Mass-transfer in the Dusty Plasma as a Strongly Coupled Dissipative System: Simulations and Experiments 109 In all the cases there was obtained a good agreement between the direct measurements of the velocity autocorrelation and mass-transfer functions and those functions, calculated from the mean-square displacement of particles using the Eqs. (15a)-(15b). That means that the stochastic model, given by the system of Langevin equations, can be used for the correct description of the motion of dust under experimental conditions The method of simultaneous determination of dusty plasma parameters, such as kinetic temperature of grains, their friction coefficient, and characteristic oscillation frequency is proposed. The parameters of dust were obtained by the best fitting the measured velocity autocorrelation and mass-transfer functions, and the corresponding analytical solutions for the harmonic oscillator. The coupling parameter of the systems under study and the minimal values of grain charges are estimated. The obtained parameters of the dusty sub- system (diffusion coefficients, pair correlation functions, charges and friction coefficients of the grains) are compared with the existing theoretical and numerical data. 9. Acknowledgement This work was partially supported by the Russian Foundation for Fundamental Research (project no. 07-08-00290), by CRDF (RUP2-2891-MO-07), byNWO (project 047.017.039), by the Program of the Presidium of RAS, by the Russian Science Support Foundation, and by the Federal Agency for Science and Innovation (grant no. MK-4112.2009.8). 10. References Allen, J.E. (1992). Probe theory - the orbital motion approach. Physica Scripta, 45, 497-503, ISSN 0031-8949 Balescu, R. (1975). Equilibrium and Nonequilibrium Statistical Mechanics, Wiley Interscience, ISBN 0-471-04600-0, Chichester Photon Correlation and Light Beating Spectroscopy. (1974). 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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 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. Advanced Topics in Mass Transfer 114 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. 2. 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, () i ii i VS V dV w n dS j dV ρ ρ τ ∂ =− ⋅ ⋅ + ∂ ∫ ∫∫ (1) where: i j - volumetric mass source of component i, kg i ·m 3 ·s -1 ; n - normal component; i w - vector velocity of component i, m·s -1 ; S - area, m 2 ; V - volume, m 3 ; i ρ - concentration of component i, kg i ·m -3 ; τ - time, s. The above equation (1) for homogenous mixture may be written in the following differential form () i ii i div w j ρ ρ τ ∂ + = ∂ (2) The vector velocity of component i, i w , can be defined by using the averaged value of momentum, ([ ] ) iav g q → ⎡⎤ ⎣⎦ = 0 lim i av g i V i q w m (3) The velocity of component i, i w , in relation to the velocity of mixture or liquid, w , is defined as follows ρ ρ =−⇒ =− ∑ 1 ii ii ii i dif w w w dif w w w (4) Introducing the relation (4) in equation (2), gives the following relationship for the mass balance of i component () i iiii div dif w w j ρ ρ τ ∂ ⎡⎤ + += ⎣⎦ ∂ (5) The above equation (5) can be rewritten as follows () () ρ ρ τ ∂ + =− + ∂ i iii div w div j j (6) Forced Convection Mass-Transfer Enhancement in Mixing Systems 115 The mass concentration of component i, c i , may be defined in the following form ρ ρ ρ ρ =⇒= i iii cc (7) Then, the differential equation of mass balance (equation 6) for the mass concentration of component i, c i , may be given by: () 0 i ii dc div j j d ρ τ + −= (8) The term i dc d ρ τ ⎛⎞ ⎜⎟ ⎝⎠ in the above equation (8) expresses the local accumulation of relative mass and the convectional mass flow rate of component i, whereas the term i j is total diffusion flux density of component i. The term (j i ) describes the intensity of the process generation of the volumetric mass flux of component i in the volume V due to the dissolution process. The resulting diffusion flux is expressed as a sum of elementary fluxes considering the concentration (c), temperature (T), thermodynamic pressure gradient (p), and the additional force interactions ()F (i.e. forced convection as a result of fluid mixing) in the following form () () () ( ) =+ ++ ii i i i jj c j T jp j F (9) A more useful form of this equation may be obtained by introducing the proper coefficients as follows ( ) ( ) ( ) ρρ ρ ρ =− − − −ln ln iiiiT ip i F j D grad c D k grad T D k grad p D k F (10) 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 τ ρρ ⎛⎞ ∂ + += ⎜⎟ ⎜⎟ ∂ ⎝⎠ ii i i jj c wgradc div (11) Introducing relation (10) in the equation (11) gives the following relationship for the balance of the mass concentration of component i where force F is generated by the mixing process ν τ ρ • = = = = ⎡⎤ ⎛⎞ ⎛⎞ ∂∂∂ ⎛⎞ ⎢⎥ ⎜⎟ ++−− − + = ⎜⎟ ⎜⎟ ⎢⎥ ⎜⎟ ∂∂∂ ⎝⎠ ⎝⎠ ⎝⎠ ⎢⎥ ⎣⎦ F F i iii iii i pconst Tconst pconst pconst j cccF w gradc div D gradc a gradT grad p div c Tp G (12) Advanced Topics in Mass Transfer 116 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 ( ) () ( ) ( ) ( ) () ( ) () () () () () () () () iiisi is i s ii i iiir i i ir dm dc d dF dc dF d d ddddVd ddcd dgradce ddrd d d gradc e d τ τρτττ β ττ β τ τττττ ρτ τ ρτ βτ β τ τττ ρτ β ττ −= ⇒−= ⇒ ⇒− = ⇒ = ⇒ ⇒= (13) 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 [] [] i i avg si avg m Fc β τ Δ = Δ Δ (14) Taking into account the above relations (equations 13 and 14) we obtain the following general relationship for the mass balance of component i [] () F F iii iii pconst Tconst pconst pconst iir avg i ccc w gradc div D gradc a gradT grad p Tp gradc e F div c G ν τ β ρ = = = = • ⎡ ⎤ ⎛⎞ ∂∂∂ ⎛⎞ ⎢ ⎥ + +− − − + ⎜⎟ ⎜⎟ ⎢ ⎥ ∂∂∂ ⎝⎠ ⎝⎠ ⎢ ⎥ ⎣ ⎦ ⎛⎞ ⎜⎟ += ⎜⎟ ⎝⎠ (15) The obtained equation (15) suggests that this dependence may be simplified in the following form: () [] () iir avg i iii i gradc e cF w gradc div D gradc div c G β τρ • ⎛⎞ ∂ ⎜⎟ +− + = ⎜⎟ ∂ ⎝⎠ (16) 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): Forced Convection Mass-Transfer Enhancement in Mixing Systems 117 - for rotational agitator ϕ ρρπ =⇒= 22 2 rot rot rot rot dw FV FVnde dt (17) - for reciprocating agitator ρρπ =⇒= 22 4 rec rec rec z dw FV FVAfe dt (18) where: A - amplitude of reciprocating agitator, m; rec d - diameter of reciprocating agitator, m; rot d - diameter of rotational agitator, m; f - frequency of reciprocating agitator, s -1 ; 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 () [] () 2 iir avg irot iii i rot gradc e cVnd w gradc div D gradc div c e G ϕ β ρ τρ • ⎛⎞ ∂ ⎜⎟ +− + = ⎜⎟ ∂ ⎝⎠ (19) () [] () 2 iir avg i iii iz rec gradc e VAf c w gradc div D gradc div c e G β ρ τρ • ⎛⎞ ∂ ⎜⎟ +− + = ⎜⎟ ∂ ⎝⎠ (20) The mass flow rate for the rotational and the reciprocating agitator can be approximated by the following equations: ρ • = rot GVn (21) ρ • = rec GVf (22) Taking into consideration the above relations (equations 21 and 22), we obtain the following relationships: () () [ ] ( ) iir avg i iii roti gradc e c w gradc div D gradc div nd c e ϕ β τρ ∂ +− + = ∂ (23) () () [] () iir avg i iii iz gradc e c w gradc div D gradc div Afc e β τρ ∂ +− + = ∂ (24) 2.1 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: () () [] [] () 00 00 0 0 00 0 2 00 0 0 000 ii ii i iii iii ir avg avg rot i rot i cwc Dc c w gradc div D grad c ll cgradce nd c div n d c e ll ϕ ττ ββ ρρ ∗ ∗ ∗∗ ∗∗ ∗ ∗ ∗∗ ∗∗∗∗ ∗ ⎡⎤ ∂ ⎡⎤ ⎡⎤ +− + ⎢⎥ ⎣⎦ ⎢⎥ ⎣⎦ ∂ ⎣⎦ ⎡ ⎤ ⎡⎤ ⎢ ⎥ += ⎣⎦ ⎢ ⎥ ⎣ ⎦ (25) Advanced Topics in Mass Transfer 118 () () [] [] () 00 00 0 0 0 0 2 00 0 00 000 ii ii i iii iii ir avg avg i iz cwc Dc c w gradc div D grad c ll cgradce Afc div A f c e ll ττ ββ ρρ ∗ ∗ ∗∗ ∗∗ ∗ ∗ ∗∗ ∗∗∗∗ ∗ ⎡⎤ ∂ ⎡⎤ ⎡⎤ + −+ ⎢⎥ ⎣⎦ ⎢⎥ ⎣⎦ ∂ ⎣⎦ ⎡ ⎤ ⎡⎤ ⎢ ⎥ += ⎣⎦ ⎢ ⎥ ⎣ ⎦ (26) The non-dimensional forms of these equations may be scaled against the convective term ⎛⎞ ⎜⎟ ⎝⎠ 0 0 0 i wc l . The dimensionless forms of the equations (25) and (26) may be given as follows: () () [] [] () 0 0 0 0 00 00 0 000 i i iii ii ir avg avg rot rot i D lc w gradc div D grad c wlw grad c e nd div n d c e ww ϕ ττ ββ ρρ ∗ ∗ ∗∗ ∗∗ ∗ ∗ ∗∗ ∗∗∗∗ ∗ ⎡⎤ ∂ ⎡⎤ ⎡⎤ +− + ⎢⎥ ⎣⎦ ⎢⎥ ⎣⎦ ∂ ⎣⎦ ⎡ ⎤ ⎡⎤ ⎢ ⎥ += ⎣⎦ ⎢ ⎥ ⎣ ⎦ (27) () () [] [] () 0 0 0 00 00 00 000 i i iii ii ir avg avg iz D lc w gradc div D grad c wlw grad c e Af div A f c e ww ττ ββ ρρ ∗ ∗ ∗∗ ∗∗ ∗ ∗ ∗∗ ∗∗∗∗ ∗ ⎡⎤ ∂ ⎡⎤ ⎡⎤ + −+ ⎢⎥ ⎣⎦ ⎢⎥ ⎣⎦ ∂ ⎣⎦ ⎡ ⎤ ⎡⎤ ⎢ ⎥ += ⎣⎦ ⎢ ⎥ ⎣ ⎦ (28) The equations (27) and (28) include the following dimensionless groups characterising the mass-transfer process under the action of the rotational or the reciprocating agitator: ττ − ⇒⇒ 0 1 0 00 00 rot rot lD S wnd (29a) ττ − ⇒⇒ 1 0 00 000 rec lD S wAf (29b) νν νν −− − ⎛⎞ ⎛⎞ ⎛⎞ ⎛⎞ ⇒⇒⇒⇒ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ⎝⎠ ⎝⎠ 00 0 11 1 00 00 0 ii i mass DD D Re Sc Pe lw lw wD (30) ⎛⎞ ⇒⇒ ⎜⎟ ⎜⎟ ⎝⎠ 00 0 00 00 1 rot rot rot nd nd wnd (31a) ⎛⎞ ⇒⇒ ⎜⎟ ⎝⎠ 00 00 000 1 Af Af wAf (31b) [...]... AIChE Journal, Vol 4, No 1, 1 14- 1 24 ISSN 0001-1 541 Geankoplis, C.J (2003) Transport processes and separation processes Principles, Pearson Education Inc., ISBN 0-13-101367-X, USA Gomaa, H.G & Al Taweel, A.M (2005) Axial mixing in a novel pilot scale gas-liquid reciprocating plate column, Chemical Engineering and Processing, Vol 44 , 1285-1295 ISSN 0255-2701 142 Advanced Topics in Mass Transfer Gomaa, H.G.,... Sherwood d −1 b ⎛x⎞ number) increase with increasing the term a ⎡{Tam } {Reω } ⎤ Sc c ⎜ ⎟ This figure shows a ⎢ x x ⎥ ⎣ ⎦ ⎝D⎠ strongly increase in mass- transfer process when the RMF is applied It was found that the 140 Advanced Topics in Mass Transfer Sherwood number increase with increase in the local magnetic Taylor number It should be noted that the mass- transfer rate is increased with the dimensionless... 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 fine- 126 Advanced Topics in Mass Transfer grained abrasive... 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 5.2 Influence of transverse rotating magnetic field (TRMF) on mass- transfer process The influence of TRMF on mass- transfer. .. 11, 143 3- 143 8 ISSN 0888-5885 Hixon, A & Baum, S.J (1 942 ) Agitation Performance of propellers in liquid-solid systems, Industrial and Engineering Chemistry, Vol 34, No 1, 120-125 ISSN 0888-5885 Hixon, A & Baum, S.J (1 944 ) Mass transfer and chemical reactions in liquid-solid agitation, Industrial and Engineering Chemistry, Vol 36, No 6, 528-531 ISSN 0888-5885 Humphrey, D.W & Van Ness H.C (1957) Mass transfer. .. transfer in a continuous-flow mixing vessels, AIChE Journal, Vol 3, No 2, 283-286 ISSN 0001-1 541 Incropera, F.P & DeWitt D.P (1996) Fundamentals of heat and mass transfer, John Wiley & Sons Inc., ISBN 0 -47 1-3 046 0-3, USA Jameson, G.J (19 64) Mass (or heat) transfer form an oscillating cylinder, Chemical Engineering Science, Vol 19, 793-800 ISSN 0009-2509 Kays, W.M & Crawford M.E (1980) Convective heat and mass. .. ( 54) The graphical illustration of the equation ( 54) is given in the coordinates ( Sh∗Sc −0.33 , Rerec ) 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... 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The processing was finished with additional smoothing of the surface with fine- Advanced Topics in Mass Transfer 126 grained abrasive. fields etc. Advanced Topics in Mass Transfer 1 14 In this work, the focus is on a mass- transfer process under various types of augmentation technique, i.e.: rotational and reciprocating mixers,. dimensionless number is equal to 2. Advanced Topics in Mass Transfer 1 24 4. Dissolutions of solid body in a tubular reactor with reciprocating plate agitator 4. 1 Literature survey Numerous