The proposed model combines electric and temperature fields, when organic dispersed hydraulic systems are subjected to electric-thermochemical treatment. This model has been applied for developing the technologies and equipment for protein production from potato juice, milk whey and for electrolytic activation of yeast productivity. The given model relations are required to establish optimal treatment parameters of media, to impose restriction on technological and design parameters, to define medium flow diagrams in the reactor, electrode and orifice material, ultimate electric field strength and current values, etc

Baraishuk, Sergey Mikhailovich

Chorny, A. D.

Krivovyazenko, D. I.

Yanko, M. V.

Zayats, E. M.

Барайшук, Сергей Михайлович

Заяц, Евгений Михайлович

Кривовязенко, Денис Иванович

Янко, Максим Валерьевич

Repository of the Belarusian State Agrarian Technical University

HEAT TRANSFER AT ELECTROCHEMICAL TREATMENT OF ORGANIC DISPERSED HYDRAULIC SYSTEMS  Zayats E. M. D. Sc. (Techn.), Belarusian State Agrarian Technical University, Belarus Chorny A. D. PhD (Physics and Mathematics), A. V. Luikov Heat and Mass Transfer Institute of the National Academy of Sciences of Belarus, Belarus Barayshuk S. M. PhD (Physics and Mathematics), Belarusian State Agrarian Technical University, Belarus Krivovyazenko D. I. Researcher, Belarusian State Agrarian Technical University, Belarus Yanko M. V.  PhD student, Belarusian State Agrarian Technical University, Belarus  Abstract The proposed model combines electric and temperature fields, when organic dispersed hydraulic systems are subjected to electric-thermochemical treatment. This model has been applied for developing the technologies and equipment for protein production from potato juice, milk whey and for electrolytic activation of yeast productivity. The given model relations are required to establish optimal treatment parameters of media, to impose restriction on technological and design parameters, to define medium flow diagrams in the reactor, electrode and orifice material, ultimate electric field strength and current values, etc.  Keywords: organic dispersed hydraulic systems, electric treatment, electric field strength, temperature, Joule heating  Agricultural products of vegetable and animal origin (vegetable and fruit juic-es, milk whey, solutions for growing microorganisms, moist and moistened forage, etc. can be conditionally referred to as organic dispersed hydraulic systems (ODHS). To improve the efficiency of using nutritive potential and increasing extraction or formation of useful materials, and to disinfect ODHSs the latter are treated in differ-ent ways – mechanical, thermal, complex [1]. The electric field applied through the medium heats it. Increasing the tempera-ture can have a positive effect on the treatment process in some cases and a negative effect in others. A temperature value depends on electric field density, medium spe-cific conduction, on design of a reactor, where a medium is treated, and on flow dia-gram. In all cases, the temperature increase means the growth of the energy intensity of the process. This is undesirable. In addition, the treatment quality is affected by a uniform temperature distribution over the medium volume. A higher temperature near the surface of current-conducting electrodes and the separating membrane may wors-en the treatment quality or in general, this may hinder the electric field applied to the medium. The present work considers a flat electrode system, whose the interelectrode space are separated by a membrane into cathode and anode regions with the length L and the widths 1δ  and 2δ , respectively. The objective of this work was to develop a mathematical model able to describe electric and temperature fields in liquid moving in the anode and cathode regions. Temperature, more precisely its uniformity be-tween electrodes, serves as a criterion in defining the character and direction of mov-ing liquid, permissible electrical conduction of membrane material, and electric field strength. As ODHSs are usually subjected to applied electric fields of relatively small strength (E < 1 kV/cm) and at rather a long interelectrode distance (le ~ 1 cm), we can consider the temperature variation in ODHS using the Ohmic conduction model to a good enough approximation [2]. Also, based on a small rate of diffusive and convec-tive transfer of a charge in comparison to a migration velocity of ions in ODHS due to an applied electric field, the equation for the total current in ODHS is of the form ( ) ( ) 0E∇ σ = ∇ σ∇Φ =  (1) We determine the ODHS temperature from the convective heat conduction equation with the consideration of Joule heating: 2( )pTC u T T E∂ ρ + ⋅∇ = ∇ λ∇ +σ ∂τ   (2) At the initial boundary condition 00T Tτ= =  and the boundary conditions ( )envT T TΓ Γλ∇ = α − , where λ, ρ, СP is the thermal conductivity, the density, and the specific heat capacity of a dispersed medium respectively, u  is the velocity vector, α is the heat transfer coefficient at the boundary Γ, T0 is the initial temperature, Tenv is the environment temperature, TΓ is the medium temperature at the boundary Γ, σ is the specific electrical conductivity of a medium, E is the electric field strength, Φ is the electric potential. The flow of the medium can be co-current and counter-current. The electric field is applied from one electrode to another. The temperature in the medium and in the membrane varies from the initial value (room temperature) to some final maxi-mum value assigned by the technological process. At that, the membrane temperature must not exceed the maximum permissible one of the treated material. For example, under coagulation of whey proteins, it is not more than 45 0С. Such a temperature range shows the heat and electrophysical parameters of the medium and the mem-brane are weakly temperature-dependent. They can be taken as constant. Thus, the temperature is the objective function. It is necessary to determine a temperature field of ODHS in the presence of internal heat sources depending on the thermophysical parameters of the moving liquid and the membrane. For the steady flow of the liquid in the interelectrode space with relative sizes 1 / 1Lδ >>  and 2 / 1Lδ >> , we can assess the medium state along the channel to a boundary layer approximation for both temperature and electric fields: ( ( ) ) ( ) 0yy E yy y y ∂ ∂ ∂Φσ = − σ = ∂ ∂ ∂ , (3) for the liquid : 2( )pT TC u y Ex y y ∂ ∂ ∂ρ = λ +σ ∂ ∂ ∂  (4) for the membrane: 20 m mT Ey y ∂ ∂= λ +σ ∂ ∂  (5) At the channel entrance, the boundary conditions are T = T and at the channel exit, the “soft” boundary condition is / = 0x LT x =∂ ∂ . In the case of counter-flow, we should take into consideration that for the anode and cathode regions, these bounda-ries are opposite to each other. At the channel (electrode) walls, the Newton condi-tions are valid for the heat flux 111 1= ( )mmenv yyT T Ty =δ +δ=δ +δ∂λ α −∂, 222 2 ( )( )= ( )mmenv yyT T Ty =− δ +δ=− δ +δ∂λ −α −∂. (6) Here mδ  is a membrane half width. We determine the coefficients of heat trans-fer αi [Wm-2K-1] from a heated fluid to an ambient medium in the cathode and anode regions using the Nusselt number as αi = Nui λi/δi [3, 4]. In the considered-type chan-nel, for the laminar flow regime we use = 7,54iNu  and for the turbulent flow regime – 0,5 0,83= 0.0155Pr Rei i iNu  valid for fluids with the Prandtl number 1 < Pri< 25 [4]. With the boundary conditions on the electrodes for the potential (10=myU=δ +δΦ , 20( ) =my U=− δ +δΦ − ) and the electric field continuity condition at the medium–membrane boundary ( my = δ  and m−δ ) included, we can show that the solution of equation (3) is the dependence: 01 21 22( )2( ) mmUE yy=δ δ δ σ + + σ σ σ   (7) Here, the electrical conductivity σ is the function of y, because it varies depending on the electrical conductivity of the medium moving above and under the membrane, as well as of the electrical conductivity of the membrane (σ1, σ2, σm respectively). Sub-stituting formula (7) into the formula for heat source term in temperature equation (4) yields: 20221 21 2(2 )( )2( ) mmUQ y Ey= σ =δ δ δ σ + + σ σ σ   (8) The regime of medium flow in the interelectrode space can be either laminar or turbulent, and determined by the Reynolds number Re /i i Hi iu D= ν  (i = 1, 2), where 2Hi iD = δ  is the hydraulic channel diameter. The analysis [4] shows that the critical Reynolds number at a transition from the laminar to the turbulent regime for a plane-parallel channel also has its traditional value of ∼ 2300. To calculate the dynamics of laminar flow, it is possible to use the theory of laminar flow of incompressible liquid in the flat channel. First, consider the cathode region. Following the approach described elsewhere in [3], for laminated flows it is possible to show that the velocity distribution in this region is  21max1 112( )( ) 1 myV y V  − δ − δ  = −   δ  , max 1132V u= . Here, it is taken into account that for considered-type channels, the pressure drop 21 11 11 2HL upDρ∆ = ξ  is assigned by the bulk velocity 1u  and the friction resistance coefficient – by the Reynolds number: 1 196 / Reξ =  [3, 4]. Similarly, for the anode re-gion of the channel we have  22max2 222( )( ) 1 myV y V  − δ − δ  = −   δ  ,  max2 232V u= .  For the dynamics of turbulent flow, bearing in mind that the Reynolds number for the problems of interest is not too high (over the range 5000–30000), the power-law velocity profile (“one seventh” law) is used to assign the velocity profile [3, 4]: 171max112( )( ) myV yV − δ =  δ , max 11 / 0.817V u=  at 1 / 2m myδ ≤ ≤ δ + δ , 172max222( )( ) myV yV − δ =  δ , max 22 / 0.817V u=  at 2( )2m myδ− δ + ≤ ≤ −δ . The last profiles are symmetric relative to the coordinates 1( ) / 2my = δ + δ  and 2( / 2)my = − δ + δ . Thus, the heat transfer problem remains the only unsolved one. Using the integro-interpolation method [5] for finding a temperature averaged over the thickness of each of the channels and the membrane, equations (4) and (5) arrive at the following system of equations: for the medium above the membrane: 211 1 1 1 1 1 1 1 1 1 1 1( ) ( )p m env envTC u n K T T K T T Ex∂ρ δ = − + − +σ δ∂, (9) for the medium under the membrane: 222 2 2 2 2 2 2 2 2 2 2 2( ) ( )p m env envTC u n K T T K T T Ex∂ρ δ = − + − +σ δ∂, (10) for the membrane: 21 1 2 20 ( ) ( ) 2m m m m mK T T K T T E= − + − +σ δ . (11) Here, we determine the bulk velocity in channels as 1,2 1,2 1,2/u Q S= , where 1,2Q  is the volumetric flowrate of the medium in the channel, 1,2S  is the flow area of the channel, 1,2n  are the subscripts of the medium direction ( 1,2 1n =  − along the х-axis, 1,2 1n = −  − opposite to the х-axis), subscripts 1, 2, m denote the parameter in the considered re-gion. The heat transfer coefficients are 11,2 1,2(1/ 2 / )m mK −= α + δ λ , 11 1 1 1(1/ 1/ / )env env e eK−= α + α + δ λ , 11 2 2 2(1/ 1/ / )env env e eK −= α + α + δ λ . Here 1,2eδ  and 1,2eλ  are the thickness and thermal conductivity of electrode material, envα  is the coefficient of heat transfer from electrode surface to room (we take 8envα =  W/m2К). The determined coefficients 1,2K  and equation (11) yield that the membrane temperature is equal to: 21 21 21 2 1 2 1 22m m mmEK KT T TK K K K K Kσ δ= + ++ + +. (12) Thus, for the medium temperature variation to be found, the system of the or-dinary differential equations must be solved 221 2 11 1 1 1 1 2 1 1 1 1 1 11 2 22 ( )m m mp env envET K KC u n T T K T T Ex K K K σ δ∂ρ δ = − + + − +σ δ ∂ +  , (13) 222 1 22 2 2 2 2 1 2 2 2 2 2 21 2 12 ( )m m mp env envET K KC u n T T K T T Ex K K K σ δ∂ρ δ = − + + − +σ δ ∂ +  . (14) We can obtain a general solution to the system of equations (13)–(14) by find-ing the characteristic equation roots and by using the variation method of arbitrary constant [6]. The present work contains the solution for a particular ODHS when protein is produced from milk whey [7]. Table 1 contains the set of characteristics needed for calculation. In our case, according to the parameters, Pr / 5i i pi iC= µ λ ≈  and the liquid flow is a laminar. Table 1 Thermophysical, Dynamic and Geometric Characteristics Parameter name Designa-tion Unit Value Voltage U V 36 Distance from cathode to membrane δ1 m 0.08 Distance from anode to membrane δ2 m 0.07 Membrane thickness 2δm m 0.00015 Channel length L m 0.5 Membrane heat capacity Сpm J kg-1К-1 1500 Membrane material density ρm kg/m3 1300 Initial membrane temperature 0mT  К 293 Initial medium temperature in cathode region  01T  К 293 Initial medium temperature in anode region 02T  К 293 Medium density in cathode region ρ1 kg/m3 995.7 Medium density in anode region ρ2 kg/m3 995.7 Medium heat capacity in cathode region Сp1 J kg-1К-1 4174 Medium heat capacity in anode region Сp2 J kg-1К-1 4174 Medium thermal conductivity in cathode re-gion λ1 W m-1 К-1 0.618 Medium thermal conductivity in anode re-gion λ2 W m-1 К-1 0.618 Membrane thermal conductivity λm W m-1 К-1 0.17 Membrane electrical conductivity σm Ohm-1m-1 0.0210 Medium electrical conductivity in cathode region σ1 Ohm-1m-1 0.48 Medium electrical conductivity in anode re- σ2 Ohm-1m-1 0.48 gion Medium bulk velocity in cathode region 1u  m⋅s-1 0.01 Medium bulk velocity in anode region (vol-umetric flowrates above and under mem-brane are the same) 2u  m⋅s-1 112u δδ Coefficient of heat transfer from electrode surface to room envα  W/m2К 8 Material thickness of electrode 1 1eδ  m 0.001 Material thickness of electrode 2 2eδ  m 0.001 Material thermal conductivity of electrode 1 1eλ  W/m К 236 Material thermal conductivity of electrode 2 2eλ  W/m К 236  For the assigned treatment parameters (Table 1) in the co-current flow of the medium in channels ( 1 2 1n n= = ), the analytical equations (13)−(14) are described by the functions 1 1141,524 12exp( 0,00213 ) exp( 0,01123,799 ) 1447,322T x x= − − − − + , (15) 2 1137,392 12exp( 0,00213 ) exp( 0,01123,845 ) 1417,547T x x= − − + − + . (16) In the case of the counter-flow of the medium ( 1 2 1n n= − = ), the solution is pre-sented by the following functions at the channel length L = 1 m: 1 310,824 843exp(0,00491 ) exp( 0,00488 ) 144,498 7,322T x x= − − − + , (17) 2 790,678 331,exp(0,00491 ) exp( 0,00488 ) 1415896 7,547T x x= − − − +  (18) at the channel length L = 0.5 m: 1 205,207 842exp(0,00491 ) exp( 0,00488 ) 144,969 7,322T x x= − − − + , (19) 2 792,025 331exp(0,00491 ) exp( 0,00488 ) 141,381 7,547T x x= − − − + . (20) The temperature variation in the membrane is 1 2 5,98930,5( ) 5mT T T= + + . (21) Figure 1 demonstrates the temperature variation according to the above analyt-ical solutions (15)-(21). There is a fairly clear trend that in the case of co-current flow, both the medium and the membrane undergo heating along the length of chan-nels (Fig. 1, а). At the same time, in the case of counter-flow, the treated medium is heated, not substantially varying the membrane temperature along the channel length (Fig. 1, b, c). It should also be mentioned that, when the channel length is the same, the medium temperature drop between the channel entrance and exit in co-current and counter-current flow of the medium (Fig. 1, а, c) is identical.   a b  c Figure 1 – Temperature variation of the treated medium along the channels and the membrane (1 − temperature T1; 2 – T2; 3 − Tm): а – co-current flow of the medium, b – co-current flow of the medium along the channel at L = 0.5 m, c – counter-current flow of the medium at L = 1 m.  The proposed model for  a relationship between electrochemical processes and electric and temperature fields, when organic dispersed hydraulic systems are sub-jected to electric-thermochemical treatment, has been utilized to develop technologies and equipment for protein production from potato juice, milk whey, for electrolytic activation of forage yeast productivity, etc. The above-mentioned relations are need-ed to set optimal treatment parameters of media, to impose resistance restrictions on technological and design parameters, to define medium flow diagrams in the reactor, electrode and orifice material, limiting values of electric field strength and density, process duration, etc.  References 1. Zayats E.M. Fundamentals of the electrothermochemical methods of pro-cessing of wet feeds. Urozhai Press, Minsk, 1997. 2. Zhakin A.I. Electrohydrodynamics / Physics-Uspekhi. 2012. Vol. 182. P. 465–485. 3. Kutateladze S.S. Heat transfer and hydrodynamic drag. Energoatomizdat Press, Moscow, 1990. 366 p. 4. Case V.M. Convective heat and mass transfer. Energiya Press, Moscow, 1972. 448 p. 5. Samarsky A.A. Theory of difference schemes. Nauka Press, Moscow, 1989. 616 p. 6. Bronshtein I.N., Semendyaev K.A. A guide to mathematics for engineers and students of technical colleges. Nauka Press, Moscow, 1967. 608 p. 7. Zayats E.M., Nikolaenok M.M. Methods and programs for calculating electri-cal technological processes and computer equipment. Technoprint, Minsk, 2003. 166 p. 

Heat transfer at electrochemical treatment of organic dispersed hydraulic sistems

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Heat transfer at electrochemical treatment of organic dispersed hydraulic sistems

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