4. Results and Discussions
Some general comments are needed before discussing the results. It must be clear to the reader how the results change (
SI,
EI,
SOI and
ENI) depending on the variation that some parameters experiment. As explained in previous sections, the Sustainability Index (
SI) varies between 0 and 1, minimum and maximum levels of satisfaction. It also consists of three partial indices (economic (
EI), social (
SOI) and environmental (
ENI)). According to
Table 1, these partial indices vary between 0 and 0.28, between 0 and 0.33, and between 0 and 0.39, respectively. Optimising the
SI is not equal to optimise each one of the partial indices separately, since there are conflicts between the indicators.
Regarding the economic dimension, there is only one indicator for assessing the total life cycle costs of each STHE design. Nevertheless, this indicator consists of two main components: investment costs (
Cinv) and the total operating costs (
Codc).
Cinv depends on the STHE surface area (
A). Consequently, minimising
A is equivalent to minimising the investment costs (
Cinv). Nevertheless, the minimum investment cost is not necessarily associated with the minimum operating costs (
Codc). The reader can verify this statement by comparing some of the different cases presented in
Table 5,
Table 6,
Table 7 and
Table 8. The operating costs increase with the shell side and tube side pressure drops, since more pumping power
Pp is needed to overcome friction losses. The pressure drop increases as the fluid velocity and tube length increase. The opposite occurs with the tube internal diameter and with the equivalent one. For the particular case of the shell side pressure drop, the baffle distance (
B) as well as the shell internal diameter (
Ds) also come into play. Consequently, one may think that by minimising the total pressure drop, the optimum operating cost (
Codc) must be obtained. It is possible to say that this idea is true in a general context. However, there can be specific cases in which a lower value of total pressure drop is not associated with a lower operating cost (
Codc). This is due to the role that densities and mass flow rates play. After analysing the results presented in
Table 5,
Table 6,
Table 7 and
Table 8, one realises that, in the case study considered in this paper, the capital investment (
Cinv) is more important than the total operating costs (
Codc), since the first one is considerably higher than the second one. Consequently, the minimisation of the capital investment will prevail over the operating costs minimisation, whether the
SI or the
EI is optimised.
In the case of the social pillar, the requirement tree is made up of three indicators. Two of them are associated with possible accidents, while the remaining one assesses the employment generation potential. The three indicators depend on the amount of material (M). However, they do not follow the same trend, since a higher value of M implies a better performance in terms of employment creation, while the opposite occurs for the accidents. Therefore, it is possible to say that optimising the SOI is not equivalent to optimising each one of the indicators separately. As a result, the optimal STHE designs will not achieve strong results from a social point of view (far from the maximum: 0.33). On the other hand, it is important to note that the minimisation of two physical magnitudes (A and M) is associated with a better performance in terms of capital investment and accidents, respectively. At first sight, one may think that a higher value of A is always equivalent to a higher value of M. Nevertheless, there are specific examples in which a higher value of A implies a lower value of M. It can therefore be said that optimising a part of the social index (SOI), in particular the results for the accidents, are, to some extent, linked to optimising the most relevant parameter (Cinv) of the economic dimension.
With regard to the environmental dimension, all the indicators depend on the amount of material (M). Furthermore, all of them present the same behaviour: a lower value of M generates a better performance (less significant environmental impact). Optimising the environmental index (ENI) implies an optimisation of a part of the social dimension, in particular the one related to accidents. Similarly, optimising the ENI will not generate good results in terms of job creation, since these two aspects are in conflict. It is also possible to say that, to a certain extent, minimising the environmental impacts will generally result in a lower capital investment.
The main objective of this study is to maximise the contribution of the STHE under design to the sustainable development. As is clear from previous paragraphs, this is not an easy task, since the mathematical model is complex and different conflicts arise. Despite this, it is possible to intuit what is going to happen with the partial sub-indices when the sustainability index (
SI) is maximised. To this end, consideration must be given to the relative importance of the requirements in
Table 1. The most important dimension of sustainability is the environmental one (39%). Regardless of the case, weights and optimisation technique, each one of the optimal or sub-optimal designs should present an environmental index (
ENI) close to the maximum (0.39), when the
SI is optimised. The reader can confirm this statement by looking at
Table 5,
Table 6 and
Table 7. On the other hand, it has been explained that optimising the environmental dimension will usually lead to cost-effective solutions in terms of capital investment. As shown in
Table 5 and
Table 6, the economic indices (
EI) linked to the optimal or sub-optimal solutions are close to the maximum possible value (0.28), which confirms the previous statement. Finally, when the
SI is maximised, the corresponding social index (
SOI) is not expected to be close to the possible maximum value (0.33). Consequently, the optimal or sub-optimal designs will present
SOIs placed in half the way between the minimum and maximum possible values. In particular, the chosen solutions will be the ones that generate a reasonably high
SOI without unduly penalising the other two dimensions of sustainability. In fact, the
SOIs contained in
Table 5 and
Table 6 take values around 0.13.
4.1. Comparisons among the Different Cases
Table 5 contains some of the most important results for the different sets of weights for the first case (linear value functions, L).
In terms of sets of weights, there are no big differences among the six options considered, not only in the SI, where the differences appear in the third decimal, reaching a maximum value of 0.0011, but also in the values adopted by the optimisation variables. Ds varies between 0.70 and 0.74 m. Something similar happens with d0 and B, varying from 0.008 to 0.0081 m, and from 0.48 to 0.50 m, respectively. This also applies to the partial indices (EI, SOI and ENI), since the differences are also found in the third and fourth decimals. The amount of material hardly changes among the six cases. In fact, the highest variation occurs when Monte Carlo is used and it takes a value of 13.1 kg of stainless steel, which can be considered as insignificant in comparison with about 1200 kg for the STHE design. The total cost (C1) experiments the highest variation when CSA is used and the difference reaches a value of 226 €. Once again, this value is negligible. Although important differences are not found in the total cost (C1), greater variations appear if one compares only the capital investment (Cinv) and the total discounted operating costs (Codc), separately. These variations are founded on small differences in the optimisation variables and, consequently, on small differences in the fluid velocities, pressure drops, tube length, etc. Nevertheless, from the integral sustainability point of view, they are not important. In other words, two similar designs can generate a comparable total cost and a similar SI even if there are differences affecting some of their constructive parameters.
On the other hand, at the time of constructing the real STHE, the different sets of weights would lead to the same design, since normalised values are always preferable to avoid cost overruns.
Regarding the optimisation techniques, all of them provided very similar SIs, being the exhaustive search and PSO the best ones, with a slight advantage over the other options. In fact, these two techniques generate the same optimal design (Ds = 0.7151, d0 = 0.008 and B = 0.5 m). The difference among the techniques for the different sets of weights varies between 0.0006 and 0.0007, in terms of SI. Differences in the fourth decimal can be considered as negligible. In fact, to place value on the techniques employed, the smallest SIs were also collected, with 0.33 being the minimum one. It is also important to note that big differences in the computation times were not find. This is because the current case study is a simple one in terms of number of optimisation variables. Once again, big differences were not found in the design parameters (Ds, d0 and B) and in the partial indices (EI, SOI and ENI) among the optimisation algorithms. Consequently, important differences did not occur in the remaining parameters. The difference in M among the optimisation techniques varies between 19.9 and 35.5 kg. Once again the variation is not relevant, since more than 1200 kg is employed in the manufacturing process of the optimal or sub-optimal solutions.
The results for the different sets of weights for the second case (non-linear value functions, NL) are shown in
Table 6. In this case, only the ones obtained with the exhaustive search approach are presented, although all the optimisation techniques were used (with similar results).
Similar comments that were made for Case 1 in terms of weighting and in terms of optimisation techniques are, again, applicable. However, it is important to compare the differences in the results between Cases 1 and 2. The maximum difference in the SI between the two cases reaches a value of 0.0135, which can be considered as insignificant. At the time of defining the non-linear value functions, the general criterion was to reward those values close to the best possible ones, and to considerably penalise those values far from the best possible ones; although with different levels of exigency. This explains why, in the second case, a highest SI was obtained. At the same time, this explains why, in the second case, the minimum SI reaches a lowest value (0.21). In terms of Ds, d0 and B, the second case values vary between 0.66 and 0.67; 0.008 and 0.0081 m; and 0.48 and 0.5 m. From three variables, two of them (d0 and B) vary between the same values in both Case 1 and 2. Ds presents bigger deviations (the highest difference takes a value under 8 cm), being the value functions’ geometries the main reason for that. On the other hand, at the time of producing the real STHE, the two cases are likely to generate the same solution, if normalised values are used.
On the basis of the above, it can be concluded that the proposed model is robust, since big differences were not found at the time of: (i) considering different sets of weights for the environmental impacts, (ii) using different optimisation techniques, and (iii) establishing alternative levels of exigency and geometries for the value functions.
4.2. General Results Discussion
Once the different cases were compared, it is possible to analyse the nexus between economics, environment and sustainability. To that end, it is necessary to separately establish the Economic Index (
EI) and the Environmental Index (
ENI) as objective functions. To do so, only the results obtained with the exhaustive search technique, for the linear case with the weights included in
Table 1 will be considered. Similar comments will be applicable if other cases are studied. The results are shown in
Table 7.
Obviously, when the EI is established as the objective function, the highest EI is obtained. This also happens with the ENI. On the other hand, looking at the design parameters, it can be concluded that d0 and B almost take the same values for the different objective functions. The difference in the value that Ds adopts reaches a maximum of 9.55 cm. Despite this, the different indices took similar values. Therefore, it can be concluded that different designs can generate similar results from economic, social, environmental and sustainability perspectives.
In this particular case, maximising the
ENI is equivalent to minimise the mass (
M) used. This is the reason why the smallest mass is obtained when the
ENI is considered as the objective function. This explains that
Ds has adopted the smallest value when
ENI is maximised. Consequently, when
ENI is the objective function, the number of tubes (
Nt) also takes the minimum value (3768). Smaller values of
Ds and
Nt lead to higher fluid velocities [
17]. This, in turn, generates higher pressure drops both for the tube and shell sides [
17]. Consequently, the operating costs (
Codc) are higher than the ones obtained when
SI and
EI are the objective functions. In fact, the increase in the operating costs is large enough to generate the largest total cost (
C1). This happens despite the fact that the capital investment (
Cinv) is the lowest one. When the
ENI is maximised,
A takes a value of 209 m
2. However, in the other two cases, the area is higher (222.5 and 226.5 m
2). This can be shocking, since the capital investment cost increases with the area, as is deducted from Equation (9). Therefore, maximising the
EI (equivalent to reduce the costs) should be associated with an area minimisation. Nevertheless, the design with the minimum area will not necessarily have the minimum total cost (
C1), even if it has the minimum investment cost, since a minimum area can lead to higher velocities and pressure drops. This is what happens in this case. When the
EI is the objective function, the total cost
C1 is the lowest one (76,462 € in comparison with 76,530 € and 78,678 € for the
SI and
ENI cases, respectively), although the investment cost is not the minor one. Therefore, maximising the
ENI and maximising the
EI are not completely equivalent, although the two are related. On the other hand, maximising the
SI is a trade-off solution between maximising the
EI and the
ENI. In other words, when the
SI is the objective function, the partial indices are slightly reduced in comparison with the possible best results, but for obtaining a better global design.
4.3. Comparisons with the Existing Literature
The results obtained here can also be compared with the unique study addressing the sustainability optimisation of an STHE [
17]. Taking into account that a considerable number of cases are addressed in this study, only the results obtained in Case 1, when an exhaustive search approach is adopted and when the weights of
Table 1 are used will be compared with the ones of [
17], in particular for the baseline case. Regarding the
SI, the maximum value presented in [
17] was close to 0.79, which is similar to the one presented in this study. The optimisation variables are also similar. In fact,
d0 and
B take exactly the same values. The shell internal diameters are different, but with a variation of only 2 cm. In fact, the shell internal diameter of [
17] is smaller than the one obtained in this paper (
Table 5). Consequently, the number of tubes is also smaller [
17]. This in turn leads to higher fluid velocities, higher pressure drops and also higher operating costs (10,312 € in contrast with the value of 9,174 € included in
Table 5). Despite this, there is only a difference of 266 € in the total cost (
C1) between this study and Ref. [
17]. This is due to the fact that
A takes a lower value in [
17], generating a smaller capital investment cost (
Cinv). In this particular case, a smaller value of
A is also linked to a smaller consumption of steel (
M). Therefore, the optimal design of [
17] presents a slightly better performance from and environmental point of view. Nevertheless, the reader should take into account that the optimal design found here and the one provided in [
17] are so similar that they will be exactly the same in a practical application, using commercial diameters. These parallels appear, although a greater number of environmental indicators was considered in this study. Nevertheless, this does not infer that the amount of environmental indicators is not important. The ideal situation would be to study all the existing environmental effects, since each impact needs different measures to be corrected.
Other authors analysed the same case study, with the total cost (
C1) as the objective function. Since the objective function is not the same as the one included in this study, considerable differences may appear in all the design parameters. As a way of comparing the results between this and other papers, the authors introduced, in the model presented here, the values of the design variables (
Ds,
d0 and
B) obtained in the existing literature (
Table 8). It is important to clarify that, in
Table 8, the values adopted by some of the parameters do not necessarily coincide with the ones presented in the corresponding references, since there can be small variations in the mathematical model. Once again, the exhaustive search approach for the linear case with the weights based on [
31] were considered.
Before comparing and discussing the results of this study and the existing literature, the reader must bear in mind that the differences among them are mainly based on: (i) variations in the mathematical model since there are alternative equations to estimate some of the STHE parameters (for example: the convective coefficients), (ii) different optimisation techniques, (iii) different values for the numerical constants of Equation (9). The latter is due to the fact that most of the authors did not update Equation (9) by taking into account the time value of money, as we did.
Some of the results are in proximity to the ones presented in this study. Even in the cases in which the differences are bigger, they are not very significant if one takes into account the ranges of possible values defined in
Section 3.1. The
SIs included in
Table 8 are always below the one presented in
Table 7 (
SI optimisation). This is a logical outcome since, in the existing literature, the objective was to minimise the total costs (
C1), instead of maximising the integral sustainability.
On the other hand, the
EIs obtained in this paper, even when they are not the objective function, are higher (
Table 7) than the ones presented in
Table 8. In other words, the results shown in this study are also better from an economic point of view. This could be a consequence of the minor differences that can exist in the mathematical model, and also in the Hall correlation among the studies, as previously alluded to. As a result, if the design parameters presented in
Table 7 are introduced in other authors’ models, the
EIs are likely to be worse than the ones obtained with the authors’ variables. Nevertheless, the same will not happen in terms of integral sustainability.
The best results of the existing literature in terms of costs ([
39],
Ds = 0.7635,
d0 = 0.01 and
B = 0.4955) and in terms of sustainability (Ref. [
42],
Ds = 0.6822,
d0 = 0.0101 and
B = 0.5) will be discussed in greater detail. In particular, the results of [
39] will be compared with the ones of
Table 7 when
EI is the fitness function (
Ds = 0.7364,
d0 = 0.008 and
B = 0.5), while the results of [
42] will be compared with the ones of
Table 5, when an exhaustive search is used and when the weights of
Table 1 are employed (
Ds = 0.7152,
d0 = 0.008 and
B = 0.5, also shown in
Table 7 when
SI is the fitness function).
When the
EI is the objective function (
Table 7), the optimisation variables (
Ds,
d0 and
B) take similar values to the ones adopted in [
39]. In fact, the higher variation occurs for the shell internal diameter and it takes a value of 2.7 cm. As the value adopted by
Ds in [
39] is higher, one might think that the number of tubes will also be higher. This may be the case if
d0 takes the same value in both studies. Nevertheless, in this case, the tube outside diameter (
d0) is also bigger in [
39], resulting in a lower number of tubes. As the velocities resulted to be similar, the difference in the total pressure drop is reduced [
17]. This results in a 427 € difference between the total operating costs of the two studies, which is not relevant. This difference is bigger for the capital investment (
Cinv), since the design of [
39] presents an area equal to 1.06 times the valued adopted in this study. Despite all these variations, the two solutions can be considered as high-performing in terms of cost.
In spite of the fact that existing studies (
Table 8) did not aim the sustainability optimisation, promising results were obtained in terms of
SI. The design of [
42] presents a lower
Ds than the one listed in
Table 5 and
Table 7. However, the tube outside diameter is bigger. Thus, the number of tubes were considerably lower, generating higher fluid velocities, pressure drops and operating costs. Nevertheless, this is one of the cases in which a larger value of
A is not associated with a higher value of
M. The design of [
42] presents a surface area of 221.73 m
2, 0.76 m
2 less than the one obtained in this study. Consequently, [
42] has obtained better results in terms of capital investment. Despite this, its total cost is still over the one included in
Table 5 and
Table 7. On the other hand, the amount of stainless steel employed in [
42] is greater. This translates into more significant environmental impacts. As the environmental dimension is the most important one, the
SI is not as big as the one obtained in this study.