.5 ‘T

space for varying values of: (a) heat of 0,; (d) reaction order, n. Below and to

activation energy, Cal/m01 function defined by eq. (6) frequency factor reaction rate constant reaction order reaction rate, mol/cm3 s ideal gas constant reactor tube radius, cm local sensitivity (a~/&$) adjoint sensitivity normalized sensitivity of U* to r$ temperature, K number of reactor tubes velocity, cm/s overall heat transfer coefficient, Cal/cm’ s K

Parametric

in tubular reactors with co-current

coolant

W, x Z

flowrate,

letters dimensionless

B

C( -

heat

A~~GIG,PT,,~I

dimensionless

of reaction

heat transfer

parameter

parameter

[2U/

P r

R,C,&(T,,,)CK’I dimensionless activation energy (E/RT,,,) enthalpy of reaction, cal/mol density, mol/cm3 ratio of reactant to coolant heat capacity

4

WK: fnPCpglW, C&J generic model input parameter

Y AH

Subscripts C

z ref *

and superscripts external coolant reactant reactor inlet reference value hot-spot value

external

1307

cooling

REFERENCES

temperature (T/T’,,) mol/s reactant conversion [(C, - C)/C, J reactor distance, cm

dimensionless

V

Greek CI

sensitivity

Adler, J. and Enig, J. W., 1964, The critical conditions in thermal explosion theory with reactant consumption. Cornbust. Flame 8, 97-103. Chemburkar, R. M., Morbidelli, M. and Varma, A., 1986, Parametric sensitivity of a CSTR. Chem. Engng Sci. 41, 1647-1654. Henning, G. P. and Perez, G. A.. 1986, Parametric sensitivitv in fixed-bed catalytic reactors. Chek. Engng Sci. 41, 83-88. Hosten, L. H. and Froment. G. F.. 1986._Parametric sensitivity in co-currently cooled tubular reactors. Chem. Engng Sci. 41, 1073-1080. Morbidelli, M. and Varma, A., 1982, Parametric sensitivity and runaway in tubular reactors. A.1.Ch.E. J. 28, 705-713. Morbidelli, M. and Varma, A., 1986, Parametric sensitivity and runaway in fixed-bed catalytic reactors. Chem. Engng Sci. 41, 1063-1071. Morbidelli, M. and Varma, A., 1987, Parametric sensitivity in fixed-bed catalytic reactors. Inter and intraparticle re-

sistances. A.1.Ch.E. J. 33, 1949-1958.

Morbidelli,

M. and Varma, A.. 1988, A generalized

criterion

for parametric sensitivity:. apphcati& to thermal explosion theory. Chem. Enana Sci. 43. 91-102. Mbrbidelli, M.,- Varma, A: and Aris, R., 1987, Reactor steady-state multiplicity and stability, in Chemical Reaction and Reactor Engineering (Edited by J. J. Carberry and A. Varma), York. Soria Lopez, Parametric

Chap. 15, pp. 973-1054.

Marcel

Dekker,

New

A., de Lasa, H. I. and Porras, J. A., 1981, sensitivity of a fixed bed catalytic reactor.

Chem. Engng Sci. 36, 285-291.