Novel Approach and Correlation for Bubble Size Distribution in a

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Thermodynamics, Transport, and Fluid Mechanics

Novel Approach and Correlation for Bubble Size Distribution in a Slurry Bubble Column Reactor Operating in the Churn-Turbulent Flow Regime Omar M Basha, and Badie I. Morsi Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b00543 • Publication Date (Web): 03 Apr 2018 Downloaded from http://pubs.acs.org on April 4, 2018

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Industrial & Engineering Chemistry Research

Novel Approach and Correlation for Bubble Size Distribution in a Slurry Bubble Column Reactor Operating in the Churn-Turbulent Flow Regime Omar M. Basha and Badie I. Morsi* Department of Chemical and Petroleum Engineering, University of Pittsburgh Pittsburgh, PA 15261, USA Keywords: Dynamic gas disengagement (DGD) technique, Slurry bubble column reactors, bubble size distribution, Sauter mean bubble diameter, bubble size correlation * Corresponding author: [email protected]

Abstract A novel approach to obtain bubble size distributions and Sauter mean bubble diameters (d32) in three-phase slurry bubble column reactors (SBCRs) is presented. This approach is based on 720 experimental runs using the dynamic gas disengagement (DGD) technique for various gasliquid-solid systems inside a pilot-scale-SBCR (0.3-m ID/3-m height), in a wide range of pressures (1-30 bar), temperatures (298–530 K) and solid concentrations (0-17 vol.%). The bubble size distributions were obtained using an iterative energy balance algorithm based on transient experimental pressure drops and forces exerted on bubbles in the disengagement cell. Bubble size distributions were described using log-normal distribution functions and the mean and variance were determined for the DGD runs. The mean and variance were correlated as functions of the system’s physical properties and operating conditions, and were used to predict d32 values with an Absolute Average Relative Error (AARE) of 11.72%. The approach was also applied to d32 values available in the literature with an overall AARE of 24.27%.

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Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1.

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Introduction and Background

The knowledge of gas bubble sizes and their holdups, among other parameters, is required for proper operation, modeling and optimization of multiphase reactors

1-2

as such parameters are

needed for solving reactor equations (mass, energy and momentum balances) whether for numerical modeling or Computational Fluid Dynamics (CFD) simulations. 3-7 Intrusive and nonintrusive experimental methods have been developed and employed to investigate such parameters. 8-9 In bubble column reactors 2, 10-18 and SBCRs, 6, 19-27 the manometric method (also known as the hydrostatic-head method) was often used to obtain the overall gas holdup; and the Dynamic Gas Disengagement (DGD) technique, coupled with a correlation relating the bubble rise velocity to the bubble diameter, such as that by Fukuma et al., 28 was employed to calculate the gas bubble sizes as well as their corresponding holdups. This paper is mainly concerned with the behavior of gas bubbles and their holdups in SBCRs. In the manometric method, the pressure change in the reactor at a given temperature, solid concentration and superficial gas velocity is monitored using multiple pressure transducers located along the height of the reactor. Whereas in the DGD technique, the feed gas to the reactor is suddenly interrupted after reaching a steady-state, and the change of the pressure drop between two points in the reactor, connected to a highly-sensitive differential pressure (dP) cell, is monitored as a function of time until all gas bubbles disengage. In general, the pressure drops in multiple sections along the reactor height are monitored, and the pressure drop profile is used to determine the gas holdup and size of the disengaged gas bubbles. Based on using the DGD technique, various authors 22, 24, 29-30 observed that the large gas bubbles move quickly upwards through the slurry in a plug-flow manner, whereas, the small gas bubbles

2


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move slowly and are often entrained within the slurry-phase. This finding led to the development of the “two-bubble” or “bimodal” class model, which has been widely used to represent the gas bubbles in the convective and dispersive flow schemes within SBCRs

31-32

in one-dimensional

(1-D) and two-dimensional (2-D) modeling of multiphase reactors. 6, 33-34 Critiques of the conventional DGD technique: It should be mentioned that despite the popularity of the bimodal class model, lumping gas bubbles into only two classes is a gross oversimplification of the complex hydrodynamics in SBCRs. Actually, (1) numerous investigators

35-41

demonstrated that the actual gas bubbles data

did not follow a bimodal size distribution, but they were in fact governed by complex breakup and coalescence mechanisms as shown in Figure 1; 36, 42-44 (2) the bimodal gas bubble distribution cannot explain the enhanced gas-phase mixing at higher superficial gas velocities, since it assumes that the increase of the volume fraction of large gas bubbles drives the gas mixing to approach a plug-flow, which contradicts experimental findings;

45-47

and (3) in a strongly

coalescing or highly turbulent flow regime, the assumptions that there are no bubble-bubble interactions and that the dispersion is axially homogenous at the moment when the gas flow is interrupted are not entirely accurate. It is important to note that in the coalescing flow regime, the bubble-bubble interactions play a significant role in bubble formation, a behavior which is oversimplified with the assumption of a bimodal bubble size distribution.

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Coalescence Mechanisms Random Collisions

Wake Entrainment

Breakup Mechanisms Turbulent Impact

Shearing off

Surface Instability

Figure 1. Bubble Coalescence and Breakup Mechanisms in SBCRs 42-43

Moreover, the DGD technique was reported to inherit many uncertainties and problems, such as: (1) Schumpe and Grund

48-49

raised some concerns about the DGD technique, including: (i) the

subjectivity involved in obtaining an accurate disengagement profile during large gas bubbles disengagement, (ii) the waterfall effect or the downward flow of liquid during the bubbles disengagement and its impact on the rise velocities of the small gas bubbles, which are retained in the liquid-phase, and (iii) the errors introduced by the bubbles entering the dispersion zone as the pressure in the plenum chamber equilibrates with the hydrostatic pressure in the reactor after the interruption of the feed gas to the reactor; (2) the technique does not account for the small gas bubbles disengaging along with the fast-moving large bubbles and the demarcation between large and small gas bubbles is subjective; (3) the technique does not differentiate between the gas bubble and the swarm properties, since the bubble rise velocity is not the same as the swarm rise 4


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velocity; and (4) it should be pointed out that the correlation by Fukuma et al., 28 which has been widely used in DGD calculations, was developed using air-tap water system in the presence of glass beads at ambient conditions. The glass beads used in their study had diameters from 0.056 to 0.46 mm and solid concentrations between 13 and 50 vol.%. This brings into question the extrapolation of such a correlation to predict the bubble rise velocity, and consequently the bubble diameters in SBCRs. This is because in a SBCR operating under industrial conditions, the gas bubbles behavior follows a probabilistic nature and the hydrodynamics are significantly more complex than those observed under ambient conditions.44, 50-51 The main focus of this paper is to develop a novel approach, which enables more accurate representation of the different gas bubble classes present in a SBCR operating under high pressures and temperatures and to develop accurate correlations for predicting the Sauter mean bubble diameter under such conditions.

2.

Experimental

2.1

Gas-liquid-solid systems used

The gas-liquid-solid systems used to develop this novel approach are shown in Table 1. A total of 720 DGD experiments, coupled with the manometric method to measure the overall gas holdup, were conducted in a high-pressure, high-temperature pilot-scale SBCR (0.3-m ID and 3m height), provided with a 6-arms spider gas distributor. Description of the SBCR and the gas distributor used is available in the literature

5

and experimental and calculation details of the

manometric method, used to measure the overall gas holdup, as well as the conventional DGD technique, used to obtain the bubble sizes and distributions, can be found elsewhere.6, 22-23, 52 A brief description of the two methods is given in the following. 5



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Table 1. Gas-liquid-solid systems and operating conditions used in this study Reference

Liquid

Solid

Inga and H2, CO, N2, Morsi 20, 53 CH4

Hexanes Mixture

H2, CO, N2, He, CH4

Isopar-M

Iron oxides Glass Beads, Al2O3

Behkish et al. 22, 52 Sehabiague and Morsi 23

Sehabiague et al. 6

Gas

N2, He N2, He

P (bar)

T (K)

1-8

Ambient

0-15

0.06–0.35

19

1.7-30 298-473

0-10

0.06-0.39

88

0- 5

0.06-0.39

247

0-17

0.1-0.3

366

C12-C13 Paraffin Iron Oil, Light F-T oxide, 1.7-30 298-530 Cut, Heavy F-T Al2O3 Cut Molten F-T Iron-based 4-31 380-500 Reactor Wax catalyst

Cs No. of uG (m/s) (vol.%) Points

2.2 Manometric method to calculate the overall gas holdup The manometric method was used to measure the overall gas holdup in the pilot-scale SBCR for the gases in the corresponding slurry (liquid-solid) systems given in Table 1. At given operating conditions, the pressure drop between the two legs of the dP cell in the SBCR was measured when the system reaches steady-state. The overall gas holdup calculation in this method is based on the following assumptions: (1) the slurry and gas phases are well mixed in the control volume between the two legs of a differential pressure cell (dP), as shown in Figure 2; and (2) the impact of the frictional forces on the pressure drop is negligible. 8, 54

6


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Figure 2. Dynamic Gas Disengagement Approach (Reproduced from 55)

The overall gas holdup calculation method is as follows. The passage of gas bubbles through the slurry-phase alters the pressure drop along a differential element (dh) of the reactor, which can be expressed as:

(1)

The density of the gas-liquid-solid system (ρGLS) in the rector is:

(2)

Where ɛG, ɛL and ɛS are the volume fractions of the gas, liquid and solids in the reactor, respectively. If the volume fraction of solids in the slurry-phase (Cs) is used, Equation (2) becomes:

7



1 1

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(3)

Where Cs is expressed as:

1

(4)

Substituting Equation (3) into Equation (1) and integrating from the bottom leg (HB) to the top leg (HT) of the dP cell gives:

1 1

(5)

If assumption (1) holds, the gas holdup and the solid concentration can be considered constant between HB and HT, leading to: 1 1

(6)

Since the distance between the two legs of the dP cell is known (∆Hcell), the overall gas holdup can be directly calculated at any pressure drop (∆Pcell) using:

1 !"## 1 $ 1 !"## 1

(7)

2.3 Conventional DGD technique to calculate the bubble sizes The conventional DGD technique was used to obtain the gas bubble sizes and their distributions as a function of time in the pilot-scale SBCR for all the gases in the corresponding slurry (liquidsolid) systems given in Table 1. In this technique, after reaching steady-state, the gas flow into the SBCR is suddenly interrupted and the dP cell continues to measure the transient pressure drop between the two legs. The increase of the pressure drop with time (t) during the DGD process is primarily due to the change in the overall density within the DGD cell.

6

The

8


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corresponding gas holdup is then calculated until all small gas bubbles ultimately disengage from the DGD cell. The calculation method of the bubble sizes and their distributions is based on the following assumptions: (1) under given conditions, the rate of gas disengagement of each bubble is constant; (2) once the gas flow to the reactor is interrupted no coalescence or breakup of gas bubbles occurs during the disengagement; and (3) the internal circulations in the reactor do not affect the bubble rise velocity. 2 Figure 3 shows a typical transient pressure drop and gas holdup (εG) profile obtained in this study under the given experimental conditions. In order to use these profiles to calculate the small and large bubbles gas holdups, two methods are commonly used. The first method assumes that the small and large gas bubbles disengage independently and that the break in the disengagement profile is due first to a complete disengagement of the large bubbles, followed by a complete disengagement of the small bubbles. It should be mentioned that the break in the disengagement profile is rather subjective, which could affect the holdups of the large and small gas bubbles.

16

The second method assumes that

the small gas bubbles disengage within the wakes of the large bubbles, and that the break in the disengagement profile is due to a complete disengagement of the large bubbles, while the small bubbles are continuing to disengage. It is important to note that the large fluctuations at the start of each run are due to the disengagement of large gas bubbles, which in turn increases the uncertainty in the estimation of the disengagement time.

19

Also, it is obvious that the second

method is more biased towards the small gas bubbles than the first method.

9



0.07

Large bubble disengagement

Small bubble disengagement

∆tLarge

∆tSmall

Pressure Drop (MPa)

0.06 0.05 0.04 0.03 0.02 0.01

T = 412 K, P = 21.1 MPa, Ug = 0.28, Cs = 10 vol.%

0 0.48

0.42

,-

0.36

εG

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.3 0.24 0.18 0.12 0

4

8 Time (s)

16

12

Figure 3. Transient Pressure and gas holdup profiles during the DGD process in the SBCR (Reproduced from 55)

The calculations of the gas holdup and corresponding bubble size using the conventional DGD technique are typically carried out as follows: The volume of the gas bubbles, which leaves the dP cell region with a height equals ∆Hcell can be represented by the decrease of the total gas holdup as follows: ()

,&

()*+

' '

(8)

Consequently, the total gas holdup is: 10


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/

. ,& &01

(9)

The rise velocity of each size of the gas bubbles can be calculated at any time (t) from: 23,&

!"## '

(10)

Since the presence of solid particles should be considered in the SBCR due to their settling velocity, the correlation by Fukuma et al. 28 is usually used to calculate the gas bubble size (db,i) from the bubble rise velocity: 3,&

4 23,& 1.69

(11)

The Sauter mean bubble diameter (d32) is then calculated using the following equation: 84

3.

8 ∑& :& 3,& 4 ∑& :& 3,&

(12)

Novel Approach to calculate the diameter and number of gas bubbles

In this novel approach, all turbulences within the SBCR were assumed to be isotropic and that the bubbles breakup occurs mainly due to the collision with eddies. Also, only eddies with length-scales longer than the bubble diameter were assumed to contribute to the bubbles breakup, whereas those with length-scales much shorter than the bubble diameter were assumed to only contribute to the transport of bubbles without having sufficient energy to contribute to bubble breakup. These latter assumptions are based on the statistical derivation by Prince and Blanch, 56 who reported that small eddies were unlikely to significantly contribute to bubbles breakup, due to their comparatively insignificant momentum when compared with that of large eddies. The calculation procedure consists of the following four steps: 11



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1. Determine the minimum and maximum stable bubble diameters: Considering the above assumptions, the minimum stable bubble diameter is equivalent to the minimum stable eddy length-scale available within the inertial subrange of the turbulent flow, which can be explained as follows. According to the principles of classical turbulence theory developed by Kolmogorov 57 and Hinze 58, a bubble with a diameter within the inertial subrange of the energy spectrum, which is immersed in an isotropic turbulence flow, will break, if the turbulent pressure fluctuations acting on its surface are greater than the surface tension stresses. The ratio between these two values is known as the turbulent Weber number and hence the turbulent breakup of a bubble will only occur, if the turbulent Weber number is greater than a critical value, which has been discussed extensively in the literature 9. Subsequent examinations of this theory, namely by Clay

59-60

, showed that the presence of a critical Weber number

designates bubble diameters greater than or equal to the Kolmogorov length-scale, which represents the minimum hydrodynamic eddy length-scale at which energy is dissipated in the turbulent flow. Therefore, the Kolmogorov’s length-scale, was subsequently used to calculate the minimum stable bubble size according to Equation (13): ;

Novel Approach and Correlation for Bubble Size Distribution in a

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