Parameter Setting Method for Control System of Cryogenic Distillation Column

Transcription

1 Journal of Nuclear Science and Technology ISSN: (Print) (Online) Journal homepage: Parameter Setting Method for Control System of Cryogenic Distillation Column Masahiro KINOSHITA & Yuji NARUSE To cite this article: Masahiro KINOSHITA & Yuji NARUSE (1981) Parameter Setting Method for Control System of Cryogenic Distillation Column, Journal of Nuclear Science and Technology, 18:8, , DOI: / To link to this article: Published online: 15 Mar Submit your article to this journal Article views: 71 View related articles Citing articles: 7 View citing articles Full Terms & Conditions of access and use can be found at Download by: [ ] Date: 30 November 2017, At: 21:03

2 Journal of NUCLEAR SCIENCE and TECHNOLOGY, 18C81, pp (August 1981). 595 Parameter Setting Method far Control System of Cryogenic Distillatwn Column Masahiro KINOSHITA and Yuji NARUSE Division of Thermonuclear Fusion Research, Japan Atomic Energy Research Institztte* Received July 23, 1980 Revised April 21, 198 I A computer code is developed for analysis of dynamic behavior of a single cryogenic distillation column. A dynamic simulation study is performed for the lead column in a typical column cascade which consists of four columns and two catalytic equilibrators. The atom fraction of protium in the bottom product and the reflux ratio are regarded as the controlled variable and the manipulated variable, respectively. Fluctuation or change of the feed composition is considered as the disturbance and the PI control scheme is introduced for a constant value control. The criteria are derived, which can be used in setting the choice of the two PI controller parameters. Investigated are the effects of the controller parameters on the response of the controlled variable to a step change in the feed composition, and the validity of the criteria is verified. The parameter setting method proposed in the present study has two advantages : it accounts for the strong nonlinearity of the column and it predicts the unstable region which must be avoided in the actual operation. KEYWORDS: hydrogen isotope separation, ergogenic distillation, dynamic characteristics, constant value control, feedback control, reflux ratio, PI control, parameter setting, proportional sensitivity, integral time, instability, step response, computer codes I. INTRODUCTION Cryogenic distillation is the most promising candidate for hydrogen isotope separation in the fuel circulation system for fusion reactors. Kinoshita et al.(l) performed a detailed analysis of steady state separation characteristics of the four cryogenic distillation columns in the column cascade proposed by Sherman et a1.(*) Design and operating conditions of each column determined by Kinoshita et al. are given in Table 1 (parametric surveys were not made for the operating pressure). The overall material balances in the column cascade are summarized in Fig. 1. Grasp of dynamic behavior of the column is indispensable in design of the control system. Davis et al.'') carried out a dynamic simulation study for a lead column in a column czscade, the configuration and specifications of which are quite different from those of the cascade shown in Fig. 1. Their study provided us with much useful information on dynamic behavior of a single column. However, the PI (proportional integral) controller parameters were not determined systematically, and a column with only 30 theoretical stages was treated to avoid long computational time. In the present study, a further analysis on dynamics and control of a single column is performed for Column(1) in Fig. 1. This column has 70 theoretical stages and more * Tokai-mura. Ibaraki-ken

3 596 ]. Nucl. Sci. Techno!., Table 1 Design and operating conditions established by computer analysism Column (1) (2) (3) (4) Flow rate of feed stream (g~mol/h) Temperature of feed stream (K) Composition of feed stream H X 10-s x lo-s x x10-7 (Mole fraction of each HD xl X IQ X IQ xl0-3 component) HT X X IQ X 10-s x1o-s X IQ DT X IQ X 10-1 T X X 10-4 Number of total theoretical stages Feed stage number Flow rate of top product (g-mol/h) Ref! ux ratio Operating pressure (atm) Condenser Total condenser Total condenser Total condenser Total condenser (I) Composition of feed stream is equilibrium composition at 25oC except Column (3). (2) Feed stream is. supplied in saturated liquid state. (3) Neither pressure drop nor decay heat of tritium is taken into account. (4) Improvements should be made for Column (3), such as increase of reflux ratio or refrigeration of stripping section, because performance is expected to be lowered due to decay heat of tritium. Raw Fuel Input loog-mol/hr 45g-mol/hr H=0.4 atom% D=99.5 atom% Neutral Beam Injector Recycle Equilibrator 2.3g-mol/hr HD (, H2) T=28.6 mci/hr 66.Sg-mol/hr D2 (, HD) D= atom% H=l atom% D=49.5 atom% T=49.5 atom% 75g-mol/hr 51. 7g-mol/hr DT (, T2, D2) H=O.l698 X l0-4 atom% D=Sl.l9 atom% T=48.81 atom9; 24. Sg -mol/in T2 (, DT) T=99.21 atom 9 i Fig. 1 Overall material balances in cryogenic distillation column cascadecn -34-

4 Vol. 18, No. 8 (Aug. 1981) 597 practical analysis can be made because most of recent estimates<ljc<j-caj for number of total theoretical stages are in the range of 70"'100. The atom fraction of protium in the bottom product from Column (1) is regarded as the controlled variable, and a constant value control scheme with a PI controller is considered. There are several conventional methods for tuning PI controllers< 9 JooJ but all of them can be applied only to linear processes. Besides, most of them use the first order plus a dead time approximation to the transfer function. Unfortunately, it is well known that distillation columns have strong nonlinearity and the first order plus a dead time approximation does not always provide a good result. The principal objective of the present study is to propose an estimation procedure of proper combination of the PI controller parameters which accounts for the strong nonlinearity. This procedure has one more advantage that it allows us to predict the unstable region which must be avoided in the actual operation. ll. CALCULATIONAL MODEL The model column for mathematical simulation is shown in Fig. 2. assumed to be composed of N theoretical stages. The first stage is the condenser and the N-th stage is the reboiler. Incorporation of the differences of latent heat of vaporization among the six molecular species (n-h 2, HD, HT, n-0 2, DT and n-t 2 ) (Factor 1) and decay heat of tritium (Factor 2) into the simulation model results in more complicated simulation procedure. If these two factors do not have significant effects on column behavior, the usual assumption of equal molal heats and equal molal overflows within the column is more favorable to avoid unnecessarily detailed analysis and long computational time. In the present study, as a help to investigation of the extent of the effects of the two factors, steady state calculations are made for Column (1), The column is ( Condenser) v, u, -Vapor -Liquid --- Vapor or Liquid or Vapor - Liquid Mixture Liquid or Vapor- Liquid Mixture Fig. 2 Model column for mathematical simulation using the following three models (Raoult's law is assumed): Model 1 Both Factors 1 and 2 are neglected. Model 2- Only Factor 1 is incorporated. Model 3 Both Factors 1 and 2 are incorporated. Liquid holdup in each theoretical stage is estimated employing the experimental data reported by Bartlit et al. <?J (the result is 1.3 g-mol/stage). Comparison among the three models is made in Table 2. Although both liquid and vapor flows decrease due to the effects of the two factors and the column performance is lowered to some extent, the differences are not significant (the atom fractions of protium in the bottom product XH N in these models are x 10-6, x 10-6 and x 10-6, respectively). Therefore, it is concluded that the assumption of equal molal heats and equal molal overflows within the column is acceptable for Column (1) to analyze its static behavior. In the present study, it is postulated that the assumption of equal molal heats and -35-

5 598 ]. Nucl. Sci. Technol., equal molal overflows within the column can be also applied to analysis of dynamic behavior of a single column. The effects of the two factors on dynamic characteristics of the column are to be investigated in further studies. However, the assumption is not expected to alter the qualitative conclusions of the present study<s). Under this assumption, flow rates of vapors and liquids are calculated from the following equations when the values of R, B, Fi and qi are given: where L1=R(V1+U1) Li=Li-1+Fiqi-Ui (j=2,..., N-1) j LN=B ( 1) V2=V1+U1+L1 1 Vi= Vi-1-F/1-qi)+Wi (j=3,..,n), Table 2 Case 1 Case 2 Case 3 Examination of calculational model Composition of bottom product (Mole fraction of each component) H2: x10-18 HD: X 10-9 HT: o x w- D2: x1o- 1 DT: T2: H2: xl0-18 HD: X 10-9 HT: X 10-6 D2: o x w- 1 DT: T2: H2: x10-18 HD: o.1967 x w- 9 HT: X 10-6 D2: x10-1 DT: T2: Flow rate of vapor stream (g-mol/h) V2=650.0 Vs 5 =650.0 V70 =650.0 V2=650.0 Vs5=618.4 V70 =590.3 V2=650.0 Vss=606.0 V7o=516.3 Vi denotes flow rate of the vapor stream leaving j-th stage. B: Prescribed value of bottom product flow rate (g-mol/h) Fi: Flow rate of feed stream supplied to j-th stage (g-mol/h) Li: Flow rate of liquid stream leaving j-th stage (g-mol/h) N: Number of total theoretical stages q i : Ratio of flow rate of liquid feed stream to flow rate of total feed stream defined for j-th stage R Reflux ratio U i : Flow rate of liquid sidestream from j-th stage (g-mol/h) Vi: Flow rate of vapor stream leaving j-th stage (g-mol/h) W i : Flow rate of vapor sidestream from j-th stage (g-mol/h). Further simplifying assumptions are made: vapor holdup is negligible in comparison with liquid holdup and liquid holdup remains constant along the time trajectory. In addition, Raoult's law is assumed in the present study because there is no sufficient data on the deviation from Raoult's law of hydrogen isotope solutions. The qualitative aspects of the present study can be updated quite readily once the experimental data are available. As a consequence, the basic equations are composed of the following simultaneous ordinary differential equations : l HL 1(dx;, 1/ dt)= V 2y;, 2 - V 1y;, 1-(L1+U1)X;, 1 (condenser) HLi(d x;. if dt)= Li-1xi. i-1 + vi+1yi. i+1-(li+u i)xi.i -(Vi+Wi)Yi.i+Fizi.i (j=2,..., N-1) HLN(dx;, N!dt)=LN-1xi, N-1- LNxi, N-V NYi.N+FNzi. N j (reboiler), (i = 1,.., m), ( 2) -36-

6 Vol. 18, No. 8 (Aug. 1981) 599 y. z,3.=ki,i.x;. z.3-pp(tj)xil _-- jlpj (j=l,... N; i=l, *..? m), where H L ~ : Liquid holdup in j-th stage (g-mol) Ki, j : m: P,"(Tj) : Pj : Tj : xi, j : Yi, j : zi, j : Vapor-liquid equilibrium ratio of component i at TjK Number of components Vapor pressure of pure component i at TjK (atm) Total pressure on j-th stage (atm) Absolute temperature on j-th stage (K) Liquid mole fraction of component i on j-th stage Vapor mole fraction of component i on j-th stage Mole fraction of component i in feed stream supplied to j-th stage. The simultaneous ordinary differential equations are numerically solved by use of Improved Euler Method in the computer code CRYDIS-4 developed for the present study. Vapor pressure is computed employing the data reported by Mittelhauser et al."') As an example, APPENDIX 1 describes the calculational procedure of the response of the composition distribution within the column to a step change in the feed composition. m. DYNAMICS AND PI CONTROL 1. Response of Controlled Variable to Step Change in Feed Composition under No Control Conditions for the fusion reaction in the plasma require the protium concentration of the fuel charge to be below a certain level. Therefore, it is important to control the protium concentration in the D-T stream which is transferred to the fuel storage for refueling. As is described in our previous report"), almost 100% of protium contained in the bottom product from Column(1) is recovered to the top side of Column(3), and the protium concentration in the bottom product from Column (4) is extremely low. Therefore, the protium concentration in the stream which is transferred to the fuel storage is essentially determined by XHN (the atom fraction of protium in the bottom product of Column(1)). In the present study, a constant value control scheme of XHN (21.0 x lo-') is considered, and the reflux ratio is chosen as the manipulated variable. The most important disturbances to be considered are fluctuations or changes of the feed conditions : the feed stream flow rate, feed temperature and feed composition. Fluctuation of the feed stream flow rate can be avoided by flow rate control. It was shown by Davis et al.'" that the feed temperature has negligible effects on column behavior. It is concluded, therefore, that the most important and unavoidable disturbance is fluctuation or change of the feed composition. The response of AXHN (=X,,(t) -XHN(0), where t denotes time from the initial upset) to a step change in the feed composition is firstly calculated under no control. The initial feed composition is given in Table 1. The upset feed composition assumed in the present study is summarized below. Upset feed composition (Mole fraction of each component) Hz : x HD : x lo-' HT : x lo-' Dz: DT: Tz: (3) (Atom fraction of protium=o.o2) Both the initial feed composition and the upset feed composition are assumed to be the equilibrium composition at 25 C established by the following isotopic exchange reactions :

7 600 J. Nucl. Sci. Technol., Hz+Tz 2HT (K,,=2.57) DzfTz 1_ 2DT (Ke,=3.82) (4) where K,, denotes the equilibrium constant. In the calculation, the liquid holdups in the condenser and in the reboiler are assumed to be five times and ten times larger than that in one theoretical stage, respectively. This postulation is quite uncertain, but it was shown by Davis et that its effects on column behavior are negligible. The calculated response line is shown in Fig. 3. The column response time (the time for AXHN to reach 98% of the steady state value x after the initial upset) is 8.54 h. This long response time eliminates need for using very complicated feedforward control. Therefore, a negative feedback control scheme illustrated in Fig. 4 is introduced with selection of a PI action for the controller, and the controlled variable XBN is assumed to be measured without time lag. Time from Initial Upset ( hr 1 Fig. 3 Response of controlled variable to step change in feed composition under no control 3 XHN : Controlled variable (atom fraction of protium in bottom product) R: Manipulated variable (reflux ratio) D: Disturbance (change in feed composition) E: Reference value (=O) G: Transfer function c: Controller (PI controller) b: Process (cryogenic distillation column) d: Disturbance Fig. 4 Block diagram of constant value control scheme 2. Transfer Function of Column and Criteria for Controller Parameter Setting Since the cryogenic distillation column is a nonlinear process, it is very difficult to find out the exact expression of the model transfer function. In the present study, several calculations which are helpful to proposal of a parameter setting method are made for the purpose of obtaining specific information on the transfer function. Responses of AXH, to a step change in the reflux ratio are calculated for the four cases : AR=l, 2,3 and 5 (AR=R(t)- (O), t>o). It is assumed that liquid holdup remains constant and as a result the flow rates in the column change instantaneously with a reflux ratio upset. The relation between t and AX,, in the case of AR=5 is shown in Fig. 5 with closed triangles, as an example. It is clear from Fig. 5 that the column is not a first order lag process because the response curve has an inflection point, and the dead time is negligible. Several forms of the transfer function (the second order lag, the third order lag and the oscillatory third order lag) are assumed and optimum values of their

8 Vol. 18, No. 8 (Aug. 1981) - parameters are searched by use of I Simplex Method(13), one of the non linear programing techniques. As a consequence, it is revealed that the second order lag approximation can provide the most satisfactory fitting to the step responses calculated for the four cases. The optimum values of the parameters in the transfer function are given in Table 3. These values are given as functions of AR because of the non- Fig. 5 linearity of the distillation column. According to Table 3, the step re sponse in the case of AR=5 is expressed by Time from Initial Upset ( hr ) Response of controlled variable to step change in reflux ratio (manipulated variable) Table 3 Optimum values of parameters in transfer function AR M(10-') Ti(h) Tz(h) Tm(h) -Ax,dt) =5.19~10-'[1+ I exp (--t/0.0424) exp (- t/0.0685)} /0.0261]. ( 5 ) The line is drawn in Fig. 5; it fits the Transfer function Gp(s)= -M/(l+T~s)/(l+Tzs) Mean delay time Tm= TI+ Tz closed triangles fairly well. Since the controller has the proportional integral action (Gc(s) = -K(l+l/T,/s), where K denotes the proportional sensitivity and T, denotes the integral time) and the process transfer function can be approximately expressed by the second order lag system (G, (s) = - M/(1+ Tls) /(l+tzs), where M denotes the static gain in the transfer function), the open loop transfer function G(s) is expressed by G(s)=Gc(s)G,(s)=KM(l+l/T,/s)/(l+Tis)/(l+ Tzs). (6) The lower limit value of the integral time Tzc to avoid instability can be derived from the following manip~lation(~)(~~). From Eq. (6), the characteristic equation (l+g(s) =O) is expressed by the following cubic equation : T,T~s~+(T,+Tz)s~+(l+~M)s+KM/T,=o. (7) Substitution of s=]wc (j is an imaginary unit and oc denotes the critical value of the angular frequency) into Eq. (7) yields -T,Tz&+~c(l+KA4)=0, -(T~+T~)w$+KM/T,~=O. (8) Elimination of wc yields the following equation for calculation of TzC from M, TI, Tz and K : Tic=KMTiTz/(l+KM)I(Ti+Tz). (9) Since the integral time T, is required to be larger than Ttc to avoid the unstable controlcg), the first criterion for controller parameter setting is Tz> I(MTiTz/(l+KM)/(Ti+T,). (10) Attention is now focused on the gain margin of the open loop transfer function to

9 602 J. Nucl. Sci. Technol., derive the second criterion. The gain margin is a measure to predict the extent of the stability of the control. The gain and the phase of the frequency transfer function G(jo) are easily derived from Eq. (6): The gain margin Y(dB) is calculated from the following two equations: According to the classical control theory, the favorable value of Y lies in the range of 3-10dB.'') This can be used as the second criterion. If the values of T1, T, and M are given, proper combination of the two controller parameters can be estimated by use of the two criteria. Because of the nonlinearity of the column, the values of T1, T, and M cannot be determined uniquely, and the response characteristics are dependent on various variables. Hence, the parameter setting method described before does not always provide the optimum choice of the controller parameters, but the instability can be avoided and the favorable control is expected to be obtained by use of this method. 3. Response of Controlled Variable to Step Change in Feed Composition under PI Control In this section, the validity of the two criteria derived in the previous section is verified. The integral of the absolute value of the error is frequently used as a criterion of control performan~e('~) : A=[mIAX,N(t)Idt. -0 (15) In the present case, one of important objectives is to keep a net protium level closer to the setpoint valcle (dxhn=o), and the value of A' defined by A' = 1-t AX,,"( t )d t (16) should be rather considered to be a measure of control performance. That is, the oscillations to go below the setpoint value are rather favorable as long as the control is sufficiently stable, because they are damped out by the fuel storage before refueling. It should be noted, however, that the control operation near the unstable region must be avoided. Parameter settings should be made from these considerations. The order of magnitude of K can be estimated from the following equation according to which the reflux ratio is manipulated : A~(t)=~AX,,(t)+~;S~dx,N(t)dt. K t (17)

10 Vol. 18, No. 8 (Aug. 1981) 603 Since the initial value of the controlled variable is approximately lo-' and the order of magnitude of AR(t) is loo, the order of magnitude of AXHN@) and KAXHN(t) should be lo-' and loo, respectively. Therefore, the value of K is around 1.Ox10'. Figure 6 is prepared for tuning PI controllers. The solid lines indicate the lower limits of Ti after K is properly established. Choice of the value of Ti near the lower limit must be avoided to assure long-term safe operation. However, if Ti is too large, the response takes on the characteristics of the proportional-only control which means the loss of significance of the PI control scheme. The broken lines indicate the assumed upper limits of Ti. In Fig. 6, the values which give 20 db to Y are regarded as the upper limits. When K is determined to be 1.0~10~, the best Ti setting seems to be around 0.025h. Although this value exceeds the upper limit in the case of AR=5, assurance of stable operation is regarded as the more important matter in the present study. The two PI controller parameters are determined as follows : Proportional Sensitivity (io*-) Fig. 6 Diagram for setting PI controller parameters K=l.Ox lo', Tt=0.025 h, (Case 1), with the result that Y varies from 3 to 25 db (Fig. 7). zlot/ v > 5 '0 3' lo0o AR (-1 R: Reflux ratio (manipulated variable) Y Gain margin K Proportional sensitivity Ti: Integral time Fig. 7 Gain margin plotted as function of AR? For comparison, two further cases are assumed as follows : K=l.Ox lo8, Ti=0.008 h, (Case 2), K=l.Ox lo8, Ti=0.05 h, (Case 3). The response of AX,, to a step change in the feed composition is calculated under PI control for each case (the calculational procedure of the step response under PI control is briefly summarized in APPENDIX 2). The series of figures (Figs. 8(a)--(c)) show the calculated response lines. The following features are observed from the figures. (1) In Case 1, the response is an oscillatory line which is damped with the increase in time from the initial upset. The overshoot is small enough because the maximum

11 604 Technol., 3, I Time from Initial Upset (hr) (a) PI control (1) I I I I I I Time from Initial Upset (hr ) (b) PI control (2) I I Time from Initial Upset ( hr ) Fig. 8(a)-(c) (c) PI control (3) Response of controlled variable to step change in feed composition under PI controls (1)-(3)

12 Vol. 18, No. 8 (Aug. 1981) 605 error IAXHNImax is only 20% of XHN(0). A successful constant value control is essentially obtained. (2) In Case 2, the control is unstable and AX,, diverges with the oscillations. (3) In Case 3, the response takes on the proportional-only control because of too large Ti. To obtain more oscillatory behavior which results in a net protium level closer to the setpoint, a value of Ti smaller than h is more favorable. However, the choice of controller parameters should be also predicated on keeping the operation far enough from the unstable region. In rough estimation, the response is expected to become near a sustained oscillation with TiZO.Ol h. It appears that the most favorable value of Ti lies in the range of h. It can be concluded that the parameter setting of Case 1 leads to very good control and the criteria derived in the previous section can be used in tuning PI controllers. IV. CONCLUSIONS (1) A computer code CRYDIS-4 is developed for analysis of dynamic characteristics of a single cryogenic distillation column for hydrogen isotope separation. By use of this code, the response of the composition distribution within the column to a step change in the feed composition can be calculated under either no control or PI control (this code can be also applied to a start-up problem of a column). (2) A dynamic simulation study is carried out for Column(1) in the column cascade shown in Fig. 1. The atom fraction of protium in the bottom product and the reflux ratio are regarded as the controlled variable and the manipulated variable, respectively. Fluctuation or change of the feed composition is considered as the disturbance. (3) The criteria which can be used in tuning the two PI controller parameters are derived. That is, a parameter setting method for PI control system is developed, The most important feature of this method is that it accounts for the strong nonlinearity of the column. This method has one more advantage that it allows us to predict the unstable region which must be avoided in the actual operation. Although the proposed parameter setting method is applied to a specific control scheme of a specific column in the present study, it is expected that parameter settings can be made for various PI control schemes of various columns, in a similar manner. ACKNOWLEDGMENT The authors wish to give their sincere thanks to Dr. Y. Obata for his continuous encouragements. Acknowledgment is also due to Dr. K. Tanaka and Mr. Y. Matsuda for their useful comments. -REFERENCES- (1) KINOSHITA, M., et al.: J. Nucl. Sci. Technol., 18:7], 525 (1981). (2) BARTLIT, J.R., SHERMAN, R.H., STUTZ, R.A., DENTON, W.H.: Cryogenics, 19, 275 (1979). (3) DAVIS, J. F., et al. : Nucl. Technol., 46, 149 (1979). (4) WILKES, W. R. : CONF , p. IV-266 (1976). (5) STACEY, W. M., et al. : ANL/FPP-77-2, (1977). (6) WILKES, W. R. : MLM-2502, p (1978). (7) BARTLIT, J.R., DENTON, W.H., SHERMAN, R.H.: LA-UR , (1978). (8) SHERMAN, R. H. : Los Alamos National Lab., Private communication, (1981). (9) IINOYA, K., HOTTA, K. : Purosesuseigyo no Kiso, (in Japanese), (1967), Asakura Syoten. (I@ COUGHANOWR, D. R., KOPPEL, L. B. : LProcess Systems Analysis and Control, (1965), McGraw

13 N d. Sci. Technol., Hill Book, New York. U1) MITTELHAUSER, H. M., THODOS, G. : CYyOgQniCS, 4, 368 (1964). U2) GUIZOUARN, L. : CEA-BIB-87, (1967). 03) NELDER, J.A., MEAD, R.: Comput. J., 7, 308 (1965). 14) HOTTA, K. : Purosesudotokusei, (in Japanese), (1975), Baihukan. [APPENDIX] 1. Calculational Procedure of Response of Composition Distribution within Column to Step Change in Feed Composition under No Control The independent variables in the simultaneous ordinary differential equations are the liquid mole fractions and absolute temperatures. If the initial conditions (the initial values of the independent variables: $1, and T, ) and the upset feed composition 21, are given, the transient analysis can be performed according to the following procedure. (1) Calculate Lj (j=1,..., N-1) and Vj (j=2,..., N) from Eq. ( 1). In the present study, Vl=O because the condenser is the total one, and qj=l because the feed is supplied in liquid state. (2) Calculate the liquid mole fraction after the time increment At, from the following equation : The other two variables m xi,, j=x!. j+at {Lj-lx!, j-1+vjtlp2p(tjtl)x!. jtllpjtl -(Lj+Uj)x!, j-(vj+wj)pp(tj)x!. jipj+fjzi, jl /HLj 2 and (3) Solve,Zpp(Tjr)xtt,j/Pj=l for Tj.. I=1 (4) Improve the value of by the following equation: (j=2,..., N-1 ; i=l,..., m). can be calculated in a similar manner. xt. j=x1. j+ At { L j-l(xp,,-i+ xi,, j-l)-( L j+uj)(xz, j+ xi,, j) -(vj+wj)p; (T?)xP, j/pj-( vj+wj)p; (TJ,)xf,, j/pj + Vjtl(PP(Tjtl)x!. jtl+pi(tt,, jtl)xl,. jtdipjtls2fjzi.jj /2/HLj 9 (j=2,..., N-1; i=i,..., m), where x!,~ denotes the improved value. The other two variables calculated in a similar manner. m (5) Solve,Xfi2p(Tj)~t,~/P~=l for Tj. %=I (6) Calculate J which is defined by the following equation: (Al) (A2) and xi, N can be (7) If J exceeds Nms (O<E<<~), return to Step (2) after substituting the values of xi., and Tj into 21, and TJ, respectively. If J does not exceed Nms, the calculation is terminated because the steady state has been reached. 2. Calculational Procedure of Response of Composition Distribution within Column to Step Change in Feed Composition under PI Control

14 Vol. 18, NO. 8 (Aug. 1981) 607 The reflux ratio R is manipulated according to the following equations: By using the approximation 5 :+ AX, N(t)dt2I {AX, N(t + At) + AXU N(t)} At/2, (A5) the value of R at the time ndt from the initial upset is numerically calculated from K n R( At)= R(0) + KAX, N( TZ At)+ - Z: {AX, N( k At)+ AX, N( k At- At)} At/2. (A6) Ti &.=I The response line under PI control is calculated according to the procedure summarized below. (1) Read the initial conditions. (2) Calculate Lj (j=l,..., N-1) and V, (j=2,..., N) from Eq. ( 1 ). (3) Calculate the composition distribution and the temperature distribution within the column after the time increment At by following the procedure described in APPENDIX 1. (4) Calculate the controlled variable XHN and determine the value of the manipulated variable R for the next time step from Eq. (A6). (5) Calculate J which is defined by s= I x,n(t)-xhn(t-at)i lxun(t>. (A7) If J exceeds E (O<E<<~), return to Step (2). If J does not exceed E, the calculation is terminated because the steady state has been reached