New rigorous one-step MILP formulation for heat exchanger network synthesis

In this paper, a rigorous MILP formulation for grass-root design of heat exchanger networks is developed. The methodology does not rely on traditional supertargeting followed by network design steps typical of the Pinch Design Method, nor is a non-linear model based on superstructures, but rather gives cost-optimal solutions in one step. Unlike most models, it considers splitting, non-isothermal mixing and it counts shells/units. The model relies on transportation/transshipment concepts that are complemented with constraints that allow keeping track of flow rate consistency when splitting takes place and with mechanisms to count heat exchanger shells and units. Several examples from the literature were tested, finding that the model usually obtains better solutions. In some cases, the model produced unknown solutions that were not found using superstructure optimization methods, even when the same pattern of matches is used. © 2005 Published by Elsevier Ltd. Keyword: Heat exchanges networks 1. Model for grass-root synthesis The problem of designing heat exchanger networks is one of the oldest problems in process synthesis and perhaps the one that has received the largest attention. The reader is referred to recent books (Biegler, Grossmann, & Westerberg, 1997; Seider, Seader, & Lewin, 1999; Shenoy, 1995; Smith, 1995) for the complete background on all the variety of methodologies developed throughout the years. In addition, the reader may consult three reviews on the topic of HENS by Furman and Sahinidis (2002), Gundersen and Naess (1988) and Jezowski (1994a, 1994b). A well-known pinch design method emerged throughout the years as the easiest response to the challenge. It relies on two steps, energy supertargeting and final network design. Energy supertargeting tries to determine the trade off between energy and area cost before attempting the design. Once this trade off is determined, a single minimum approach temperature (HRAT) is established and a design is performed, by starting to place matches at the pinch and using a tick-off rule (Linnhoff and Hindmarsh, 1983; Smith, 1995). Designs obtained using the pinch design methodology have been shown to be non-optimal. To ameliorate some of the shortcomings of the pinch design method, an alternative minimum temperature difference, the exchanger minimum approach temperature (EMAT) was introduced and used. At the same time, superstructure-like non-linear mathematical programming models started to be proposed. A large variety of methodologies have been developed after these initial approaches using several alternative objective functions in sequential and one step, as well as iterative procedures. All these formulations are thoroughly reviewed by Furman and Sahinidis (2002). ∗ Corresponding author. Tel.: +1 405 3255458; fax: +1 405 3255813. E-mail address: [email protected] (M.J. Bagajewicz). 0098-1354/$ – see front matter © 2005 Published by Elsevier Ltd. doi:10.1016/j.compchemeng.2005.04.006 1946 A. Barbaro, M.J. Bagajewicz / Computers and Chemical Engineering 29 (2005) 1945–1976 Nomenclature Sets B {(i, j)| more than one heat exchanger is permitted between hot stream i and cold stream j} Cz {j|j is a cold stream present in zone z} C n {j|j is a cold stream present in temperature interval n in zone z} CUz {j|j is a cooling utility present in zone z} (CUz⊂Cz) Hz {i|i is a hot stream present in zone z} H m {i|i is a hot stream present in temperature interval m in zone z} HUz {i|i is a heating utility present in zone z} (HUz⊂Hz) Mz {m|m is a temperature interval in zone z} mi {m|m is the starting temperature interval for hot stream i} m f i {m|m is the final temperature interval for hot stream i} M i {m|m is a temperature interval belonging to zone z, in which hot stream i is present} nj {m|m is the starting temperature interval for cold stream j} n f j {m|m is the final temperature interval for cold stream j} N j {n|n is a temperature interval belonging to zone z, in which cold stream j is present} NIH {i| non-isothermal mixing is permitted for hot stream i} NIC {j| non-isothermal mixing is permitted for cold stream j} P {(i, j)| a heat exchange match between hot stream i and cold stream j is permitted} P im {i| heat transfer from hot stream i at interval m to cold stream j is permitted} P jn {j| heat transfer from hot stream i to cold stream j at interval n is permitted} SH {i| splits are allowed for hot stream i} SC {j| splits are allowed for cold stream j} Z {z|z is a heat transfer zone} Parameters Azijmax maximum shell area for an exchanger matching hot stream i and cold stream j in zone z c ij variable cost for a new heat exchanger matching hot stream i and cold stream j c ij fixed charge cost for a heat exchanger matching hot stream i and cold stream j c i cost of heating utility i c j cost of cooling utility j Cpim heat capacity of hot stream i at temperature interval m Cpjn heat capacity of cold stream j at temperature interval n Fi flow rate of hot process stream i Fj flow rate of cold process stream j F i upper bound for the flow rate of heating utility i F j upper bound for the flow rate of cooling utility j hjn film heat transfer coefficient for cold stream j in interval n him film heat transfer coefficient for hot stream i in interval m H z,H im enthalpy change for hot stream i at interval m of zone z H z,C jn enthalpy change for cold stream j at interval n of zone z q ijm lower bound for heat transfer from hot stream i at interval m to cold stream j q ijn lower bound for heat transfer from hot stream i to cold stream j at interval n Ti temperature range of stream i Tj temperature range of stream j T m upper temperature of interval m T m lower temperature of interval m TML mn mean logarithmic temperature difference between intervals m and n A. Barbaro, M.J. Bagajewicz / Computers and Chemical Engineering 29 (2005) 1945–1976 1947 Variables Azij area for an exchanger matching hot stream i and cold stream j in zone z  z,k ij area of the kth exchanger matching hot stream i and cold stream j in zone z G z,k ijm auxiliary binary variable that determines whether the k-th exchanger between hot stream i with cold stream j in zone z exists at interval m of when (i, j)∈B. K z,H ijm determines the beginning of a heat exchanger at interval m of zone z for hot stream i with cold stream j. Defined as binary when (i, j)∈B and as continuous when (i, j) / ∈B. K z,C ijn determines the beginning of a heat exchanger at interval n of zone z for cold stream j with hot stream i. Defined as binary when (i, j)∈B and as continuous when (i, j) / ∈B. K̂ z,H ijm determines the end of a heat exchanger at interval m of zone z for hot stream i with cold stream j. Defined as binary when (i, j)∈B and as continuous when (i, j) / ∈B. K̂ z,C ijn determines the end of a heat exchanger at interval n of zone z for cold stream j with hot stream i. Defined as binary when (i, j)∈B and as continuous when (i, j) / ∈B. q im,jn heat transfer from hot stream i at interval m to cold stream j at interval n in zone z q̄ z,H inm non-isothermal mixing heat transfer for hot stream i between intervals m and n in zone z q̄ z,C jmn non-isothermal mixing heat transfer for hot stream i between intervals m and n in zone z q̂ z,H ijm heat transfer from hot stream i at interval m to cold stream j in zone z q̂ z,C ijn heat transfer to cold stream j at interval n from hot stream j in zone z q̃ z,H ijm auxiliary continuous variable utilized to compute the hot side heat load of each heat exchanger when several exchangers exist between hot stream i and cold stream j in zone z q̃ z,C ijn auxiliary continuous variable utilized to compute the cold side heat load of each heat exchanger when several exchangers exist between hot stream i and cold stream j in zone z q z im,jn auxiliary continuous variable utilized to compute the area of individual heat exchangers between hot stream i with cold stream j in zone z when (i, j)∈B. U ij number of shells in the heat exchanger between hot stream i and cold stream j in zone z U z,k ij number of shells in the kth heat exchanger between hot stream i and cold stream j in zone z X im,jn auxiliary continuous variable equal to zero when an exchanger ends at interval m for hot stream i and at interval n for cold stream j. A value of one corresponds to all other cases. Y z,H ijm determines whether heat is being transferred from hot stream i at interval m to cold stream j. Defined as binary when (i, j) / ∈B and as continuous when (i, j)∈B. Y z,C ijn determines whether heat is being transferred from hot stream i to cold stream j at interval n. Defined as binary when (i, j) / ∈B and as continuous when (i, j)∈B. α z,H ijm auxiliary continuous variable equal to one when heat transfer from interval m of hot stream i to cold stream j occurs in zone z and it does not correspond to the beginning nor the ending of a heat exchanger. A value of zero corresponds to all other cases. α z,C ijn auxiliary continuous variable equal to one when heat transfer from hot stream i to interval n of cold stream j occurs in zone z and it does not correspond to the beginning nor the ending of a heat exchanger. A value of zero corresponds to all other cases. Furman and Sahinidis (2002) discuss the “strong need for the development of approximation algorithms”. This stems from the realization that heat exchanger network design is an NP-Hard problem (Furman & Sahinidis, 2001). They also suggest that the simplifying assumptions that have been used (“isothermal mixing, no split stream following through more than one exchanger and no stream bypass”) diminish the merits of some successful one-step methods. They call for a “truly complete formulation of the HENS problem without any simplifying assumptions.” Some efforts in this direction have been made by Jezowski, Shethna, & Castillo (2003), who proposed linear models. We believe that we are responding to that challenge to a good extent, both on the modeling aspect of limiting the simplifying assumptions to a minimum and proposing a MILP formulation that can be attractive from a computational standpoint. This MILP model is based on the transportation–transshipment paradigm and it has the following features: • counts heat exchangers units and shells; • approximates the area required for each exchanger unit or shell; 1948 A. Barbaro, M.J. Bagajewicz / Computers and Chemical Engineering 29 (2005) 1945–1976 • controls the total number of units; • implicitly determines flow rates in splits; • handles non-isothermal mixing; • identifies bypasses in split situations when convenient; • controls the temperature approximation (HRAT/EMAT or Tmin) when desired; • can address block-design through the use of zones; • allows multiple matches between two streams. All the above features are the result of a special transshipment/transportation scheme that is capable of precisely describing the structure of the network using different sets of binary variables. Consequently, the model has a remarkable ability to produce cost-optimal networks. The one-step structure of the formulation also presents important advantages in terms of user intervention demand and allows achieving a high degree of design flexibility. Contrasting with to the traditional two-step structure (Targeting/Supertargeting and Network Design) of most of the approximate methods, this new formulation directly gives cost-effective solutions at once. Although there have been attempt to establish one-step procedures based on mathematical programming (complete list provided by Furman & Sahinidis, 2002), our proposed procedure does not rely on any the simplifying assumptions used so far. In addition, unlike others, it is MILP and is reasonably fast. Several examples from the literature were tested, finding that the model usually obtains better solutions in terms of cost-optimality. In some cases, the model produced unknown solutions that were not found using superstructure optimization methods, even when the same pattern of matches is used. 2. Mathematical model 2.1. Set definitions We now proceed to outline the general philosophy of the model. For this purpose, let us define a number of different sets that w