Pergamon Chmtiol Engineering Srirnce. Vol. 49. No. 13. pp. 2035-2045, 1994 Copyright 0 1994 Elsetier Science Ltd

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OOOS2509(94)EOO30-T

A COMPARISON OF TWO-PHASE AND WELL-MIXED MODELS FOR FLUIDIZED-BED POLYETHYLENE

REACTORS

K. B. MCAULEY, J. P. TALBOT’ and T. J. HARRIS Department of Chemical Engineering, Queen’s University, Kingston, Ontario, Canada K7L 3N6

(First received 12 duly 1993; accepted in reuised ,form 17 January 1994)

Abstract-A steady-state model incorporating interactions between separate bubble and emulsion phases in a fluidized-bed polyethylene reactor was developed by Choi and Ray [Chem. Engng Sei. 40,2261-2279

(1985a)]. Correlations for maximum stable bubble size indicate that bubbles within the bed arc consider- ably smaller than those in their original model. In the paper, the influence of bubble size and superficial velocity on reactor operation are examined. It is shown that bubble size critically influences the rate of heat and mass transfer within the bed, and when the bubbles are as small as those predicted by the maximum stable bubble size correlations, there is little or no resistance to the transfer of monomer and heat between the phases. A simplified welt-mixed mode1 is developed to describe reactor operation in the limiting case where there is no difference between bubble and emulsion gas temperatures and concentrations. The differences between the predictions of temperature and monomer concentrations of the two-phase and simplified models are less than 2 or 3 K and 2 mol%, respectively, in the operating range of industrial interest. Therefore, a simple back-mixed model is appropriate for predicting temperature and concentration in the gas phase of industrial fluidized-bed polyethylene reactors.

1. INTRODUCTION

The modelling of polyethylene production in fluidized-bed reactors has received considerable at- tention in recent years. At the molecular level, kinetic models have been developed to predict the molecular weight and short chain branching distributions of the polymer molecules (Galvan and Tirrell, 1986; de Carvalho et al., 1989; McAuley et al., 1990). At the larger scale of the individual polymer particles, resist- ances to heat and mass transfer within the particles and in the boundary layer surrounding the particles have been analysed by Galvan and Tirrell (1986), Floyd et al. (1986a, b, 1987) and Hutchinson and Ray (1987). It has been established that, under many con- ditions, heat transfer and diffusional resistances do not play an important role at the particle level in gas-phase polyethylene reactors. Nevertheless, for very young particles with highly active catalysts, heat and mass transfer resistances can become significant, leading to multiple steady states and particle over- heating. Modelling at the particle level is useful for predicting polymer particle growth and morphology (Hutchinson et al., 1992) and, if diffusional or temper- ature gradients within the particles are significant, for predicting the effects of intraparticle temperature and concentration gradients on the molecular properties of the polyethylene produced.

For many catalyst systems in which heat and mass transfer resistances do not influence monomer con- centrations and temperatures within the polymer par-

‘Current address: Northern T&corn, 185 Corkstown Road, Nepean, Ontario, Canada K2H 8V4.

titles, the monomer concentration at the catalyst sites is determined by the equilibrium sorption of the monomer within the polymer particles. Hutchinson and Ray (1990) have developed thermodynamic models to predict equilibrium monomer concentra- tions at the catalyst sites from external gas-phase monomer concentrations in the vicinity of the poly- mer particles.

The research in the current article focuses on the modelling of gas-phase polyethylene reactors at an even larger scale, that of fluidization phenomena and heat and mass transfer within the bed. It is important to determine whether heat and mass transfer resist- ances between the bubbles and the emulsion phase are significant in typical industrial reactors, or whether the entire contents of the fluidized-bed reactor can reasonably be treated as well-mixed. If the well-mixed assumption is valid, experimental results and correla- tions developed from well-stirred gas-phase laborat- ory reactors [e.g. Lynch and Wanke (199111 can be applied directly to industrial-scale fluidized-bed reac- tors. McAuley et al. (1990) and McAuley and Mac- Gregor (1992, 1993) assumed a well-mixed bed in the development of the reaction rate expressions used in their models to predict gas compositions, temperature and polymer properties in fluidized-bed polyethylene reactors.

Choi and Ray (1985a) developed the only fluidized- bed polyethylene reactor model in the literature that considers temperature and concentration gradients within the gas in the bed. Their model considers the interaction of separate emulsion and bubble phases within the reactor bed. The current article updates

2036 K. B. MCAULEK et al.

and extends their work. The existence of a maximum stable bubble size, and revised assumptions regarding heat and mass interchange mechanisms within the bed are incorporated in the mathematical model. A sensi- tivity analysis is performed to determine the influence of key parameters, such as particle size, bubble size, superficial velocity and catalyst feed rate, on predicted steady-state reactor operating conditions. A simpli- fied model of the process, corresponding to the well- mixed assumption, is developed, and the predictions of the two-phase model and simplified model are compared.

Z MODEL DEVELOPMENT

To obtain a clear understanding of the fluidized- bed model, the assumptions that have been made in its development are summarized below. With the ex- ception of the assumption regarding the maximum stable bubble size, these assumptions were also made by Choi and Ray (1985a).

2.1. Model assumptions

(9

(ii)

(iii)

(iv)

(v)

(vi)

The fluidized bed comprises two phases, the bubble phase and the emulsion phase. Reac- tions occur only in the emulsion phase which is at minimum fluidizing conditions. All gas in excess of that required for minimum fluidization passes through the bed in the bubble phase. The bubbles grow only to their maximum stable size. Because the bubbles reach this maximum size near the base of the bed, all bubbles are assumed to have a uniform size throughout the bed. The bubbles travel up through the bed in plug flow at constant velocity. Heat and mass interchange coeffl- cients are average values over the height of the bed. The emulsion phase is perfectly back-mixed. In reality there will be a temperature gradient over the first 10 cm above the distributor (Wagner et al., 1981), but above this level the axial temperature gradient is very small. Radial concentration and temperature gradi- ents within the bed are negligible because the emulsion phase is well-mixed and the bubbles are distributed uniformly across the bed. The assumption of uniform radial distribution of bubbles is supported by the work of Glicksman et al. (1987), who observed uni- form bubble distribution across the bed sur- face in similar large-particle systems with small bubbles relative to the bed diameter. There is negligible resistance to heat and mass transfer between the gas and solids in the emulsion phase. This assumption is valid when the catalyst particles are sufficiently small and the catalyst activity is not ex- tremely high (Floyd et al., 1986a).

(vii) A mean size has been used for the polymer particles and neither elutriation of fines nor particle agglomeration are considered in the model.

(viii) The only gas-phase component considered in the model is ethylene. Talbot (1990) de- veloped a more comprehensive model that also considers inert components in the gas phase.

2.2. Bubble-phase material and energy balances Since the bubbles are assumed to be non-itter-

acting spheres, consideration of an individual bubble can be used to infer the behaviour of the entire bubble phase. The only mechanism by which mass can be transferred from the bubbles to the emulsion phase is by diffusion through the bubble clouds. Energy is transferred between the phases due to the temperature gradient between the bubbles and the emulsion and also by the diffusing gas. Writing the material and energy balances for a single bubble (Kunii and Levenspiel, 1969) gives

dcb - = $(C, - C,), dz

dTb -= dz

(2)

where the concentration of the monomer is expressed in moles of ethylene per unit volume of bubble. The variables in equations (1) and (2) are defined in the notation.

2.3. Emulsion-phase material and energy balances The steady-state mass and energy balances on the

emulsion phase are described by algebraic rather than differential equations, since the emulsion phase is per- fectly back-mixed. A mass balance on the interstitial gas in the emulsion phase may be written as

+ mn(Cb - C-2) (1 - 8)&,, = O. (3)

The first term considers the flow of gas into the emulsion phase at the base of the bed and the outflow of gas at the top. The second and third terms are the rates of ethylene outflow with the product stream and ethylene consumption by polymerization, respect- ively. The final term is the rate of mass transfer of ethylene from the bubble phase to the emulsion phase.

c is an average concentration of ethylene in the bubbles, over the height of the bed.

The solid-phase mass balance is shown below:

R, + qcat - QU - +-)p, = 0. (4)

The three terms in eq. (4) are the solids production rate by polymerization, the catalyst feed rake to the reactor and the rate of solids outflow in the product

Comparison of two-phase and well-mixed models 2037

stream. A mass balance on the catalyst gives

4cnr - X,&(1 - =%I&% = 4 (5)

where X,,, is the mass fraction of catalyst in the solid phase.

The steady-state energy balance on the emulsion phase is

+ RPC - AH - (c, - c,)(T, - T,,r)l

- 7cDHU,,(T, - T,) = 0. (6)

The first term is the enthalpy associated with gas Row to the emulsion phase at the base of the bed and the enthalpy of the gas stream leaving the emulsion at the top of the bed. We assume that the heat capacity of the gas is the same in the inflow and outflow streams. The second term is the enthalpy associated with the mass transfer of gas from the bubble phase into the emulsion. The third term is the convective and con- ductive heat transfer between the emulsion and bubble phases, and the fourth is the rate of heat generation by the polymerization reaction. The final term is the rate of heat loss to the surroundings through the reactor wall. The enthalpy associated with the catalyst feed to the reactor has been neglected because it is small relative to the other terms in eq. (6). All symbols in eqs. (l)-(6) are defined in the notation The volumes of the emulsion and bubble phases in the bed can be calculated from the total volume of the bed and the bubble fraction, 6:

v, = AH(1 -a), (7)

V, = AH& (8)

Reaction rates in gas-phase ethylene polymer- ization are generally modelled as first-order in both monomer concentration and in the quantity of cata- lyst in the reactor (Choi and Ray, 1985a; de Carvalho et al., 1989; McAuley et al., 1990), so that

R, = &C,X,,,PJH(~ - @(I - ~,f)Mw- (9)

This rate expression assumes that the ethylene con- centration at the catalyst sites is proportional to C,, the concentration in the emulsion gas. Since monomer diffusion rates, both to the polymer particles and within them (Floyd et al., 1986a, b, 1987), are fast relative to the reaction kinetics, the ethylene dissolved in the polymer phase is in equilibrium with the ethy- lene in the emulsion gas. Hutchinson and Ray (1990) have shown that Henry’s law applies to the solubility of ethylene in polyethylene at the conditions in the reactor. Hence, the proportionality assumption is valid. The rate constant, k,, includes the Henry’s law constant for ethylene partitioning between the two phases. We neglect the mild effect of temperature on the Henry’s law constant and assume that k, displays Arrhenius behaviour. The effect of catalyst deactiva- tion is not included in the model.

Correlations used to predict the bubble fraction in the bed, the voidage of the emulsion phase, bubble- phase and emulsion-phase gas velocities, and mass and heat interchange coefficients in eqs (l)-(9) are listed in Table 1. The correlation for the minimum fluidization velocity, U,/. assumes that the particles are spherical. The majority of these correlations were used by Choi and Ray (1985a), although it appears that the dimensions of the overall material and energy

Table 1. Fluidization and interchange correlations

Minimum fluidizing velocity (Lucas et al., 1986)

Bubble velocity’ (Davidson and Harrison, 1963)

Re,, = (29.F + 0.0357 Ar)“.’ - 29.5

u,=u,- u,,+0.711&z

Bubble fraction’ (Kunii and Levenspiel, 1969)

Emulsion velocity’ (Bukur et al., 1974)

Mass interchange’ (Kunii and Levenspiel, 1969)

Heat interchange’ (Kunii and Levenspiel, 1969)

‘Used by Choi and Ray (198%).

2038 K. B. MCAULEY et al.

interchange coefficient correlations that they used are inconsistent with their model.

Both Choi and Ray (1985a) and Talbot (1990) in- cluded a cloud to emulsion heat interchange coeffi- cient, H,,, in their models:

‘.’ . (10)

This interchange coefficient is similar in structure to K,, and was meant to account for the resistance to heat transfer between the cloud and the emulsion. In bubbling fluidized beds, the mechanism for mass transfer between the bubbles and the emulsion in- volves gas diffusion through the cloud. Hence, the interchange coefficient K,, in Table I is required to account for this diffusional resistance. Use of H,, as in eq. (10) incorrectly implies that the main path for heat transfer is by conduction through the gas in the cloud. Since the clouds contain particles with large thermal mass, and the transfer of heat between gas and the particles is fast, heat transfer through the cloud is rapid. It is governed primarily by heat transfer be- tween the cloud gas and polymer particles and by subsequent movement of the particles between the cloud and the emulsion. Hence, the resistance to heat transfer between the cloud and the emulsion is small compared to the bubble to cloud resistance (Kunii and Levenspiel, 1969) and is neglected in our model.

Choi and Ray used Broadhurst and Becker’s (1975) correlation for the voidage in the emulsion phase. This correlation can be inaccurate for large particles, giving values of E,~ near 0.38 for this system (Talbot, 1990). Calculations of the voidage in settled polyethy- lene particles using measured bulk densities (Wagner et al., 1981) give values ranging from 0.46 to 0.65. Hence, we have used a more reasonable value of E,,,~ = 0.5 in the model. Other physical parameters used in the mode1 are listed in Table 2.

Table 2. Standard operating conditions and parameters

Parameter Value

flc, (poise) P# Wcm3) ppol Wm3) PEPt w~31 cp (J/g/K) k,0 (cm”/8 cat/s) & (J/mol) AH (J/g) d, (cm) db (cm) H (cm) D (cm) P (atm) &ml Co (mol/cm3) To (K) i’-, (K) T6.r (K) U. @m/s)

o.oc0115 0.024 0.95 2.37

4.004 4.16 x 106

37,260 3829 0.05

15 1097 396

20.42 0.5

0.00085 316 293

273.15 34.8

2.4. Maximum stable bubble size When choosing a correlation for bubble size, it is

important to consider the maximum size that a bubble can reach before it becomes unstable and breaks up into smaller bubbles. Davidson and Harrison (1963) established that the settling velocity of the bed particles should be greater than or equal to the bubble rise velocity, U,. The maximum bubble size could then be obtained from

d 2tJ:

hmal= -3 9

(11)

where Ur is the terminal velocity of particles of dia- meter d,. Grace (1986a) postulated the same equation form, but used instead the terminal velocity of a par- ticle of diameter 2.7d,, resulting in a larger stable maximum bubble size than Davidson and Harrison. Using the correlation of Flemmer and Banks (1986) to predict terminal velocity, and the maximum stable bubble size criteria of Davidson and Harrison (1963) and Grace (1986a), polyethylene particles with a mean diameter of 0.05 cm would result in maximum stable bubble sizes of 9.77 and 24.2 cm, respectively.

The use of maximum stable bubble size correlations is a major difference between this model and that of Choi and Ray (1985a), who used a mean bubble size predicted by Mori and Wen (1975) for a bubble half- way up the reactor bed. Mori and Wen (1975) caution in their paper that bubble stability was not taken into consideration. For the reactor dimensions in Table 2, the Mori and Wen correlation predicts an average bubble diameter of 89 cm (Talbot, 1990), which is 22% of the reactor diameter. This predicted bubble size is clearly larger than the maximum stable sizes predicted by either the Davidson and Harrison or the Grace correlation. As we will demonstrate in this article, bubble size is a critical parameter that affects the predicted heat and mass transfer rates in fluidized- bed reactor models. Our model assumes that the bubbles rapidly reach their maximum stable size near the base of the bed, and that this bubble diameter is the average bubble size within the bed.

2.5. Consequences ofthe model assumptions and poten- tial improvements

The current model assumes that the emulsion phase is perfectly back-mixed. According to Grace (1986a), at least eight different assumptions have been adopted in various fluidized-bed reactor models to describe the axial mixing of the gas in the dense phase. These assumptions range from upward plug flow of the emulsion gas, through perfect mixing, to downward flow of the emulsion gas. Some models, for example that by Shiau and Lin (1993), divide the emulsion phase into a series of well-mixed sections with both upward and downward flow of emulsion gas between adjacent stages. In many cases, the predicted conver- sions are relatively insensitive to the emulsion-phase gas mixing pattern assumed, especially when conver- sions are low (Grace, 1986a). Therefore, adopting a more complicated approach to mixing of the gas in

Comparison of two-phase and well-mixed models 2039

the emulsion phase is not warranted when modelling fluidized-bed polyethylene reactors. In models used to predict the properties of polymer produced in fluidized-bed reactors, the assumptions made regard- ing the mixing and residence time distribution of the particles in the bed are more important issues. In product property models, especially those used for product grade transition studies (McAuley et al., 1990; McAuley and MacGregor, 1991, 1992; Ramanathan and Ray, 1991), it is common to assume that the solid phase is well-mixed. Imperfect mixing could have a dramatic effect on the time required to change from one polyethylene grade to the next. This problem is beyond the scope of the current article.

The formulation of the interface mass and energy interchange coefficients in Table 1 (Kunii and Lcvenspiel, 1969) considers the existence of a cloud phase between the bubble and emulsion which pro- vides the main resistance to mass transfer. In the current model, and in that proposed by Choi and Ray (1985a), only a bubble phase and an emulsion phase are considered in detail. The importance of the cloud phase is largely neglected, although Grace (1971) pos- tulated that the cloud can play an important role in such large-particle systems. As a result, the current model formulation ignores the effects of cloud shed- ding and is expected to underpredict mass transfer rates at the cloud boundary. Talbot’s (1990) calcu- lations of the volume of the cloud phase in gas-phase polyethylene reactors suggest that it could be as large as half of the bubble volume and that the cloud comprises roughly 25% of the dense phase. Jones and Ghcksman (1984) observed in their experiments with large particles that the variation from two-phase theory was due, not to the presence of clouds, but rather to low-resistance paths for gas flow through the bed provided by sequences of closely spaced bubbles or bubble chains. Therefore, possible refinements to the two-phase bubbling bed model developed in this article include consideration of the clouds as a separ- ate phase or consideration of gas through-flow effects due to bubble chains. Still more complex models, based on fundamental mass, momentum and energy conservation equations, could be derived to predict heat and mass transfer between the bubbles and the emulsion (Kuipers et al., 1992) without resorting to correlations for predicting bubble behaviour. Such detailed models require considerable computation, and are not justified by our current purpose, which is to determine whether the well-mixed assumption is appropriate for the gas phase in industrial fluidized- bed polyethylene reactors.

In some recent articles, Kunii and Ievenspiel (1991a, b) have shown that in systems wherein the fluidizing gas is absorbed by the particles in the bed, larger bubbleemulsion mass interchange rates are experienced than are predicted by the correlations for K, and H,,, in Table 1. Particles that rain through the bubbles significantly enhance mass transfer rates by absorbing gas from the bubbles wherein gas concen- trations are higher than in the emulsion. Similarly,

heat transfer can also be enhanced when heat generat- ing particles traverse the cooler bubble phase. Since large quantities of ethylene are first absorbed and then consumed within the polyethylene particles to pro- duce heat, industrial fluidized-bed polyethylene reac- tors provide an ideal example of a system with gas absorbing, heat generating particles. Therefore, we expect that our model may underpredict the actual heat and mass interchange rates between the bubbles and emulsion, and overpredict concentration and temperature differences between the two phases.

An additional reason that our model may over- predict the true temperature and concentration differ- ences between the bubble and emulsion phases is that we have assumed the bubbles to be totally non-inter- acting spheres of constant size. Both bubble breakup and coalescence can act to enhance heat and mass transfer (Grace, 1986a). When bubbles coalesce, they momentarily assume the local dense-phase composi- tion. Coalescence between two bubbles can lead to a complete and sudden renewal of the gas in the bubbles by gas in the emulsion phase (Pereira et al., 198 1).

2.6. Method OS solution Differential equations (1) and (2) can be integrated

analytically to determine both Cb and T, as functions of z, the vertical position in the bed. These expressions can then be used to determine the following equations

for CI, and T,,, which are “average” values required to achieve the same total mass and energy transfer over the height of the bed

E=&[l -exp($+)]. (12)

Tb - Te :=~[l--crp(=-E=& (13) To - Te

Using the parameters in Table 2 and the correlations in Table 1, eqs (3)-(6), (9), (12) and (13) form a set of seven equations with the following eight unknowns:

C,, c, T,, %, Q, R,, X,,, and qol. Talbot (1990) developed a simple non-iterative technique to solve this system to obtain steady-state operating condi-

tions. First, values of Tb ranging from 325 to 400 K are chosen, and then each of the other variables is evaluated in turn. The analysis has been limited to this temperature range because below 325 K the polym- erization rate is very low and above 400 K polymer particles melt and agglomerate.

3. PARAMETRIC STUDIES

In this section, we investigate the effects of model parameters and reactor operating conditions on model predictions.

3.1. Efects of superficial gas velocity Industrial fluidized-bed polyethylene reactors are

operated at superficial gas velocities ranging from 3 to

2040 K. B. MCAULEY et al.

6 times the minimum fluidization velocity (Wagner

Using a bubble diameter of 15 cm, we observe that

et al., 1981). The major effect of an increase in the

the model predicts temperatures in the bubble phase

superficial velocity, U,, is a reduction in tlae time required for a given quantity of gas to pass through

that are 0.5-5 K lower than in the emulsion phase,

the bed. For a specified catalyst feed rate, this smaller residence time leads to a lower reaction rate, a lower

and ethylene concentrations that are 0.555% higher.

single-pass conversion and a lower emulsion temper- ature. Figure 1 shows that as U, decreases from 6U,/

The size of these temperature and concentration dif-

to 3&f f the safe regime, in which the reactor can be

ferences depends to a great extent on the catalyst feed

operated without a danger of particle melting, de- creases accordingly. A major function of the gas pas-

rate, and hence on the rate of reaction. For low

sing through the bed is to remove the heat of reaction. Small values of U. lead to high sensitivity of the

reaction rates, corresponding to emulsion temper-

emulsion temperature to changes in catalyst feed rate and to other potential operating disturbances. High gas velocities are required to reduce the risk of particle melting, agglomeration and subsequent reactor shut- down. However, high gas velocities reduce the conver- sion of monomer per pass through the reactor, and can lead to greater elutriation of small particles from the bed. Elutriated particles are prevented from pas- sing out of the reactor and into the gas recycle system using a velocity reduction zone at the top of the reactor where entrained particles are given the op- portunity to drop back into the bed. Particle return may also be aided by a cyclone (Wagner et al., 1981).

320

(a)

i I.4 0.1 0.2 0.3 c

Catalyst Feed Rate [Q/S]

Y,

(c) O 0:1 012 0:3 0:4

Cata,,‘st Feed Rate [Q/S]

atures below 340 K and conversions less than 2.5% per pass, the temperature difference is less than 1.5 K. At high polymerization rates corresponding to emul- sion temperatures above 380 K and conversions above 7% per pass, the temperature difference is ap- proximately 4 K. This is true regardless of the superfi- cial velocity at which the reactor is operated.

3.2. Eflects of bubble size

tween the bubble and emulsion phases increases, lead- ing to a smaller resistance to heat and mass transfer. As shown in Fig. 2(b) and 2(d), the temperature and concentration differences between the bubble and

Grace (f986a) and Davidson and Harrison (1963)

emulsion phases are much greater for large bubbles

developed correlations for the maximum stable bubble size for a given particle size, based on the

than for small bubbles. Larger differences are pre-

terminal velocity. For polyethylene particles 0.5 cm in diameter, the bubble diameters predicted from these

dicted for higher catalyst feed rates and higher operat-

correlations are 24.42 and 9.77 cm, respectively. Larger bubble diameters are predicted by bubble size

ing temperatures, and can be as large as 20 K and

correlations that do not consider bubble stability. To investigate the importance of using an accurate bubble size, simulations were performed to determine the effects of changing bubble size on model predic- tions. Figure 2 shows the variation of the steady-state emulsion temperature when bubbles of sizes 5, 10, 20 and 40 cm were present in the bed. A superficial velo- city of 34.8 ems or 6U,,,I was used in these simulations.

As bubble size decreases, the interfacial area be-

0.1 D.2 0.3 , Catalyst Feed Rat13 [Q/S]

(4 Catalyst FE& Rate [Q/S]

Fig. 1. The effect of superficial gas velocities ranging from 3U,, to 6C& on steady-state reactor operation. & = 15 cm.

Comparison of two-phase and well-mixed models

as

L z

-MB-=--rc*10*5

z 15 c

1 10

8 t 5 & E

0

(W Catalyst Feed Rate [O/S]

0 0.1 0.2 0.3 0.4

W Catam Feed Rate [g/s]

Fig. 2. The effect of bubble diameters ranging from 5 to 4Ocm on steady-state reactor operation, uo = 6U,,.

(d)

0.4 Catdysl Feed Rate [g/s]

15%, respectively, for 40 cm bubbles, but are less than 1 K and 1% for 5 cm bubbles.

Figure 2(a) and (c) shows the effects of catalyst feed rate and bubble size on the emulsion temperature and on the single-pass conversion of ethylene. As expected, higher catalyst feed rates lead to higher polymer- ization rates and to higher emulsion temperature and conversion. The effects of bubble size on emulsion temperature and conversion are less obvious. Smaller bubbles have a lower velocity through the bed com- pared with larger bubbles. As a result, smaller bubbles lead to a larger bubble fraction, 6, in the bed and to a reduction in the volume of the emulsion phase for a given expanded bed height. Therefore, the residence time for both the solid phase and the catalyst de- creases with decreasing bubble size, reducing the quantity of catalyst in the reactor for a given catalyst feed rate. This reduction in the quantity of catalyst tends to reduce R,, the rate of polymerization, thereby reducing the rate of heat generation in the emulsion phase.

As shown in Fig. 2(a), at low catalyst feed rates, the effect of a reduced bubble size is to reduce the emul- sion temperature because smaller bubbles lead to im- proved heat removal rates, to less catalyst in the reactor and to reduced heat generation. At high cata- lyst feed rates, the emulsion temperature for 4Ocm bubbles falls below that for 20 and 10 cm bubbles. This cross-over results when mass transfer effects be- come dominant for the large bubbles. As shown in Fig. 2(d) for 40 cm bubbles, the ethylene concentra- tion in the emulsion phase falls rapidly at catalyst feed rates above 0.35 g/s. The lower concentration of ethy- lene in the emulsion reduces the polymerization rate,

thereby reducing the heat generation rate. This mass transfer effect for larger bubbles becomes more signi- ficant at higher catalyst feed rates and eventually causes the T, vs qc., curve for 4Ocm bubbles to fall below the curves for 20 and 10 cm bubbles.

At low catalyst feed rates and low conversions, Fig. 2(c) shows that smaller bubbles lead to lower single- pass conversions. This may seem counter-intuitive because smaller bubbles lead to enhanced mass trans- fer and will result in higher ethylene concentrations in the emulsion phase. However, small bubbles also lead to less catalyst in the reactor and to lower emulsion temperatures. At low catalyst feed rates and low con- versions, the net effect is a reduction in the polymer- ization rate, and a reduction in the single-pass conver- sion as the bubbles become smaller. Only at higher catalyst feed rates and high conversions does the mass transfer effect become dominant, and the conversion vs catalyst feed rate curve for 40 cm bubbles falls below that for 10 cm bubbles. Typical single-pass con- versions of ethylene in industrial fluidized-bed poly- ethylene reactors are on the order of 225% (Choi and Ray, 1985b), but overall conversions can exceed 98% (Sinclair, 1983), since most of the unreacted monomer is recompressed, cooled and recycled to the bottom of the reactor.

3.3. Eficts of particle size The two-phase bubbling-bed model developed in

this article assumes that all particles have the same mean diameter of 0.05 cm, whereas in actual fact the continuous addition of catalyst and removal of prod- uct leads to a distribution of particle sizes ranging from 0.005 to 0.2 cm. Talbot (1990) used population

2042 K. B. MCAULEY et al.

balances and statistical distributions to develop math- ematical models for predicting the particle size distri- bution in fluidized-bed polyethylene reactors.

If all particles in the reactor are of the same size, then an increase in particle size leads to an increase in terminal velocity and to a larger maximum stable bubble size. A secondary effect is to increase the min- imum fluidization velocity which allows a greater superficial velocity without causing elutriation. Par- ticle populations with a distribution of sizes can be summarized using a number of different average par- ticle diameters (Allen, 198 1); for example, the number- length mean diameter, the number-volume mean diameter, the length-surface mean diameter and the surface-volume mean diameter. Each of these mean diameters uses a single particle size to match the original distribution in two properties. For example, using the number-volume mean diameter, the original distribution is approximated by a population of uni- form particles with the same number of particles and the same total volume. The total surface area and the sum of all particle diameters will not match that of the original distribution. Each of the different means weights the contributions of small and large particles differently.

The small particles in the distribution have a lower terminal velocity and are expected to influence the maximum stable bubble size more than larger par- ticles. Thus, a particle population with a broad distri- bution of particle diameters may lead to a smaller maximum stable bubble size than predicted using the number-length or number-volume mean diameter. These smaller bubbles would then lead to smaller differences between emulsion and bubble-phase con- ditions than predicted by our model. The large par- ticles in the distribution have a higher terminal velo- city than the average particle, and have a greater tendency to rain through the bubbles, thereby en- hancing heat and mass transfer rates between the bubble and emulsion phases. Thus, we expect that a model that properly accounts for the effects of a broad particle size distribution on the maximum stable bubble size will predict enhanced heat and mass transfer rates compared with our model for particles raining through the bubbles, and would give predic- tions closer to those of the simplified well-mixed model described in the next section.

4. A SIMPLIFIED WELL-MIXED MODEL

In the limiting case, where bubbles are very small or interphase mass and energy transfer rates are very

high, c = C, and z = T,. To examine the validity of this assumption, made by McAuley et al. (1990), we derive a back-mixed model and compare its predic- tions with those of the detailed two-phase model.

4.1. SimpliJied model development In the simplified model, there is unrestricted gas

and heat flow between the bubble and emulsion phases, and the temperature and composition are uni-

form in the gas phase throughout the bed. This type of model could be derived simply by setting K,,, and H, in eqs (3), (6), (11) and (12) to very large values. However, this approach leads to numerical difficulties

as (c, - C,) and (z - T,) approach zero. Rederiving the model equations with the assumption of unre- stricted mass and energy transfer between the bubble and emulsion phases provides a more reliable numer- ical solution. Assuming that the gas composition, C, is uniform in the emulsion and bubble phases, the fol- lowing steady-state mass balance can be written for all of the monomer in the bed:

0 = AU,(C, - C) - QCE,, - Rp/Mw. (14)

As in eq. (3), we assume that the product stream leaving the reactor is at its minimum fluidizing condi- tions. Since QC&,,,, is much smaller than the other terms in eq. (14), the validity of this assumption is not crucial.

Assuming that the bubbles and the emulsion are at the same temperature, T, we can write the following energy balance on the reactor contents:

0 = AU,C&(T,, - T)

+ RpC - AH - (c, - c,) (T - Tr,,)l - xDHU,(T - T,). (15)

As in eq. (6), we have neglected the effect of concentra- tion change on the enthalpy of the gas leaving the reactor. We have also neglected the enthalpy as- sociated with the catalyst feed stream because it is small compared with the other terms in eq. (15). The remainder of the simplified model consists of eqs (4), (5) and (9). This set of five equations in six unknowns can be solved for any steady-state temperature be- tween 325 and 400 K by the following steps. First, choose a value of T and solve for R, in eq. (15). Next, assuming that q.., in eq. (4) is small, solve for Q. Given R, and Q, solve eq. (14) for C and eq. (9) for X,,,. Finally, obtain qea, from eq. (5).

The values of S and E,,,/ in eq. (9) must be the same as those in the full two-phase model in order to obtain the same solid-phase residence time in the bed. In- formation about the bubbIe fraction and voidage is not required in back-mixed models that assume a con- stant mass of polymer in the bed rather than a con- stant bed height.

4.2. Comparison of two-phase and well-mixed models For given monomer inlet conditions, particle size

and catalyst feed rates, the differences between the two models depend on the superficial gas velocity and the bubble diameter.

4.2.1. The effect of super$cial gas velocity. At low to moderate emulsion temperature, Fig. 3 shows that the difference between the predictions of the two models is very small, approximately 3 K near temper- atures of 370 K and less than 2 K near 360 K. At the same operating conditions, the differences between the predicted concentrations in the gas phase sur-

Comparison of two-phase and well-mixed models 2043

+ 0 0.1 0.2 0.3 0.4

Catalyst Feed Rate [g/s]

,,h___ _~~ ----- , 0.1 0.2 0.3 0.4 (

Catalyst Feed Rate [g/s] 5

Fig. 3. Comparison of the predicted reaction temperatures Fig. 4. Comparison of the predicted reaction temperatures for the simplified and two-phase models at superficial for the simplified and two-phase models with bubble dia-

velocities of 3U,, and 6U,,,,, assuming 15 cm bubbles. meters of 10 and 40 cm, using a superficial velocity of 6U,,,,.

rounding the particles are on the order of 1 or 2%. Both models predict that multiple steady states can occur. Like the model of Choi ad Ray (1985a), if temperatures beyond the melting point of the polymer

are considered, both models predict the existence of up to three steady states for a given catalyst feed rate.

The difference between the simplified model predic- tions for the 3U,,,, and 6U,,,f cases is a result of the change in the relative quantities of gas and solids in the bed required to maintain a constant expanded bed height. As the gas feed rate to the bed increases, the

solid-phase residence time is reduced as is the

quantity of catalyst in the reactor.

4.2.2. The effect of bubble diameter. As shown in Fig. 4, there is very little difference between the well- mixed and two-phase model temperature predictions if the bubble size is 10 cm. However, for large bubbles at high operating temperatures, the difference can be significant. The same result applies to the monomer concentration in the gas surrounding the polymer particles.

5. CONCLUSIONS

The model of Choi and Ray (1985a) describing ethylene polymerization in a fluidized-bed polyethy- lene reactor has been revised to account for a max- imum stable bubble size in the reactor, and to better model the resistances to heat transfer between the bubble and emulsion phases. Our model does not account for bubble coalescence and breakup, cloud shedding, the influence of small particles in the par- ticles size distribution on the maximum stable bubble size, nor the presence of particles raining through the bubbles. We expect that all of these phenomena lead to smaller heat and mass transfer resistances between the bubble and emulsion phases than is predicted by our two-phase model, and that the true behaviour of the gas concentration and temperature lies between the predictions of the two-phase and simplified models. Simulations show only minor deviations be- tween the predictions of the two-phase and back- mixed models for superficial velocities in the range of

3U,, to 6U,,,I and for the small bubble sizes predicted by the maximum stable size correlations. Based on this analysis we recommend the use of a well-mixed

gas assumption when modelling commercial fluidized-bed polyethylene reactors.

Acknowledgements-The authors wish to acknowledge the financial support of the Natural Sciences and Engineering Research Council and Queen’s University.

A Ar

6 5 C

G

d D 9

9 H

HbC

H,,

H,

AH k

k, L MW P

ClCPl Q Re

R, T

z u u#%

NOTATION

reactor cross-sectional area, cm2 Archimedes number [ = dip&, - pg)g/&] heat capacity, J/gmol/K heat capacity, J/g/K concentration of ethylene, mol/cm3

average ethylene concentration in bubble phase diameter, cm reactor diameter, cm self-diffusion coefficient for ethylene, cm2/s gravitational acceleration, cm/s’ bed height, cm bubble to cloud heat interchange coefficient, J/cm3/s/K cloud to emulsion heat interchange coeffi- cient, J/cm’/s/K overall heat interchange coefficient, J/cm 3/s/K heat of reaction, J/g thermal conductivity, J/cm/s/K rate constant, cm3/g cat/s mass transfer coefficient, s-l molecular weight, g/mol pressure, atm catalyst feed rate, g/s product rate, cm’/s Reynolds number (Remf = peU,,,,D/u,) reaction rate, g/s temperature, K

average bubble temperature, K velocity, cm/s wall heat transfer coefficient, J/cm’/K/s

2044

V volume, cm3 X -x1 catalyst mass fraction Z height above distributor, cm

Greek letters S volume fraction bubbles in bed & voidage of emulsion phase

P viscosity, poise

P density, g/cm’

Subscripts b bubble property c cloud property cat catalyst property e emulsion property

B gas propew 4 minimum fluidization conditions

PO1 polymer property ref reference value s polymer property T terminal 0 entrance or superficial conditions 03 ambient condition

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