Features of the method 

Some of the features of simulated annealing as  found from literature [22] are as follows: 

1。 The quality of the final solution is not affected by the initial guesses, except that the computational effort may increase with worse starting designs。 

2。 Because of the discrete nature of the function and constraint evaluations, the convergence or transition characteristics are not affected by the continuity or differentiability of the functions。 

3。 For problems  involving  behavior constraints  (in addition to lower and upper bounds on the design variables), an equivalent unconstrained function  can  be formulated as in the case of genetic algorithms。 

Figure 2。 Simulated annealing procedure [22]。 

SA implementation 

The objective function is the minimization of HE cost Obj(x) given in Eq。 (44) and x is a solution string representing a design configuration。 

The design variable x1 takes 12 values for tube outer diameter in the range of 0。00635m to 0。0635m (corresponds to 0。25–2。5”; refer to Table 1 for exact discrete values); the variable x2 takes eight values of the various tube lengths in the range 1。2192m to 7。3152m represented by numbers 1 to 8; x3 takes seventeen values for the variable baffle spacing, in the range 0。2 to 1 times the shell diameter; x4 takes number of tube passes 1-1, 1-2, 1-4, 1-6, 1-8 repre- sented by numbers from 1 to 5; x5 represents the tube pitch - either triangular or square - taking two values represented by 1 and 2; x6 takes the shell head types: fixed tube sheet or U tube, outside packed head, split ring floating head, and pull through floating head re- presented by the numbers 1, 2, 3 and 4, respectively; x7  takes four values for the baffle cut in the range  15 to 45%。 

In the present study, the geometric constraints,  velocity and pressure drop on the fluids exchanging  heat is considered to be the feasibility constraint。   For a given design configuration, whenever any of the above constraints exceeds the specified limit, a high value for the heat exchanger cost is returned through penalty(x) function (refer Eq。 (44)) so that as an in- feasible configuration it will be eliminated in the next iteration of the optimization routine。 The total number of design combinations with these variables are  128175244 = 261120。 This means that if an  exhaustive search is to be performed it will take at the maximum 261120 function evaluations before arriving at the global minimum heat exchanger cost。 So the strategy which takes few function evaluations is the  best one。 Considering minimization of heat exchanger cost as the objective function, simulated annealing technique is applied to find the optimum design confi- guration with geometry, velocity and pressure drop as the constraint。 

The code was developed in a Matlab environ- ment。 The main steps of the approach are shown in Figures 1 and 2。 

Case studies 

The effectiveness of the present approach using SA algorithm is assessed by analyzing three case studies。 

Case 1: 4。34 MW duty, methanol- brackish wa- ter exchanger [15]。 

Case 2: 1。44 MW duty, kerosene-crude oil ex- changer [14]。 

Case 3: 4。54 MW duty, oil-cooling water ex- changer [19]。 

The first two case studies were analyzed by Caputo et al。 [6] using GA approach and taken from  literature [15,14]。 The third case studies was analyzed by Ponce-Ortega et al。 [16] using GA approach and taken from the literature [19-21]。 

The original design specifications, shown in Table 2, are supplied as inputs to the described SA  algorithm for each of the three cases。 The resulting optimal exchanger architectures obtained by SA are  compared with the results obtained by Caputo et  al。 [6] using GA approach and with original design so- lution given by Sinnot [15] and Kern [14]。 In order to allow a consistent comparison, cost functions of first two approaches are computed as described (Eq。 (29)-