HEN Retrofitting Methodology

Methodology for solving HEN retrofitting problems can be divided into a series of steps: representation/model of HEN, formulation of optimization problem, and assessment of optimization results. The present study employs nodal representation of HEN (Bochenek and Jezowski, 2010). In this, ‘nodes’ or potential sites for heat exchange, stream splitting or mixing are placed on both hot and cold process streams, and these are used to represent the HEN structure. For example, a heat exchanger between 4^th node of hot stream 2 and 6^th node of cold stream 1 is represented as one row: (2 4 1 6) in the structural matrix. The entire HEN is thus represented as a number of rows in the structural matrix. This representation has provision for user to choose number of nodes on a stream based on existing HEN topology. Nodal representation of HEN uses two other matrices for splits and their fractions. Readers are referred to Sreepathi and Rangaiah (2016) for details on these matrices.

HEN model calculations employ the following basic equations for each heat exchanger in the network, to calculate nodal temperatures. The assumptions in these calculations are constant heat capacity and heat transfer coefficient, counter-current flow and steady state.

Q = CP T − T = CP T − T = UA∆T (6.13)

∆T =^{( ) ( )}

( )

( )

(6.14)

Here, CP is the product of flow rate and heat capacity, T is the stream temperature of hot/cold stream at in/out (indicated by super/subscripts) of the exchanger, U is the overall heat transfer coefficient and A is the heat transfer area. For calculating stream outlet temperatures for known

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stream inlet temperatures, Equations 6.13 and 6.14 can be combined as follows.

Fig. 6.1 Flowchart of the MODE-R program; see Section 6.2 for details

Set values of F, Cr, NP, MNG and TR

Set generation no., G = 1

Set individual no., i = 1

Generate a mutant individual (v), using three randomly selected individuals from initial/current/parent

population and v_i = x_r0 + F(x_r1 – x_r2).

Generate a trial individual (u) as follows.

If rand (0,1) ≤ Cr or j = jrand, then ui,j = vi,j, else ui,j = xi,j

Check the trial individual for violation of decision variable bounds; if there is any violation, randomly re-initialize that

particular decision variable inside the bounds.

Evaluate values of objective and constraints for trial

individual.

Is trial individual near to any individual in tabu

list, by TR?

Combine parent and child populations

Select population for next generation by non-dominating sorting of combined population and

crowding distance calculations, if required.

Yes

Yes

No

Include trial individual into child population. Update tabu

list with the trial individual.

Randomly initialize population using xj = xjL + rand (0,1)(xjU – xjL) Evaluate values of objective and constraints for all individuals (x),

and randomly fill the tabu list.

i = i + 1 Start

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T = α T + 1 − α T (6.15a)

T = 1 − α T + α T (6.15b)

where

α = ^{( )}

( ) (6.16a)

α = ^{( )}

( ) (6.16b)

γ = and γ = (6.16c)

Using the above equations and an iterative procedure, temperatures at all nodes in the HEN are calculated. Then, existing exchangers are assigned suitably (e.g., considering area) to exchangers in the HEN design generated by the optimizer. This is to re-use existing exchangers in order to reduce investment needed. More details on all these with (flow charts) and illustrative example are available in Section 7.4 in Sreepathi and Rangaiah (2016).

Equations 6.15 and 6.16 are used to solve problems involving streams with constant heat capacity. For handling variable heat capacity problems, Sreepathi and Rangaiah (2015) have developed an approach based on the following cubic equation and fixed point iteration.

H = H + aT + bT + cT (6.17) The present chapter considers both constant and variable heat capa-city problems, and the respective procedures for calculating nodal temperatures are taken from Sreepathi and Rangaiah (2015, 2016). It also employs reassignment of existing exchangers along with a practical limitation on the additional area of 15%, i.e., for an existing heat exchanger of 100 m², maximum additional area is 15 m². For full details on this, readers are referred to Sreepathi and Rangaiah (2014b).

The two objectives used for all HEN retrofitting case studies are utility cost after retrofitting and investment cost for retrofitting. Utility cost is given by:

Utility cost = (∑ Q C ) + (∑ Q C ) (6.18) Here, Chot and Ccold are the unit cost of hot and cold utilities respectively;

Nheater and Ncooler are respectively the number of heaters and coolers in the

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HEN; Qj and Qk are respectively the amount of hot and cold utilities required by j^th heater and k^th cooler. The investment cost is given by:

Investment cost = (∑ A C ) + ∑ A C (6.19) Here, Al is the additional area required for l^th existing heat exchanger and C^add is the unit cost of additional area; Nex is the number of exchangers requiring area addition; Nnew is the number of new heat exchangers installed; Am and Cnew are respectively the area of m^th new heat exchanger and its cost, respectively. Utility and investment costs are expected to be conflicting since suitable changes in HEN and consequently investment for it are needed to reduce utilities required and their cost. Results presented later in Figs. 6.2 to 6.6 confirm this.

Decision variables in HEN retrofitting optimization are elements of the structural matrix, areas of all heat exchangers and split fractions (if there are split streams). Of these, elements of the structural matrix are integer variables, and others are continuous variables. All these variables have suitable bounds. In addition, constraints on approach temperatures are present to satisfy MAT of an exchanger. Thus, for a network with N exchangers (excluding heaters and coolers), there will be 2N constraints on approach temperatures.

MOO for HEN retrofitting is performed using MODE-R, and the results are compared with those using NSGA-II program in MATLAB (NGPM). For each case study, MOO program was run 5 times with a population (NP) of 200 and maximum number of generations (MNG) of 1000; this was carried out by both MODE-R and NGPM programs.

The graphs presented in this chapter show only the non-dominated solutions at particular generations obtained in the 5 runs by the indicated program. For example, the Pareto-optimal front after 200 generations of MODE-R for a particular case study, is obtained by combining all the solutions of the 5 runs of MODE-R program after 200 generations, and then selecting only the non-dominated solutions. Note that optimization calculations for all 5 runs were performed for MNG of 1000 in order to obtain Pareto-optimal front at intermediate generations such as after 200, 400, 600 and 800 generations.

For comparison of the results at different generations, the best Pareto-optimal front (or known Pareto-optimal front) for each case study is obtained by selecting the non-dominated solutions from all the solutions found in four additional runs. These are two runs of NGPM with a larger population of 2000 and MNG of 2000, and two runs of

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MODE-R with a larger population of 1000 and MNG of 1000. The larger values chosen for NGPM are based on the results presented later, which showed that the Pareto front obtained by NGPM is sub-optimal compared to that by MODE-R.

Three multi-objective performance metrics are used in this work, to compare the non-dominated solutions obtained by MODE-R and NGPM programs. They are convergence metric (CM), spread extreme (SPe) and spread distribution (SPd). CM is calculated between the non-dominated

solutions obtained in the current generation and the best/known Pareto-optimal front for the case study (Gong and Cai, 2009).

CM = ^∑ (6.20)

Here, NDS is the number of non-dominated solutions in the current generation, and di is the Euclidean distance of each of these solutions to its nearest non-dominated solution in the best Pareto-optimal front.

CM indicates progression of the Pareto front over generations, and its ideal value is zero when the obtained Pareto-optimal front overlaps the best Pareto-optimal front.

SPe is the sum of the Euclidean distances between the extreme solution (of an objective function) of the current generation and corresponding extreme solution of the best/known Pareto-optimal front.

SP = ∑ δ(e , S) (6.21) Here, M is the number of objective functions (2 for all case studies in this work), and (em, S) is the Euclidean distance between the extreme solution of m^th objective function in the current generation and that in the best/known Pareto-optimal front. SPe measures how well the extreme solutions of the obtained Pareto-optimal front are found by the MOO technique. Ideal value of SPe is zero, when the extreme solutions obtained in the current generation coincide with those of the best Pareto-optimal front.

SPd is the sum of absolute differences between the Euclidean distance of two neighboring points and the average of such distances for all non-dominated solutions in the current generation. Hence, it does not require the best/known Pareto-optimal front.

SP = ∑ δ − δ (6.22)

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Here, j is the Euclidean distance of j^th non-dominated solution to its closest solution/neighbor, and δ is the average of j for all NDS non-dominated solutions of the current generation. SPd captures how evenly the non-dominated solutions in the current generation are distributed, and its ideal value is zero when all j are equal. Performance metric, ‘Spread’ in the literature (Deb, 2001) includes both SPd and SPe

in its definition; our experience showed that a lower value of ‘Spread’

does not necessarily mean that the obtained non-dominated solutions are closer to the extreme points in the best Pareto-optimal front, and have a wider range of objective function values. Hence, both these terms are individually calculated and compared for better understanding of the obtained fronts.