Transshipment models

Step 2. Formulate and solve the LP transshipment model of Papoulias and Grossmann (1983) for each period of operation to determine the minimum utility cost and pinch location for each period. 

Step 3. Formulate and solve the multiperiod MILP transshipment model of Floudas and Grossmann (1986) to determine a minimum set of stream matches for feasible operation in all the periods and the heat transferred in each match in each period. 

Step 5. Derive the multiperiod superstructure based upon the matches and heat transferred in each match predicted by the multiperiod MILP transshipment model. Formulate an NLP to optimize the superstructure (Floudas and Grossmann, 1987a) to give the HEN structure and exchanger sizes which minimize investment cost. 

Al) Energy balances on each hot process and utility stream in each temperature interval (TI) of the multiperiod MILP transshipment model. Each energy balance involves the residuals (heat cascaded) to and from the TI and the heat transferred in each stream match in the TI. 

By formulating and solving the multiperiod MILP transshipment model for these two periods of operation, the following set of matches is identified ... 

Target (i) without constraints on the matches was addressed via the feasibility table (Hohmann, 1971), the problem table analysis (Linnhoff and Flower, 1978a), and the T - Q diagram (Umeda et al., 1979). For unconstrained and constrained matches rigorous mathematical models were developed, namely the LP transportation model of Cerda et al. (1983), its improvement of the LP transshipment model of Papoulias and Grossmann (1983) and the modified transshipment model of Viswanathan and Evans (1987). 

Target (ii) was addressed rigorously by Cerda and Westerberg (1983) as a Mixed Integer Linear Programming MILP transportation model and by Papoulias and Grossmann (1983) as an MILP transshipment model. Both models determine the minimum number of matches given the minimum utility cost. 

The target of minimum utility cost in HENs can be formulated as a linear programming LP transshipment model which corresponds to a well known model in operations research (e.g., network problems). The transshipment model is used to determine the optimum network for transporting a commodity (e.g., a product) from sources (e.g., plants) to intermediate nodes (e.g., warehouses) and subsequently to destinations (e.g., markets). 

Papoulias and Grossmann (1983) drew the analogy between the transshipment model and the HEN, which is shown in Table 8.2. Using this analogy, heat is considered as a commodity which is transferred form the hot process streams and hot utilities to the cold process streams and cold utilities via the temperature intervals. The partitioning procedure discussed in the previous section allows only for feasible transfer of heat in each temperature interval (see also the remarks of section 8.3.1.3). 

Figure 8.4 shows pictorially the analogy between the transshipment model and the heat exchanger network. The nodes on the left indicate the sources while the nodes on the right denote the destinations. The intermediate nodes, shown as boxes, are the warehouses. The simple arrows denote the heat flow from sources to warehouses and from the warehouses to destinations, while the highlighted arrows denote the heat flow from one warehouse to the one immediately below. 

Having presented the pictorial representation of the transshipment model we can now state the basic idea for the minimum utility cost calculation. 

Then the transshipment model for minimum utility cost is... 

Remark 11 Cerda et al. (1983) first proposed the transportation model for the calculation of the minimum utility cost, and subsequently Papoulias and Grossmann (1983) presented the transshipment model PI which requires fewer variables and constraints than the transportation model. 

This example is a modified version of illustration 8.3.2, and its data are shown in Table 8.3. Since we have two hot utilities and HP steam is the hottest hot utility we treat the intermediate hot utility (i.e., hot water) as a hot stream for the partitioning into temperature intervals. Then the pictorial representation of the transshipment model becomes (see Figure 8.7) ... 

In the previous section we discussed the minimum utility cost target and its formulation as an LP transshipment model. The solution of the LP transshipment model provides ... 

Papoulias and Grossmann (1983) proposed an MILP transshipment model for the formulation of the minimum number of matches target. This model is applied to each subnetwork of the HEN problem. 

The basic idea in the transshipment model for the minimum number of matches target is to model explicitly the potential heat exchange between all pairs of streams (excluding hot utilities to cold utilities) with respect to... 

Figure 8.8 Graphical representation of TJ - k for the MILP transshipment model...

The appropriate definition of the lower and upper bounds can have a profound effect on the computational effort of solving the model P2. In fact, the tighter the bounds, the less effort is required, even though the same solution can be obtained for arbitrarily large Uij. Finally, the nonnegativity and the top and bottom residual constraints are also linear. The variables are a mixed set of continuous and binary variables. Therefore, P2 corresponds to a mixed-integer linear programming MILP transshipment model. 

The MILP transshipment model P2 can be modified so as to correspond to a minimum utility cost calculation with restricted matches. The key modification steps are... 

In the top temperature interval only the matches between the hot process streams and the cold process streams can take place. In the bottom interval we have the matches of Hi, H2, HZ with Cl and CW only since C2 does not participate in this TI. As a result we need to introduce 12 instead of 18 continuous variables Qijk. We need to introduce nine binary variables for the aforementioned potential matches. The MILP transshipment model P2 is ... 

Solving this MILP transshipment model with GAMS/CPLEX and applying integer cuts, the following four global solutions are obtained each with five matches, but different distributions of heat loads ... 

Another approach is the modify the MILP transshipment model P2 so as to have preferences among multiple global solutions of model P2 according to their potential of vertical heat transfer between the composite curves. Such an approach was proposed by Gundersen and Grossmann (1990) as a good heuristic and will be discussed in section 8.3.3. 

A number of research groups suggested that in certain cases it is desirable to allow for matches between hot-to-hot, and cold-to-cold process streams (Grimes etai, 1982 Viswanathan and Evans, 1987 Dolan et al., 1989). It is worth noting that the MILP transshipment model of Papoulias and... 

Note that these expressions are piecewise linear in Tcf4 and T< and hence they will not change the nature of the transshipment model formulations for... 

As we discussed on Section 8.3.2.2, the MILP transshipment model may have several global solutions which all exhibit the same number of matches. To establish which solution is more preferable, Gundersen and Grossmann (1990) proposed as criterion the vertical heat transfer from the hot composite to the cold composite curve with key objective the minimization of the total heat transfer area. 

The basic idea in discriminating among multiple global solutions of the MILP transshipment model of Papoulias and Grossmann (1983) is to use as criterion the verticality of heat transfer,... 

To formulate the modified transshipment model based on the above idea we introduce the variables Sij which should satisfy the constraints... 

The modified MILP transshipment model that favors vertical heat transfer is of the following form ... 

This example is the same example used as illustration of the MILP transshipment model P2 in section 8.3.2.2. The constraints of P2 were presented, and hence we will focus here on the ... 

In section 8.3.2.2 we discussed the MILP transshipment model for the minimum number of matches target. For a given minimum utility cost solution (i.e., given HRAT s, QSi, QWj, location of pinch points and hence subnetworks), the solution of the MILP transshipment model, which is applied to each subnetwork provides ... 

The same type of information is also provided by the vertical MILP transshipment model discussed in section 8.3.3.3 which discriminates among equivalent number of matches using the assumption of vertical heat transfer. 

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