Solid effects on hydrodynamics and heat transfer in an external loop airlift reactor

Solid effects on hydrodynamics and heat transfer in an external loop airlift reactor

Chemical Engineering Science 61 (2006) 1300 – 1311 www.elsevier.com/locate/ces Solid effects on hydrodynamics and heat transfer in an external loop a...

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Chemical Engineering Science 61 (2006) 1300 – 1311 www.elsevier.com/locate/ces

Solid effects on hydrodynamics and heat transfer in an external loop airlift reactor H. Dhaouadia,∗ , S. Poncinb , J.M. Hornutb, c , G. Wildb a Faculté des Sciences, Dpt. de Chimie, Laboratoire de Chimie Appliquée et Environnement, Bvd. de l’Environnement, 5000 Monastir, Tunisia b Laboratoire des Sciences du Génie Chimique, CNRS ENSIC/INPL Nancy, France c Université Henri Poincaré de Nancy I, IUT Le Montet, France

Received 19 November 2004; received in revised form 25 June 2005; accepted 18 August 2005 Available online 6 October 2005

Abstract The effect of a solid presence on global hydrodynamic parameters and heat transfer in an external loop airlift reactor has been experimentally investigated. Results obtained in both two- and three-phase flow are presented in this study. Two different external loop airlift reactor sizes have been used and local hydrodynamic characteristics including local gas hold-up and bubble velocity have been obtained in two-phase flow. Optical and ultrasound probes have been used to obtain this information, respectively. It was found that an increase of solid hold-up leads to a decrease of liquid velocity and heat transfer coefficient. Measured in a two- and three-phase reactor using a horizontal-heating probe, a correlation of the average gas hold-up and heat transfer coefficient is proposed. Correlation parameters are identified in homogeneous and heterogeneous flow regimes, which have been derived from the gas slip velocity concept. The experimental liquid velocity and gas hold-up in the riser have been represented in a satisfactory way by a hydrodynamic model, either in the absence or in the presence of solid particles. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Airlift reactor; Hydrodynamics; Heat transfer; Two-phase; Three-phase

1. Introduction Airlift reactors present the same advantages as conventional bubble columns; they are characterised by a simple construction, good mixing and low shear rate as compared with stirred tanks. They are particularly well suited for processes with demands for rapid and uniform distribution of the reaction components, and for multiphase systems for which high mass and heat transfer are necessary. Many installations employ this type of reactor in various sectors of industrial activities. They are mainly used as bioreactors in fermentation processes (Al-Qodah and Lafi, 2001) and in the biotransformation of many substances (Sánchez Mirón et al., 2002; Klein et al., 2002; Shu and Yang, 1996; Siegel and Robinson, 1992). Some particular applications to the

∗ Corresponding author. Tel.: +216 73 500 276; fax: +216 73 500 278.

E-mail address: [email protected] (H. Dhaouadi). 0009-2509/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2005.08.024

wastewater treatment process are increasingly being developed (Beun et al., 2002; Van Benthum et al., 1997, 2000; Frijters et al., 1997; Heijnen et al., 1997). Airlift reactors also have several chemical applications. They are used in ore leaching, high tonnage heterogeneous catalyst industries as well as in the ethylene chlorination process (Orejas, 1999). Several works describe globally and locally the fluid dynamic in external loop airlift reactors (Freitas et al., 1999, 2000; Dhaouadi et al., 1996; Douek et al., 1994). Many models, generally derived from energy balances, are used to predict hydrodynamics and mass transfer in these reactors (García-Calvo et al., 1999; Livingston and Zhang, 1993; Kochbeck et al., 1992). In most published papers the three-phase hydrodynamic behaviour is an extension of the two-phase behaviour. Glass beads several millimetres in diameter are often used as the solid phase. Some works are conducted with marine sediments (Tobajas et al., 1999) or active biomass (Schügerl, 1997) as the solid phase. Few works are conducted with very small diameter particles (Oey et al., 2001).

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In all these applications, the heat exchange aspect is important. However, works that describe the heat transfer in airlift reactors are very rare (Kawase and Kumagai, 1991). In a review paper Kim and Kang (1997) studied the effect of gas velocity and density and also that of the solid and liquid properties on the heat transfer coefficients in three-phase fluidised beds and discussed the analogy between the heat and mass transfer in this type of reactor. Indeed, many gas–liquid and gas–liquid–solid reactions in chemistry or biochemistry are either endothermical or exothermical. The removal or the supply of the heat by an indirect way is an important aspect in the design of these reactors in order to ensure safe process operation or to maintain optimal conditions for reaction. In order to design the heat transfer surfaces, information is required on heat transfer coefficients between the two- or three-phase bed and heating or cooling surfaces. It is well known now that the heat transfer coefficient is highly influenced by the hydrodynamic behaviour. Several published papers have focused on hydrodynamics of these reactors but only a few of them have studied heat transfer, especially in three-phase airlift reactors.

Airlift reactors may be operated in homogeneous and heterogeneous regimes. Under the usual operating conditions of an airlift reactor, the heterogeneous regime would prevail in the riser and the homogeneous regime would prevail in the downcomer. While some qualitative observations have been made on flow regimes in such reactors, the correlations of the hydrodynamic parameter (mean gas hold-up and liquid velocity) usually do not take into account differences in flow regime. Here, we shall consider the gas slip velocity based on the classical approach of Wallis (1969): Ug Ul − . g 1 − g

The basic equations of the model used are the following:   gH pm Ug,m ln 1 + sl,r Pat    2 1 Ad Ad = Kf,r sl,r + Kf,d sl,d (1 − s,d )3 2 Ar Ar 0.64(2)3N/2 sl,r H (Vl,o − V l,r )3 3 × Vl,d + Dr   1 1 21/2 × + − 2(3N − 1) (3N + 1) (3N ) + g Vgl sl,r gH, g =

Ugm , Vgl + 0.5Vl,c + Vl,r

where Vlc =

(Vlo − V lr ) (1 − g − sr )



 N . N +2

(2) (3)

(4)

It has to be kept in mind that the model used does assume that there is no gas present in the downcomer, which is the main weakness especially for slurry-gas systems, where, experimentally, it has been observed that the presence of gas in the downcomer is relatively important.

2. Theoretical aspect

Vgl =

1301

(1)

For the slurry system, considered here as a pseudo-homogeneous phase, the same concept for gas slip velocity will be used. Many hydrodynamic models have been developed for airlift reactors. Experimental results obtained in this work will be compared to the García-Calvo et al. (1999) model, based, like almost all published papers in this field, on an energy balance. For induced liquid velocity in the downcomer the fitting parameters are the friction coefficient in both riser and downcomer parts of the external loop airlift reactor and the  parameter. As proposed by the authors the value of the number N (liquid profile parameter) is taken to be equal to 2. For slip velocity calculation, the Zuber and Findlay model is used rather than the fixed value (0.25 m s−1 ) proposed by Garcia-Calvo. From the application of the mass conservation principle between the downcomer and the riser, liquid velocity in the riser is easily deduced from the one in the downcomer.

3. Experimental The investigation of global hydrodynamics (regime transitions, overall gas hold-up, liquid circulation velocity and the effect of small particle with volume fractions up to 7% vol.) has been conducted in two external loop airlift reactors of 10 cm ID, 3 m height (EL1) and 15 cm ID, 6 m height (EL2). The heat transfer measurements have been conducted only on an EL1 reactor. The experimental set-up is shown in Fig. 1. The sparger used represented in Fig. 1 consists of perforated tubes: 60 holes of 1 mm for the EL1 reactor and 64 holes of 1 mm for the EL2 reactor. Glass beads of 90 m mean diameter (measured by a MALVERN laser granulometer) having a density of 2.537 have been used as solid particles, and water and air as liquid and gas phases. Pressure probes as well as sampling taps are mounted at different axial positions of the riser and the downcomer. The velocity of the samplings was chosen to ensure isokinetic conditions in the sampling taps and the riser (or the downcomer) so that the flow pattern will not be disturbed. An ultrasound probe based on the Doppler effect has been used to measure both the bubble rising velocity and frequency at different axial and radial positions of the riser. Volume-averaged gas hold-ups have been measured using the usual manometric method in two-phase flow, which has been combined to the sampling of slurry suspension through the sampling taps in three-phase flow; both measurements are performed at different axial positions in the riser. The solid hold-up distribution in different sections of the riser has also been determined using the Wenge method (see Fig. 2). This method is as follows: after shutting off the gas feeding, the pressure measured at the

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Legend:

2nd floor

6 1, 2, 3, 4

7

1, 2, 3,4 7

riser

1, 2, 3 & 4 probe

gas – liquid separator

1, 2

downcomer

1 Conductivity probe 2 Pressure probe 3 Ultrasonic probe 4 Monofiber probe 5 Heat probe 6 Septum for tracer injection 7 Sampling tap

5 7

3, 4

1, 2, 3,4

2

Acquisition card

1st floor 7 1, 2, 3, 4

3, 4

sparger

7

downcomer to riser junction

5 7

7

1, 2, 3, 4

1, 2

1, 2, 3, 4 6

ground level

Fig. 1. Experimental set-up.

7 measuring time

Pressure data acquisition Signal (volt)

6.9 6.8 6.7 6.6

Pressure

6.5

gasing off

gasing on

6.4 6.3 2 phases

6.2 gas desengagement

6.1

Time

6 0

20

40

60

80

100

120

Time (s) Fig. 2. Wenge method application: sample of pressure data acquisition.

140

H. Dhaouadi et al. / Chemical Engineering Science 61 (2006) 1300 – 1311

1303

During the experimentation, the probe is implanted horizontally in the column and a chosen temperature difference is adjusted by varying the electrical power. The probe can be placed at different axial positions. The heat probe temperatures is usually between 24 and 29 ◦ C for liquid temperatures of 23 to 26 ◦ C. This difference reduces the free thermal convection effects and variations of fluid properties in the heat transfer film. Tests showed that an increase of the probe temperature by a few degrees has no effect on the measured heat transfer coefficient. Also, it is verified that a steady-state value of the probe temperature is reached within 2 s of the application of heating current. 4. Results and discussion

Fig. 3. Home-made heat probe used in this work.

bottom of the studied section will increase within the air disengagement period. According to the Wenge method (Wenge et al., 1995), we have nearly 10–20 s (for the particles used in this study) of stabilised pressure within which measurements could be done. Pressure Measurements then enable solid and gas hold-up calculations using the following relationships:   p l s = 1− , (5) l − s p0   s − l p g = 1 − . (6) + s p0 l Superficial gas supply rates were varied up to 0.1 m s−1 . The global liquid circulation velocity in the downcomer is determined using a tracer technique. More details concerning the hydrodynamic measurement techniques and calculations can be found elsewhere (Dhaouadi, 1997). For heat transfer measurements, the probe used (see Fig. 3) allows to estimate the transfer coefficient between the probe and the bulk at various superficial gas, and by consequent liquid, velocities. The probe is constituted of a brass cylinder of 15 mm external diameter. This cylinder is electrically heated by the interior with a resistance rolled on a central brass core of 5 mm diameter. The length of the resistance is 1 m, its diameter 1 mm and the power 12–500 W under 12–80 V. The probe is provided with six thermocouples installed in different positions on the external surface of the brass tube. Two other thermocouples are installed in the column: one to measure the temperature of the heated volume element near the probe and the other, installed at the bottom of the column, to control the average temperature of the liquid. The heat transfer from the resistance to the external probe cylinder is ensured by an interstitial mixture which has to present a high thermal conductivity and be as homogeneous as possible to avoid hot points on the resistance. A mix of 50 m diameter copper particles with water and a wetting agent was selected.

A transition from the homogeneous to the heterogeneous flow regime is observed in the three-phase airlift reactor (see Fig. 4). Compared to the gas–liquid–solid system, and for the EL2 reactor, the transition in flow regime for an air–water system is delayed to higher velocities because of the relatively high liquid velocity in the riser. For the EL1 reactor, and in comparison to the air–water system case, this transition is observed for lower velocities with the GLS system; this shift towards smaller transition velocities increases slightly with an increase of the solid quantity present in the reactor (see Fig. 5). A solid presence would favour the formation of larger bubbles even in a homogeneous regime. The extension of the slip velocity concept to the three-phase systems clearly shows that the presence of a solid leads to bubble coalescence by increasing the turbulence. In the airlift reactor case, liquid and gas velocities are interdependent and the liquid movement in the riser delays the flow regime transition. Also, It appears that a plot of the slip velocity vs. the overall gas+liquid velocity (or vs. the liquid velocity) yields a clear distinction between flow regimes. Finally, in a homogeneous flow, the slip velocity decreases with increasing liquid velocity UL while in a heterogeneous regime it is an increasing function of UL . Within the range of solid concentrations investigated, only the initial solid addition has an influence on liquid velocity, which has also been reported by Lu et al. (1994). Indeed, the liquid velocity decreases rapidly when changing from a water to a slurry system with 3% vol solid content; it decreases very feebly when the solid passes from 3 to 5 and 7% vol (see Fig. 6). By itself, the slurry viscosity variation (not important within the solid content range studied) does not seem to justify the liquid velocity variations. This slowing can be linked to the upward flow driving force variation as follows: Compared to the water system, the riser gas hold-up in a three-phase reactor is slightly lower. Simultaneously, the downcomer gas hold-up is somewhat higher in the three-phase system than in the two-phase system (see Figs. 7 and 8). Adding solid to water therefore leads to a decrease in the driving force and consequently to a decrease in the liquid circulation velocity. It was experimentally found that the supplementary solid addition does not induce any change in gas hold-up in the reactor. The solid phase is homogeneously distributed in the reactor and does not affect the driving force any more. This has

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0.7 water water + 3% vol. solid water + 5% vol. solid water + 7% vol. solid

0.65 0.6

Vgl (m/s)

0.55 0.5 0.45 0.4 0.35 0.3

EL2

0.25 0.2 0

0.02

0.04

0.06

0.08

0.1

0.12

Ug,r (m/s) Fig. 4. Solid effect on slip velocity and flow regime transition (EL2).

0.55

0.5

heterogeneous regime (coalesced bubbles)

EL1

Vgl (m/s)

0.45 water water + 4% vol. solid water + 7% vol. solid

0.4

0.35

0.3 homogeneous r egime (dispersed bubbles)

0.25 0

0.03

0.06

0.09

0.12

0.15

Ug,r (m/s) Fig. 5. Solid effect on slip velocity and flow regime transition (EL1).

been experimentally verified for all solid hold-ups studied and even for low gas superficial gas velocities in an EL2 reactor and from superficial gas velocities of about 0.02 m s−1 in EL1. Results obtained for a solid hold-up of 5% and for the EL2 reactor have been reported in Fig. 9 which shows the uniformity of the solid hold-up in the downcomer and at different axial positions of the riser. This figure also compares results obtained by both Wenge and sampling methods in the riser. Experimental data are well described using the model of Calvo-Garcia. For the liquid velocity, the fitting parameters are the friction loss coefficients and the  parameter (see theoretical aspect paragraph and Figs. 6 and 7).

A change in gas hold-up reflects a change in bubble velocity and often a change in bubble size. An increase of bubble velocity corresponds to a decrease of the gas hold-up. Variations of bubble velocity are often related to their size change. Liquid velocity in the riser is more affected by a solid presence than the gas hold-up. The increase of solid hold-up increases the system turbulence, which leads to a decrease of the velocity of the rising bubbles (see Fig. 10) and therefore that of the bubble size. Indeed, the bubble frequency increases with increasing solid hold-up (see Fig. 11 ).The wider frequency spectra obtained in a three-phase system as compared to the two-phase one indicates a larger bubble size distribution.

H. Dhaouadi et al. / Chemical Engineering Science 61 (2006) 1300 – 1311

1305

0.5 0.45 EL2

0.4

Ul,r (m/s)

0.35 0.3 0.25 0.2

water water + 3% vol solid

0.15

water + 5% vol solid

0.1

water + 7% vol solid

0.05

E. Garcia-Calvo model

0 0

0.02

0.04

0.06

0.08

0.1

0.12

Ug,r (m/s) Fig. 6. Solid effect on liquid velocity in the riser—E. Garcia-Calvo model application (EL2).

14 12 EL2

εg,r (%)

10 8 6

Water Water + 3% vol. solid Water + 5% vol. solid Water + 7% vol. solid E. Garcia-Calvo model

4 2 0 0

0.02

0.04

0.06

0.08

0.1

0.12

Ug,r (m/s)

Fig. 7. Solid effect on gas hold-up in the riser—E. Garcia-Calvo model application (EL2).

Variations in global and local hydrodynamic parameters in three-phase airlift reactors can be the consequence of one, or of a combination of the following mechanisms:

• the presence of a solid in the bubble wakes, stabilises trails of bubbles which increase their upward velocity and reduce the gas hold-up.

• bubble coalescence producing larger and faster bubbles, • turbulence generating smaller bubbles with lower rising velocity, • changes in physico-chemical slurry properties. In fact, when particles are distinctly smaller than bubbles, and their terminal velocity is lower than the liquid velocity, the slurry can be considered as a pseudo-homogeneous phase. The solid presence increases both the density and the slurry viscosity. The resulting effect is a decrease in the liquid velocity, even though it has the opposite effect on bubble rise velocity, and

The hydrodynamic behaviour of airlift reactors depends strongly on the reactor geometry, dimension, shape and riser to downcomer junction (and vice versa). On the other hand, the hydrodynamic behaviour is also strongly dependent on the flow regime. Thus, correlating the mean hydrodynamic parameters according to the flow regime will be helpful. Within the range of solid concentrations investigated a correlation is proposed which takes into account the slurry viscosity rather than the solid concentration (the slurry is supposed to be a pseudo-homogeneous phase) for the estimation of both

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6

EL2

5

εg,d (% )

4

3

2 water 3% vol. solid 5% vol. solid 7% vol. solid

1

0 0

0.02

0.04

0.06

Ug,r

0.08

0.1

0.12

(m.s-1)

Fig. 8. Solid effect on gas hold-up in the downcomer (EL2).

6

5

εs,r,d

4

EL2 ; εS = 5 %

3

Z=0.25 m Z=1.35 m Z=3.80 m mean solid holdup (wenge method) downcomer

2

1

0 0

0.02

0.04

0.06

0.08

0.1

0.12

Ug,r (m/s) Fig. 9. Solid hold-up in the riser and the downcomer (EL2).

superficial liquid velocity and gas hold-up in the riser. The proposed interpolation correlations are available for EL1 reactor configuration. The solid presence modifies the slurry viscosity which can be easily estimated by Thomas’ formula (1965) ((cP ) = 1 + 2.5s + 10.052s + 2.73 × 10−3 e16.6s ). The correlations proposed for both gas hold-up and liquid velocity are of the following type:   sl c b Ulr or gr = aU gr . (7) water

In the same way, the external heat transfer coefficient to a tube will be linked to the superficial gas velocity in the riser, Ugr , according to the corresponding flow regime. The parameters of the proposed correlations differ according to the flow regime; the correlations yield a good agreement with experiments (see Figs. 12 and 13). The values of the parameters of these correlation developed here for both a three-phase system and viscous media (ranging between 0.001 and 0.033 Pa s) for the estimation of both liquid velocity and gas hold-up in the riser and taking

H. Dhaouadi et al. / Chemical Engineering Science 61 (2006) 1300 – 1311

1307

0.8 EL2

0.7 0.6

Vb,r (m/s)

0.5 0.4 0.3 water 3% vol. solid 5% vol. solid 7% vol. solid

0.2 0.1 0 -0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

r/R (-) Fig. 10. Solid effect on radial profile of rising bubble velocity in the riser.

300

250 water 3% vol.solid

Frequency

200

150

100

50

0 -100

-80

-60

-40

-20

0

20

40

60

80

100

Vb,r (cm/s) Fig. 11. Solid effect on bubble size distribution.

into account the flow regime can be found in the following tables. For superficial liquid velocity in the riser Parameter

Homogeneous

Incertitude

Heterogeneous

Incertitude

a b c

0.71 0.44 −0.12

0.057 0.022 0.009

0.42 0.26 −0.12

0.021 0.021 0.006

And for the gas hold-up in the Parameter

Homogeneous

Incertitude

Heterogeneous

Incertitude

a b c

1.59 0.90 −0.05

0.146 0.026 0.007

0.45 0.46 −0.08

0.034 0.032 0.008

Concerning the heat transfer, measurements in EL1 were conducted in both a two- and three-phase flow system. It was found

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Ul,r (m/s)

1

0.1

0.01 0.001

0.01

0.1

1

Ug,r (m/s) Fig. 12. Riser liquid velocity experimental data correlation with regard to the flow regime.

1

ε g,r

0.1

0.01

0.001 0.001

0.01

0.1

1

Ug,r (m/s) Fig. 13. Riser gas hold-up experimental data correlation with regard to the flow regime.

that the heat transfer coefficient, measured with the homemade probe, increases with the gas flowrate augmentation (see Fig. 14). From the data obtained from the two-phase system, and taking into account the flow regime, we propose the following correlation relating the external heat transfer coefficient and the superficial gas velocity in the riser: 0.48 he (W/m2 K) = 26.09 × 103 Ugr ,

(m s

−1

Ugr expressed in

), for the homogeneous regime.

0.33 hc (W/m2 K) = 15.95 × 103 Ugr ,

Ugr expressed in

(m s−1 ), for the heterogeneous regime.

Tests conducted with two different values of solid hold-up (1% and 3% vol) in the EL1 reactor clearly show that the increase of solid concentration induces a decrease of the heat transfer performance of the reactor (see Fig. 15). For superficial gas velocities less than 0.018 m s−1 (1% solid content case) and 0.036 m s−1 (3% solid content case) a negligible effect on he is observed so that the results are quite similar to the water case. This is due to the fact that for small gas velocities solid hold-up is lower in the EL1 reactor as previously indicated and the presence of particles has a negligible effect. This effect is furthermore reinforced by the negligible effect of the gas velocity on the liquid velocity and by the higher imprecision of

H. Dhaouadi et al. / Chemical Engineering Science 61 (2006) 1300 – 1311

1309

10000

hce = 15.95.103Ug,r0.33 heterogeneous regime

2

hce (W/m .k)

hce = 26.09.103Ug,r0.48 homogeneous regime

EL1 reactor

1000 0.001

0.01

0.1

1

Ug,r (m/s) Fig. 14. External heat transfer coefficient correlation with regard to the flow regime.

8000 7000 Ug,r=0.036 m/s

5000

2

hce (W/m .K)

6000

4000 3000 Ug,r=0.018 m/s

2000

water 1% vol. solid 3% vol. solid

EL1

1000 0 0

0.02

0.04

0.06

0.08

0.1

0.12

U g,r (m/s) Fig. 15. Solid effect on the external heat transfer coefficient.

measurement in these gas velocity ranges. Complex phenomena occur when particles, gas and liquid are in contact. Especially, the heat amount transferred in each phase is still difficult to evaluate. 5. Conclusion The effect of the presence of a solid on hydrodynamics and heat transfer characteristics has been investigated. It is experimentally shown that transition in a flow regime occurs at smaller velocities in a three-phase system than in a two-phase system. Measurements of global and local hydrodynamic parameters show that the presence of a solid phase leads to many changes in bubble size and velocity distribution, liquid velocity and gas hold-up. Within the riser section of the reactor, the liquid velocity decreases notably, compared to gas hold-up,

with increasing solid hold-up. The upward force acting on liquid circulation decreases with increasing gas hold-up in the downcomer. The increase of solid hold-up increases the system turbulence; thus, the bubble size and rising velocity decrease. Because of the relatively high liquid circulation rate in airlift reactors, bubble coalescence also occurs, but is counter-balanced by the system turbulence, the gas holdup in the riser is less affected by the presence of solid than liquid velocity. External heat transfer coefficients measured in the airlift reactor show that increasing the solid hold-up decreases the heat transfer. Correlations of global hydrodynamic parameters with operating variables and taking into account the flow regime transition are proposed. Up to now these correlations are, however, only valid for the geometry and size of the airlift reactor on which they have been based.

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Notation a, b, c A D g h H Kf N p P r R S T U V Vgl Vlo Vlc V

correlation parameter cross-sectional area, m2 diameter, m gravitational constant, m s−2 heat transfer coefficient, w m−2 K height of the reactor, m friction coefficient, dimensionless liquid profile parameter pressure, Pa power, W radius, m column radius, m surface, m2 temperature, K superficial velocity, m s−1 interstitial (linear) velocity, m s−1 gas slip velocity, m s−1 interstitial velocity on the axis of column, m s−1 mean interstitial velocity in the core region, m s−1 mean linear velocity, m s−1

Greek letters     

parameter related to two-phase flow, dimensionless difference hold-up, dimensionless viscosity, Pa s density, kg m−3

Subscripts at b c d e g h l m r s sl 0

atmospheric bubble column downcomer external gas heat liquid middle riser solid slurry time zero

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