Importance of Bubble size distribution in Gas-Liquid Flows

Ravindra Aglave, Thomas Eppinger & Ashkan Davanlou *

*Presenting Author

ABSTRACT

Gas-liquid system are found abundantly in the chemical & processes industries. Knowledge of gas volume fraction, its variation in throughout the volume and eventual influence on mass transfer and reactions is absolutely essential for good design. Experimental measurements have allowed to look in to such systems to a good level of details, resulting in to generation of correlations. These correlations are commonly used in the design process. However, they are very much limited to the size, type and character of the lab or pilot scale system from which they were created. This leaves process engineers with a task of ascertaining if a particular design will meet all the process conditions or not. CFD simulations have allowed process engineers to look at plant scale designs due to the ability to investigate them virtually using computer models.

INTRODUCTION

Gas injection in reactors, bubble columns etc. is common unit operation in the chemical industry for many processes such as fermentation, hydrogenation, chlorination etc., to realize mass transfer, reaction and heat transfer. In this work, we are focusing on gas sparged systems. Typical injection system consists of a sparger that injections bubbles. As bubble move upwards, they coalesce or break-up depending up on the material properties of the fluids (viscosity, surface tension, density) involved as well as the hydrodynamic behavior of the system (velocities, shear, recirculation etc.). Modeling has been increasing used in scale-up and design of such reactors.

In many such models, certain assumptions are made. As more research and numerical techniques are developed, these assumptions are relaxed to gain more accuracy. In gas liquid flows, it is commonly assumed that the bubble sizes are uniform or have a mean value. However in reality, there is a varying size of bubbles. This assumption can introduce large error in the calculation of gas volume fraction. Moreover, it can give a misrepresentation of the distribution as well. This means there could be area where the gas volume fraction could be critically lower than process requirement, destabilizing the process or even resulting in an unsafe condition. In this paper we present the results from a simpler method, known as S-gamma model, which accounts for coalescence and break up of bubbles to understand the size distribution of bubbles.

There has been considerable work done to capture the bubble size distributions in academia. Advanced methods such as QMOM or DQMOM can calculate the coalescence and breakup and can account for the transition of bubbles from one size class to another [1]. Such methods are highly compute intensive (several orders of magnitude higher) while the incremental accuracy may not be very high.

In this paper we are presenting the results from a simpler method, known as S-gamma model [2], which accounts for coalescence and break up of bubbles and uses only a distribution

function. Due to its simple nature, the compute costs of this method are not high. However, one can get much higher fidelity in prediction of integral gas holdup as well as its distribution.

GAS-LIQUID FLOWS IN REACTORS

DISSOLVED GAS

In gas-liquid reactors, dissolved gas concentration in the fluid is of great importance. Reaction rates and heat release is sensitive to the dissolved gas concentration affecting the yield and performance of the reactor. The supply of the gas takes place via gas sparging and the concentration of dissolved gas in the liquid depends inter alia on the generated bubble size, gas phase fraction, pressure and temperature. The bubble sizes can be locally very different in the reactors and impair performance.

Experimental Measurements

Experimental measurements are common to determine the bubble sizes. These measurements then form the basis of all subsequent design calculations including gas volume fraction and mass transfer. Different sparger designs can give a different baseline injected bubbles. The measurement location and direction (vertical or horizontal) also affects the bubble size distribution. That is because bubbles coalesce or break-up from the point of injection to the point of measurement. One such laboratory system is shown in Figure 1. A horizontal pipe with holes is

Figure 1: Experimental set-up with a sparger and measurement probe (left). Comparison of bubble size distribution measurements with model results (top right). Frequency distribution of bubble sizes (bottom right)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10 510 1010 1510

cdf Q0 [‐]

particle diameter dp [µm]

Experiment

STAR‐CCM+

used as a sparger. The bubble sizes are measured at few centimeters above the sparger using an optical probe. In such a situation, one has to guess the size of the bubbles to us in a simulation at the sparger to match those measured at a certain distance above. These results are shown in Figure 1. However, the shape of this distribution is different in different measurement planes asshown in Figure 2.

Figure 2: Various measurement directions of optical probe (left). Corresponding bubble size distribution shape.

It becomes clear from this investigation that bubble sizes change significantly even in small distances due to coalescence and breakup. If one were to rely on a single measurement and use it for scale-up, it may result in a suboptimal performance of the equipment. TO investigate such multiphase system more rigorously, computational fluid dynamics simulation can be used to investigate the variation in bubble size distribution, its effect on gas holdup variance and consequently on mass transfer and reaction rates.

Modeling Methods

There are two ways to model such multiphase systems:

In the Euler-Lagrange method individual elements of the disperse phase are modeled explicitly for each bubble. The advantages of this method are the availability of detailed information on each blister. The disadvantage is that for a larger number of bubbles the computing time increases extremely.

Measurement Position Could Have a Significant Influence on the result.

Vertical

Horizontal

In the plane

In the Euler-Euler model a complete set of Navier-Stokes equation is solved for each phase, weighted and coupled by the phase fractions. In contrast to the Euler-Lagrange modeling, one can get only averaged values for each phase in a cell which means that information related to an individual bubble is 'lost'. But, the number of bubbles has no influence on the computing times. This method is well suited for high number of bubbles. Also population balance models can be considered easily, which is not trivial in Euler-Lagrange methods

SIMULATION CASE STUDY EXAMPLE

A multi-stage fermentation reactor at production scale with four Rushton turbines and a bottom mounted sparger is shown in Figure 3 based on the work of Vrabel [4]. The geometric dimensions are listed in the adjoining table. The simulation of the air / water multiphase flow was performed with the Euler-Euler method using STAR-CCM+. Turbulence is treated with the k-ε model and the movement of the impeller is modeled with the Moving Reference Frame approach. For the simulations, some simplifications have been made, for example, only a 60 degree segment was modeled with periodic boundaries. This simplification is based on a previous study [2] which showed negligible effects on the results.

Figure 3: Geometrical sketch and details of the system investigated

Property Value Rushton Impellers 4 Blades per impeller 6 Impeller Diameter (D) 0.7 m Liquid Height (H) 6.55 m Liquid Volume 22 m3 Tank Diameter (T) 2.09 m Baffles 4 Density of water 997 kg/m3 Dynamics viscosity of water 8.84 e-4 Pa-sDensity of air 1.18 kg/m3 Dynamic viscosity of air 1.85 e-5 Pa-sImpeller frequency 115 rpm Volumetric air flow rate 0.0263 m3/s Superficial air velocity 0.77 cm/s

SIMULATION RESULTS

The results of the liquid and the gas phase distribution are shown in (Figure 5, left). In the case of an unsparged vessel, a significant compartmentalization can be seen comparable to experimental observations. Once the vessel is gassed, the ascending gas bubbles influence the flow field, which leads to an additional mixing between the compartments [5]. The gas phase distribution shows areas with varying gas content. An accumulation takes place in the centers of the recirculation zones and in the wake of the agitator blades which is confirmed by experimentally observed gas hauling.

Figure 4: (Left) Rising gas bubbles in an aerated vessel cause additional vertical mixing, thereby reducing the size of circulation cells. (Right)Accounting for coalescence and breakup in modeling (purple column) gives better comparison with experimental measurements (red column) than making a single size assumption

Integral Gas Hold-up

The integral gas holdup, which is an important design parameter, can be significantly overestimated if the bubble size is assumed to be constant 1mm (Figure 4, right). This is because the small bubbles do not rise so quickly and also better follow the flow of liquid. A constant bubble size of 2 or 3 mm however underestimates the gas holdup. It can be concluded that with a constant bubble size somewhere between 1 and 2 mm the experimental gas holdup can be matched well. However, then the simulation is no longer predictive. At production scale, simulation results should be reasonably accurate in comparison to experiment to be of value.

However when a bubble size distribution along with coalescence and breakup is also modeled, then no adjustment or tuning of the simulation is needed to get a match with the experiments. It is regardless of what bubble size distribution is specified at inlet. It automatically adapts the 'right' bubble size distribution as soon as the gas bubbles come into the vicinity of the first agitator. There, the bubbles are broken due to the local liquid flow conditions and therefore becomes independent of the specified bubble size at the inlet. The inlet boundary condition only affects the bubble size distribution in the small area between sparging and impeller.

MODELING BUBBLE SIZE DISTRIBUTION AND THE S-GAMMA PRINCIPLE

One method to model size distribution and population balance is by defining different classes and solve for each of these classes transport equations with exchange terms. This can be numerically expensive. In STAR-CCM+, a simplified model, the so-called S-Gamma model is implemented. The S-Gamma model consists of only two additional equations.

In this model it is assumed that the bubble size distribution can be described by a log-normal distribution (1).

√exp (1)

Such a distribution is determined by only two parameters: the mean and variance. A balance

quantity Sγ is defined as follows which is dependent on the moments of the distribution.

(2)

Then transport equations for S-Gamma (eq. 3) can be formulated and solved as:

⁄

∙ ⁄ u (3)

In STAR-CCM+ a total of 3 equations for the 0th, 2nd and 3rd moment are used as follows: 0th moment, S0: it is the particle number density, n.

0Sn (4)

2nd moment, S2: related to the interfacial area density, A

0

2)(2 SdddPdn

iA (5)

3rd moment, S3: is the dispersed volume fraction, α

3

0

3

6)(

6SdddP

dn

(6)

Since the 3rd moment represents the volume fraction, which is already calculated in the Eulerian multiphase framework only two additional equations have to be solved. On the left side of the equation is a storage term and a convective term while the right side has terms for bubble break-up & coalescence and for any other changes in the size distribution. The Sauter mean diameter (SMD) d32 is

22

332

16

SS

Sd

(7)

This can be e.g. an increase in size due to addition of mass. An expression of the break up Sbr

and coalescence Scl  is still needed and is given by:

(8)

∬ , ∆ ,,

(9)

Both break up and coalescence expressions are each a function of the material properties and the flow conditions, as well as the turbulent dissipation and / or shear rate or the frequency of collisions and coalescence probability. The details of these so called coalescence and break-up kernels, their model parameters as well as the exact formulation for the transport variables are described in detail elsewhere [2], [3].

MODELING MASS TRANSFER

The transfer of mass is modeled with a mass transfer coefficient “kl’, the interfacial area “al” and the driving concentration difference.

(10)

(11)

(12) The interfacial area “al” and the bulk concentration are by calculated STAR-CCM+. The mass

transfer coefficient is defined using a Sherwood-correlation (Ranz-Marshall form) (8) depending on the flow. Henry's law is assumed for the equilibrium concentration of gas at the interface.

As briefly mentioned in the introduction, the fermentation reactor should be operated in the

region of the plateau. Under this condition, the consumption of oxygen by the micro-organisms in the entire reactor can be assumed to be constant. Accordingly, we have assumed a constant oxygen consumption rate 0.22 gm/m3-s. In addition, we examine only the steady state. This means that the integral oxygen transfer rate (OTR) must be identical to the integral consumption of oxygen.

MASS TRANSFER RESULTS

Before discussing the results, it is worthy to note that the mean value of gas bubbles given at the inlet condition is immaterial to the results. This is because as soon as the bubbles come near

the first set of impellers, they will break-up and later coalesce in the quiescent regions. Therefore only minor errors are introduced in the region between the sparger and the first set of impellers. We used a mean size (SMD) of 2 mm with a log normal distribution. The simulations were run in steady state and no time history was followed. Our focus was a faster turnaround time for industrial scale while capturing coalesce and break up effects for higher accuracy.

Gas Hold-up

The proof with several examples of lab scale measurements and corresponding S-gamma model comparisons can be found in [6] and [7]. The purpose of the paper is to prove industrial application of S-gamma model. This can make it an enabler to address the scale-up process with higher accuracy with faster turnaround time for the lab, bench and production scales.

This section shows the differences between a simulation with a constant bubble size and a simulation with a bubble size distribution (BSD) including coalescence and break up. The constant bubble size has been chosen to be more or less equal to the Sauter mean diameter (SMD) with bubble size distribution. In Figure 5 (left), the local gas phase volume fraction for the constant bubble size case is shown while Figure 5(right) shows for the case with BSD. It is to be noticed that the fluid flow is not able to carry gas bubbles below the sparger. Also, the distribution is significantly more homogeneous and the gas hold up value is lower (Integral value of 12.8%) with a constant single bubble size in comparison to the case with BSD (integral value of 17.1%)

Figure 5: Gas volume fraction for case a) constant bubble size of 3.5mm, gas holdup= 12.8% b)bubble size distribution with SMD=3.3mm, gas holdup=17.1%

The results for the case with bubble size distribution (Figure 6) are shown somewhat in more detail. The Sauter mean diameter (SMD) in the radial direction has a very small value in the vicinity of the agitating elements. In this region, the turbulent dissipation and the shear rates are quite large, causing breakup of bubbles and lead to small bubble sizes.

Figure 6: Sauter mean diameter and particle size variance for case with bubble size distribution

Figure 7: Mass transfer rate and interfacial area density for (left) case with constant bubble size of 3.5mm and (right) bubble size distribution case with SMD=3.3mm

In the area between the stirring elements, the bubble size and the SMD increase due to

coalescence. In the right part of the section plane, the variance is shown: In the region of high

 

Constant Bubble Size                               Bubble Size Distribution with SMD=3.5mm 

turbulent dissipation, the size distribution is relatively narrow. Then the bubble start coalescing in the region of low shear rates, a significant spread in the fields occurs in which the bubbles coalesce again. Mass Transfer

Figure 7 compares the results between constant bubble size and BSD cases. In each plot, left half of the sectional plane shows oxygen mass flow density, while interfacial area density is shown on the right half of the figure. In both cases, no mass transport is seen below the sparger, since no bubbles are indeed available. Figure 7 (left) shows a fairly uniform distribution (or little variation) above the sparger ring for both variables. Figure 7(right) however is completely different. Due to the break-up of the bubbles in the impeller region and the resulting small bubbles contribute to a fairly large interfacial density of up to 1000 m2/m3. This very large phase interface then also leads to a correspondingly high local mass transfer, which gradually decreases with height. Dissolved Gas

The dissolved oxygen (right) and net oxygen flux (left) for the two cases is shown in Figure 10 and 11. Again, there is a higher inhomogeneity (but total dissolved oxygen is also higher) for the case with BSD while constant bubble size assumption case shows a more regular pattern.

The left portions of each figure displays all cells in which there is an oxygen-net flow (transfer minus the local consumption by the microorganisms) of the bubbles in continuous phase. The transparent areas represent regions where the consumption by the MO is greater than the local mass transfer. Since oxygen is dissolved sufficiently everywhere and diffusion can be neglected, the oxygen must be transported convectively there. This indicates that hydrodynamics (convection) plays a significant role in transport of oxygen and the bubble size distribution affects this transport. Therefore it should not be neglected.

Figure 8: Net oxygen flux and dissolved oxygen (DO) for the case of constant bubble size of 3.5mm.

In both cases, enough oxygen is present in the reactor, so that no limitation of the reactions rate is to be expected due to oxygen deficiency. However, in the case of bubble size distribution, the dissolved oxygen level is significantly higher.

Figure 9: Net oxygen flux and dissolved oxygen (DO) for the case of bubble size distribution with SMD = 3.5 mm

This means either a higher safety factor is available (overdesign) in case of failure of sparger. Or alternatively, a reduced sparging/air flow can be used and thus allow a reduced pump performance and lower energy costs without compromising the performance. This again indicates that accounting for bubble size distribution gives more accurate representation of the oxygen distribution and can help in cost reduction by removing unnecessarily high safety factors.

CONCLUSIONS

Coalescence and break-up of bubbles are an essential part of gas-liquid flows. Accounting for these two phenomenon is important when simulating such flows. We have shown that when consideration of bubble size distributions with coalescence and breakup for the simulation of multiphase stirred tanks is given, it leads to significantly different results and more accurate results.

Generally accounting for coalescence and break-up had been done using compute intensive models such as QMOM/DQMOM. In our work we have used a relatively simple S-gamma model for accounting for these phenomenon. The S-gamma model is able to give higher accuracy. It is not as compute intensive as QMOM/DQMOM models. This makes the S-gamma model industrially more attractive to use. Its simplicity also makes it useful for performing multiple simulation studies to explore better designs.

REFERENCES

[1] K. Swiderski, D. Caviezel, M. Labois, D. Lakehal, “Computational modelling of gas–liquid multiphase flows with DQMOM and the N-phase Algebraic Slip Model”, Computational Methods in Multiphase Flow VIII, p299, WIT Transactions on Engineering Sciences, Vol 89, 2015

[2] S. Lo, D.S. Zhang, J. Comp. Multiphase Flows, 1(1), pp. 23-38, 2009

[3] STAR-CCM+ V11.02 users guide, CD-adapco

[4] P. Vrabel et al. (2000), “COMPARTMENT MODELAPPROACH: Mixing in Large Scale Aerated Reactors with Multiple Impellers”, Chem. Eng. Sci. 55

[5] T. Eppinger, N. Tracy, G. Bastide, R. Fitzsimmons & R Aglave, “Investigation of the Influence of Different Mesh Types on Accuracy and Calculation Time in Stirred Slim Multi-Stage Impeller Vessels”, AIChE Annual Meeting, 2014

[6] S. Lo, A .Splawski & B.J. Yun, “The importance of correct modeling of bubble size and condensation in predicting sub-cooled boiling flows”, JThe Journal of Computational Multiphase Flows pp299-308, 2012

[7] M. Tandon, A. Khanolkar, A. Splawski & S. Lo, “Validation of Numerical Simulations of Gas-Liquid Systems in a Vertical Pipe”, 8th International Conference on Multiphase Flows (ICMF 2013), Jeju, S. Korea, 2013, available in pdf format on the web via https://www.researchgate.net/publication/261876116_Validation_of_Numerical_Simulations_of_Gas-Liquid_Systems_in_a_Vertical_Pipe