Conventional reaction engineering technique may be applied to electrolytic process with emphasis on residence time distribution (RTD) metho

Basha, C.A.

Vasu, K.I.

IR@CECRI

B. Electrochem 4(5) May 1988, pp 475479 MODELLING AND SIMULATION OF ELECTROCHEMICAL PROCESSES C AHMED BASHA and K I VASU I Central Electrochemical Research Institute, Karaikudi 623 006, INDIA [Received: 1987 July; Accepted: 1988 March] Conventional reaction engineering technique may be applied to electrolytic process with emphasis on residence time distribu- tion (RTD) method. Key words: Modelling and simulation, residence t ime distribution model, reactor model I INTRODUCTION The residence time distribution function may be written as athematical modelling and simulation is a recognised techni- E (t) = C(t) - % - C I M q u e  in many process design, evaluation and control. The er- t C, c c n t  (3) tent to which modelling is used varies throughout the process in- Where A t  is the sampling Once average residence time dustries. Electrolytic reactors are in no way more difficult to model is known, the - diagram may be described by an equation of than the conventional chemical reactors. The main problem in the the form application of reaction engineering technique to electrochemical pkesses  is that the kinetic data are in the form of polarization curves; reaction mechanism involves intermediates and current CCO- c = .- = t E(t) . . . (4) distribution may be nonuniform. Co An attempt is made as to how conventional reaction engineer- where Co is the initial concentration at the inlet of the system. ing technique may be applied to an electrolytic process with em- The spread or variance of a given variable about its means in phasis on residence time distribution (RTD) method 11.1. the case of sample from a continuous distribution for a definite number of equidistance points, (equal sampling intervals) is given by I Residence Time Diitribution (RTD) function The structure of the mathematical model for any process (either Cxli f(xi) - Z xif(xi) chemical or electrochemical) involving fluid flow is affected by the s,' = Z f(xi) C f ( x i )  . . .(6) flow pattern prevailing in the system and is reflected in the RTD of fluid elements in their passage. The system may be an elec- trochemical flow cell. This distribution obeys statistical laws and variance in time units ( St 2, can be ascertained from the form of a test signal passing through the system. A test signal may be simulated by injecting an amount C f (xi) = C Ci of a tracer into the feed stream as a step input or as an impulse or a sine wave. C xi f (xi) = C ti  Ci With a tracer injected instantaneously, its concentration is observed at the outlet of the system for series of interval, and these data are plotted to give a C - diagram. This is the RTD based on an impulse stimulus and can be defined as a, C = J t E(t) dt ...( I )  0 The RTD function E(t) gives the fraction of the tracer in exit stream of the reactor that has spent a time less than t within the reactor. The average residence time 7 is given by h -- T =  set E(t) dt Z t c  e. - .. . . 7 a t )  dt L C  0 The function E(t) gives the fraction of material in the reactor exit stream which has spent a time between t and t + dt within the reactor, multiplying by it by the concentration of material, CA that has been in the fluid element of the age t + dt gives the average concentration CA of the material leaving the reactor Ahmed Basha and Vasu - Modelling and simulation of electrochemical processes The determination of residence times for fluid element from the above equation is based on the concept of statistical moments and probability density function. The zeroth moment of a probability density function is unity. The first moment is the expectation or mean value of the ensemble which gives a measure of location of the centre of the distribution. The second moment is variance which gives a measure of the spread of the distribution. The third mo- ment is a measure of the symmetry or skewness of the distribution with respect to mean. The fourth moment characterises the sharp- ness of the peak about the mode. Theoretically, the moments are calculated in the range of zero to infinity. For a continuous distribu- tion they have the form: 03 Mo = j E(t). dt 0 03 t E(t) dt 0 Since experimental distribution curves are step . wise approximated, it is assumed that the resultant distribution is discrete, and the expressions for the respective moments are as follows: If the experimental C - diagram nas s u ~  extended tail. Eq. (9) lacks in accuracy. A more accurate method for estimation of Bistribution parameter is given below. It is based on the use of werage ordinates of the C - diagram with the limit of integration specified in advance. The procedure is as follows: I.  The moment for the experimental C - curve are found by the trapezoidal rule. men for the zeroth moment where CWsi = (Ci+i + Ci)/2 is the average of the ordinate diagram in the interval 0 to a. For any moment then a j ~ d t  a o J Cdt Ca,i (ti+~-ti) i = l  with the interval chosen to be ~t ( 0.02 (t/ r), the estimation of distribution parameters from, eq. (1 1) yields results practically the same as the exact expressions i.e. Eq. (8). 2. Moments are then found for models with the limit of integra- tion from 0 to a and not from 0 to c-, as is usually done, and a system of equations of the form: is solved where the moment M a a M = t j C d t /  C d t  ...( 13) i 0 0 is found from a model. The choice of the break point on the curve, t = a, depends on the conditions of measurement (th accuracy and sensitivity of elements) because the experimental curve is used to estimate the correctness of the model itself. The number of equations in the system equation (12) depends on the number of model parameters. For some models, such as cascade type or a tank with a stagnant zone, MjM can be expressed analytically. In cases where this can- not be done because of complexity of analytical solution for equa- tion of the model, MjM should be found by directly integrating the model equations. Reaction envlronmeat and transfer function A knowledge of the reaction environment l q i n  different electrolytic cells is important in the selection of the most appropriate cell 476 B. Electrochem 4(5) May 198B Ahrned Basha and Vasu - Modelling and simulation of electrochemical processes depending on the required contacting pattern and mode of opera- TABLE-I: Transfer function of standard idealized model tion to get high selectivity and useful current density. It is provid- ed by RTD studies using tracer technique. The marker pulse techni- Standard model Transfer function que is an analogue of the dye-injection technique of conventional chemical reaction engineering [3_] with added advantage that the marker species may be generated at the electrode and therefore 1. Continuous stirred tank reactor 1 exactly experiences the same environment as the intermediate in (CSTR) sT1 + 1 the real reaction. This is not possible with dyes. For example, with 2. Plug flow reactor (PFR) e- s ~ d  well injection the momentum of the jet electrodes breaks through the sublayer so that the stream lines closest to the wall cannot be 3. m equal-size CSTR in series 1 marked. The technique has been used successfully in parallel chan- ( ST, + 1 L  nel W, battery model a, packed bcd[q, fluidized bed [TI, trickle 4. unequal-size bed [a], disc stack cell [91, cell with differentially or co-rotating CSTR in series 1 disc electrode [8,41 and cells consisting of f d n g  or moving coax- ( ST, + 1) ( sT2 + 1) ... ( sT,+l) ial cylinders. A pulse of current injects marker ions into the stream, e-  ST^ and their direct arrival is detected by a pair of electrodes main- 5. PFR and CSTR in series tained at the potential of the limiting current; the current in the Remarks (  TI+ 1) detector circuit is then a measure of distribution of marker con- centration in time, which reflects residence time. V , where V is the reactor volume TI  = Q q is the fluid space velocity Delta function, step function or any wave form (like sine wave) can be used. When a tracer material is injected into the reactor L under study as a pure sine wave at a particular frequency, the Td = -where L is the length of reactor response is like a pure sine wave of the frequency and differs from u U is the superficial velocity the input stimulus only in magnitude and phase as dictated by the behaviour of the reactor. The resultant curve is called a frequency response. The frequency response is usually interpreted in the fre- T, = V , andp is the Laplace operator. quency rather than the time domain through the Laplace transform - method. The ratio of the Laplace transform of an output time func- 9 tion to the Laplace transform of the respective input time func- tion gives the transfer function of a system. From the transfer func- tion developed for a wide range of frequencies, one can readily plot a frequency response diagram which can then be compared : A with that of the known frequency responses for standard reactor - B models. The transfer function of standard rea4tor models are sum- marized in Table I. Reaction model MASS t R A r n W R t  For developing a transfer function of a continuous stirred tank As -Ab electrochemical reactor (CSTER), consider the reaction ne Y 0 A-B 0 e * Although the electrode reactions vary in their complexity, it can u Y be represented by the diagram in Fig. 1, in which the reactant A a is transported from the bulk electrolyte to the electrode surface Y where it undergoes reaction via a number of reactive intermediates to product B. Then, B is transported back to the bulk. Assuming that the reaction is of the "Tafel" type, the rate of reaction will / B~ - eb be given by / MASS TRANS- / - r  = - 1 ~ N A  - -_ - i~ S -  A d t n F (1 4) Fig. 1: Mass transfer and reaction at an electrode B. Electrochem 415) May 1988 477 Ahmed -ha and Vasu - Modelling and simulation of electrochemical processes Further where NA is the number of moles of A transported to the elec- trode surface for reaction, t time, iA current density, S the elec- trode surface, kL the mass transfer coefficient, kh electrochemical rate constant, CA the concentration reactant A and b,s the subscript Elimination of CAs from eqs. (IS) and (16) leads to where and k is over all &s converbion constant. For hydrogen evolution where kH and a are polarization constants for hydrogen evolution. md the current efficiene, C, for the primary reaction will be k current distribution is assumed to be uniform h t h e  cell. The bllowing procedure is used in the simulation. 1) The all operates at a fmed current and flow condition so that i and kL are known. 2) Using eq.(19), E is found, which requires a numerical technique. (3) The value for E obtained is then used in (17) to calculate iA (4) Knowing i and i, C is calculated (5) Vary i and repeat the procedure 2 to 4 (6) Vary kL and repeat the procedure 2 to 5.The results are shown in Table 11. TABLE-II: Variation of current density and current efficiency on maas transfer coefficient ne Reaction A d B, 2H + 2e ' H2 2Fk = 10, 2FkH = 0.1, I,a = 19V-1 [A],, = 1 kmol/m3 - i Current efficiency (To) kA.m-2 kL X 10+3 crn.sec-1 1 2 3 4 Reactor model In case of CSTER, other quantities involved will be designated as follows: the volumetric flow rate or space velocity q; the reactor volume hold up V, the composition of A in the feed q, the wm- position of A in the exit stream as Co which is same as CA - the bulk concentration of A in reactor. Material balance gives: The Laplace transform of Eq. (21) gives: q Ci (s) - q Co (s) - k S Co (S) = VSCo where s is the Laplace operator. Dividing the output by the input, the resulting equation is on putting V/q = T, the space time and S/V = a, the specific electrode area, Ahmd Basha and Vaw - Modelling and simulation of electrochemical proclrrrres starting point for electrochemical reaction modelling which can be mxuiily expanded and modified to incorporate more complex reac- tion schemes or more accurate reaction kinetics. 1 REFERENCES I The above expression describes changes in the state of the system with time, i.e. its transient response. Since CdCl measured under steady state condition under which s approaches zero, expression (23) may be reduced to the following form. CONCLUSION From a comparison of Eq. (24) with transfer function of a CSTR (see Table I) it is readily seen that reactor model for an elec- trochemical reaction under steady state condition can be derived from transfer function by replacing s with k. This offers means of evaluating the kinetics of a process from its frequency response where there is steady state prevailing at the electrode surface, i.e. the rate of trailsport of reactant to the surface is equal to its rate of reaction. The RTD model equations presented should provide a useful 1. 0 Levenspiel, Chemical Reaction Engineering, 2nd Edition, Wiley Eastern Ltd., (1974) p 253 2. R E W Jansson, Dechema - Monographien Bond 94 - Verlag Chem. (1983) p 30 3. P V Dankwaerts, Chem Eng Sci, 2 (1953) 1 I 4. R E W Jansson and R J Marshall, Electrochim Acta.27 (1982) 823 5. R J Marshall, Ph. D. Thesis, Southampton University (1981) I 6. F Sarfarazi, Ph. D. Thesis, Southampton University (1979) 7. I H Justinijanvovic, Ph. D. Thesis, Southampton University (1975) 8. Z Ibnizicgic, Ph. D. Thesis, Southampton University (1978) 1 9. J Ghoroghian, Ph. D. Thesis, Southampton University (1978) B. Electrochem 4(5) May 1988 479 

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