A Simple Approach to Highly Sensitive Tubular ...

SIAM J. APPL. MATH.

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Vol. 48, No. 5, October 1988

008

A SIMPLE APPROACH TO HIGHLY SENSITIVE TUBULAR REACTORS* P. S. HAGAN, M. HERSKOWITZ$,

AND

C.

PIRKLE

Abstract. The steady state operation of a long tubular fixed-bed reactor in which a single highly exothermic reaction is occurring is analyzed. To avoid temperature runaways, such reactors must be operated so that their temperature rises AT are small fractions of the adiabatic temperature rise A T..d. So here asymptotic methods based on A T/A T,,d TCcr(X) for some X then the solution (2.13) pertains only until X(z) reaches a value at which Tr(x)< Tc (which may be X =0). At this point the radial diffusion of heat can no longer balance the heat generation at reasonable temperatures, so T increases on the rapid z/7 scale toward a stable but extremely high temperature solution of (2.11). That is, a thermal runaway occurs (see Fig. 2). So the leading order analysis predicts that a reactor runaway occurs if Tc> Tr(x) for some X, and that the reactor operates stably according to (2.13) otherwise.

2.4. Explicit asymptotic solutions. Reaction rates typically increase exponentially in T over limited temperature ranges. For example, for Arrhenius kinetics

(2.14)

S(X, T)= a(X)

where 0= E(T- T)/RT, and where e consider the expansion

(2.15)

a(X) e -z/eL e -2+’’" RT/E is usually between 0.03 and 0.15. So

e -z/eT=

.

S(X, T)--S(X, )e A(T-’)+B(T-’)2+’’"

around some fixed temperature To obtain the radial equilibrium profiles T explicitly, we solve (2.11) for the reaction rate (2.15) with the B(T- )2+... terms set to zero. With (2.13), this then yields the solution of the reactor equations (2.6a)-(2.6d) to leading order in r/ and in B. T(O,X,T)

Reactor Runaway

Stable Operation

I, FIG. 2. Qualitative bifurcation diagram of the local equilibrium solutions at a fixed X. The arrows indicate domains-of-attraction. As shown, if 7,. < T?(X) and T < T2(r, X, T,.) for all r, then T evolves rapidly to the stable solution T(r, X, T,.). However, if either T,. > TY(X), or T,. < Tr(x) but T> T2(r, X, T,.) for all r, then

T evolves rapidly to an extremely high temperature solution (not shown) and a thermal runaway occurs (see

For S(X, T)

[22] (2.16)

]).

S(X, T) exp {A(T- T)}, the exact solutions of (2.11) are given by T(r, ce)= T +[4ce/ y-Z log (1- ce + cerZ)]/a

where the constant a must satisfy

8a(1-a) e-4"/r=AQS(X, ) e -A(-L). The left side of (2.17) has a single maximum at ce a,,(7), where 2 (2.18) Orn (7) =1a(T-I-2-JT +4) (2.17)

U(&) the standard model requires higher reaction rates to attain a given temperature rise b; (3.17) and (3.20) show that it allows a higher 4 before predicting runaway. A zero-order reaction is essentially the worst case for comparing the two models, since the reactor’s "hot spot" will be localized for both positive order and negative order reactions (see Fig. 6). So the o model and the standard model will agree for reactors with small Biot numbers, say 3’_-< 1. This agreement occurs because when 3" rrh/A is small, the diffusion of heat across the reactor radius is easy compared to the heat transfer resistance at the wall; thus the radial distribution of the heat generation is relatively unimportant.

4. Multiple reactions. Suppose that n reactions are occurring in the reactor. Then all concentrations can be expressed in terms of n variables X (X1, X n), each measuring the extent of one of the reactions. Similar to (2.6a)-(2.6d), the dimensionless reactor equations can be written as

(4.1a) (4.1b)

X-- r/ lfe

x{r

r

X[ + 1

(X, T),

j

1,’’

,

"riTz T,.r +-r Tr + QJsJ(x, T)

for 0 < r < 1; with the boundary conditions

(4.1c) (4.1d)

X=0, Tr=O X{=O, -Tr=3"(T-T)

at r=0, at r=l,

",

n,

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A SIMPLE APPROACH TO TUBULAR REACTORS

1097

and the initial conditions X--0, T= Tc at z=0. Here SJ(X, T) and QJ are the dimensionless reaction rate and heat release of reaction j. To extend the a model to this case, define the total heat release H as

H(X, T)=Y QJS2(X, T)

(4.2) and the averages

j(z) as

(z)-- (X(z, r)). (4.3) Now in general there is no average temperature T(z) such that Sg(, )= (SJ(, T(z, r)))

(4.4)

holds exactly for all reactions. Therefore, to obtain a simple 1D model we define the heat-averaged temperature T(z) by

H(X, T)=(H(X, T(z, r)))

(4.5) and approximate

d

(4.6a)

S(X’ T),

dz

j

1,.

., n.

So the XJ(z) may be less accurate than in the single reaction case. We now approximate the temperature dependence of H as exponential at each point z, and follow the Karman-Pohlhausen procedure of Appendix A. This yields dT

r/-d-z -8c/a+ Qs(, T)

(4.6b)

where a is defined implicitly by

A(T- T.)=4a/y-log (1 ce) +. (B/A 2) log 2 (1 a),

(4.6C) and where now

a(, )= [log H(, )]f, B(, )=1/2[log H(, )]Tw. (4.6d) Once (4.6a)-(4.6d) have been solved, the approximate solution is (4.7)

X(z,r)=-(z),

1

T(z,r)= Tc+{4a/y-Zlog(1-a+ar2)}.

So to handle multiple reactions, we have essentially just replaced S(X, T) by the total heat release H(X, T) everywhere except (4.6a). For the special case of n Arrhenius reactions S (X, T)

a

(X)

e -/RT,

j

1,

n

note that

(4.8a) (4.8b)

A=/R2, B/A2 -R/ ff

+- (EJ-/) 1

where E(X, T) is simply the average

(4.8c)

E (X, T)

2E

2

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1098

HAGAN, HERSKOWITZ, AND PIRKLE

weighted by the heat release of each reaction

wJ(:, )= QJaJ(’)e-E"/R’/H(Y(, ).

(4.8d)

Commonly, the activation energies E are all less than twice the minimum activation energy. In this case the last term in (4.8b) is always less than 1/16. So the last term in (4.6c) is still usually very small and can often be neglected. To test this model, consider two first-order Arrhenius reactions A1-->A2 and A1-+ A3, with the second reaction having twice the activation energy of the first. We assume that both reactions release equal amounts of heat, so Q= Q2._ Q, and choose the reaction rates

S(X, T)= .80(1-X-X 2) e -/e e e/eL, S2(X, T)= .20(1-X-X 2) e -2’/’ e 2E/e so that the second is 25% of the first at T T.. As in Figs. 5 and 7, in Fig. 8 we graph

(4.9a) (4.9b)

the maximum value of T(z) as a function of /with Q/’rl ATao held fixed. As before, the solid point in Fig. 8 is where / first exceeds the runaway criterion of [2]. Additionally, Fig. 9 shows the full curves .l(z), 2(z), and :(z) for a particular / chosen so that the reactor is very near to a runaway.

max(OK)

600

550

FIG. 8. The peak value

of 7"(z)

in the reactor as a function

of l for two simultaneous first-order reactions

(see equation (4.9)). Shown are the predictions of the model (o), the exact two-dimensional equations (e), and the standard one-dimensional model (s). There is no visible difference between the predictions of the c model and the exact equations. The parameters are EATa/RT,2 =20, RT,./E =.10, y--4.8, T,. 500 K and Le

1.25.

Our experience with multiple reactions has been very limited. However, in the few examples considered, the c model seems to be no less accurate than we would expect for a single reaction. In particular, we have not seen any appreciable loss of accuracy in the .,J(z), despite the additional approximation used to obtain (4.6a). $. Conclusions. The examples presented here have all been mathematical exercises chosen to test the range over which the c model accurately simulates the original two-dimensional equations. The a model is used to analyze specific physical reactors elsewhere [24], [25]. In [2] we note that the maximum feasible reaction rates occur

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A SIMPLE APPROACH TO TUBULAR REACTORS

1099

+ (z) e 540

50

.25

.50

.75

z

X (z)

e .2

.25 .75 .50 z FIG. 9. The axial profiles (z), J(z) and X2(z) for two simultaneous first-order reactions. The value r/= .078 has been chosen so that the reactor is very near to a runaway (see Fig. 8). The other parameters are the same as in Fig. 8.

when the reactor reaches runaway. So after deriving the runaway criterion, in [2] we use it to obtain the theoretical limits on the reactor’s productivity and efficiency. We have found the c model, the runaway criterion and the theoretical limits to be useful in screening potential reactor designs and operating conditions to assess relative reactor performances and sensitivities. Once the final design (tube diameter, tube depth,...) and operating conditions (flow rate, pressure,...) have been settled, we have then solved the two-dimensional equations to verify the predictions of the c model. Since axial dispersion can be significant in short laboratory reactors, we intend to extend the c model to include axial dispersion. Similarly, transient two-dimensional calculations are far too slow to be used for real time control, so we intend to devise an accurate time-dependent one-dimensional model. If the Karman-Pohlhausen procedure and approximations used here still suffice, then these extensions can be made by simply adding the X,, T,, X,.z, and Tzz terms from the two-dimensional equations to the c model. Appendix A. Here we derive the c model by substituting the radial profiles (2.21) into the reactor equations (2.6a), (2.6b) and then averaging across the reactor’s cross section (see (2.22)). We first need to expand the reaction rate as

(Ala)

S(J, T)= S(, ’)e A(T-’)+B(7"-)2+’’’,

where

(Alb)

a(J, )= [log S(J, )],

B(J, )=1/2[log S(J, )],

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1100

HAGAN, HERSKOWITZ, AND PIRKLE

around some mean temperature T. To obtain the simplest model, at each z we choose T(z) to be the reaction-averaged temperature defined by

(A2)

S(X, T)=(S(X, T(z, r))). Then the cross-sectionally averaged equations (2.22) immediately reduce to (A3a) Xz S(X, T),

(A3b)

Tr +- T + QS(X, T).

r{ r}

Moreover, with the radial profile (2.21b),

T+- T

(A4)

2(rT)dr=-8o/A.

So we only need to relate c and (T) to the reaction-averaged temperature T(z). Using the expansion (Ala) in (A2), we find that (ea(T-)[1 + B( Z- )2 +. .])= 1. (m5) Substituting T(z, r) from (2.21b) and integrating then yields A(T- L) 4a/r-log (1-a)

(A6)

+ B/a2

{8 +

(1- /2) lg (1- )+ lg2 (1-

Now recall from (2.20) that 0< < (y) 8 8+- (- /2)

(A7)

[El < .00253 for all 0<