MuIticomponent Dist iIIation Column Design: A Semi-rigorous Approach S.N. Maiti, S. Ganguly‘, A.K. Das’ and D.N. Saraf Department of Chemical Engineering Indian lnstitufe of Technolorn Kanpur 208 076, INDIA A new semi-rigorous approach for the design and simulation of multicomponent distillation columns is presented which combines a rigorous Naphtali-Sandholm algorithm with Edmister’s shortcut method. The column is assumed to comprise of single and multi-tray units with the number of trays inside the multi-tray unit treated as an iterative variable. The assumption of linear variation of flows inside a multi-tray unit made by earlier workers (which often lead to convergence d.iBculties or erroneous results) has been replaced with a more rigorous calculation procedure. Examples are presented to demonstrate the usefilness of the new method for both design and simulation calculations. The semi-rigorous model equations can be solved directly to obtain a reavonubly accurate and practical design, whereas most other available methodr require repeated calculations. Computer memory requirement is drastically reduced in this semirigorousformulation as compared to rigorous methods. Introduction Multistage distillation methods are readily available in the journal literature and will not be considered here. The two basic approaches are a rigorous design method and a column rating procedure. Naphtali and Sandholm [ 11 used stagewise grouping of the distillation equations with subsequent linearization to give the resulting set of equations a bIock mdiagonal structure, permitting solution by a simple technique. Ohmura and Kasahara [2] reported a semi-tray-by-trayapproach to solve simulation problems. It combined the tray-by-tray procedure of Tomich [3] with the shortcut method of Edmister 141, using an effective stripping factor for a group of trays which were considered as a single unit. The method of Ohmura and Kasahara [2] had several shortcomings. There are a large number of dependent variables which lead to the difficulty of modulation inside the two loops, especially for large problems. A number of simplifying approximations were made, which may not always be justified. Linearity of flow profiles inside a multi-tray unit is often objectionable. In this algorithm, there is no provision for inclusion of nonideal thermodynamics and plate efficiencies. Most of the shortcomings of Ohmura and Kasahara‘s work have subsequently been removed by Ganguly [51; Ganguly et al., [61 and Maiti [7l. Based on the approach of Ohmura and Kasahara, a semi-rigorous algorithm has been developed in this study which includes the my-by-my model of Naphtali and Sandholm and the shortcut model of Edmister. A more rigorous calculation I. Presently at Engineers India Ltd. (R & D), Gurgaon, India. 2. Presently at Indian Oil Corporation (R & D). Faridabad, India. *Authorfor correspondence. 37 SN.Maiti, S. Ganguly, A.K. Das and DN. Sarrrf p e d u r e replaces the assumed linearity offlow profiles inside the multi-tray units. Provision has been made for use of nonideal thermodynamics and stage efficiencies. The concept of incorporating the number of trays inside a multi-tray unit as an independent variable was exploited to convert the original simulation method into an effective design tool. Development of the Model The general procedure of solving multicomponent, multistage separation problems is based on the solution of the MESH (Mass,Equilibrium, Summation and Enthalpy) equations, and using appropriate convergence techniques. The use of a NewtonRaphson type convergence method requires selection and ordering of the unknown variables and the corresponding functions (MESH equations). Figure 1 shows a general equilibrium stage. The corresponding MESH equations in the NaphtaliSandholm formulation are given in Table 1 for a column with N equilibrium stages and C components. * i V.i+1, j 'i, j Figure 1. Schematic representation of a general single tray unit. Table 1. The MESH Equations 1. Component materid balmcc: Mi,= (l+S,L)lij+(l+S~)~;j-li-,j-ui+~j-f;j=~; 2. Fqiilibrium relationship: 38 i = I ,....N ; ~ =,..., I c (I) Midticomponent Distillation Colwnn Design: A Semi-rigorousApproach Semi-rigorous Method The column model incorporates two types of units - single tray and multi-tray. In Figure 2, the single tray units are the simple equilibrium trays, whereas the multi-tray units are a group of single trays. In the single tray units, the iterated variables are lb YO and Ti.in the multi-tray unit, instead of the stage temperature, the number of equilibrium stages inside the unit (Mi) is taken as an independent variable in addition to li. and vij The MESH equations for single my units are the same as in the Napf&ili-Sandholrnmethod. The M-, H- and S- equations for multi-tray units are also the same as for the single trays but the E- equations are different. Condenser> El H product Top p G product ream Side stream Feed Single tray units b Re boikr Bottom Bottom product product (b) Semi rigorous model with (a) Conventional traysingle and multitray units by-tray model Figure 2, Schematic represeluation of a complex multistage distillation column. 39 SN. Maiti, S . Ganguly. A.K. Das and DN. Sard Effective Stripping Factor and Equilibrium Relation for Multi-tray Units In the multi-tray unit i, the vapor rising from the unit is not in simple equilibrium with the descending liquid as in single tray units (see Figures 2 and 3). Ohmura and Kasahara obtained a relationship between the vapor flowrate and the liquid flowrate leaving the unit by using effective stripping factor of Edmister [4], namely: where mister's effective slripping factor (Se$ is defined by: 'i1,{1 + S'M~) + 0.25 - 0.5 (9) with S'ilj = K'il,{l + SiV)Vi /L'ii (10) and (11) The K'i1. and K > M ~values are the equilibrium constants of component j at the first ) inside the ith multi-tray and the &t plate temperatures (pi1 and T ' ~ Mrespectively, unit (see Figure 3). Primes refer to variables/parameters inside the multi-tray units, and an additional subscript is used to specify the tray number within these units. For the Naphtali-Sandholm formulation, the equilibrium relation for the ith multi-tray unit was derived as follows: 40 MulticomponewDistillation Column Design: A Semi-rigorousApproach Hence, the E-equation is Equations (9) to (14) show that the effective stripping factors depend on the flow rate of liquid from the first plate (L'il) and flow rate of vapor from the last plate (V'M) inside the ith multi-tray unit These values are calculated as described below. vi+l,j \i,j Figure 3 . Schematic representation of a general multi-tray unit. Calculation of L;I and V$M Ohmura and Kasahara [2] incorporated L'i1 as an independent variable. Calculation of v ' i ~assumed that the load changed linearly inside the multi-tray unit, as given by: v '=~ Vi+l - ( [(1+ SiL, Li -L'i1] / (Mi - 1 ) ) (16) 41 SN.Maiti, S.Ganguly. A.K.Das and D . N.Saraf In the proposed method these two profiles are calculated by solving component material balance, enthalpy balance and equilibrium relations inside the multi-tray unit as discussed below. The vapor leaving the first plate of the multi-tray unit is at its dew point temperature (T;1) as given by: The above equation is solved for T'i1. Hence, the component flow rates of liquid from the first plate in the unit are given by the equilibrium relation: (vii I Vi)- K'ilj (l'ilJ/L'il)= 0 (18) The enthalpy and total material balance equations for this plate are given by Equations (19) and (20) respectively: Since H'i2 is a function of T'i2, and is given by the dew point relation at the second plate in the unit as: then Equations (18) to (21) can be solved simultaneouslyto obtain L i l . Similarly, the vapor rising from the last plate in the multi-tray unit ( v ' i ~is) calculated using the material balance, equilibrium relation and enthalpy balance for the bottom plate in the unit The bubble point relation at the last plate in the unit is given by: and T;.M is obtained by solving Equation (22). The equilibrium relation at this plate is: 42 Multicomponent Distillation Column Design: A S m X g o r o u s Approach The total material balance equation is: L'iM-1= (I+sf)Li + v ' M - v i + l =f(V'iM) (25) From Equations (24) and (Z) the ,only unknowns are V'N and h ' i ~ - l=f(TiM-1)1. [ The latter is obtained by considering the bubble point relation at the last but one plate in the unit, which is given by: Hence, V ' ~ Mcan be obtained by solving Equations (23) to (26) iteratively. Computation Technique The model equations are solved simultaneously by a modified Newton-Raphson method making use of a damping factor whiIe correcting the solution vector. The convergence criterion used is described by Henley and Seader [8], namely: where s=N(2C+1) ("C F . 2 ) 10-10 i=l 1 fv, Specifying N , Tf Pf,Pi, Sf, S/ and Qi, the remaining N(2C + 1 ) unknown variables ( l b Vij and Ti or Mi) can be obtained by solving the N(2C + 1) model equations simultaneously. Design Specifications For the design of a two-product column, only two specifications can be accepted. These are usually given by specifying one composition each in the top and the bottom products. In the Naphtali-Sandholm method, such specification is included by dropping out one MESH equation and including a dummy equation in its place. For example, in order to specify the top product composition o)oi>, replace the enthalpy balance for the condenser by an equation of the form: v1j - & (29) v1j) YDj = 0 In the semi-rigorous formulation, consideration of the variables and the equations for each unit revealed that an additional specification in each multi-tray unit was possible. Since there are one or two single trays between the multi-tray unit and top product, either the same specification can be demanded at the multi-tray unit resulting in a conservative design, or the composition at the multi-tray unit can be 43 SN.Mairi. S.Ganguly.A.K. Dar and D . N.Swaf estimated from the given product specification. Therefore, unlike the NaphtaliSandholm formulation, the enthalpy balance at the condenser can be retained and an additional product composition specification can be included in the multi-tray unit. In the case of a column with side-stream product draw-off, only the fraction of the liquid and/or the vapor flow can be specified in the Naphtali-Sandholm method. Replacement of equations with other specifications is not possible. However, the semi-rigorous method allows the side-seeam product composition as well as flow rate to be specified. The semi-rigorous method for rating of existing columns allows the number of trays inside the multi-tray unit as an additional specification. Computer Program A computer program called CADPRO has been developed by the authors (contact DN. Saraf for details) which includes distillation calculations in both the rigorous design and rating approaches. In the design mode, one design specification at each multi-tray unit is provided and the number of trays, the feed location and product distribution are calculated. In the ratings approach, the number of trays grouped in each multi-tray unit is specified, and the temperature of each tray and product distribution are calculated. A thermodynamic package is also provided to account for nonideal systems. Figure 4 shows the flow chart of the design algorithm in CADPRO. This program was executed for some test problems using an HP 9000/850 computer. Results and Discussion The following three problems have been solved to demonstrate the application of the semi-rigorous algorithm. Problem I : A 16-plate column including a partial reboiler and a total condenser. The saturated liquid feed is introduced at the 81h plate from the top at a rate of 100 kmol/h containing propane 23% n-butane 37% and the rest of n-pentane at 280 K and 17 am. The distillate rate and reflux ratio are 22.6 kmoVh and 5 respectively. Problem 2: A 20-plate column including a partial reboiler and a total condenser. The feed flow rate, composition, temperature and pressure are same as in Problem 1. The distillate rate and reflux ratio are also the same, but the saturated liquid feed was introduced at the 10th plate. Problem 3: The column consisted of 17 trays with a partial condenser and a partial reboiler. The feed is a mixture of ethane 3%,propane 20% n-butane 37%. n-pentane 35% and the rest of n-hexane, temperature 374 K at 17 am, introduced at the 91h tray at 100 kmol/h, reflux = 150kmolh, and distillate = 23 kmol/h (Henley and Seader, 1981). Problems 1 and 2 have been solved both in the design and ratings modes, whereas Problem 3 has been solved for design only. 44 Multicomponent Distillation Column Design: A Semi-rigorous Approach Specify one liquid or vapor flow in each multitray unit 1 Specify No. of trays in each multitray unit Mi . Compute initial , gueasesof vij and l i j r I Compute internal flows: L:, and V& in multitray units Compute sum of the squares of the discrepency functions Is s u m L 5e? 1 No I Compute correction vector using Newton-Raphson Compute optimal step length and new values of v ; j , I i j , or Mi from summation Equation Figure 4. Flowchartfor semi-rigorourdesign method. 45 SN.Maiti, S.Ganguly. A.K. Das and D. N.Saraf Design In the design approach, one specificationequation in each multi-tray unit is required. In principle, it should be possible to specify any combination of liquid and vapor flow rates for the top and bottom sections, although a solution may not always be achieved. It was observed for the present case, that if the liquid flow rates in both the units were specified,the problem did not converge. Problem 1 was configured as an 11 unit, 10 unit and 9 unit column with two multi-tray units - one each in the rectifying and stripping sections. These were solved by the semi-rigorous method for three different design specifications. The design results obtained are given in Table 2. For all the column configurations and product specifications,the calculated number of stages in each multi-tray unit (when rounded to the nearest integer) correspond to the actual number of trays (given in the sixth column for comparison). units unit multi-tray units MI M2 11 6 042, 082 3.1284 3.2803 3.2125 4.4249 4.1824 4.3279 4.2619 4.2302 3.9391 3.8202 3.7581 3.7010 3,9007 3.8155 3.7517 5.1019 4.6449 4.9282 v42, vK3 v41, %2 10 5 v3lr v72 v32s v72 v32, 1173 9 5 1131, 172 w3ar +a v33, h 2 multi-tray units 3, 4 iterations 4 4 495 5 3 2 3 4 5 4 3 3 * M,and M2are the number of trays in the top and bottom multi-tray unit respectively. Figure 5 shows the progress of convergence of the number of trays inside the multi-tray units as iterations proceed. Figures 6, 7 and 8 show the convergence of liquid and vapor flows and temperature profiles respectively for the same case.The convergence can be seen to be quite fast and reasonably smooth. To investigate the effect of different design specifications and initial guesses, Problem 1 was solved considering a total of 9 units (case3 in Table 2). The results of the computations are shown in Table 3. It was observed that both the design specification and the initial guess had a profound effect on the final design, the latter having most influence. For the same design specifications, different initial guesses resulted in markedly different designs. Clearly, the design problem has a multiplicity of solutions. This is to be expected because Edmister's relation (Equation 15) is highly nonlinear in the number of stages (Mi)inside the multi-tray unit. To check if the various solutions thus obtained would indeed meet the design requirements, the column equations were solved in rating mode using the values for A41 and M2 as obtained from the design calculations (without rounding off). Both the designs gave essentially the same product distribution. It is well known that different combinations of total number of stages and feed location can lead to the same product specifications. The mathematical multiplicity of solutions observed in the present study is indeed a manifestation of the above physical reality. 46 Multkomponent Distillation Column Design: A Semi-rigorousApproach 6 5Unit-9 1- 0; 1 I I 1 2 3 4 No. of Iterations I 5 5. Convergence inside a multi-tray unit for problem I . 47 SN.Maiti. S. Ganguly. A.K. Das and D . N.Saraf Iturotion no.1 Final solution (iteration no.' 5) I I I 3 1 - I I I I I 5 7 U nit number I I 9 11 Figure 6. Convergence of liquidflow profilefor problem 1. Table 3: Resultsfor various design specifications and dferent initial guesses. Case Specifications Second initial gums First initial gums No. in multi-tray units Mi 1 2 mi. ~ .I .n. 4.26 Ma 5.10 %l, 17, 3.93 4.93 3 ~ 4.76 5.62 4.42 4 4 3 4 48 i 17s, VJI, c*3 4.48 Iterations A& 4 2.76 2.49 2.27 . -% 7.77 8.26 14.7 diverged Ikrdions 4 4 7 Multicomponent Distillation Column Design: A Semi-rigorousApproach 160 140 - 4 r : 120 0 E Y - :loo c 0 L -z 80- .c -Initial guess L 60> 40 Iteration no.1 Fino1 solution (iteration no. 5 ) o 0 6 - 20 3 1 5 7 9 11 Unit number Figure 7 . Convergence of vaporflow profile for problem I . Although all designs obtained for a given product specification are valid, an optimum is required. Numerous procedures for calculating optimum feed stage location are available in the literature [9,10]. The simplest and most commonly used is an empirical rule for feed-plate location where the ratio of key component liquid mole fraction on the feed stage should be as close as possible to that in the liquid portion of the feed s m m . For design Problem 1, the key component ratio in the feed is 0.62. In cases 1 and 2 (Table 3) it was found that this ratio is 0.62 and 0.61 respectively for the first initial guess, and 0.89 and 0.93 respectively for the second initial guess. Therefore, the designs obtained by the first initial guess are the optimal designs. However, as mentioned by Hanson and Newman [lo], empirical rules do not always work well, and hence a more general procedure must be evolved. For a particular reflux and product specification, the design which yields the minimum number of stages is the optimum. Therefore, an optimization problem can be formulated as: NmR Nms minf= C Mi+ C Mi i= 1 i= 1 subject to all model equations as constraints. Here N d and Nms are the number of multi-tray units in the rectifying and stripping sections respectively. 49 SN.Maiti, S.Ganguly, AX. Das and D.N.Swaf Computations were performed to compare the designs obtained by using a linear vapor profile inside the multi-tray unit [2] and the present method. In all the 11 cases studied (not presented here), the present method converged to desired solutions, whereas a linear profile failed to converge in five cases. Problems 2 and 3 were solved in the design mode and the results were found to be similar to those of Problem 1, and thus vindicated the conclusions already drawn. Further details on these and other computationsare available from the authors. 410 3 90 *u.370 L a c 2 u E r-" 35c 3 30 310 1 I 3 I I I I I 5 7 Unit number Figure 8. Convergence of temperature profilefor problem 2. 1 I 9 I 11 Rating The number of plates inside each multi-tray unit was varied to reduce the total number of units. The rating results for Problem 1 are shown in Table 4. It was observed that the product compositions for different configurations remained almost the Same with the separation between components slightly decreasing as the total number of units decreased. Similar results were obtained for Problem 2. 50 MulticomponeruDistillation Column Design: A Semi-rigorousApproach Table 4. Rating results by semi-rigorous method compared with Naphtali-Sandholm methodfor Problem 1 . S1. NO. of Feed NO. units 16' Trays in Top product . Bottom product multi-tray units cornposition composition 1 21.5363 1.4637 -9 8 1.0614 35.9385 39.9978 0.0022 2 11 6 3, 4 21.4573 1.5427 35.8604 1.1396 0.0030 39.9970 1.8406 3 7 4 5, 6 21.1594 35.587 1.4130 39.973 0.0269 * Calculation by Naphtali-Sandholm method. unit NO. of iterations 2 3 4 Since the number of units is smaller compared to the total number of trays in the column, the computer memory requirement is significantly reduced for the semirigorous method. Conclusions A computer-aided design package, CADPRO, has been developed which incorporates a semi-rigorous design procedure. It enables calculation of the total number of stages and the feed location for a given set of design specifications without repeated simulations. However, different initial guesses may lead to a multiplicity of solutions because of the nonlinear nature of the design equations. Selection of an optimal design using an empirical relationship works well in the examples cited. Further investigations are required to establish the usefulness of the suggested optimization approach. The semi-rigorousmethod when used for rating problems gives results which are comparable to those obtained from the rigorous Naphtali-Sandholm procedure, but requiring much less computer memory. When the number of trays inside a multi-tray unit exceeds 5 or 6, the validity of the effectivestripping factor approximation begins to break down. Hence for very large columns it may become necessary to provide more than one multi-tray unit in each section. Acknowledgement Financial support from the Department of Science and Technology, New Delhi, under Project No. In - 5 (67) /87 - ET, is gratefully acknowledged. References D.P.1971. Multicomponent separation calculation by Linearization.AIChE J.. 17(1), 148-153. 2. Ohmura, S. and Kasahara. S. 1978. New distillation calculation method utilizing salient features of both short-cut and tray-by-tray method. J . Chem.Eng. Japan. ll(3). 185-193. 3. Tomich, J.F. 1970. A new simulation method for equilibrium stage processes. AIChE J.. 16(2). 229-232. 4. Edmister, W.C.1943. Design for hydrocarbon absorption and stripping. Id.Eng. Chem., 35(8), 837-839 1. Naphtali, L.M. and Sandholm, SH. Mairi. S. Ganguly, A.K. Das and D. N.Saraf 5. Ganguly, S. 1985. M.Tech. Thesis, Indian Inrtitute of Technology, Kanpw. 6. Ganguly, S., Das, A.K. and Saraf,DN. 1985. Paper presented at National Symposium on Modeling and Simulation in Chemical Engineering, Indian Institute of Science, Bangalore, India, August 22-24 7. Maiti. S.N. 1989. M.Tech. Thesis, Indian Institute of Technology, Kanpw. 8. Henley, E.J. and Seader, J.D. 1981. Equilibrium-Stage Separation Operations in Chemical Engineering, John Wiley & Sons,New York. 9. King, C.J. 1980. Seperation Processes,McGraw-Hill., New York. 10. Hanson, D N . and Newman, J.S. 1977. Calculation of distillation columns at the optimum feed plate location. I d . Eng. Chem. Process. Des. D o . , 16(2), 223-227. Received: 18 Janurary 1993; Accepted gfter revision: 2 August 1993. 52

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