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- 1. FOR A CONTINUOUS WORKING OF THE VASILESCU-KARPEN’SCONCENTRATION PILEMihai DOGARU – Society for Promotion of Renewable, Inexhaustible and New Energies –SPERIN, mihdogaru@yahoo.comMircea Dimitrie CAZACU – University POLITEHNICA, Bucharest, cazacu@hydrop.pub.ro1. The imperious actuality of Vasilescu Karpen concentration pile.The global warming in the two last centuries, due to the low efficiencies of thethermodynamical cycles, as well as the gas emission with green-house effect, followed in thelast time of the important climate changes, manifested by strong storms and torrential rains,caused by the water vapor excess, resulted by burning, after 1850 years, also of thehydrocarbons too; will lead in the same time, in our opinion, at the situation that the Earthwill lose gradually its seasons [1].To avoid this effect, besides the arising of afforestated surfaces, on the first place of thetechnical proceedings the concentration pile of the famous Romanian scientist NicolaeVasilescu Karpen [2]÷[7] situate as the only proceeding known by us, which will produce theelectric energy by diminishing of electrolyte and consequently environment temperature.2. Historical antecedents of the electrochemistryAlthough the founder of the interdisciplinary science of the Electro-magneto-dynamicsin 1831 and of the Electrochemistry in 1833 is the famous physicist and chemist MichaelFaraday (1791-1867), galvanizing applications are known yet from the asiro-babilonianepoch, when the statues are gilded means by an electric pile.In an previous paper [8] we tried to obtain the continuous working of the K pile,applying a magnetic field created by permanent magnets, in the aim to intensify theconnective heat transfer between the electrolyte and the environment through the pile walls.In this work we shall present a mathematical model of the K pile working in absence ofmagnetic field, but considering the thermal-siphon effect of the electrolyte density variation.3. Two-dimensional mathematical model of the K pile and equation transformation3.1. Electric field equations [9] with the component JZ = 0 according with figure 1, are:- electric charge conservation equation, unstable in the iterative numerical calculus,which can be identically verified by the introduction of the electric current line function(X,Y) = const. and which, with respect of the potential linesI ( ),X Y∈ are in the relations:X Ydiv 0J JJX Y∂ ∂= + ≡∂ ∂ur→ X YJ X′ ′= =∈I and Y XJ ′ ′Y= − =∈I , (1)which permits the introduction of complex variable function( ) ( ) ( ), ,Z X Y i X YΦ =∈ + I , (2)whose real part, representing by the equation (1) verification, the harmonic potential,2 2X Y,0X Y′′ ′′∆ ∈=∈ +∈ = , (3)the imaginary part, representing the electric current line function
- 2. ( )X YX Yd dd d d 0 , const.X YX Y X YJ J′ ′= → = + = → =I I I I , (4)supposing the spectral line orthogonality ( ), const.X Y∈ = and ( ), const.X Y =I- magnetic circuit law in the case of quasistationary regime,Y YX Y ZX Y Zrot Xi j kH H H HH J iJ jJ kJ i j k XX Y Z Z Z X YH H H∂ ∂ ∂ ∂∂ ∂ ∂ ⎛ ⎞= = + + = = − + + −⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠r r ruur ur r r rrr r(5)and which, in the case of made hypothesis, one reduces of the equations:Z YX1 YH H BJY Z Z∂ ∂ ∂= − − ⋅∂ ∂ µ ∂, X ZY1 XH H BJZ X Z∂ ∂ ∂= − ⋅∂ ∂ µ ∂, Y XZ 0H HJX Y∂ ∂= − =∂ ∂, (5’)the last of these equations demonstrating the two-dimensional magnetic field dependence on ascalar potential ( , )X Yℵ , in accordance with the equations:( ) ( )X Y X Y X X, grad ,H X Y i H jH X Y i j H′ ′ ′= + = ℵ = ℵ + ℵ → =ℵr r r r rand , (5”)YH ′=ℵY- Ohm’s law, where σ is the electric conductivity and ( ),X Y∈ the electric potential( ) ( )x Y X Ygrad , =J iJ jJ X Y E iE jE= + = ∈ σ = σ +ur urr r r r, → X XJ EX′=∈ = σ , . (3)Y Y YJ E′=∈ = σYT0 C Ψ = 0c Ψ = 0+ _ gη = 0X A a 0T0 T0 = const.JXFig. 1. Two-dimensional configuration of a K pile, boundary conditions and co-ordinate axes3.2. The scalar temperature field equation, considering the heat connective transferdue to the K pile cooling in permanent regime, with the constant specific heat and thermalconduction coefficient, is( ) ( ) ( ) ( ) ( )2 22 2X Y X YX Y,T c T U T V T T R J J K J X Y′ ′ ′′ ′′ρ + = λ + + + − ⎡ ⎤⎣ ⎦ , (6)- liquid state equation of variable density in accordance with the expression [10]( ) ( ) ( )0 0 01T T Tρ = ρ −β⋅δ = ρ −ρ β − 0T Twith X,Y T X,Y 0 X,YT′ ′ ′ ′ρ = ρ = −ρ β , (7)where the influence of the relative volume variation δW/W as function of the temperaturevariation,
- 3. ( )T 000T-T 3W WWTW W−δβ = δ = = α being the volume dilatation coefficient, three time greaterthan the linear coefficient ( )T 000L LLT TL L−δα = δ = −T .3.3. The equation system of the velocity hydrodynamic field, composed by:- motion equations on the two directions, of a liquid with constant viscosity( ) ( ) ( ) ( )2 2X Y X X YUU VU T P U U T′ ′ ′ ′′ ′′+ ⋅ρ + = υ + ⋅ρ , (8)( ) ( ) ( ) ( )2 2X Y Y X YUV VV T P V V g T⎡′ ′ ′ ′′ ′′+ ⋅ρ + = υ + − ⋅ρ⎣⎤⎦ , (9)- electrolyte mass conservation equation, unstable in the iterative numerical calculus,is according the estimation from [10]( ) ( ) ( ) ( ) ( )X Y 0 X YX Y1 0U V T U T V T T U V′ ′ ′ ′ ′ ′ρ + ρ = −β + + −β − + =⎡ ⎤⎣ ⎦ , (10)but identically verified, introducing the streamline function Ψ (X,Y) = CT. by the relations:( )( ) Y1,,U X YT X Y′= Ψρ⎡ ⎤⎣ ⎦, ( )( ) X1,,V X YT X Y′= − Ψρ⎡ ⎤⎣ ⎦. (11)By introduction of the thermal-current line function and pressure functionelimination in virtue of Schwarz’s relation, of second order mixed derivative commutativity, one obtains a nonlinear with partial differential equation up to 4X,Y Y,XP P′′ ′′= thorder forthermal-current line function and up to 2ndorder for temperature function.4. The equation system transformation for the numerical treatmentFor more generality of the numerical solving, we shall use the dimensionless variablesand functions, denoting by:,X Yx yA A= = andm m m 0 0 0,, , , , ,U V T JjU U U A T Ju vΨ ∈= = ψ = θ = = ε =∈ 0η =II,(12)with which the relation (2) becomes in dimensionless form ( ) ( ) (, ,z x y i xϕ = ε + η )y .4.1. The electric charge conservation equation, replacing the partial differentialswith the expressions in finite differences, in the case of a quadratic grid δy = δx = δ, becomes2 2x y,0x y′′ ′′∆ ε = ε + ε = , → 1 0 3 2 0 42 22 20ε − ε + ε ε − ε + ε+ =δ δ, (13)Thus, the solving of the algebraic relation (13) makes the calculation of the harmonicelectric potential40 ii=114ε = ε∑ , (13’)associated to the equation with partial differentials (13), representing the very known formulaof the mean value.The numerical solution stability is assured always by the error propagation relationin any both calculus directions in the considered domain [11]x,yn+1 n14±δε = δε . (14)4.2. Temperature field equation, utilizing the (11) relations, becomes indimensionless form( ) ( )2 22 2 2y x x y x y xx y1o KaPéj j j j′ ′ ′ ′ ′′ ′′ψ θ −ψ θ = θ + θ + ℑ + − + 2y , (15)
- 4. in which one denoted the similitude numbers Péclet, Joule and Karpen of the specific thermal-electric considered phenomenon, by the expressions:m0 0Pé=U AaT ρ,20mo=R J AcUℑ , 0mKa =K A JcU. (16)Replacing in (15) the expressions of partial differentials, deduced by the finitedifference method and expliciting the θ0 value from the linear part, one obtain the associatealgebraic relation( )( ) ( )( )2 2420 i 1 3 2 4 2 4 1 3 0i=11 Pé Pé o PéKa4 16 4 4j jℑ δ δθ = θ + ψ − ψ θ − θ − ψ − ψ θ − θ + −⎡ ⎤⎣ ⎦∑ 0 , (17)from which we can deduce the error propagation relation, for instance on the calculusdirection ± x( ) ( ) 2 22 4 2 4x 2n+1 n n n+1 n+1Pé - Pé -1 Pé o4 16 16 4 4j± ψ ψ θ θ⎛ ⎞ ℑ δ ℑ δδθ = ± δθ δψ + δ − δ⎜ ⎟⎝ ⎠mPé oj , (18)which assure numerical solution stability in the conditions:( )2 4Pé -114 16ψ ψ⎛ ⎞±⎜ ⎟⎝ ⎠p , ( )2 4Pé - 16θ θ p , and .(19)2Pé o 4ℑ δ p 2Pé o 4ℑ δ p5. The specific boundary conditions of the studied problem in the case of steady state:5.1. For the hydrodynamic field, ΨC = 0 on the whole vessel outline containing theelectrolyte, inclusive at its free surface and on the electrode axis, with unknown values ± ΨEon the two electrodes, which will result follows the iterative numerical calculus and of theelectric current resulted by closing of the circuit on the exterior charge resistor.5.2. For the thermal field, TC = T0 = the environment temperature, on the wholevessel outline supposed as thermal perfect conductor, inclusive at the electrolyte free surface,with unknown but constant values TE = const. on the two electrodes, good thermal conductorsand depending of the pile working regime intensity,5.3. For the electric field, on the axis perpendicularly on the two electrodes A E⊥ andwith unknown values ± Cη at their two ends, with the boundary conditions on electrodes face0η =( )FEFEJ J Y= and back ( )BEBEJ J Y= , resulted by the relation (14) and solving in theoundary conditions concerning to the voltage values on the both electrodes .b E±εR J 2K JStableworkingUnstableworking0 Jcritical J
- 5. 6. A final observationThe calculus program being for the moment in course of elaboration, we shall permitonly a consideration to explain the experimental result of the electrode polarization in thesituation of a intense working of the Vasilescu Karpen pile, considering the graphicrepresentation (fig.2) of the thermal energy functions of pile heating and cooling respectively,given by the equation (17), limiting the two zones of continuous working, stable of physicalpoint of view and specific of pile reduced charge, with respect of the intense one, when thedeveloped energy by heating overtakes, by our estimation, its cooling one, resulting by thenormal working.7. References[1] M.D.Cazacu. Tehnologii pentru o dezvoltare durabilă (Technologies for a sustainabledevelopment). Al II-lea Congres al Academiei Oamenilor de Ştiinţă din România “Dezvoltarea înpragul mileniului al III-lea”, 27-29 septembrie 1998, Bucureşti, Ed.EUROPA NOVA, 1999, 533-539.[2] N. Vasilescu-Karpen. Piles à oxygène empruntant leur énergie au milieu ambiant. Acad. deSci., Paris, 1944, T. 218, p.228.[3] N. Vasilescu-Karpen. Piles à hidrogène empruntant leur énergie au milieu ambiant. Acad. deSci., Paris, 1948, T. 226, p.1273.[4] N. Vasilescu-Karpen. La variation de la f.é.m. de la pile à gas avec la pression. F.é.m. de lapile de concentration à oxygène et la thermodynamique. C.R. Acad.de Sci., Paris, 1949, T. 228, p.231.[5] N. Vasilescu-Karpen. Pila electrică de concentraţie cu oxigen şi termodinamica (The electricconcentration pile with oxygen and the thermodynamics). Bul.Şt. Acad. RPR, Secţ.de Şt.Mat.şi Fiz.,T. VII, nr. 1, Ian.-Febr.-Martie, 1955, p. 161.[6] N. Vasilescu-Karpen. Afinitatea electronilor pentru oxigenul dizolvat în apă şi termodinamica(Electron affinity for the dissolute oxygen in water and the thermodynamics). Bul.Şt.Acad. RPR,Secţ.de Şt.Mat.şi Fiz., T. VII, nr. 2, April-Mai-Iunie, 1955, p. 491.[7] N. Vasilescu-Karpen. Fenomene şi teorii noi în electrochimie şi chimie fizică (Newphenomena and theories in electrochemistry and physical chemistry). Ed.Academiei RPR, 1957[8] M.D.Cazacu, M.Dogaru, A.Stăvărache. Vasilescu-Karpen pile as a proceeding to diminish theEarth warming. Conf.Naţională pt.Dezvoltare Durabilă, 13 iunie 2003, Univ.Politehnica, Bucureşti,273-280.[9] A.Timotin, V.Hortopan. Lecţii de bazele electrotehnicei (Lessons of electrotechnics basis).Vol. I, Editura Didactică şi Pedagogică, Bucureşti, 1964.[10] M.D.Cazacu, M.D.Staicovici. Numerical solution stability of the Marangoni effect. The 5thInternat.Conf. OPROTEH November 2002, University of Bacău, Modelling and Optimization in theMachines Building Field – MOCM – 8, Romanian Academy 2002, 161-166.[11] D.Dumitrescu, M.D.Cazacu. Theoretische und experimentelle Betrachtungen über dieStrömung zäher Flüssigkeiten um eine Platte bei kleinen und mittleren Reynoldszahlen. Zeitschr. fürAngew. Math. und Mechanik, 1, 50, 1970, 257- 280.

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