APPLICATION OF THE RELAXATION METHOD FOR SOLVING REACTING DISTILLATION PROBLEMS

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1 APPLICATION OF THE RELAXATION METHOD FOR SOLVING REACTING DISTILLATION PROBLEMS Hiromasa KOMATSU Numazu College of Technology, Numazu 410 A new method for correcting liquid compositions by the relaxation method has been developed. This method consists of the normalization between liquid compositions calculated at w-th trial and these resulting from the material balance at the hypothetical steady state. It has been confirmed that calculated results by use of this correction method converge to the actual values given by experimental data and that changes of the system from the start-up to the final steady state is simulated. This method is not only useful for the reacting distillation problems, but also it may be useful in the absence of chemical reactions. Introduction In recent years, several computational methods have been proposed for solving nonideal multicomponent distillation problems. In paticular, the method of convergence and the relaxation multi-# method may be familiar1>2). As both methods may be applied to distillation problems at unsteady state operation, they are very interesting. It is considered that the most important point in the convergencemethodis howto approach to the actual values of experimental data, however, there is little confirmation that the converged results are equivalent to the actual results. Thus, experiments on the reacting distillation of the esterification system of acetic acid and ethanol were made in order to compare calculated results with experimental data. Thereafter the application of the relaxation method for solving the reacting distillation problems was examined by use of the new procedure based on the material balance at the hypothetical steady state. The relaxation method has been available for solving the reacting distillation problems at unsteady state operation. 1. Experimental Section A diagram of the experimental apparatus is shown in Fig. 1. It consisted of seven units of single bubble cap plates. Each single bubble cap plate unit was made of pyrex glass of 140 mminside diameter and 140mm height with pyrex glass tubes of 140mm height and 70mm inside diameter. A schematic diagram of a single bubble cap plate unit is shown in Received June 25, Fig. 2. The liquid hold-ups were about 0.61 in the bottom and about 0.4/ in the other plates. The bottom plate and the plate below the top were equivalent to the lower feed plate and the upper feed plate, respectively. The main heater was located at the bottom, and the column wall was warmed by the flexible heater and it was insulated with asbestos covering. Tempera- Fig. 1 Diagram of the apparatus Fig. 2 Diagram of the single bubble cap unit JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

2 ture measurements were madewith a recording Ptresistance thermometer. The flow operation was carried out after the total reflux operation. It required an average of 5-6 hrs to reach steady state from the start of the flow operation. Following the steady state operation for 3-4 hrs, liquid samples from each plate, distillate, waste and feed were collected through each valve, then the samples were analyzed by the gas chromatography. Since a flash reacting distillation is equivalent to the elemental reacting distillation with one plate operation, these experiments were carried out as reference experiments. This apparatus was essentially the same as the modified Othmerstill which was used to measure the vapor liquid equilibrium data3) and the experimental procedure was the same as that of the reacting distillation as mentioned above, except the one plate operation. A photograph of the apparatus is shown in Fig. 3. Experimental results are shown in Table 1 for the flash reacting distillation and in Table 2 for the multiplate reacting distillation. 2. Development of Equations for the Reacting Distillation For convenience in deriving the general working equations, consider the schematic reacting distillation column shown in Fig. 4 with N stage counted upward from the bottom denoted as zero. First, consider the flash distillation. The well known equations describing the continuous distillation are derived from the requirements for the conservation of the material and for the phase equilibria. Equations for the reacting distillation are essentially the same as those for the distillation without reaction, except the term of reacted moles. -^-=Zzi -(Dyi+ Wxt-nRl) (1) Equation (1) represents the rate of accumulation of z-th component; therefore it is not zero during the unsteady state. An implicit form of the finite difference approximation for the differential equation is as follows. dd" Ad (2) For the convenience of calculations, consider xit0+a0 to be a calculated value (*?+1) at N+l-th trial and xit0 to be a calculated value (xf) at 7V-th trial, it reduces to Eq. (3). do~ ao y} Substituting Eq. (3) into Eq. with some rearrangements. (1), Eq. (4) is obtained x^^+iadlhxzzt-dklfxy- Wxt+nl%) (4) VOL. 10 NO Fig. 3 Photograph of the apparatus Fig. 4 Schematic diagram of the reacting distillation column Equations for the multi-stage distillation are analogous to those for the flash distillation; thus, according to the law of the conservation of mass, the following equations are obtained during the unsteady state. For the bottom, Xfr^Xtt + WIHiXLtft - VK?t à" *?«- Wxtt+ZtZH+ntu) (5) For the middle plate, x^=xl + {AdlH3){Ux]+Ui -xl) - ViKJiXl -K^xUJ+nhi) (6) At the upper feed plate7=/, -ViK^ -K^^^+ZfZf.+nl^ (7) At the top stagej=t, ^i+1=x7i + (^IHt)(VKt_l t ixu ti -Lxh-DKZxZi+nl+tt) (8) The first hand in parenthesis of the second term on the right these equations correspond to the relaxation factor. If liquid compositions at any iteration (n-th) are 201

3 Run 6 Run 9 Exp. hr AcOH EtOH Z=4.7m//min Exp. hr Z>=3.2m//min AcOH W=lAml/mm EtOH wat er Z=9.0m//min Exp. hr Z>=2.5m//min AcOH 0^6.4m//min EtOH water Z=4.6 m//min Z>=0.9 m//min W=3J ml/mm Run ll Zf=13.0 m//min AcOH Upper feed Z&=0.0 m//min EtOH plate operation ^==10.6 m//min water D=2.4m//min R=2.1 Run 12 Z/=0.0, m//min AcOH Lower feed Z6=12.5 m//min EtOH plate operation W^IO.5 m//min water D=2.0m//min i?=7.1 Run 14 Z/=11.4 m//min AcOH Two feed Z6=8.21 m//min EtOH plates operation W=\1.0mlmm water D=2.6m//min 7?=5.8 Run 15 Zf=3Am//min AcOH Two feed plates Zb=3Jm//min EtOH operation J^=6.5 m//min water Z)=0.6m//min Run 16 Total reflux operation Table 1 Experimental results for the flash reacting distillation Comp. Zi Xi yt xt yt xt yt xt yt hr 3.0hr hr 2.0 hr hr 3.0hr water Compositions are expressed as mole fraction. 4.5 hr hr hr 6.0 hr O.349 _ Table 2 Experimental results for the multi-plates reacting distillation Xj X± X2 XS Xi X5 XQ X7 Xd Zf Zb 7?=3.1 L=15.5m//min AcOH EtOH water Compositions are expressed as mole fraction given, Eqs. (4) through (8) may be used to calculate liquid compositions at the next trial (n+l-th) by use of the vapor liquid equilibrium relation and the reaction rate equation as shown in Table 33>4). The thermal condition of the feed, feed compositions and all flow rate are given, then calculations may be initiated by using feed compositions as the initial values of the liquid compositions for all stages of the column. 3. A Correction Method for Liquid Compositions between n-th Trial and w+l-th Trial Even if convergence is obtained, it might converge 202 to an interrest result. In order to avoid this, the following correction method has been proposed. nmis defined as the iteration number necessary to reach steady state, if 6mrepresents the time necessary to reach steady state and 6A is a time increment, nm may be evaluated as dm\ad. It may be better that 0m be selected as five times the residence time which is the total holdup divided by total feed rate {ZH\ZZ\ xxmaji is defined as the liquid composition which is evaluated from material balances as formulated by Eq. (9) through Eq. (13). For the flash distillation, xl^ izzt+nlt -Dyfil W (9) JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

4 For the reacting multi-stage distillation, at the bottom step, xl^imuzz.+ Zn^-Dytd/W (10) At middle steps, xla,5i={v(yu,i-yl)-lxu,i+nhi}il (ll) At the upper feed plate, xl.i/i={k(y;_1 <-^?<)-Lx«+Z/z/i+/i«/i}/L (12) At the top, xla^iyyu^+nlu-dy^ll (13) Then corrected values of liquid compositions (x?+)i) are represented as follows, :Cii =*?#i{^/(»^ (14) where x^ji denotes calculated liquid compositions from Eq. (4) through Eq. (8). Thus values of variables resulting from Eq. (4) through Eq. (8) may be controlling at the beginning of the trials, but values of the variables resulting from the hypothetical steady state by Eq. (9) through Eq. (14) will govern as trials progress. The values in the left hand of Eq. (14) may be used as liquid compositions for the next trial during unsteady state. 4. Computational Procedure The computational procedure is as follows : Step 1 Determine reflux ratio (R), feed rate (Z), distillate rate (Z>), waste rate (W) and feed compositions (Zi). Step 2 Specify the time increment {Ad), liquid hold-up of each stage (Hd), necessary time to reach the steady state (0m), relaxation factor (a) and convergence criterion (s). Step 3 Calculate the vapor flow rate (V3) and the liquid flow rate (L3). Step 4 Use the overall feed compositions as the initial liquid compositions in each stage. Step 5 Evaluate vapor compositions (yjt) and reacted moles (n%jt) by use of the vapor-liquid equilibrium relation accompanied with a chemical reaction and the reaction rate equation. Step 6 Solve basic equations Eq.(4) through Eq. (8) and normalize liquid compositions. If the values of compositions are negative, let them be equal to zero, then normalize them. Step 7 Compare liquid compositions calculated at step 6 with those at a previous trial. If the difference between them is less than a convergence criterion, stop, if it is not satisfied, proceed to the succeeding step. Step 8 Correct liquid compositions by Eq. (14) according to Eq. (9) through Eq. (13). Replace x% by x*+)t9 then repeat step 5 through VOL. 10 NO Table 3 Vapor-liquid rate (r) equilibrium ratio (Kt) and reaction Values of Kt may be evaluated by following equations log Ki =AilT(0K) -\-Bi for i-th component except / EtOH (2) water (3) (4) At -2.3X X xlO3 AcOH Bi=ClXk(l -Xif+C2xk(l -xt)+cbxk +C4(l -^)3+C5(l -x,)2+c6(l -^)+^0 i-j-k C± C2 C3 C4 C5 C6 B i 0 B2 ethanol Use the liquid composition at the hypothetical conversion 0. Bs water _4_ Bi ethylacetate _3_ Use the liquid composition at the hypothetical conversion 1. For acetic acid (K±) #!=().022x/[ q-l.667 Reaction rate may be evaluated by the following equation4) r=4.76xlo^dcw.63 x10-4c3c4 where Ct denote the molarity of /-component (mol//) Fig. 5 Calculated results without the correction method for Run 6 step 8 until convergence is obtained. 5. Calculated Results In order to confirm the utility of this method, comparisons of calculated results with experimental data were made. Calculated results without the correction method of Eq. (14) for Run 6 of the flash reacting distillation experiment are shown in Fig. 5 as a typical example. In this case, the relaxation factor (Jdrj/H) is assumed to be about 2xlO~4, where 7] denotes efficiency. Although convergence has been obtained, it seem to be uncorrect, because the difference between experimental data and calculated results is too large. In Fig. 6, results corrected by Eq. (14) for Run 6 is shown, there is a good agreement between experimental data and calculated results in this case. It has been recognized that the behaviours of the changes of the system from the start-up to the final steady state may be simulated. 203

5 Fig. 6 Calculated results by use of the correction method for Run 6 Fig. 7 Calculated results by use of the correction method for Run ll of the multi-stage reacting distillation Table 4 Calculated results for Run 6 of the reacting flash distillation Compositions [mol. fract.] Trialno. AcOH EtOH Water 0 x y x y x y x y x y x y Relaxation factor (Adr]IH) = min/mol Since the reacting-distillation with multi-stage is analogous to reacting flash distillation except being one stage operation, essentially this calculation procedure can be applied to reacting multi-stage distillation. Calculations for experimental data of the multistage operation have been carried out, and comparison of calculated results with experimental data for Run ll by the one upper feed plate operation is shown in Fig. 7 as a typical example. Good agreement was obtained. It has been confirmed that the correction method by Eq. (14) is available for the reacting distillation and that the calculated results converge to the actual value. As typical examples, calculated results for Run 6 of the reacting flash distillation and for Run ll of the multi-stage reacting distillation are shownin Tables 4 and 5. Conclusi ons A new method for correcting liquid compositions in the relaxation method has been developed. This method has depended on the normalization between values of variables calculated at «-th trial and those resulting from the material balance at the hypothetical steady state; it is expressed by Eq. (14). The relaxation method is useful in calculating reacting distillation problems by means of this correction method. It may be available for solving distillation problems in the absence of chemical reactions. Acknowledgment The author is grateful to professors, Dr. M. Hirata (Tokyo Metropolitan Univ.), Dr. I. Yamada(Nagoya Inst. of Tech.) and Dr. C. D. Holland (Texas A & MUniv.), for useful suggestions. Nomenclature D H = distillate =holdup rate [moles] L nr = overflow = reacted rate moles nm = trial number necessary to reach steady state R = reflux ratio V = vapor rate Table 5 Calculated results for Run ll of the reacting distillation I Xi X2 X3 X4 X6 Xq X7 Xd Attrialno.=120 AcOH EtOH water Attrialno.=210 AcOH EtOH water Feed compositions are used as intitial values as follow, *acoh=0.2559, zetoh=0.6159, zwater=0.0743, z= JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

6 Ad V waste rate liquid compositions vapor compositions feed rate feed compositions time increment effici ency convergencecriterion <Subscripts> ca = calculated values CO = corrected values i =component j = plate number [mini ma = material balance at the hypothetical steady state Literature Cited 1) Holland, C. D. : "Multicomponent Distillation", Prentice- Hall, Inc., Engelwood Cliffs, New Jersey (1966). 2) Ishikawa, T. and M. Hirata: /. Chem. Eng. Japan, 5, 125 (1972). 3) Komatsu, H. and M. Hirata: Kagaku Kogaku, 30, 989 (1966). 4) Smith, J. M. : "Chemical Engineering Hill Book Co. Inc., New York (1956). Kinetics", McGraw- (Presented in of Chem. Engrs., part at Japan, the 39th Annual Meeting at Kobe, April 1974.) of The Soc. VOL. 10 NO