Gas–Liquid Mass Transfer in Unbaffled Multi-Impeller Stirred Tank Reactors for Vinylidene Fluoride Emulsion Polymerization under Supercritical Conditions: Experimental Pressure-Based Approach and CFD Simulations of Hydrodynamics
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Gas–Liquid Mass Transfer in Unbaffled Multi-Impeller Stirred Tank Reactors for Vinylidene Fluoride Emulsion Polymerization under Supercritical Conditions: Experimental Pressure-Based Approach and CFD Simulations of Hydrodynamics
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Industrial & Engineering Chemistry Research

Cite this: Ind. Eng. Chem. Res. 2025, 64, 51, 24538–24550
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https://doi.org/10.1021/acs.iecr.5c04184
Published December 10, 2025

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Abstract

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A nonisothermal method to measure gas–liquid volumetric mass transfer coefficients (kLa) as well as the gas solubility (in water and polymer particles) has been developed to investigate the effects of monomer pressure (ranging from gas to supercritical conditions), agitation rate (and agitator form), and volume of reactor contents on this important quantity. The method was applied to two different autoclaves with different agitation systems and used to study mass transfer in polyvinylidene fluoride (PVDF) latexes. Computational fluid dynamic (CFD) simulations of the two reactors were performed to estimate the vortex surface area and the turbulent energy dissipation rate. For the two reactors considered, estimates of kLa are demonstrated to be strongly correlated with these two quantities. This suggests that, first, gas–liquid (G/L) mass transfer is in large part governed by the headspace–liquid interfacial area and that a more pronounced vortex enhances mass transfer. Then, the incorporated bubbles are broken into smaller bubbles at a rate that is determined by the turbulent energy dissipation rate. Enhancing bubble breakage increases their specific area and leads to faster dissolution of the gaseous monomer. It is shown that this simplified CFD approach, providing these key factors, can be used to predict global mass transfer rates for this polymerization system when the reactor volume and agitation systems vary.

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Published as part of Industrial & Engineering Chemistry Research special issue “Celebrating the Legacy of Prof. Jose M. Asua: Emulsion Polymerization and Polymer Reaction Engineering”.

1. Introduction

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Stirred tanks are widely used in chemical and process industries for many different types of operations, and the geometry of the vessel and agitators may vary according to the process requirements. (1) For gas–liquid (G/L) reactors, one may encounter reactors where the gas is bubbled directly in the liquid as well as stirred vessels containing compressed gas in the headspace on the top of a liquid where the gas is incorporated into the liquid by agitation across the interface. In the first category, where the gas is bubbled in the vessel, a high length/diameter (L/D) ratio is desirable because this helps increase the time spent by the bubbles in the liquid phase and thus mass transfer between the two phases. (2) In the second category of stirred vessels, there are two sources of G/L mass transfer: the top headspace–liquid interface (or eventually vortex) and the interface between the incorporated bubbles and the liquid phase. In this context, G/L mass transfer can be improved using multiple impellers. Indeed, multi-impeller configurations enhance the mixing efficiency, increase the interfacial surface area within the mixture and improve gas dissolution and utilization. (3−5) Also, the creation of a vortex can be helpful. The vortex depth depends on the agitation intensity, the form of the impellers and their distance from the interface, the possible presence of baffles or other inserts, as well as the viscosity and density of the gas and liquid phases. (6,7)
In conventional emulsion polymerization (i.e., involving a liquid monomer), the stirring intensity is usually not required to be high since even with a relatively low level of mixing, mass transfer limitation of monomer from liquid droplets to water is generally considered to be negligible, and the concentration of monomer in water and in the polymer particles quickly reach saturation. Also, the reduced mixing intensity helps avoid the coagulation of particles. However, in cases where the monomer is gaseous or supercritical, such as the emulsion polymerization of vinylidene fluoride (VDF), the agitation geometry and intensity greatly influence G/L transfer. In the specific case of VDF polymerization, the design and operation of the agitation system as well as the volume of latex in the reactor were found to play determining roles in the kinetics and molecular weights. (8−10) This is true in general for any G/L polymerization system, such as ethylene–vinyl acetate emulsion copolymerization, where the agitation was found to influence the copolymer composition. (2) More recently, Merlin and Schork (2024) (11) demonstrated that in emulsion polymerization of gaseous monomers, mass transfer limitations are strongly dependent on the reactor pressure and used Henry’s law relationship for determining the solubility of the gaseous monomer in the aqueous phase.
The aim of this work is 2-fold: (1) to develop an experimental method for the estimation of the gas–liquid mass transfer coefficient, kLa, and the solubility of monomer in water and in the polymer particles, under different reactor conditions (temperature, pressure, latex volume, and impeller type) for different polymerization reactors. The pressure-drop method is adapted to cases where temperature is not kept constant during the analysis and accounts for possible change in the liquid volume due to polymer swelling by the monomer; (2) the second objective is to explore the possibility of using a simplified CFD approach to predict the turbulent energy dissipation rate and the vortex area and evaluate how they are correlated to the kLa under different operating conditions, thereby allowing us to bypass extensive experimentation each time we change the reactor or agitation system.

1.1. Mass Transfer in G/L Systems

In emulsion polymerization involving a gaseous or supercritical monomer, bubbles are entrained from the headspace of the vessel to the liquid bulk by mechanical agitation. Then, monomer is transferred to the water phase and polymer particles in 2 steps: (i) transfer from the bubbles to the water phase and (ii) then transfer from the water to the particles where the main polymerization takes place. Since the specific surface area of particles (m2 m–3 latex) is much greater than that of bubbles, as their characteristic size is much smaller, mass transfer of monomer from the water phase to the particles can be considered to be instantaneous, i.e., we can assume that polymer particles are in equilibrium with the monomer depending on its concentration in the aqueous phase (i.e., the ratio of monomer concentration in particles to that in water is constant). Hence, the dominant step of monomer mass transfer is the flux of monomer from the bubbles to water. (8,12) The following equation describes the variation of the monomer concentration in water (13)
Vwd[M]wdt=kLaVw([M]wsat−[M]w)−RpVw−Qw[M]w
(1)
where kL is the mass transfer coefficient between the gas bubbles and the water (m3 m–2 s–1), a is the G/L interfacial surface area per unit volume between the two phases (m2 m–3), Rp is the molar reaction rate per volume of water (mol s–1 m–3), Vw is the volume of water, Qw is the exit flow rate of liquid phase (null in batch and semibatch operations), [M]w is the real concentration of monomer in water and [M]wsat is the equilibrium concentration of monomer in water which is a thermodynamic property only affected by pressure and temperature, not agitation. (13) Ultimately, for the estimation of the real concentration of monomer in the water (and thus in the particles), one must determine the values of the product kLa as it is very difficult to measure each quantity separately.

1.2. Volumetric Mass Transfer Coefficient

Several methods are proposed in the literature to experimentally determine kLa in stirred tanks, such as the dynamic pressure method; (5,14,15) the oxidation of sulfite; (14,16) the dehydrogenation of styrene; (16) or image analysis. (17) However, the pressure-drop method is the most frequently used because kLa can be determined by simply monitoring the changes in the pressure caused by the absorption of gas into the liquid phase. (16,18)
In kLa, the mass transfer coefficient, kL, represents the rate of molecular diffusion through the G/L interface, which has been shown to be a function of the intensity of turbulent mixing. (19) To estimate kL from kLa, the total mass transfer area a is required (i.e., bubbles size and gas fraction). (13) Coupling computational fluid dynamics (CFD) with a population balance model of the bubble size distribution can provide an estimate of a. (3) However, such coupling is complex due to the moving free surface and the different magnitude between the vortex area and the bubble size, which are, however, to be estimated simultaneously, so usually a number of approximations must be imposed to get a solution with a reasonable computational cost. For these reasons, it is more common to estimate the product kLa.

1.3. Correlations of kLa

Traditionally, kLa is expressed by empirical correlations based on the experimental data. Factors such as the properties of the liquid and gas phases, gas feeding mode, geometry of the reactor and impellers, agitation rate, stirrer position, and the presence of solid particles can influence mass transfer in G/L systems. (20)
Chaudhari et al. (16) proposed a correlation for kLa (c.f. eq 2) as a function of the agitation speed for a stirred autoclave. Using the agitation rate N instead of the turbulent energy dissipation rate makes the model specific to the impeller design. Other geometric parameters are included to reflect the effect of the autoclave size, the ratio of the gas volume (VG) to the liquid volume (VL) and the different impeller placements (i.e., ratio of the height of the impeller from bottom (hi,1) to the total height of the liquid (hi,2), and the ratio of the impeller diameter (Di) to that of the reactor (Dr)). The correlation, originally developed for a 0.6 L reactor was used successfully to estimate the values of kLa obtained by catalytic hydrogenation for geometrically similar, single impeller reactors of 2 and 5 L.
kLa=1.48×10−3N2.18(VGVL)1.88(DiDr)2.16(hi,1hi,2)1.16
(2)
Recently, Gelinski et al. (9) proposed and identified a comparable correlation for kLa as Chaudhari et al. (16) for VDF absorption in a latex at a high-pressure for a stirred tank with several impellers mounted on a central shaft. kLa was defined as a function of the agitation rate, the ratio of volume of gas to liquid for different orientations of a bottom hydrofoil impeller (down-pumping or up-pumping) (i.e., different tuning parameters were identified for each case) as follows:
kLa=A(VGVL)BNC
(3)
where A, B, and C are tuning parameters. By accounting for the agitation speed, both bubble breakage and the size of the vortex are accounted for. However, given that this reactor was equipped with several impellers, the vortex shape depended strongly on the volume of the liquid. It was found that correlation 3 was valid for only very small changes in the liquid volume in the reactor. Experimental polymerizations showed that changing the liquid levels (at constant agitation rate) in semibatch polymerizations corresponded to large oscillations in the polymerization rate as a function of time, most likely due to changes in kLa. This suggests that a single value of kLa would not be sufficient if we want to describe the polymerization rate and the evolution of the polymer properties for an entire production run in a reactor with multiple impellers with large changes in the liquid volume.
It has been suggested that the turbulent energy dissipation rate can be correlated with mass transfer performance and is more generalizable to other reactor and agitator systems than is simply the agitation speed. (14,20) For this reason, we propose to use this quantity to develop a correlation for mass transfer that can be used in different reactors with different volumes and agitator setups, combined with the information about the gas–liquid interfacial area.

2. Material and Methods

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2.1. Experimental Setup

Experiments were carried out in 2 unbaffled multi-impeller stirred tank reactors with 4 and 4.8 L of capacity, which will be referred to as R4L and R5L, respectively. A schematic of the R4L reactor (diameter = 10 cm with conical bottom) is shown in Figure 1. The reactor was equipped with the agitation system shown and referred to as Setup2. (8) It contains three units of pitched blade impellers (6 blades 45°) of 5 cm diameter and one hydrofoil A345 of 8 cm diameter in the pumping-up position at 3.6 cm from the bottom. The second polymerization unit, R5L (diameter = 12 cm), presented in Figure 2, is slightly larger than R4L, with a different height-to-diameter (H/D) ratio (R4L and R5L have H/D ratios of 5.3 and 3.7, respectively). In addition, the bottom is hemispherical rather than a truncated cone as is the case with R4L. The reactor was equipped with the mixing Setup MN1 composed of one propeller AX1 of 10 cm of diameter placed at 0.8 cm from the bottom of the reactor and 7 units of propeller AX1 of 8 cm diameter stacked above each other.

Figure 1

Figure 1. Scheme of the R4L reactor and mixing Setup2.

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Figure 2

Figure 2. Scheme of R5L reactor and mixing Setup MN1.

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While it is common in G/L reactors to sparge the gas from the bottom of the reactor or via a dip tube, these options are not practical in our system due to coagulation and fouling inherent to VDF emulsion polymerization. Hence, the monomer was fed into the reactor in the headspace and entrained into the liquid phase by agitation with multiple impellers.
Both reactors are 316 stainless steel jacketed reactors, equipped with thermal baths (Lauda PL1 for R4 and Lauda RP2045 for R5L) to control the reactor temperature by oil circulation in the jacket (no oil circulates in the reactor cover or the bottom). The reactor temperature was measured by a thermocouple (J Atex for R4L and an Atex PT100 for R5L), protected by a metal tube, and placed inside the reactor. The temperature probes and dip tubes served as baffles. The inlet and outlet temperatures of the circulating oil in the jacket were measured with a platinum resistance PT100, and a thermometer is used to measure the exterior reactor temperature for both reactors. Reactor pressure was monitored with an ATEX pressure sensor (Keller PA-23EB for R4L and Keller PA-23SYEI for R5L). For experiments with pressure lower than 60 bar, vinylidene fluoride monomer was injected directly from a storage bottle to the reactor. For experiments with higher pressure, it was first liquefied in a condenser maintained at −20 °C by thermic bath (PL1, Lauda); then, it was fed to each reactor via a dual diaphragm pump (Metering pump Novados H1, SPXflow). The electrical signals (temperature and pressure) were sent to a PC for data acquisition and reactor temperature control.

2.2. Dynamic Monomer Absorption Experiments

The monomer absorption experiments were done using a methodology derived from the original work of refs (16,18). The reactor was charged with deionized water or latex at the desired solid content (SC) (10%, 20%, and 38%), and the liquid was agitated at 150 rpm (this low speed prevents coagulation). When the reactor reached the desired temperature (83 °C for R4L and 74 °C for R5L), agitation was maintained for 30 min. The agitation was then turned off, and we waited for 30 min for T and P equilibrium. Monomer was injected in the system until the desired pressure (30–90 bar). After monomer injection, the system was kept without agitation until the pressure reached equilibrium (30–60 min). The agitation was turned on at the desired speed, and the pressure drop was monitored (30–60 min).

2.3. CFD Simulations

CFD computations are done with NiceFlow software (LEMMA), a mixed finite-volume finite-element vertex-based solver dedicated to unstructured simplicial meshes. The core technologies consist of an advanced MUSCL-like formulation based on the Edge-Based Reconstruction formalism (21,22) and a powerful adaptive anisotropic remeshing toolbox emanating from the work of the Inria γ laboratory. (23,24)
For the current study, a diphasic liquid–gas incompressible model is used. In the CFD calculations, the liquid phase was assumed to have the same properties as water. The fact that the mass transfer experiments were run with a 10 wt % solid content latex (5.6 vol %) should not lead to significant errors. The gas phase stands for the gaseous or supercritical monomer, so in some cases, it is nearest to a liquid than a gas (the critical pressure of VDF is 44 bar at 83 °C). Indeed, some of the experiments are run with a true gas, while others are run with a supercritical fluid. As the density of supercritical VDF is approximately 300 kg m–3 at 88 bar and 83 °C, it still remains lighter than water and must be drawn into the liquid phase by agitation. A SIMPLE-like method is used for the velocity/pressure resolution. The underlying formulation is robust enough to enable computations without the need to iterate the velocity/pressure coupling. The impeller motion is accounted for using an original Immersed Boundary method operating in the rotating frame, meaning that the immersed moving objects are not the impellers but the baffles. This choice enables us to maintain good resolution at the impeller surface, which is mandatory in order to compute the dissipated power accurately.
The free surface dynamics are resolved using an interface tracking method based on the so-called Level-Set formalism. (25−27) Here, it is fundamental to highlight the fact that this method is not able to capture simultaneously the continuous part of the free surface (i.e., vortex surface) and the dispersed part (i.e., bubbles incorporated by the violent motion of the free surface due to the interaction with the impellers). This is due to an unavoidable lack of spatial resolution: assuming bubbles of size ∼1 mm and having in mind that the Direct Numerical Simulation (DNS) of an accurately resolved bubble requires at least a mesh size ∼diameter/10, we end up with a computational requirement of a mesh of several billions of cells which is still above current capabilities in CFD. Since our objective is to restrict ourselves to conventional CFD resources, we deliberately choose not to pursue the DNS approach. With this in mind, conversely, we can reasonably assume that the Level-Set method provides sufficient accuracy to capture the continuous part of the interface, namely, the vortex surface area. A direct implication of the Level-Set method’s limitation in capturing fine-scale features, such as bubbles, should be highlighted: this drawback, inherent to all interface tracking methods which operate on under-resolved meshes, will have, as primary consequence, a mass conservation problem. To avoid long-time divergence of the mass balance, a crude global conservation method inherited from the work of Smolianski (28) is used. Note that such a correction can have an impact on the continuous-dispersed topology of the interface. In some sense, the mechanism of the global conservation algorithm is as follows: small bubbles that cannot be tracked by the Level-Set method are reinjected automatically in the continuous gas phase. Also, it should be emphasized that restricting the CFD analysis to the continuous part of the interface is a strong assumption given that the dynamics of monomer bubbles play a key role in mass transfer. In fact, the overall transfer coefficient can be expressed as the sum of the transfer through the vortex and bubbles areas. So, the energy dissipation, which governs bubble breakage, will also be considered in the correlation of the mass transfer coefficient. Our approach deliberately avoids the costly computation of bubble areas. Specifically, the objective is to assess whether, to some extent─at the expense of generality, simple CFD simulations can be used to derive meaningful correlations of the form of the mass transfer coefficient, while still achieving good agreement with experimental data. Here, the exponents A, B, and C are treated as tuning parameters that compensate for the lack of detailed knowledge on bubble dynamics.
Finally, intensive usage of the adaptive anistropic remeshing technology is done in order to obtain an acceptable accuracy (notably for the kinetic energy balance law between the dissipated power, the pressure force work, and the viscous force work). All of the simulations are run for 100 periods of rotation. Remeshing based on the Hessian of |U| is performed at each period. A mesh complexity increase of growth rate of 1.8 is applied every 10 periods. The first mesh is very crude (a few hundred thousand tetrahedra, i.e., not more than a representation of the surface domain) and is permanently improved up to a final mesh with around one hundred million tetrahedra. The simulations are run on 64 AMD Milan cores. The time step is around 1° of rotation per iteration. BDF2 temporal integration scheme and a V4MUSCL spatial scheme (29) is used. Computations took around 4 days.

3. New Pressure-Drop Approach to Estimate kLa and Monomer Solubility

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Figure 3 shows the normalized pressure profiles for monomer absorption experiments at approximately 30 bar (± 1 bar) and 10% solid content for both R4L and R5L reactors (i.e., divided by the initial pressure). It can be seen that the pressure decreases once the agitation is turned on (at time 0). A higher agitation speed leads to faster stabilization of the pressure, thus indicating faster monomer dissolution. There is no correlation between the stabilization time and the latex level, since the vortex amplitude changes nonlinearly with the latex volume. As the glass transition temperature is on the order of −40 °C, there should be no significant diffusion limitations in particles on the order of 100 to 200 nm in diameter. A larger volume of latex leads to the absorption of a larger amount of monomer and thus to a more pronounced drop in pressure (but the initial pressure is not exactly the same in all cases, which also influences the final pressure and the solubility). In all cases, the equilibrium information can be exploited to estimate the solubility of the monomer in water and polymer. The pressure measurements in R5L oscillate less, due to a better control of temperature as can be seen in Figure 4.

Figure 3

Figure 3. Normalized pressure during VDF absorption in a latex of 10% SC at 30 bar for (a) R4L at 83 °C, 400 rpm; (b) R4L at 83 °C, 550 rpm; (c) R5L at 74 °C, 400 rpm; and (d) R5L at 74 °C, 500 rpm.

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Figure 4

Figure 4. Temperature during VDF absorption in a latex of 10% SC at 30 bar for (a) R4L at 83 °C, 400 rpm; (b) R4L at 83 °C, 550 rpm; (c) R5L at 74 °C, 400 rpm; and (d) R5L at 74 °C, 500 rpm.

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It can be seen from Figure 4 that the temperature decreases as the agitation is turned on, especially for low latex volumes. Note that feeding monomer at reactor temperature (74 °C for R5L and 83 °C for R4L) is not possible because the pressure would exceed the feeding line security seal limit (60 bar). So, the injection of monomer at a temperature lower than the reactor temperature results in the cooling of the reactor contents. In addition, the absence of agitation during this filling step can lead to the formation of cold spots in the reactor and the reduction of the reactor temperature. Note that in the conventional pressure-based methodology, the temperature is assumed to be constant. So, the first adaptation of the methodology concerns handling the variation in temperature during the measurement due to cooling by monomer injection. Even though the temperature is better controlled in R5L, it cannot be considered constant during the treatment of pressure decay. Second, the nature of the liquid phase used in our work differs from the usual pressure-based methodology. In a latex, the monomer will be absorbed by water but also by the polymer particles, with a much higher solubility. Hence, we adapted the methodology of refs (16,18) to account for the monomer absorbed by both water and the polymer particles.
Table 1 presents the model used to evaluate pressure decay due to the absorption of a gaseous monomer into a latex. Knowing that the PVDF produced by emulsion polymerization is semicrystalline (approximately χ = 50%), the monomer only dissolves in the amorphous polymer. The model is simulated in the MATLAB, and the optimization is done using the function lsqnonlin, to provide the mass transfer coefficient kLa and the monomer saturation concentration (i.e., solubility) in water (using experiments with only water) or in the amorphous polymer (using experiments with latex). The minimization criterion is the difference between the experimental and modeled pressures. The temperature used throughout the model is the measured value (as a function of time.) By this way, the model accounts for possible cooling and compression effects when injecting the monomer.
Table 1. Model for Absorption of the Monomer into a Latex
monomer mass balance
(4)
concentration of monomer in polymer (mol m–3)
(5)
concentration of monomer in amorphous polymer (mol m–3 am pol)
(6)
monomer partitioning coefficient (−) (9)
(7)
number of moles of monomer in amorphous polymer (mol)
(8)
number of moles of monomer in water (mol)
(9)
number of moles of monomer in the gas phase (m3)
(10)
volume of swollen particles (m3)
(11)
volume of monomer in water (m3)
(12)
volume of the latex (m3)
(13)
volume of gas (m3)
(14)
reduced pressure (−)
(15)
compressibility factor for VDF (−)
(16)
pressure (Pa)
(17)
The algorithm used to evaluate kLa, [M]wsat and [M]psat consists of first giving initial estimates of kLa, [M]wsat or [M]psat. Then, [M]w is calculated by the integration of eq 4, and [M]p is calculated using eq 5 and Kwp (eq 7). The number of moles of monomer in the particle, np, (eq 8) and in water, nw, (eq 9) can then be calculated. The change in the latex volume due to polymer swelling is accounted for in Ve (eq 13), from which the headspace volume is calculated to be VG (eq 14). The pressure Psim is then calculated (eq 17), using the measured temperature. The minimization criterion is then the difference between the experimental and the simulated pressures (Pexp – Psim). The optimization function repeats these steps until ∑|Pexp – Psim|2 < 10–3 Pa.

4. Results and Discussion

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4.1. Estimation of Monomer Solubility in Water and Polymer Particles by Model Fitting to Experimental Pressure

Based on the experiments like those presented in the previous section but with only water, the solubility of monomer in water was estimated by fitting the model (Table 1) to the measured pressure. This information is mainly present in the equilibrium period, but we use all periods as we estimate kLa simultaneously. Figure 5 presents the effect of pressure on the solubility of VDF in water at 74 and 83 °C, as measured in two distinct reactor setups. Despite variations in the experimental conditions (including the liquid levels of 2.2, 2.5, and 2.6 L and agitation rates of 400 and 500 rpm) both reactors yielded similar solubility values across the tested pressure range. The values for the equilibrium concentration of VDF obtained were similar, with the solubility at 74 °C being slightly higher than at 83 °C, as expected. We believe that the values are similar due to the low solubility of VDF in water (for example, the solubility of styrene in water at 1 bar is 6.4 mol m–3 versus 0.72 mol m–3 for the VDF estimated in this work through extrapolation of the curve). Indeed, the equilibrium concentration of VDF in water is governed by the pressure and temperature; therefore, the consistency of both estimations confirms the validity of the proposed method. The estimated solubility of VDF in water was then used to estimate the solubility in polymer by redoing the same experiments with a latex.

Figure 5

Figure 5. Solubility of VDF in water at 74 and 83 °C estimated by model fitting to the measured pressure. Experimental conditions for R4L reactor: 400 rpm, 2.2 L of water (30 and 60 bar) and 2.5 L (90 bar). Experimental conditions for R5L reactor: 500 rpm, 2.6 L of water (30, 50, and 90 bar).

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Figure 6 illustrates the pressure dependence of the solubility of VDF in the amorphous polymer phase. The solubility increases with pressure, as was observed for the solubility in water. Note that the curve covers gaseous and supercritical conditions (the VDF critical pressure being 44 bar). Indeed, at elevated pressures the gas density increases and becomes closer to that of the bulk, and in the present system, this increases its solubility in water and in the polymer. The correlations for the solubility of the monomer in water and in polymer as a function of pressure are presented in Table 2. Note that while there is a slight deviation from a direct linearity, there is no sound reason to suspect a strong deviation from Henry’s Law behavior in this case.

Figure 6

Figure 6. Solubility of VDF in amorphous polymers estimated by model fitting to the measured pressure. Experimental conditions: R4L reactor, 83 °C, 550 rpm, and 2.5 L latex at 10% SC.

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Table 2. Correlations of VDF Solubility as a Function of Pressure
concentration of monomer in water at saturation (mol m–3)T (°C)P (bar)
R4L[M]wsat = 0.7299 P830–90
R5L[M]wsat = 0.7462 P740–90
concentration of monomer in amorphous polymer at saturation (mol m–3 am. p.)T (°C)P (bar)
R4L[M]p,amsat = 20.762 P830–90

4.2. Estimation of the Volumetric Mass Transfer Coefficient kLa by Model Fitting to Experimental Dynamic Pressure and Estimation of the Energy Dissipation and the Vortex Area by CFD

Model fitting with the measured pressure also provides an estimate of kLa based on the transition period (although the solubility is revealed mainly from the equilibrium period). It can be estimated both in water or in the latex, but estimations in the latex are more precise due to the more pronounced pressure decay. Figure 7 presents the influence of the latex volume and agitation rate on kLa in reactors R4L and R5L, respectively. In both reactors and across all tested latex volumes, an increase in the agitation rate consistently led to an increase in the kLa. Indeed, increasing the agitation rate has two effects. First, as the agitation rate increases, the turbulent energy dissipation increases, which increases the rate of bubble breakage, and so the G/L interfacial area increases. Also, the vortex may deepen, especially in unbaffled vessels, thus increasing mass transfer surface area from the headspace to the liquid. (6,30)

Figure 7

Figure 7. Experimental volumetric mass transfer coefficient (kLa) as a function of the latex volume at 30 bar and 10% SC (a) R4L reactor at 83 °C and (b) R5L reactor at 74 °C (vertical dashed lines = midpoint of the impeller blades).

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The effect of the liquid volume exhibits a more intriguing effect on kLa. Remember that kLa is governed by mass transfer from the headspace to the liquid (with a vortex in our case) as well as mass transfer from the bubbles to the liquid (which is influenced by the turbulent energy dissipation rate). First, increasing the volume leads to a linear decrease in the turbulent energy dissipation rate (given in W kg–1). More importantly, in an unbaffled stirred tank, kLa is dependent on the vortex magnitude, which itself depends on the distance between the highest immersed impeller and the top G/L interface. This correlation is not linear in a multi-impeller vessel. Indeed, Figure 7 shows oscillations in kLa with the volume of the latex in both reactors. This reveals a nonlinear change in the vortex amplitude when the latex volume. As illustrated in Figure 8, the area of the vortex (av) obtained by CFD increases when the liquid level is close to the midpoint of the impeller blades, highlighted in the figures by vertical dashed lines, and decreases when the liquid level is far from the impeller. So, kLa reaches its maximum when the liquid level is near the midheight of the impellers, which corresponds to the maximal vortex area. Conversely, kLa decreases when the latex level is significantly above the highest immerged impeller due to the reduction in vortex area. The optimal interaction between the impeller-generated flow and the free surface occurs; therefore, when the liquid level aligns with an impeller center, and as we have several impellers, we get oscillations in kLa. This phenomenon further supports the importance of the impeller choice and spacing between impellers, as demonstrated experimentally by Ecosia et al. (2022) and Aladro et al. (2024). (8,10)

Figure 8

Figure 8. Estimation of the vortex area av (by CFD) as a function of volume of water for (a) R4L reactor and (b) R5L reactor (vertical dashed lines = midpoint of the impeller blades).

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The reactor R4L is equipped with the multi-impeller that is referred to as Setup2, which has poor performance due to the large spacing between the impellers (8 cm) and the flux direction of the bottom hydrofoil being in a pump-up position. (8) Regarding reactor R5L, the mixing Setup MN1 is developed to enhance mass transfer by decreasing the distance between impellers (4.5 cm) and to reduce shear by using AX1 propellers. As seen in Figure 7, the mixing performance of Setup MN1 (in reactor R5L) is superior with kLa values ranging from 0.0140 to 0.1505 s–1, while kLa values for Setup2 (in reactor R4L) ranges from 0.0023 to 0.0575 s–1 at the best. This is in part because reactor R5L has a higher av in all cases than R4L (Figure 8) and in part due to a higher energy dissipation (due to a higher number of impellers).
The energy dissipation values were obtained by CFD at different volumes of the liquid phase (water) and agitation rates. Figure 9 shows the volume-averaged energy dissipation. As expected, an increase in the energy dissipation is observed with the agitation speed. Regarding the effect of the volume, two phenomena occur when increasing the liquid volume. First, the same agitator power is dissipated into a larger volume, so the energy dissipation is expected to decrease continuously with the latex volume. However, when a new impeller is immerged into the liquid as the latex level increases, this leads to a sudden increase in energy dissipation. This effect is more pronounced for R4L at 550 rpm, where we can see a change in the slope (i.e., the rate at which the average energy dissipation decreased with the volume slightly increases) when a new impeller gets immersed (e.g., 2.2 L for the fourth impeller). Also, it can be seen that the decrease in the energy dissipation with the latex volume is more pronounced in the reactor R4L, while for R5L energy dissipation was almost constant with the latex volume. First, in a multi-impeller system, the energy dissipation is proportional to the number of impellers, hence Setup MN1 provides a higher energy dissipation than Setup2 at the same rate of agitation. Also, the differences in impeller diameter (Di) and power number (Np) between the pitched blades (Di = 5 cm) in Setup2 and AX1 (Di = 8 cm) in Setup MN1 play a big role in energy dissipation. An approximation of ε in a multi-impeller agitator is given by the following equation:
ε=NiNpN3Di2
(18)
where Ni is the number of immersed impellers. In conclusion, this means that with Setup MN1 a given value of kLa can be reached at a lower agitation speed than with Setup2.

Figure 9

Figure 9. Average turbulent energy dissipation rate obtained by CFD for (a) reactor R4L and (b) reactor R5L (vertical dashed lines = midpoint of the impeller blades).

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Higher pressures, above the critical pressure of VDF, were then investigated. Figure 10 shows the effect of the pressure on the kLa. As the pressure increases from 30 to 88 bar, kLa increases from 0.0061 s–1 to 0.0255 s–1. In addition to solubility effects that change the bulk viscosity and interfacial tension, increasing the pressure increases the gas density. This consequently influences the hydrodynamic behavior, particularly the vortex formed during agitation. Figure 11 shows that as the pressure increases from 30 to 88 bar, the vortex area increases, and av is doubled in the case of 550 rpm. A less significant effect on av is observed at 400 rpm. The pressure effect on the vortex is a complex nonlinear process. For instance, mixing two immiscible liquids of similar density would lead to a huge vortex. In addition to influencing the vortex, the increase in the gas density may influence the bubble breakage rate. Indeed, an example of a breakage kernel of the bubbles is given by the following equation. (31)
Ω(d)=C1d−2/3ε1/3exp⁡[−C2σρGε2/3d5/3]
(19)
where d is the bubble diameter, ρG is the gas density, σ is the G/L interfacial tension, and C1 and C2 are tuning parameters. So, the breakage frequency increases with the bubble diameter, the energy dissipation, the gas density, and with a lower interfacial tension. Smaller gas bubbles will have a larger specific mass transfer exchange area. Both the increased vortex and breakage frequency contribute therefore to an increase in the mass transfer area, which leads to an increase in kLa with pressure, as depicted in Figure 10.

Figure 10

Figure 10. Volumetric mass transfer coefficient at 83 °C estimated by model fitting of the pressure absorption. Experimental condition for R4L reactor, Setup2, 550 rpm, and 2.5 L latex.

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Figure 11

Figure 11. Vortex (a) area and (b) shape obtained by CFD at R4L reactor, Setup2, and 2.2 L of water for different pressure.

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4.3. New Correlations for kLa

Based on the experimental observations, the following correlation of kLa is proposed. It is described as a function of both the vortex area and the average turbulent energy dissipation rate.
kLa=AεBavC
(20)
where A, B, and C are tuning parameters. Equation 20 of kLa was simulated using ε and av obtained by CFD. Figures 12 and 13 show that kLa estimations are very close to the experimental measurements. The identified parameters for the two reactors at different temperatures are listed in Table 3. First, it is important to remember that kLa contains the area, which is composed of the vortex area av plus the bubble surface area, with the latter being importantly influenced by ε. Another influence of ε is on kL (i.e., the rate of molecular diffusion through the G/L interface). So, the effects cannot be completely decorrelated. However, increasing ε by 5% implies 6% change in kLa, while a 5% change in av implies 5% change in kLa for R4L at 400 rpm for instance. So ε has a larger effect on kLa.

Figure 12

Figure 12. (a) Volumetric mass transfer coefficient (kLa) (experimental and by eq 20) and (b) correlation versus experimental values of kLa at 83 °C, 30 bar, and 10% SC for the R4L reactor (vertical dashed lines = midpoint of the impeller blades).

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Figure 13

Figure 13. (a) Volumetric mass transfer coefficient (kLa) (experimental and by eq 20) and (b) correlation versus experimental values of kLa at 74 °C, 30 bar, and 10% SC for the R5L reactor (vertical dashed lines = midpoint of the impeller blades).

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Table 3. Parameters of the Volumetric Mass Transfer Coefficient (kLa) Correlation (eq 20)
parameterR4L 83 °CR5L 74 °C
A0.73960.2476
B1.19701.5014
C0.96520.6058

5. Conclusions

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We proposed a new pressure-based methodology to determine mass transfer of a gaseous monomer into a latex based on the pressure and temperature measurements, which allows to estimate both kLa and the equilibrium concentrations of monomer (i.e., solubility) in water and in the polymer particles. A variable experimental range of pressure was considered, covering gaseous and supercritical conditions. The experimental results were coupled with CFD simulation of the vortex area and energy dissipation to propose correlations of kLa as a function of the latex level in the reactor. The study of mass transfer of VDF in emulsion has shown that

  • The pressure has a positive effect on the solubility of VDF in water.

  • kLa increases with pressure, since the difference between density of gas and liquid decreases, so the area of vortex increases.

  • Increasing the rate of agitation enhances mass transfer since it increases both the energy dissipation (so bubble breakup) and the vortex.

  • When volume of latex (i.e., position of the headspace–liquid interface with respect to the impeller) is close to the middle of impeller, kLa increases, due to the increased vortex. However, the energy dissipation generally decreases with the volume.

The pressure-based methodology proposed in this work to determine kLa was validated in 2 reactors. It is advantageous compared to other methods since it accounts for variations in temperature and can provide the solubility values simultaneously. However, it is important to point out that kLa was found to be correlated with the air–liquid interfacial area in the current work, besides the energy dissipation. While there are 2 dip tubes in this reactor (that reduce the size of the vortex), there are no baffles per se. In reactors equipped with baffles that reduce the size and persistence of a vortex in a given reactor, the interfacial area is almost limited to the reactor section. In this case, it is possible that another quantity (e.g., energy dissipation in the surface layer as opposed to the volume average energy dissipation which clearly does not work well) that can be correlated with kLa will need to be identified if baffles are used to significantly reduce the size of the vortex in a specific reactor configuration. This is not particularly different from the approach used here as such quantities are also calculated in the CFD simulations.

Author Information

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  • Corresponding Author
  • Authors
    • Micheli Nolasco Araujo - Universite Claude Bernard Lyon 1, CP2M UMR 5128, CNRS, 69616 Villeurbanne, France
    • Thomas Boucherès - Arkema, Centre de Recherche Rhône-Alpes Rue Henri-Moissan, 69149 Oullins-Pierre-Bénite, France
    • Nida Sheibat-Othman - Universite Claude Bernard Lyon 1, CNRS, LAGEPP UMR 5007, 69616 Villeurbanne, France;  Orcidhttps://orcid.org/0000-0002-2822-9566
  • Notes
    The authors declare no competing financial interest.

Nomenclature

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a

mass transfer area (m2)

av

vortex area (m2)

d

bubble diameter (m)

Di

diameter of impeller (m)

Dr

diameter of reactor (m)

kL

mass transfer coefficient (s–1 m–2)

Kwp

monomer partitioning coefficient (−)

[M]w

concentration of monomer in the water (mol m–3)

[M]wsat

concentration of monomer in the water at saturation (mol m–3)

[M]p

concentration of monomer in the polymer (mol m–3)

[M]p,am

concentration of monomer in amorphous polymer (mol m–3)

[M]psat

concentration of monomer in the polymer at saturation (mol m–3)

Mw,m

molecular weight (kg mol–1)

ng

number of moles of monomer in reactor headspace (mol)

np

number of moles of monomer in polymer (mol)

nw

number of moles of monomer in water (mol)

N

agitation rate (rps)

Ni

number of immersed impellers

P

gas pressure (Pa)

Pc

critical pressure (Pa)

Pr

reduced pressure (−)

Qw

exit flow rate of liquid phase (mol s–1)

R

ideal gas constant (J mol–1 K–1)

Rp

rate of polymerization (mol m–3 s–1)

t

time (s)

T

reactor temperature (K)

VG

volume of gas (m3)

VL

volume of liquid (m3)

Vmp

volume of monomer in polymer (m3)

Vmw

volume of monomer in water (m3)

Vp

volume of polymer (m3)

Vpart

volume of swollen particle (m3)

Vw

volume of water (m3)

Z

compressibility factor

Greek Letters

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ε

energy dissipation (W kg–1)

ρ

density (kg m–3)

σ

interfacial tension (N m–1)

χ

crystallinity (%)

Ω

bubble breakage frequency

References

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  • Abstract

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    Figure 1

    Figure 1. Scheme of the R4L reactor and mixing Setup2.

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    Figure 2

    Figure 2. Scheme of R5L reactor and mixing Setup MN1.

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    Figure 3

    Figure 3. Normalized pressure during VDF absorption in a latex of 10% SC at 30 bar for (a) R4L at 83 °C, 400 rpm; (b) R4L at 83 °C, 550 rpm; (c) R5L at 74 °C, 400 rpm; and (d) R5L at 74 °C, 500 rpm.

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    Figure 4

    Figure 4. Temperature during VDF absorption in a latex of 10% SC at 30 bar for (a) R4L at 83 °C, 400 rpm; (b) R4L at 83 °C, 550 rpm; (c) R5L at 74 °C, 400 rpm; and (d) R5L at 74 °C, 500 rpm.

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    Figure 5

    Figure 5. Solubility of VDF in water at 74 and 83 °C estimated by model fitting to the measured pressure. Experimental conditions for R4L reactor: 400 rpm, 2.2 L of water (30 and 60 bar) and 2.5 L (90 bar). Experimental conditions for R5L reactor: 500 rpm, 2.6 L of water (30, 50, and 90 bar).

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    Figure 6

    Figure 6. Solubility of VDF in amorphous polymers estimated by model fitting to the measured pressure. Experimental conditions: R4L reactor, 83 °C, 550 rpm, and 2.5 L latex at 10% SC.

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    Figure 7

    Figure 7. Experimental volumetric mass transfer coefficient (kLa) as a function of the latex volume at 30 bar and 10% SC (a) R4L reactor at 83 °C and (b) R5L reactor at 74 °C (vertical dashed lines = midpoint of the impeller blades).

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    Figure 8

    Figure 8. Estimation of the vortex area av (by CFD) as a function of volume of water for (a) R4L reactor and (b) R5L reactor (vertical dashed lines = midpoint of the impeller blades).

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    Figure 9

    Figure 9. Average turbulent energy dissipation rate obtained by CFD for (a) reactor R4L and (b) reactor R5L (vertical dashed lines = midpoint of the impeller blades).

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    Figure 10

    Figure 10. Volumetric mass transfer coefficient at 83 °C estimated by model fitting of the pressure absorption. Experimental condition for R4L reactor, Setup2, 550 rpm, and 2.5 L latex.

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    Figure 11

    Figure 11. Vortex (a) area and (b) shape obtained by CFD at R4L reactor, Setup2, and 2.2 L of water for different pressure.

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    Figure 12

    Figure 12. (a) Volumetric mass transfer coefficient (kLa) (experimental and by eq 20) and (b) correlation versus experimental values of kLa at 83 °C, 30 bar, and 10% SC for the R4L reactor (vertical dashed lines = midpoint of the impeller blades).

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    Figure 13

    Figure 13. (a) Volumetric mass transfer coefficient (kLa) (experimental and by eq 20) and (b) correlation versus experimental values of kLa at 74 °C, 30 bar, and 10% SC for the R5L reactor (vertical dashed lines = midpoint of the impeller blades).

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  • References

    This article references 31 other publications.

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      Moucha, T.; Linek, V.; Prokopová, E. Gas Hold-up, Mixing Time and Gas-Liquid Volumetric Mass Transfer Coefficient of Various Multiple-Impeller Configurations: Rushton Turbine, Pitched Blade and Techmix Impeller and Their Combinations. Chem. Eng. Sci. 2003, 58 (9), 1839– 1846,  DOI: 10.1016/S0009-2509(02)00682-6
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