HUNEK, József

KOLLÁR-HUNEK, Klára

SAWINSKY, János

English

Periodica Polytechnica (Budapest University of Technology and Economics)

I. THE USE OF SPLINE·FUNCTIONS TO CALCULATE THE: STEADY . STATE CONCENTRATION PROFILE IN COUNTERCURRENT EXTRACTION COLUMNS By K. KOLL_iR-Hu~EK, J. HU;'IEK* and J. SAWI;'ISKY* Department of Mathematics of the Faculty of Chemical Engineering Received March 6, 1980 Presented by Prof. Dr. D. KR . .\.LIK In previous works [1-3] the one dimensional diffusion model has been used to describe the steady-state concentration profile along an extraction column. The equations of the model arc as follows: d 2x dx Al - + B 1x + Cl)' = 0 dz2 dz d2x I A d)' B C -- T !:1? - 2'V + ?x d ? --d -.I -o Z... Z The boundary conditions are: dx dz and d)' = 0 if z = 0 dz d)' dx - + A?)' = D., and dz - - dz o if z=l (1 ) (2) In the present work an approximate solution to the above differential equation system has been obtained by using Spline functions. In our calculations the coefficients AI'" D 1 , A 2 ••• D2 have been fixed constants. However, the calculations may be carried out for coefficients which are functions of column length, z. If the condition of linearity is given up, the calculations become more complicated [4]. The approximate solution of the differential equation system was ob-tained by cubic polynomial univariant Spline functions. The Spline function was defined in terms of its first derivative [5] [6]: A distribution interval between [0; 1] may be 'where and h=_l_ n-1 *Department of Chemical Engineering, Technical University, Budapest (3) 260 KOLLAR-HUNEK, K. et al. In the case the form of Spline function is: - z-z· S HI - 2S j) ( )2 + h J (4) if (j = I, 2, ... (n - I» A Spline function in this fOl'm (4) can be applied for X(z) as well as for Y(z) concentration functions. Thus, altogether 4n (j = 1,2, .. , n) pal'ameters have to be determined with the symbols Xj and Yj for the constants gj' and SXj and SYj for Sr The determining equations are the follo,.,.-ing: 1 I 4 1 3 3 - -SX '-1 T - . SX ' ..L - . SX '.1.1 + - X '-I - - . X '+ = 0 h J h J I hi, h2 i h2 J 1 ~. SY'--l + i.. SY + ~. SY-'-l h j h i h j, 3 3 -y, --·Y·.1.=O h2 i-I h2 i ,1 j = 2, 3, .. . (n - I) (Sa) ~ ,SX'+l + (- ~1+ B1) ·X· + h i h2 I i +-~ ,X'1+- C1' Y '=O 'h2 iT' i j = 1,2, .. . (n - 1) 6 ..L ~ . X 1 + Cl' Y = 0 'h2 n- n (5b) . SY, - ~ . SY '+l..L (-~ ..L Bz) . Y J, + i It i I h2 I ..:.. ~ . Y'+l + Co' X ' = 0 'h2 j ~ i j = 1,2, .. . (n - 1) [i. +A J. SY ..L 2 . Sy _1..L (-~ + Bo) . Y + It 2 n I It n I h2 - n 6 + - . Y -1 + Co' X = 0 h2 n - n (5c) and CALCULATING THE STEADY·STATE CONCENTRATION PROFILE I SXI + Al . Xl = DI SXn = 0 SYI = 0 261 (5d) Among these equations the continuity of the first and second derivatives of the Spline functions is described by (5a). The differential equation system (1) gives (5b) and (5c) and the boundary conditions (2) results in (5d). Equation system (5) obviously consists of 4n equations having 4n un-known parameters. This linear equation system has been solved by the Gaus-sian method. Table I No. of Pe~ Pev Note dataset 1 2 2 In all 2 2 8 2 cases the 3 2 4 1 2 if = I 4 2 4 4 2 I xin = 1 5 2 4 16 2 I 0 )'in = 6 8 4 1 2 7 8 8 8 2 condition 8 4 2 8 4 is fulfilled 9 4 4 1 4 10 4 4 4 4 Table n h= 0.1 z X X .1X Y Y LlY Spline Literature . 10' SpUne Literature ·10' 0.00 0.7218 0.7218 0 0.2977 0.2970 7 0.05 0.6947 0.6947 0 0.2967 0.2960 7 0.15 0.6445 0.6445 0 0.2895 0.2888 7 0.50 0.5031 0.5032 -1 0.2372 0.2323 4 0.85 0.4167 0.4172 -5 0.1562 0.1559 3 0.95 0.4068 0.4073 -5 0.1328 0.1326 2 1.00 0.4055 0.4059 -4 0.1208 0.1206 2 , L'ldXI = 0.0015 L'rdYI = 0.0032 262 KOLLAR-HUNEK, K. et al. h = 0.04 X X X LlX Y Y LlY Z SpUne Literature . 10' SpUne Literature . 10' 0.00 0.7219 0.7218 1 0.2971 0.2970 1 0.05 0.6948 0.6947 1 0.2961 0.2960 1 0.15 0.6445 0.6445 0 0.2890 0.2888 2 0.50 0.5032 0.5032 0 0.2324 0.2323 1 0.85 0.4171 0.4172 -1 0.1560 1 0.95 0.4073 0.4073 0 0.1326 0 1.00 ! 0.40.')9 0.4059 0 0.120i 1 i I -I L'1~Xl = 0.0003 L'ILlYi = O.OOOi The calculation of the concentration profile "was done hy the approxi-mate method using the Spline functions for the model parameter values given in Table 1. The calculation results have been compared with those published by l\iCl\fULLEl', MrYAUCHI and VERMEULEN [7]. This comparison is shown in Table 2 for the results of dataset No 1. with two different values of the distrihution interval h. Summary The axial mixing has a considerable effect on the concentration profile and conse-quently on the effectivity of an extraction column. The present work presents the Spline-approxi-mative solution of the differential equation system of the one dimensional diffusion model for calculating concentration profiles. The accuracy of the method has been shown by comparing the results to the ones cal-culated by the exact solution [I]. The advantage of our present method is that it can be extended for models -with non-constant coefficients, too. References 1. MIYAUCHI, T.-VER~lEULEK, T.: lnd. Eng. Chem. Fundls 1963, 2 113. 2. SLEICHER, C. A.: A.I.Ch.E.Jl. 1959, 5 145. 3. MECKLENBURGH, J. C.-HARTLAND, S.: The Theory of Backmixing. J. Wiley, London 1975. 4. CHU, S. CH.: J. Eng. ~Iath. 1970, 4 65. 5. PRENTER, P. M.: Splines and Variational Methods. J. WHey, New York 1975. 6. AHLBERG, J. H.-NILSON, E. N.-WALsH, J. L.: The Theory of Splines and Their Appli-cations. Academic Press, New York 1967. 7. McMuLLEN, A. K.-MIYAUCHI, T.- VER~rEULEK, T.: U. S. Atomic Energy Comission Rept. UCRL-3911-Suppl. 1958. Kliira KOLL . .\.R-HuNEK dr. J ozsef H UNEK dr. Jimos SAWINSKY 1 H·!52! Budap"t CALCULATING THE STEADY-STATE CONCENTRATION PROFILE I Notation f phase ratio gj = SI.! (Zj) value of SI.! (z) at Zj h lengtli of the distribution interval k distribution coefficient n number of dividing points of the length Z Nox true Number of Transfer Units Pe Peclet number S (z) Spline function for distribution SX first derivative of X (z) Spline function SY first derivative of Y (z) Spline function x concentration in the raffinate phase y concentration in the extract phase z dimensionless length of the column X dimensionless concentration in the raffinate phase Y dimensionless concentration in the extract phase Subscripts in inlet j serial number of dividing point of the length z Q overall x raffinate phase y extract phase 263 

I. THE USE OF SPLINE-FUNCTIONS TO CALCULATE THE: STEADY-STATE CONCENTRATION PROFILE IN COUNTERCURRENT EXTRACTION COLUMNS

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I. THE USE OF SPLINE-FUNCTIONS TO CALCULATE THE: STEADY-STATE CONCENTRATION PROFILE IN COUNTERCURRENT EXTRACTION COLUMNS

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Periodica Polytechnica (Budapest University of Technology and Economics)