The K-L reactor model for circulating fluidized beds

Chemical Engineering Science 55 (2000) 4563}4570

The K-L reactor model for circulating #uidized beds Daizo Kunii , Octave Levenspiel
* 1-25-16 Nakamachi, Meguro-ku, Tokyo 153-0065, Japan
Chemical Engineering Department Oregon State University, 103 Gleeson Hall, Corvallis, OR 97331, USA Received 18 November 1999; received in revised form 15 February 2000; accepted 28 February 2000

Abstract In this paper we present a model for determining the performance behavior of a CFB reactor. This model is not sophisticated and does not require computer calculations, but is realistic and convenient for engineers engaged in the development of new processes. A number of parameters must be evaluated to use this model. This paper shows what they are, and thus suggests where future research of CFB should be. Four examples in the di!erent #ow regimes illustrate the use of this model.  2000 Elsevier Science Ltd. All rights reserved. Keywords: CFB reactors; Conversion; Two-region two-zone model

1. Introduction

2. Vertical distribution of solids

Today the CFB plays the dominant role in FCC and in a few other large-scale industrial catalytic processes. It most likely will play an important role in future processes for converting natural gas produced in remote sites to transportable liquids. Much e!ort has gone into studying its behavior, and many models have been proposed to explain how gas and solid contact and react with each other. The "nal goal of these studies is to predict reactor behavior, as shown in Fig. 1. The models that have been proposed to date are invariably computer-based and require the user to accept the developed programs to make behavior predictions. This paper develops a model which is transparent in that it is based on assumptions which the user can check and modify at will. The reactor behavior can be determined directly, and then its predictions are compared with the ideal of straight plug #ow of gas through the CFB. We "rst deal with the vertical distribution of solids in the reactor, then the radial distribution. We follow this by developing the performance equations for gas reactions on catalytic solids, and we end up with some illustrative examples to show how the predictions of this model match the experiment.

For a super"cial gas velocity u , and solid mass velo city G , a reasonable representation of observation shows Q that the solids are distributed in two regions in the vessel, a constant solid fraction f in the lower dense region of B height H , and an upper lean region of height Hl in B which the solid fraction fl falls exponentially with height from f towards the saturation carrying capacity of the B gas f H. Fig. 2 shows the symbols used to describe the bed. For a "xed gas #ow rate Fig. 3 shows how the height of the two regions depend on the solid #ow rate. As mentioned, the fraction of solids in the lean region decreases exponentially to the saturation carrying capacity of the gas f H, thus at any point zl in the lean region we have

* Corresponding author. Tel.: 1-541-753-9248; fax: 1-541-737-4600. E-mail address: [email protected] (O. Levenspiel).

fl "f H#( f !f H)exp[!az ], (1) B D where a is called the decay constant for the solids. At the top of the vessel Hl the fraction of solids becomes f , and zl becomes Hl , in which case Eq. (1)  becomes, when rearranged, 1 f !f H Hl " ln B . (2) a f !f H  Also the mean fraction of solids in the lean region is 1 fM l " Hl

0009-2509/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 0 7 3 - 7



&l



fl dz.

(3)

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Fig. 1. The CFB reactor problem. Fig. 4. The shape of the vessel exit in#uences the solid fraction at the top of the vessel.

reported correlations of f or u with the material  Q balance expression

(4)

G "o (u !u ) f , (5) Q Q  Q  where u is the slip velocity of the solids. Unfortunately, Q the reported values di!er widely, so which should we trust? Finally, if the vessel has a smooth exit we may be able to approximate the slip velocity by the terminal velocity of the particles u , in which case R G "o (u !u ) f . (6) Q Q  R  Fig. 4a shows a sharp ine$cient exit which causes a signi"cant percolation of solids down the vessel. Fig. 4c shows a smooth exit. With a large through#ow rate of solids

To predict the reactor behavior of the CFB we need to know the fraction of solids at the top of the vessel f ,  see Fig. 4. It is best to have a direct measurement of f . If  this information is not available one may have to use

Hl (calculated)(H , (7) R In this situation the vessel will have a lower dense region and an upper lean region as shown by the curves in Figs. 3a}c. These curves represent the fast yuidization regime.

Fig. 2. Symbols used to describe the geometry of CFBs.

Combining with Eq. (1) gives on integration f !f  fM l "f H# B aHl

Fig. 3. Behavior of a CFB for various solid #ow rates.

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3. Ideal plug 6ow of gas Consider a "rst-order catalytic reaction 1 dN  "kC APR, !r"!   < dt Q 1 dN  "kC . or !r "!   = dt

(9)

In general, for plug #ow of gas through a catalyst bed of height H having a constant solid fraction f, the performance equation is C kf H  " . (10) C u CV  But in a bed where the solid fraction changes with height z, but is uniform across the cross section, the performance equation is

ln

Fig. 5. Measured solid distributions of a CFB, for runs A and G of Schoenfelder et al. (1996).

When the through#ow rate of solids is very small then the vessel will not have a lower dense region. The whole vessel becomes lean, with maybe a shallow dense solid acceleration zone by the solid inlet. Fig. 3d and e illustrate this behavior, which occurs when Hl (calculated)'H . R

(8)

The dense solid acceleration zone is di$cult to represent. In any case, we have here what is called the pneumatic transport regime. All nine runs reported by Schoenfelder, Kruse and Werther (1996) are in this low solid #ow regime, and Fig. 5 shows two of their results. Now that we have developed the equations for the vertical distribution of solids in a vessel, we need numerical values for the various #ow constants: u Q u R

the slip velocity of particles the terminal velocity of a solid particle in the gas stream f the mean fraction of solids in the dense region of B a CFB f the fraction of solids in the wall zone of a CFB U f H the saturation carrying capacity of solids in the gas stream a the solid decay constant in the lean region of a CFB Values for some of these constants were collected and summarized from the literature by Kunii and Levenspiel (1991, 1997). This will not be repeated here. We now develop the equations in turn for plug #ow of gas in the reactor, for the FF regime, then for the PT regime.



C k &  " f (z) dz . (11) C u CV   We use these equations, obtained from Levenspiel (1999), in deriving our equations. The fast #uidized reactor has a lower dense region and an upper lean region, as shown in the curves of Figs. 3a}c. For the dense region ln

C kf H ln  " B B u C B  and for the lean region

(12)





C kfM H k f !f H B " l l " f H# B Hl (13) C u aHl u CV   So for the whole vessel either add Eqs. (12) and (13), or else "nd fM overall experimentally and use the following equation: ln

ln

C kfM H  " R C u CV 

(14)

4. The CFB reactor model - FF regime Observations and measurements show that the lower region has a lean core which becomes progressively richer in solids as one approaches the wall. Let us approximate this by two distinct zones, a central core zone with solid fraction f , and a wall zone rich in solids f . BA BU Also let the volume fraction of core be d (m core/m B reactor). The upper lean region exhibits clumps of rising and falling solids plus solids at the wall. Let us again approximate this by two distinct zones: a core zone f , and a wall JA zone f which includes all rising and falling clumps of JU solids. Again let d be the volume fraction of this core. Of A course, the wall zone gets thinner on rising up the vessel.

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Fig. 7. Mechanism of reaction in the dense region of a CFB.

This represents a series/parallel process. So the performance equation for the dense region is





Fig. 6. The distribution of solids in the CFB model.

1 C H B . ln  " f Hd k# B C 1/d K #1/f (1!d )k u B B AU U B  (21)

In addition, let us make the following two simplifying assumptions:

For the lean region we have a similar expression except that d is replaced by dM l , which is given by Eq. (19), B C 1 H J ln B " f HdM l k# C 1/dM l K #1/f (1!dM l )k u CV AU U  (22)

f "f "f BU JU U and

(15)

f "f "f H. (16) BA JA If actual data or if better assumptions are available, certainly use them in the place of Eqs. (15) and/or (16). This represents our model, and Fig. 6 shows the situation visualized. A material balance at any level of the lower region gives f !f B f "f Hd #f (1!d ) or d " U B B U B B f !f H U and at any level of the upper region gives fl (z)"f Hdl (z)#f (1!dl (z)) U For the upper region as a whole

(17)

(18)

f !fM l or dM l " U . (19) f !f H U In addition, assume that gas only rises in the core zones, but is stagnant in the wall zone. Finally, let K be AU the gas interchange coe$cient between core and wall zones, de"ned as fM l "f HdM l #f (1!dM l ) U

m gas going from c to w, or w to c/s K " . AU m of reactor

(20)





For the reactor as a whole C C C  "ln  #ln B , (23) C C C CV B CV where the two terms on the RHS are given above. ln

5. For lean solids reactors + pneumatic transport For a low solid circulation rate, Eq. (8) applies and the whole reactor is in the pneumatic transport regime, as shown in Figs. 3d}e, or Fig. 8. If no solid fraction data is available estimate f from  Eqs. (5) or (6), then fM l from (4), and dM l from Eq. (19). Finally, apply Eq. (22) to "nd the conversion of gas. If some solid fraction data is available, such as f or fM l ,  certainly use it in preference to the unreliable u or f . Q  This analysis assumes that there is no solid accumulation near the feed injection point where solids are accelerating up the bed. The experimental results of Schoenfelder et al. (1996) of Fig. 5 show that this is not a good assumption for their vessel. Probably this is due to the design of their solid distributor.

4.1. Conversion equations

6. Illustrative examples

In the lower dense region the rising gas reacts with solids in the core. It also transfers to the wall zone where it also reacts. This is shown in Fig. 7.

The following examples, based on reactor experiments reported by Schoenfelder et al. (1996), show how to use this model for the various #ow regimes (see Fig. 9). But

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Table 1 Outline of the conditions used in the four examples Example Solid #ow rate, Representative Solid distribution data number G curves of Fig. 3 Q 1

17 kg/m s

e

2

24

between c and d

3

100

b

4

300

a

Fig. 8. Solid distribution in a lean solids CFB.

Represents run A of Schoenfelder et al. (1996); the data is shown in Fig. 5 Represents run G of Schoenfelder et al. (1996), the data is shown in Fig. 5 No data taken by Schoenfelder et al. (1996) at these G Q values

Example 1. G "17 kg/m s - run A of Schoenfelder et Q al. (1996) For G "17 kg/m s and u "3.1 m/s, Schoenfelder Q  et al. (1996) measured the vertical distribution of solids shown in Fig. 5a, and a conversion of gas of X "16%.  They had to choose k"2.72 m/m cat s to get these results. Let us see what our model gives. Since G is very small the bed is lean. So use the solid Q distribution data directly. The additional data needed to use our model is f "0.023 (see Fig. 5a)  fM "0.027 (see Fig. 5a) f "0.4 (*). U Eq. (19) then gives Fig. 9. Experimental reactor setup.

0.4!0.027 dM l " "0.9564, 0.4!0.01 "rst, Table 1 outlines the regimes studied. From Schoenfelder et al. (1996) we have d "50;10\ m, o " N Q 1420 kg/m, 203C, 100 kPa, u "1.9;10\ m gas/m KD bed s and K "0.3 s\. Reaction is APR, "rst order. AU To "t their measured conversion data, Schoenfelder et al. (1996) had to pick di!erent k values for each of their nine runs. We picked one value for all four of our examples k"ko "1.5 m/m cat s. Q The calculated values and their sources: u "0.0990 m/s (from Kunii & Levenspiel, 1997, p. 2478) R f H"0.01 (from Kunii & Levenspiel, 1997, p. 2475)

Eq. (22) gives ln

C



"0.1645,

Therefore C CV "0.8483 or X "15.13%.  C  For plug yow, Eq. (14) gives

f "0.2 (from Kunii & Levenspiel, 1997, p. 2474) B a"0.4 m\(*).



1 14  " 0.01(0.9564)(1.5)# C 1 3.1 1 CV # 0.9564(0.3) 0.4(1!0.9564)(1.5)

C fM kH 0.027(1.5)14 R" ln CV " "0.1829. C u 3.1  

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Therefore

Next, the height of the lean region is given by Eq. (2)

C  "0.8328, X "16.72%.   C CV

1 0.2!0.01 Hl " ln "6.33 m, 0.4 0.0251!0.01

Example 2. G "24 kg/m s - run G of Schoenfelder et Q al. (1996) The data of Fig. 6b show that the bed is barely completely in the lean regime. Also u "3.0 m/m bed s(given by Schoenfelder et al., 1996),  f "0.035 (from Fig. 5b)  fM "0.07 (from Fig. 5b), f "0.40 (*) U Then Eq. (19) gives

Therefore H "14!6.33"7.67 m. B This result shows that curve b of Fig. 3 represents the reactor. Then Eq. (4) gives 0.2!0.0251 "0.0791. fM l "0.01# 0.4(6.33) Eq. (19) gives 0.45!0.0791 dM l " "0.8430 0.45!0.01 and Eq. (17) gives

0.4!0.07 dM " "0.8462. 0.4!0.01

0.45!0.2 d " "0.5682. B 0.45!0.01

Eq. (22) gives the reactor behavior as

We are now ready to use the reactor performance equations. For the dense region, Eq. (21) gives

ln





14 C 1  " 0.01(0.8462)(1.5)# 1 3.0 C 1 CV # 0.8462(0.3) 0.4(1!0.8462)(1.5) "0.3751.



Therefore C CV "0.6872 X "31%.  C  For plug yow, Eq. (14) gives C 0.07(1.5)14 " "0.4900. 3.0 C CV Therefore

Example 3. G "100 kg/m s. Q This solid #ow rate is higher than any used by Schoenfelder et al. (1996). Without solid distribution data means that we must estimate more constants so the data used is as follows: u "3.0 m/m bed s (selected condition), f "0.2(*),  B u Q f "0.45 (*), "2(*). U u R Then Eq. (5) gives G 100 Q f " " "0.0251.  o (u !u ) 1420(3.0!0.099;2) Q  Q

(i)

"0.2966.

Therefore C B "0.7433. C  For the lean region, Eq. (22) gives

ln

C CV "0.6126 C  or X "39%.  



C 1 7.67 ln " 0.01(0.5682)1.5# C 1 3.0 1 B # (0.5682)(0.3) 0.45(1!0.5682)1.5

ln





6.33 C 1 B " 0.01(0.8430)1.5# 1 3.0 C 1 CV # 0.8430(0.3) 0.45(1!0.8430)1.5 (ii)

"0.1843.

Therefore C CV "0.8317. C B Finally Eq. (23) gives C C C CV " CV ) B "0.8317(0.7433)"0.6182. C C C  B  Therefore X "38%.  For plug yow we need fM . Since this is not available put K "R. Then Eq. (i) becomes AU C 1 7.67 ln  " 0.0085# "0.767. C 3.4309 3.0 B





D. Kunii, O. Levenspiel / Chemical Engineering Science 55 (2000) 4563}4570

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Therefore

Therefore

C B "0.4645. C  Also, Eq. (ii) becomes

C CV "0.5664 or X "43%.  C  For plug yow, fM "f "0.2, so Eq. (14) gives B C 0.2(1.5)14 ln CV " "1.4, C 3.0  Therefore





1 C 6.33 B " 0.0126# "0.251. C 9.4362 3.0 CV Therefore

ln

C CV "0.2466 or X "75%.   C 

C CV "0.7786. C B Combining gives, for plug #ow

6.1. Final comments

C C C CV " CV ) B "0.7786(0.4645)"0.3618. C C C  B  Therefore

1.

X "64%.  

2.

Example 4. G"300 kg/m s This G value represents a very high solid through#ow Q rate. Additional data needed are 3.

u "3.0 (selected condition),  f "0.2 (*), B f "0.5 (*), U u Q "2 (*). u R First of all Eq. (5) gives

Eq. (2) gives 1 0.2!0.01 Hl " ln "3.03 m. 0.2 0.1888!0.01 This is practically a dense bed throughout, as sketched in Fig. 3a. So assume that this is a total dense bed. Then Eq. (17) gives 0.5!0.2 d " "0.6122. B 0.5!0.01

FF G H k k

decay constants for solids, m\ concentration of reactant A, mol/m circulating #uidized bed volume fraction of solids, m solids/m vessel saturation carrying capacity of gas, m solids/m vessel fast #uidization regime mass velocity, kg/m s height, m "rst-order reaction rate constant, m/kg cat s "rst-order reaction rate constant, m/m cat s

Table 2 Performance from the model, from plug #ow and from mixed #ow

Then the reactor performance expression of Eq. (21) gives



14 C 1 ln CV" 0.01(0.6122)1.5# 3 1 C 1  # 0.6122(0.3) 0.5(1!0.6122)1.5 "0.5684.

a, f H, f , f , K and k B U AU These four examples illustrate reactors operating in various #ow regimes Example 1: lean pneumatic #ow. Example 2: lean FF regime alone. Example 3: both lean and dense FF regime. Example 4: dense FF regime alone. Table 2 and Fig. 10 compare the performance of these reactors with that of plug #ow and of mixed #ow. These clearly show that the CFB reactor behaves worse than both the plug- and mixed-#ow reactors.

Notation a C or C  CFB f fH

300 f " "0.0754.  1420(3.0!0.099;2)



To use this model we need information or reasonable assumptions about

kfM H R "y u  0.1829 0.4823 1.0174 1.4

X (%)  

X (%)  

X (%)   "y/(1#y)

15.17 31.0 38.0 43.3

16.72 38.8 64.0 75.3

15.46 32.5 50.4 58.3

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D. Kunii, O. Levenspiel / Chemical Engineering Science 55 (2000) 4563}4570

u Q

Fig. 10. Comparison of the conversion from a CFB reactor with that from plug #ow and from mixed #ow.

u R d o *

slip velocity of a particle with respect to the rising gas, m/s terminal velocity of a falling particle, m/s volume fraction of core, m core/m bed density, kg/m indicates that the value is arbitrarily chosen

Subscripts c d ex l s t top w z

core zone lower dense region leaving the vessel upper lean region solid total height at the top of the vessel wall zone height

References

K AU PT r  u 

interchange coe$cient between c and w, m transferred from c to w/m vessel s, or vice versa pneumatic transport regime reaction rate, m converted/m cat s super"cial gas velocity, m gas/m bed s

Schoenfelder, H., Kruse, M., & Werther, J. (1996). Two-dimensional model for circulating #uidized bed reactors. A.I.Ch.E. Journal, 42, 1875}1888. Kunii, D., & Levenspiel, O. (1991). Fluidization engineering, (2nd edn). Boston, MA, USA: Butterworth-Heinemann. Kunii, D., & Levenspiel, O. (1997). Circulating #uidized beds. Chemical Engineering Science, 52, 2471}2488. Levenspiel, O. (1999). Chemical reaction engineering (3rd Ed.). (p. 396) New York, NY, USA: Wiley.