- Email: [email protected]
Multiphase model of flow and separation phases in a whirlpool: Advanced simulation and phenomena visualization approach
Journal of Food Engineering 274 (2020) 109846
Contents lists available at ScienceDirect
Journal of Food Engineering journal homepage: http://www.elsevier.com/locate/jfoodeng
Multiphase model of flow and separation phases in a whirlpool: Advanced simulation and phenomena visualization approach Marta Stachnik, Marek Jakubowski * Koszalin University of Technology, Faculty of Mechanical Engineering, Department of Food Industry Processes and Facilities, Poland
A R T I C L E I N F O
A B S T R A C T
Keywords: Whirloool CFD Simulation Ekman’s spiral Multiphase rotating flow
In whirlpool, a separator used in the brewing industry, the deposits accumulate in the central zone at the bottom of the separator. The deposit assumes a shape that resembles a flattened cone, referred to as the “tea leaf paradox.” It is so-named because the deposit accumulates seemingly against the influence of the centrifugal force that develops from the rotating movement of the mixture. Previous studies have performed a simulation of the two-phase volume of fraction (VOF) model with swirling liquid and air. In the present study, we suggest an expansion of the existing model with a dispersed phase of the hot trub. A three-phase VOF model was built to analyze the movement and formation of the cone. Material data obtained referred to the wort and hot trub after boiling with hops. The simulated shape of the cone was compared to its counterparts obtained under industrial conditions. The values of the velocity of particles of deposit were also analyzed. Results were compared with the visualization of the cone’s formation for rapidly sedimenting particles and for hot trub and wort flow under laboratory conditions. The analysis deems the VOF three-phase model suitable to predict sedimentation and accumulation of sediment.
1. Introduction Whirlpool (Fig. 1a in Section 2.3) is a separator used to remove dispersed solid phase from the liquid. It is a cylindrical, empty container that is filled tangentially. Filling of the whirlpool tangentially results in the rotating movement of the separated mixture. The separation is enabled by the rotating movement and sedimentation is sustained by the gravitation. It is an elegant and effective approach for liquid clarifica tion. It has a low requirement of energy, as it is a mix of spontaneous naturally occurring phenomena. Whirlpool has been universally applied to separate hot deposit from the wort. When wort is boiled with hops, a hot trub, i.e., protein-tanning conglomerates, is formed; this trub is a waste and must be removed. The empty container of the separator is filled through the inlet hole located tangentially to the wall. The mixture is thus set into a rotating movement. The primary flow that occurs is due to the tangential filling of the container. A system of secondary flow appears during slowing down of the rotating movement of the mixture. The dominating whirls are responsible for the formation of the deposit cone in the central zone at the bottom of the separator. This effect was reported by Einstein in
1926 and named as tea leaf paradox (Long et al., 2014). Even though the name suggests that this phenomenon is a paradox, it is quite simple to explain. The centrifugal force is formed due to the rotation of the separated mixture. Seemingly, one should expect the particles to move outside, closer to the joints of the bottom and the wall of the tank. As this does not occur, the phenomenon is termed as paradox. The hampering influence of the container’s bottom and the walls cause the deposit to follow a spiral movement to the central zone of the bottom of the container. Additionally, this phenomenon is supported by a gradient of the hydrostatic pressure resulting from the difference in the height of the column of liquid in the container. This difference results from the shape of the free surface, which is formed by the rotating movement of the separated mixture. The problem of multiphase modeling for the tea leaf paradox is related to both modeling of the multiphase flow with the separation and specificity of the rotating movement. A literature review shows various studies on the Computation Fluid Dynamics models of the multiphase flow or on the CFD models of the rotating movement. However, few studies have linked both these issues. The problem of modeling of the sedimenting dispersed solid phase, hot trub in this case, required prac tice definition and the selection of parameters of the dispersed phase
* Corresponding author. Eng Koszalin University of Technology Faculty of Mechanical Engineering Department of Food Industry Processes and Facilities, 15-17 Raclawicka st., PL-75620, Koszalin, Poland. E-mail address: [email protected] (M. Jakubowski). https://doi.org/10.1016/j.jfoodeng.2019.109846 Received 16 March 2019; Received in revised form 11 November 2019; Accepted 27 November 2019 Available online 5 December 2019 0260-8774/© 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
M. Stachnik and M. Jakubowski
Journal of Food Engineering 274 (2020) 109846
Nomenclature Cd D dd Dmd F fd g H hb I KH Khb Kub L lT LT mdc b n P Pk
α t
tk trx u u0 ub uslide uv uτ v β δw
mass density of dispersed phase, kg m 3 vessel diameter, m average diameter of particles, μm diffusion coefficient of dispersed phase in a fluid, m s 2 additional volumetric forces, N kg 1 coefficient of flow resistance for dispersed phase gravitational acceleration, m s 2 height of the fluid after filling, m height of the secondary flow, m; identity matrix dimensional proportions of the vat (H/D) the number of the height of the vortex at the bottom of vat the number of the velocity of the vortex at the bottom of the vat distance in the tunnel, m turbulence intensity turbulence length scale, m mass transfer between phases (if the particles do not dissolve, then: mdc ¼ 0), kg s 1 normal to surface unit vector pressure, Pa turbulence production volume fraction - time, s
ε η μ μT ν ρ σk σω ω Index c d m r
Kolmogorov time scale relaxation time of particles, s velocity, m s 1 normal velocity on inlet, m s 1 maximum velocity of the secondary flow, m s velocity in-between phases, m s 1 free fall velocity of particle, m s 1 friction velocity, m s 1 volume, m3 model constant distance between wall and grid, m dissipation rate of turbulent energy Kolmogorov length, m dynamic viscosity, Pa⋅s turbulent viscosity, Pa⋅s kinematic viscosity, m2 s 1 density, kg m 3 model constant model constant turbulent frequency
1
continuous phase dispersed phase mixture - phase representation, 1 – water, 2 – air, 3 – particles
Fig. 1. Whirlpool: a) one of the biggest capacity (1020 hL) vessel in Europe (Warka brewery brewhouse by GEA–Huppmann); b) geometrical model with the finite element grid and boundary conditions for the mathematical model.
that would diligently represent particle movement and accumulation in the form of a cone. Many CFD studies on the multiphase flow discuss drying, fluidiza tion, and bubble columns but mainly focus on spray drying. Azmira et al. (2019) developed a CFD-DEM (Discreet Method Element) model of grain drying that included reduction of grain size. By using this model, they
investigated the influence of temperature and velocity on the rate of drying and the size of grains. Kim and Chung (2018) analyzed the granulation of food powders, including a change in their rheological properties and water content. Wang et al. (2018) studied the effective ness of an air curtain to prevent dust propagation in a rock tunnel during drilling. Liu et al. (2019) investigated a ventilation system. Cai et al. 2
M. Stachnik and M. Jakubowski
Journal of Food Engineering 274 (2020) 109846
(2019) studied the effect of air flow rate on pollutant dispersion pattern of coal dust particles at fully mechanized mining face. Jaskulski et al. (2018) built a CFD model of a counter-current spray dryer. They analyzed collisions of particles, including the interaction between drops and that between the drop and the dry particle. For simulation, they suggested a moisture evaporation model based on the concept of a characteristic drying curve. For the same issue, i.e., counter-current spray drying process Jubaer et al. (2019) compared the efficiency of different turbulence models for the simulation of the flow in the spray dryer. The best model was the k ω SST model that was also utilized by the authors of the present study. Modeling of a rotating flow is not easy. However, because this kind of rotating movement is universally present, many studies have been conducted on this subject, but are rather loosely related to the food in dustry. Anufriev et al. (2014) analyzed the spatial structure of whirl pools in a rotary stove by using CFD. Rotary flow improves mixing of air with fuel and the distribution of the mixture in the entire volume of the stove. The simulation results formed the base for follow-up studies and Particle Image Velocimetry (PIV) measurements in a four-vortex furnace for the burning of coal dust (Anufriev et al., 2017). Another example is the use of Large Eddy Simulation (LES) to analyze the influence of a conical obstacle on the flow in a pipe and reduction of whirls in the flow (Kharoua and Alshehhi, 2018). The LES model is particularly interesting for the subsequent development of simulation analyses of a rotating flow with sedimentation of the dispersed phase, as presented in this study. However, this model is highly demanding in terms of computational power, particularly for transient analysis. The modeling of a rotating movement in cyclones was the subject of investigation in the subsequent studies. Huang et al. (2018) presented a study of guide vanes to create whirlpools and uniflow stream for the preliminary separation of the mixture. Cyclone’s efficiency was assessed with a computer model with Eulerian–Lagrangian approach with the RSM (Stress Reynolds Model). Another example is the study by Liu et al. (2018). A vane-separator performance was simulated with a two-phase model to analyze the size of gas bubbles at the inlet, the propagation of the gaseous phase, and the intensity of the turbulence. The obtained model correctly predicts the size of gas bubbles based on the rate of flow at the inlet and fractions of phases and viscosity of a liquid. An example of a study of dedusting equipment is a numerical analysis of the air flow in a fully mechanized tunneling face. The simulation allowed assessing the migration of dust during excavation. The model served to assess the possibility to convert a multiradial swirling flow field produced by the generator into an effective dust-control air curtain where the tunneler driver is located (Hua et al., 2018). However, particular attention should be given to the study of the movement of the cell colonies in bioreactors. Here, the tea leaf paradox is undesirable, and mixing prevents the colonies from forming a sedi ment cone at the bottom. Yamamoto et al. (2018) used numerical model was to study stirring and orbital shaking tanks. This work presents the results of a 3D simulation of a three-phase model with two liquid phases: air and wort and a dispersed phase: hot trub. Addition of the dispersed phase is aimed to allow full flow analysis and separation in equipment with a tea leaf paradox phenomenon, e.g., whirlpool. In existing studies, we did not find a CFD analysis for the agglomeration and formation of the sediment cone. Thus, the present study is unique. This study is related to both the tea leaf paradox and the sedimentation of particles into a concentrated deposit.
2.1. Mathematical model
2. Materials and methods
3 fd ρ ρd ρ juslide juslide ¼ m rp: 4 dd c ρm
This section presents equations of a mathematical model to describe a rotating flow of the wort in the tank and the equation to describe the movement of the hot trub. The next subsection describes the assump tions for the computer model, i.e., initial and boundary conditions, and simplifications.
The correct laminar to turbulent transfer is ensured by the limitation of the Reynolds number as follows:
A flow of a fluid is described by the conservation of mass (continuity) equation and the conservation of momentum—momentum equation, also known as the Navier–Stokes equations. The equation of continuity (1) for the mixture of the dispersed solid and continuous phase is obtained by applying the principle of mass conservation to a small volume of fluid. The standard (general) form for Cartesian coordinates is as follows (Misra et al., 2018):
∂ ðρ Þ þ r ⋅ ðρm um Þ ¼ 0 ∂t m
(1)
where:
ρm ¼
n X
(1.1)
α r ρr
r 1
and (1.2)
α1 þ α2 þ α3 ¼ 1
Equations for the conservation of momentum are derived by applying Newton’s law of motion to a small volume of fluid. Newton’s law states that the rate of change of momentum for a volume of fluid is equal to the sum of all external forces acting on that volume. The equation of momentum balance (2) takes the following form:
ρm
∂um þ ρm um ⋅r ¼ ∂t
� r
p þ ðμm þ μT Þ rum þ ðrum ÞT
þρm g þ F;
��
r⋅f½ρd Cd ð1
Cd Þ�uslide uslide g (2)
where: Cd ¼
α3 ρd ρm
(2.1)
and Pn um ¼
αρ ρm
(2.2)
α r μr
(2.3)
r 1 r r ur
and
μm ¼
n X r 1
For a Newtonian fluid (in this case, it is wort), viscosity is constant. The left side of the equation represents a temporal variation in mo mentum and fluid acceleration, whereas the right side represents the pressure gradient force (the normal stress) and the viscous force (the tangential shear stress). Gravity force is necessary for sedimentation (Jakubowski et al., 2014; Misra et al., 2018). The flow of a dispersed phase is represented by equation (3): � � ∂α3 Dmd mdc rϕd ¼ (3) þ r ⋅ α3 um þ α3 ð1 Cd Þuslide ∂t ϕd ρd The Schiller–Neumann model was applied because a fluid-fluid drag function is used. It is described by equation (4):
3
(4)
M. Stachnik and M. Jakubowski
Journal of Food Engineering 274 (2020) 109846
8 9 � 24 � > > < = 1 þ 0:15Re0:687 p Re < 1000 for p fd ¼ Rep ; Re > 1000 > > p : ; 0:44
δþ w ¼
(5)
dd ρc juslide j
b n ⋅Nα3 ¼ 0
(5.1)
μm
�
2 ρkr⋅u; 3
2 k ¼ ðu0 lT Þ2 3 by:
k
�
ρm βω2 :
(8)
Turbulence by the wall (10) is described as follows: (10)
Dissipation of the energy by the wall (11) is given as: k ; kυδþ wμ
(16)
The simulation was conducted with software Ansys CFX 18.1 and was based on the Finite Volume Method. The modeled case was turbu lent, and the tank’s geometry was regular; thus, it was suitable to use the structured–hexahedral–grid. The quality of the mesh was judged ac cording to the shape of the free surface. A VOF of 0.5 was set for the free surface test level. The shape of the free surface natural conditions is of the paraboloid of revolution. The choice of the number of the grid’s element was based on a previous analysis (Jakubowski et al., 2014) and was set to around 2,000,000 elements (Fig. 1b). Mesh quality was validated by testing the volume of wort in the tank after filling enclosed by free surface for VOFwort equal to 0.1, 0.5, and 0.9. The volumes varied by 5%; therefore, the mesh was deemed suitable. In the CFX the volume of control is assembled around the nodes (Cell Vertex), where each element is divided in sub volumes. In term of grid quality CFX is more permissive. The CFX solver calculates 3 important measures of mesh quality at the start of a run: mesh orthogonality – minimum 56.1: good; aspect ratio – maximum 12: good; expansion factor – maximum 7: good. Mean skewness is 0.99 and deems mesh bad. This high skew is a result of fitting hexahedral elements of mesh along a curve of the cylindrical geometry. This is unavoidable issue. However, hexahedral is the only mesh that allows the simulations and grants correct shape of the free surface; any other type or combination provides wrong results or divergence. However, in terms of overall mesh quality minimum quality is 0.73, maximum ¼ 1, and mean ¼ 0.97. The inlet surface was refined to 30 � 30 nodes. In earlier simulations, it was noted that the lower number of nodes at the inlet leads to incorrect calculation of the rate of flow. This results in much slower filling of the tank than in the experiment. The top limiting surface was considered as an open condition, considering that at time zero, the tank was empty—filled with air (volume of fraction (VOF) for air ¼ 1,VOF for wort ¼ 0, and VOF for residue ¼ 0). The open condition enabled the air to be pushed out by the liquid but also to flow back. Opening (outflow)
The relationship between shear stress and the viscosity of liquid (9) is shown by: � � � 2 2 ðμc þ μT Þ rum þ ðrum ÞT ðμc þ μT Þðr ⋅ um ÞI ρkI bn ¼ 3 3 u ρc þτ ðum ðum ⋅ bn Þbn Þ (9) δw
ρm
(15)
2.3. Computational model
(7)
The velocity by the wall is described by equation (8):
ω¼
LT
where p0 is the surrounding—constant pressure.
2.2. Boundary conditions
rk ⋅ b n¼0
1=4
The stress continuity condition on the free boundary—the free sur face is as follows (17): �� n; (17) n ¼ p0 b pI þ ηm ru þ ðruÞT ⋅ b
The SST model allows avoiding miscalculations of the turbulence at the wall. This model uses the near-wall formulation of the κ-ω model by switching automatically from low Reynolds number formulation to the wall function treatment at the wall boundary. The SST model correctly calculates the turbulence in cases such as sudden pipe expansions. In the analyzed case, we have a similar situation—the diameter of the filling pipe is much smaller than that of the container.
um ⋅ b n¼0
ðβ*0 Þ
α3 ¼ α30
(6.2)
� ω ∂ω þ αr ρum ⋅ rω ¼ r ⋅ ½αr ðμm þ σω μT Þrω� þ αr ϕ Pk ∂t k
k1=2
The concentration of the dispersed phase at the inlet (16) is expressed by:
- for energy dissipation rate (7):
α r ρ;
(14)
Dissipation rate of the turbulence energy at the inlet (15) is described
ω¼
(6.1)
ω
(13)
The kinetic energy of the turbulence at the inlet (14) is given as follows:
and:
μT ¼ ρ
n u0 b
um ¼
where: 2 ðr⋅uÞ2 3
(12)
The boundary condition at the inlet, which defines the velocity of the mixture, is represented by equation (13):
An SST model was used for turbulence. This two-equation model is the simple and popular choice for modeling turbulence. This is because in this model, two different transport equations characterize two inde pendent properties of the turbulent flow. Moreover, it is robust, economical, and reasonably accurate for a large variety of turbulent flows (Jakubowski et al., 2014; Zhang and Fu, 2019). To solve turbu lence, a homogenous turbulence model was used. Turbulence equations were written originally only for single-phase flows. Turbulence for each volume fraction (αr) of the phase is modeled separately by the following equations: - for kinetic energy (6): � � � � ∂k μ αr ρ þ αr ρu ⋅ rk ¼ r ⋅ αr μ þ T rk þ αr ðPk ρβ* kωÞ; (6) ∂t σk
� � Pk ¼ μT ru : ru þ ðruÞT
(11.1)
The movement of the dispersed phase by the wall is defined by equation (12):
where Reynolds number is described by equation (5.1): Rep ¼
ρm u τ δ w μ
(11)
where:
4
M. Stachnik and M. Jakubowski
Journal of Food Engineering 274 (2020) 109846
Fig. 2. Geometrical forms of the deposit cone and volume fraction changes during the course of filling and whirling at the chosen time steps: (a) 7.5 s; (b) 10 s (c) 15 s; (d) 45 s; (e) 65 s; (f) 70 s; (g) 80 s; (h) 135 s; seen from a side (on the left) and from the bottom (on the right).
condition prevents numerical overflow and provides numerical stability. The flow direction of air was set to normal to boundary condition with medium turbulence intensity. The filling velocity was set as 2 m s 1. Velocity was calculated from Denk (1998) suggestions for correct filling velocity. At the inlet, VOF for wort was set as 0.99 and for residue as 0.01. This proportion represents the real concentration of residue in the wort before clarification. Mass and momentum at the inlet were defined by Cartesian velocity components. To enforce the tangential clockwise manner of filling, Z and X components were 0 and the Y component was set to be 2 m s 1. The turbulence intensity was medium (5%). The wall and the bottom of the tank were defined as a stationary, smooth wall with the no-slip condition. To achieve correct results near the wall, the wall distance of the first element was set at 3 � 10 6 m, which ensured that the dimensionless wall distance (yþ) was 1. The wall function in the simulation was set as automatic. To calculate sedimentation, it was necessary to provide data such as the viscosity of the residue, its diameter, and density. Thus, it is possible to use the presented model to analyze sedimentation of different vari eties of the residue. This is relevant as beer production utilizes many different raw materials that produce different hot trubs. Given the fact that the suspended residue in the wort constitutes a small fraction, viscosity for the hot trub was set the same as that for the wort. Particles were modeled by an equivalent diameter (0.41 mm) and density (980 kg m 3), and the data were based on the measurements of the hot trub. During filling and whirling, particles of the hot trub collide inelastically with the wall and with each other and create bigger con glomerates. As conglomerates become bigger and reach critical mass,
they fall out of the suspension and form sediment at the bottom. Resti tution coefficient for the particles was set to 0.2. Particles are forced to stop at the wall by the restitution coefficient of 0.1. The time step of the simulation varied depending on the level of filling the whirlpool. Numerical stability in CFX is measured by Courant number and solution convergence. Courant number should be less than 1 and solution should converge after less than 6 iterations to be considered correct. The time step was set to keep courant number at 0.57. This thus led to solution converging after 3 to 6 iterations (six it erations occurred in the early state of filling). Moreover, the RMS was monitored to not exceed 1 � 10 4. For the initial phase of filling, time step was set to 0.5 � 10 3 s. In the later phase of filling (after exceeding the height of the inlet by the free surface), time step was increased to 3 � 10 3 s. After filling (for the stage of whirling), time step was set to 1 � 10 3 s. Ansys CFX is an element-based finite-volume with second-order discretization schemes in space and time simulating software. Dis cretized governing equations are solved by coupled algebraic multigrid €hne, 2019). The momentum equation was solved using the algorithm (Ho 2nd order upwind scheme. 3. Results and discussion 3.1. Simulation of the formation of the hot trub cone This section presents results obtained from the CFD simulation. The focus is on the residue behavior and hot trub cone shape. The process of 5
M. Stachnik and M. Jakubowski
Journal of Food Engineering 274 (2020) 109846
Fig. 3. Velocity distribution of particles in the deposit cone, during the course of filling, and whirling, in selected time steps: (a) 7.5 s; (b) 15 s; (c) 65 s; (d) 70 s; (e) 90 s; (f) 135 s.
wort clarification can be divided into two characteristic periods: filling and whirling. Fig. 2 illustrates the hot trub cone formation. The results from CFD simulation reflect the data from experiments and industrial process. Deposit is identified for every mesh element, for which the participation of the particle faction was estimated above 0.5 VOF. Simulation of clarification was carried for a total of 135 s. The filling was carried out for 75 s, and it was continued until the height of the fluid was equal to the diameter of the vat (counted volumetrically for the stand still). This proportion is based on earlier studies and reflects the indus trial conditions of the process. At this moment, the inflow of wort and hot trub mixture was cut off. The process of whirling lasted from 76th to 135thsecond; at that time, the wort slowed down and deposit fell to the bottom, thereby forming a cone. Following the progression of the tea leaf paradox, described origi nally by Einstein (1926) and reported by other authors (TandonMar shall, 2010; Long et al., 2014; Rahman and Andersson, 2018), hot trub should accumulate in the center zone of the whirlpool’s bottom in the form of a dense cone. Unlike previous descriptions, the cone is more flattened, as shown in the data from CFD presented in Fig. 2, and confirmed with the observation of experimental visualization of this study. Fig. 2 presents movement of the hot trub during clarification process, seen from the side and from the bottom. CFX VOF model consisting of two continuous phases and one dispersed does not allow for particle tracking. Phases are tracked based on their volume fraction, what is suitable for the sediment accumulation. During filling, the deposit fol lows the horizontal primary flow because of the tangential manner of the filling process. At the beginning of the filling process, 7.5th second, the hot trub moves in the outside space, near the joint between the bottom and walls of the tank (Fig. 2 a). Hot trub volume fraction at the inlet was set to 1%, thus VOF of hot trub in the beginning was 0.01. Particles are small and dispersed. However, some particles begin to form conglom erates of VOF ¼ 0.6. Most of the particles were suspended in full volume of fluid. The rotating movement of the mixture slows down due to the natural dispersion of the energy. As the filling continues, 10th second, particles move closer to the tank’s axis. Conglomerates with VOF>0.2 are more numerous in the central part, while the dispersed small
particles are still being pumped near the wall. The central zone is still empty, thus no hot trub is identified there. Particles spiral toward the center of the tank (Fig. 2 b). In 15th second of filling the bottom is fully covered with the fluid. Hot trub particles are identified across the plain and conglomerates of VOF>0.5 are numerous and VOF ¼ 1 conglom erates are identified (Fig. 2 c). After 45 s of filling – halfway through – the hot trub moves toward the central zone of the container (Fig. 2 d). At this point the cone is fragmented and primary flow drags it across the bottom. Majority of particles formed bigger conglomerates with VOF ¼ 0.25–0.50. In 45th second sediment cone had a diameter of 85 mm. At the end of the filling process, 65thand 70thsecond, the volumetric frac tion of the dispersed solid phase stabilizes. Volume of fluid in the tank hampers the influence of the incoming fluid on the cone’s shape. Hot trub forms denser cone in the central zone of the tank, whit only pe ripheries staying loose (Fig. 2 e-f). Small particles are still pumped in the tank and can be identified near the wall. In 65th second cone had diameter in the widest part of 125 mm and on the thinner side of 75 mm. Upon the end of filling process, the shape of the hot trub suspended in the fluid is stabilized. In 70th second particles suspended in the fluid were mostly small with few bigger conglomerates, VOF ¼ 0.01–0.5. The shape of the cone resembles a circle with the mean diameter of 95 mm. At the 80th second the filling is stopped, but small particles are still present near the walls, at this moment hot trub’s cone is strongly dis rupted (Fig. 2 g). This behavior is associated with a sudden change in velocity, which causes short-lasting pulsatile flow near the bottom of the tank. High-velocity flow caused by filling is still present in the fluid but it is now followed by low-velocity flow after the inlet is closed. Majority of the particles sedimented to the bottom, and very few particles were still suspended in the fluid. Cone’s diameter at this time was 135 mm. Cone’s core has VOF ¼ 1, whereas the surface has the VOF ¼ 0.5–0.75. Toward the end, 135th second, the flow comes to a standstill, and the hot trub obtains its peculiar shape (Fig. 2h.). Surface VOF of the cone shows VOF ¼ 1 in some places, thus indicating overall denser cone, with mean diameter of 130 mm. Lower values of VOF at the surface indicates presence of smaller, lighter particles that can be easily moved by the fluid. The edges of the cone formed by smaller particles take the longest to stop (Fig. 3 – velocity profiles) and are thus susceptible to disruption 6
M. Stachnik and M. Jakubowski
Journal of Food Engineering 274 (2020) 109846
Fig. 4. Graph of the change in the share of the volume of hot trub compared to the change in VOF share in the test element (above: chart of the change in the total volume during whirlpool filling).
and scouring. Subsequently, only the cone’s periphery undergoes a change in the shape; these are the hallmarks of the spiral form (more widely described later in the study). Particle velocity change during the sedimentation was analyzed. Fig. 3 presents particle velocity distribution for selected time steps. According to the natural course of the phenomena, at first, maximum values of the velocity appear in the area of the joint between the bottom and the wall of the tank, and this is the reason why the deposit is localized only in this area (Fig. 3 a). Around the 65th second of filling, the hot trub cone starts to form in the middle zone, and the minimum
values of the velocity appear in this zone (Fig. 3b and c). At the end of the filling process, maximum values of the velocity are identified at the edges of formation of the cone (Fig. 3 d). With the passage of time, the primary flow slows down, and hot trub cone assumes its ultimate shape; this is also reflected in the location change of maximum velocity values (Fig. 3 e 3 f). Extremely low values of the particle velocity are identified in the central zone of the cone, which is typical for the nature of the phenomenon. Simulation is thus in compliance with observations in real conditions. Apart from the visual data related to the phenomenon of the cone
Fig. 5. Shapes of deposit cones: a) industrial hot trub cone—whirlpool capacity 30 hL, b) industrial hot trub cone–whirlpool capacity 530 hL, c) hot trub cone from the CFD simulation, d) quartzsand (visualization of the formation of the cone for rapidly sedimenting deposit). 7
M. Stachnik and M. Jakubowski
Journal of Food Engineering 274 (2020) 109846
formation, the simulation provides information about the volume of hot trub and changes in time. In Fig. 4, the graph presents the change in the volume of the hot trub in the liquid phase compared to the participation of the hot trub in the test volume of the fluid element. The distribution of test elements in which particles are identified explains the situations introduced in Fig. 2, where a greater volume of deposit is identified during whirling than in the final cone. That results from VOF of the deposit in each mesh element, rather than the actual participation of deposit in the volume of the fluid, for which the distribution was pre sented in the form of a graph in Fig. 4 (color red). For the formed cone, the participation of the fraction of liquid in the test element is smaller than the total volume of deposit. Experimental data show that the participation of the liquid phase in the hot trub deposit is (depending on raw material composition) from 74% to 82%. In the simulation, the two volumes of hot trub are 0.28 � 10 3 m3 (total volume of hot trub) and 0.11 � 10 3 m3 (volume of hot trub in the cone). The difference (60%) represents approximately the participation of the liquid phase in the cone. The graph shows that the total volume of hot trub in the tank stabilizes, and the deposit becomes denser. For comparison, a change in wort’s total volume is presented (Fig. 4—upper chart) during the filling and whirling (in the graph, stages are separated with a red dotted line).
simulation model with conditions of the flow in the laboratory whirl pool. More details on this subject can be found in Jakubowski et al. � ska and Jakubowski (2017). (2014) and Sterczyn While scaling the tea leaf paradox effect from a laboratory whirlpool to an industrial scale, the authors derived a set of similitude numbers. More details on this subject can be found in Diakun and Jakubowski (2014). One such number is the duration of the main flow in the whirlpool (it is termed as time break): Kt ¼
ut 1 D ð1 þ KH Þ
(18)
Whirling time depends on the inertial mass of the rotating wort and the forces that retard the flow: t¼
D ð1 þ KH Þ u
(19)
The formula for the dissipation force of the flow in the Ekman Boundary layer is u 2 D Hρ ¼t 2 FS2 ub μ Dhb
KF1=
(20)
3.4. Cone forming flow is given as
3.2. Correspondence with the real conditions Numerically obtained shape of the cone, its consecutive geometrical forms, and the time of sedimentation were compared with states appearing in real conditions. Hot trub in industrial conditions forms a flattened cone independent of the working volume of the whirlpool (Fig. 5 a, b). The same geometrical form is obtained from numerical calculations (Fig. 5 c). For substitute deposit in the form of the solid dispersed phase with the high rate of sediment velocity (e.g., quartz sand), the cone has distinctive shoulders resulting from the movement of particles in the so-called Ekman spiral (Fig. 5 d). This phenomenon is well known in the field of meteorology in studies of seas and oceans (Sukhanovskii et al., 2017; Farmanara et al., 2018; Constantin and Johnson, 2018). This form of the cone provided grounds to conduct follow-up works in the optimization of the whirlpool’s bottom con struction. This subject is discussed in details in section 3.5. Other authors also identified the appearance of spiral patterns in a rotating movement. Gonz� alez-Vera et al. (2018) conducted experi mental studies in the scope morphodynamics of the sedimentary layer in a cylinder filled with liquid during spin-down experiments. Spiral for mation of the sedimenting liquid was identified as an outcome of the experiment.
Kb ¼
tμ D2 ρ
(21)
Adjusted Reynolds number for whirlpool tank for primary flow is noted as � � uDρ 1 Rew ¼ 1þ (22) μ KH Adjusted Reynolds number for whirlpool tank for secondary flow is noted as Reb ¼
uDρ Khb μ Kub
(23)
Apart from the originally derived criteria numbers, the work of other authors was implemented. The authors selected universal numbers (adapted for the specifics of the whirlpool condition) to scale the phe nomenon of sedimentation. An appropriate dimensionless number is the Driftage Elements Number (Kunes, 2012). It describes drifting and sedimentation of fine particles suspended in a liquid: ND ¼
uv u gH
(24)
The next dimensionless number suggested to describe the phenom enon in the whirlpool is Inertia Parameter (Wessel and Righi, 1988). This number determines the relationship of the inertia with the friction forces between particles in the liquid flow:
3.3. The experimental verification of the model and scaling the phenomenon PIV (Particle Image Velocimetry) experimental studies were pre sented in earlier reports. The investigations were used to verify the CFD model of a rotating flow of the wort in the whirlpool. The verification was performed by comparing values of the velocity of the primary and secondary flows. The research study on PIV included PMMA tank (plexi glass), rotary piston pump NOVALOBE (GEA Hilge, Germany), magnetic-inductive volumetric flow meter (SM-7500, IFM electronics, Germany), and a table with a set of mirrors. The tank was of the same size as that used in the simulation. Velocity measurements were per formed with double-pulsed laser (EverGreen 532 nm, 145 mJ; Quantel, USA) and a PIV camera FlowSense4M (DantecDynamics, Denmark). Acquired data were analyzed with DynamicStudio software (v. 6.7, DantecDynamics, Denmark). Flow tracing particles were made of a polymer and had a similar density to the fluid. The cone formation was visualized with quartz sand as a rapidly sedimenting material. Velocity, flow rate and size of the tank were the same as in the simulation. The investigations showed a high level of the compliance of the
Ninert ¼
d2d ρd u 18ηH
(25)
To quantify the relationship of the volume of the continuous phase (beer wort) with the density of the dispersed phase (hot trub), a Mass Number was suggested (Iverson and Denlinger, 2001). For Nm > 1, the momentum transfer of the dispersed phase is dominant. For whirlpool Nm < 1, the phenomenon occurs due to a low volume of hot trub sus pended in the wort during the filling. Nm ¼
vd ρd v c ρc
(26)
Stokes Particle Number is universally used to describe the movement of particles under the influence of the inertia in the turbulent flow (Lau and Graham, 2015):
8
M. Stachnik and M. Jakubowski
Journal of Food Engineering 274 (2020) 109846
Fig. 6. Selected variants of the whirlpool’s bottom modification with reference to the fragment of the spiral (data expressed in quarters of its course): a) the second quarter, b) the third quarter, c) the fourth quarter, d) second and third quarter, e) third and fourth quarter (on the left, red indicates fragment of the spiral; on the right, appropriate modification in three repeats).
St ¼
� �2 trx R d ¼ ; tk 18 η
4. Conclusion
(27)
The developed CFD model is an accurate and reliable representation of both the flow and sedimentation in whirlpool during the filling and whirling phases. Expanding the existing model with information about properties of the dispersed phase enabled a full analysis of wort clari fication. The experimental results allowed verification of the CFD model and proved its suitability. This model can be used to test bottom mod ifications to create denser hot trub cone. Simulation analysis showed that hot trub starts to cluster in bigger conglomerates halfway through the filling process, and preliminary formation of the cone appears at half-time of whirling. When the cone is formed, the maximum values of the velocity linger on edges of the cone. Analysis of the movement and formation of the hot trub cone confirms that the particles follow the trajectory of the Ekman spiral. It is therefore justified to consider fragments of spiral geometry to direct the primary flow to intensify the tea leaf paradox in the separator. Particle movement is mostly studied by DPM, DEM, or DDPM models. However, the DPM model is incapable to simulate sedimenta tion. Both DEM and DDPM showed problems with turbulence conver gence and the computation is time-consuming. The VOF model used for all three phases had reasonable computational time and had very good convergence for continuity, momentum, and turbulence equations. Ve locity profiles and cone shape were also appropriate.
where trx ¼
Rd2 ; 18ν
(27.1)
rffiffi � �2 trx R d Rd2 ν 3 1 ; η ¼ ν4 ε4 ¼ ; trx ¼ ; tk ¼ ε tk 18 η 18ν
(27.2)
and St ¼ and 3
1
η ¼ ν4 ε4
(27.3)
3.5. Prospect of follow-up works Currently, the authors are conducting simulations and PIV experi ments to assess the possibility to modify the whirlpool’s bottom con struction based on fragments of the Ekman spiral. Fig. 6 shows some variants of the modifications. Altogether, there are over 30 different variants that are the subject of a patent from the authors of the study (Jakubowski and Stachnik, 2019). The suggested alteration in the shape of the bottom is aimed to improve conditions of the formation of the hot trub cone in whirlpool through an intensified direction of the flow toward the central zone at the bottom of the separator. An important assumption while making modifications is that they should not make it difficult to remove deposit after the process and should not disrupt secondary processes, e.g., whirlpool washing, which is essential to maintain the microbiological �rka-Stepuk, 2018). safety of beer production (Piepio
Author contributions Marta Stachnik: Software, Investigation, Validation, Resources, Writing - Original draft preparation, Visualization, Writing - Reviewing and Editing, Marek Jakubowski: Conceptualization, Methodology, Investigation, Resources, Validation, Writing - Original draft prepara tion, Visualization, Writing - Reviewing and Editing.
9
M. Stachnik and M. Jakubowski
Journal of Food Engineering 274 (2020) 109846
References
Jubaer, H., Afshar, S., Xiao, J., Chen, X.D., Selomuly, C., Woo, M.W., 2019. On the effect of turbulence models on CFD simulations of a counter-current spray drying process. Chem. Eng. Res. Des. 141, 592–607. https://doi.org/10.1016/j.cherd.2018.11.024. Kharoua, N., Alshehhi, M., 2018. The interaction of confined swirling flow with a conical bluff body: numerical simulation. Chem. Eng. Res. Des. 136, 207–218. https://doi. org/10.1016/j.cherd.2018.04.034. Kim, J.E., Chung, Y.M., 2018. CFD-DEM simulation of the fluidized-bed granulation of food powders. Biotechnol. Bioproc. Eng. 1–15. https://doi.org/10.1007/s12257018-0382-6. Kunes, J., 2012. Dimensionless Physical Quantities in Science and Engineering, first ed. Elsevier, ISBN 9780123914583, p. 96. Lau, T., Graham, N., 2015. The influence of Stokes number on particle clustering within a two-phase turbulent jet. In: 7th Australian Conference on Laser Diagnostics in Fluid Mechanics and Combustion, at Melbourne, Australia. Liu, S., Yang, L-l, Zhangm, D., Xu, J.-Y., 2018. Separation characteristics of the gas and liquid phases in a vane-type swirling flow field. Int. J. Multiph. Flow 107, 131–145. https://doi.org/10.1016/j.ijmultiphaseflow.2018.05.025. Liu, Q., Nie, W., Hua, Y., Peng, H., Liu, C., Wei, C., 2019. Research on tunnel ventilation systems: dust diffusion and pollution behaviour by air curtains based on CFD technology and field measurement. Build. Environ. 147, 444–460. https://doi.org/ 10.1016/j.buildenv.2018.08.061. Long, D.F., Perivilli, S.V., MaugerJ, V.W., 2014. Einstein’s tea leaf paradox and its relevance to dissolution testing. Dissolution Technol. 21 (3), 17–18. https://doi.org/ 10.14227/DT210314P17. Misra, A., Bonamy, C., de Souza, L.M., Hohl, L., Illner, M., Kraume, M., Repke, J.-U., Th�evenin, D., 2018. A multi-fluid approach to simulate separation of liquid-liquid systems in a gravity settler. Comput. Aided Chem. Eng. 43, 31–36. https://doi.org/ 10.1016/B978-0-444-64235-6.50008-5. Piepi� orka-Stepuk, J., 2018. Analysis of physical impurities in regenerated solutions used in cleaning brewing systems. J. Inst. Brew. https://doi.org/10.1002/jib.545 (in press). Rahman, M., Andersson, H.I., 2018. Revolving flow of a fluid-particle suspension with suction. Alexandria Eng. J. 57 (4), 2567–2572. https://doi.org/10.1016/j. aej.2017.08.017. Sterczy� nska, M., Jakubowski, M., 2017. Research on particles’ velocity distribution in a whirlpool separator using the PIV method of measurement. Int. J. Food Eng. 13, 6. https://doi.org/10.1515/ijfe-2016-0316. Sukhanovskii, A., Evgrafova, A., Popova, E., 2017. Non-axisymmetric structure of the boundary layer of intensive cyclonic vortex. Dyn. Atmos. Oceans 80, 12–28. https:// doi.org/10.1016/j.dynatmoce.2017.08.001. Tandon, A., Marshall, J., 2010. Einstein’s tea leaves and pressure systems in the atmosphere. Phys. Teach. 48 (5), 292–295. https://doi.org/10.1119/1.3393055. Wang, H., Nie, W., Cheng, W., Liu, Q., Jin, H., 2018. Effects of air volume ratio parameters on air curtain dust suppression in a rock tunnel’s fully-mechanized working face. Adv. Powder Technol. 29, 230–244. https://doi.org/10.1016/j. apt.2017.11.007, 2018. Wessel, R.A., Righi, J., 1988. Generalized correlations for inertial impaction of particles on a circular cylinder. Aerosol Sci. Technol. 9 (1), 29–60. https://doi.org/10.1080/ 02786828808959193. Yamamoto, T., Yano, M., Okano, Y., Kino-oka, M., 2018. Numerical investigation for the movement of cell colonies in bioreactors: stirring and orbital shaking tanks. J. Chem. Eng. Jpn. 51 (5), 423–430. https://doi.org/10.1252/jcej.17we112. Zhang, J., Fu, S., 2019. An efficient approach for quantifying parameter uncertainty in the SST turbulence model. Comput. Fluids 181, 173–187. https://doi.org/10.1016/j. compfluid.2019.01.017.
Anufriev, I.S., Krasinsky, D.V., Shadrin, E.Y., Sharypov, O.V., 2014. Visualization of flow structure in a vortex furnace. Tech. Phys. Lett. 40 (10), 879–882. https://doi.org/ 10.1134/S1063785014100022. Anufriev, I.S., Sharypov, O.V., Dekterev, A.A., Shadrin, E.Yu, Papulov, A., 2017. Study of flow structure in a four-vortex furnace model. Thermophys. Aeromechanics 24 (6), 849–855. https://doi.org/10.1134/S0869864317060038. Azmira, J., Houa, Q., Yuab, A., 2019. CFD-DEM simulation of drying of food grains with particle shrinkage. Powder Technol. 343 (1), 792–802. https://doi.org/10.1016/j. powtec.2018.11.097. Cai, P., Nie, W., Chen, D., Yang, S., Liu, Z., 2019. Effect of air flow rate on pollutant dispersion pattern of coal dust particles at fully mechanized mining face based on numerical simulation. Fuel 239, 623–635. https://doi.org/10.1016/j. fuel.2018.11.030. Constantin, A., Johnson, R.S., 2018. Large-scale oceanic currents as shallow-water asymptotic solutions of the Navier-Stokes equation in rotating spherical coordinates. Deep Sea Res Part 2 Top Stud Oceanogr. https://doi.org/10.1016/j. dsr2.2018.12.007 (in press). Denk, V., 1998. The whirlpool state-of-the-art 1997. Brauwelt Int. 1, 31–43. Diakun, J., Jakubowski, M., 2014. Dimensionless numbers of structural and process similitude of a whirlpool hot trub separator. J. Food Process. Eng. 36 (6), 748–752. https://doi.org/10.1111/jfpe.12043. Einstein, A., 1926. The cause of the formation of meanders in the courses of rivers and of the so-called Baer’s law. Naturwissenschaften 14 (2), 223–224 (in German). Farmanara, M., Malakooti, H., Hassanzadeh, S., 2018. Large eddy simulation of the Ekman transport in a stratified coastal sea: a case study of the Persian Gulf. Estuar. Coast Shelf Sci. 212, 372–386. https://doi.org/10.1016/j.ecss.2018.08.003. Gonz� alez-Vera, A.S., Duran-Matutem, M., van Heijsmt, G.J.F., 2018. Morphodynamics of a sediment bed in a fluid-filled cylinder during spin-down: an experimental study. Phys Rev Fluids 3, 1–22. https://doi.org/10.1103/PhysRevFluids.3.124306, 124306. Hua, Y., Nie, W., Wei, W., Liu, Q., Liu, Y., Peng, H., 2018. Research on multi-radial swirling flow for optimal control of dust dispersion and pollution at a fully mechanized tunnelling face. TunnUndergr Sp Tech 79, 293–303. https://doi.org/ 10.1016/j.tust.2018.05.018. Huang, L., Deng, S., Chen, Z., Guan, J., Chen, M., 2018. Numerical analysis of a novel gas-liquid pre-separation cyclone. Separ. Purif. Technol. 194, 470–479. https://doi. org/10.1016/j.seppur.2017.11.066. H€ ohne, T., 2019. Evaluation of safety of light-water-cooled reactors using CFD. In: Joshi, J.B., Nayak, A.K. (Eds.), Advances of Computational Fluid Dynamics in Nuclear Reactor Design and Safety Assessment. In Woodhead Publishing Series in Energy, pp. 337–386. https://doi.org/10.1016/B978-0-08-102337-2.00004-3. Iverson, R.M., Denlinger, R.P., 2001. Flow of variably fluidized granular masses across three dimensional terrain: 1. Coulomb mixture theory. Papers on Chemistry and Physics of Minerals and Rocks Volcanology 106 (B1), 537–552. https://doi.org/ 10.1029/2000JB900329. Jakubowski, M., Stachnik, M., 2019. Whirlpool tub for use in brewing industry to remove hot sediment from beer wort after cooking, has cylindrical tank equipped with bottom sealing and formed in octagonal shape, and arms connected with middle part of cylindrical tank. National patent. PL419602-A1. Jakubowski, M., Sterczy� nska, M., Matysko, R., Poreda, A., 2014. Simulation and experimental research on the flow inside a whirlpool separator. J. Food Eng. 133, 9–15. https://doi.org/10.1016/j.jfoodeng.2014.02.011. Jaskulski, M., Wawrzyniak, P., Zbici� nski, I., 2018. CFD simulations of droplet and particle agglomeration in an industrial counter-current spray dryer. Adv. Powder Technol. 29 (7), 1724–1733. https://doi.org/10.1016/j.apt.2018.04.007.
10
Contents lists available at ScienceDirect
Journal of Food Engineering journal homepage: http://www.elsevier.com/locate/jfoodeng
Multiphase model of flow and separation phases in a whirlpool: Advanced simulation and phenomena visualization approach Marta Stachnik, Marek Jakubowski * Koszalin University of Technology, Faculty of Mechanical Engineering, Department of Food Industry Processes and Facilities, Poland
A R T I C L E I N F O
A B S T R A C T
Keywords: Whirloool CFD Simulation Ekman’s spiral Multiphase rotating flow
In whirlpool, a separator used in the brewing industry, the deposits accumulate in the central zone at the bottom of the separator. The deposit assumes a shape that resembles a flattened cone, referred to as the “tea leaf paradox.” It is so-named because the deposit accumulates seemingly against the influence of the centrifugal force that develops from the rotating movement of the mixture. Previous studies have performed a simulation of the two-phase volume of fraction (VOF) model with swirling liquid and air. In the present study, we suggest an expansion of the existing model with a dispersed phase of the hot trub. A three-phase VOF model was built to analyze the movement and formation of the cone. Material data obtained referred to the wort and hot trub after boiling with hops. The simulated shape of the cone was compared to its counterparts obtained under industrial conditions. The values of the velocity of particles of deposit were also analyzed. Results were compared with the visualization of the cone’s formation for rapidly sedimenting particles and for hot trub and wort flow under laboratory conditions. The analysis deems the VOF three-phase model suitable to predict sedimentation and accumulation of sediment.
1. Introduction Whirlpool (Fig. 1a in Section 2.3) is a separator used to remove dispersed solid phase from the liquid. It is a cylindrical, empty container that is filled tangentially. Filling of the whirlpool tangentially results in the rotating movement of the separated mixture. The separation is enabled by the rotating movement and sedimentation is sustained by the gravitation. It is an elegant and effective approach for liquid clarifica tion. It has a low requirement of energy, as it is a mix of spontaneous naturally occurring phenomena. Whirlpool has been universally applied to separate hot deposit from the wort. When wort is boiled with hops, a hot trub, i.e., protein-tanning conglomerates, is formed; this trub is a waste and must be removed. The empty container of the separator is filled through the inlet hole located tangentially to the wall. The mixture is thus set into a rotating movement. The primary flow that occurs is due to the tangential filling of the container. A system of secondary flow appears during slowing down of the rotating movement of the mixture. The dominating whirls are responsible for the formation of the deposit cone in the central zone at the bottom of the separator. This effect was reported by Einstein in
1926 and named as tea leaf paradox (Long et al., 2014). Even though the name suggests that this phenomenon is a paradox, it is quite simple to explain. The centrifugal force is formed due to the rotation of the separated mixture. Seemingly, one should expect the particles to move outside, closer to the joints of the bottom and the wall of the tank. As this does not occur, the phenomenon is termed as paradox. The hampering influence of the container’s bottom and the walls cause the deposit to follow a spiral movement to the central zone of the bottom of the container. Additionally, this phenomenon is supported by a gradient of the hydrostatic pressure resulting from the difference in the height of the column of liquid in the container. This difference results from the shape of the free surface, which is formed by the rotating movement of the separated mixture. The problem of multiphase modeling for the tea leaf paradox is related to both modeling of the multiphase flow with the separation and specificity of the rotating movement. A literature review shows various studies on the Computation Fluid Dynamics models of the multiphase flow or on the CFD models of the rotating movement. However, few studies have linked both these issues. The problem of modeling of the sedimenting dispersed solid phase, hot trub in this case, required prac tice definition and the selection of parameters of the dispersed phase
* Corresponding author. Eng Koszalin University of Technology Faculty of Mechanical Engineering Department of Food Industry Processes and Facilities, 15-17 Raclawicka st., PL-75620, Koszalin, Poland. E-mail address: [email protected] (M. Jakubowski). https://doi.org/10.1016/j.jfoodeng.2019.109846 Received 16 March 2019; Received in revised form 11 November 2019; Accepted 27 November 2019 Available online 5 December 2019 0260-8774/© 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
M. Stachnik and M. Jakubowski
Journal of Food Engineering 274 (2020) 109846
Nomenclature Cd D dd Dmd F fd g H hb I KH Khb Kub L lT LT mdc b n P Pk
α t
tk trx u u0 ub uslide uv uτ v β δw
mass density of dispersed phase, kg m 3 vessel diameter, m average diameter of particles, μm diffusion coefficient of dispersed phase in a fluid, m s 2 additional volumetric forces, N kg 1 coefficient of flow resistance for dispersed phase gravitational acceleration, m s 2 height of the fluid after filling, m height of the secondary flow, m; identity matrix dimensional proportions of the vat (H/D) the number of the height of the vortex at the bottom of vat the number of the velocity of the vortex at the bottom of the vat distance in the tunnel, m turbulence intensity turbulence length scale, m mass transfer between phases (if the particles do not dissolve, then: mdc ¼ 0), kg s 1 normal to surface unit vector pressure, Pa turbulence production volume fraction - time, s
ε η μ μT ν ρ σk σω ω Index c d m r
Kolmogorov time scale relaxation time of particles, s velocity, m s 1 normal velocity on inlet, m s 1 maximum velocity of the secondary flow, m s velocity in-between phases, m s 1 free fall velocity of particle, m s 1 friction velocity, m s 1 volume, m3 model constant distance between wall and grid, m dissipation rate of turbulent energy Kolmogorov length, m dynamic viscosity, Pa⋅s turbulent viscosity, Pa⋅s kinematic viscosity, m2 s 1 density, kg m 3 model constant model constant turbulent frequency
1
continuous phase dispersed phase mixture - phase representation, 1 – water, 2 – air, 3 – particles
Fig. 1. Whirlpool: a) one of the biggest capacity (1020 hL) vessel in Europe (Warka brewery brewhouse by GEA–Huppmann); b) geometrical model with the finite element grid and boundary conditions for the mathematical model.
that would diligently represent particle movement and accumulation in the form of a cone. Many CFD studies on the multiphase flow discuss drying, fluidiza tion, and bubble columns but mainly focus on spray drying. Azmira et al. (2019) developed a CFD-DEM (Discreet Method Element) model of grain drying that included reduction of grain size. By using this model, they
investigated the influence of temperature and velocity on the rate of drying and the size of grains. Kim and Chung (2018) analyzed the granulation of food powders, including a change in their rheological properties and water content. Wang et al. (2018) studied the effective ness of an air curtain to prevent dust propagation in a rock tunnel during drilling. Liu et al. (2019) investigated a ventilation system. Cai et al. 2
M. Stachnik and M. Jakubowski
Journal of Food Engineering 274 (2020) 109846
(2019) studied the effect of air flow rate on pollutant dispersion pattern of coal dust particles at fully mechanized mining face. Jaskulski et al. (2018) built a CFD model of a counter-current spray dryer. They analyzed collisions of particles, including the interaction between drops and that between the drop and the dry particle. For simulation, they suggested a moisture evaporation model based on the concept of a characteristic drying curve. For the same issue, i.e., counter-current spray drying process Jubaer et al. (2019) compared the efficiency of different turbulence models for the simulation of the flow in the spray dryer. The best model was the k ω SST model that was also utilized by the authors of the present study. Modeling of a rotating flow is not easy. However, because this kind of rotating movement is universally present, many studies have been conducted on this subject, but are rather loosely related to the food in dustry. Anufriev et al. (2014) analyzed the spatial structure of whirl pools in a rotary stove by using CFD. Rotary flow improves mixing of air with fuel and the distribution of the mixture in the entire volume of the stove. The simulation results formed the base for follow-up studies and Particle Image Velocimetry (PIV) measurements in a four-vortex furnace for the burning of coal dust (Anufriev et al., 2017). Another example is the use of Large Eddy Simulation (LES) to analyze the influence of a conical obstacle on the flow in a pipe and reduction of whirls in the flow (Kharoua and Alshehhi, 2018). The LES model is particularly interesting for the subsequent development of simulation analyses of a rotating flow with sedimentation of the dispersed phase, as presented in this study. However, this model is highly demanding in terms of computational power, particularly for transient analysis. The modeling of a rotating movement in cyclones was the subject of investigation in the subsequent studies. Huang et al. (2018) presented a study of guide vanes to create whirlpools and uniflow stream for the preliminary separation of the mixture. Cyclone’s efficiency was assessed with a computer model with Eulerian–Lagrangian approach with the RSM (Stress Reynolds Model). Another example is the study by Liu et al. (2018). A vane-separator performance was simulated with a two-phase model to analyze the size of gas bubbles at the inlet, the propagation of the gaseous phase, and the intensity of the turbulence. The obtained model correctly predicts the size of gas bubbles based on the rate of flow at the inlet and fractions of phases and viscosity of a liquid. An example of a study of dedusting equipment is a numerical analysis of the air flow in a fully mechanized tunneling face. The simulation allowed assessing the migration of dust during excavation. The model served to assess the possibility to convert a multiradial swirling flow field produced by the generator into an effective dust-control air curtain where the tunneler driver is located (Hua et al., 2018). However, particular attention should be given to the study of the movement of the cell colonies in bioreactors. Here, the tea leaf paradox is undesirable, and mixing prevents the colonies from forming a sedi ment cone at the bottom. Yamamoto et al. (2018) used numerical model was to study stirring and orbital shaking tanks. This work presents the results of a 3D simulation of a three-phase model with two liquid phases: air and wort and a dispersed phase: hot trub. Addition of the dispersed phase is aimed to allow full flow analysis and separation in equipment with a tea leaf paradox phenomenon, e.g., whirlpool. In existing studies, we did not find a CFD analysis for the agglomeration and formation of the sediment cone. Thus, the present study is unique. This study is related to both the tea leaf paradox and the sedimentation of particles into a concentrated deposit.
2.1. Mathematical model
2. Materials and methods
3 fd ρ ρd ρ juslide juslide ¼ m rp: 4 dd c ρm
This section presents equations of a mathematical model to describe a rotating flow of the wort in the tank and the equation to describe the movement of the hot trub. The next subsection describes the assump tions for the computer model, i.e., initial and boundary conditions, and simplifications.
The correct laminar to turbulent transfer is ensured by the limitation of the Reynolds number as follows:
A flow of a fluid is described by the conservation of mass (continuity) equation and the conservation of momentum—momentum equation, also known as the Navier–Stokes equations. The equation of continuity (1) for the mixture of the dispersed solid and continuous phase is obtained by applying the principle of mass conservation to a small volume of fluid. The standard (general) form for Cartesian coordinates is as follows (Misra et al., 2018):
∂ ðρ Þ þ r ⋅ ðρm um Þ ¼ 0 ∂t m
(1)
where:
ρm ¼
n X
(1.1)
α r ρr
r 1
and (1.2)
α1 þ α2 þ α3 ¼ 1
Equations for the conservation of momentum are derived by applying Newton’s law of motion to a small volume of fluid. Newton’s law states that the rate of change of momentum for a volume of fluid is equal to the sum of all external forces acting on that volume. The equation of momentum balance (2) takes the following form:
ρm
∂um þ ρm um ⋅r ¼ ∂t
� r
p þ ðμm þ μT Þ rum þ ðrum ÞT
þρm g þ F;
��
r⋅f½ρd Cd ð1
Cd Þ�uslide uslide g (2)
where: Cd ¼
α3 ρd ρm
(2.1)
and Pn um ¼
αρ ρm
(2.2)
α r μr
(2.3)
r 1 r r ur
and
μm ¼
n X r 1
For a Newtonian fluid (in this case, it is wort), viscosity is constant. The left side of the equation represents a temporal variation in mo mentum and fluid acceleration, whereas the right side represents the pressure gradient force (the normal stress) and the viscous force (the tangential shear stress). Gravity force is necessary for sedimentation (Jakubowski et al., 2014; Misra et al., 2018). The flow of a dispersed phase is represented by equation (3): � � ∂α3 Dmd mdc rϕd ¼ (3) þ r ⋅ α3 um þ α3 ð1 Cd Þuslide ∂t ϕd ρd The Schiller–Neumann model was applied because a fluid-fluid drag function is used. It is described by equation (4):
3
(4)
M. Stachnik and M. Jakubowski
Journal of Food Engineering 274 (2020) 109846
8 9 � 24 � > > < = 1 þ 0:15Re0:687 p Re < 1000 for p fd ¼ Rep ; Re > 1000 > > p : ; 0:44
δþ w ¼
(5)
dd ρc juslide j
b n ⋅Nα3 ¼ 0
(5.1)
μm
�
2 ρkr⋅u; 3
2 k ¼ ðu0 lT Þ2 3 by:
k
�
ρm βω2 :
(8)
Turbulence by the wall (10) is described as follows: (10)
Dissipation of the energy by the wall (11) is given as: k ; kυδþ wμ
(16)
The simulation was conducted with software Ansys CFX 18.1 and was based on the Finite Volume Method. The modeled case was turbu lent, and the tank’s geometry was regular; thus, it was suitable to use the structured–hexahedral–grid. The quality of the mesh was judged ac cording to the shape of the free surface. A VOF of 0.5 was set for the free surface test level. The shape of the free surface natural conditions is of the paraboloid of revolution. The choice of the number of the grid’s element was based on a previous analysis (Jakubowski et al., 2014) and was set to around 2,000,000 elements (Fig. 1b). Mesh quality was validated by testing the volume of wort in the tank after filling enclosed by free surface for VOFwort equal to 0.1, 0.5, and 0.9. The volumes varied by 5%; therefore, the mesh was deemed suitable. In the CFX the volume of control is assembled around the nodes (Cell Vertex), where each element is divided in sub volumes. In term of grid quality CFX is more permissive. The CFX solver calculates 3 important measures of mesh quality at the start of a run: mesh orthogonality – minimum 56.1: good; aspect ratio – maximum 12: good; expansion factor – maximum 7: good. Mean skewness is 0.99 and deems mesh bad. This high skew is a result of fitting hexahedral elements of mesh along a curve of the cylindrical geometry. This is unavoidable issue. However, hexahedral is the only mesh that allows the simulations and grants correct shape of the free surface; any other type or combination provides wrong results or divergence. However, in terms of overall mesh quality minimum quality is 0.73, maximum ¼ 1, and mean ¼ 0.97. The inlet surface was refined to 30 � 30 nodes. In earlier simulations, it was noted that the lower number of nodes at the inlet leads to incorrect calculation of the rate of flow. This results in much slower filling of the tank than in the experiment. The top limiting surface was considered as an open condition, considering that at time zero, the tank was empty—filled with air (volume of fraction (VOF) for air ¼ 1,VOF for wort ¼ 0, and VOF for residue ¼ 0). The open condition enabled the air to be pushed out by the liquid but also to flow back. Opening (outflow)
The relationship between shear stress and the viscosity of liquid (9) is shown by: � � � 2 2 ðμc þ μT Þ rum þ ðrum ÞT ðμc þ μT Þðr ⋅ um ÞI ρkI bn ¼ 3 3 u ρc þτ ðum ðum ⋅ bn Þbn Þ (9) δw
ρm
(15)
2.3. Computational model
(7)
The velocity by the wall is described by equation (8):
ω¼
LT
where p0 is the surrounding—constant pressure.
2.2. Boundary conditions
rk ⋅ b n¼0
1=4
The stress continuity condition on the free boundary—the free sur face is as follows (17): �� n; (17) n ¼ p0 b pI þ ηm ru þ ðruÞT ⋅ b
The SST model allows avoiding miscalculations of the turbulence at the wall. This model uses the near-wall formulation of the κ-ω model by switching automatically from low Reynolds number formulation to the wall function treatment at the wall boundary. The SST model correctly calculates the turbulence in cases such as sudden pipe expansions. In the analyzed case, we have a similar situation—the diameter of the filling pipe is much smaller than that of the container.
um ⋅ b n¼0
ðβ*0 Þ
α3 ¼ α30
(6.2)
� ω ∂ω þ αr ρum ⋅ rω ¼ r ⋅ ½αr ðμm þ σω μT Þrω� þ αr ϕ Pk ∂t k
k1=2
The concentration of the dispersed phase at the inlet (16) is expressed by:
- for energy dissipation rate (7):
α r ρ;
(14)
Dissipation rate of the turbulence energy at the inlet (15) is described
ω¼
(6.1)
ω
(13)
The kinetic energy of the turbulence at the inlet (14) is given as follows:
and:
μT ¼ ρ
n u0 b
um ¼
where: 2 ðr⋅uÞ2 3
(12)
The boundary condition at the inlet, which defines the velocity of the mixture, is represented by equation (13):
An SST model was used for turbulence. This two-equation model is the simple and popular choice for modeling turbulence. This is because in this model, two different transport equations characterize two inde pendent properties of the turbulent flow. Moreover, it is robust, economical, and reasonably accurate for a large variety of turbulent flows (Jakubowski et al., 2014; Zhang and Fu, 2019). To solve turbu lence, a homogenous turbulence model was used. Turbulence equations were written originally only for single-phase flows. Turbulence for each volume fraction (αr) of the phase is modeled separately by the following equations: - for kinetic energy (6): � � � � ∂k μ αr ρ þ αr ρu ⋅ rk ¼ r ⋅ αr μ þ T rk þ αr ðPk ρβ* kωÞ; (6) ∂t σk
� � Pk ¼ μT ru : ru þ ðruÞT
(11.1)
The movement of the dispersed phase by the wall is defined by equation (12):
where Reynolds number is described by equation (5.1): Rep ¼
ρm u τ δ w μ
(11)
where:
4
M. Stachnik and M. Jakubowski
Journal of Food Engineering 274 (2020) 109846
Fig. 2. Geometrical forms of the deposit cone and volume fraction changes during the course of filling and whirling at the chosen time steps: (a) 7.5 s; (b) 10 s (c) 15 s; (d) 45 s; (e) 65 s; (f) 70 s; (g) 80 s; (h) 135 s; seen from a side (on the left) and from the bottom (on the right).
condition prevents numerical overflow and provides numerical stability. The flow direction of air was set to normal to boundary condition with medium turbulence intensity. The filling velocity was set as 2 m s 1. Velocity was calculated from Denk (1998) suggestions for correct filling velocity. At the inlet, VOF for wort was set as 0.99 and for residue as 0.01. This proportion represents the real concentration of residue in the wort before clarification. Mass and momentum at the inlet were defined by Cartesian velocity components. To enforce the tangential clockwise manner of filling, Z and X components were 0 and the Y component was set to be 2 m s 1. The turbulence intensity was medium (5%). The wall and the bottom of the tank were defined as a stationary, smooth wall with the no-slip condition. To achieve correct results near the wall, the wall distance of the first element was set at 3 � 10 6 m, which ensured that the dimensionless wall distance (yþ) was 1. The wall function in the simulation was set as automatic. To calculate sedimentation, it was necessary to provide data such as the viscosity of the residue, its diameter, and density. Thus, it is possible to use the presented model to analyze sedimentation of different vari eties of the residue. This is relevant as beer production utilizes many different raw materials that produce different hot trubs. Given the fact that the suspended residue in the wort constitutes a small fraction, viscosity for the hot trub was set the same as that for the wort. Particles were modeled by an equivalent diameter (0.41 mm) and density (980 kg m 3), and the data were based on the measurements of the hot trub. During filling and whirling, particles of the hot trub collide inelastically with the wall and with each other and create bigger con glomerates. As conglomerates become bigger and reach critical mass,
they fall out of the suspension and form sediment at the bottom. Resti tution coefficient for the particles was set to 0.2. Particles are forced to stop at the wall by the restitution coefficient of 0.1. The time step of the simulation varied depending on the level of filling the whirlpool. Numerical stability in CFX is measured by Courant number and solution convergence. Courant number should be less than 1 and solution should converge after less than 6 iterations to be considered correct. The time step was set to keep courant number at 0.57. This thus led to solution converging after 3 to 6 iterations (six it erations occurred in the early state of filling). Moreover, the RMS was monitored to not exceed 1 � 10 4. For the initial phase of filling, time step was set to 0.5 � 10 3 s. In the later phase of filling (after exceeding the height of the inlet by the free surface), time step was increased to 3 � 10 3 s. After filling (for the stage of whirling), time step was set to 1 � 10 3 s. Ansys CFX is an element-based finite-volume with second-order discretization schemes in space and time simulating software. Dis cretized governing equations are solved by coupled algebraic multigrid €hne, 2019). The momentum equation was solved using the algorithm (Ho 2nd order upwind scheme. 3. Results and discussion 3.1. Simulation of the formation of the hot trub cone This section presents results obtained from the CFD simulation. The focus is on the residue behavior and hot trub cone shape. The process of 5
M. Stachnik and M. Jakubowski
Journal of Food Engineering 274 (2020) 109846
Fig. 3. Velocity distribution of particles in the deposit cone, during the course of filling, and whirling, in selected time steps: (a) 7.5 s; (b) 15 s; (c) 65 s; (d) 70 s; (e) 90 s; (f) 135 s.
wort clarification can be divided into two characteristic periods: filling and whirling. Fig. 2 illustrates the hot trub cone formation. The results from CFD simulation reflect the data from experiments and industrial process. Deposit is identified for every mesh element, for which the participation of the particle faction was estimated above 0.5 VOF. Simulation of clarification was carried for a total of 135 s. The filling was carried out for 75 s, and it was continued until the height of the fluid was equal to the diameter of the vat (counted volumetrically for the stand still). This proportion is based on earlier studies and reflects the indus trial conditions of the process. At this moment, the inflow of wort and hot trub mixture was cut off. The process of whirling lasted from 76th to 135thsecond; at that time, the wort slowed down and deposit fell to the bottom, thereby forming a cone. Following the progression of the tea leaf paradox, described origi nally by Einstein (1926) and reported by other authors (TandonMar shall, 2010; Long et al., 2014; Rahman and Andersson, 2018), hot trub should accumulate in the center zone of the whirlpool’s bottom in the form of a dense cone. Unlike previous descriptions, the cone is more flattened, as shown in the data from CFD presented in Fig. 2, and confirmed with the observation of experimental visualization of this study. Fig. 2 presents movement of the hot trub during clarification process, seen from the side and from the bottom. CFX VOF model consisting of two continuous phases and one dispersed does not allow for particle tracking. Phases are tracked based on their volume fraction, what is suitable for the sediment accumulation. During filling, the deposit fol lows the horizontal primary flow because of the tangential manner of the filling process. At the beginning of the filling process, 7.5th second, the hot trub moves in the outside space, near the joint between the bottom and walls of the tank (Fig. 2 a). Hot trub volume fraction at the inlet was set to 1%, thus VOF of hot trub in the beginning was 0.01. Particles are small and dispersed. However, some particles begin to form conglom erates of VOF ¼ 0.6. Most of the particles were suspended in full volume of fluid. The rotating movement of the mixture slows down due to the natural dispersion of the energy. As the filling continues, 10th second, particles move closer to the tank’s axis. Conglomerates with VOF>0.2 are more numerous in the central part, while the dispersed small
particles are still being pumped near the wall. The central zone is still empty, thus no hot trub is identified there. Particles spiral toward the center of the tank (Fig. 2 b). In 15th second of filling the bottom is fully covered with the fluid. Hot trub particles are identified across the plain and conglomerates of VOF>0.5 are numerous and VOF ¼ 1 conglom erates are identified (Fig. 2 c). After 45 s of filling – halfway through – the hot trub moves toward the central zone of the container (Fig. 2 d). At this point the cone is fragmented and primary flow drags it across the bottom. Majority of particles formed bigger conglomerates with VOF ¼ 0.25–0.50. In 45th second sediment cone had a diameter of 85 mm. At the end of the filling process, 65thand 70thsecond, the volumetric frac tion of the dispersed solid phase stabilizes. Volume of fluid in the tank hampers the influence of the incoming fluid on the cone’s shape. Hot trub forms denser cone in the central zone of the tank, whit only pe ripheries staying loose (Fig. 2 e-f). Small particles are still pumped in the tank and can be identified near the wall. In 65th second cone had diameter in the widest part of 125 mm and on the thinner side of 75 mm. Upon the end of filling process, the shape of the hot trub suspended in the fluid is stabilized. In 70th second particles suspended in the fluid were mostly small with few bigger conglomerates, VOF ¼ 0.01–0.5. The shape of the cone resembles a circle with the mean diameter of 95 mm. At the 80th second the filling is stopped, but small particles are still present near the walls, at this moment hot trub’s cone is strongly dis rupted (Fig. 2 g). This behavior is associated with a sudden change in velocity, which causes short-lasting pulsatile flow near the bottom of the tank. High-velocity flow caused by filling is still present in the fluid but it is now followed by low-velocity flow after the inlet is closed. Majority of the particles sedimented to the bottom, and very few particles were still suspended in the fluid. Cone’s diameter at this time was 135 mm. Cone’s core has VOF ¼ 1, whereas the surface has the VOF ¼ 0.5–0.75. Toward the end, 135th second, the flow comes to a standstill, and the hot trub obtains its peculiar shape (Fig. 2h.). Surface VOF of the cone shows VOF ¼ 1 in some places, thus indicating overall denser cone, with mean diameter of 130 mm. Lower values of VOF at the surface indicates presence of smaller, lighter particles that can be easily moved by the fluid. The edges of the cone formed by smaller particles take the longest to stop (Fig. 3 – velocity profiles) and are thus susceptible to disruption 6
M. Stachnik and M. Jakubowski
Journal of Food Engineering 274 (2020) 109846
Fig. 4. Graph of the change in the share of the volume of hot trub compared to the change in VOF share in the test element (above: chart of the change in the total volume during whirlpool filling).
and scouring. Subsequently, only the cone’s periphery undergoes a change in the shape; these are the hallmarks of the spiral form (more widely described later in the study). Particle velocity change during the sedimentation was analyzed. Fig. 3 presents particle velocity distribution for selected time steps. According to the natural course of the phenomena, at first, maximum values of the velocity appear in the area of the joint between the bottom and the wall of the tank, and this is the reason why the deposit is localized only in this area (Fig. 3 a). Around the 65th second of filling, the hot trub cone starts to form in the middle zone, and the minimum
values of the velocity appear in this zone (Fig. 3b and c). At the end of the filling process, maximum values of the velocity are identified at the edges of formation of the cone (Fig. 3 d). With the passage of time, the primary flow slows down, and hot trub cone assumes its ultimate shape; this is also reflected in the location change of maximum velocity values (Fig. 3 e 3 f). Extremely low values of the particle velocity are identified in the central zone of the cone, which is typical for the nature of the phenomenon. Simulation is thus in compliance with observations in real conditions. Apart from the visual data related to the phenomenon of the cone
Fig. 5. Shapes of deposit cones: a) industrial hot trub cone—whirlpool capacity 30 hL, b) industrial hot trub cone–whirlpool capacity 530 hL, c) hot trub cone from the CFD simulation, d) quartzsand (visualization of the formation of the cone for rapidly sedimenting deposit). 7
M. Stachnik and M. Jakubowski
Journal of Food Engineering 274 (2020) 109846
formation, the simulation provides information about the volume of hot trub and changes in time. In Fig. 4, the graph presents the change in the volume of the hot trub in the liquid phase compared to the participation of the hot trub in the test volume of the fluid element. The distribution of test elements in which particles are identified explains the situations introduced in Fig. 2, where a greater volume of deposit is identified during whirling than in the final cone. That results from VOF of the deposit in each mesh element, rather than the actual participation of deposit in the volume of the fluid, for which the distribution was pre sented in the form of a graph in Fig. 4 (color red). For the formed cone, the participation of the fraction of liquid in the test element is smaller than the total volume of deposit. Experimental data show that the participation of the liquid phase in the hot trub deposit is (depending on raw material composition) from 74% to 82%. In the simulation, the two volumes of hot trub are 0.28 � 10 3 m3 (total volume of hot trub) and 0.11 � 10 3 m3 (volume of hot trub in the cone). The difference (60%) represents approximately the participation of the liquid phase in the cone. The graph shows that the total volume of hot trub in the tank stabilizes, and the deposit becomes denser. For comparison, a change in wort’s total volume is presented (Fig. 4—upper chart) during the filling and whirling (in the graph, stages are separated with a red dotted line).
simulation model with conditions of the flow in the laboratory whirl pool. More details on this subject can be found in Jakubowski et al. � ska and Jakubowski (2017). (2014) and Sterczyn While scaling the tea leaf paradox effect from a laboratory whirlpool to an industrial scale, the authors derived a set of similitude numbers. More details on this subject can be found in Diakun and Jakubowski (2014). One such number is the duration of the main flow in the whirlpool (it is termed as time break): Kt ¼
ut 1 D ð1 þ KH Þ
(18)
Whirling time depends on the inertial mass of the rotating wort and the forces that retard the flow: t¼
D ð1 þ KH Þ u
(19)
The formula for the dissipation force of the flow in the Ekman Boundary layer is u 2 D Hρ ¼t 2 FS2 ub μ Dhb
KF1=
(20)
3.4. Cone forming flow is given as
3.2. Correspondence with the real conditions Numerically obtained shape of the cone, its consecutive geometrical forms, and the time of sedimentation were compared with states appearing in real conditions. Hot trub in industrial conditions forms a flattened cone independent of the working volume of the whirlpool (Fig. 5 a, b). The same geometrical form is obtained from numerical calculations (Fig. 5 c). For substitute deposit in the form of the solid dispersed phase with the high rate of sediment velocity (e.g., quartz sand), the cone has distinctive shoulders resulting from the movement of particles in the so-called Ekman spiral (Fig. 5 d). This phenomenon is well known in the field of meteorology in studies of seas and oceans (Sukhanovskii et al., 2017; Farmanara et al., 2018; Constantin and Johnson, 2018). This form of the cone provided grounds to conduct follow-up works in the optimization of the whirlpool’s bottom con struction. This subject is discussed in details in section 3.5. Other authors also identified the appearance of spiral patterns in a rotating movement. Gonz� alez-Vera et al. (2018) conducted experi mental studies in the scope morphodynamics of the sedimentary layer in a cylinder filled with liquid during spin-down experiments. Spiral for mation of the sedimenting liquid was identified as an outcome of the experiment.
Kb ¼
tμ D2 ρ
(21)
Adjusted Reynolds number for whirlpool tank for primary flow is noted as � � uDρ 1 Rew ¼ 1þ (22) μ KH Adjusted Reynolds number for whirlpool tank for secondary flow is noted as Reb ¼
uDρ Khb μ Kub
(23)
Apart from the originally derived criteria numbers, the work of other authors was implemented. The authors selected universal numbers (adapted for the specifics of the whirlpool condition) to scale the phe nomenon of sedimentation. An appropriate dimensionless number is the Driftage Elements Number (Kunes, 2012). It describes drifting and sedimentation of fine particles suspended in a liquid: ND ¼
uv u gH
(24)
The next dimensionless number suggested to describe the phenom enon in the whirlpool is Inertia Parameter (Wessel and Righi, 1988). This number determines the relationship of the inertia with the friction forces between particles in the liquid flow:
3.3. The experimental verification of the model and scaling the phenomenon PIV (Particle Image Velocimetry) experimental studies were pre sented in earlier reports. The investigations were used to verify the CFD model of a rotating flow of the wort in the whirlpool. The verification was performed by comparing values of the velocity of the primary and secondary flows. The research study on PIV included PMMA tank (plexi glass), rotary piston pump NOVALOBE (GEA Hilge, Germany), magnetic-inductive volumetric flow meter (SM-7500, IFM electronics, Germany), and a table with a set of mirrors. The tank was of the same size as that used in the simulation. Velocity measurements were per formed with double-pulsed laser (EverGreen 532 nm, 145 mJ; Quantel, USA) and a PIV camera FlowSense4M (DantecDynamics, Denmark). Acquired data were analyzed with DynamicStudio software (v. 6.7, DantecDynamics, Denmark). Flow tracing particles were made of a polymer and had a similar density to the fluid. The cone formation was visualized with quartz sand as a rapidly sedimenting material. Velocity, flow rate and size of the tank were the same as in the simulation. The investigations showed a high level of the compliance of the
Ninert ¼
d2d ρd u 18ηH
(25)
To quantify the relationship of the volume of the continuous phase (beer wort) with the density of the dispersed phase (hot trub), a Mass Number was suggested (Iverson and Denlinger, 2001). For Nm > 1, the momentum transfer of the dispersed phase is dominant. For whirlpool Nm < 1, the phenomenon occurs due to a low volume of hot trub sus pended in the wort during the filling. Nm ¼
vd ρd v c ρc
(26)
Stokes Particle Number is universally used to describe the movement of particles under the influence of the inertia in the turbulent flow (Lau and Graham, 2015):
8
M. Stachnik and M. Jakubowski
Journal of Food Engineering 274 (2020) 109846
Fig. 6. Selected variants of the whirlpool’s bottom modification with reference to the fragment of the spiral (data expressed in quarters of its course): a) the second quarter, b) the third quarter, c) the fourth quarter, d) second and third quarter, e) third and fourth quarter (on the left, red indicates fragment of the spiral; on the right, appropriate modification in three repeats).
St ¼
� �2 trx R d ¼ ; tk 18 η
4. Conclusion
(27)
The developed CFD model is an accurate and reliable representation of both the flow and sedimentation in whirlpool during the filling and whirling phases. Expanding the existing model with information about properties of the dispersed phase enabled a full analysis of wort clari fication. The experimental results allowed verification of the CFD model and proved its suitability. This model can be used to test bottom mod ifications to create denser hot trub cone. Simulation analysis showed that hot trub starts to cluster in bigger conglomerates halfway through the filling process, and preliminary formation of the cone appears at half-time of whirling. When the cone is formed, the maximum values of the velocity linger on edges of the cone. Analysis of the movement and formation of the hot trub cone confirms that the particles follow the trajectory of the Ekman spiral. It is therefore justified to consider fragments of spiral geometry to direct the primary flow to intensify the tea leaf paradox in the separator. Particle movement is mostly studied by DPM, DEM, or DDPM models. However, the DPM model is incapable to simulate sedimenta tion. Both DEM and DDPM showed problems with turbulence conver gence and the computation is time-consuming. The VOF model used for all three phases had reasonable computational time and had very good convergence for continuity, momentum, and turbulence equations. Ve locity profiles and cone shape were also appropriate.
where trx ¼
Rd2 ; 18ν
(27.1)
rffiffi � �2 trx R d Rd2 ν 3 1 ; η ¼ ν4 ε4 ¼ ; trx ¼ ; tk ¼ ε tk 18 η 18ν
(27.2)
and St ¼ and 3
1
η ¼ ν4 ε4
(27.3)
3.5. Prospect of follow-up works Currently, the authors are conducting simulations and PIV experi ments to assess the possibility to modify the whirlpool’s bottom con struction based on fragments of the Ekman spiral. Fig. 6 shows some variants of the modifications. Altogether, there are over 30 different variants that are the subject of a patent from the authors of the study (Jakubowski and Stachnik, 2019). The suggested alteration in the shape of the bottom is aimed to improve conditions of the formation of the hot trub cone in whirlpool through an intensified direction of the flow toward the central zone at the bottom of the separator. An important assumption while making modifications is that they should not make it difficult to remove deposit after the process and should not disrupt secondary processes, e.g., whirlpool washing, which is essential to maintain the microbiological �rka-Stepuk, 2018). safety of beer production (Piepio
Author contributions Marta Stachnik: Software, Investigation, Validation, Resources, Writing - Original draft preparation, Visualization, Writing - Reviewing and Editing, Marek Jakubowski: Conceptualization, Methodology, Investigation, Resources, Validation, Writing - Original draft prepara tion, Visualization, Writing - Reviewing and Editing.
9
M. Stachnik and M. Jakubowski
Journal of Food Engineering 274 (2020) 109846
References
Jubaer, H., Afshar, S., Xiao, J., Chen, X.D., Selomuly, C., Woo, M.W., 2019. On the effect of turbulence models on CFD simulations of a counter-current spray drying process. Chem. Eng. Res. Des. 141, 592–607. https://doi.org/10.1016/j.cherd.2018.11.024. Kharoua, N., Alshehhi, M., 2018. The interaction of confined swirling flow with a conical bluff body: numerical simulation. Chem. Eng. Res. Des. 136, 207–218. https://doi. org/10.1016/j.cherd.2018.04.034. Kim, J.E., Chung, Y.M., 2018. CFD-DEM simulation of the fluidized-bed granulation of food powders. Biotechnol. Bioproc. Eng. 1–15. https://doi.org/10.1007/s12257018-0382-6. Kunes, J., 2012. Dimensionless Physical Quantities in Science and Engineering, first ed. Elsevier, ISBN 9780123914583, p. 96. Lau, T., Graham, N., 2015. The influence of Stokes number on particle clustering within a two-phase turbulent jet. In: 7th Australian Conference on Laser Diagnostics in Fluid Mechanics and Combustion, at Melbourne, Australia. Liu, S., Yang, L-l, Zhangm, D., Xu, J.-Y., 2018. Separation characteristics of the gas and liquid phases in a vane-type swirling flow field. Int. J. Multiph. Flow 107, 131–145. https://doi.org/10.1016/j.ijmultiphaseflow.2018.05.025. Liu, Q., Nie, W., Hua, Y., Peng, H., Liu, C., Wei, C., 2019. Research on tunnel ventilation systems: dust diffusion and pollution behaviour by air curtains based on CFD technology and field measurement. Build. Environ. 147, 444–460. https://doi.org/ 10.1016/j.buildenv.2018.08.061. Long, D.F., Perivilli, S.V., MaugerJ, V.W., 2014. Einstein’s tea leaf paradox and its relevance to dissolution testing. Dissolution Technol. 21 (3), 17–18. https://doi.org/ 10.14227/DT210314P17. Misra, A., Bonamy, C., de Souza, L.M., Hohl, L., Illner, M., Kraume, M., Repke, J.-U., Th�evenin, D., 2018. A multi-fluid approach to simulate separation of liquid-liquid systems in a gravity settler. Comput. Aided Chem. Eng. 43, 31–36. https://doi.org/ 10.1016/B978-0-444-64235-6.50008-5. Piepi� orka-Stepuk, J., 2018. Analysis of physical impurities in regenerated solutions used in cleaning brewing systems. J. Inst. Brew. https://doi.org/10.1002/jib.545 (in press). Rahman, M., Andersson, H.I., 2018. Revolving flow of a fluid-particle suspension with suction. Alexandria Eng. J. 57 (4), 2567–2572. https://doi.org/10.1016/j. aej.2017.08.017. Sterczy� nska, M., Jakubowski, M., 2017. Research on particles’ velocity distribution in a whirlpool separator using the PIV method of measurement. Int. J. Food Eng. 13, 6. https://doi.org/10.1515/ijfe-2016-0316. Sukhanovskii, A., Evgrafova, A., Popova, E., 2017. Non-axisymmetric structure of the boundary layer of intensive cyclonic vortex. Dyn. Atmos. Oceans 80, 12–28. https:// doi.org/10.1016/j.dynatmoce.2017.08.001. Tandon, A., Marshall, J., 2010. Einstein’s tea leaves and pressure systems in the atmosphere. Phys. Teach. 48 (5), 292–295. https://doi.org/10.1119/1.3393055. Wang, H., Nie, W., Cheng, W., Liu, Q., Jin, H., 2018. Effects of air volume ratio parameters on air curtain dust suppression in a rock tunnel’s fully-mechanized working face. Adv. Powder Technol. 29, 230–244. https://doi.org/10.1016/j. apt.2017.11.007, 2018. Wessel, R.A., Righi, J., 1988. Generalized correlations for inertial impaction of particles on a circular cylinder. Aerosol Sci. Technol. 9 (1), 29–60. https://doi.org/10.1080/ 02786828808959193. Yamamoto, T., Yano, M., Okano, Y., Kino-oka, M., 2018. Numerical investigation for the movement of cell colonies in bioreactors: stirring and orbital shaking tanks. J. Chem. Eng. Jpn. 51 (5), 423–430. https://doi.org/10.1252/jcej.17we112. Zhang, J., Fu, S., 2019. An efficient approach for quantifying parameter uncertainty in the SST turbulence model. Comput. Fluids 181, 173–187. https://doi.org/10.1016/j. compfluid.2019.01.017.
Anufriev, I.S., Krasinsky, D.V., Shadrin, E.Y., Sharypov, O.V., 2014. Visualization of flow structure in a vortex furnace. Tech. Phys. Lett. 40 (10), 879–882. https://doi.org/ 10.1134/S1063785014100022. Anufriev, I.S., Sharypov, O.V., Dekterev, A.A., Shadrin, E.Yu, Papulov, A., 2017. Study of flow structure in a four-vortex furnace model. Thermophys. Aeromechanics 24 (6), 849–855. https://doi.org/10.1134/S0869864317060038. Azmira, J., Houa, Q., Yuab, A., 2019. CFD-DEM simulation of drying of food grains with particle shrinkage. Powder Technol. 343 (1), 792–802. https://doi.org/10.1016/j. powtec.2018.11.097. Cai, P., Nie, W., Chen, D., Yang, S., Liu, Z., 2019. Effect of air flow rate on pollutant dispersion pattern of coal dust particles at fully mechanized mining face based on numerical simulation. Fuel 239, 623–635. https://doi.org/10.1016/j. fuel.2018.11.030. Constantin, A., Johnson, R.S., 2018. Large-scale oceanic currents as shallow-water asymptotic solutions of the Navier-Stokes equation in rotating spherical coordinates. Deep Sea Res Part 2 Top Stud Oceanogr. https://doi.org/10.1016/j. dsr2.2018.12.007 (in press). Denk, V., 1998. The whirlpool state-of-the-art 1997. Brauwelt Int. 1, 31–43. Diakun, J., Jakubowski, M., 2014. Dimensionless numbers of structural and process similitude of a whirlpool hot trub separator. J. Food Process. Eng. 36 (6), 748–752. https://doi.org/10.1111/jfpe.12043. Einstein, A., 1926. The cause of the formation of meanders in the courses of rivers and of the so-called Baer’s law. Naturwissenschaften 14 (2), 223–224 (in German). Farmanara, M., Malakooti, H., Hassanzadeh, S., 2018. Large eddy simulation of the Ekman transport in a stratified coastal sea: a case study of the Persian Gulf. Estuar. Coast Shelf Sci. 212, 372–386. https://doi.org/10.1016/j.ecss.2018.08.003. Gonz� alez-Vera, A.S., Duran-Matutem, M., van Heijsmt, G.J.F., 2018. Morphodynamics of a sediment bed in a fluid-filled cylinder during spin-down: an experimental study. Phys Rev Fluids 3, 1–22. https://doi.org/10.1103/PhysRevFluids.3.124306, 124306. Hua, Y., Nie, W., Wei, W., Liu, Q., Liu, Y., Peng, H., 2018. Research on multi-radial swirling flow for optimal control of dust dispersion and pollution at a fully mechanized tunnelling face. TunnUndergr Sp Tech 79, 293–303. https://doi.org/ 10.1016/j.tust.2018.05.018. Huang, L., Deng, S., Chen, Z., Guan, J., Chen, M., 2018. Numerical analysis of a novel gas-liquid pre-separation cyclone. Separ. Purif. Technol. 194, 470–479. https://doi. org/10.1016/j.seppur.2017.11.066. H€ ohne, T., 2019. Evaluation of safety of light-water-cooled reactors using CFD. In: Joshi, J.B., Nayak, A.K. (Eds.), Advances of Computational Fluid Dynamics in Nuclear Reactor Design and Safety Assessment. In Woodhead Publishing Series in Energy, pp. 337–386. https://doi.org/10.1016/B978-0-08-102337-2.00004-3. Iverson, R.M., Denlinger, R.P., 2001. Flow of variably fluidized granular masses across three dimensional terrain: 1. Coulomb mixture theory. Papers on Chemistry and Physics of Minerals and Rocks Volcanology 106 (B1), 537–552. https://doi.org/ 10.1029/2000JB900329. Jakubowski, M., Stachnik, M., 2019. Whirlpool tub for use in brewing industry to remove hot sediment from beer wort after cooking, has cylindrical tank equipped with bottom sealing and formed in octagonal shape, and arms connected with middle part of cylindrical tank. National patent. PL419602-A1. Jakubowski, M., Sterczy� nska, M., Matysko, R., Poreda, A., 2014. Simulation and experimental research on the flow inside a whirlpool separator. J. Food Eng. 133, 9–15. https://doi.org/10.1016/j.jfoodeng.2014.02.011. Jaskulski, M., Wawrzyniak, P., Zbici� nski, I., 2018. CFD simulations of droplet and particle agglomeration in an industrial counter-current spray dryer. Adv. Powder Technol. 29 (7), 1724–1733. https://doi.org/10.1016/j.apt.2018.04.007.
10