Simulation and Design Methods for Multiphase Multistream Heat Exchangers

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1 Preprint, th IFAC Symposium on Dynamics and Control of Process Systems, including Biosystems Simulation and Design Methods for Multiphase Multistream Heat Exchangers Harry A. J. Watson * Paul I. Barton ** * Massachusetts Institute of Technology, Cambridge, MA 0239 USA ( hwatson@mit.edu). ** Massachusetts Institute of Technology, Cambridge, MA 0239 USA ( pib@mit.edu) Abstract: Multistream heat exchangers (MHEXs) are found in many energy intensive and industrially relevant cryogenic processes. However, design and optimization of such processes is rendered difficult by the inability to simulate MHEXs robustly, as most current flowsheet-level MHEX models solve only an energy balance with no constraint on second law feasibility. The new model described herein combines an extension of the classical pinch analysis algorithm with explicit dependence on the heat exchange area to formulate a nonsmooth equation system which can be solved for up to three unknown variables in an MHEX. The resulting model is further augmented to simulate realistic thermodynamic phenomena, such as phase change and equilibrium, which are also naturally described by additional nonsmooth equations. Keywords: Heat exchangers, Process models, Process simulators, Numerical methods, Energy. INTRODUCTION Multistream heat exchangers in cryogenic processes are notoriously difficult to model, simulate, and design. Such processes are by nature extremely energy intensive, and therefore stand to benefit greatly from accurate process optimization; however, without robust and flexible models for heat exchange unit operations, simulation and optimization of these processes cannot be performed. Taking liquefied natural gas (LNG) production as an example, the streams in the process MHEXs are usually both multicomponent and multiphase, which creates an even more challenging simulation problem, especially if the phases traversed in the exchangers are not known ahead of time. The use of process simulation software is common in the literature involving processes with multistream heat exchangers. Commercial simulators employ proprietary models which generally permit solving for a single unknown variable, afforded by the energy balance, typically taken as the one of the exchanger outlet temperatures. However, as an example, the first author s experience using the MHEATX block from Aspen Plus R (Aspen Technology Inc., 204) suggests there are no rigorous checks in place to avoid heat exchange between two streams at very similar temperatures, or prevent temperature crossovers. A more rigorous approach to modeling MHEXs involves the use of a superstructure concept (Hasan et al., 2009). The major disadvantage of this methodology is that simulating the MHEX involves the solution of a very complex mixed-integer nonlinear program (MINLP) model, which is extremely challenging to find globally. The authors are grateful to Statoil for providing financial support for this research. Another method for robust modeling of MHEXs borrows heavily from pinch analysis and the analysis of composite curves. Kamath et al. (202) showed how to create a fully equation-oriented (EO) model for MHEXs by considering the unit operation as a heat exchanger network (HEN) that requires no external utilities. Applied to the Poly Refrigerant Integrated Cycle Operations (PRICO) process, the formulation results in a moderately-sized mathematical program with complementarity constraints (MPCC) (3426 equations using Soave-Redlich-Kwong thermodynamics) which requires completing a complicated initialization procedure to obtain a suitable initial guess from which to solve the problem, as well as a solution method suitable for MPCCs. Both of these rigorous approaches require the solution of a hard optimization problem to ensure the feasibility of a MHEX. It is also notable that the MHEX modeling literature rarely makes mention of the dependence of heat exchange area on the performance of the operation. Rather, the size is usually only calculated following determination of the output stream states, if at all. In contrast, this article describes a model and solution procedure for MHEXs which avoids returning non-physical solutions, doesn t rely on approximations or solving a hard optimization problem, incorporates information about the available heat exchange area directly into the procedure, and handles phase changes without needing to fix conditions a priori. 2. PRELIMINARIES This paper considers the modeling of countercurrent MHEXs in which a set of hot streams, indexed by a set H, exchange heat with a set of cold streams, indexed by a set C. Each hot stream i H has a constant molar heat capacity flowrate F Cp,i, enters at temperature T in and exits at temperature T out i, with T in i T out i i,. Similarly, Copyright 206 IFAC 839

2 each cold stream j C has a constant molar heat capacity flowrate f Cp,j, enters at temperature t in j, and exits at temperature t out j, with t in j t out j. The overall heat transfer coefficient of the MHEX is denoted by U and the available heat transfer area is denoted by A. The product UA is often used as the metric for the physical ability of a MHEX to exchange heat to avoid the need for accurate estimation of the value of U in early-stage design. The energy balance around this unit operation is given by: F Cp,i(Ti in Ti out ) f Cp,j(t out j t in j ) = 0. () i H j C However, it is not sufficient for an MHEX to satisfy only an energy balance all heat transfer must be both feasible in the sense of the second law and also possible in terms of the available area. While it is straightforward to write equations for these conditions in the two-stream heat exchanger case, such constraints do not immediately generalize to the multistream case. However, two equations, and the necessary algorithms for their evaluation, can be formulated to enforce these requirements, as detailed in Watson et al. (205) and summarized in the following section. The equations in this MHEX model are nonsmooth and require specialized algorithms to solve. Facchinei et al. (204) developed a linear program (LP) based Newton method that works well for such systems. At each step, this method solves the following LP to find the next iterate: min γ,u γ s.t. f(u k ) + G(u k )(u u k ) γ f(u k ) 2, (u u k ) γ f(u k ), u X, where G(u k ) is a generalized derivative of f at u k, γ is a variable introduced to drive convergence to the solution, and X is a set of known bounds on the solution. The u part of the solution is then used as the next iterate for the algorithm. In practice, the infinity norms are replaced by their equivalent linear reformulations. When f is piecewisecontinuously differentiable (PC ) in the sense of Scholtes (202), and G(u k ) is an element of the B-subdifferential (Qi, 993) of f at u k, the method can achieve local Q-quadratic convergence to the solution. The automatic method of Khan and Barton (205b) can be used to evaluate B-subdifferential elements for PC functions using a vector forward mode of automatic differentiation. Additionally, the LP-Newton method will not necessarily fail if G(u k ) is singular, unlike methods based on linear equation solving. This makes the method particularly well suited for the application of MHEX simulation. 3. A NEW MODEL FOR MULTISTREAM HEAT EXCHANGERS (2) This section summarizes the novel model formulation of Watson et al. (205). 3. Formulation of MHEX feasible heat transfer constraint As shown in Kamath et al. (202), a multistream heat exchanger can be viewed as a special case of a HEN where the external utilities are not present. Therefore, as in classical pinch analysis, heat transfer is considered feasible if the hot and cold composite curves corresponding to the streams in the MHEX are separated by at least some minimum temperature difference, ΔT min. With the additional goal of maximizing the energy transferred in the MHEX, a feasible solution corresponds to the vertical separation between the hot and cold composite curves being exactly ΔT min. This is equivalent to the more readily solved problem of reducing the horizontal distance between the composite curves at the pinch point to zero after applying a temperature shift of ΔT min to the cold curve. Adding nonphysical extensions of the curves which extend to the maximum and minimum temperatures existing in the heat exchanger allows this distance to be evaluated everywhere in this range, even if both of the true composite curves were not defined at every pinch candidate temperature. With these modifications, the hot and cold composite curve enthalpy values corresponding to each pinch candidate are defined using the following expressions: EBP p H = [ F Cp,i max{0, T p Ti out } max{0, T p Ti in } i H max{0, T min T p } + max{0, T p T max } ], (3) p P, EBP p C = [ f Cp,j max{0, (T p ΔT min ) t in j } j C max{0, (T p ΔT min ) t out j } (4) + max{0, (T p ΔT min ) t max } max{0, t min (T p ΔT min )} ], p P, where T max/min is the maximum/minimum hot stream temperature, t max/min is the maximum/minimum cold stream temperature, and P = H C is the set of pinch point candidates with temperatures, T p, defined by: { T p T in = i, p = i H, (5) + ΔT min, p = j C. t in j In (3) and (4), the last two terms correspond to the nonphysical extensions of the curves. After calculating EBP p C EBP p H, p P, the minimum horizontal distance between the extended composite curves is found by searching through the finite set P : min {EBP p C EBP p H } = 0. (6) p P The expression inside the min statement of (6) is negative at any pinch temperature where the hot composite curve lies to the right of the shifted cold composite curve, and positive where it lies to the left. 3.2 Formulation of MHEX area constraint An algorithm for calculating the heat exchange area for a MHEX on each iteration of an equation-solving procedure is described in this section. Let D be the index set for the points at which the composite curves are nondifferentiable, as well as their endpoints. For d D, let Q d denote the enthalpy value at this point, and construct the triple (Q d, T d, t d ), where T d is the temperature on the hot composite curve at Q d, and t d is the temperature on the cold composite curve at Q d. If all such triples are sorted in 840

3 order of nondecreasing Q d value, then an adjacent pair of triples in the list demarcates an interval of the composite curves in which part of the MHEX can be modeled as a two-stream heat exchanger. The total MHEX heat transfer area can then be found by summing over all such intervals: UA = d D d D ΔQ d ΔT d LM, (7) where ΔQ d = Q d+ Q d is the width of enthalpy interval d, D = 2( H + C ), and ΔTLM d is a modified version of the log-mean temperature difference across this same enthalpy interval which is defined later in this section. In standard practice, heat transfer area is calculated after heat integration is performed, and the the integrated composite curves are used to define the enthalpy intervals. However, by including (7) in the system of equations being solved simultaneously, an additional unknown for simulation is freed as the area can be specified in the problem input. However, the full sorted list of (Q d, T d, t d ) data must then be calculated at each iteration, starting from an incomplete list of just the inlet and outlet temperatures which are arranged in an arbitrary order. Sorting this list into nondecreasing order based on enthalpy value sets up the intervals for (7) correctly. If the sort is performed using bubble sort, then the only operations involve taking the max or min of two values. If the for loops in bubble sort run over the entire length of the list to ensure the same number of operations are performed for a given input size (regardless of how well-sorted the input is), then this sorting operation is a composite PC function which maps the unsorted input to the sorted output. The missing temperatures in the triples (e.g. T d ) are calculated as follows given the value of Q d : Q d ( F Cp,i max{0, T d Ti out } max{0, T d Ti in } ) = 0. i H (8) An analogous equation implicitly defines unknown cold curve temperatures. The solution value of these equations can be determined by searching and interpolating over the piecewise affine segments of the relevant composite curve. Calculating the correct generalized derivatives of the unknown temperature at the solution of (8) requires application of an implicit function result for PC functions from Khan and Barton (205a). Finally, the standard definition of the log-mean temperature difference must be modified so that evaluating the function never results in undefined behavior, as follows (based on Zavala-Río et al. (2005)): ΔTLM d = 2 (ΔT d +ΔT d+ ) if ΔT d = ΔT d+, ΔT d+ ΔT d ln(δt d+ ) ln(δt d ) otherwise, (9) where ΔT d = max{δt min, T d t d } is the temperature difference at the start of enthalpy interval d, and ΔT d+ = max{δt min, T d+ t d+ } is the temperature difference at the end of the interval. The if statement in the definition of the log-mean temperature difference is necessary to make the calculation defined for all possible inputs ΔT d > 0 and ΔT d+ > 0. Fortunately, since this function is continuously differentiable on the positive quadrant of R 2 (Zavala-Río et al., 2005), the if statement in (9) does not introduce complications for calculating correct derivatives. In summary, the set of equations describing the multistream heat exchanger now consists of (), (6), and (7). This is a nonsmooth equation system involving three equations in three unknowns. In particular, the left-hand side of each equation is a composite PC function, as each can be expressed as a finite composition of continuouslydifferentiable functions and PC functions such as the binary max and min functions. Including bounds on the temperature variables is also recommended to avoid nonrealistic solutions where hot streams have gained heat as if they were cold streams and vice-versa. 4. MODELING PHASE CHANGES IN MULTISTREAM HEAT EXCHANGERS In the context of heat integration and MHEX simulation problems, the challenge associated with a stream changing phase is modeling the change in its heat capacity flowrate. Pinch analysis techniques assume every stream has a constant heat capacity flowrate, and therefore an affine temperature-enthalpy relationship, but naïvely assuming this for a liquefying multicomponent refrigerant or natural gas stream would introduce significant error. Therefore, the temperature-enthalpy behavior of such streams must be approximated by a series of affine segments. A good choice is to approximate each of the three individual phase regions (superheated vapor, two-phase, and subcooled liquid) with one (or possibly more) affine segments. However, this is not straightforward in the general case where the process stream inlet/outlet temperatures, pressures, and compositions are possibly variables in the simulation. Kamath et al. (202) propose a disjunctive model in which the physical streams in the process are subdivided into substreams corresponding to superheated (sup), two-phase (2p), and subcooled (sub) regions. The heat integration calculations for the MHEX are then performed using these substreams instead of the physical process streams. However, instead of using either binary variables or complementarity constraints to assign the phase substream inlet/outlet temperatures as in Kamath et al. (202), the values of Tsup in/out, T in/out sub, and T in/out 2p can be expressed as continuous nonsmooth functions of the process stream temperature (T IN/OUT ), the dew point (T DP ), and the bubble point (T BP ) as follows: T in/out sup = max{t DP, T IN/OUT }, (0) T in/out 2p = mid{t DP, T IN/OUT, T BP }, () T in/out sub = min{t IN/OUT, T BP }, (2) where the function mid : R 3 R maps to the median of its three arguments and is also a PC function. The possible appearance and disappearance of phases from iteration to iteration also causes problems for vapor-liquid equilibrium calculations. Consider a typical steady-state flash operation, in which a feed stream with molar flowrate F with n c components at molar composition z separates into a liquid stream with molar flowrate L at molar composition x and a vapor stream with molar flowrate V at molar composition y. In the classical approach to flash 84

4 calculations, it is assumed that both liquid and vapor outlet streams exist and are in equilibrium, and the material balance and equilibrium relationships (expressed in terms of k i, the distribution coefficient for component i) are combined into the well-known Rachford-Rice equation (Rachford Jr. and Rice, 952). When the heat duty of the flash, Q flash, is specified (as will be the case in the MHEX simulations in this work), an energy balance is also needed. These two equations can then be solved via a Newton-type method for the vaporized fraction of the feed, V F, and the flash temperature, from which all other relevant quantities in the formulation can be recovered. However, under conditions where only one outlet stream exists, the equilibrium conditions cannot be satisfied. A new nonsmooth model which accounts for the appearance and disappearance of phases while performing flash calculations is given by solving the following equation instead of the Rachford-Rice equation: mid { V F, V nc F, i= z i (k i ) + V F (k i ) } = 0. (3) Here, the three arguments in the mid function correspond to finding an all liquid outlet, an all vapor outlet and a twophase outlet, respectively. Note that the third term is just the negative of the standard Rachford-Rice residual. The working mechanism of the equation is as follows. When the outlet is all vapor, V/F = and the Rachford-Rice expression is positive, so that here, the first term is equal to, the second term is equal to zero and the third term is negative. Thus, the mid expression picks the second term and evaluates to zero, satisfying (3) with V/F =. The arguments for the other phases are analogous. A final provision can be added in order to simulate streams that undergo very small (or no) temperature changes, as is the case with pure component phase changes or when phase substreams disappear based on the current value of the model variables. The inclusion of nonsmooth functions in this work allows this behavior to be modeled more precisely than in approaches which rely on smoothing approximations, while avoiding the need for binary variables or disjunctions. For instance, if it is likely that a cold substream will undergo a very small or no temperature change in the two-phase region during iteration, () can be replaced with the following expressions based on the discussion in Kamath et al. (202): t in 2p =min t out 2p { mid{t DP, T IN, T BP }, mid{t DP, T IN, T BP } + mid{t DP, T OUT, T BP } ε }, 2 (4) { =max mid{t DP, T OUT, T BP }, mid{t DP, T IN, T BP } + mid{t DP, T OUT, T BP } + ε }, 2 (5) where ε is a user-defined fictitious temperature change. Conventionally this approach is considered undesirable; however, in the specific context of nonsmooth equations, nonsmooth equation solvers will not be affected by the usual ill-conditioning caused by making such a parameter too small. Analogous equations can also be written for a hot two-phase substream, as well as the substreams in the other regions of the phase diagram. These nonsmooth model elements can be combined to simulate multiphase MHEXs in complex processes. In many cases, splitting a physical stream into just three substreams will be insufficient to capture its true temperatureenthalpy relationship. Therefore, as described in Kamath et al. (202), the superheated, subcooled, and two-phase substreams are further discretized into n sup, n sub, and n 2p affine segments, respectively, to improve the estimation of the nonlinear behavior. The substreams are subdivided into segments of equal enthalpy difference, and their inlet/outlet temperatures are therefore implicitly defined by the solution of energy balances. Given the temperature and heat load for each segment, a constant heat capacity flowrate can be determined by dividing the heat load of the segment by the difference between its inlet and outlet temperature. Note that the use of (4) and (5) will never allow this quotient to become undefined. These heat capacity flowrates are then used in evaluating (), (6), and (7). The size of this MHEX model is given by: n var = 3 + 2n str + n str [(n sup ) + (n sub ) + 2(n 2p )], (6) where n var is the number of variables/equations needed to model the MHEX, and n str is the number of physical process streams that enter the MHEX in the flowsheet. The first term of (6) accounts for the base MHEX model consisting of (), (6), and (7), the second term accounts for calculating the dew and bubble points of each stream involved in the heat exchanger, and the last term accounts for all the substream temperatures (and vapor fractions in the two-phase region) which must be calculated. Despite the reduction in model complexity as compared to other approaches, the problem remains difficult to solve. Improved robustness of the LP-Newton algorithm was observed by replacing all the infinity norms in (2) with the - norm, and changing the expression on the right-hand side of the first constraint to γ max { f(u k ), f(u k ) 2 }. The change of norm was motivated by the observation that the step size calculated by (2) at each iteration is bounded by unity if the largest residual value corresponds to an equation with a zero row in the generalized derivative. This occurs often with (6), as any temperature variables in the simulation will only influence the minimum distance between the composite curves over a limited range of values. The addition of the max term to the first constraint allows the method to take larger steps when f(u k ) >, which helps mitigate sensitivity to problem scaling since this method does not share the invariance to affine scaling exhibited by classical Newton methods. Finally, it is important that the initial guess provided be as near to the solution as possible to aid convergence, which necessitates a robust initialization procedure. In the procedure developed by the authors, only guesses for the three MHEX model variables and the bubble/dew point temperatures of each stream are required from the user. The bubble and dew point estimates are then refined first by solving their defining equations. Initial guesses for the remaining temperatures are obtained by assuming a linear relationship between temperature and enthalpy in each phase, and then improved by solving each of the energy 842

5 balances independently to generate a better estimate. Example shows the method of this paper applied to simulate the PRICO process under the assumption of ideal thermodynamic behavior and Raoult s Law. Example. Figure shows the PRICO (Maher and Sudduth, 975) process for producing liquefied natural gas. The PRICO process is a single-stage mixed refrigerant (MR) process. The MR stream serves as both a hot stream, high-pressure refrigerant (HPR), and as a cold stream, low-pressure refrigerant (LPR) by means of intermediate expansion and compression operations. Tables and 2 give Case I. In this case, the pressures and compositions in the flowsheet are held fixed. Let u THPR OUT, u 2 t OUT LPR, and u 3 UA be the unknown variables afforded by the base MHEX model consisting of (), (6), and (7). The throttle valve outlet temperature, u 4 t IN and vapor IN LPR fraction, u 5 V F, are given by performing a fixed LPR pressure/enthalpy flash calculation around the valve. The remainder of the variable set is comprised of the unknown temperatures and vapor fractions given by the solution of the energy balances around each affine segment. Using CPLEX v2.5 (IBM, 205) as the LP solver, the simulation converges to the solution with u * = K, u * 2 = K, u * 3 = 8.83 MW/K, u * 4 = 6.6 K, and u * 5 = 0. with f(u * ) < 0 9 after 56 iterations (0.63 seconds) starting from u 0 = 8.5 K, u 0 2 = 275 K, u 0 3 = 20 MW/K, u 0 4 = 6.95 K, u 0 5 = 0.0, and the rest of the initial guess calculated by the initialization subroutine. Figure 2 shows the composite curves at the solution. In spite of the piecewise-affine approximations, the composite curves show much of the true curvature, particularly around the natural gas stream bubble point (95.59 K) Fig.. Flowsheet of the idealized PRICO liquefaction process for natural gas. the data for the streams involved in the MHEX for the PRICO process under three sets of simulation conditions. In each case, n sup = 3, n 2p = 5, and n sub = 3 for all three process streams entering the heat exchanger. From (6), the MHEX is modeled with 45 equations and variables. Table. Natural gas stream data. Property Value Flowrate (kmol/s).00 Pressure (MPa) Inlet temperature (K) Outlet temperature (K) 8.5 Composition (mol %) N 2 CH 4 C 2 H 6 C 3 H 8 n-c 4 H Table 2. Refrigerant stream and MHEX data. Property Case I Case II Case III Flowrate (kmol/s) High pressure level (MPa) Low pressure level (MPa) u HPR inlet temp. (K) HPR outlet temp. (K) u LPR inlet temp. (K) u 4 u 4 u 4 LPR outlet temp. (K) u 2 u 2 u 2 Composition (mol %) Nitrogen Methane Ethane Propane u 0.4 n-butane u UA (MW/K) u T min (K).2 u 3 u 3 Temperature (K) Enthalpy (kw) Fig. 2. Composite curves for the MHEX in the PRICO process simulated under the conditions of Case I. Case II. In this case, the composition of the refrigerant mixture is allowed to vary while the U A value of the MHEX is fixed. Let u be the mole fraction of n-butane in the refrigerant, u 2 TLPR OUT and u 3 ΔT min with u 4 and u 5 as before. The simulation converges to the solution with u * = , u * 2 = K, u * 3 = 0.99 K, u * 4 = K, and u * 5 = with f(u * ) < 0 9 after 38 iterations (0.44 seconds) starting from u 0 = , u 0 2 = K, u 0 3 =.2 K, u 0 4 = 6.95 K, u 0 5 = 0.0, and the rest of the initial guess calculated by the initialization subroutine. Figure 3 shows the composite curves at the solution, which are pinched together more closely than in Case I due to the larger available heat transfer area. Case III. In this case, the pressure of the MR stream is allowed to vary while the UA value of the MHEX is fixed. Let u be the discharge pressure of the throttle valve, u 2 t OUT LPR and u 3 ΔT min with u 4 and u 5 as before. The simulation converges to the solution with u * = 0.7 MPa, u * 2 = K, u * 3 = 2.77 K, u * 4 = 2.67 K, and u * 5 = 0.25 after 4 iterations (.3 seconds) starting from u 0 = 0.7 MPa, u 0 2 = K, u 0 3 =.2 K, u 0 4 = 6.95 K, u 0 5 = 0.0, and the rest of the initial guess calculated by the initialization subroutine. Figure 4 shows 843

6 Temperature (K) Enthalpy (kw) Fig. 3. Composite curves for the MHEX in the PRICO process simulated under the conditions of Case II. the composite curves at the solution. Satisfying the low UA value specification requires the pressure at the throttle valve exit to approach atmospheric pressure, leading to greater temperature change across the valve and more separation of the composite curves. Temperature (K) Enthalpy (kw) Fig. 4. Composite curves for the MHEX in the PRICO process simulated under the conditions of Case III. In each case, the model is successfully able to simulate the PRICO process under ideal thermodynamics. Cases II and III are challenging simulation problems since many of the equations in the model are sensitive to the composition and pressure of the refrigerant stream. However, the new simulation strategy converges to the solution in each case without requiring significant computational time or effort. 5. CONCLUSION A method for simulating multistream heat exchangers both with and without phase changes has been presented which differs substantially from those found in the literature to date. The final formulation is a compact nonsmooth model which is solved with equation-solving methods, in contrast to those which require solving a difficult optimization problem. Accordingly, the model is significantly less complex and allows for realistic simulation of process flowsheets involving multiphase MHEXs outside of an optimization framework. The method can be extended to incorporate cubic equations of state using the EO formulation from Kamath et al. (200) and a physical property calculation package; however, performing such a simulation for processes larger than PRICO is extremely challenging, and future work involves investigating both a modular approach and improvements to the nonsmooth algorithm for nonideal flash calculations. ACKNOWLEDGEMENTS The authors wish to thank Kamil Khan for many helpful discussions regarding the mathematical background of this work. The authors are also grateful to Truls Gundersen at the Norwegian University of Science and Technology (NTNU) for his input on the project. REFERENCES Aspen Technology Inc. (204). Aspen Plus v8.4. Aspen Technology Inc., Cambridge, MA. Facchinei, F., Fischer, A., and Herrich, M. (204). An LP- Newton method: nonsmooth equations, KKT systems, and nonisolated solutions. Mathematical Programming, 46, 36. Hasan, M.M.F., Karimi, I.A., Alfadala, H.E., and Grootjans, H. (2009). Operational modeling of multistream heat exchangers with phase changes. AIChE Journal, 55(), IBM (205). IBM ILOG CPLEX v ibm.com/software/commerce/optimization/cplexoptimizer/index.html. Kamath, R.S., Biegler, L.T., and Grossmann, I.E. (202). Modeling multistream heat exchangers with and without phase changes for simultaneous optimization and heat integration. AIChE Journal, 58(), Kamath, R.S., Biegler, L.T., and Grossmann, I.E. (200). An equation-oriented approach for handling thermodynamics based on cubic equation of state in process optimization. Computers & Chemical Engineering, 34(2), Khan, K.A. and Barton, P.I. (205a). Generalized derivatives for hybrid systems. Submitted. Khan, K.A. and Barton, P.I. (205b). A vector forward mode of automatic differentiation for generalized derivative evaluation. Optimization Methods & Software, 30(6), Maher, J. and Sudduth, J. (975). Method and apparatus for liquefying gases. US Patent 3,94,949. Qi, L. (993). Convergence analysis of some algorithms for solving nonsmooth equations. Mathematics of Operations Research, 8(), Rachford Jr., H.H. and Rice, J.D. (952). Procedure for use of electronic digital computers in calculating flash vaporization hydrocarbon equilibrium. Journal of Petroleum Technology, 4(0), Scholtes, S. (202). Introduction to Piecewise Differentiable Equations. SpringerBriefs in Optimization, New York, NY. Watson, H.A.J., Khan, K.A., and Barton, P.I. (205). Multistream heat exchanger modeling and design. AIChE Journal, 6(0), Zavala-Río, A., Femat, R., and Santiesteban-Cos, R. (2005). An analytical study of the logarithmic mean temperature. Revista Mexicana de Ingeniería Químca, 4,