Forced Convection Mass-Transfer Enhancement in Mixing Systems

The design, scale-up and optimization of industrial processes conducted in agitated systems require, among other, precise knowledge of the hydrodynamics, mass and heat transfer parameters and reaction kinetics. Literature data available indicate that the mass-transfer process is generally the rate-limiting step in many industrial applications. Because of the tremendous importance of mass-transfer in engineering practice, a very large number of studies have determined mass-transfer coefficients both empirically and theoretically. Agitated vessels find their use in a considerable number of mass-transfer operations. They are usually employed to dissolve granular or powdered solids into a liquid solvent in preparation for a reaction of other subsequent operations (Basmadjian, 2004). Agitation is commonly used in leaching operations or process of precipitation, crystallization and liquid extraction. Transfer of the solute into the main body of the fluid occurs in the three ways, dependent upon the conditions. For an infinite stagnant fluid, transfer will be by the molecular diffusion augmented by the gradients of temperature and pressure. The natural convection currents are set up owing to the difference in density between the pure solvent and the solution. This difference in inducted flow helps to carry solute away from the interface. The third mode of transport is depended on the external effects. In this way, the forced convection closely resembles natural convection expect that the liquid flow is involved by using the external force. Mass-transfer process in the mixing systems is very complicated and may be described by the non-dimensional Sherwood number, as a rule is a function of the Schmidt number and the dimensionless numbers describing the influence of hydrodynamic conditions on the realized process. In chemical engineering operations the experimental investigations are usually concerned with establishing the mass-transfer coefficients that define the rate of transport to the continuous phase. One of the key aspects in the dynamic behaviour of the mass-transfer processes is the role of hydrodynamics. On a macroscopic scale, the improvement of hydrodynamic conditions can be achieved by using various techniques of mixing, vibration, rotation, pulsation and oscillation in addition to other techniques like the use of fluidization, turbulence promotes or magnetic and electric fields etc.


Introduction
The design, scale-up and optimization of industrial processes conducted in agitated systems require, among other, precise knowledge of the hydrodynamics, mass and heat transfer parameters and reaction kinetics. Literature data available indicate that the mass-transfer process is generally the rate-limiting step in many industrial applications. Because of the tremendous importance of mass-transfer in engineering practice, a very large number of studies have determined mass-transfer coefficients both empirically and theoretically. Agitated vessels find their use in a considerable number of mass-transfer operations. They are usually employed to dissolve granular or powdered solids into a liquid solvent in preparation for a reaction of other subsequent operations (Basmadjian, 2004). Agitation is commonly used in leaching operations or process of precipitation, crystallization and liquid extraction. Transfer of the solute into the main body of the fluid occurs in the three ways, dependent upon the conditions. For an infinite stagnant fluid, transfer will be by the molecular diffusion augmented by the gradients of temperature and pressure. The natural convection currents are set up owing to the difference in density between the pure solvent and the solution. This difference in inducted flow helps to carry solute away from the interface. The third mode of transport is depended on the external effects. In this way, the forced convection closely resembles natural convection expect that the liquid flow is involved by using the external force. Mass-transfer process in the mixing systems is very complicated and may be described by the non-dimensional Sherwood number, as a rule is a function of the Schmidt number and the dimensionless numbers describing the influence of hydrodynamic conditions on the realized process. In chemical engineering operations the experimental investigations are usually concerned with establishing the mass-transfer coefficients that define the rate of transport to the continuous phase. One of the key aspects in the dynamic behaviour of the mass-transfer processes is the role of hydrodynamics. On a macroscopic scale, the improvement of hydrodynamic conditions can b e a c h i e v e d b y u s i n g v a r i o u s techniques of mixing, vibration, rotation, pulsation and oscillation in addition to other techniques like the use of fluidization, turbulence promotes or magnetic and electric fields etc.
In this work, the focus is on a mass-transfer process under various types of augmentation technique, i.e.: rotational and reciprocating mixers, and rotating magnetic field. According to the information available in technical literature, the review of the empirical equations useful to generalize the experimental data for various types of mixers is presented. Moreover, the usage of static, rotating and alternating magnetic field to augment the mass process intensity instead of mechanically mixing is theoretical and practical analyzed.

Problem formulation of mass diffusion under the action of forced convection
Under forced convective conditions, the mathematical description of the solid dissolution process may be described by means of the integral equation of mass balance for the component i, namely, where: i j -volumetric mass source of component i, kg i ·m 3 ·s -1 ; n -normal component; The velocity of component i, i w , in relation to the velocity of mixture or liquid, w , is defined as follows Introducing the relation (4) in equation (2), gives the following relationship for the mass balance of i component The above equation (5) (7) Then, the differential equation of mass balance (equation 6) for the mass concentration of component i, c i , may be given by: The term A more useful form of this equation may be obtained by introducing the proper coefficients as follows where: i D -coefficient of molecular diffusion, m 2 ·s -1 ; T k -relative coefficient of thermodiffusion, kg i ·kg -1 ; p k -relative coefficient of barodiffusion, kg i ·kg -1 ; F k -relative coefficient of forced diffusion, kg i ·m 2 ·kg -1 ·N -1 . In the case of the experimental investigations of mass-transfer process from solid body to its flowing surrounding dilute solution, the boundary layer around the sample is generated. This layer is dispersed in the agitated volume by means of the physical diffusion process and the diffusion due to the forced convection. Then, the differential equation of mass balance equation for the mass concentration of component i diffuses to the surrounding liquid phase is given as follows Introducing relation (10) in the equation (11)  where: ν -kinematic viscosity, m 2 ·s -1 ; • G -mass flow rate, kg·s -1 . The above relation (12) may be treated as the differential mathematical model of the dissolution process of solid body. The right side of the above equation (12) represents the source mass flux of component i. This expression may be represented by the differential kinetic equation for the dissolution of solid body as follows where: i β -mass-transfer coefficient in a mixing process, kg·m -2 ·s -1 ; i m -mass of dissolving solid body, kg i ; s F -surface of dissoluble sample, m 2 . The above equation (13) cannot be integrated because the area of solid body, F s , is changing in time of dissolving process. It should be noted that the change in mass of solid body in a short time period of dissolving is very small and the mean area of dissolved cylinder may be used. The relation between loss of mass, mean area of mass-transfer and the mean driving force of this process for the time of dissolving duration is approximately linear and then the mass-transfer coefficient may be calculated from the simple linear equation The obtained equation (15) suggests that this dependence may be simplified in the following form: The agitated vessels find their use in a considerable number of mass-transfer operations. Practically, the intensification of the mass-transfer processes may be carried out by means of the vertical tubular cylindrical vessels equipped with the rotational (Nienow et al., 1997) or the reciprocating agitators (Masiuk, 2001). Under forced convective conditions, the force F in equation (16) may be defined (Masiuk, et al. 2008 n -rotational speed of agitator, s -1 ; V -liquid volume, kg·m -3 ; ρ -liquid density, kg·m -3 . Introducing the proposed relationships (17) and (18) in equation (16), give the following relations for the agitated system by using the rotational and reciprocating agitator, respectively The mass flow rate for the rotational and the reciprocating agitator can be approximated by the following equations:

Definition of dimensionless numbers for mass-transfer process
The governing equations (23) and (24) may be rewritten in a symbolic shape which is useful for the dimensionless analysis. The introduction of the non-dimensional quantities denoted by sign ( ) * into these relationships yield: The non-dimensional forms of these equations may be scaled against the convective term ⎛⎞ ⎜⎟ ⎝⎠ The equations (27) and (28) Table 1 summarizes all essential and independent dimensionless parameters met in mass-transfer process under the action of the agitated systems.

Mass-transfer characteristics
Under convective conditions a relationship for the mass-transfer similar to the relationships obtained for heat-transfer may be expected of the form (Incropera & DeWitt, 1996): The two principle dimensionless groups of relevance to mass-transfer are Sherwood and Schmidt numbers. The Sherwood number can be viewed as describing the ratio of convective to diffusive transport, and finds its counterpart in heat transfer in the form of the Nusselt number (Basmadjian, 2004). The Schmidt number is a ratio of physical parameters pertinent to the system. This dimensionless group corresponds to the Prandtl number used in heat-transfer. Moreover, this number provides a measure of the relative effectiveness of momentum and mass transport by diffusion. Added to these two groups is the Reynolds number, which represents the ratio of convectiveto-viscous momentum transport. This number determines the existence of laminar or turbulent conditions of fluid flow. For small values of the Reynolds number, viscous forces are sufficiently large relative to inertia forces. But, with increasing the Reynolds number, viscous effects become progressively less important relative to inertia effects. Two additional dimensionless groups, the Péclet number and the Stanton number (see Table  1) are also used and they are composed of other non-dimensional groups. Evidently, for relation (37) to be of practical use, it must be rendered quantitative. This may be done by assuming that the functional relation is in the following form (Kay & Crawford, 1980) bc Sh a Re Sc = The mass-transfer coefficients in the agitated systems can be correlated by the combination of Sherwood, Reynolds and Schmidt numbers. Using the proposed relation (38), it has been found possible to correlate a host of experimental data for a wide range of operations. The coefficients of relation (38) are determined from experiment. For mass-transfer under natural-convection conditions, where the Reynolds number is unimportant, the mass-transfer may be described by using the following general expression (Bird et al., 1966) ( ) The Grashof number, Gr , plays the same role in free convection that the Reynolds number plays in forced convection. Recall that the Reynolds number provides a measure of the ratio of the inertial to viscous forces acting on a fluid element. In contrast, the Grashof number indicates the ratio of the buoyancy force to the viscous force acting on the fluid (Fox & McDonald, 1993). Under forced convection conditions where the Grashof number is unimportant the boundary layer theory suggests the following form of the relation (38) (Garner & Suckling, 1958) 0.83 0.44 The exponent upon of the Schmidt number is to be 0.33 (Noordsij & Rotte, 1968;Jameson, 1964;Condoret et al., 1989;Lemcoff & Jameson, 1975) as there is some theoretical and experimental evidence for this value (Sugano & Rutkowsky, 1968), although reported values vary from 0.56 (Wong et al., 1978) to 1.13 (Lemlich & Levy, 1961

Mass-transfer correlations in rotationally agitated liquid-solid systems
We have compiled a list of the most frequently used correlations and tabulated them in Table 2. The following correlations have been found useful in predicting transport coefficients in the agitated systems. These equations may be successfully applied to analyze the case of the dissolution of solid bodies in a stirred tank. As it mentioned above, mass-transfer process is very complicated and may be described by the non-dimensional Sherwood number, as a rule is a function of the Schmidt number and the dimensionless numbers describing the influence of hydrodynamic conditions on the realized process. Use of the dimensionless Sherwood number as a function of the various non-dimensional parameters (see relation 37 or 39) yields a description of liquid-side mass-transfer, which is more general and useful. In majority of the works, both theoretical and practical (see Table 2), the correlations of mass-transfer process have the general form where the Sherwood number is a function of the Reynolds number, the Schmidt number, and differ by the fitting parameters a, b and c. www.intechopen.com

References Correlation
Situation Comments (Hixon & Baum, 1941)  Instead of Reynolds number, it would be more natural to express the Sherwood number in terms of the Pèclet number, which is the product of the Reynolds and Schmidt numbers (see Table 2). When this number is small, transport takes place due to mass diffusion, and the fluid velocity, density and viscosity do not affect the transport rates. In this case, the Sherwood number is a constant, dependent only on the configuration. For laminar flow past a spherical body, the limiting value of this dimensionless number is equal to 2. www.intechopen.com

Dissolutions of solid body in a tubular reactor with reciprocating plate agitator 4.1 Literature survey
Numerous articles concern with the effect of vibration on mass-transfer in a gas-liquid contradicting in a reciprocating plate column Baird et al., 1992;Gomma & Al Taweel, 2005;Gomma et al., 1991;,2000, 2003Sundaresan & Varma, 1990). In these papers is stated that the oscillatory motion of the gasliquid system in reciprocating plate columns assures much higher interfacial contacting areas than in conventional bubble column with the lower power consumption. The dissolution of solid particles into water and other solutions was investigated by Tojo et al. , where agitation realized by using circular flat disc without perforation. The mass-transfer coefficient is calculated by measuring the slope of the concentration time curve in the first second of particle dissolution. The breakage of chalk aggregates in both the vibrating and rotating mixers and the analyse of the model of breakage which relates the pseudo-equilibrium aggregate size to the energy dissipation rate in the stirred vessel has been investigated by Shamlou et al. (Shamlou et al., 1996). The breakage of aggregates in both the vibrating and the rotating mixers occurs by turbulent fluid stresses, but it is not depended on the source generating the liquid motion.

Experimental details
The intensification of the dissolution under the action of reciprocating agitators were carried out by means of the reactor presented by (Masiuk, 2001). It is consisted of the vertical vessel, the system measuring mass of dissolving solid body (rock-salt), the device for measuring concentration in the bulk of mixed liquid (distilled water) and the arrangement for reciprocating plate agitator at various amplitudes and frequencies. The agitation was carried out with a single reciprocating plate with flapping blades oriented horizontally where it reciprocated in a vertical direction  and different number of the multihole perforated plates agitators. The detail schemes of the vibratory mixers are given in the relevant references (Masiuk, 1996;Masiuk b , 1999;Masiuk, 2000). Additionally, the main geometrical dimensions of mixers and the operating ranges of the process parameters are collected in the Table 3. The reciprocating agitator was driven by the electric a.c. motor coupled through a variable gear and a V-belt transmission turned a flywheel. A vertical oscillating shaft with the perforated plates and a hardened steel ring through a sufficiently long crank shaft were articulated eccentrically to the flywheel. An inductive transducer mounted inside the ring and a tape recorder was used to measure the inductive voltage directly proportional to the total force straining of the shaft. The average mass-transfer coefficient was calculated from a mass balance between a dissolving solid cylindrical sample and no flowing surrounding dilute solution (see equation 14). Two conductive probes connected to a multifunction computer meter were used to measuring and recording of the concentration of the achieve solution of the salt. The mass of the rock salt sample decreasing during the process of dissolution is determined by an electronic balance that connected with rocking double-arm lever. On the lever arm the sample was hanging, the other arm connected to the balance. In the present investigation the change in mass of solid body in a short time period of dissolution is very small and the mean area of dissolved cylinder of the rock salt may be used. Than the mean mass-transfer 0.14·10 -3 -6.83·10 -3 Mean driving force, kg NaCl ·(kg solvent ) -1 0.002-0.0137 Table 3. The main geometrical dimensions of mixer and operating conditions coefficient may be calculated from the linear kinetics equation using the mean concentration driving force of the process determined from two time response curves. Raw rock-salt (>98% NaCl and rest traces quantitative of chloride of K, Ca, Mg and insoluble mineral impurities) cylinders were not fit directly for the experiments because their structure was not homogeneous (certain porosity). Basic requirement concerning the experiments was creating possibly homogeneous transport conditions of mass on whole interfacial surface, which was the active surface of the solid body. These requirements were met thanks to proper preparing of the sample, mounting it in the mixer and matching proper time of dissolving. As an evident effect were fast showing big pinholes on the surface of the dissolved sample as results of local non-homogeneous of material. Departure from the shape of a simple geometrical body made it impossible to take measurements of its area with sufficient precision. So it was necessary to put those samples through the process of socalled hardening. The turned cylinders had been soaked in saturated brine solution for about 15 min and than dried in a room temperature. This process was repeated four times.
To help mount the sample in the mixer, a thin copper thread was glued into the sample's axis. The processing was finished with additional smoothing of the surface with finewww.intechopen.com grained abrasive paper. A sample prepared in this way had been keeping its shape during dissolving for about 30 min. The duration of a run was usually 30 sec. The rate of mass-transfer involved did not produce significant dimensional change in diameter of the cylinder. The time of a single dissolving cycle was chosen so that the measurement of mass loss could be made with sufficient accuracy and the decrease of dimensions would be relatively small (maximum about 0.5 mm). Before starting every experiment, a sample which height, diameter and mass had been known was mounted in a mixer under the free surface of the mixed liquid. The reciprocating plate agitator was started, the recording of concentration changes in time, the weight showing changes in sample's mass during the process of solution, and time measuring was started simultaneously. After finishing the cycle of dissolving, the agitator was stopped, and then the loss of mass had been read on electronic scale and concentration of NaCl (electrical conductivity) in the mixer as well. This connection is given by a calibration curve, showing the dependence of the relative mass concentration of NaCl on the electrical conductivity (Rakoczy & Masiuk, 2010).

Results and discussion
In order to establish the effect of the Reynolds number on mass-transfer coefficient data obtained from the experimental investigations are graphically illustrated in a ( ) −0.33 log Sh Sc versus ( ) log Re systems for the in figure 1. These experimental results were obtained for the single reciprocating plate with flapping blades. Figure 1 demonstrates that, within the limits of scatter among the plotted data represented by the points, the non-dimensional Sherwood number increase in the Reynolds number. The . It is believed that the proposed dimensionless equation (44) is useful to generalize the experimental data in this work for the whole region of the reciprocating Reynolds number without the break of the correlating graphical line. Moreover, figure 1 presents a graphical form of equation (43), as the full curve, correlated the data very well with standard deviation σ = 0.66 . The difference between the predicted and calculated values of the non-dimensional Sherwood number is less than ±15% for approximately 80% of the data points. In present experimental investigations of the influence of the perforated plate reciprocating agitators on the mass-transfer process was additionally investigated for different number of plates varying from 1 to 5. The results of the mass-transfer experiments for different number of plates should be correlated using the relationship similar to the expression obtained for the reciprocating agitator with single perforated plate with flapping blades (equation 43). In the present report the mass-transfer process is described by the similar somewhat modified relationship (37) In the case of this work, the experimental data have been correlated using the new mean modified dimensionless Sherwood number, * Sh , as a ratio of the dimensionless Sherwood, Sh , number and the dimensionless maximum density of mixing energy, ρ * max E . In the selection of the suitable agitator for the transfer process, it is not sufficient to take into consideration the power consumption and mixing time separately. The density of the maximum mixing energy, ρ max E , is defined as the product of the maximum power www.intechopen.com consumption, max P , and the mixing time, mix t , relate to the volume of mixed liquid, L V (Masiuk & Rakoczy, 2007) The values of the density of the maximum mixing energy (47) may be determined by using the relationships describing the maximum power consumption The graphical illustration of the equation (54) is given in the coordinates ( ) * − 0.33 , rec Sh Sc Re log-log system in figure 2.

Fig. 2. Modified mass-transfer characteristics for reciprocating agitator with different number of plates
The rate of mass-transfer increases with increasing number of plates. The modified dimensionless Sherwood number, * Sh , increases with increasing number of plates. Within the laminar region of the flow, the mass-transfer characteristics obtain the extreme (see figure 3). In the region of large Reynolds number, the curves of the modified dimensionless Sherwood number versus the reciprocating Reynolds number decreases sharply. Additionally, the change of plates of the reciprocating agitator has considerable profitable influence on the mass-transfer process.

Enhancement of solid dissolution process under the influence of transverse rotating magnetic field (TRMF)
The transverse rotating magnetic field (TRMF) is a versatile option for enhancing several physical and chemical processes. Studies over the recent decades were focused on application of magnetic field (MF) in different areas of engineering processes (Rakoczy & www.intechopen.com Masiuk, 2009;Rakoczy, 2006). Static, rotating or alternating MFs might be used to augment the process intensity instead of mechanically mixing. Practical applications of RMF are presented in (Volz & Mazuruk, 1999;Melle et al., 1999;Walker et al. 2004;Nikrityuk et al. 2006;Yang et al., 2007;Fraňa et al. 2006;Spitzer, 1999). Recently, TRMF are widely used to control different processes in the various engineering operations. This kind of magnetic field induces a time-averaged azimuthal force, which drives the flow of the electrical conducting fluid in circumferential direction. The magnetic filed lines rotate in the horizontal direction with the rotation frequency of the field, ω TRMF . It is obvious that this parameter is equal to the frequency of alternating current. An electrical field, E , is generated perpendicular to the magnetic field, B . Perpendicular to the electric field, the Lorenz magnetic force, m F , is acting as the driving force for the liquid rotation. The TRMF may be generated by cylindrical inductor resembling the stator of a three-phase asynchronous electrical engine. The similar inductor of TRMF was applied in (Lu & Li, 2000;Nekrassov & Chekin, 1961;Nekrassov & Chekin, 1962). In the case of the electrical conducting fluid, the TRMF induces currents inside this liquid. These interact with the field of the inductor and generate electromagnetic force inside the electrical conducting liquid. The pattern of hydrodynamic flows due to TRMF in a cylindrical volume depends on the number of pole pair, TRMF p , of TRMF inductor. The distribution of TRMF may be considered as the quasi-uniform or non-uniform for the number of pole pair equal to 1 and > 1 TRMF p , respectively. The application of TRMF may be generated the specific flow in a liquid cylindrical volume. The liquid flow is an azimuthal motion of medium around the central axis of the cylindrical volume. This flow is directed as the magnetic field rotates and the azimuthal velocity is maximal at the cylindrical vessel's walls. Obviously, this movement may determine the heat or mass-transfer inside the rotating liquid. According to available in technical literature, research has not been focused on the mass-transfer during dissolution of a solid body to the surrounding liquid under the action of RMF. In the present report, the experimental investigations has been conducted to explain the influence of this kind of MF on the mass-transfer enhancement. Moreover, the influence of localisation of a NaCl-cylindrical sample localisation on the mass-transfer rate was experimentally determined.

Influence of transverse rotating magnetic field (TRMF) on mass-transfer process
The influence of TRMF on mass-transfer process is manifested by the magnetic force defined as the divergence of Maxwell's stress tensor and expressed by the following relation (Rakoczy & Masiuk, 2010) where B -magnetic induction vector, kg·A -1 ·s -1 ; H -magnetic field strength vector, A·m -1 ; V -liquid volume exposed to TRMF, m 3 ; μ m -magnetic permeability, kg·m·A -2 ·s -2 .
In the above equation (55) The mass flow rate for this case may be approximated by using the following relation where τ TRMF -time of magnetic field diffusion, s.
This parameter may be defined basing on the well-known advection-diffusion type equation (Möβner & Gerbeth, 1999;Guru & Hiziroğlu, 2004) ( ) where w -velocity, m·s -1 ; σ e -electrical conductivity, A 2 ·s 3 ·kg -1 ·m -3 ; ( ) νσ μ − = 1 me m -effective diffusion coefficient (magnetic viscosity), m 2 ·s -1 The above equation (58)  describes the relative importance of advection and diffusion of the MF. Taking into account the above definitions of the non-dimensional groups, we obtain the following general relationship of the magnetic induction equation It should be noticed that the time of magnetic field diffusion, τ TRMF , may be defined as Taking into consideration the above relations (Eqs 56 and 65), we obtain the following relationships: The governing equation (66) The non-dimensional forms of the above equation (67) where ω TRMF -angular velocity of magnetic field, rad·s -1 .
The interaction between of the TRMF and the used liquid may be also described by means of the non-dimensional Hartmann number:

Experimental details
All experimental measurements of mass-transfer process using the TRMF were carried out in a laboratory set-up including electromagnetic field generator. A schematic of the experimental apparatus is presented in figure 3. This setup may be divided into: a generator of the rotating electromagnetic field (1), a glass container (2) with the conductivity probes (3-4), an electric control box (5) and an inverter (6) connected with multifunctional electronic switch (8) and a personal computer (7) loaded with special software. This software made possible the electromagnetic field rotation control, recording working parameters of the generator and various state parameters. From preliminary tests of the experimental apparatus, the glass container is not influenced by the working parameters of the stator. The TRMF was generated by a modified 3-phase stator of an induction squirrel-cage motor, parameters of which are in accordance with the Polish Standard PN-72/E-06000. The stator is supplied with a 50 Hz three-phase alternating current. The transistorized inverter (4)  For the experimental measurements, MF is generated by coils located axially around of the cylindrical container. As mentioned above, this field is rotated around the container with the constant angular frequency, ω TRMF . The TRMF strength is determined by measuring a magnetic induction. The values of the magnetic induction at different points inside the glass container are detected by using a Hall probe connected to the personal computer.
In the present work, the local mass-transfer rates were correlated with the magnetic field measurements. The aim was to investigate a dissolution process of the NaCl-cylinder probe The TRMF is produced by coils located around the cylinder, and the axes are directed along the radius. When the alternating current is supplied to the windings, the generated MF rotates around the cylinder column. An a.c. transistorized inverter controlled the rotating frequency (equal to the alternating frequency current in the range of A typical example of dependence relationship between the spatial distribution of the magnetic induction and various current values of level is presented in figure 5. The TRMF with the magnetic induction, B , is controlled by using an a.c. alternating current frequency equal to the frequency of TRMF. As follows form the experimental measurements, the values of a magnetic induction are spatially distributed and time independent. It is seen that in the area occupied with this container, the TRMF distribution depends strongly on the spatial coordinates (see Fig.5). A local mass-transfer rate was obtained in three radial locations (I for x = 0.08 m; II for x = 0.11 m; III for x = 0.14 m) form the wall of the cylindrical container. It is seen that the magnetic induction at locations III and II is greater than an location I (see Fig.5). Typical profiles of the averaged magnetic induction at different locations of a NaCl-cylinder sample are graphically shown in figure 6. In order to establish the effect of a current level on the averaged values of a local magnetic induction, the obtained data shown in this figure may be analytically described by: The enhancement of the dissolutions process under the action of TRMF was carried out by using the experimental set-up shown in Fig.3 and described in section 4.2.

Result and discussion
Under the TRMF conditions, a relationship for the mass-transfer can be described in the general form: In the presented report, the mass-transfer process under the action of TRMF is described by using the modified magnetic Taylor number, which defines the magnitude of the magnetic effect in the tested experimental set-up. . Figure 7 gives an overview results, in the form of the proposed analytical description (see Eqs 84a-84c) for the experimental investigations. The first conclusion drawn from the inspection of this graph is that the proposed relationships fit the analysed experimental data very well. Figure 7 demonstrates that, within the scatter limits among the plotted data represented by points, the dimensionless Sherwood number increases with increasing the magnetic local Taylor number. It was found that as the intensity of the MF increases, the relative velocity of liquid inside the cylindrical container increases. This liquid rotation may be defined by The graphical presentation of Eq.(87) is given in figure 8. The proposed form of Eq.(87), which is presented in figure 8

Conclusion
The present considerations show interesting features concerning the effects of forced convection on the mass-transfer in the mixing systems. Based on the experimental results, the effects of the reciprocating agitators for the wide range of Reynolds number are correlated by the suitable equations. With the respect to the other very useful mass-transfer relationships given in the pertinent literature, the theoretical description of problem and the equations predicted in the present work is much more attractive because it generalizes the experimental data taking into consideration the various parameters, which defined the hydrodynamic state and the intensity of forced convection effects in the tested mixing systems. Moreover, the list of empirical equations for the prediction of mass-transfer coefficients for the rotational agitated systems has been presented. The tabulated correlations in Table 2 may be successfully applied to study the dissolution of solid bodies in the rotational stirred tank. It should be noticed that the novel approach to the mixing process presented and based on the application of transverse rotating magnetic field (TRMF) to produce better hydrodynamic conditions in the case of the mass-transfer process. From practical point of view, the dissolution process of solid body is involved by using the turbulently agitated systems. In previous publications are not available data describing the mass-transfer operations of the dissolution process under the TRMF conditions. Therefore, the experimental investigations have been conducted to explain the influence of this kind of magnetic field (MF) on the mass-transfer enhancement. Moreover, the influence of localization of NaCl-cylindrical sample on the mass-transfer was determined. The influence of the TRMF on this process is described using the non-dimensional parameters formulated on the base of fluid mechanics equations. These dimensionless numbers allow quantitative representation and characterization of the influence of hydrodynamic state under the TRMF conditions on the mass-transfer process. The dimensionless groups are used to establish the effect of TRMF on this operation in the form of the novel type dimensionless correlation.
The study of mass-transfer process under the action of TRMF results in significant enhancement of the solid dissolution rate. The mass-transfer measurements obtained in the presence of this kind of MF indicates a strong dependency on dissolution rate per localization of NaCl-cylindrical sample. In the present report, the mass-transfer under the action of TRMF is described by novel type of dimensionless group (the local magnetic Taylor number).