1164 W. Xiao et al.

1164 W. Xiao et al. September, 2009 be heated. Initial temperatures Tin and target temperatures Tout are given. Heat capacity flow rates f and heat transfer film coefficients of stream k are also specified. A set of hot utilities, HU, and a set of cold utilities, CU, and their corresponding temperatures as well as their relevant cost data are also given. Cost data are given for possible heat transfer equipment, including fixed and area-dependent cost factors that may also include piping and installation costs. Then the objective is to determine the HEN configuration that minimizes the annual cost. And utility loads, heat exchange areas, the number of units, heat loads and operation temperatures of every heat exchanger, stream matches and flow rates of every branch stream are all included. The superstructure mathematical model of HEN can be formulated as an MINLP problem; moreover, it does not depend on the pinch technology, so simultaneous optimization of all the economic indicators of HEN can be achieved. The superstructure is a stagewise representation of a structure embedded with all possible matches between any two streams within a stage. And in the superstructure the number of stages is Nk=max{NH, NC}. Only one match is defined for the same hot and cold stream pair in a stage, hence in each stage the maximum allowable number of heat exchangers is NH× NC. The heaters and coolers are outside the superstructure stages for simplification. Fig. 3 shows an illustration of the superstructure for a synthesis problem with two hot streams and two cold streams. To formulate the mathematical model of HEN for the proposed superstructure described previously, the following definitions are necessary. qcu, i is cold utility load of every hot stream i, and qhu, j is the hot utility load of every cold stream j. Outlet temperatures of the splits of every hot process stream i and cold stream j of a match in the stage k matching heat exchange are represented as thi, j, k and tci, j, k respectively, the related heat capacity flow rates are fhi, j, k and fci, j, k and the heat load of heat exchange matches is qi, j, k. thi, k represents inlet temperature of hot stream i in k stage, while tcj, k represents outlet temperature of cold stream j in k stage after whose splits are all mixed together. THin and TCin are initial temperatures of hot and cold streams, THout and TCout are objective temperatures of hot and cold streams, respectively. Therefore, superstructure mathematical model of HENs can be established as follows: HEN synthesis subject to: (1) Overall heat balance of each stream (9) (2) Heat balance of every heat exchanger (10) (3) Mass and heat balance at stage k For each hot stream: (11) For each cold stream: (12) (4) Assignment of superstructure inlet temperature (13) (5) Feasibility of temperatures (14) (6) Cold and hot utilities loads (15) (7) Minimum approach temperature constraints For each heat exchanger: (16) For hot utility: (17) Where thu, j, in and thu, j, out are inlet and outlet temperatures of hot utility matching with cold stream j, respectively. For cold utility: (18) Where tcu, i, in and tcu, i, out are inlet and outlet temperatures of cold utility matching with hot stream i, respectively. (8) Other constraints The continuous variables (thi, j, k, tci, j, k, fhi, j, k, fci, j, k, qi, j, k, thi, k, tcj, k, qcu, i, qhu, j) should be nonnegative. (thi, j, k, tci, j, k, fhi, j, k, fci, j, k, qi, j, k, thi, k, tcj, k, qcu, i, qhu, j)≥0 (19) The binary variables (yi, j, k, ycu, i, yhu, j) denote the existence of exchangers, coolers and heaters. (yi, j, k, ycu, i, yhu, j)∈(0, 1) (20) Objective function of HEN synthesis: To simultaneously optimize HEN, the objective function is written as total annual cost, which includes utilities cost, fixed capital cost and area cost of heat exchangers. The cost equation of heat exchange equipments (including heat exchangers, coolers and heaters) is Cf+C·AB, where the first item Cf is fixed cost of heat exchanger, the second one is area cost of heat exchanger, and C, A and B are area cost coefficient, heat transfer area and exponent for area cost, respectively. The objective function of simultaneous synthesis is presented as follows. (21) Where ccu and chu are per unit costs for cold and hot utilities, respectively. The area of any match (i, j, k) (including heaters and coolers) can be calculated according to Eq. (22). Ai, j, k=qi, j, k/(Ki, j·LMTDi, j, k) (22) Kij is the overall heat transfer coefficient of the match in between (THin, i − THout,i) fhi = qi, j, k + qcu, i, i∈NH j Σ k Σ ⋅ (TCout,i − TCin, i) fcj = qi, j, k + qhu, j,j∈NC i Σ k Σ ⋅ (thi, k − thi,j,k)⋅fhi, j, k = qi,j, k (tci, j, k − tcj, k+1)⋅ fci, j, k = qi, j, k i∈NH, j∈NC, k∈NK fhi,j,k = fhi, thi,j,k ⋅ fhi,j, k = thi, k+1 ⋅ fhi, i∈NH, k∈NK j Σ j Σ fci, j, k = fcj, tci, j, k ⋅fci,j,k = tcj, k ⋅fcj, j∈NC, k∈NK i Σ i Σ THin,i = thi,0, i∈NH; TCinj = tcj,Nk, j∈NC thi, k≥thi, j, k, tcj, k+1≤tci,j,k, THout,i≤thi,Nk, TCout,j≥tcj,0 i∈NH, j∈NC, k∈NK (thi,Nk − Tout, i)⋅fhi = qcu, i, i∈NH; (TCout,j − tcj,0)⋅fcj = qhu, j, j∈NC thi, k − tci,j,k≥Δtmin, thi,j,k − tcj, k+1≥Δtmin, i∈NH, j∈NC, k∈NK thu, j, in − TCout,j≥Δtmin, thu,j,out − tcj,0≥Δtmin, j∈NC thi,Nk − tcu, i,out≥Δtmin, THout,i − tcu, i, in≥Δtmin, i∈NH min Ccu i Σ qcu, i + Chu j Σ qhu,j + Cfi,j k Σ j Σ i Σ yi, j, k + Cfcu,i i Σ ⋅ ⋅ ⋅ ⋅ycu, i + Cfhu, j j Σ yhu, j + Ci,j k Σ j Σ i Σ Ai, j, k Bi, j ⋅ ⋅ ⋅yi, j, k + Ccu, i i Σ Acu,i Bcu,i ycu, i + Chu,j j Σ Ahu,j