Inﬂuence of Modiﬁcation of the Geometry of the Wave-Plate Mist Eliminators on the Droplet Removal Efﬁciency— CFD Modelling

: This study is concerned with droplet separation using wave-plate mist separators. The inﬂuence of continuous phase velocity, droplet size, and geometry on droplet removal efﬁciency has been investigated. The following modiﬁcations were analysed: drainage channel presence, length, angle, and modiﬁcation to the streamlined shape. Furthermore, the inﬂuence of the physical parameters of the separated substances on separator efﬁciency was investigated. Calculations were conducted using computational ﬂuid dynamics (CFD). The results were compared with the experimental data from the literature. Based on the results obtained, a new shape of drainage channels was proposed, which is characterised by high droplet removal efﬁciency with relatively low pressure drop.


Introduction
The necessity of removing liquid droplets from the process gas streams is a common problem in industrial technologies. Rising raw materials prices and strict environmental protection plans have become challenging to design more innovative and efficient systems for the droplet removal from process gases and vapour streams. The presence of gasliquid aerosols (mist) can result from spontaneous condensation or desublimation in supersaturated gas-vapour mixtures or chemical reactions. It is undesired in many gasliquid contact devices such as absorbers, quench coolers, or condensers [1]. Eliminating droplets from the gaseous phase aims to prevent the failure of process equipment requiring the use of gases or vapours with high purity, protecting the environment against pollution and degradation, recovering valuable substances, or realising the desire to obtain a product of the highest purity. Wave-plate mist eliminators are applied when necessary to remove droplets of size ranging from a few to tens of micrometres. However, removing very fine droplets requires different solutions such as a glass fibre bed separator intended for removing droplets of a diameter smaller than 2 µm [1]. Wave-plate mist eliminators need to have appropriate geometry (in practice, the channel through which the gas flow has a zig-zag shape, which causes a sharp change in the direction of the flowing fluid stream). The gas containing liquid droplets flows across the bends, forcing it to change the flow direction a few times. Due to their high density compared to gas, liquid droplets are subject to inertial forces, which causes the deflection of the droplet trajectories from the gas motion patch and separation of droplets on the walls. Those forces depend on droplet mass and velocity as well as the geometry of mist eliminator [2]: where Fs represents inertial force, R is a bending place curvature radius, and m p , u p , d p, and ρ p are the mass, velocity, diameter, and density of the droplets, respectively. It can be seen that inertial force is proportional to the d 3 p . Due to this fact, wave-plate mist eliminator efficiencies decrease sharply for smaller droplets. For example, in the same conditions, for 37.5 µm droplets, the separation efficiency of the wave-plate separator was over 90%, but for droplets of diameter less than 25 µm, it dropped sharply to less than 40% [3]. Droplets that have not been re-entrained by the gas can adhere, coalesce, form a film, and flow down under the influence of gravity to be removed from the separator space. The advantage of inertial droplet separators is a low-pressure drop. They can be used to remove foam-forming liquids and high-viscosity liquids, which in the case of mesh and fibre separators is ineffective due to the robust clogging spaces between wires or fibres by liquid droplets. Furthermore, most wave-plate separators have relatively higher resistance to droplet re-entrainment than wire mesh separators, although they cannot obtain high efficiency under conditions of low gas velocities or/and fine droplets [3].
Three essential parameters affect the efficiency of the inertial droplet separator [4]: • The linear velocity of the gas; • Separator channels' width; • Presence of drainage channels.
As the inertial force value is proportional to the square of the velocity, increasing the gas velocity has a complex influence on droplet removal efficiency. Below 6 m/s, the increase in velocity is beneficial for efficiency; however, for velocities above this value, the influence of secondary droplet entrainment increases, reducing droplet removal efficiency. Nevertheless, for some wave-plate separators with multiple bends, re-entrainment will not occur as long as the gas velocity is lower than 8 m/s [3]. High velocities can also cause flooding phenomenon if a mist eliminator is working in the vertical position. In that case, an upward gas flow prevents the gravitational downflow of droplets [2]. The presence of drainage channels allows maintaining high efficiency even at high linear gas velocities. The shape and type of drainage channels can significantly affect the droplet removal efficiency of the separator.
The problem of optimising the operation of an inertial mist eliminator has been widely described in the literature [5][6][7][8][9][10]; however, an interesting issue is the influence of drainage channel geometry on droplet removal efficiency. Computational fluid dynamics has a unique role in understanding this process. It allows understanding the system's conditions better and easily estimates the influence of process parameters and geometry changes on droplet removal efficiency.
Some successful CFD analyses of airflow and droplet transport wave-plate mist eliminators are described in the literature [11,12]. There is also proof that the length of drainage channels has a crucial influence on droplet removal efficiency and pressure drop [11]. The novelty of this paper is presenting CFD simulations of the influence of the different aspects of the geometry of drainage pockets such as length, angle, and shape on the droplet removal efficiency and different types of droplets and process medium velocity. The first part of the work is focused on the selection of turbulence and turbulent dispersion models. Next, the crucial part of the work presents the influence of the presence, width, and slope angle of drainage channels. The results concern not only the efficiency but also the pressure drop in the system. Based on the results obtained, a new shape of drainage channels was proposed, which is characterised by a high efficiency of droplet removal with reasonably low pressure drop. In addition, droplet flows of media used in the chemical industry such as n-octane, diesel fuel, sulphuric acid, and aqueous ammonia solution were simulated in the new system.

Materials
The present work assumed that air is a continuous phase and water is a discrete phase. Section 3.4 also included parameters of three other media used to simulate discrete phases.

Experimental System
In the present work, different configurations of commercial wave-plate mist eliminators have been investigated using numerical methods. In the first stage, two types of geometries described by [5] were analysed: one without drainage channels (geometry A) and another with zig-zag shaped drainage channels (geometry B_10_0). Visualisations of considered geometries are presents in Figures 1 and 2. Those results were used to validate chosen numerical methods for further calculations.  In the second stage of work, variants with analogical drainage channels of different lengths (a), which reduce the width of the main channels (L), were simulated. Secondly, there were modelled variants with drainage channels of 10 mm length at different angles (α). Geometrical details of every configuration are listed in Table 1 and presented in Figure 3.

Simulation Parameters and Assumptions
In order to analyse the effects of using different geometries, medium velocities, and droplet diameters, the commercial software Fluent developed by Ansys Inc. was used. Calculations were carried out using the 2D domain. Calculations were carried out for two different average velocities on the inlet: 2.0 m s −1 and 4.0 m s −1 , which are typical values in industrial practice. The upper limit of velocity was determined to avoid dragging droplets.
In order to conduct simulation, the following were assumed: • Mass fraction of water was lower than 0.1.
The design assumptions presented above are commonly used to determine the effect of geometry on system efficiency [3][4][5]11]. In particular, this refers to the value of the mass fraction of water, which below 0.1 should not lead to the formation of a liquid film on the wall, and the phenomenon of droplet coalescence should not significantly affect the system efficiency. The agreement between simulation results and experimental data presented later in this paper confirms the applicability of the model assumptions.

Continuous Phase Modelling
The SIMPLE method was used for the pressure-velocity coupling, and second-order discretisation schemes were used for all variables to minimise numerical diffusion effects [13]. Computations were regarded as satisfactory converged when total normalised residuals were smaller than 10 −6 . Computational meshes were generated using Ansys Meshing software and for every variant of geometry. The size of numerical grids was approximately 850,000 quadrilateral cells. The mesh was densest in the wave-plate region, and the near-wall cell sizes were set accordingly to satisfy the y+~1 condition. Mesh independence was checked at the highest tested Reynolds numbers using two quantities: average wall shear stress value at the walls and average turbulence energy dissipation rate in the system. The results of both these two quantities were constant (less than 3% difference), even with using denser meshes than those described. The obtained mesh was validated by comparing calculated results with experimental data. The final mesh for the B_10_0 variant was presented in Figure 4. In the first stage, simulation of B_10_0 geometry was conducted, and obtained results were compared with accurate experimental data with a maximum uncertainty below 5% presented in [5] to select the appropriate turbulence model for further calculations. The gas-phase velocity in a broad part of wave-plate separators is relatively high, and turbulent flow occurs, which means that the k-ε model should be accurate. However, wave-plate channels are narrow, and the flow is less turbulent, making the k-ω model more precise than k-ε in this part of the device. Furthermore, the SST k-ω includes the turbulence shear stress transport, and it is accurate for simulating flow under near-wall conditions [2]. In the present work, three turbulence models based on Reynolds-averaged Navier-Stokes (RANS) were compared: • k-ε with standard and enhanced wall treatment; Reynolds Stress Model (RSM) with enhanced wall treatment.
Many works have been published in recent years presenting the possibility of using more advanced mathematical models, so-called large eddy simulation (LES), to simulate industrial processes [13][14][15][16], enabling the simulation of processes in a wide range of Reynolds numbers. However, it should be remembered that such a solution increases both the calculation time and the demand for computing power many times. It is recommended to use it only when the calculation results are significantly more accurate than those obtained using lower-order models. In a similar process, [17] carried out calculations using RANS and LES models. They observed that differences between models results are 3% and less than 7% on average for the pressure drop and the removal efficiency, respectively, and the RANS approach can be used successfully significantly enhanced with a turbulent dispersion of droplets model.
The following equations govern the continuity and momentum in RANS models: ∂ρ ∂t For the k-ε model, the turbulent kinetic energy, k, and the specific dissipation rate of k, ε, are obtained from the following equations: For the SST k-ω model, the turbulent kinetic energy, k, and the specific dissipation rate of k, ω, are obtained from the following equations: The Reynolds Stress Model is more complicated than the other turbulence models mentioned above because of using anisotropic Reynolds stress tensors. It requires most computational resources, and it was used only as a reference.

Discrete Phase Modelling
Once continuous phase flow is computed, the droplet behaviour can be calculated. In order to simulate droplet motion, a Discrete Phase Model (DPM) was applied. This model uses a Lagrangian approach for modelling droplet trajectories. Droplet motions are tracked using Newton's second law [2]. The following equation governs forces acting on droplets: where u and u p are gas and droplet velocities, respectively; ρ and ρ p are gas and droplet densities, respectively; g is gravitational acceleration, and F x is an additional acceleration.
In this work, one-way coupling between continuous and discrete phases was considered. It is assumed that droplet movement does not affect the continuous phase flow pattern. In Equation (8), F D (u − u p ) is a resistant force of continuous phase per droplet mass (m s −2 ), and it can be calculated from the Reynolds number and droplet parameters. Coefficient F D can be expressed as: where Re is the relative Reynolds number, and C D is the drag force coefficient, which is obtained from the following expressions [18]: for Re d ≤ 1000 (11) where Re d is the droplet Reynolds number defined by: For efficiency calculations, we used droplets of size within the range from 1 µm to the size when the calculated efficiency equals 100%. In all cases, droplets were injected in a perpendicular direction to the inlet plane. Efficiency is defined as a ratio of the number of droplets removed from continuous phase (c in − c out ) to the total injected droplets (c in ): The droplet motion was calculated using two variants of the DPM model: With turbulent dispersion affecting droplet motion.
To include the phenomenon of turbulent dispersion in simulating droplet motion, stochastic tracking model was used, which allows calculating droplet motion trajectories by using instantaneous velocity: Many particles are injected at one point in a stochastic model, and their motion trajectories are calculated using a fluctuation module.
The results of all RANS models described in Section 2.4 were verified using experimental data [5]. Figure 5 and Tables 2-5 show a summary of results and relative errors for geometry B_10_0, which has the same structure and size as used in the work of [5]. In this case, the turbulent dispersion model was implemented to improve turbulent models used for calculations. The influence of including this model in calculations will be presented in the next section.  Analysing Figure 5 and Tables 2-5, one can assume that all used models predict the trend of increase in efficiency with droplet size. However, there are significant differences in expected values. The most accurate results were observed using the SST model, especially for higher efficiencies. Furthermore, the SST model is the only one that accurately predicts almost 100% efficiency for droplets with a diameter of 18 µm (2 m s −1 ) and 12 µm (4 m s −1 ), which are very close results to experimental data. This conclusion is similar to [5] and [15]. As a result, the SST model was found as the most suitable for simulating the gas phase.

Comparing Calculations with and without Turbulent Dispersion
In Figure 6 and Tables 6-9, the results and relative errors with and without using turbulent dispersion for the SST model are presented and compared with experimental data.   Analysing Figure 6 and Tables 6-9, one can observe that more accurate results, compared to the experimental results, were received using the SST model with turbulent dispersion, particularly for droplets of small diameter obtaining higher efficiency, which is following expectations, because for smaller droplets, the model predicts a higher acceleration of droplets, impacting fluctuations of the continuous phase. Based on the obtained results, all subsequent calculations will be done using the SST model with turbulent dispersion as the most suitable model describing the work of wave-plate mist eliminator.

Non-Dimensional Numbers
Calculated values of Reynolds number on the inlet, for hydraulic diameter d h = 0.68 m, are presented in Table 10. The Reynolds number is the most important non-dimensional number considered case, because it illustrates the nature of gas flow (Re) and droplet flow (Re d ). It allows one to select the appropriate turbulence model for CFD calculations and parameters used in the equations. However, the Stokes number, representing the ratio between droplet relaxation time and gas characteristic time, is also loosely related to droplet removal efficiency, even though it is not related to turbulent dispersion effects [5]. Since the characteristic length is channel width, the Stokes number can be calculated by the following equation: where u in is gas velocity on inlet and α c is the bend angle between wave-plates. The relation between Stokes number and removal efficiency is presented in Figure 7. It can be noticed that the regression curve shape is similar to the droplet diameter-efficiency curve obtained from calculations.

Boundary Conditions
For the continuous phase, the following boundary conditions were used: • "Velocity-inlet" with constant value on the inlet to the computational domain; • "Outflow" on the outlet of the computational domain; • "Stationary wall" without slip on the edges of the wave-plate mist eliminator and drainage channels; • "Symmetry" on the edges of the computational domain outside the wave-plate mist eliminator excluding inlet and outlet.
For the discrete phase, the following boundary conditions were used: • "Trap" on the edges of the wave-plate mist eliminator and drainage channels; • "Escape" on the rest of the edges of the computational domain.
Droplets of the dispersed phase were put into the domain through the inlet by the uniform injections of spheres separately for every diameter from investigated range (1 to 40 µm, every 3 µm) with corresponding velocity and constant total mass flow rate of 0.00355 kg s −1 . Every injection was calculated separately.

Results
From a practical and industrial point of view, three parameters were crucial to simulate flow fields of analysed geometries:

•
The removal efficiency of droplets; • Pressure drop on wave-plate mist-eliminator; • The mean force acting on the single drainage channel.
Pressure drop was calculated as a difference between pressures on the inlet and outlet of investigated mist eliminators. The mean force acting on the drainage channel was calculated as an arithmetic average from forces acting on every drainage channel. Calculation results for both parameters are listed in Tables 11 and 12. Based on the above results, it can be assumed that according to Darcy-Weisbach law, the pressure drop (∆p) increases proportionally to the square of the medium velocity (u): The results of pressure drop are following the literature data [2,14]. From an economic point of view, the pressure drop is an undesirable phenomenon. As a result, geometry variant B_30_0 most likely has no practical usage. Furthermore, it can be observed that the force acting on the drainage channel is proportional to the pressure drop on the mist eliminator. In the following chapters, the influence of various parameters was considered in detail.

Influence of Gas Inlet Velocity on Droplet Removal Efficiency on Wave-Plate Mist Eliminator
The calculations were carried out for two inlet velocities, 2 m s −1 and 4 m s −1 , and two characteristic geometries were investigated: The obtained results are shown in Figure 8. The results confirm that the increase of the inlet velocity of the gas increases the droplet separation efficiency. In B_10_0, it can be seen that for the speed of 4 m s −1 , almost 100% efficiency is obtained for the droplet diameter of 12 µm (18 µm for the lower velocity). Nevertheless, it should be noted that this is done at the cost of more than four times higher pressure drop. The results also confirm that drainage channels positively affect separation efficiency for larger droplet diameters, increasing it by about 20-30%. For droplet sizes smaller than 9 µm for 2 m s −1 and 7 µm for 4 m s −1 , the drop separation efficiency is higher for a system without channels, which is obviously due to the inertia forces that act on the drops. In a system with drainage channels, the cross-section near the channels decreases, which causes an increasing velocity, and smaller drops follow the gas rather than deposition on the walls.

Influence of Drainage Channels Length on Droplet Removal Efficiency on Wave-Plate Mist Eliminator
The next step was to investigate the influence of the length of drainage channels crucial for the mist eliminator. In the present work, we investigated geometry variants with drainage channels with the following lengths: The obtained results are shown in Figure 9. Based on the results, the droplet removal efficiency increases with drainage channel length. This phenomenon is associated with the increased velocity of the continuous phase, decreased area of flow, and increased area of the drainage channel. Figures 10 and 11 show velocity fields and vectors of investigated geometries. It can be observed that the highest gas velocities are obtained in the close neighbourhood to drainage channels and that the velocity increases with the next channel, which means that self-similarity flow conditions for the subsequent cross-sections near the channel have not been achieved. These results are similar to those of [5].     With areas with high-velocity values, stagnation areas and recirculation areas are observed in and just behind drainage channels. All observed phenomena cause the drainage channels with higher lengths to provide better results, although it also generates higher pressure drops, which are undesirable. Those variants of geometries can be used if highpressure drops are acceptable.

Influence of Drainage Channels Angle on Droplet Removal Efficiency on Wave-Plate Mist Eliminator
Simulations were conducted for geometry variants with different angles of drainage channels. Geometry B_10_0 was modified to obtain three different angles of drainage channels: The results in Figures 12-14 show the velocity fields of the investigated geometries.   Droplet removal efficiency increases with drainage channel angle (α). This phenomenon can result from moving drainage channels to areas with higher gas velocities, which causes higher inertial force acting on droplets in those areas and generates higher local pressure drops. The maximum velocity increases with the slope of the channels. In B_10_0, the maximum velocity is 5.5 times higher than the system inlet velocity, and in B_10_90, it is 8.5 times higher. Simultaneously, the separation efficiency for drainage channel slopes of 0 and 30 degree increases by about 40%, with the pressure drop by about 25%. Higher angles can also increase secondary droplet entrainment; however, the model does not include this effect.

Influence of Physicochemical Properties of the Droplet on Removal Efficiency on Wave-Plate Mist Eliminator
Mist eliminators are used for removing droplets of media with different physicochemical properties. In the present work, we analysed the influence of different types of droplet materials on the work of the mist eliminator listed in Table 13. Calculations were carried out for geometry variants A and B_10_0. The results are shown in Figure 15.
Based on the plots shown in Figure 15, it can be observed that the density of the medium has a strong influence on the droplet removal efficiency. This phenomenon is since the inertia force acting on droplets is directly proportional to droplet mass. A higher density of medium causes better droplet removal efficiency. A similar effect can be observed by analysing the change in droplet size. Higher diameters of droplets cause higher droplet mass and hence inertia force, making them easier to remove by a mist eliminator. The presented results again show a better droplet removal efficiency when using a system with drainage channels, where the efficiency increases by about 20%. However, it should be remembered that this happens at the cost of an increased pressure drop of about 6%. A more visible effect is observed for larger droplet sizes.

Modification of Wave-Plate Mist Eliminator with Streamline Drainage Channels
The results presented in the previous sections show that the computational fluid dynamics correctly predict the influence of process and geometric parameters on the removal efficiency. After analysing the results, the authors proposed a new drainage channel to achieve high process efficiency with a slight pressure drop. When analysing the pressure distribution for the cases under consideration, e.g., geometry B_15_0 shown in Figure 16, it can be seen that high-pressure drop values appear near the sharp edges of the drainage walls, predominantly perpendicular to the flow direction. For longer edges, this effect is even more visible. Therefore, it was decided to propose drainage channels with a much more streamlined shape. The geometry of the new system is shown in Figure 17, and the main dimensions are shown in Table 14.  The results of droplet removal efficiency are shown in Figures 18 and 19. The authors decided to compare the results obtained for the new geometry for the most similar previous cases: B_10_0 and B_15_0.   Based on the plots presented in Figure 16, geometry variant C_15_0 provides the highest droplet removal efficiency. Figure 19 shows velocity fields and vectors of the investigated new geometries, and Tables 15 and 16 present the pressure drop and the mean force acting on drainage channels for the three, comparing geometry. For comparison, Figure 20 shows velocity fields and vectors for the analogous case with non-streamline drainage channels.  Analysing the velocity distributions of the considered cases for 2 m s −1 , one can observe that the new geometry gets higher maximum velocities than the corresponding variant with non-streamlined drainage channels. This phenomenon is related to the shape of the external drainage channel walls. In C_15_0, the main gas stream is flattened, which is mainly visible on the third channel flows. A critical phenomenon is the occurrence of vortexes inside the channels. In the case of simple channels, the vortex is flattened in an oval shape. Around the channel's edges, the fluid from inside comes into contact with a gas with a nearly maximum velocity. Although, in the case of the profiled channels, a different phenomenon occurs. The forming vortex is round in shape, and the channel's edge disturbs the flow to a lesser extent, which will substantially impact the entrapment of deposited droplets-this can be well observed analysing the droplet trajectories presented in Figures 21 and 22. Based on the visible results, the droplet removal efficiency is about 100% for droplets in the size of 14 µm. This new variant also has over two times higher droplet removal efficiency than geometry variant B_10_0 for droplets of 6 µm and 80% higher droplet removal than variant B_15_0. However, the pressure drop is about 23% higher than variant B_10_0 and similar to variant B_15_0 for the inlet velocity of 2 m s −1 . One can observe that the results are even better for higher velocity. The streamline construction of drainage channels should also minimalise the phenomenon of secondary droplet entrainment. Naturally, experiments should validate every result obtained through simulation.

Conclusions
The work of the inertial droplet separator was modelled using computational fluid mechanics. The influence of practical important parameters on the droplet removal efficiency was investigated: the inlet velocity of the continuous phase, the droplet size, the effect of the presence of drainage channels, and their geometry, i.e., the length and angle of the slope. The simulation results show that drainage channels increase the pressure drop, which is undoubtedly a disadvantage of using drainage channels in inertial droplet separators. On the other hand, the removal efficiency values of the separator with drainage channels are much higher than those without drainage channels. Drainage channels cause the velocity of droplets to increase, which results in increased removal efficiency. At the same time, it should be remembered that the literature data show that the effect of droplet entrainment (not included in the calculation) will be much higher in the separator without drainage channels, which will further reduce the separation efficiency.
Three turbulence models and two variants of turbulent dispersion modelling were used in the present work. The best results were obtained for the transition SST model with the turbulent dispersion model attached. Such conclusions are consistent with the literature, and an excellent agreement with the experimental literature data allows simulating other parameters such as changes in slope angle and the size of drainage channels.
The simulation results show that small changes in geometry significantly improve the separator efficiency with a relatively small increase in pressure drop. The analysis of the obtained results allowed authors to propose a new geometry of drainage channels with streamlined shapes. This solution allows achieving a significant increase in droplet removal efficiency while keeping a comparable pressure drop for lower velocities and even decreasing for higher velocities than the basic geometry of channels of similar length. These results should be confirmed by extending the theoretical model, particularly with the effects of secondary entrapment of droplets and forming a liquid film on the wall, and comparing these with the result of the experimental data.

Conflicts of Interest:
The authors declare no conflict of interest.

Symbols
The following symbols are used in this manuscript: