Mass Transfer in a Centrifugal Turbine Agitator-Pump

This article is a continuation of the research, centering on a vacuum-filtration system, which is designed to reduce the concentration of calcium in water; a process is also known as—water softening. The problem of solving the concentration distribution of the initial (embryonic) particles of CaCO3-particles, which were introduced into the limited volume of the apparatus with a turbine agitator-pump, is addressed through the use of diffusion and deterministic-stochastic models of mass transfer. The solution of the extreme problem allows determining the most important process parameters, such as time of dispersions homogenization and the dispersion mass flow rate to the surface of a special filter. For these parameters a comparative analysis of the adequacy of the theory was found through experiments, performed in the study. We found that uniform distribution of concentrations along the height of the apparatus is achieved by the angular velocity of the rotation 400 rpm for the turbine with 6 7 blades at the time of homogenization 14s. In this case, the dispersion mass flow to the surface of the cylindrical filter is ≥ 50 mg/s at an average concentration of the introduced CaCO3 particles, which is equal to 10 g/L. We determined that the accuracy of the results depends on: the coordinates of the material input in the apparatus volume, the surface shape of the filter and the volumetric flow rate of the liquid (water), being discarded by the turbine blades in the normal direction to their surface.


Introduction
A vacuum-filtration system, designed to reduce the concentration of calcium in the water during the water sof-tening process was developed at the Applied Research Institute of Ben-Gurion University of the Negev (Israel) (Figure 1) [1].The most important part of this system is centrifugal turbine agitator-pump, sucking and discarding flows of liquid (water) with suspended particles of CaCO 3 (dispersed flow, dispersion, suspension) to the surface of a special filter.Hydrodynamics and mass transfer in such a turbine agitator-pump were studied and presented in [2] [3].We calculated the main hydrodynamic parameters, which characterize the dynamic interaction of turbine blades with the flows of a viscous liquid.Based on the empirical Equation: 1 6 1 3

Sh 2 Re Pe
T α = + ⋅ , the coefficients of mass transfer k and diffusion (dispersion) coefficients D * of the CaCO 3 substance by mass transfer from liquid (water) to the surface of the initial (embryonic) spherical particles CaCO 3 , introduced in the apparatus with turbine agitator-pump, were calculated.The purpose of this paper is: • To present a solution to problem of distribution of CaCO 3 particles in a limited volume of the apparatus with a turbine agitator-pump; • To determine the most important technological process parameters, such as time of dispersion homogeni- zation h t and the dispersion mass flow rate to the filter surface f q ; • To use experimentation to conduct an analysis of the adequacy of the theory.

Diffusion Model
To determine the function of the concentration distribution W for dispersed particles of CaCO 3 in a limited volume of the apparatus with a turbine agitator-pump, in the absence of mass exchange, we use the following three-dimensional Equation: ( ) with initial and boundary conditions in the following form: represents the relative velocity of the particle; i D , which is the diffusion (dispersion) coefficients; i x , which are cylindrical coordinates ( 1, 2,3 i = and match , , r z ϕ ).The z axis is directed ver- tically upward from the center of the apparatus bottom.M is the mass of all CaCO 3 particles, introduced into the apparatus volume L V at point ( ) in order to be distributed.k -represents specific dispersion fluxes to the bounding surface ( ) kg m s i s ⋅ ; δ -being the Dirac delta function [4].The function ( ) The conclusion and solution of the equation ( 1) in the case of wandering particles with constant speeds i V * and coefficient i D * in the infinite volume are given, for example, by Chandrasekhar [5].A similar approach to the description of the diffusion in the liquid suspended particles was used by Landau and Lifshitz [6].Thus, the solution of the Equation (1) with the above mentioned initial conditions is represented in form [7]: ( ) The dimension of the distribution function of the concentration W is ( ) Here n is the number of displacements of particles per unit of time; 2 , i i x x -respectively, the average displacements and mean square displacements of the diffusing particles.It is known, that they represent the first and second initial moments of the regulatory function k r for displacement of individual particles.

Deterministic-Stochastic Model
The problem of determining such a function is reduced to the solution of the stochastic differential Langevin Equation [8] for the motion of dispersed particles in an external force field: ( ) This is derived from the most general Meshcherskij Equation [9].Here j F are active forces, acting on the particles; , Φ Φ , which are the inertial forces in the portable motion of particles with the liquid and Coriolis force of inertia, respectively; ( ) A t -is the perturbing acceleration, which is characteristic of the random effects on the particle by the surrounding liquid.The function ( ) A t has the following statistical properties: it does not depend on the relative velocity i V and it changes quite rapidly, when compared with the i V .η -change correction vector, which theoretically depends on the geometry of the apparatus with the turbine, the kinematical and the physical parameters of the process.( ) 4) models the Brownian motion of dispersed particles in the presence of external force fields as a Markov process in phase velocity space i V .We write further the Equation ( 4) in the projection on the axis of the moving coordinate system , , r z ϕ showing it to be rigidly attached to a rotating turbine and neglecting the Coriolis force, as shown below: the relative velocity of the particle, projected on the axis , , .
The first two Equations in ( 5) are identical in appearance.From these equations we must find the probability distributions ( ) which respectively mean the probability of finding the relative displacement i x and the relative velocity ( ) of the particle at a time t , if initially at 0 t = the particle was in position 0 i x and had an initial speed ( ) . To get this distribution we must first find a formal solution of the Equation ( 5).This can be done, for example, by the method of variation of parameters [5], considering Equation ( 5) as an ordinary differential equation.We can then bring the solution to the following form: , where ( ) A η has the same properties as ( ) -is the function, that is specified by a decision (5).Then, according to (5), the probability distribution of the vector R is given by: ( ) where , q kT m β = ( ) -is the Boltzmann constant [8].Using the above findings and omitting the intermediate calculations, we write the expression for the displacement of the individual particles i x and their relative velocities and corresponds to the direction of the axes , .r ϕ ( ) ( ) 1, 2,3 i i η = -are correction vector projections on axes of cylindrical coordinate system , , .
r z ϕ

Calculation of Diffusion Coefficients. Relative Velocities of Particles
We first determine the number of positive displacements of particles per unit of time i n along the axes , , r z ϕ as follows: . Using Equations ( 3) and ( 7), for the diffusion (dispersion) coefficient i D we have: -is the coefficient of Brownian diffusion, having, for example, the value: Due to the smallness of the D′ , second summands in ( 8) can be neglected.Then the formulas i D will take the form of: The combination of Equation (9) with Equation (7) shows, that, if the initial velocities i D = This is consistent with the physical meaning of the process.Formula (9) are valid for the values of the coefficients 0, Therefore for a water softening process the value of the particle diameter p d and the correction factor k η can be estimated.For the data from [3] we have: V u − from Equation ( 7) and for i D from Equation (9) should be averaged over time.We take as an averaging time interval the time of dispersion homogenization h t , that is, the time to reach an average concen- tration av c of the diffusing component throughout the apparatus volume, while: where с η -is the coefficient of heterogeneity of the concentration of the component in the volume of the appa- ratus.We have: Replace the exponential terms in Equations ( 7) and ( 9) at their approximate values [9]: ( ) A further task is to determine the time of dispersion homogenization h t .

Time Homogenization of Dispersion
Consider the following extreme problem: . Substituting the function W from Equation (2), the dispersion homogenization time h t we obtain the fol- lowing expression: ( ) ( ) The diffusion coefficient i D * and the relative velocities i V * in Equation ( 13) also contain unknown parameter h t .Therefore, the calculation of h t is required in order to carry out an iterative method.As a first ap- proximation we can use parameter values from [3] and the dimensionless Equation for the time of homogenization h t′ , obtained by study of the suspensions homogenization process in apparatuses with stirrers [7], as fol- lows: t tends to its average value av c throughout the apparatus volume and does not depend on the coordinates shown in (Figure 2).The exponential term in Equation ( 2) can be neglected and for controlled precision of calculation W we can use the following simple formula:     14) and ( 15) and the analytical formulas shown in Equations (12) and (13).

Time of homogenization
As seen from Table 1, using the formulas in Equations ( 13) and ( 14) for calculating the homogenization time h t , gives almost perfectly matched results for the above presented values of the geometrical and phy-sical para- meters.An important role is played by the coordinates ( ) of the substance input point in the volume of the apparatus.Calculation results for average concentration av n  , according to formula in Equation ( 15), are consistent with the experimental data, presented in Figure 2, for turbine agitator-pumps numbered 1 ( ) correspond to a turbine agitator-pump number 3 with the number of blades shown as ( )

Mass Flows of Dispersion
We describe in Equation ( 16) the dynamic boundary conditions for the dispersed flow of particles, flowing through the turbine agitator-pump, with the achievement of the homogenization time h t : Here ( ) -is the lateral surface of the turbine; -is the cross-section area of the turbine suction tube; 3 1, -is the volumetric flow of liquid (water) discharged by the turbine blades in the normal direction to their surface, ( ) , r q q ϕ of the turbine agitator-pump.At the same time should be valid the law of mass conservation, then: 1 z r r q q q ϕ ϕ ξ ξ = + → + = .Using the results of the experiments, presented in [2] [3], we can determine the mass flow rate of the dispersion to the filter surface f q with use of the formula, shown in the Equation (17): , where Wf Q -is the volumetric flow rate of water, cleaned from CaCO 3 -particles, passing through the filter surface and f ξ represents the proportionality factor for the filter;   From Table 2 and Equation (17) we see, that in order to increase the mass flow rate of dispersion to the surface of the filter f q , we must first increase its surface f s and increase the normal flow rate of liquid n Q .The boundary conditions, expressed through Equation ( 16), can be used for analytical determination of the correction , , r z ϕ η η η .For that to take effect, we need to substitute into Equation ( 16) the distribution function of the con- centration W from Equation (2) and the relative velocity i V * and the diffusion coefficient i D * from Equation (12).However, the expressions for , , r z ϕ η η η are too cumbersome.Therefore, the calculation of W should be guided by the values of the coefficients in Table 1.

Conclusions
The result of this research shows that the diffusion and deterministic-stochastic models of the substance transfer (particles of CaCO 3 ) in a limited volume of apparatus with the turbine agitator-pump can be used as an attachment to the issue of the removal of calcium from water, a process often called water softening.The main results of the use of the developed mathematical and physical models are: • Optimal distribution of CaCO 3 particles concentrations within the volume of the apparatus is achieved by the angular velocity of rotation 400 rpm ω = for the turbine with 6 -7 blades and homogenization time of 14 s.• The accuracy of the calculation process parameters for homogenization time h t and mass flow rate f q affects the coordinates of the input point ( ) x x − , the shape and surface of the filter f s and the volumetric flow of liquid (water) , n Q as discarded by turbine blades in the normal direction to their surface.Some hydro- dynamic parameters involved in the calculation, such as the speed of the carrier medium (water) L u , volumetric flow rates , in n Q Q were calculated on the basis of the vortex hydrodynamic model, described in [2].Mass transfer parameters, described as the average diffusion (dispersion) coefficients D * and the average diameters of the diffusing particles av d , used in the calculations are taken from [3], preceding to this study.
• Some formulas, such as Equations ( 13)-(15), can be used in calculations of homogenization time for sus- pensions.They can also be used to determine average concentrations of diffusing components in various types of mixing devices and dispersion homogenizers.ϕ -coordinates, m .Greek symbols: ε -the concentration of the initial (embryonic) particles of CaCO 3 , introduced into the vo- lume of the apparatus, kg kg ; µ -the dynamic viscosity coefficient, kg m s ⋅ ; ν -the coefficient of kine- matical viscosity, 2 m s ; ρ -density, 3 kg m ; ω -angular velocity of the turbine, rpm .
Indices: a -apparatus; av -the average value; in -the entrance to a suction tube of the turbine agitator- pump; b -lateral surface; c -Coriolis force of inertia; e -portable motion together with the turbine; ffilter; h -homogenization; L -liquid (water); n -normal direction; p -particle of CaCO 3 R -resistance; s -surface; T -turbine; W -water; , , r z ϕ -radial, tangential and axial directions; 0-the initial value; * - the averaged value.

r
is the probability of finding the particles group, moved and do not interacting with each other at time t at point , represents the movements of individual particles.W serves as a diffusing component (particles) concentration ( ) 3 kg m at point R in the time t .
force, involved in the relative motion of the particles.-being the resistance coefficient, which depends on the con- centration of the dispersed phase (particles of CaCO 3 ) and the mass transfer from the liquid to the surface of the particles (for the Stokes resistance force 3π τ -represents the centripetal acceleration and rotation acceleration of the carrier medium respec- distribution function W of Equation (2) consists of the relative velocity with the diffusion coefficients having constant value, the expression obtained for ( ) pi Li the average value of the diffusion (dispersion) coefficient; Due to component CaCO 3 concentration W at the time of homogenization h the empirical data of[2] [3], Equations ( the flow rate at the inlet to the turbine suction pipe; and i ξ -represents proportionality factors.Expressions in the Equa- tion (16) are dispersion mass flows in ( ) kg s in the inlet ( ) surface of the filter; b T T s b H n = ⋅ -represents the surface of n turbine blades and T b -represents the distance between the centers of macro vortices, formed in the rear region after screws [3].
At that dispersion, mass flows to the surface of the cylindrical filter are 50

Table 1 ,
we present the values of homogenization time , = which Figure 2. Distribution of

Table 1 .
Time of homogenization and average concentration.

Table 2 ,
below, is an example of calculating the mass flows of dispersion , , , . ϕ

Table 2 .
Mass flows of dispersion: example of calculation. in