Hydrodynamic Flow Regimes, Gas Holdup, and Liquid Circulation in

Ind. Eng. Chem. Res. 1998, 37, 1251-1259

1251

Hydrodynamic Flow Regimes, Gas Holdup, and Liquid Circulation in Airlift Reactors Mohamed E. Abashar,*,† Udi Narsingh, Andre E. Rouillard, and Robin Judd Reaction and Fluidization Engineering Group (RFEG), Department of Chemical Engineering, University of Durban-Westville, Private Bag X54001, Durban, South Africa

This study reports an experimental investigation into the hydrodynamic behavior of an externalloop airlift reactor (ALR) for the air-water system. Three distinct flow regimes are identifiedsnamely homogeneous, transition, and heterogeneous regimes. The transition between homogeneous and heterogeneous flow is observed to occur over a wide range rather than being merely a single point as has been previously reported in the literature. A gas holdup correlation is developed for each flow regime. The correlations fit the experimental gas holdup data with very good accuracy (within (5%). It would appear, therefore, that a deterministic equation to describe each flow regime is likely to exist in ALRs. This equation is a function of the reactor geometry and the system’s physical properties. New data concerning the axial variation of gas holdup is reported in which a minimum value is observed. This phenomenon is discussed and an explanation offered. Discrimination between two sound theoretical modelssnamely model I (Chisti et al., 1988) and model II (Garcia Calvo, 1989)sshows that model I predicts satisfactorily the liquid circulation velocity with an error of less than (10%. The good predictive features of model I may be due to the fact that it allows for a significant energy dissipation by wakes behind bubbles. Model I is now further improved by the new gas holdup correlations which are derived for the three different flow regimes. 1. Introduction Airlift reactors are known to be efficient contactors for processes involving gases, liquids, and solids. Their relatively simple mechanical design, low shear rate, high capacity, good mixing, absence of mechanical agitators, and low cost make them a versatile type of bioreactor. Applications of airlift reactors in biotechnology and chemical industry and their advantages and differences over bubble columns are given by many investigators (Verlaan, 1987; Chisti, 1989; Al-Masry, 1993). The main difficulty in the mathematical modeling and design of airlift reactors has been the lack of information on the hydrodynamics (Ho et al., 1977; Merchuk et al., 1980; Merchuk and Stein, 1981; Moresi, 1981; Verlaan, 1987; Chisti et al., 1988; Joshi et al., 1990; Garcia Calvo and Leton, 1991, Hatch, 1993). An important aspect in the modeling of the hydrodynamics of airlift reactors is the relationship between the dependent variables of gas holdup and liquid circulation rate and the independent variables of superficial gas velocity (independently controllable), the physical properties of the fluids, and reactor geometry. This last variable has a strong influence on the hydrodynamics and makes it difficult to compare the results from different sources. Several empirical correlations are reported in the literature (Chisti, 1989), but these do not give any fundamental understanding about the hydrodynamics and they are thus of limited use for extrapolation. Notable theoretical models are those of Chisti et al. (1988), Garcia Calvo (1989), and Garcia Calvo et al. (1991) which are based on a view of the energy balance and those of Hsu and Dudukovic (1980), Merchuk and * Author to whom all correspondence should be addressed. † E-mail: [email protected].

Stein (1981), Verlaan (1987), Joshi et al. (1990), and Young et al. (1991) which follow from a consideration of the momentum balance. The purpose of the present study is to investigate the hydrodynamics experimentally and to examine the validity of the two existing hydrodynamic models based on the energy balance approach. 2. Circulation and the Energy Balance The energy balance approach considers that the driving force for circulation in the reactor is produced by the change in energy as gas bubbles rise and expand up the riser. This energy is dissipated by the internal friction losses in the fluids and the friction losses against the reactor wall. The energy balance over an airlift reactor loop is given by the following equation:

{

Rate of energy input due to isothermal gas expansion

or

}{

Rate of energy dissipation due to internal turbulence ) and friction between the gas-liquid interface

{

Rate of energy losses due to friction between the fluids and the reactor

Ein )

∑E¨ + ∑Eˆ

∑E¨ ) E¨ r + E¨ d + E¨ t + E¨ b ∑Eˆ ) Eˆ r + Eˆ d + Eˆ t + Eˆ b

}

}

+

(1)

(2) (3) (4)

where r, d, t, and b refer to riser, downcomer, top, and bottom sections of the reactor. Because of the negligible

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1252 Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998

drift of gas with respect to the liquid in both the top ¨ b are essentially and bottom sections, the terms E ¨ t and E zero. The energy input due to the isothermal expansion of an ideal gas (supplied at a pressure of Ph and a superficial gas velocity of Ugr) is given by the following (Chisti, 1989):

(

Ein ) UgrArPh ln 1 +

)

FLgh Ph

(5)

2.1. Model I (Chisti et al., 1988). The model of Chisti et al. makes the following assumptions: (1) Steady-state conditions. (2) Isothermal conditions. (3) Negligible mass transfer between the gas and the liquid. (4) The energy losses terms due to the skin friction in the riser and the downcomer negligible (E ˆr ≈ 0, E ˆ d ≈ 0) in comparison to the other dissipation terms. This assumption is justified by the experimental evidence of Lee et al. (1986) for low-viscosity Newtonian fluids (e.g., water). (5) The pressure drop due to acceleration negligible (Wallis, 1969). The energies associated with turbulence and internal ¨ d are obtained by an energy balance friction, E ¨ r and E on the riser and on the downcomer. The energy balance for the riser is given by the following:

Ein + pressure energy loss ) potential energy gain + E ¨ r (6) or (neglecting the mass of the gas compared to that of the liquid):

˘ Lr(1 - ro)gh ) m ˘ Lrgh + E ¨r Ein + m

(8)

The energy balance on the downcomer is written as

potential energy loss ) pressure energy gain + E ¨ d (9) or

˘ Ld(1 - do)gh + E ¨d m ˘ Ldgh ) m

(10)

E ¨d ) m ˘ Lddogh

(11)

hence

For the case where there is no liquid draw off, the continuity equation gives the following

˘ Ld m ˘ Lr ) m

(12)

Substitution of eqs 8, 11, and 12 into eq 3 gives

∑E¨ ) Ein - m˘ Lr(ro - do)gh

(13)

The energy losses due to friction in the top and the bottom of the reactor (caused by expansion, contracting,

VLr2 E ˆt ) m ˘ LrKt 2

(14)

VLd2 ˘ LdKb E ˆb ) m 2

(15)

where V is the linear velocity and K is the friction loss coefficient. The relations between the linear velocities of the liquid in the riser and the downcomer and the superficial liquid velocity in the riser are obtained as follows:

VLr ) VLd )

ULd (1 - do)

ULr

(16)

(1 - ro) )

()

Ar ULr Ad (1 - do)

(17)

Substitution of eqs 14-17 in eq 4 gives

[

∑

E ˆ )m ˘ Lr

Kt

(1 - ro)

+ 2

]

Kb(Ar/Ad)2 ULr2 (1 -  )2 2 do

(18)

Substitution of eq 13 for ΣE ¨ and eq 18 for ΣE ˆ in eq 2 gives the superficial liquid velocity in the riser as

ULr )

[

2gh(ro - do) Kt

(1 - ro)2

(7)

hence

˘ Lrrogh E ¨ r ) Ein - m

and changes in flow direction) are given by

+

]

0.5

Kb(Ar/Ad)2 (1 - do)2

(19)

A reliable model based on the overall momentum balance was also developed by Hsu and Dudukovic (1980) for the prediction of the liquid recirculation velocity in gas-lift reactors. This momentum equation can also be reduced to eq 19, when friction losses are neglected. 2.2. Model II (Garcia Calvo, 1989). The main differences between this model and model I are that in this model the energy losses are effectively ascribed to skin friction against the reactor wall and to friction due to gross slip of the bubbles and the liquid in the riser. Energy dissipated by internal circulation in the bubble wakes, for example, is excluded. The model assumptions for model II are the following: (1) Steady-state conditions. (2) Isothermal conditions. (3) Negligible internal recirculation. (4) Negligible mass transfer between the gas and the liquid. (5) The gas holdup in the downcomer negligible (do ) 0). (6) The average density of the gas-liquid equal to the liquid density. (7) The pressure drop due to acceleration negligible (Wallis, 1969). (8) Constant slip velocity in the riser. (9) Gas holdup in the riser considered to be the mean gas holdup. ¨ r becomes From assumptions 3 and 5 E ¨ d ≈ 0 and E only the energy dissipated at the gas-liquid interface in the riser and is given by the following (Richardson and Higson, 1962):

∑E¨ ) E¨ r ) ∫P

P2 1

VsrAr dP ) VsArro(P2 - P1) ) VsArroFLgh (20)

Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998 1253

where Vs is the slip velocity (Wallis, 1969):

Vs )

Ugr Ugr ULr - VLr ) ro ro (1 - ro)

(21)

From eqs 17 and 21:

()

VLd ) ULd ) (1 - ro)

Ar V ) Ad Lr (1 - ro)

( )(

)

Ar Ugr - Vs (22) Ad ro

Following Garcia Calvo (1989) and expressing all the friction in terms of the liquid velocity in the downcomer, the friction losses between the fluids and the reactor is given by

VLd2 ULd3 ΣE ˆ ) Kf m ) KfFLAd ) ˘ Ld 2 2 Ar Ugr 1 KfFLAd (1 - ro) - Vs 2 Ad ro

( )(

[

)]

3

(23)

Substitution of eqs 5, 20, and 23 in eq 2 gives

(

) ( )[

FLgh - VsroFLgh Ph Ar 2 Ugr 1 (1 - ro) - Vs KfFL 2 Ad ro

UgrPh ln 1 +

(

)]

3

) 0 (24)

The corresponding superficial liquid velocity in the riser is obtained from the definition of the slip velocity in eq 21:

(

ULr ) (1 - ro)

)

Ugr - Vs ro

(25)

3. Experimental Section The pilot plant external loop airlift reactor used in this work is shown in Figure 1. The reactor was made of borosilicate glass with an approximated working volume of 0.725 m3. The major dimensions are summarized in Table 1. Air was introduced through a circular perforated plate sparger containing 193 holes of 1-mm diameter on a 11-mm square pitch. The sparger was designed according to the criteria given by Ruff et al. (1978) and was located about 0.94 m from the base of the reactor. The flow rate of the air was measured by a turbine flowmeter and controlled by a needle valve just downstream of it. A set of rotameters was included for visual indications only. The liquid circulation velocity in the downcomer was measured by an electromagnetic flowmeter. The pressure drop along the riser was measured by a set of inverted U-tube manometers connected to four taps 1.1 m apart in three sections of the riser. The first tap was located at 1.36 m above the sparger which is greater than the maximum distance necessary for the equilibrium bubble size, 5 times the column diameter (Joshi et al., 1990). The pressure measurement system enabled the calculation of the local gas holdup as well as the overall gas holdup for individual and a combination of sections. A single inverted U-tube manometer was used to measure the gas holdup in the downcomer. For all experiments air and filtered tap water were used and the reactor was

Figure 1. Experimental setup for the external loop airlift reactor: (1) air compressor; (2) pressure regulator and filter; (3) turbine flowmeter; (4) rotameters; (5) sparger; (6) water drainage; (7) pressure tapping; (8) electromagnetic flowmeter; (9) water inlet; (10) inverted U-tube manometer; (11) disengagement tank. Table 1. Major Reactor Dimensions

riser downcomer disengager

diameter (m)

height (m)

dispersion height (m)

0.225 0.225 1.58 × 0.38 × 0.50

6.75 6.75

6.06

operated at room temperature and atmospheric pressure. The dispersion height was kept at 6.06 m. 4. Results and Discussion 4.1. Characterization of Flow Regimes. The characterization of various flow regimes has been described by Shah et al. (1982) for bubble columns. The bubbly (homogeneous) flow regime is characterized by almost uniform sized bubbles with equal radial distribution. The heterogeneous (churn turbulent) flow regime is characterized by large bubbles moving with high rise velocities in the presence of small bubbles. Heterogeneous flow is typically characterized by a nonuniform radial gas holdup profile. The third regime (slug flow) occurs only in small diameter columns (up to 0.15

1254 Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998

Figure 2. Local gas holdup and mean gas holdup vs superficial gas velocity in the riser.

m). At high gas flow rates the large bubbles are stabilized by the column wall, leading to the formation of bubble slugs. Krishna and Ellenberger (1996) reported a transition regime between homogeneous and heterogeneous regimes. The same characterization is used for airlift reactors (Verlaan, 1987; Chisti, 1989). Verlaan (1987) identified a transition point between two regimes (at approximately 0.05 m/s) in airlift reactors by the presence of a discontinuity in a plot (double logarithmic scale) of the superficial liquid velocity vs the superficial gas velocity. He also showed that the two-phase drift model of Zuber and Findlay (1965) was only applicable up to a maximum value of the total flow and beyond this value the required plug flow behavior (i.e., bubbles distributed homogeneously in a radial sense) no longer existed. This maximum value also corresponded to Verlaan’s transition point. This was not in contradiction with the conclusion of Merchuk and Stein (1981) who reported that the two-phase drift model of Zuber and Findlay satisfactorily fitted all their experimental data for the whole range, but a careful analysis of their range of experimental data as well as their value of the distribution coefficient (1.03) show clearly that their experimental range of parameters is below the transition point (i.e., in the bubbly flow regime). Joshi et al. (1990) proposed different criterion to differentiate between the homogeneous and the heterogeneous flow regimes. They detected a sharp increase in the slope of the drift flux vs the gas holdup plot. This sharp increase was taken as the transition between regimes. They also showed that the location of the transition point depends upon several factors such as sparger design, superficial liquid velocity, reactor geometry, and the fluid physical properties. The local gas holdup profiles for the three sections in the riser are shown in Figure 2. The overall gas holdup is also shown by a dotted line. Three separate flow regimes are both suggested by the data and also confirmed by visual observation, namely the bubbly flow (homogeneous), transition, and heterogeneous (churn turbulent) flow regimes are identified. The flow regimes cannot be sharply identified in this figure, but the data

Figure 3. Superficial liquid velocity vs superficial gas velocity in the riser: (a) normal scale and (b) double logarithmic scale.

is more suggestive of three, rather than two regimes with a single transition. Figure 3 shows plots of ULr vs Ugr used by many workers to identify the transition point. Figure 3a shows that the plot of the power law dependence is not of much help in this respect. In the double logarithmic plot (Figure 3b) there are clear discontinuities in the slope of the function and the function itself. These discontinuities represent the transition points (i.e., the start and the end of the transition regime). Transition is clearly over a range. Figure 4 shows that the two-phase drift model of Zuber and Findlay fits the experimental data rather well until some transition begins. The simplest form of the drift model used to fit our experimental data up to the start of transition (using the method of least squares) is

Vgr ) 1.07(Ugr + ULr) + 0.538 0 e (Ugr + ULr) e 0.8 (26) The value of the distribution coefficient (1.07) confirms that there must be relatively flat radial profiles of the velocity and the gas holdup (homogeneous regime) in the column before the transition starts. Some values for the drift flux parameters found by other investigators for external loop airlift reactors are compared with

Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998 1255

Figure 4. Linear gas velocity vs the total flux in the riser.

Figure 5. Local gas holdup profiles in the riser for various superficial gas velocities.

Table 2. Parameters for the Drift Flux Model

system

dr (m)

air-water air-water air-water air-water

0.225 0.200 0.240 0.140

drift distribution velocity coefficient (m/s) 1.07 1.20 1.13 1.03

0.538 0.260 0.280 0.330

ref this work Verlaan (1987) Nicol and Davidson (1988) Merchuk and Stein (1981)

our measurements in Table 2. Apparently, the consideration of the change in the flow pattern reported in this study may be responsible for this deviation. Our estimates of superficial gas velocity at the start and end of the transition regime are 0.02 and 0.043 m/s, respectively. This data may be compared with the single transition point measurement of 0.05 m/s reported by Verlaan (1987) and Joshi et al. (1990). It is, of course, not really possible to compare the exact location of the transition, since it is profoundly affected by the geometry of the reactor, fluid properties, and operating conditions. Another important result is that the local gas holdup profile measured in the second (middle) section of the column intersects and becomes lower than the local gas holdup profile for the first section of the column. It is interesting that this crossover occurs in the vicinity of the start of the transition regime, as shown clearly in Figure 2. This strange behavior is as a result of the minimum in the gas holdup which is observed along the length of the riser, as shown in Figure 5. This minimum in the gas holdup along the riser may be explained by considering the balance of the two opposing influences of the hydrostatic pressure and bubble coalescence. The decrease of hydrostatic pressure along the column increases the bubble size and thus the total bubble volume and consequently the gas holdup. But the increase of the bubble size due to coalescence acts to increase bubble velocity, and this decreases the residence time and hence tends to decrease the gas holdup. Of course, these complicated interrelated factors happen simultaneously, but the dominant one gives the overall result. It is the change in the balance of these factors that possibly contributes the start and finish of the transition regime. Clearly, the balance is affected by

Figure 6. Mean gas holdup vs the overall gas holdup in the riser.

other physical factors such as the location of the sparger and of course the physicochemical properties of the fluids which will affect the bubble dynamics. 4.2. Gas Holdup Correlations. Figure 6 shows that, despite the axial variation of the gas holdup, the arithmetic mean of the local gas holdup for the three sections is approximately equal to the overall gas holdup in the riser. However, we now propose separate correlations to predict the overall gas holdup in the riser for each of the three regimes rather than correlating the whole range by a single equation. The form of the correlations is arrived at from the drift gas model as follows:

Ugr )

ro(C1ULr + C2) (1 - roC1)

(27)

1256 Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998

Figure 7. Parity plot for the overall gas holdup in the riser.

The Maclaurin expansion of the term 1/(1 - roC1) is

1 ) 1 + roC1 + (roC1)2 + (roC1)3 + ... (28) (1 - roC1) Definitely this expansion is singular at roC1 ) 1. Assuming a power law dependence as shown in eq 29,

ULr ) aUbgr

(29)

then eq 27 becomes

Ugr ) (C3Ubgr + C2)(ro + ro2C1 + ro(roC1)2 + ro(roC1)3 + ...) (30) Neglecting the high-order terms (since ro is typically rather small) and also the term roC2 gives

ro ) RUβgr

(31)

The coefficients (R, β) are obtained for all flow regimes by analyzing our experimental data using nonlinear regression analysis. A correlation for each regime is obtained as follows: (a) homogeneous regime

ro ) 0.29U0.74 gr

0 e Ugr e 0.02

(32)

0.02 < Ugr e 0.043

(33)

(b) transition regime

ro ) 0.37U0.81 gr

holdup in the riser is shown in Figure 8. The following correlation is obtained:

do ) 0.174ro

Ugr > 0.043

(34)

Figure 7 shows the predicted values of the overall gas holdup from these correlations vs the experimental values. The power law correlations are in good agreement with the experimental values ((5%). The plot of the gas holdup in the downcomer vs the overall gas

0.007 e ro e 0.06

(35)

A similar linear relationship with a slope of 0.79 and intercept of -0.057 was reported by Bello et al. (1985). The differences between eq 35 and the correlation of Bello et al. (1985) is due to the fact that the overall gas holdup in the downcomer depends on the geometry and the efficiency of the disengagement tank. 4.3. Models Application. These gas holdup correlations developed in this study for the three flow regimes are now implemented, mainly in model I. 4.3.1. Model I (Chisti et al., 1988). The reactor geometry affects the values of Kb and Kt. In this study the friction coefficients at the bottom and the top of the reactor are estimated using the engineering correlations as Kb ≈ Kt ) 1.8 (Streeter and Wylie, 1979). Different values of Kb are reported in the literature, but this value (1.8) is almost the same as the value obtained by Verlaan (1987). Figure 9a shows a comparison between experimental and predicted values of the superficial liquid velocity in the riser using model I (eq 19). The developed correlations (eqs 32-34) for the gas holdup are implemented in this model. It is clearly shown in Figure 9a that this model predicts the superficial liquid velocity in the riser with satisfactory accuracy and the difference between experimental and predicted values is lower than (10%. Figure 9b shows the prediction of model I when the well-known gas holdup correlation of Hills (1976) is used for the different flow regimes.

ro )

(c) heterogeneous regime

ro ) 1.58U1.232 gr

Figure 8. Gas holdup in the downcomer vs the overall gas holdup in the riser.

Ugr 0.24 + 1.35(Ugr + ULr)0.93

ULr > 0.3 m/s (36)

Substitution of this correlation in eq 19 allows the calculation of the superficial liquid velocity in the riser (ULr) for each value of the superficial gas velocity in the riser (Ugr). The resulting nonlinear algebraic equation is solved numerically by an IMSL (International Mathematics and Statistics Library) subroutine called ZSPOW

Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998 1257

Figure 9. Parity plot for the superficial liquid velocity in the riser (model I): (a) using different gas holdup correlations for various flow regimes from this work and (b) using gas holdup correlation of Hills (1976) for various flow regimes.

Figure 10. (a) The overall gas holdup in the riser vs the square superficial liquid velocity in the downcomer and (b) parity plot for the superficial liquid velocity in the riser (model II).

based on a variation of Newton’s method which uses a finite difference approximation to the Jacobian and takes precautions to avoid large step sizes or increasing residuals. It is obvious that the deviation of the model prediction becomes significant ((21%) as shown in Figure 9b and the overall predictiveness is unacceptable. The prediction of model I is very sensitive to the value of ro. 4.3.2. Model II (Garcia Calvo, 1989). Hsu and Dudokovic (1980) developed a relation between gas holdup and the liquid velocity based on momentum balance in which the pressure difference between the riser and the downcomer is considered to be due to the friction losses in the reactor.

square of the superficial liquid velocity in the downcomer. The best fit was made by using the method of least squares. From the slope of the line the total friction coefficient (Kf) was found according to eq 37 to be equal to 4.43. From this figure, as far as the linear relationship exists, the total friction coefficient is independent of the liquid velocity and the gas holdup. A comparison between the total friction coefficient (Kf) and the total value of Kb and Kt (Kb + Kt ) 3.6) shows that most of the energy losses due to the friction between the fluids and the reactor are dissipated at the top and the bottom of the reactor. The slip velocity was calculated according to Harmathy’s equation (1960):

Kf U 2 ro ) 2gh Ld

(37)

This relation has been used by many workers (Verlaan, 1987; Merchuk and Stein, 1981). Figure 10a shows the experimental measured gas holdup in the riser vs the

[

Vs ) 1.53

]

σLg(FL - Fg) FL2

1/4

(38)

where the surface tension of the water (σL) is 0.075 N/m. Equation 24 is a nonlinear algebraic equation. Knowing Kf and Vs this equation is solved by ZSPOW to

1258 Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998

predict ro for each value of the superficial gas velocity (Ugr) and the corresponding superficial liquid velocity is obtained from eq 25. Figure 10b shows that the difference between the experimental and the predicted values of the superficial liquid velocity in the riser is less than (16%. It is obvious that model I is more accurate than model II; this may be due to consideration of energy dissipated by the wakes behind the bubbles and then may be considered different for very different flow regimes. Despite the lower accuracy of model II in this study, it still has an attractive feature of predicting the liquid superficial velocity and the gas holdup on the basis of the total friction coefficient (Kf), the slip velocity (Vs), and the superficial gas velocity (Ugr).

j ) mean gas holdup, dimensionless F ) density, kg/m3 σ ) surface tension, N/m

5. Conclusions

Literature Cited

Three flow regimes, rather than two flow regimes separated by a transition point, have been observed in an ALR. New correlations for gas holdup in the riser in the three regimes are proposed. Measurements in the riser show that there is an axial variation in the gas holdup and a minimum may exist. This minimum may be as a result of the competing influences of hydrostatic pressure vs bubble coalescence. The model of Chisti (1989) is used to predict the measured liquid circulation rates. The predictions are very sensitive to the value of ro used, and the predictions of this model are considerably enhanced by using the separate correlations for the gas holdup for the three flow regimes. Chisti’s model is a better fit to the present data than Calvo’s model. Further studies are needed to develop a method of estimating the transition points in ALRs. Nomenclature a ) coefficient in eq 29 A ) cross-sectional area, m2 b ) coefficient in eq 29 C1, C2, C3 ) constants in eqs 27-30 Ein ) rate of energy input due to isothermal gas expansion, W E ¨ ) rate of energy dissipation due to internal turbulence and friction between the gas-liquid interface, W E ˆ ) rate of energy losses due to friction between the fluids and the reactor, W g ) gravitational acceleration, m/s2 h ) dispersion height, m K ) friction coefficient, dimensionless Kf ) total friction coefficient, dimensionless m ˘ ) mass flow rate, kg/s Ph ) reactor headspace pressure, Pa U ) superficial velocity, m/s V ) linear velocity, m/s Vs ) slip velocity, m/s Z ) dimensionless height of three sections Greek Symbols R ) coefficient in eq 31 β ) coefficient in eq 31  ) local gas holdup, dimensionless ro ) overall gas holdup in the riser, dimensionless do ) overall gas holdup in the downcomer, dimensionless

Abbreviations ALR ) airlift reactor Subscripts b ) bottom of the reactor d ) downcomer g ) gas L ) liquid o ) overall r ) riser s ) slip t ) top of the reactor

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Received for review July 2, 1997 Revised manuscript received November 14, 1997 Accepted November 15, 1997 IE9704612