Optimization of a Non-Isothermal, Non-Adiabatic Fixed-Bed Catalytic

28 Optimization of a Non-Isothermal, NonAdiabatic Fixed-Bed Catalytic Reactor

Downloaded by UNIV OF MINNESOTA on August 14, 2013 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1974-0133.ch028

Model THOMAS GORDON SMITH and JAMES J. CARBERRY Department of Chemical Engineering, University of Notre Dame, Notre Dame, Ind. 46556

A mathematical model of a non-isothermal, non-adiahatic fixed bed catalytic reactor was optimized as a function of six variables using the Hookes-Jeeves algorithm. Two objective functions were used: yield and productivity. Productivity is more useful for preliminary design. High productivities can be obtained with little sacrifice in yield by using small catalyst particles, large tubes, low feed concentrations, and high gas velocities. Operating conditions for high productivity exhibit relatively low parametric sensitivity. Although long reactors are required for the highest productivity, optimization has established a goal against which the results of new strategies for improving reactor performance can be compared. A more active catalyst limits maximum productivity because of increased parametric sensitivity. Parametrically sensitive conditions can be discovered and thus avoided by this approach.

A lthough various aspects of fixed bed catalytic reactors have been studied JLM. during the past decade, specific guidelines are needed to design such reactors, as well as base conditions against which proposals for improving their performance can be measured. This information is particularly desirable for non-isothermal, non-adiabatic fixed bed catalytic reactors because of the many interdependent parameters which determine their local and overall behavior. This, together with the nonlinear nature of the system, renders intuitive design approaches suspect. This work discusses results which may be useful to both the plant engineer and the design specialist in this regard. The oxidation of naphthalene to phthalic anhydride on a vanadium pentoxide ( V 0 ) catalyst was analyzed to determine the optimum combination of feed concentration, temperature, and flow rate, coolant temperature, tube diameter, and catalyst particle diameter. These negotiable parameters, together with the reaction kinetics, fix reactor performance. Industrially, it is 2

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362

In Chemical Reaction Engineering—II; Hulburt, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1975.

28.

SMITH AND CARBERRY

Non-Isothermal, Non-Adiabatic Reactor

363

necessary to achieve 99% conversion of the naphthalene which in turn specifies reactor length. Although the production of phthalic anhydride is a well estab lished commercial process, detailed operating conditions are not available in the literature. However, the following general articles are of interest. Froment (I) investigated the stability of fixed bed catalytic reactors for phthalic anhydride production from o-xylene on a V 0 catalyst, using the following parameters:


2

Γ

0

D

t

= 357°C, c = Q

.0092, Re* =

5

120, T ' = c

357°C,

= 2.5 cm, d = .3 cm. O

Computations using a steady-state, two-dimensional model showed that the maximum phthalic anhydride concentration occurs at a reactor length of 3m for these conditions. However, this length did not achieve the desired o-xylene conversion. Furthermore, an increase in, say, inlet temperature, in an attempt to increase conversion, caused runaway—i.e., complete reaction of the phthalic anhydride and large thermal excursions. Froment concludes that no one would risk running the reactor at the conditions chosen and asserts that the reactor performance could probably be improved by reducing the magnitude of the hot spot using some form of catalyst dilution (see, e.g., Ref. 2). The question arises as to the basis on which the conditions were chosen and whether a different choice would lead to more desirable results. Along similar lines Carberry and White (3), using kinetic constants from a fluidized bed reactor, simulated the production of phthalic anhydride from naphthalene over V 0 . They used as a base case T ' = 3 2 0 ° C , c = .005, Re* = 75, T ' = 3 1 0 ° C , D = 5 cm, and d = 0.5 cm. The reactor length for this and other nearby conditions varied between 100 and 1000 cm. Their twodimensional model was similar to Froment's but with the important inclusion of terms to account for inter-intraphase mass diffusion and interphase heat diffusion. Also, most of the calculations were based on 99% naphthalene conversion. While Froment looked at severe parametric sensitivity leading to reactor instability, Carberry and White illustrated milder forms of parametric sensitivity of conversion and yield with respect to Pe , Pe , T ' , c , and activa tion energy. They also demonstrated that mass transfer limitations within the catalyst seriously limited the expected yield of phthalic anhydride for the op erating conditions and kinetics chosen. As with Froment, these conditions were chosen arbitrarily. However, regardless of the conditions, it is unlikely that absolute results can ever be attained, and design techniques must thus rely on relative simulations. 2

c

5

0

t

0

p

h

m

c

0

Obviously a more detailed study is needed to determine if higher con version and yield and decreased sensitivity can be obtained, and how it can be couched in terms of an approach useful to the design engineer. The results of such a study follow. Kinetics Our kinetics are based on the efforts of D'Alessandro and Farkas (4) and Drott (5). The former obtained experimental data for the naphthalene oxida tion in a fixed bed reactor over pure V 0 cylindrical pellets. Using their data and a computerized direct search algorithm, Drott "backed out" the five activation energies and the five frequency factors associated with the following 2

5


364

CHEMICAL REACTION ENGINEERING

II

scheme naphthequinone

Ni

k\

naphthalene /c ^ 3

> maleic anhydride

C0 ,H 0 2

2

Since fc is relatively small and k >> 4

this can be reduced to

k

5

l9

k\ naphthalene Downloaded by UNIV OF MINNESOTA on August 14, 2013 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1974-0133.ch028

ki

> phthalic anhydride

Jb

k

2

> phthalic anhydride 3

• maleic anhydride

C0 ,H 0 2

2

with / *t = exp [22Λ k

2

-

28000N

( = exp ( 20.35 -

30000\ )

{

23000\

kz = exp I 15.74 —

RT' ) k

These kinetics result in inlet operating temperatures of 4 0 0 ° - 4 5 0 ° C which are those used industrially. Reactor Model The mathematical model is essentially the same as that used by Carberry and White with the following changes: (1) The Peclet number for heat is computed as a function of reactor conditions using the Beek (6) modification of the Argo-Smith equation. (2) Since appreciable axial temperature gradients exist, the gas density is calculated as a function of length. Any variables or properties (e.g., u, Sc, Pe , K , h, etc.) which depend on density are similarly adjusted. For reference, the equations solved on the digital computer are listed below: h

e

dC 1_ / c W dz ~ Pe \dr* m

1 dC\ _ d (ki + kz)C r dr / u pV

+

dz ~ Pe \ar h

2

r dr )

+

+

W

C uT'

9

p

( 5 )

0

Equations 4 and 5 are dimensionless with respect to d , c , and T ' . The point yield of the intermediate, phthalic anhydride, is calculated as p

dC dC

p&

_ ~

πΐιφ tanh φ / C β \ ηι φ tanh φ, \ C + β - 1/ " 2

7

2

2

p a

τ

ι

7

0

/ W

0

β \ - 1/

(

j

where dfC represents the amount of naphthalene converted to phthalic anhy dride. The following boundary conditions were used: at ζ = 0: Τ - 1, C = 1, C dT

p a

= 0

dC


(7)

28.

SMITH AND CARBERRY

Non-Isothermal, Non-Adiabatic Reactor dC ^ =0

at r « D /2d : t

365

p

where T is not a function of length. At the surface of the particle w

i?' (fci + *s) C = (ki + k )C = fc a (C - C.) 8

v

g

3

(10) (11)

# = ha (T' - T) B


In addition, the following equations were needed:

(12)

l-^+ïg^+tfà^ -h w =

or

= 0.2 0 1

cr ι

40

ι

80

ι

ι

ι

ι

120

160

200

240

ο

Reactor

Figure 2.

I 280

Length at

99%

Ι

320

Re* T' c

dr,

I

440

ι

480

t

1

520

560

600

Optimization using yield as the objective function *

Co

I

400

Conversion, cm

Initial Operating Conditions 440

I 360

.005

120 440 2.5 .25

Operating Conditions at h = 400 cm σ

σ 440

.01 120 440 2.5 .25

420

.004 128 420 1.5 .25

460

.0082 130 420 1. .35



28.

SMITH AND CARBERRY


369

concerned. By trial and error a set of starting conditions must be found which will result in 99% conversion (yield or productivity need not be high) in a length considerably less than the maximum tolerable during optimization. For Figure 1, this maximum length was chosen as 3000 cm; for Figure 2, it was 600 cm. The choice of maximum length is a function of the length at which the optimum is reached or, as is usually the case, the available computing time. Recall that a single simulation of a 600-cm reactor took about 1 minute of computing time. A typical optimization to a maximum length of 600 cm requires on the order of 40 separate simulations. As shown in Figures 1 and 2, the starting operating conditions for the optimization have a considerable effect on the subsequent pattern leading to the optimum and the number of simulations needed to arrive there. Ideally, the optimization algorithm would operate on all starting conditions in such a way that the final optimum set of operating conditions would be identical. If yield is the objective function, the results of all optimization runs, regardless of starting conditions, arrived at an apparent plateau of between 0.85 and 0.88 at 600 cm, but the final "optimum" operating conditions were not the same. If a sufficiently long reactor were used, all optimization patterns would eventually result in an optimum yield approaching one and similar operating conditions (an isothermal reactor). The same reasoning holds when productivity is the objective function. However, a much greater length is now needed to even approach the (tilted) plateau. Reactor Length at 99% Conversion, cm 0

200

40

600

80

IOOO

1400

1800

2200

2600

120 160 200 240 280 320 360 400 440 480 520 560 600 Reactor Length at 9 9 % Conversion, cm

Figure 4.

Variation of productivity and yield during an optimization. Same conditions as Figures 1 and 2.

Figure 4 shows the progress of yield when productivity is the objective function and vice-versa. When productivity is the objective function, the yield eventually increases to an acceptable value, but when yield is optimized, productivity drops as optimization proceeds (other results show that the productivity generally varies in a random manner). Also, the conditions which favor the highest yield invariably result in low productivity. Thus, if a single objective function had to be used, productivity would be better.


370


g 0.91 »0

1 100

1 200

1 300


Reactor

T 200

1 400

L__ 600

500

I 600

Length, cm

i

l 300

l 400

Reactor

I 000

1 500

L e n g t h , cm

r

0.250 -

0.125

-

0

100

200

300

400

500

600

Reactor Length, cm

Figure 5. Base conditions: TO = 468°C, c = 0.0054 mole fraction, Re* = 460, T = Tw = 462°C, D, = 2.85 cm, dp = 0.15 cm. Top: gas and catalyst tem perature profiles; middle: conversion and yield profiles; bottom: effectiveness factor profiles. 0

c


Π


28.

SMITH AND CARBERRY


371

Analysis of Figures 1 and 4 shows that high productivities can be obtained with only a modest sacrifice in yield; also long reactors are needed to approach the optimum if one insists on 99% conversion. If this constraint is relaxed, the reactor length at which the plateau is reached decreases dramatically. For example, if 98% conversion were tolerable, a reactor 25% shorter would be sufficient and would simultaneously result in a slightly higher yield. The precise conditions the design engineer would choose would be a function primarily of feed cost and pressure drop and would likely represent a compromise between optimum productivity and optimum yield. Assume that a 600-cm reactor is designed for maximum productivity and 99% conversion. For the upper curve in Figure 1, note the operating conditions at 600 cm. With these as inlet conditions to the reactor, Figure 5 shows the progress of some of the dependent variables with length. In essence, the optimization has manipulated the six independent variables so that the maximum temperature is not so high that further reaction of the phthalic anhydride can occur to a great extent, yet it is high enough that the desired conversion can be achieved in the given length. It seems evident that the model should take into account both catalytic effectiveness and radial influences even near the optimum. The calculated particle diameter is quite small (see caption, Figure 1). This prevents the catalyst from becoming totally ineffective around the hot spot (where most of the reaction is occurring). The optimization also favors high Reynolds numbers which correspondingly reduce the magnitude of the radial temperature gradient and permit larger tube diameters—a desirable feature. High Reynolds numbers also maintain the catalyst temperature near the fluid temperature

Figure 6. Effect of 10° increase in coolant temperature, T ' i r , on reactor temperature, conversion, and yield


372


II


as shown in Figure 5. Some industrial processes operate with low residence times which probably indicates high gas velocities and Reynolds numbers. Thus, it may be profitable to tolerate much higher pressure drops than are presently considered acceptable. Finally, the optimization recommends a low feed concentration. We found, in contrast to Froment but in agreement with Drott, that lower feed concentrations lead to operating conditions which exhibit less parametric sensitivity.

0

100

200

300

400

500

600

Reactor Length, cm

ο οa.

Reactor Length, cm

Figure 7. Effect of 10% increase in inlet feed con centration, C o , on reactor temperature, conversion, and yield Some of these trends, such as those involving temperature, Reynolds number, or particle diameter may appear obvious on an individual basis (this is in fact how the various starting operating conditions for the optimization were chosen), but selecting the optimum set of six independent variables from the many possible combinations is another matter, particularly since the results change considerably (not necessarily in an unstable manner) for relatively small changes in the parameters in certain regions of the parameter space. Furthermore, a given parameter change may either increase or decrease the objective function depending on the state of the other five parameters, as illustrated in Figure 3 for the concentration. Thus, an unsystematic search, as might occur from random observations of plant data, could prove frustrating in locating the optimum operating conditions. Most important, Figures 1 and 2 show that the choice of the objective function itself requires care. For example, one could reasonably look only at yield; in that case the highest yields are obtained with smaller tube diameters together with larger particle diameters and lower inlet temperatures and Reynolds numbers. However, Figures 1 and 4 show that much higher productivities, with only a small sacrifice in yield, In Chemical Reaction Engineering—II; Hulburt, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1975.


28.

SMITH AND CARBERRY


373

can be secured by using larger tube diameters and smaller particles, in direct contrast to the trend indicated with yield as the objective function. Immunity to parametric sensitivity is of course not guaranteed for the particular "optimum" conditions obtained. However, as Froment notes, this may be as critical a factor as any in selecting an operating strategy. Figures 6 and 7 show the sensitivity of the results of Figure 5 to a 10° perturbation in coolant temperature, and a 10% increase in naphthalene concentration, respectively, these representing the two most crucial operating conditions. The resulting temperature profiles are impressively stable. Note also the yield deterioration of approximately 10% which accompanies the enlarged hot spot. During some optimizations, with the kinetics we have used, we have noticed that there are combinations of the operating parameters which result in sensitive behavior. Higher feed concentration is one. Large catalyst particle diameters also exhibit increased parametric sensitivity. Thus, as a side benefit, the algorithm seems to lead to conditions which are not only in some sense optimum but also more stable. Indeed, this might indicate that parametric sensitivity, as it relates to design and simulation, is not as major a problem as suspected. The effect of dealing with a more active catalyst particle—i.e., a particle containing more actual catalytic material—can be studied by carrying out the simulation with pre-exponential factors increased by a factor of 5. Optimization with productivity as the objective function showed that the more vigorous kinetics enables the reactor to operate at a lower temperature for greater yield, in agreement with the results of Chomitz (9) and Drott (5). In fact, yields >90 mole % were possible with these kinetics. This advantage is secured at the price of greater parametric sensitivity which in turn reduces maximum productivity by restricting inlet concentration and tube diameter. In addition, reactor control would be more difficult. Regardless of the kinetics, one might inquire as to the nature of the solutions at lengths preceding those of the plateau. Actually, in Figures 1 and 2, the points prior to the plateau are not necessarily optimal, and there seems to be no straightforward way of obtaining the optimum for these lengths. This is because if one proceeds from given starting operating conditions and perturbs each independent variable in turn, those successful perturbations invariably result in a reactor which requires additional length to achieve the desired 99% conversion. Since optimization involves changing the character of the hot spot, the above observation makes sense. However, as a result, there is no obvious method, short of superimposing an optimization of the starting operating conditions themselves, to ensure an absolute optimum for shorter reactors. The variety of starting operating conditions tried, although speculative, were guided by the parameter trends and ensure that the magnitude of the plateau itself is correct. Conclusions The major results of the optimization are as follows: (a) Productivity is the most useful objective function for preliminary design work. (b) Very high productivities can be obtained with little sacrifice in yield by using small catalyst particles, large tubes, low feed concentrations, and high Reynolds numbers (i.e., high gas velocities). In contrast, optimization based on yield alone calls for large catalyst particles, small tubes, and lower temperatures and gas velocities and results in low productivities.



374


II

(c) The operating conditions for high productivity exhibit relatively low parametric sensitivity. (d) Although long reactors are necessary to reach the highest produc tivity (not the case with yield), the optimization has established a goal against which the results of new strategies to improve reactor performance can be compared. Also a slight relaxation in the 99% conversion specification results in much shorter reactors. (e) The design engineer, in light of previous plant experience, can sug gest additional constraints (e.g., pressure drop limitations) which will reduce the size and limits of the search space. He can also construct a more mean ingful objective function in terms of total capital and operating costs rather than productivity per tube. This would settle the question of whether the higher pumping costs resulting from the small catalyst particles and high gas velocities are outweighed by the associated large increase in productivity. (f) A more active catalyst limits maximum productivity because of in creased parametric sensitivity. (g) The question of parametric sensitivity as it relates to invalidating computer oriented design and simulation now appears academic. If a certain combination of operating conditions renders the reactor parametrically sensi tive, then, as shown by the optimization, said conditions will also result in productivities and yields which are far from optimum. It is therefore suffi cient that a simulation can alert one to the existence of parametrically sensitive conditions so they can be avoided. Nomenclature a Bi c C C d D D D e E h /i AH ι

external surface-to-volume ratio of catalyst particle Biot number for mass transfer, k L/D naphthalene concentration in air, mole fraction concentration, dimensionless with respect to c heat capacity of gas, cal/gram-mole °C catalyst particle diameter, cm inside tube diameter, cm effective mass diffusivity in radial direction, cm /sec molecular diffusivity, cm /sec emissivity activation energy of ith reaction, cal/gram-mole interphase heat transfer coefficient, cal/sec cm °C wall heat transfer coefficient, cal/sec cm °C heat released by i reaction, cal/gram-mole

H

total rate of heat release, Σ Δ Η ^ , cal/cm

ki k k k K L

reaction rate constant of ith reaction, sec" frequency factor of ith reaction, sec" interphase mass transfer coefficient, cm/sec thermal conductivity of solid, cal/sec cm °C thermal conductivity of gas, cal/sec cm °C 1/a

O

O

t

e r

i

w

s

Q

2

2

2

2

t h

4

sec

1

1

0i e

sa s

Pe Pe Pr r R Re

3

h m

Peclet number for effective heat transfer in radial direction, d u*/λ Peclet number for effective mass transfer in radial direction, a u/D Prandtl number, ο μ/Κ radial coordinate, dimensionless with respect to d ideal gas law constant, 1.987 cal/gram-mole °K Reynolds number, d^u/v, at reactor inlet O

O

Ό

ε

O


eT

28.

Re* Ri Sc Τ Τ T" u w* ζ k

SMITH AND CARBERRY


Reynolds number based on superficial velocity, d u*/v, rate of ith reaction, gram-mole/cm sec Schmidt number, μ/pD temperature, dimensionless with respect to T' temperature, °C temperature, °K interstitial gas velocity, cm/sec superficial velocity, EU, cm/sec axial coordinate, dimensionless with respect to d O

375

at reactor inlet

3

0

O


Greek Letters β γ Δ ε η

ratio of rate constants, k /k intraphase diffusivity ratio, D /D step change void fraction of catalyst bed catalytic effectiveness factor, tanh

Optimization of a Non-Isothermal, Non-Adiabatic Fixed-Bed Catalytic

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