- Email: [email protected]
Flexibility analysis leads to a sizing strategy in distillation columns
Computers chem. Engng Vol.20, Suppl., pp. SI39-SI~-,J,.1996
Pergamon
S0098-1354(96)00034-8
Copyright© 1996Elsevier ScienceLtd Printed in Great Britain. All rights reserved 0098-1354/96$15.00+0.00
FLEXIBILITY ANALYSIS LEADS TO A SIZING STRATEGY IN D I S T I L L A T I O N C O L U M N S Patricia M Hoch and Ana M Eliceche Chemical Engineering Department -Universidad National dei Sur- CONICET 12 de Octubre 1842 - 8000 Bahia Bianca- Argentinaemail: prelicec@criba, edu. ar
Abstract - A sizing strategy for distillation columns is presented to ensure a feasible operation for any possible realization of the uncertain parameters. The flexibility analysis of a given design identifies the worst set of uncertain parameters for each equipment size of the column. A systematic way of associating sizes with critical subspaces and the corresponding active constraints leads to the sizing strategy presented. The evaluation of sizes for a given flexibility requires the solution of a set of algebraic equations derived from the flexibility analysis.
Keywords: distillation column, design, flexibility, uncertainty.
INTRODUCTION The design of distillation columns with nominal parameters has been extensively studied in the literature. Recently, the design of distillation columns with uncertain parameters has been addressed. Fisher et al. (1985) studied the effect of overdesign in the operability of distillation columns when changes in the feed composition are expected. Kubic and Stein (1988) considered random and fuzzy uncertainties in the design of distillation columns. The sensitivity of the column design to uncertainties in feed, model and cost parameters was studied by Hoch (1993) and Hoch and Eliceche (1994). A general framework to evaluate the operational flexibility of chemical processes was presented by Swaney and Grossmann (1985), where a quantitative index called flexibility index is proposed. For a given column design the evaluation of the flexibility index was presented by Hoch et al (1995). The flexibility index measures the size of the parameter space over which feasible steady-state operation of the column can be obtained by proper adjustment of the control variables. Flexibility of a given column design represents the ability to adjust the different conditions that may be found during operation. The main objective of this work is to consider uncertain data at the design stage so that the column design ensures feasible operation for any possible realization of the uncertain parameters. Equipment sizes are evaluated so that their capacities do not become a column bottleneck. A rigorous modeling of the distillation column is used. A methodology for a flexible design is developed, so that the column can accommodate feed perturbations and parametric uncertainty. A feasible operation can be guaranteed for the uncertain space considered at the design stage. External sources of variations include changes in component feed flowrates and cooling water temperature. Internal sources of variations or uncertainty include parameters such us maximum allowed vapor velocity and heat transfer coefficients of the condenser and reboiler. The control variables will be chosen during the operation for any particular realization of uncertain parameters.
FLEXIBILITY EVALUATION The flexibility represents the ability of a process to accommodate a set of uncertain parameters. A bounded set of uncertain parameters is assumed, where each parameter varies independently of the remaining ones. The base point is defined by the nominal values of the uncertain parameters ON, while expected deviations from the nominal values in the positive (Ae+) and negative (AO) directions for each parameter are assumed at the design stage. The uncertain space is represented by a hyperrectangle centered at the nominal point. The flexibility index was defined by Swaney and Grossmann (1985) as the maximum hyperrectangle T that can be expanded around the nominal point in the feasible region. The sides of the hyperrectangle are proportional to the expected deviations. The hyperrectangle can be defined in terms pea scalar variable 8 as follows: SI39
S 140
European Symposiumon Computer Aided Process Engineering---6. Part A T(~i) = {0/0 N -~iA0- ~0 <0 N +8A0+},8_> 0
(1)
The constraints of the feasible region are inequality constraints (fj(d,z,0)_<0) associated with separation objectives and equipment capacities. The design variables d are associated with equipment capacities and z are the control variables. The equality constraints associated with the steady state modeling of the column are solved within a procedure which includes mass and energy balances, equilibrium relationships and thermodynamic properties.The modeling of the distillation column can be as rigorous as desired. Thus, alternative procedures can be used for the steady state modeling of the column. The flexibility index F for a given design d, is determined by the maximum value of 8 in the semi-infinite programming problem: F= max8 s.t. max minmax fj(d,z,0)_<0 O~T(8) z
(2)
jd
T(8) = {0/O N - ~ A 0 - _<'0 *( ON +~A0 + } F corresponds to the maximum deviation of uncertain parameters from the nominal values for which feasible operation can be guaranteed by proper manipulation of the control variables. If F is equal to one, the design can accommodate the expected deviations assumed A0+ and A0-. Solution of problem (2) is in general very difficult. For the special case that the constraint functions f(d,z,0) are jointly quasi-convex in z and one-dimensional quasi-convex in 0, the solution of problem (2) will lie at a vertex of the hyperrectangle T(8) as shown in Swaney and Grossmann (1985). If these conditions are satisfied, problem (2) can be decomposed in a two-level optimization problem as follows: F = min8 k keV
(3)
with k e V={k/l _< k < 2m}, m being the number of uncertain parameters considered. Each ~' involves the solution of the following non linear programming problem: ~ik =max~ &z
S.t. fj(d,z,0k)-<0
jeJ
(4)
Ok =0 N +GA0 k where 8k is the maximum displacement of the uncertain parameters from the nominal values in the direction of the vertex k. The separation objectives are related to minimum purities and recoveries or maximum impurities allowed in the final products. For conventional columns, there are two control variables z, one associated with the feed distribution in top and bottom products and the remaining one associated with the ratio of vapour and liquid flowrates circulating in the column. A nonlinear programming code should be used to solve each nonlinear programming (NLP) problem (4).
SIZING STRATEGY Constraint functions versus uncertain parameters
The matrix of partial derivatives of the constraint functions with respect to the uncertain parameters can be partitioned in dense, nule and sparse submatrices as shown in table 1. All the constraints depend on the control variables (R, reflux flowrate and B, bottom product, for conventional columns) and the feed perturbations (Ik, light key feed flowrate: hk, heavy key feed flowrate), generating a dense submatrix. There is no interaction between the separation objectives (S1,$2,$3) and the uncertain parameters of the capacity equations (Uc, condenser heat transfer coefficient,Ur reboiler heat transfer coefficient, Tw, cooling water temperature; Ga, maximum allowed vapour
European Symposium on Computer Aided Process Engineering--qS.Part A
SI41
velocity) generating a nule submatrix. The remaining sparse submatrix indicates the dependence of sizes (Ac, condenser area; Ar, reboiler area; De, column diamete0 on the uncertain parameters involved in the design equations. The sizes depend on the parameters involved on their corresponding capacity equations but are independent of the parameters associated to the remaining design equations. Thus, the dimension of the subspace of uncertain parameters which contribute to modify a given size is smaller than the dimension of the initial space of uncertain parameters. The sparsity of the matrix showing the functionality of the constraint functions with respect to the uncertain parameters was exploited by Kabatec and Swaney (1992) to reduce the computational work of the branch and bound algorithm developed to search through the vertices in the context of heat exchangers. A given size depends on a given subset of uncertain parameters. Thus, each size has an associated set of uncertain parameters. The maximum displacements from the nominal point in this subspace do not depend on the values assumed for the remaining uncertain parameters. The first step is to identify sizes with their corresponding subspaces reducing the search space and then to analize the vertices of the corresponding subspace. There will be a worst combination of uncertain parameters for each size. Table 1- Separation and capacity constraints versus uncertain parameters.
R B lk hk Uc Ur Tw Ga
SI x x x x 0 0 0 0
$2 x x x x 0 0 0 0
$3 x x x x 0 0 0 0
Ac x x x x x 0 x 0
Ar x x x x 0 x 0 0
Dc x x x x 0 0 0 x
Solution of a system of nonlinear equations If nz is the number of control variables, there will be at least (nz+l) active constraints at the solution of problems (3) and (4) as shown by Swaney and Grossmann(1985). For a conventional column, there will be two active constraints associated with separation objectives and one active constraint related with the maximum capacity of the equipment which is limiting the flexibility in a given direction of the uncertain space. The association of sizes with subspaces and the identification of their corresponding worst combination of uncertain parameters leads to the following sizing strategy. For a column in operation, each size can accommodate a different degree of uncertainties related to its associated parameters. The flexibility factor of each capacity is the minimum of the maximum displacements in which that capacity is an active constraint. In that sense the flexibility index coincides with the minimum of the flexibility factors. Therefore the design strategy would be to size the equipments in such a way that they all have a flexibility factor equal to one. The equipment sizes required to accommodate variations of a set of uncertain parameters are evaluated solving a set of nonlinear algebraic equations. The nonlinear equations to be used correspond to the active constraints in the solution of problem 4. There should be as many active constraints related to separation objectives as control variables, and the remaining ones are related to capacity constraints. For example, the condenser area is an active constraint when its heat transfer coefficient (Uc) has a negative deviation from the nominal point and the cooling water temperature (Tw) has a positive deviation. This behavior can be explained with the design equation of the condenser. The minimum displacement from the nominal point will occur for this combination of uncertain parameters (Uc and Tw) in addition to positive displacements for the light and heavy key feed flowrates. This direction corresponds to the worst combination of uncertain parameters for the condenser area. Thus, we can solve the set of nonlinear equations in this direction fixing the flexibility factor equal to one. The solution of the nonlinear equations will give the required area of the condenser. The set of nonlinear equations include the active constraints of problem 4 and the procedure to solve the steady state model of the distillation column. The worst combination of parameters for the reboiler area corresponds to positive displacements of the key component feed flowrates and a negative deviation of the heat transfer coefficient (Ur). In this direction the
SI42
European Symposiumon Computer Aided Process Engineering---6. Part A
solution of the corresponding equations allows the determination of the reboiler area to accommodate the assumed variations on these parameters. Similarly, the diameter of the column can be evaluated solving the set of nonlinear equations for the direction with positive displacements of the key component feed fiowrates and negative deviations of the maximum allowed vapour velocity (Ga). Alternatively, the direction with positive deviations of the key component feed flowrates and negative deviations of the heat transfer coefficients and maximum allowed vapour velocity, allows the simultaneous determination of the three sizes.
NUMERICAL EXAMPLE In the example presented in this work, the idea is to evaluate the sizes of the debutanizer column of an ethylene plant in such a way that the equipment capacities do not become a bottleneck for any realization of the uncertain space assumed. The control variables, R and B, will be chosen during the operation for any particular combination of the uncertain parameters. The feed is saturated liquid at 15 bar and the column operates at 4 bar. The component feed flow rates in kgmol/h are: propylene, 0.199, propane, 0.190, n-butane (lk) 24.706 and n-pentane (hk) 9.875. The total feed flow rate is 34.97. The column has a total condenser. The separation objectives (St, $2, $3) are shown in table 2. The nominal values and expected deviation for the uncertain parameters are shown in table 3.
Table 2: Design specifications St - Maximum molar fraction of butane allowed in the bottom product (Xc,.B) $2- Maximum amount of pentane in the top product (Xcs.D) $3 - Minimum pentane recovery in the bottom product (bcs)
0.01786 0.025 0.97
Table 3: Nominal values for the uncertain parameters and their 10% expected deviations. Nominal value Condenser heat transfer coefficient (Uc) Reboiler heat transfer coefficient (Ur) Inlet cooling water temperature (Tw) Maximum allowed vapor velocity (Ga) Butane feed flow rate (lk) Pentane feed flow rate (hk)
0.473 0.552 20 0.380 24.706 9.875
Expected deviations i-0.0473 :t0.0552 _+.2 :t0.0380 _-t2.4706 ~0.9875
[kW/m2°C] [kW/m2*C] [*el [nVsl [kmol/h] [kmol/hl
Design for a given flexibility A rigorous steady-state modeling of the column based on the Naphtali-Sandholm algorithm is used. The SRK equations were used in this example to predict the vapour-liquid equilibrium. Analytical derivatives of the separation objectives with respect to the control variables are derived by the chain rule using elements of the jacobian matrix generated in the column procedure. For a debutanizer column the constraints of the NLP problem (4) are convex on the control variables. The succesive quadratic programming code OPT of Biegler and Cuthrell (1985) is used to solve problem 4 for all the vertices. The hyperrectangle generated by the deviations from the nominal point of six parameters has 64 vertices. For a given design the analysis of the solutions in the 64 vertices provides valuable information. The first equipment becoming a bottleneck and the scenarios where it can happen are determined. It is important to identify each capacity with the corresponding worst combination of parameters to evaluate its flexibility factor. With the information provided by the flexibility analysis, coupled with a rigorous modeling of the column, the design strategy is to size each capacity with a flexibility factor equal to one. The active constraints and scenarios in which the system of nonlinear equations can be solved are identified in the flexibility analysis, where the active constraints related to separation objectives and their corresponding scenarios in terms of component feed flowrates perturbations are determined.
European Symposium on Computer Aided Process Engineering--6. Part A
S143
In this particular example, the column diameter and the condenser and reboiler areas are evaluated solving a set of nonlinear equations. The sizes corresponding to flexibility factors equal to one are shown in table 4. These sizes can accommodate the deviations assumed, for the worst combinations of parameters. The overdesign is evaluated for a given flexibility. At the design stage, the flexibility increases the investment cost. The sizes were evaluated for different degrees of flexibility, and the behaviour of investment cost versus flexibility is shown in Figure 1. Table 4: Dimensions of the column Flexibility Number of rectification sta[~es (Nr) Number of strippin8 sta~es (Ns) Diameter of the column (Dc) Condenser area (Ac) Reboiler area (Ar)
Figure 1: Increment of the investment cost versus flexibility
1 9 11 0.6224 m 30.515 m2 24.069 rn2
Figure 2: Flexibility vs. feed location (FL)
1
F
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.-'/
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./
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~ 5
--~
0.88 7
F
9
~ 10
) 11
~ 12
13 FL
The nominal values used for the number of ideal stages in the rectification and stripping sections were obtained designing the column for the nominal point. A NLP problem is solved where the objective function is to minimize the total cost as shown by Hoch and Eliceche (1991). To quantify the incidence of the feed location in the column flexibility, some alternative feed locations were explored. The flexibility was evaluated locating the feed in four alternative locations, one or two stages above and one or two stages below the actual feed. The results are shown in Figure 2. The flexibility is quite sensitive to feed location. For the perturbations on the component feed flowrates considered, the optimal feed location has not changed. The feed location for the nominal point (FL--I 1) remains being the best one.
CONCLUSIONS
AND SIGNIFICANCE
A strategy for the design of conventional distillation columns considering parametric uncertainty has been presented. This is an important contribution considering that there are no restrictions in the column model to be used. The association of sizes, scenarios and their corresponding flexibility factors leads to a sizing strategy which only requires the solution of a set of algebraic equations, thus, the methodology presented is quite simple. The evaluation of the increment in investment cost due to a flexibility expansion facilitates a decision at the design stage. The methodology can be extended to nonconventional columns and other separation processes, allowing the equipment sizing in the presence of uncertain data when rigorous models are available.
SI44
European Symposium on Computer Aided Process Engineering----6.Part A NOTATION
Ac Ar B d Dc f F FL Ga hk lk Nr Ns R Sj Tw Uc Ur z
Condenser area Reboiler area Bottom molar flow rate Design variables Column diameter Nonlinear inequality constraints Flexibility index Feed location Maximum allowed vapor velocity Heavy key molar flow rate Light key molar flow rate Number of rectification ideal stages Number of stripping ideal stages Reflux molar flow rate Separation objective j Cooling water temperature Condenser global heat transfer coefficient Reboiler global heat transfer coefficient Control variables
Greek letters: 0 8 A
Uncertain parameter Scalar representing the maximum variation from the nominal point Variation REFERENCES
Biegler L.T. and Cuthrell J., 1985, Improved Unfeasible Path Optimization for Sequential-Modular Simulators II: The Optimization Algorithm. Comp. and Chem. Eng. 9, 3. Fisher W. Doherty M. and Douglas J., 1985, Effect of Overdesign on the Operability of Distillation Columns, I&EC Proc. Des. Dev. 24, pp 593-598. Grossmann I.E. and Straub D.A, 1991, Recent Developments in the Evaluation and Optimization of Flexible Processes, Computer Oriented Process Engineering, Voi 10, pp. 49-59. Halemane K. and Grossmann I.E., 1983, Optimal Process Design under Uncertainty, AIChEJ. 29, 3. Hoch P. (1993). Disefio Optimo y Flexibilidad de Columnas de Destilaci6n, Phi) Thesis, Universidad Nacional del Sur, Bahia Blanca, Argentina. Hoeh P. and Eliceche A.M., 1991, Optimal Design of Non-Conventional Distillation Columns, Computer Oriented Process Engineering, Vol 10, pp 369-374. Hoch P. and Eliceche A.M., 1994, Sensitivity of Distillation Column Design to Uncertain Parameters, 1ChemE Symp. Set. 133, pp. 459-.466. Hoch, P.M., A. M. Eiiceche and I. E. Grossmann, 1995, Evaluation of Design Flexibility in Distillation Columns, Comp. and Chem. Eng., Vol 19, pp. $669-674. Kabatek, and Swaney R., 1992, Worst-case Identification in Structures Process Systems, Comp. and Chem. Engn. 3, p 473. Kubic W.L. and Stein F.P., 1988, A Theory of Design Reliability using Probability and Fuzzy Sets, AIChE J. 34, 4, pp. 583-601. Swaney R. and Grossmann I.E., 1985, An Index for Operational Flexibility in Chemical Process Design. Part I: Formulation and Theory. AIChEJ. 31, pp. 621-630.
Pergamon
S0098-1354(96)00034-8
Copyright© 1996Elsevier ScienceLtd Printed in Great Britain. All rights reserved 0098-1354/96$15.00+0.00
FLEXIBILITY ANALYSIS LEADS TO A SIZING STRATEGY IN D I S T I L L A T I O N C O L U M N S Patricia M Hoch and Ana M Eliceche Chemical Engineering Department -Universidad National dei Sur- CONICET 12 de Octubre 1842 - 8000 Bahia Bianca- Argentinaemail: prelicec@criba, edu. ar
Abstract - A sizing strategy for distillation columns is presented to ensure a feasible operation for any possible realization of the uncertain parameters. The flexibility analysis of a given design identifies the worst set of uncertain parameters for each equipment size of the column. A systematic way of associating sizes with critical subspaces and the corresponding active constraints leads to the sizing strategy presented. The evaluation of sizes for a given flexibility requires the solution of a set of algebraic equations derived from the flexibility analysis.
Keywords: distillation column, design, flexibility, uncertainty.
INTRODUCTION The design of distillation columns with nominal parameters has been extensively studied in the literature. Recently, the design of distillation columns with uncertain parameters has been addressed. Fisher et al. (1985) studied the effect of overdesign in the operability of distillation columns when changes in the feed composition are expected. Kubic and Stein (1988) considered random and fuzzy uncertainties in the design of distillation columns. The sensitivity of the column design to uncertainties in feed, model and cost parameters was studied by Hoch (1993) and Hoch and Eliceche (1994). A general framework to evaluate the operational flexibility of chemical processes was presented by Swaney and Grossmann (1985), where a quantitative index called flexibility index is proposed. For a given column design the evaluation of the flexibility index was presented by Hoch et al (1995). The flexibility index measures the size of the parameter space over which feasible steady-state operation of the column can be obtained by proper adjustment of the control variables. Flexibility of a given column design represents the ability to adjust the different conditions that may be found during operation. The main objective of this work is to consider uncertain data at the design stage so that the column design ensures feasible operation for any possible realization of the uncertain parameters. Equipment sizes are evaluated so that their capacities do not become a column bottleneck. A rigorous modeling of the distillation column is used. A methodology for a flexible design is developed, so that the column can accommodate feed perturbations and parametric uncertainty. A feasible operation can be guaranteed for the uncertain space considered at the design stage. External sources of variations include changes in component feed flowrates and cooling water temperature. Internal sources of variations or uncertainty include parameters such us maximum allowed vapor velocity and heat transfer coefficients of the condenser and reboiler. The control variables will be chosen during the operation for any particular realization of uncertain parameters.
FLEXIBILITY EVALUATION The flexibility represents the ability of a process to accommodate a set of uncertain parameters. A bounded set of uncertain parameters is assumed, where each parameter varies independently of the remaining ones. The base point is defined by the nominal values of the uncertain parameters ON, while expected deviations from the nominal values in the positive (Ae+) and negative (AO) directions for each parameter are assumed at the design stage. The uncertain space is represented by a hyperrectangle centered at the nominal point. The flexibility index was defined by Swaney and Grossmann (1985) as the maximum hyperrectangle T that can be expanded around the nominal point in the feasible region. The sides of the hyperrectangle are proportional to the expected deviations. The hyperrectangle can be defined in terms pea scalar variable 8 as follows: SI39
S 140
European Symposiumon Computer Aided Process Engineering---6. Part A T(~i) = {0/0 N -~iA0- ~0 <0 N +8A0+},8_> 0
(1)
The constraints of the feasible region are inequality constraints (fj(d,z,0)_<0) associated with separation objectives and equipment capacities. The design variables d are associated with equipment capacities and z are the control variables. The equality constraints associated with the steady state modeling of the column are solved within a procedure which includes mass and energy balances, equilibrium relationships and thermodynamic properties.The modeling of the distillation column can be as rigorous as desired. Thus, alternative procedures can be used for the steady state modeling of the column. The flexibility index F for a given design d, is determined by the maximum value of 8 in the semi-infinite programming problem: F= max8 s.t. max minmax fj(d,z,0)_<0 O~T(8) z
(2)
jd
T(8) = {0/O N - ~ A 0 - _<'0 *( ON +~A0 + } F corresponds to the maximum deviation of uncertain parameters from the nominal values for which feasible operation can be guaranteed by proper manipulation of the control variables. If F is equal to one, the design can accommodate the expected deviations assumed A0+ and A0-. Solution of problem (2) is in general very difficult. For the special case that the constraint functions f(d,z,0) are jointly quasi-convex in z and one-dimensional quasi-convex in 0, the solution of problem (2) will lie at a vertex of the hyperrectangle T(8) as shown in Swaney and Grossmann (1985). If these conditions are satisfied, problem (2) can be decomposed in a two-level optimization problem as follows: F = min8 k keV
(3)
with k e V={k/l _< k < 2m}, m being the number of uncertain parameters considered. Each ~' involves the solution of the following non linear programming problem: ~ik =max~ &z
S.t. fj(d,z,0k)-<0
jeJ
(4)
Ok =0 N +GA0 k where 8k is the maximum displacement of the uncertain parameters from the nominal values in the direction of the vertex k. The separation objectives are related to minimum purities and recoveries or maximum impurities allowed in the final products. For conventional columns, there are two control variables z, one associated with the feed distribution in top and bottom products and the remaining one associated with the ratio of vapour and liquid flowrates circulating in the column. A nonlinear programming code should be used to solve each nonlinear programming (NLP) problem (4).
SIZING STRATEGY Constraint functions versus uncertain parameters
The matrix of partial derivatives of the constraint functions with respect to the uncertain parameters can be partitioned in dense, nule and sparse submatrices as shown in table 1. All the constraints depend on the control variables (R, reflux flowrate and B, bottom product, for conventional columns) and the feed perturbations (Ik, light key feed flowrate: hk, heavy key feed flowrate), generating a dense submatrix. There is no interaction between the separation objectives (S1,$2,$3) and the uncertain parameters of the capacity equations (Uc, condenser heat transfer coefficient,Ur reboiler heat transfer coefficient, Tw, cooling water temperature; Ga, maximum allowed vapour
European Symposium on Computer Aided Process Engineering--qS.Part A
SI41
velocity) generating a nule submatrix. The remaining sparse submatrix indicates the dependence of sizes (Ac, condenser area; Ar, reboiler area; De, column diamete0 on the uncertain parameters involved in the design equations. The sizes depend on the parameters involved on their corresponding capacity equations but are independent of the parameters associated to the remaining design equations. Thus, the dimension of the subspace of uncertain parameters which contribute to modify a given size is smaller than the dimension of the initial space of uncertain parameters. The sparsity of the matrix showing the functionality of the constraint functions with respect to the uncertain parameters was exploited by Kabatec and Swaney (1992) to reduce the computational work of the branch and bound algorithm developed to search through the vertices in the context of heat exchangers. A given size depends on a given subset of uncertain parameters. Thus, each size has an associated set of uncertain parameters. The maximum displacements from the nominal point in this subspace do not depend on the values assumed for the remaining uncertain parameters. The first step is to identify sizes with their corresponding subspaces reducing the search space and then to analize the vertices of the corresponding subspace. There will be a worst combination of uncertain parameters for each size. Table 1- Separation and capacity constraints versus uncertain parameters.
R B lk hk Uc Ur Tw Ga
SI x x x x 0 0 0 0
$2 x x x x 0 0 0 0
$3 x x x x 0 0 0 0
Ac x x x x x 0 x 0
Ar x x x x 0 x 0 0
Dc x x x x 0 0 0 x
Solution of a system of nonlinear equations If nz is the number of control variables, there will be at least (nz+l) active constraints at the solution of problems (3) and (4) as shown by Swaney and Grossmann(1985). For a conventional column, there will be two active constraints associated with separation objectives and one active constraint related with the maximum capacity of the equipment which is limiting the flexibility in a given direction of the uncertain space. The association of sizes with subspaces and the identification of their corresponding worst combination of uncertain parameters leads to the following sizing strategy. For a column in operation, each size can accommodate a different degree of uncertainties related to its associated parameters. The flexibility factor of each capacity is the minimum of the maximum displacements in which that capacity is an active constraint. In that sense the flexibility index coincides with the minimum of the flexibility factors. Therefore the design strategy would be to size the equipments in such a way that they all have a flexibility factor equal to one. The equipment sizes required to accommodate variations of a set of uncertain parameters are evaluated solving a set of nonlinear algebraic equations. The nonlinear equations to be used correspond to the active constraints in the solution of problem 4. There should be as many active constraints related to separation objectives as control variables, and the remaining ones are related to capacity constraints. For example, the condenser area is an active constraint when its heat transfer coefficient (Uc) has a negative deviation from the nominal point and the cooling water temperature (Tw) has a positive deviation. This behavior can be explained with the design equation of the condenser. The minimum displacement from the nominal point will occur for this combination of uncertain parameters (Uc and Tw) in addition to positive displacements for the light and heavy key feed flowrates. This direction corresponds to the worst combination of uncertain parameters for the condenser area. Thus, we can solve the set of nonlinear equations in this direction fixing the flexibility factor equal to one. The solution of the nonlinear equations will give the required area of the condenser. The set of nonlinear equations include the active constraints of problem 4 and the procedure to solve the steady state model of the distillation column. The worst combination of parameters for the reboiler area corresponds to positive displacements of the key component feed flowrates and a negative deviation of the heat transfer coefficient (Ur). In this direction the
SI42
European Symposiumon Computer Aided Process Engineering---6. Part A
solution of the corresponding equations allows the determination of the reboiler area to accommodate the assumed variations on these parameters. Similarly, the diameter of the column can be evaluated solving the set of nonlinear equations for the direction with positive displacements of the key component feed fiowrates and negative deviations of the maximum allowed vapour velocity (Ga). Alternatively, the direction with positive deviations of the key component feed flowrates and negative deviations of the heat transfer coefficients and maximum allowed vapour velocity, allows the simultaneous determination of the three sizes.
NUMERICAL EXAMPLE In the example presented in this work, the idea is to evaluate the sizes of the debutanizer column of an ethylene plant in such a way that the equipment capacities do not become a bottleneck for any realization of the uncertain space assumed. The control variables, R and B, will be chosen during the operation for any particular combination of the uncertain parameters. The feed is saturated liquid at 15 bar and the column operates at 4 bar. The component feed flow rates in kgmol/h are: propylene, 0.199, propane, 0.190, n-butane (lk) 24.706 and n-pentane (hk) 9.875. The total feed flow rate is 34.97. The column has a total condenser. The separation objectives (St, $2, $3) are shown in table 2. The nominal values and expected deviation for the uncertain parameters are shown in table 3.
Table 2: Design specifications St - Maximum molar fraction of butane allowed in the bottom product (Xc,.B) $2- Maximum amount of pentane in the top product (Xcs.D) $3 - Minimum pentane recovery in the bottom product (bcs)
0.01786 0.025 0.97
Table 3: Nominal values for the uncertain parameters and their 10% expected deviations. Nominal value Condenser heat transfer coefficient (Uc) Reboiler heat transfer coefficient (Ur) Inlet cooling water temperature (Tw) Maximum allowed vapor velocity (Ga) Butane feed flow rate (lk) Pentane feed flow rate (hk)
0.473 0.552 20 0.380 24.706 9.875
Expected deviations i-0.0473 :t0.0552 _+.2 :t0.0380 _-t2.4706 ~0.9875
[kW/m2°C] [kW/m2*C] [*el [nVsl [kmol/h] [kmol/hl
Design for a given flexibility A rigorous steady-state modeling of the column based on the Naphtali-Sandholm algorithm is used. The SRK equations were used in this example to predict the vapour-liquid equilibrium. Analytical derivatives of the separation objectives with respect to the control variables are derived by the chain rule using elements of the jacobian matrix generated in the column procedure. For a debutanizer column the constraints of the NLP problem (4) are convex on the control variables. The succesive quadratic programming code OPT of Biegler and Cuthrell (1985) is used to solve problem 4 for all the vertices. The hyperrectangle generated by the deviations from the nominal point of six parameters has 64 vertices. For a given design the analysis of the solutions in the 64 vertices provides valuable information. The first equipment becoming a bottleneck and the scenarios where it can happen are determined. It is important to identify each capacity with the corresponding worst combination of parameters to evaluate its flexibility factor. With the information provided by the flexibility analysis, coupled with a rigorous modeling of the column, the design strategy is to size each capacity with a flexibility factor equal to one. The active constraints and scenarios in which the system of nonlinear equations can be solved are identified in the flexibility analysis, where the active constraints related to separation objectives and their corresponding scenarios in terms of component feed flowrates perturbations are determined.
European Symposium on Computer Aided Process Engineering--6. Part A
S143
In this particular example, the column diameter and the condenser and reboiler areas are evaluated solving a set of nonlinear equations. The sizes corresponding to flexibility factors equal to one are shown in table 4. These sizes can accommodate the deviations assumed, for the worst combinations of parameters. The overdesign is evaluated for a given flexibility. At the design stage, the flexibility increases the investment cost. The sizes were evaluated for different degrees of flexibility, and the behaviour of investment cost versus flexibility is shown in Figure 1. Table 4: Dimensions of the column Flexibility Number of rectification sta[~es (Nr) Number of strippin8 sta~es (Ns) Diameter of the column (Dc) Condenser area (Ac) Reboiler area (Ar)
Figure 1: Increment of the investment cost versus flexibility
1 9 11 0.6224 m 30.515 m2 24.069 rn2
Figure 2: Flexibility vs. feed location (FL)
1
F
•
I
•
/
17.00 t
/
/ 13.00 //
/
•
/
0.96
/
•
/ "~
/ 9.00
0.92
.-'/
I~I
./
/ 5.00
~ 3
~ 5
--~
0.88 7
F
9
~ 10
) 11
~ 12
13 FL
The nominal values used for the number of ideal stages in the rectification and stripping sections were obtained designing the column for the nominal point. A NLP problem is solved where the objective function is to minimize the total cost as shown by Hoch and Eliceche (1991). To quantify the incidence of the feed location in the column flexibility, some alternative feed locations were explored. The flexibility was evaluated locating the feed in four alternative locations, one or two stages above and one or two stages below the actual feed. The results are shown in Figure 2. The flexibility is quite sensitive to feed location. For the perturbations on the component feed flowrates considered, the optimal feed location has not changed. The feed location for the nominal point (FL--I 1) remains being the best one.
CONCLUSIONS
AND SIGNIFICANCE
A strategy for the design of conventional distillation columns considering parametric uncertainty has been presented. This is an important contribution considering that there are no restrictions in the column model to be used. The association of sizes, scenarios and their corresponding flexibility factors leads to a sizing strategy which only requires the solution of a set of algebraic equations, thus, the methodology presented is quite simple. The evaluation of the increment in investment cost due to a flexibility expansion facilitates a decision at the design stage. The methodology can be extended to nonconventional columns and other separation processes, allowing the equipment sizing in the presence of uncertain data when rigorous models are available.
SI44
European Symposium on Computer Aided Process Engineering----6.Part A NOTATION
Ac Ar B d Dc f F FL Ga hk lk Nr Ns R Sj Tw Uc Ur z
Condenser area Reboiler area Bottom molar flow rate Design variables Column diameter Nonlinear inequality constraints Flexibility index Feed location Maximum allowed vapor velocity Heavy key molar flow rate Light key molar flow rate Number of rectification ideal stages Number of stripping ideal stages Reflux molar flow rate Separation objective j Cooling water temperature Condenser global heat transfer coefficient Reboiler global heat transfer coefficient Control variables
Greek letters: 0 8 A
Uncertain parameter Scalar representing the maximum variation from the nominal point Variation REFERENCES
Biegler L.T. and Cuthrell J., 1985, Improved Unfeasible Path Optimization for Sequential-Modular Simulators II: The Optimization Algorithm. Comp. and Chem. Eng. 9, 3. Fisher W. Doherty M. and Douglas J., 1985, Effect of Overdesign on the Operability of Distillation Columns, I&EC Proc. Des. Dev. 24, pp 593-598. Grossmann I.E. and Straub D.A, 1991, Recent Developments in the Evaluation and Optimization of Flexible Processes, Computer Oriented Process Engineering, Voi 10, pp. 49-59. Halemane K. and Grossmann I.E., 1983, Optimal Process Design under Uncertainty, AIChEJ. 29, 3. Hoch P. (1993). Disefio Optimo y Flexibilidad de Columnas de Destilaci6n, Phi) Thesis, Universidad Nacional del Sur, Bahia Blanca, Argentina. Hoeh P. and Eliceche A.M., 1991, Optimal Design of Non-Conventional Distillation Columns, Computer Oriented Process Engineering, Vol 10, pp 369-374. Hoch P. and Eliceche A.M., 1994, Sensitivity of Distillation Column Design to Uncertain Parameters, 1ChemE Symp. Set. 133, pp. 459-.466. Hoch, P.M., A. M. Eiiceche and I. E. Grossmann, 1995, Evaluation of Design Flexibility in Distillation Columns, Comp. and Chem. Eng., Vol 19, pp. $669-674. Kabatek, and Swaney R., 1992, Worst-case Identification in Structures Process Systems, Comp. and Chem. Engn. 3, p 473. Kubic W.L. and Stein F.P., 1988, A Theory of Design Reliability using Probability and Fuzzy Sets, AIChE J. 34, 4, pp. 583-601. Swaney R. and Grossmann I.E., 1985, An Index for Operational Flexibility in Chemical Process Design. Part I: Formulation and Theory. AIChEJ. 31, pp. 621-630.