1 Introduction

A heat exchanger (HE) is a device which is used for recovering the thermal energy between two or more fluids kept at different temperature. Various types of heat exchangers are available in the industry i.e. shell-and-tube heat exchangers (STHE), plate-fin heat exchangers (PFHE), fin-and-tube heat exchanger (FTHE), gasket heat exchanger, and various configurations of heat exchanger networks (HENs) etc. The design of these HEs is based on various geometric and operational parameters while meeting with certain specified objective(s) such as minimum total cost, maximum heat transfer rate, minimized weight etc. with certain specified constraint(s) such as constrained associated with maximum pressure drop, minimum and maximum fluid flow velocity, structural constraint etc. These HEs are used in various industries as process equipment e.g. heats recovery, refrigeration, cryogenics, food processing and many other industries.

PFHEs and STHEs are the widely used heat exchangers for industrial and as well as commercial applications. The STHE is the HE in which on fluid is carried by tube and the other fluid is carried by shell. The heat transfer takes place by mean mutual heat transfer between the tube and shell side fluid. The design of STHE depends upon many geometrical parameters i.e. shell diameter, tube diameter, number tubes, baffle spacing, number of tube passes, tube length, tube layout etc. and operational parameters i.e. specification of heat pump, fluid flow velocity inside the tube and shell, fouling resistance etc. PFHEs are the HE in which transfer of heat takes place between two fluids by means of plates and finned chambers. The design of PFHE depends upon many structural parameters i.e. fin height, length and thickness, length of fluids flow, no flow length, number of fins, fin frequency etc. and many operational parameters. Similarly, the design of other HEs depends upon many structural and operational parameters.

The HENs design uses a detailed analysis for the optimum design of heat exchangers. It employs three concepts: the composite curves, the grid diagram of process stream and the pinch point; and these are applied to minimize the energy use in the process. This process takes the necessary streams information from the user and decides the best arrangement of heat exchangers, heaters and coolers so that the amount of utilities needed such as cooling water and steam is minimized.

It is clear from the above discussion that the design of heat exchangers is based upon many geometrical and operational parameters with high complexity. Hence, the design of a cheap and effective HE becomes a complicated task. To ensure the finest performance, usage and low cost of the HE, the design optimization techniques are applied in the product development stage. In the process of design optimization of HEs various studies are carried out with different objectives. This work summarises these studies.

Traditional techniques such as steepest decent, linear programming and dynamic programming usually fail to solve such as non-linear large-scale problems. Most of the traditional techniques require gradient information and hence it is not possible to solve non-differentiable functions with the help of such traditional techniques. Moreover, such techniques often fail to solve optimization problems that have many local optima. To overcome these problems, advanced optimization algorithm are developed which are gradient free.

During the last two decades, many advanced optimization techniques such as genetic algorithm (GA), non-nominated sorting GA (NSGA-II), simulated annealing (SA), artificial bee colony (ABC), imperialist completive algorithm (ICA), bio-geography based optimization (BBO), cuckoo search algorithm (CSA), firefly algorithm (FFA), ant colony optimization (ACO), particle swarm optimization (PSO), teaching–learning-based optimization (TLBO), Jaya algorithm, etc. had been used for the design optimization heat exchangers. These algorithms are having their own merits and demerits i.e. GA can solve every optimization problem which can be described with the chromosome encoding and solves problems with multiple solutions but GA requires tuning of many algorithmic-specific parameters such as mutation probability, selection operator, crossover probability, etc. NSGA-II is having explicit diversity preservation mechanism and elitism does not allow an already found Pareto optimal solution to be deleted but crowded comparison can restrict the convergence and it requires the tuning of algorithmic-specific parameters such as mutation probability, crossover probability etc. BBO is an efficient algorithm for optimization and it prevents the degradation of the solutions but BBO is poor in exploiting the solutions. There is no provision for selecting the best members from each generation and it requires the tuning of many algorithmic-specific parameters. PSO is a derivative-free technique just like as other heuristic and it is having the character of memory but it requires the tuning of algorithmic specific parameters and multiplicity of population is not enough to reach the global optimal solution. Similarly, other advanced optimization algorithms needs the tuning of their own algorithmic specific parameters expect TLBO and Jaya algorithms which are algorithmic-specific parameter-less algorithms.

Figure 1 shows the publication statistics for the design optimization of HEs using advanced optimization techniques. These algorithms have already proved their significance in the field of design optimization of various types of heat exchanger designs. Design optimization of heat exchangers through advanced optimization techniques is now proving as a milestone for the heat exchanger design and hence various researchers are trying to make use of these advanced optimization techniques for the heat exchangers design. This paper makes an efforts to identify all such works in which the use of various advanced optimization techniques are involved till now for the design optimization of different types of heat exchangers. The next section summarizes the applications of advanced optimization techniques for the design optimization of heat exchangers.

Fig. 1
figure 1

Publication statistics for the design optimization of HEs using advanced optimization techniques

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2 Review of the Applications of Advanced Optimization Algorithms for Design Optimization of Heat Exchangers

The literature review of the design optimization of HEs using advanced optimization is organized into three parts:

  1. (a)

    Design optimization of plate-fin heat exchangers.

  2. (b)

    Design optimization of shell-and-tube heat exchangers.

  3. (c)

    Design optimization related to some miscellaneous heat exchangers i.e. fin-and-tube, heat exchanger networks (HENs), printed circuit heat exchanger (PCHE), vertical U-tube ground heat exchanger, wavy-fined-and-elliptical tube heat exchanger and U-shaped square duct heat exchanger.

The next section presents the application of advanced optimization algorithms for the design optimization of PFHEs.

2.1 Review of the Application of Advanced Optimization Algorithms for Design Optimization of Plate-Fin Heat Exchangers

Table 1 presents the summary of the applications of advanced optimization algorithms for the design optimization of PFHEs with design variables, constraints and objective function(s) [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35]. The PFHEs are widely used in the industries of chemical, petroleum, petrochemical and power generation. In this extended surfaces are added for enhancing the heat transfer rate. The major design parameters of PFHE are: hot and cold fluid flow length, no-flow length, fin thickness, fin height, fin pitch, fin frequency etc. The selection criterion of suitable combination of these parameters for a particular application depends upon allowable pressure drops, ranges of the temperatures, thermal stress of fluid and heat exchanger material and dynamic properties of fluids.

Table 1 Review of the applications of advanced optimization algorithm for PFHE design
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In the recent years, the application of advanced optimization algorithms for the design optimization of PFHEs has got much momentum since the last decade [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35]. It can be observed from the literature review that the various algorithms has been applied for the design optimization PFHEs and these are: GA, SA, PSO and its variants, NSGA-II, ICA, HSA, Bees algorithm, DE algorithm and its variants, BBO algorithm, TLBO algorithm and Jaya algorithm and its variants. The research has been carried by aiming various objectives i.e. minimization of total annual cost [1, 3,4,5,6, 8, 9, 12,13,14,15,16,17,18,19, 22, 23, 28, 29, 34], maximization of effectiveness [3, 7, 12, 13, 16, 19, 29, 31, 34], minimization of pressure drop [7, 17, 25, 27, 34], maximization of heat transfer rate [5, 6, 14, 17, 22, 26], minimization of entropy generation units [2, 4, 8, 13, 23, 32, 33], minimization of total weight or volume of the PFHE [1, 4, 8, 9, 15], minimization of heat transfer area [4, 27, 34], layer pattern optimization [10, 20, 21] and optimization of friction factor and Colburn factor [24, 30, 35]. Manny researchers had carried out multi-objective optimization [3, 5, 6, 12, 28, 29, 34] by considering the different combination of the above mentioned objectives.

It can be observed from the above literature review that the advanced optimization algorithms have been applied mainly aiming the minimization of total cost and maximization of effectiveness. Furthermore, very few researchers have applied multi-objective optimization [3, 5, 6, 12, 28, 29, 34] for the design optimization of the PFHE. It can also be observed from Table 1 that few researchers have integrated surrogate models along with the advanced optimization algorithms [3, 23, 24] and few researchers have used computational fluid dynamics with advanced optimization algorithms [6, 35]. Moreover, the analytical models used in design optimization of PFHE contain many assumptions such as thermal and physical properties are independent of temperature gradient, steady state analysis, constant heat transfer coefficient etc. while developing the models which may lead errors as compared to the actual situations.

2.2 Review of the Application of Advanced Optimization Algorithms for Design Optimization of Shell-and-Tube Heat Exchangers

Table 2 presents the summary of the applications of advanced optimization algorithms for design optimization of STHEs with design variables, constraints and objective function(s) [12, 28, 29, 36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74]. The STHEs are widely used in the process industries, steam generators in pressurized and water reactor plants, feed water heaters, in conventional and nuclear power stations as condenser etc. In this, one fluid flows through the tube and other fluid flow through the shell and the mutual heat transfer takes place between the fluids. The major design parameters of STHEs are: shell diameter, tube diameter, tube length, number tubes, number of tube passes, tube layout pattern, tube pitch, baffle spacing, etc. The selection criteria of the suitable combination of these parameters for a particular application depends upon allowable pressure drops, ranges of the temperatures, thermal stress of fluid and heat exchanger material, dynamic properties of fluids, fouling factor, maintenance cost and clean ability.

Table 2 Review of the applications of advanced optimization algorithm for STHE design
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In the recent years, the application of advanced optimization algorithms for the design optimization of STHEs has got much momentum [12, 28, 29, 43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75]. It can be observed from the literature review that the various advanced optimization algorithms have been applied for the design optimization STHEs and these are: GA, SA, PSO and it variants, NSGA-II, ICA, HSA, Bees algorithm, DE algorithm and its variants, BBO TLBO, ABC, FFA, GSA, I-ITHS algorithms and Jaya algorithm and its variants. The research has been carried by aiming various objectives i.e. minimization of total annual cost [12, 28, 36,37,38,39, 41,42,43,44,45, 48, 49, 52, 53, 55, 57, 60,61,62,63,64,65,66,67, 69,70,71,72,73, 75], maximization of effectiveness [12, 28, 42, 62, 71], minimization of pumping power [50, 54, 68], maximization of heat transfer rate [54, 56, 57, 64, 68], minimization of pressure loss [56], minimization of entropy generation units [47, 51, 68], minimization of total weight or volume of the STHE [59, 74], minimization of heat transfer area [50] and maximization of thermal efficiency [46, 58]. Manny researchers had carried out multi-objective optimization [12, 28, 42, 46, 47, 50, 51, 56, 57, 62, 64, 68, 71] by considering the different combination of the above mentioned objectives.

It can be observed from above literature review that the advanced optimization algorithms have been applied mainly aiming the minimization of total cost. Furthermore, few researchers had applied multi-objective optimization for the design optimization of the STHE. It can also be observed from Table 2 that few researchers have used discrete parameter optimization [36, 50, 63, 70] in order to get the advantage of standard sizes available for STHE parts. A few researchers had used computational fluid dynamics (CFD) with advanced optimization algorithms [56, 64]. A Kriging response model was integrated with MOGA [64]. Furthermore, PROMETHEE (preference ranking organization method) method [72] was used to select the best optimal solution among the Pareto optimal solutions. A misappropriation of parametric costing approaches for optimal equipment design was revealed [67]. Moreover, the analytical models used in design optimization of STHEs contain many assumptions while developing the models which may lead errors as compared to the actual situations.

2.3 Review of the Application of Advanced Optimization Algorithms for Design Optimization of Some Miscellaneous Heat Exchangers

Table 3 presents the applications of advanced optimization algorithms for the design optimization of heat exchanger networks (HEN) [76,77,78, 80, 87, 92, 93, 95, 98], fin-and-tube heat exchangers (FTHEs) [6, 79, 81, 82, 85, 91, 94], printed circuit heat exchangers (PCHEs) [83, 89], vertical U-tube ground HE [96], wavy fined-and-elliptical tube HE [96] and U-shaped square duct HE [97]. The algorithms used for the design optimization of these heat exchangers [75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98] includes: GA, SA, NSGA-II, PSO, ABC, DE and CSO algorithms. The objectives of these heat exchangers were: total cost (HENs); effectiveness, total cost, heat transfer rate and thermal resistance (FTHEs); effectiveness and pressure drop (PCHE); heat transfer rate, pressure drop and entropy generation (vertical U-tube ground heat exchanger), size of bore hole HE [90]; Colburn and friction factor of wavy fined-and-elliptical tube heat exchanger [96] and pressure drop and heat transfer rate (U-shaped square duct HE). A few researchers had applied multi-objective optimization of HEs [77, 82, 83, 86, 96]. The CFD was used with advanced optimization algorithm [96]. In [97] several surrogate models were integrated with the advanced optimization algorithms. It can be seen from this literature survey [76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98] that there is still a lot of scope for the application of advanced optimization algorithms for the design optimization of these heat exchangers. There are many efficient and most power algorithms such as biogeography-based optimization (BBO), gravitational search algorithm (GSA), firefly algorithm (FFA), cuckoo search (CS), bat algorithm (BA), TLBO, Jaya etc. are available in the literature which can be used. Furthermore, the cost model used for FTHE is very old and contains many assumptions.

Table 3 Review of the applications of advanced optimization algorithm for some miscellaneous heat exchanger design
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3 Conclusions

This work presents the optimization aspects of the widely used heat exchangers such as shell-and-tube heat exchangers, plate-fin heat exchangers, fin-and-tube heat exchangers, heat exchangers network and some other heat exchangers such as printed circuit heat exchanger, vertical U-tube ground heat exchanger, bore hole heat exchanger, wavy fin-and-elliptical tube heat exchanger and U-shaped square duct heat exchanger. A thorough literature review related to design optimization of these heat exchangers are made and summarised. The objective function(s), design variables and constrained imposed to the design are listed. A critical remark on various research works is also presented. The following observations are made based on this review work:

  • A lot of work is conducted on the parameter optimization of STHEs and PFHEs as compared to the other heat exchangers.

  • The objective function(s) considered in most of the cases was total cost and effectiveness in case of PFHE and total cost in case of STHEs.

  • A few multi-objective optimization algorithms were applied to STHEs PFHEs.

  • A few researches had used surrogate models along with advanced optimization algorithms in order to minimize the error of empirical relations.

  • A few researchers had used computational fluid dynamics along with advanced optimization algorithms.

  • The cost models used for the optimization are old and do not include the direct relation with the heat exchanger parts and maintenance aspects in most of works were not included.

  • The cost models used were developed from the vendor point of view. The accurate cost models from vendor as well as manufacturing point of view are not available.

  • In most of the research works, the design optimization of HEs (STHE and PFHE) were carried out by considering the design variables as continuous variables or semi-discrete variables. A few works were carried out by considering all the design variables as discrete variables (standard size of the parts).

  • The application of advanced fluid such as nano-fluid and smart fluid along with the optimization were taken up.

  • A very few research work was carried out in the field of design optimization of fin-and-tube heat exchangers. Application many powerful and more advanced optimization algorithms are missing which may provide good designs as compared to the others.

  • In case of design optimization of heat exchanger networks, the application of advanced optimization is limited which needs to explore this field.

  • Similarly, the application of the advanced optimization algorithms to other heat exchangers such as printed circuit heat exchanger (PCHE), vertical U-tube ground heat exchanger, wavy-fined-and-elliptical tube heat exchanger and U-shaped square duct heat exchanger is limited. The field of design optimization of these heat exchangers using advanced optimization still needs to be explored from the point of view of application of various advanced optimization algorithm, objectives as well as mathematical models.