Introduction

HEs are the equipment used for heat transfer between two or more fluids with different temperatures [1, 2]. In the majority of the HEs, the streams are separated by using a surface to avoid mixing of the fluids. These devices are utilized for several intentions such as air conditioning, heating and cooling, renewable energy systems and chemical and petrochemical process [3,4,5]. Typically, HEs are composed of streams’ inlets and outlets in addition to surface of heat transfer as the core of device. In addition, some other components such as fins, pipes and tanks can be used in HEs. The main advantage of HEs is no requirement for moving parts in their structure. There are various criteria that can be applied for categorizing the HEs such as number of phases, architecture and degree of surface compactness [6].

Similar to other heat transfer media, the performance of HEs depend on different elements [7,8,9]. The architecture of the components is one of the factors that can influence both pressure drop and heat transfer rate. Arani et al. [10] investigated pressure drop of shell and tube HE with various types of tube bundles. They found that for the same mass flow rate, twisted oval tubes with segmental baffle had much lower pressure loss compared to the ones with elliptical and circular tubes bundles. Abd et al. [11] investigated effects of baffle space on the performance of a HE for different conditions. They found that increment in the cutting space could decrease the heat transfer coefficient and pressure drop. In another work, Son et al. [12] found that the HE with spiral baffle plates could outperform conventional HE from heat transfer point of view. Specification of the streams is one of the most influential factors in performance of HEs. Using nanofluids, with improved thermos-physical properties [13, 14], is one of the attractive approaches implemented in recent years for heat transfer enhancement. For instance, Fares et al. [15] observed that applying graphene/water nanofluid in a HE can enhance the thermal efficiency significantly. It should be noted that the performance of nanofluids as working fluid of HEs depends on different factors such as the concentration, operating condition and configuration of the system [16]. New types of nanofluids, known as hybrid nanofluids that contain more than one nanomaterial [17, 18], are attractive alternatives for the conventional heat transfer fluids. Despite the advantages of using nanofluids in heat transfer improvement in HEs, the pressure loss could be increase as an unfavorable consequence [19]. In a helical shell and tube HE, Singh and Sarkar [20] evaluated the exergy, economics, and energy of a 580 MW nuclear power plant’s shell and tube condenser using hybrid nanofluids as coolants. Al2O3 + TiO2, Al2O3 + Cu, Al2O3 + Ag, and Al2O3 + MWCNT were among the hybrid nanofluids studied. They also looked at how the concentration of nanoparticles affected coolant demand, operational costs, and pumping power. CFD was used by Ouellette et al. [21] to perform a detailed analysis on solar-geothermal shell and tube HEs. They also looked at the fluid velocity and temperature in the proposed geometrical designs. They discovered that increasing the mass flow rate on the shell side boosted fluid velocity while minimizing the temperature difference through the HEs. They also discovered that elevating the mass flow rate of the shells-side improves exergetic performance and heat transfer, while decreasing the HE's functionality. With a combination of Organic Rankine Cycle (ORC) and parabolic trough solar collector, Erdogan and Colpan [22] presented a thermal design for the formulation of shell and tube HEs. The mathematical operations on the formulated equations were performed using the Engineering Equation Solver (EES). According to their findings, non-continuous baffles provided a higher heat transfer coefficient, which improved heat transfer across the entire shell side. Abed et al. [23] investigated the existence of electromagnetism in shell and tube HEs using a new optimization method. When comparing different situations of water, kerosene, oil and methanol, they discovered that the area of heat transfer declined by up to 68.4, 17 and 23%, respectively. For all of the case studies, the total expenses are likewise reduced. Shirvan et al. [24] used experimental results to investigate the performance of shell and tube HEs using a cosine wave-like shape. The optimization process was determined using the response surface technique. They discovered that increasing the flow velocity of hot water reduces thermal performance. When compared to smooth tubes, wavy tubes have a larger factor of thermal performance. Shirvan et al. [25] evaluated corrugated-wall shell and tube HEs experimentally. They discovered that in the presence of cold water, heat transfer and effectiveness can be improved. They also discovered that smooth tube is less robust than corrugated tube. Marzouk et al. [26] investigated the efficiency of shell and tube HEs by inserting a circular rod into a tube with an unusual nail shape. The acquired data revealed that thermodynamics and thermal performance had significantly improved, whereas hydraulic performance had suffered a significant disadvantage. In shell and tube HEs, Zahid et al. [27] implemented CFD simulation to optimize performance of the included parameters. They discovered that increased material thermal conductivity, reduced inlet velocity, reduced baffle spacing and the use of triangular tube have a significant impact on the reduction of condensate temperature.

In addition to heat transfer rate and pressure drop as measures of HEs, exergy analysis is a powerful tool to evaluate their performance more deeply [28, 29]. Dizaji et al. [30] examined helical tube-in-tube coiled HEs and gave a detailed analysis of the second law. They investigated the effects of flow behavior, geometrical and thermodynamic parameters on exergetic features (including efficiency and exergy loss) for these types of HEs through experiments. They found that coil pitch had a negligible effect on exergy loss. Furthermore, parallel flow patterns showed the greatest increase in exergy loss. With a dual-pipe HE, Geete [31] presented a thorough examination of entransy-based thermal resistance, entransy and exergy in a variety of pipe materials, including copper, brass, cast iron, aluminum and steel. They discovered that copper tubes have the best performance due to their efficacy. The heat transfer mechanism through circular tubes through internally repeated ribs in ring form was quantitatively studied by Ahmed et al. [32]. They also looked at the exergy and entropy of each of the tubes. They discovered that for ring-type tubes, increased thermal hydraulic efficiency was associated with lower entropy and higher exergy. Shabgard et al. [33] constructed a thermal network model to evaluate HE efficiency, solar thermal power heating, a unit of latent thermal energy storage, and a hot and cold water system made up of tube collectors. They also looked at exergy analysis to see how well the second law performed. They discovered that increasing the number of pipes minimizes temperature swings and improves exergy efficiency due to the reduced temperature drop. Hosseinizadeh et al. [34] investigated the exergy and energy of Ferrofluids in the presence of an extrinsic magnetic field using triple HEs. They discovered that the magnetic field could enhance overall performance. Because of greater Reynolds numbers, the impact of the magnetic field on the thermos-fluid characteristics was reduced. When the magnetic field affects all locations, maximum performance is attained. However, because to the existence of magnetic, which tends to increase entropy generation, the effectiveness of the second law is diminished.

Higher entropy generation in a component of a system means more exergy destruction. Due to the dependency of exergy efficiency on the entropy generation of the components, it would be useful to investigate HEs based on entropy generation. Till now, some review studies have been provided on the shell and tube HEs; however, the focus of these works have not been on the exergy and entropy generation. For instance, in a review study by Silaipillayarputhur and Khurshid [35], design of these HEs and the related formulation were reviewed. In another article [36], improvement in shell and tube HEs with helical baffles was reviewed. Salahuddin et al. [37] reviewed advancement made in the field of helical baffles applied in shell and tube HEs and discussed effects of different factors such as spacing and inclination angle. Due to the lack of a comprehensive review on the exergy analysis of these HEs and their entropy generation, this study is designed to focus on this aspect in order to provide required information for the scholars working on this field. In this work, studies on the exergy efficiency of shell and tube HEs, with focus on entropy generation, are reviewed and the findings are represented and discussed. In the following sections, the studies are divided into two groups, conventional and nanofluidic HEs.

Performance evaluation of HEs

Different expressions have been proposed for evaluating the performance of the HEs. Assuming there are two streams, cold and hot, in a HE, temperature of cold stream increases by receiving heat from the hot stream and the temperature of hot stream decreases as shown in Fig. 1. In this section, some of the most important expressions are explained and provided based on Ref [6]. Equation (1) can be used to express the overall heat transfer rate as a differential equation.

$$dQ=U\left({T}_{\mathrm{h}}-{T}_{\mathrm{c}}\right)dA=U\Delta TdA$$
(1)

where A, U, \({T}_{\mathrm{c}}\) and \({T}_{\mathrm{h}}\) are heat transfer area, overall heat transfer coefficient, and temperatures of cold and hot streams, respectively. In order to determine heat transfer of streams, Eq. (2) can be used as follows [39]:

Fig. 1
figure 1

Temperatures of hot and cold streams in a HE [38]

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$$Q=\int CdT={C}_{\mathrm{h}}\left({T}_{\mathrm{h},\mathrm{i}}-{T}_{\mathrm{h},\mathrm{o}}\right)={C}_{\mathrm{c}}\left({T}_{\mathrm{c},\mathrm{o}}-{T}_{\mathrm{c},\mathrm{i}}\right)$$
(2)

o and i as subscripts refer to outlet and inlet conditions, respectively. C refers to the heat capacitance that equals to the multiplication of specific heat of the fluid and its mass flow rate. Applying heat transfer coefficient, the heat transfer rate can be determined as follows:

$$Q=\int U\Delta TdA=UA{\Delta T}_{\mathrm{lm}}$$
(3)

where U and \({\Delta T}_{\mathrm{lm}}\) are the HE heat transfer coefficient and log mean temperature difference (LMTD), respectively. Effectiveness is one of the criteria widely applied for evaluation of HEs, refers to the maximum possible rate of heat that is thermodynamically obtainable. It can be determined as follows [39]:

$$\varepsilon =\frac{Q}{{Q}_{\mathrm{max}}}$$
(4)

where

$$Q_{{{\text{max}}}} = C_{{{\text{min}}}} \left( {T_{{{\text{h}},{\text{i}}}} - T_{{{\text{c}},{\text{i}}}} } \right)\; = C_{{{\text{min}}}} \Delta T_{{{\text{max}}}}$$
(5)

In Eq. (5), \({C}_{min}\) refers to the minimum value of \({C}_{\mathrm{c}}\) and \({C}_{\mathrm{h}}\). By applying both Eqs. (4) and (5), it can be written:

$$\varepsilon =\frac{{C}_{\mathrm{h}}({T}_{\mathrm{h},\mathrm{i}}-{T}_{\mathrm{h},\mathrm{o}})}{{C}_{\mathrm{min}}{\Delta T}_{\mathrm{max}}}=\frac{{C}_{\mathrm{c}}({T}_{\mathrm{c},\mathrm{o}}-{T}_{\mathrm{c},\mathrm{i}})}{{C}_{\mathrm{min}}{\Delta T}_{\mathrm{max}}}$$
(6)

Applying Eqs. (3) and (6) provides following equation:

$$\varepsilon =\frac{UA}{{C}_{\mathrm{min}}}\frac{{\Delta T}_{\mathrm{lm}}}{{\Delta T}_{\mathrm{max}}}$$
(7)

NTU (number of transfer units), indicative of the size of the HE and also a good indicative of heat transfer rate in HEs, can be expressed as follows:

$$NTU=\frac{UA}{{C}_{\mathrm{min}}}$$
(8)

Exergy destruction due to temperature difference between the streams can be determined by using Eq. (9) as follows [40] (Fig. 2):

$${I}_{\mathrm{\Delta T}}=\left(1-\frac{{T}_{0}}{{T}_{1}}\right)Q-\left(1-\frac{{T}_{0}}{{T}_{2}}\right)Q$$
(9)
Fig. 2
figure 2

Exergy flow in a HE [40]

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Exergy destruction in a HE is due to the heat transfer of streams with different temperatures and pressure loss. As shown in Fig. 3, the temperatures of streams significantly influence the exergy loss due to heat transfer and consequently the exergy destruction of the HE.

Fig. 3
figure 3

Exergy destruction in heat transfer process [40]

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Exergy analysis of shell and tube HEs

One of the mostly used kinds of HEs is shell and tube type. This type of HE composed of shell with one or more tubes inside it as shown in Fig. 4. The streams flow through the shell and tubes. Regarding the wide applications of shell and tube HEs, this study focuses on the exergy analysis and entropy generation of these types.

Fig. 4
figure 4

Schematic of a shell and tube HE [6]

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The exergy efficiency of the HEs and entropy generation are influenced by different variables [41] as shown in Fig. 5. Mert and Badak [42] investigated the performance of a 1–1 shell and tube HE and used the COMSOL tool to run an exergetic simulation. They calculated three-dimensional pressure, temperature, and velocity profiles and discovered that exergy destruction is reduced when the shell side velocity temperature is higher, and the tube side velocity temperature is lower. Mert et al. [43] experimentally assessed exergy efficiency of a HE by considering different mass flow rates of the streams and tube side inlet temperature. In their setup, hot stream flows through the tubes while the cold stream flows via the shell side. They found that increase in tube side inlet temperature from 20 to 55 °C leads to increase in the exergy efficiency by around 40% as shown in Fig. 6. Some modifications in the structure of HEs can lead to improvement both in energy and exergy efficiencies. For instance, Wang et al. [44] installed sealers on the shell side of a HE in order to block the gaps between the shell and baffle plats. It was observed that installation of the sealers increased exergy coefficient by up to more than 10% in cases of high shell side flow rate as shown in Fig. 7. Similar to simple shell and tube HEs, exergy analysis can be applied for shell and helical coil tube HEs. For example, Abbas et al. [45] applied exergy analysis for this type of HE by considering the effects of mass flow rates of streams on exergy destruction. They observed that increase in the mass flow rates caused higher exergy destruction, as illustrated in Fig. 8, which was attributed to the higher heat transfer or in other word, higher temperature difference. In another work [46], effect of temperature at the inlets of shell (cold stream) and tube (hot stream) sides of a shell and coiled tube HE was investigated. They observed that exergy loss increased by increment in the temperature at the inlet of coil side and reduction in the inlet temperature of shell side. Moreover, they found that increase in the coil pitch of the HE led to higher exergy loss. In another work [47], focused on helically coiled HE by applying exergy analysis, it was found that using the coil with the highest number of turns and the lowest diameter was the most efficient structure among the applied coils with the same length. Exergy analysis has been performed on the other types of shell and tube HEs in some studies. As an example, Kumar et al. [48] investigated as HE with triple meshed helical coil in different operating conditions. They observed that by increase in the inlet temperature of hot, and consequently increase in the temperature difference between the streams, exergy loss increased. Moreover, they noticed that increase in the mass flow rate of hot stream caused increment in the exergy loss.

Fig. 5
figure 5

The most important factors influencing exergy efficiency of HEs

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Fig. 6
figure 6

Effect of tube side inlet temperature on the efficiency and outlet temperatures [43]

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Fig. 7
figure 7

Effect of installing sealer on the exergy coefficient [44]

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Fig. 8
figure 8

Effect of mass flow rates of streams on the exergy destruction [45]

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Modification on the structure of components is one of the approaches useful for improving the heat transfer rate of HEs; however, it may cause exergy losses. In a study by Dizaji et al. [49], investigated the effect of corrugating the shell and tube of a HE on the exergy loss. They observed that corrugation existence on the shell and tube sides of the HE led to increase in exergy loss. They observed that existence of corrugation causes secondary flow creation which reduces the thickness of the boundary layer on the outer surface of the tube. Moreover, the turbulence of the fluid and mixing of thermal boundary layer along the HE increases. All of these phenomena led to improvement in heat transfer coefficient which results in higher Number of Transfer Unit (NTU). Increment in NTU causes increase in irreversibility of heat transfer which means higher exergy loss of the HE. In another study [50], HE with helically plained coil tube and helically corrugated coil tubes were compared by applying exergy analysis and it was observed that using helically corrugated coiled tube led to more than 20% reduction in exergy loss compared with helically plained coiled tube. The structure and configuration of baffle are other factors that could affect the exergy efficiency of HEs. In a study by Said et al. [51], different baffle configurations were investigated in HE and the exergy efficiency of the system was compared under various operating conditions. The considered baffle configurations in their study were conventional single segmental baffle (CSSB), flower segmental baffle (FSB), staggered single segmental baffle (SSSB) and hybrid segmental baffle (HSB). As shown in Fig. 9, structure of baffle affects the exergy efficiency of the HE and using HSB led to the highest exergy efficiency among the applied baffles. In another work [52], effect of fin geometry on the exergy loss of a shell and helically coiled finned tube was investigated. They found that for constant velocity and number of fins, higher height of fins led to increment in exergy loss. This increment in the exergy loss was attributed to the higher pressure loss in cases of increased fins height. The same trend was observed for increasing the number of fins in the HE. In another work [53], Taguchi analysis was applied on the basis of exergy loss for a HE with helically grooved shell. They found that the minimum exergy loss obtained in case of lowest height of groove and flow rate and the maximum input temperature (cold stream). According to their findings, the height of the grooves influence exergy loss of the HE. DETHE software was used by Zueco and Ayala- Miñano [54] to determine the exergy analysis and irreversibility on shell and tube HEs. They discovered that changing tube pitch had a higher impact on irreversibility, implying that increased tube pitch causes greater irreversibility.

Fig. 9
figure 9

Exergy Efficiency of HE for different configurations of baffle [51]

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Exergy concept and entropy generation are applicable for optimization of HEs [55]. Minimization of entropy generation, which means higher exergy efficiency, is widely used for modeling and optimization of thermal mediums such HEs [56, 57]. By applying optimization on the HEs, it would be possible to improve the effectiveness and decrease the pumping power [58]. In addition to exergy, other factors such as economic indicators can be used for multi objective optimizations of HEs [59]. In a study by Hajabdollahi et al. [60], both exergy and cost were considered as the objectives of an optimization problem on a HE. They observed conflict between the objective function which means that any modification on the geometrical parameters that reduces the exergy loss led to higher cost of the system. Moreover, they found that pressure drop and temperature difference between the streams caused irreversibility which means higher exergy destruction. In another work [57], minimization of entropy generation was applied for multi objective optimization of a HE. In their work, objectives were minimization of the dimensionless entropy generation that was related to the heat conduction with finite temperature difference and friction of fluid for finite pressure drop. Compared with single objective optimization, this approach led to reduction in pumping power for the same effectiveness of the HE. Exergy and entropy concepts can be applied for optimization of HEs with more complex structure. For instance, Arivazhagan et al. [61] optimized a HE with porous medium on its tube based on entropy generation. According to their optimization, there was an optimum value for Reynolds number and higher values led to increment in the irreversibility.

Heat transfer rate in the HEs could be enhanced by applying nanofluids [9, 62]. Generally, increment in the concentration of solid structures of nanofluids lead to heat transfer and pressure loss improvement [63, 64]. As an example, Sridhar et al. [65] investigated heat transfer efficiency in shell and tube HEs employing Ag-water and SnO2-water nanofluids. Because of the unique intrinsic properties of the nanoparticles, they discovered that thermal conductivity was enhanced by 29 and 39 percent, respectively. Increases in thermal conductivity tend to increase the coefficient of heat transfer but rises in nanofluid density and viscosity boosted the friction factor. Furthermore, they discovered that Ag had a sharper pressure drop than SnO2 due to the nanofluid’s stronger thermos-physical characteristics. Improvement in the performance of HEs in cases of using nanofluids is dependent on different specifications of nanofluids such as shape of nanomaterials [66]. Existence of nanostructures in the working fluid can cause reduction in entropy generation and consequently lower exergy destruction. The reduction in the entropy generation of nanofluidic HEs could be under the influence of different constituents such as the particle shape or concentration of solid phase. For instance, Elias et al. [67] compared entropy generation of a HE with boehmite alumina with various particle shapes including brick, cylindrical, platelet, blade and spherical shapes and found that using the spherical shape particles causes the lowest entropy generation. In another work, Esfahani et al. [68] compared the exergy destruction of a HE by using water and graphene oxide/water in 0.01% and 0.1 mass% concentrations in hot stream while water was used as the cold stream. They found that increase in mass flow rate of hot stream caused increment in exergy loss while increase in the concentration of the nanofluid resulted in lower exergy loss. Bahiraei et al. [69] investigated nanofluid flow through shell and tube HE with new type of ladder helical baffles using the second law. They employed Boehmite nanofluids with a variety of nanoparticle shapes contained in a heated fluid with a constant Reynolds number. They discovered that platelet nanoparticle suspension produces the most thermal entropy, whereas oblate spheroid particles suspended in fluid produce the least thermal entropy.

In Table 1, the reviewed works are summarized.

Table 1 Summaries of the studies on exergy analysis and entropy generation of shell and tube HEs
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Recommendations for upcoming studies

In the previous parts of the manuscript, works on the exergy and entropy generation of the shell and tube HEs are reviewed. Despite the existence of several valuable works in this field, there are some suggestions for the next studies in this area. First of all, it will be useful to develop some models based on data-driven methods. By using these methods, due to their ability in forecasting and modeling of complicated problems [70,71,72,73,74], it would be possible to predict exergy efficiency of the HEs in a more time-saving way. In addition to benefits of data driven methods in exergy analysis of HEs in term of time consumption, it would be possible to develop comprehensive models based on these approaches by applying proper inputs [75]. Regarding the better performance of nanofluidic HEs, it is recommended to apply exergy analysis for HEs with other types of nanofluids, especially the ones with carbonic nanostructures. Moreover, the HEs with hybrid nanofluids, as promising heat transfer fluids [76,77,78], would be attractive cases for exergy analysis. In addition, in cases there is significant difference between the inlet and outlet temperatures of the streams, considering the properties of the fluids as a function of temperature would lead to more accurate and realistic outputs.

For the optimization studies, it would be useful to develop the optimization algorithms and applying more recent approaches [79]. Hybridization of optimization approaches would be useful to provide novel algorithms [80]. In addition, different optimization methods can be applied and compared their performance [81,82,83]. Moreover, it is recommended to consider more objectives for optimization of these HEs by using different criteria such as environmental ones in addition to exergy efficiency and entropy generation.

Conclusions

In the current article, works on the exergy analysis and entropy generation of shell and tube HEs are reviewed and their results are discussed and represented. The main findings from these studies are summarized as follows:

  • Different operating conditions such as inlet temperature and mass flow rates of the streams affect the exergy efficiency of the HEs. For instance, by increase in inlet temperature of tube side of a HE from 20 to 55 °C, exergy efficiency can be increased significantly.

  • Modifications on the structure of the HEs, such as installing sealers, can be useful in term of exergy coefficient; as an example, up to more than 10% increase in exergy coefficient has been observed in a study.

  • Modification on the geometry of shell and tubes could lead to improvement in exergy efficiency.

  • In addition to the cores of HEs, the additional components of the HEs such as the baffles and their architecture can influence the exergy efficiency.

  • Applying nanofluids in HEs can decrease the entropy generation and consequently enhance the exergy efficiency.

  • Exergy concept and entropy generation are applicable for optimal design of shell and tube HEs.

  • In addition to single objective optimization, other objectives such as cost, can be used for optimal design of HEs.

  • Optimal design of HEs by employing exergy concept can lead to reduction in pumping power and increase in effectiveness.