Analysis and Design of Fixed Bed Catalytic Reactors - Advances in

1 Analysis and Design of Fixed Bed Catalytic Reactors

Downloaded by UNIV OF CALGARY on November 26, 2012 | http://pubs.acs.org Publication Date: August 1, 1974 | doi: 10.1021/ba-1972-0109.ch001

G. F . F R O M E N T Laboratorium voor Petrochemische Techniek, Rijksuniversiteit, Ghent, Belgium

The models used to describefixedbed catalytic reactors are classified in two broad categories: pseudo-homogeneous and heterogeneous models. In the former the conditions on the catalyst are considered to equal those in the fluid phase; in the latter this restriction is removed. The pseudo-homogene ous category contains the ideal one-dimensional model, the one-dimensional model with effective axial transport, and the two-dimensional models with axial and radial gradients. Particular emphasis is placed on such problems as para metric sensitivity, runaway, and instabilities induced by axial mixing. In the heterogeneous category attention is given to the effect of transport phenomena around and inside the catalyst particle on the behavior of the reactor. Finally, a new, general two-dimensional heterogeneous model is set up and compared with previously discussed models. ' • y h i s brief review of the analysis and design of fixed bed catalytic reactors does not allow us to concentrate on specific cases and proc esses. Instead, an attempt is made to discuss general models and the principles involved i n the design of any type of reactor, no matter what the process is. In Table I the models are grouped i n two broad categories: pseudohomogeneous and heterogeneous. Pseudo-homogeneous models do not account explicitly for the presence of the catalyst, i n contrast to hetero geneous models, which lead to separate conservation equations for fluid and catalyst. Within each category the models are classified according to increasing complexity. T h e basic model, used i n most of the studies until now, is the pseudo-homogeneous one-dimensional model, which only considers transport b y plug flow i n the axial direction ( A . I ) . Some type of mixing in the axial direction may be superposed on the plug flow A

1 In Chemical Reaction Engineering; Bischoff, K.; Advances in Chemistry; American Chemical Society: Washington, DC, 1974.

2

C H E M I C A L REACTION ENGINEERING

Table I.

Classification of Fixed Bed Reactor Models A . Pseudo-Homogeneous (t = t ;c = c ) 8

One-dimensional

A.I Basic, ideal A . I I + axial mixing

8

8

B . I + interfacial gradients B . I I + intraparticle gradients

Two-dimensional A . I l l + radial mixing


B. Heterogeneous (t 5* t ;c c)

8

B . I I I + radial mixing

to account for nonideal flow conditions ( A . I I ) . If radial gradients must be accounted for, the model becomes two-dimensional ( A . I I I ) . The basic heterogeneous model considers only transport by plug flow, but it dis tinguishes between conditions i n the fluid and on the solid (B.I). The next step towards complexity is to take the gradients inside the catalyst into account ( B . I I ) . Finally, the most general models used today—viz, the two-dimensional heterogeneous models—are discussed i n Β. III. Even within this framework the paper does not give a complete bib liographic survey. It focuses on what the author believes are the essential points or on some aspects which have received extensive coverage i n recent years and upon which our viewpoints need clarification or correction. Pseudo-Homogeneous

Models

The Basic One-Dimensional Model. The basic or ideal model assumes that concentration and temperature gradients occur only i n the axial direction. The only transport mechanism operating i n this direction is the over-all flow itself, and this is considered to be of the plug flow type. The conservation equations may be written, for the steady state and a single reaction carried out i n a cylindrical tube. dc u

s

Us9fC ^ P

= (-AH)

=

P B

(1)

ÇBTA

r

A

- 4

(Γ -

T) w

(2)

with initial condition : at ζ = 0, c = c Τ = T Q

0

In most cases the pressure drop i n the reactor is relatively small so that a mean value for the total pressure is used i n the calculations. Pres sure drop correlations for packed beds were set up b y L e v a ( I ) and Brownell (2). They lead to predictions which are i n excellent agreement. The correlations for the heat transfer coefficient, U, show considerable spread (3, 4, 5 ) . Recently, D e Wasch and Froment set up correlations

In Chemical Reaction Engineering; Bischoff, K.; Advances in Chemistry; American Chemical Society: Washington, DC, 1974.


1.

FROMENT

3

Fixed Bed Reactors

for U which are linear with respect to the Reynolds number and which have also a static term (6). Integration of Reactions 1 and 2 is straight forward, either on a digital or an analog computer. Questions which can be answered by such simulation and which are important in catalytic reactor design are: what is the tube length required to reach a given conversion; what w i l l the tube diameter have to be; or the wall tempera ture? A n important problem encountered with exothermic reactions is how to limit the hot spot in the reactor and how to avoid excessive sensi tivity to variations in the parameters? This problem was treated analyti cally by Bilous and Amundson (9) and more empirically, but more directed towards practical appplication, by Barkelew (7). Barkelew's results are represented in Figure 1, which is based on many numerical integrations and which has general validity for single reactions. N/S is the ratio of the rate of heat transfer per unit volume at τ = 1, where τ =

N/S

Figure 1.

Barkelew plot for the parametric sensitivity of an ideal tubular reactor


4


(E/RT )(T — T ), to the rate of heat generation per unit volume at τ = 0 and zero conversion—i.e., at the reactor inlet. The ratio T /S is that of the dimensionless maximum temperature to the adiabatic tem perature rise above the coolant temperature. A set of curves is obtained with S as a parameter. They have an envelope, occurring very close to the knee of an individual curve. Above the contact point with the enve lope, T changes rapidly with N/S but not below. Therefore, Barkelew proposed a criterion according to which the reactor is stable to small fluctuations if its maximum temperature is below the value at the contact point to the envelope. Recently, V a n Welsenaere and Froment analyzed the problem in a different way (8). B y inspecting temperature and partial pressure profiles i n a fixed bed reactor they concluded that extreme parametric sensitivity and runaway may be possible (1) when the hot spot exceeds a certain value and (2) when the temperature profile devel ops inflection points before the maximum. They transposed the peak temperature and the inflection points into the p-T phase plane. The locus of the maximum temperatures, called the "maxima curve" and the locus of the inflection points before the hot spot are shown as p and ( pi ) ι respectively. The symbol ( pi ) represents the locus of the inflection point beyond the hot spot, which is of no further interest i n this analysis. 2

W

w

max


m a x

m

2

625

650

T(°K)

675

700

Figure 2. p-T phase plane showing trajectories, maxima curve, and foci of inflection points according to Van Wel senaere and Froment


1.

FROMENT

5

Fixed Bed Reactors


0.04

600

650

700

T (°K) W

Figure 3. Upper and lower limits, mean and exact critical values for the inlet partial pressure predicted by the Van Welsenaere and Froment cri terion. Filled points are values derived from Barkelew's criterion. T w o criteria were derived from this. T h e first is based on the obser vation that extreme sensitivity is found for trajectories—the p-T relations in the reactor—intersecting the maxima curve beyond its maximum. Therefore, the trajectory going through the maximum of the maxima curve is considered as critical and as the locus of the critical inlet condi tions for ρ and Τ corresponding to a given wall temperature. This is a criterion for runaway based on an intrinsic property of the system—not on an arbitrarily limited temperature increase. The second criterion states that runaway w i l l occur when a trajectory interesects (Pi)i, the locus of inflection points arising before the maximum. Therefore, the critical tra jectory is tangent to the (pt) curve. A more convenient version is based on an approximation for this locus represented b y p in Figure 2. Representation of the trajectories i n the p - T plane requires numeri cal integration, but the critical points involved i n the criteria—the maxis


6


mum of the maxima curve and the point where the critical trajectory is tangent to p are located easily by elementary formulas. T w o simple extrapolations from these points to the reactor inlet lead to upper and lower limits for inlet partial pressures and temperatures above which safe operation is guaranteed. Figure 3 shows some results for a specific reaction. They are compared with those obtained from Barkelew's cri terion (filled dots), which is more complicated to use, however. In Figure 3 p°i,i; p° ,i and p°i, ; P°u,2 are lower and upper limits, based upon the first and second criteria respectively; p° ,\ and p° ,2 are their mean values; p° ,i and p° ,2 are the exact values obtained by numerical back integration from the critical points defined by the first and second criteria. These latter values are given here to show the error introduced by simple extrapolation methods. The mean values p°m,i and P°m,2 agree remarkably well with p ° , i and ρ ° , . The reaction con sidered here has pseudo-first-order kinetics and a heat effect suggested by gas phase hydrocarbon oxidation. F o r a specific set of conditions the first criterion limits the hot spot to 31.6°C, with p = 0.0135 atm and p = 0.0197 arm; the second criterion limits ΔΤ to 29.6°C, while p = 0.0142 atm and p = 0.0195 atm. B y numerical integration of the system of differential equations, what could be called "complete" runaway is ob tained with pa = 0.0183 atm. The above criteria are believed to be of great help in first stages of design since they permit a rapid and accurate selection of operating con ditions before any computer calculations are done. Their application is limited, however, to single reactions. Nothing like this is available for complex reactions which have many parameters. Complex cases w i l l probably always be handled individually. Objections may be raised against this model. First, it can be argued that the flow in a packed bed reactor deviates from the ideal plug flow pattern because of radial variations i n flow velocity and mixing effects. Second, it is an oversimplification to assume that temperature is uniform in a cross section. The first objection led to a development which is discussed in the next section, the second to models discussed later. One-Dimensional Model w i t h A x i a l M i x i n g . Accounting for the velocity profile is practically never done since it immediately complicates the computation seriously. In addition very few data are available to date, and no general correlation could be set up for the velocity profile (10,11, 12, 13). The mixing in an axial direction which is caused by turbulence and the presence of packing is accounted for by superposing an "effec tive" mechanism upon the over-all transport by plug flow. The flux arising from this mechanism is described by a formula analogous to F i c k s law for mass transfer or Fourrier's law for heat transfer. The propor tionality constants are "effective" diffusivities and conductivities. Because 8

u

2

m


cr

m

cr

cr

σ

2

l t l

U t l

h

2

Ut2


1.

FROMENT

7

Fixed Bed Reactors


of the assumptions involved i n their derivation they contain implicitly the effect of the velocity profile. This field has been reviewed and organized by Levenspiel and Bischoff (14). The principal experimental results con cerning the effective diffusivity i n axial direction are shown i n Figure 4 {12,15,16,17,18,19).

^

1

6 V

7 ι

4a 3a 3b

-

.01

10

100

1000

Axial mixing in packed beds—Peclet vs. Reynolds numbers diagram McHenry and Wilhelm (15) Ebach and White (16) Carberry and Bretton (17) Strong and Geankoplis (18) Cairns and Prausnitz (12) Hiby (19) Hiby, without wall effect (19)

Figure 4. Curve 1: Curve 2: Curve 3: Curve 4: Curve 5: Curve 6: Curve 7:

For design purposes P e based on d may be considered to lie be tween 1 and 2. Little information is available on \ . Yagi, K u n i i , and Wakao (20) determined X experimentally, while Bischoff derived it from the analogy between heat and mass transfer i n packed beds (21). The continuity equation for a component A may be written i n the steady state: a

p

ea

ea

^

dc 2

e

"

a

U

s

T

dc ~ dz

"

(3) r

A

p

B

=

and the energy equation: (4) The boundary conditions have given rise to extensive discussion (29, 30, 31,32). Those generally used are

U (CQ S

—

C)

=

—

zD

e

dc dz

for ζ = 0


8


ç> u c (T f

s

p

0

1 ]

-

K e a

(5)

dz


dc dz

for ζ = L

This leads to a two-point boundary value problem requiring trial and error in the integration. For the flow velocities used i n industrial practice the effect of axial dispersion of heat and mass upon conversion is negli gible when the bed depth exceeds about 100 particle diameters (27). Despite this the model has received great attention recently, more par ticularly the adiabatic version. The reason is that the introduction of axial mixing terms into the basic equations leads to an entirely new feature —namely, the possibility of more than one steady-state profile through the reactor (28). Indeed, for a certain range of operating conditions three steady-state profiles are possible with the same feed conditions, as shown i n Figure 5. The outer two steady-state profiles are stable, at least to small pertubations while the middle one is unstable. W h i c h steady state profile w i l l be predicted by steady-state computations depends on the initial estimates of c and Τ involved in the integration of this two-point boundary value problem. Physically this means that the steady state actually experienced depends on the initial profile in the reactor. For all situations where the initial values are different from the feed conditions transient equations must be considered to make sure the correct steady-state profile is pre dicted. To avoid those transient computations when they are unnecessary, it is useful to know a priori if more than one steady-state profile is pos sible. Figure 5 shows that a necessary and sufficient condition for unique ness of the steady-state profile in an adiabatic reactor is that the curve

T(L) Figure 5.

T(L)

One-dimensional tubular reactor with axial mix ing. Outlet vs. inlet temperature.


1. t

in

FROMENT

9

Fixed Bed Reactors

= / [ i ( L ) ] has no hump. Mathematically this means that Equation 6 g - P e '

^

a

+

/ ( T )

= 0

Pe'

= γτ—

where z' = \-


and/(D =

Ρ Β

(Τ

Λ ί

tt

- Γ) e x p ^ l

(6)

-

has no bifurcation point, whatever the length of the reactor. Luss and Amundson (29) to the following conditions:

This led

S \f'(T) - PeV/4] < 0

(7)

up

T

Analysis and Design of Fixed Bed Catalytic Reactors - Advances in

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