1 Introduction

Iran's geography and climate are highly suitable for solar energy-based technologies. Solar energy can play an important role in Iran’s energy security and help to reduce greenhouse gas emissions and fossil fuel use, which is the largest source of Iran’s carbon dioxide emissions. In this context, combining solar collectors with the heat pumps in a single combined system, known as the solar-assisted heat pumps (SAHPs), have been developed rapidly during the last decade. SAHPs use solar radiation and ambient energy as the heating source, which can be considered ‘free’ energy [1, 2]. In a SAHP system, the solar thermal panels act as a low-temperature heat source and the produced heat is utilized to feed the evaporator of the heat pump [3]. The aim of SAHPs design is to produce energy more efficiently and less expensively to reach a high COP. SAHPs are widely used for producing domestic hot water (DHW) and can efficiently serve up to 80% of hot water needs with no fuel cost or pollution and with minimal operation and maintenance (O&M) expense [4]. SAHPs may be categorized into two types such as direct solar-assisted heat pump (DX-SAHP) and indirect solar-assisted heat pump (IDX-SAHP) [5]. In recent years, several researches have been conducted on the DX-SAHPs and IDX-SAHPs. Shi et al. [3] presented the advancements and the current status of DX-SAHPs. Further, Cao et al. [6] simulated a thermodynamic model of a DX-SAHP and optimized its performance. Newly, Song et al. [7] proposed a novel DX-SAHP with a hybrid compound parabolic concentrator/photovoltaic/fin evaporator to make full use of solar and air energy in a limited space. Mohanraj et al. [8] experimentally analyzed the economic of a heat pump water heater-assisted regenerative solar still using latent heat storage. Moreover, Duarte et al. [9] carried out an experimental analysis of the influence of environmental conditions in a small size DX-SAHP operating with CO2 as the refrigerant to heat water. Singh et al. [10] designed, built, and experimentally studied a convective closed-loop SAHP for both simple heat pump drying and SAHP drying modes. In another research, Kong et al. [11] examined the efficiency of a DX-SAHP using R290 with micro-channel heat transfer technology during the winter period, experimentally. Belmonte et al. [12] simulated two different SAHPs able to cover a significant part of the space heating demand of a single-family house located in Madrid (Spain) using TRNSYS. Besides, Sezen et al. [13] investigated the effect of ambient conditions on heating modes, and identify the preferable ambient condition ranges for each SAHP system depends on their heating modes. Song et al. [14] analyzed the optimal structure of evaporator, three direct-expansion solar-assisted heat pumps with Fresnel PV evaporator (FPV-SAHP), hybrid Fresnel PV plus thermoelectric generator (TEG) evaporator (FPV/TEG-SAHP), and traditional PV evaporator (PV-SAHP). Singh et al. [15] developed a batch-type solar-assisted heat pump dryer (SAHPD) and experimented in both simple heat pump dryer (HPD) and SAHPD modes for closed system drying of banana chips. Furthermore, Simonetti et al. [16] presented a yearly energetic and economic assessment of three different solar-assisted heat pump concepts integrated with electrical and thermal storages and applied to a single-family house. Recently, Chen et al. [17] presented a continuing research on the direct expansion solar-assisted ejector-compression heat pump cycle for water heater aiming to reveal the system dynamic characteristic by experimental method. Yang et al. [18] modeled the serial, parallel, and dual-source indirect expansion solar-assisted air source heat pumps and simulated under the weather conditions in London using TRNSYS to investigate the operating performance over a typical year. Additionally, Li and Mao [19] utilized TRNSYS to optimize the solar thermal and heat pump combisystems under five distinct climatic conditions. Zhou et al. [20] presented the investigation of a novel mini channel Photovoltaic/Thermal (PV/T) combined heat pump system using both experimental and theoretical methods. Kim et al. [21] carried out a comparative investigation of various IDX-SAHPs. Recently, Huan et al. [22] simulated an IDX-SAHP using TRNSYS for a university bathroom in Xi'an and validated it through experimental measurements. Ma et al. [23] conducted the performance analysis of an IDX-SAHP with CO2 refrigerant. Singh et al. [24] performed an experimental performance analysis of an IDX-SAHP dryer for banana chips. Also, Liu et al. [25] developed a novel low-concentrating photovoltaic/thermal solar-assisted water source heat pump (LCPV/T-WWHP) system to satisfy both electricity and thermal demand of a university building. Youssef et al. [4, 26] experimentally studied an IDX-SAHP with latent heat storage. Bridgeman et al. [27] constructed an IDX-SAHP with the glycol–water solution (GWS) in the collector loop and verified the obtained experimental results with the results of the investigation conducted by Freeman et al. [28]. Besides, Oztop et al. [29] reviewed the previously conducted studies and applications in terms of design, performance assessment, heat transfer enhancement techniques, experimental and numerical works, thermal heat storage, effectiveness compassion and recent advances. In another work, Bayrak et al. [30] assessed and compared the performance of five types of solar air heaters and determined the energy and exergy efficiencies. Also, Zhang et al. [31] presented an auspicious method, the multi-island genetic algorithm (MIGA), to optimize the configuration of a universal selective solar absorber, which can avoid the premature phenomenon and superior to traditional genetic algorithm.

In the process of system design, the best system parameters shall be opted in an efficient and cost-effective way. Accordingly, besides the system design, the parametric analysis, optimization, and decision makings of the system must be implemented before any prototype construction begins. The literature overview divulges although some studies have been conducted on the solar-based energy systems theoretically and experimentally, no research has been done on the Energy–Exergy analyses, performance-based SOO and MOO processes, and MCDA of the IDX-SAHPs, altogether. Because these systems are new, studies on them are not enough, and thus, there is still a lot of potential for research on them. The main objectives of this research are:

  • Developing the numerical modeling of the IDX-SAHP performance

  • Carrying out the SA of the design parameters using OPAT

  • Performing the energy–exergy analyses on the IDX-SAHP

  • Performing the SOO and MOO approaches on the IDX-SAHP performance using NSGA-II

  • Selecting the optimal layout of the IDX-SAHP using AHP MCDA coupled with the NSGA-II

  • Comparing the performance of the maximized IDX-SAHP with the initial system

2 Materials and Methods

2.1 IDX-SAHP Description

Figure 1 depicts the schematic view of the proposed IDX-SAHP system. The IDX-SAHP under investigation has three separate loops. The collector loop includes the collector, pump, and an evaporator utilizing the \(50-50\%\) GWS by volume as the working substance. The evaporator is the joint equipment put between the solar collector and the heat pump loops. The heat pump loop includes a compressor, a condenser, a filter-drier, a thermostatic expansion valve (TEV).

Fig. 1
figure 1

The schematic view of the proposed IDX-SAHP system

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During the IDX-SAHP operation, the GWS passes through the solar collector by the pump and obtains the ambient and solar energies (Fig. 1: from 7 to 5). The GWS passing through the evaporator shell delivers its energy to the R-134a refrigerant in the evaporator tubes (Fig. 1: from 5 to 6). Accordingly, the R-134a superheats after obtaining the GWS heat (Fig. 1: from 4 to1) and then enters the compressor to increase its pressure as much as the condenser pressure (Fig. 1: from 1 to 2). Thereupon, the superheated R-134a passes through the condenser tubes and is condensed by the water passing through the condenser shell (Fig. 1: from 2 to 3). The condensed R-134a then enters the TEV to decrease its pressure as much as the evaporator pressure (Fig. 1: from 3 to 4). Ultimately, the low-pressure and low-temperature R-134a re-enters the evaporator tubes and the IDX-SAHP cycle is ended. Table 1 provides the design parameters of the IDX-SAHP.

Table 1 Design parameters of the IDX-SAHP
Full size table

2.2 Assumptions

  • Steady-state process.

  • The process of passing the R-134a through the compressor is taken into account polytropic.

  • The process of passing the R-134a through the TEV is taken into account isenthalpic.

  • No pressure drops in the IDX-SAHP cycle.

  • \({T}_{W,i}\) \(=\) \({T}_{a}\).

2.3 Collector Modeling

The heat rate absorbed by the solar collector (\({{\dot{Q}}_{cl}}\)) is determined as per the difference between the average temperatures of the collector plate (\({{T}_{p}}\)) and the ambient air (\({{T}_{a}}\)) as follows [32]:

$${\dot{Q}_{cl}} = A_{cl} S - {\dot{Q}_{\mathrm{loss}}}$$
(1)
$${\dot{Q}_{\mathrm{loss}}} = A_{cl} {U_{L}} \left( {{T_{p}} - {T_{a}} } \right)$$
(2)

Also, \({{\dot{Q}}_{cl}}\) can be defined as per the difference between the average temperatures of the GWS (\({{T}_{gw}}\)) and the ambient air (\({{T}_{a}}\)):

$${ \dot{Q}_{cl}} = {A_{cl}} {F^{\prime}}S - A_{cl} {F^{\prime}}{U_{L}} \left( {{T_{gw}} - {T_{a}} } \right)$$
(3)

where

$$S = {I_{T}} \left( {\tau \alpha } \right)$$
(4)
$${T_{gw}} \approx \frac{{{T_{5}} + {T_{7}} }}{2}$$
(5)

If \({{N}_{gl}}=0\), then \(\tau =1\). Also, \({{U}_{L}}\) is defined as:

$${U_{L}} = \left( {\frac{{N_{gl} }}{{\frac{C}{{T_{pm} }}\left[ {\frac{{\left( {T_{p} - T_{a} } \right)}}{{N_{gl} + f}}} \right]^{e} }} + \frac{1}{{h_{w} }}} \right)^{ - 1} + \frac{{\sigma \left( {T_{p} + T_{a} } \right)\left( {T_{p}^{2} + T_{a}^{2} } \right)}}{{\frac{1}{{\varepsilon_{p} + 0.0059N_{gl} h_{w} }} + \frac{{2N + f - 1 + 0.133\varepsilon_{p} }}{{\varepsilon_{g} }} - N_{gl} }}$$
(6)

here,

$${h_{w}} = 5.7 + 3.8{u_{w}}$$
(7)
$$C = 520 \times \left( {1 - 0.000051{\beta^{2}} } \right)$$
(8)
$$e = 0.43 \times \left( {1 - \frac{100}{{T_{pm} }}} \right)$$
(9)
$$f = \left( {1 + 0.089{h_{w}} - 0.1166{h_{w}} {\varepsilon_{p}} } \right)\left( {1 + 0.07866{N_{gl}} } \right)$$
(10)

Additionally, the collector efficiency factor \(({{F}^{{\prime}}})\) is calculated as:

$${F^{\prime}} = \left[ {\left( {1 - F} \right)\frac{{D_{cl} }}{W}} \right] + F$$
(11)

where

$$F = \frac{{tanh\left( {R\left( {W - {D_{cl}} } \right)/2} \right)}}{{R\left( {W - {D_{cl}} } \right)/2}}$$
(12)
$$R = \left( {\frac{{U_{L} }}{{K_{p} \delta }}} \right)^{0.05}$$
(13)

here, \(F\) is called the absorber fin efficiency and R is a fixed coefficient. Furthermore, the \({\dot{Q}}_{cl}\) can be calculated as follows:

$${\dot{Q}_{cl}} = {\dot{m}_{gw}} {C_{p,gw}} \left( {{T_{5}} - {T_{7}} } \right)$$
(14)

Considering \({T}_{gw}\approx \frac{{T}_{5}+{{T}_{7}}}{2}\) and the given mass flow rate of GWS (\({\dot{m}}_{gw}\)) and specific heat of GWS (\({{C}_{p,gw}}\)) in Table 2, \({{T}_{5}}\) and \({{T}_{7}}\) can be obtained.

Table 2 Equations of exergy analysis for each component of the proposed IDX-SAHP
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2.4 Evaporator Modeling

The first law of thermodynamics for the solar collector loop may be presented as:

$${ \dot{Q}_{\mathrm{eva}}} = {\dot{Q}_{cl}} + {\dot{W}_{\mathrm{pump}}}$$
(15)

The heat transfer rate through the evaporator based on the glycol side was calculated from the \({{\dot{m}}_{gw}}\), \({{C}_{p,gw}}\), and the GWS temperature at each side of the evaporator as follows:

$${\dot{Q}_{\mathrm{eva}}} = {\dot{m}_{gw}} {C_{p,gw}} \left( {{T_{5}} - {T_{6}} } \right)$$
(16)

Also, the heat rate transferred from GWS to the R-134a (\({{\dot{Q}}_{eva}}\)) can be obtained by the energy balance across the evaporator as follows [30]:

The \({\dot{Q}}_{eva}\) can also be determined through Newton’s law of cooling as [33]:

$${ \dot{Q}_{\mathrm{eva}}} = {U_{eva}} {A_{s,eav }} \Delta {T_{m}}$$
(17)
$$\Delta {T_{m}} = \frac{{\left( {{T_{5}} - {T_{4}} } \right) - \left( {{T_{6}} - {T_{4}} } \right)}}{{{\text{ln}}\left( {\frac{{{T_{5}} - {T_{4}} }}{{{T_{6}} - {T_{4}} }}} \right)}}$$
(18)

where \({U}_{eva}\) is the overall heat transfer coefficient, \({A}_{s,eav}\) is the surface area of heat transfer, of the evaporator, and \({\Delta T}_{m}\) is the logarithmic average of the temperature difference between the GWS and R-134a streams at each end of the evaporator. It should be noted that the phase of R-134a changes from two-phase (point 4) to saturated vapor (point 1), at a constant temperature and constant pressure, namely, \({T}_{4}={T}_{1}\) and \({P}_{4}={P}_{1}.\) In this regard, \({T}_{1}\) and \({P}_{1}\) (the temperature and pressure at the inlet of the compressor in saturated vapor state) may be obtained. Moreover, the \({\dot{Q}}_{eva}\) can be calculated as:

$$\dot{Q}_{eva} = \dot{m}_{r} \left( {h_{1} - h_{4} } \right)$$
(19)
$$\mathop {m_{r} }\limits^{.} = \frac{{\omega \eta_{v} V_{d} }}{{60v_{1} }}$$
(20)

2.5 Compressor Modeling

The compressor power (\({\dot{W}}_{cm}\)) is determined as:

$$\dot{W}_{\mathrm{cm}} = \mathop {m_{r} }\limits^{.} \frac{{P_{1} v_{1} }}{{\eta_{\mathrm{cm}} }}\frac{k}{k - 1}\left( {\left( {\frac{{P_{2} }}{{P_{1} }}} \right)^{{\frac{k}{k - 1}}} - 1} \right)$$
(21)

where \({P}_{1}\) and \({P}_{2}\) are the pressures of R-134a at the inlet and outlet of the compressor.

2.6 Condenser Modeling

The heat rate delivered from R-134a in the condenser tubes to the water in the condenser shell (\({\dot{Q}}_{\mathrm{cond}}\)) can be obtained as follows:

$$\dot{Q}_{\mathrm{cond}} = \dot{m}_{r} \left( {h_{2} - h_{3} } \right)$$
(22)

In addition,\({\dot{Q}}_{\mathrm{cond}}\) equals the heat rate absorbed by the water in the condenser shell from R-134a in the condenser tubes (\({\dot{Q}}_{W}\)):

$$\dot{Q}_{W} = M_{W} C_{p,w} \frac{{T_{W,o} - T_{W,i} }}{\Delta t} = \dot{m}_{r} \left( {h_{2} - h_{3} } \right)$$
(23)

here \({M}_{W}\) and \({C}_{p,w}\) are, respectively, the water mass passing through the condenser shell and the specific heat capacity of the water. Moreover, \({T}_{W,i}\) and \({T}_{W,o}\) are the mean water temperature at the inlet and outlet of the condenser shell, respectively. As per the first law of thermodynamics in heat pump loop:

$$\dot{Q}_{\mathrm{cond}} = \dot{Q}_{W} = \dot{Q}_{\mathrm{eva}} + \dot{W}_{cm}$$
(24)

Additionally, \(\Delta t\) is the working hour of the IDX-SAHP to maintain the \({T}_{W,o}\) fixed:

$$\Delta t = \frac{{\dot{m}_{r} \left( {h_{2} - h_{3} } \right)}}{{M_{W} C_{p,w} \left( {T_{W,o} - T_{W,i} } \right)}}$$
(25)

2.7 TEV Modeling

As TEV is isenthalpic, thus:

$$h_{3} = h_{4}$$
(26)

here \({h}_{3}\) and \({h}_{4}\) are, respectively, the enthalpies of R-134a at the input and output of the TEV.

2.8 COP and CE

The COP and CE of the system are determined as:

$${\text{COP}} = \frac{{\dot{Q}_{\mathrm{cond}} }}{{\dot{W}_{\mathrm{cm}} + \dot{W}_{\mathrm{pump}} }}$$
(27)
$${\text{CE}} = \frac{{\dot{Q}_{cl} }}{{I_{T} A_{cl} }}$$
(28)

2.9 Computational Flow Diagram

In the current research, the thermodynamic traits of the R-134a are utilized from the formulas presented by Cleland [34]. Additionally, energy equation solver (EES) software [35] is utilized to obtain the thermodynamic traits of the GWS. Figure 2 depicts the flow diagram to compute the COP and CE. As is evident in Fig. 2, first, an initial guess is made for \({T}_{gw}\) to estimate the \(\dot{{Q}_{\mathrm{cl}}}\) using Eqs. (1–14). In this regard, \({\dot{Q}}_{eva}\) is achieved using Eq. (15). Also,\({T}_{5}\) and \({T}_{7}\) can be obtained through Eqs. 3 and 14. Next, \({T}_{4}\) that is equal to \({T}_{1}\) is obtained using Eqs. (16–18). As the temperature at the inlet of the compressor (\({T}_{1}\)) is in saturated vapor state, thus, the pressure and the enthalpy of the R-134a at the inlet of the compressor (\({P}_{1}, {h}_{1}\)) can be extracted. Besides, \({h}_{4}\) is obtained by Eq. (19). Similarly, as \({h}_{3}={h}_{4}\) (Eq. (26)) and also state 3 is the saturated liquid state, thus, all of R-134a properties at state 3, especially \({P}_{3}\), can be extracted. Because \({{P}_{2}=P}_{3}\), thus \(\frac{{P}_{2}}{{P}_{1}}\) is obtained to calculate \({\dot{W}}_{\mathrm{cm}}\) by Eq. (21). Moreover, \({\dot{Q}}_{cond}\) may be determined by Eqs. (24), and thus, the \({T}_{w,i}\) can be obtained for a constant \({T}_{w,o}\) by Eqs. 22 and 23. For the reason that the obtained \({T}_{w,i}\) is equal to \({T}_{a}\), hence if \(\left|\mathrm{obtained} {T}_{w,i} -{T}_{a}\right|\le tol\), then the \({T}_{gw}\) is admissible and the flowchart loop ends.

Fig. 2
figure 2

Computational flow diagram to solve the proposed IDX-SAHP model

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Consequently, the COP and CE of the system are defined. Otherwise if \(\left|\mathrm{obtained} {T}_{w,i} -{T}_{a}\right|>tol\), the artificial neural network (ANN) is utilized to guess a new \({T}_{gw}\). ANNs are computing systems inspired by the biological neural networks that constitute animal brains [36, 37]. In an ANN, there are three basic layers comprising input layer, hidden layer, and output layer, each of which include neurons. The neurons number of the input (output) layer equals to the number of inputs (outputs), while the neurons number of the hidden layers varies during the training process to obtain the optimum accuracy. The input layer is the first ANN layer and consists of input neurons that receives input data in the form of text, numbers, audio files, image pixels, and so on. These neurons send data to the deeper layers, which in turn will send the final output data to the last output layer. The hidden layers are in the middle of the ANN model. These hidden layers perform various types of mathematical calculations on the input data and identify patterns in it. In the output layer, the result is obtained by accurate calculations of the middle layer. Neurons are connected between layers through weights. In each neuron, all inputs from the preceding layer are weighted by the weights and added together, along with a threshold. The weights are modified by learning. The relationship between the outputs (\(y)\) and inputs (\({x}_{i})\), known as transfer function, is defined as [38]:

$$y = f\left( {\mathop \sum \limits_{i = 1}^{n} w_{i} x_{i} - b} \right)$$
(29)

here \({w}_{i}\) are the weights of connection from the ith neuron in the previous layer to the neuron and \(b\) is a threshold. Then, the activation function was applied to inculcate nonlinearity in the model and help the network learn any complex relationship between inputs and outputs. In this work, a tansig activation function was used as:

$$f\left( x \right) = \frac{1}{{1 + {\text{exp}}\left( { - x} \right)}}$$
(30)

After getting the predictions from the output layer, the error is calculated, namely, the difference between the actual output (\({T}_{a}\)) and the predicted output (\(achieved {T}_{w,i}\)). As ANN gathers its knowledge by detecting the patterns and relationships in data and learns through experience, thus, it can detect the \({T}_{gw}\) as per its intelligent algorithm and the information obtained in previous iterations in such a way that \(\left|achieved {T}_{w,i} -{T}_{a}\right|\le tol\) (tolerance of 0.01). Accordingly, all of the thermodynamic traits of GWS and R-134a throughout the IDX-SAHP cycle are extracted, and therefore, CE and COP are defined.

2.9.1 Thermodynamic Exergy Analysis

The exergy of a system (\(\dot{E}\chi\)) is the maximum useful work possible during a process that brings the system into equilibrium with a heat reservoir, reaching maximum entropy [39]. The real thermodynamic inefficiencies in an energy conversion system are related to exergy destruction and exergy loss. An exergy analysis determines the system components with the largest exergy destruction and the processes that cause them. The equations for input exergy, output exergy, exergy destruction (irreversibility) as well as the exergy efficiency of each component of the proposed IDX-SAHP are defined in Table 3 [33].

Table 3 Ratio scale [56]
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In the equations presented in Table 3, \({T}_{0}\) is the dead state temperature and is considered to be equal to the ambient temperature (\({T}_{a}\)). Also, \({\psi }_{r}\) is the specific stream exergy of the refrigerant and \({\dot{E\chi }}_{rad}\) is the exergy of solar radiation entering the solar collector defined as [40]:

$$\mathop E\limits^{.} \chi_{rad} = A_{cl} I_{T} \left[ {1 + \frac{1}{2}\left( {\frac{{T_{0} }}{{T_{sr} }}} \right)^{4} - \frac{4}{3}\left( {\frac{{T_{0} }}{{T_{sr} }}} \right)} \right]$$
(31)

Here, \({T}_{sr}\) is the temperature of solar radiation which is 6,000 Kelvin.

The exergy efficiency of the system (\({\eta }_{ex-sys}\)) is defined as the ratio of the exergy rate obtained from the heating of the water in the condenser to the sum of the electrical work of the compressor and pump plus the exergy of the solar radiation entering the collector, as follows [33]:

$${\eta _{{ex - sys}} } = \frac{{{\dot{Q}_{{cond}}} \left( {1 - \frac{{2{T_{0}} }}{{{T_{{W,o}}} + {T_{{W,i}} }}}} \right)}}{{{\dot{W}_{{cm}}} + {\dot{W}_{{pump}}} + {\dot{E}\chi _{{rad}} }}}$$
(32)

The exergy destruction of the system (\({\dot{E\chi }}_{des-sys}\)) is also defined as the sum of the irreversibilities of all system components as follows:

$$\dot{E}\chi _{{{\text{des - sys}}}} = \dot{E}\chi _{{{\text{des - cl}}}} + \dot{E}\chi _{{{\text{des - cond}}}} + \dot{E}\chi _{{{\text{des - eva}}}} + \dot{E}\chi _{{{\text{des - cm}}}} + \dot{E}\chi _{{{\text{des - TEV}}}}$$
(33)

2.9.2 Simulation-Based Optimization Approach

Figure 3 depicts the integration framework of the simulation-based optimization approach. As depicted, the thermodynamic model of the proposed system is derived and programmed using MATLAB programming language. The inputs of the MOO problem are the design parameters, decision parameters, and boundaries of decision parameters (constraints) [41, 42]. NSGA-II algorithm [43, 44, 45] was programed in MATLAB and coupled with the IDX-SAHP code to run the thermodynamic model and extract the outputs of the system, i.e., COP and CE (fitness functions). Afterward, NSGA-II gathers the simulation results [\(F({X}_{i})\)] and their relevant decision variables (\({X}_{i}\)), and then sorts them as per its computational intelligence algorithm to identify the better \({X}_{i}\) through an efficient heuristic search [46, 47, 48]. Next, MATLAB enters the new achieved \({X}_{i}\) into the IDX-SAHP model and makes it run to extract the new \(F({X}_{i})\). This process runs on till the termination criteria are reached, namely, the maximum number of iterations of 100 is met or the average rate of change in the spread of the Pareto curve becomes less than the tolerance of 0.001. These termination criteria are chosen as per several pre-tests to attain the best trade-off between the running time and the accuracy of the Pareto curve [43, 49, 50].

Fig. 3
figure 3

Integration framework of simulation-based optimization approach

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The NSGA-II coupled to the IDX-SAHP model leads to achieving the Pareto curve which represents the optimal non-dominated solutions [49, 51]. Finally, the best solution among all optimal solutions on the Pareto curve is chosen by the analytic hierarchy process (AHP) MCDA [52, 53] programmed under the MATLAB environment [54]. The principle of the AHP method is to make pairwise comparisons of the diverse alternatives and criteria to achieve their relative weights and to make the best decision among the alternatives [55]. The steps of AHP implementation are as follows [56]:

Step 1: The decision-making problem is organized as a hierarchy and is divided into three main levels including:

  • First level: the objective of the decision-making problem lies at the first of the hierarchy structure. In this work, the objective is the selection of a single optimum solution for the final configuration.

  • Second level: the criteria lie at the second level of the hierarchy, which are the fitness functions including COP (F1) and CE (F2)

  • Third level: the alternatives lie at the third level of the hierarchy structure, which are the Pareto-optimal solutions.

Step 2: The fitness functions are compared to the pairwise to evaluate their weights through the ratio scales listed in Table 4. The pairwise comparison matrix A is organized as per the decision maker’s judgments (\({A}_{xy}\)), as follows:

$$\begin{aligned}A& = \left[ {\begin{array}{*{20}l} 1 \\ {A_{21} } \\ {\begin{array}{*{20}l} \ldots \\ {A_{n1} } \\ \end{array} } \\ \end{array} \begin{array}{*{20}l} {A_{12} } \\ 1 \\ {\begin{array}{*{20}l} \ldots \\ \ldots \\ \end{array} } \\ \end{array} \begin{array}{*{20}l} \ldots \\ \ldots \\ {\begin{array}{*{20}l} 1 \\ \ldots \\ \end{array} } \\ \end{array} \begin{array}{*{20}l} {A_{1n} } \\ {A_{2n} } \\ {\begin{array}{*{20}l} \ldots \\ 1 \\ \end{array} } \\ \end{array} } \right] \mathrm{where} \\ A_{xy} &= 1/A_{yx} ; x,y = 1,2,3, \ldots ,n \end{aligned}$$
(34)
Table 4 Decision parameters and their boundaries
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Step 3: Define the maximum eigenvalue (\({\lambda }_{\mathrm{max}}\)) of the judgment matrix (\({A}_{xy}\)) as follows:

$$\det \left( {A - \lambda A} \right) = 0$$
(35)

Step 4: Compute the consistency index \((CI)\) as follows:

$$CI = \frac{{\lambda_{\mathrm{max}} - n}}{1 - n}$$
(36)

Here, and n is the order of the \({A}_{xy}\) matrix.

Step 5: Calculate the consistency ratio \((CR)\) to check judgment consistencies. Judgments are acceptable for \(CR\) values less than \(0.1\).

$$CR = \frac{CI}{{RI}}$$
(37)

RI is the average random consistence indicator of the \({A}_{xy}\) matrix.

Step 6: Calculate the weight vector, as follows:

$$M_{i} = \mathop \prod \limits_{x = 1}^{n} A_{xy}\, x = 1, 2, 3, \ldots ,n$$
(38)
$$\overline{{w_{N} }} = \sqrt[n]{{M_{x} }} \,x = 1, 2, 3, \ldots ,n$$
(39)

Step 7: Normalized weight vector (\(\overline{{w }_{N}}\)), as follows:

$$W_{x} = \frac{{\overline{{w_{N} }} }}{{\mathop \sum \nolimits_{x = 1}^{n} \overline{{w_{N} }} }}$$
(40)

\({W}_{i}\) is the relative weight coefficient of each factor.

$${\text{W }} = {\text{ }}\left[ {{\text{w}}_{1} ,{\text{ w}}_{2} ,...,{\text{w}}_{n} } \right]^{{\text{T}}}$$
(41)

Step 8: Evaluate the solutions on the Pareto curve as per the \({W}_{x}\) of each criterion, and then obtain a single optimum solution for the final configuration.

It should be noted because of the variations range of the decision variables, the number of feasible configurations of the IDX-SAHP is countless, while the proposed simulation-based multiple-criteria optimization method leads to lessening the running time substantially compared to numerous searches.

2.9.3 Fitness Functions and Decision Parameters

In the current research, CE and COP are chosen as two fitness functions with equally important weight. Additionally, the decision parameters and their boundaries are presented in Table 5. It should be noted that the decision parameters and their boundaries were opted as per several grounds comprising the working experiences of the authors in heating, ventilation, and air conditioning (HVAC) laboratories and companies, scrutinizing a good deal of simulations and experimental investigations [27, 28, 32, 71] and published books [57–60] concerning SAHPs, the available equipment in the market, and design restrictions.

Table 5 Climate characteristics of Tehran, Iran [66]
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3 Climate Characteristics of the Studied City

Tehran is the capital of Iran with a population of around 8.7 million in the city and 15 million in the larger metropolitan area of Greater Tehran and is the most populous city and the most energy consuming city in Iran [61, 62]. In the current investigation, Tehran with a temperate climate characteristic is opted to analyze and optimize the IDX-SAHP performance. The climate characteristics of Tehran are presented in Table 6.

Table 6 Results achieved by the SOOs and MOO
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4 Results and Discussion

This section compromises five parts. In the first part, the performance evaluation of the developed IDX-SAHP performance is performed. In the second part, the SA of the design parameters (pre-optimization) [63–65] comprising \({T}_{a}\), \({I}_{T}\), \({T}_{W,o}\), \(\omega\), \({A}_{cl}\), \({k}_{p}\), \({N}_{gl}\), \({D}_{cl}\), \({L}_{\mathrm{{cond}}}\), and \(\delta\) is carried out using OPAT technique. Then in the third part, the exergy analysis of the IDX-SAHP is investigated. In addition, in the fourth part, the SOO and MOO on the IDX-SAHP performance are implemented using NSGA-II. Afterward, the best configuration for the IDX-SAHP is opted using AHP MCDA. Finally, in the fifth part of the work, performance of the maximized IDX-SAHP is compared to the initial one.

4.1 Verification

To verify the IDX-SAHP, the parameters of the developed IDX-SAHP are opted as the specifications of IDX-SAHP system built by Bridgeman [27]. Then, the \(\mathrm{COP}\), \({\dot{W}}_mathrm{cm}\), \({\dot{Q}}_mathrm{cond}\), and \({\dot{Q}}_mathrm{eva}\) of the developed IDX-SAHP were compared to the Bridgeman experimental work [27]. Figures 4 and 5 indicate the values achieved by the numerical model and the experimental values achieved by Bridgeman [27]. As shown, the experimental values of the \(COP\), \({{\dot{W}}_{\mathrm{cm}}}\), \({{\dot{Q}}_{\mathrm{cond}}}\), and \({{\dot{Q}}_{\mathrm{eva}}}\) are almost the values achieved by the numerical model with the relative errors of \(6\%\), \(12\%\), \(15\%\), and \(11\%\), respectively. The reason for the small difference between numerical modeling and experimental values may be due to the losses in the IDX-SAHP elements, in particular in the piping system.

Fig. 4
figure 4

COP obtained by the numerical modeling and the experiment

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Fig. 5
figure 5

\({\dot{W}}_{cm}\), \({\dot{Q}}_{cond}\), and \({\dot{Q}}_{eva}\) obtained by the numerical modeling and the experiment

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4.2 Performance Evaluation

To evaluate the monthly performance of the developed IDX-SAHP, \({T}_{W,i}\) and \({T}_{a}\) were taken into account the same, and \({T}_{W,o}\) was kept constant at \(50^\circ\) C. Other IDX-SAHP characteristics are presented in Tables 1 and 4. Figures 6 and 7 present the COP and CE values per month. The highest COP is \(6.3\) in July with the highest values for \({T}_{a}\) (\(31\) °\(C\)), \({I}_{T}\) (\(570.552\, {{\mathrm{W/{m}}}^{2}}\)), and MSHs (\(343\) \(\mathrm{hours}\)) throughout the year. Additionally, the smallest COP is \(2.4\) in December with the least values for \({I}_{T}\) (307.8684 \(\mathrm{W/{m}^{2}}\)), MSHs (\(161\) hours), and \({T}_{a}\) (\(5\) \(^\circ C\)) throughout the year. On the contrary, the largest CE is \(86\%\) in September with a small value for of \({I}_{T}\) (\(524.7188\) \(\mathrm{W/{m}^{2}}\)), and almost high values for \({T}_{a}\) (\(26\) \(^\circ C\)) and MSHs (\(320\) hours) throughout the year. Furthermore, the least CE is \(51\%\) in February with a high value for \({{I}_{T}}\) (\(391.942\) \({\mathrm{W/{m}^{2}}}\)), and almost small values for \({T}_{a}\) (\(6\) \(^\circ C\)) and MSHs (\(189\) hours) throughout the year.

Fig. 6
figure 6

COP per month

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Fig. 7
figure 7

CE per month

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Also, Figs. 8 and 9 present, respectively, the working hour of the IX-SHHP and \({\dot{Q}}_{\mathrm{{cond}}}\) per month. To have \({T}_{W,o}\) of \(5\) 0 \(^\circ\) C, the total working hour is \(1319\) hours throughout the year and ranges from \(74\) hours in July to \(143\) hours in December. In addition, \({\dot{Q}}_{cond}\) ranges from \(122 \mathrm{kWh}\) in July to \(342 \mathrm{kWh}\) in December. In the warm months, the working hour and \({\dot{Q}}_{\mathrm{cond}}\) is smaller because \({T}_{W,i}\), \({T}_{a}\), \({I}_{T}\), and MSHs are larger and vice versa.

Fig. 8
figure 8

Working hour per month

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Fig. 9
figure 9

\({\dot{Q}}_{\mathrm{cond}}\) per month

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4.3 Sensitivity Analysis (SA)

This part of the research gives the SA of main IDX-SAHP parameters such as \({T}_{a}\), \({I}_{T}\), \({T}_{W,o}\), \(\omega\), \({A}_{cl}\), \({k}_{p}\), \({N}_{gl}\), \({D}_{cl}\), \({L}_{\mathrm{cond}}\), and \(\delta\) using the OPAT technique [67, 68]. According to the OPAT, each time, one parameter changes over its boundaries and the other parameters are fixed at their initial values [69, 70] (as provided in Table 1) and the IDX-SAHP performance is assessed.

4.3.1 Influence of \({{\varvec{T}}}_{{\varvec{a}}}\)

Figure 10 depicts the influence of \({T}_{a}\) on the IDX-SAHP performance. Increasing \({T}_{a}\) at constant \({I}_{T}\) enhances \({T}_{p}\) and \({T}_{gw}\). Therefore, \({\dot{Q}}_{\mathrm{loss}}\) decreases which leads to increase in CE. Additionally, by reducing \({\dot{Q}}_{\mathrm{loss}}\), more heat rate is delivered to the GWS, and accordingly, to the R-134a passing through the evaporator tubes. In this regard, \({\dot{W}}_{\mathrm{cm}}\) decreases which leads to increase in COP [32]. The COP-CE diagram of the developed IDX-SAHP in terms of \({T}_{a}\) variations is in accordance with the results of the investigation conducted by Kong et al. [32].

Fig. 10
figure 10

Influence of \({T}_{a}\) on the IDX-SAHP performance

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4.3.2 Influence of \({{\varvec{I}}}_{{\varvec{T}}}\)

Figure 11 depicts the influence of \({I}_{T}\) on the IDX-SAHP performance. With the increase in \({I}_{T}\) at constant \({T}_{a}\), the \({T}_{p}\) is enhanced. The higher \({T}_{p}\) leads to increase \(\left|{T}_{p}-{T}_{a}\right|\), and accordingly, \({\dot{Q}}_{\mathrm{loss}}\) increases and thus CE reduces. Moreover, by increasing the \({I}_{T}\), the \({\dot{Q}}_{cl}\) is enhanced. Consequently, \({T}_{\mathrm{gw}}\) increases and the GWS obtains more energy from the solar collector, and accordingly, the heat is delivered more to the R-134a passing through the evaporator tubes. Consequently, \({\dot{W}}_{\mathrm{cm}}\) reduces which leads to increase in COP [32, 71]. The COP-CE diagram of the developed IDX-SAHP in terms of \({I}_{T}\) variations is in accordance with the results of the investigation conducted by Kong et al. [32].

Fig. 11
figure 11

Influence of \({I}_{T}\) on the IDX-SAHP performance

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4.3.3 Influence of \({{\varvec{N}}}_{{\varvec{g}}{\varvec{l}}}\)

Figure 12 depicts the influence of \({N}_{gl}\) on the IDX-SAHP performance. With the increase in \({N}_{gl}\) at constant \({T}_{a}\) and \({I}_{T}\), the COP and CE enhance a little. The glass cover of the solar collector plays as a thermal insulation which traps the solar radiation and decreases \({\dot{Q}}_{\mathrm{loss}}\), and accordingly, CE increases [72]. Furthermore, with the reduction of \({\dot{Q}}_{\mathrm{loss}}\), the \({T}_{p}\) is enhanced, and the GWS obtains more energy from the solar collector, and accordingly, the heat is delivered more to the R-134a passing through the evaporator tubes. Consequently, \({\dot{W}}_{\mathrm{cm}}\) reduces which leads to increase in COP.

Fig. 12
figure 12

Influence of \({N}_{gl}\) on the IDX-SAHP performance

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4.3.4 Influence of \({{\varvec{T}}}_{{\varvec{W}},{\varvec{o}}}\)

Figure 13 depicts the influence of \({T}_{W,o}\) on the IDX-SAHP performance. With the increase in \({T}_{W,o}\) at constant \({T}_{a}\) and \({I}_{T}\), the \({\dot{W}}_{\mathrm{cm}}\) is enhanced, and accordingly, COP lessens. Besides, the larger the \({\dot{W}}_{\mathrm{cm}}\) the higher refrigerant enthalpy at the evaporator inlet. In this regard, the refrigerant capacity lessens in obtaining the energy from the GWS in the evaporator shell. Consequently, the GWS comes back to the solar collector at a larger temperature which leads to reducing in CE [32, 73].

Fig. 13
figure 13

Influence of \({T}_{W,o}\) on the IDX-SAHP performance

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4.3.5 Influence of \({\mathbf{A}}_{\mathbf{c}\mathbf{l}}\)

Figure 14 depicts the influence of \({\mathrm{A}}_{\mathrm{cl}}\) on the IDX-SAHP performance. Increasing \({\mathrm{A}}_{\mathrm{cl}}\) at constant \({T}_{a}\) and \({I}_{T}\) enhances \({\dot{Q}}_{cl}\) which leads to increase in \({T}_{\mathrm{gw}}\). Accordingly, the GWS obtains more energy from the solar collector, and accordingly, the heat is delivered more to the R-134a passing through the evaporator tubes. Consequently, \({\dot{W}}_{\mathrm{cm}}\) reduces which leads to increase in COP [32, 71]. Besides, the higher \({\mathrm{A}}_{cl}\) the higher \({\dot{Q}}_{loss}\), which leads to reduction of CE. The COP-CE diagram of the developed IDX-SAHP in terms of \({\mathrm{A}}_{cl}\) variations is in accordance with the results of the investigation conducted by Kong et al. [32].

Fig. 14
figure 14

Influence of \({\mathrm{A}}_{cl}\) on the IDX-SAHP performance

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4.3.6 Influence of \({\varvec{\omega}}\)

Figure 15 depicts the influence of \(\omega\) on the IDX-SAHP performance. By increasing the \(\omega\) at constant \({T}_{a}\) and \({I}_{T}\), the temperature of R-134a at the compressor outlet and \({\dot{W}}_{\mathrm{cm}}\) enhance which leads to reduction in COP. Additionally, the larger \(\omega\) the larger R-134a mass flow rate (\({\dot{m}}_{r}\)), and accordingly, the R-134a obtains more energy from the GWS passing through the evaporator tubes. In this regard, the GWS comes back to the solar collector at a lower temperature which leads to reduction in \({\dot{Q}}_{\mathrm{loss}}\), and consequently the CE is enhanced. It is worth noting that decrease in COP and increase in CE by increasing \(\upomega\) indicate that there is an optimum \(\upomega\) to achieve an IDX-SAHP with the highest performance. The COP-CE diagram of the developed IDX-SAHP in terms of \(\upomega\) variations is in accordance with the results of the investigation conducted by Kong et al. [32].

Fig. 15
figure 15

Influence of \(\upomega\) on the IDX-SAHP performance

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4.3.7 Influence of the \({{\varvec{L}}}_{{\varvec{c}}{\varvec{o}}{\varvec{n}}{\varvec{d}}}\)

Figure 16 depicts the influence of \({L}_{\mathrm{cond}}\) on the IDX-SAHP performance. Increasing \({L}_{\mathrm{cond}}\) at constant \({T}_{a}\) and \({I}_{T}\) leads to increase in \({A}_{\mathrm{coil}}\). In this regard, \({\dot{Q}}_{\mathrm{cond}}\) is enhanced, and accordingly, \({\dot{W}}_{\mathrm{cm}}\) lessens which leads to increase in COP. Further, the lower \({\dot{W}}_{\mathrm{cm}}\) the lower R-134a enthalpy at the evaporator inlet. As a result, the refrigerant capacity is enhanced in obtaining the energy from the GWS in the evaporator shell. Consequently, the GWS comes back to the solar collector at a lower temperature which leads to increase in CE [32, 73]. Moreover, as depicted, the IDX-SAHP has the best performance in \({L}_{\mathrm{cond}}\) of \(50 (m)\). If \({L}_{\mathrm{cond}}>50 (m)\), COP and CE would lessen since \({L}_{\mathrm{cond}}\) is larger enough, even redundantly, to deliver the thermal energy from the R-134a to the water. The COP-CE diagram of the developed IDX-SAHP in terms of \({L}_{\mathrm{cond}}\) variations is in accordance with the results of the investigation conducted by Zhang et al. [74].

Fig. 16
figure 16

Influence of \({L}_{\mathrm{cond}}\) on the IDX-SAHP performance

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4.3.8 Influence of \({{\varvec{D}}}_{{\varvec{c}}{\varvec{l}}}\)

Figure 17 depicts the influence of \({D}_{cl}\) on the IDX-SAHP performance. Increasing \({D}_{cl}\) at constant \({T}_{a}\) and \({I}_{T}\) leads to increase the side surface of the collector tubes, and accordingly, the GWS gains more energy from the collector, resulting in increase of \({\dot{Q}}_{\mathrm{cl}}\). In this regard, \({T}_{\mathrm{gw}}\) is enhanced, and therefore, the thermal energy is delivered more to the R-134a passing through the evaporator tubes. Consequently, \({\dot{W}}_{\mathrm{cm}}\) lessens, and thus COP increases. Besides, the higher \({T}_{\mathrm{gw}}\) the lower \({\dot{Q}}_{\mathrm{loss}}\) that leads to increase in CE. Also, as indicated, the IDX-SAHP has the best performance in \({D}_{\mathrm{cl}}\) of \(13 \mathrm{(mm)}\). Because \({D}_{cl}>13 \mathrm{(mm)}\) is too big for the IDX-SAHP, resulting in \({\dot{Q}}_{\mathrm{{loss}}}\) increase and GWS charge decrease, and consequently the COP and CE would lessen [74]. The COP-CE diagram of the developed IDX-SAHP in terms of \({D}_{\mathrm{cl}}\) variations is in accordance with the results of the investigation conducted by Zhang et al. [74].

Fig. 17
figure 17

Influence of \({D}_{cl}\) on the IDX-SAHP performance

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4.3.9 Influence of \({\varvec{\delta}}\) and \({{\varvec{k}}}_{{\varvec{p}}}\)

Figures 18 and 19, respectively, depict the influences of \(\delta\) and \({k}_{p}\) on the IDX-SAHP performance. By increasing the \({k}_{p}\) and \(\delta\) at constant \({T}_{a}\) and \({I}_{T}\), the \({T}_{\mathrm{gw}}\) is enhanced due to the higher \({\dot{Q}}_{cl}\). Thus, the GWS obtains more energy from the collector, and accordingly, the heat is delivered more to the R-134a passing through the evaporator tubes. Consequently, \({\dot{W}}_{cm}\) reduces which leads to increase in COP [32, 71]. Besides, the higher \({T}_{gw}\) the lower \({\dot{Q}}_{\mathrm{loss}}\) which leads to increase in CE [72, 74]. Also, the higher \({k}_{p}\) the lower temperature gradient in the collector plate. In this regard, by increasing the \(\delta\) slightly, the accumulation of thermal energy in the collector plate would enhance a little, and consequently, COP and CE would enhance a little. The COP-CE diagram of the developed IDX-SAHP in terms of \(\delta\) variations is in accordance with the results of the investigation conducted by Zhang et al. [74].

Fig. 18
figure 18

Influence of \(\delta\) on the IDX-SAHP performance

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Fig. 19
figure 19

Influence of \({k}_{p}\) on the IDX-SAHP performance

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The SA indicated that the performance of the IDX-SAHP would be noticeably affected by the design parameters. Consequently, the performance may be largely enhanced by selecting the appropriate IDX-SAHP parameters. In this regard, the parametric analysis reveals the behaviors of the system performance with the diverse input parameters and helps the designers and engineers to select the most appropriate materials for the system. However, to achieve the optimum IDX-SAHP, the MOO coupled to the MCDAs necessitates.

4.4 Exergy Analysis Results

In this section, the influence of various factors such as \({T}_{a}\), \({I}_{T}\), \({A}_{cl}\) and \(\omega\) on exergy efficiency (\({\eta }_{ex-sys}\)) and exergy destruction \(({\dot{E\chi }}_{des-sys})\) of the IDX-SAHP system is investigated using the OPAT technique.

4.4.1 Influence of \({{\varvec{T}}}_{{\varvec{a}}}\) on \({{\varvec{\eta}}}_{{\varvec{e}}{\varvec{x}}-{\varvec{s}}{\varvec{y}}{\varvec{s}}}\) and \({\dot{{\varvec{E}}{\varvec{\chi}}}}_{{\varvec{d}}{\varvec{e}}{\varvec{s}}-{\varvec{s}}{\varvec{y}}{\varvec{s}}}\)

Figure 20 depicts the influence of \({T}_{a}\) on exergy efficiency and exergy destruction of the system. Since \({T}_{a}={T}_{w,i}\), thus, by increasing the \({T}_{a}\) in constant \({I}_{T}\), the \({T}_{w,i}\) increases. Therefore, as \({T}_{w,o}\) is constant at \(50^\circ\) C, the difference between \({T}_{w,i}\) and \({T}_{w,o}\) is reduced and the amount of heat loss from the condenser is reduced, accordingly. In this regard, the exergy efficiency increases and the exergy destruction reduces. As observed, \({T}_{a}\) had a very small influence on the system. On the other hand, by increasing \({T}_{a}\), the temperature difference between the collector surface and the environment decreases and heat loss from the collector surface decreases; this is beneficial in increasing exergy efficiency. As seen in Fig. 20, by increasing \({T}_{a}\) from \(-5\) to \(30\) \(^\circ\)C in constant \({I}_{T}\), the system's exergy efficiency increases from 17.3 to 18.2% and the system's exergy destruction decreases from 2.25 to 2.15 kW.

Fig. 20
figure 20

Influence of \({T}_{a}\) on the exergy efficiency and exergy destruction of the IDX-SAHP

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4.4.2 Influence of \({{\varvec{I}}}_{{\varvec{T}}}\) on \({{\varvec{\eta}}}_{{\varvec{e}}{\varvec{x}}-{\varvec{s}}{\varvec{y}}{\varvec{s}}}\) and \({\dot{{\varvec{E}}{\varvec{\chi}}}}_{{\varvec{d}}{\varvec{e}}{\varvec{s}}-{\varvec{s}}{\varvec{y}}{\varvec{s}}}\)

Figure 21 depicts the influence of \({I}_{T}\) on exergy efficiency and exergy destruction of the system. With the increase in \({I}_{T}\) at constant \({T}_{a}\), the \({T}_{p}\) is enhanced. The higher \({T}_{p}\) leads to increase \(\left|{T}_{p}-{T}_{a}\right|\), and accordingly, \({\dot{Q}}_{loss}\) increases, and thus, CE reduces. Subsequently, the exergy destruction increases and the exergy efficiency reduces by increasing the \({\dot{Q}}_{\mathrm{loss}}\). Moreover, as observed from Fig. 21, the influence of \({I}_{T}\) on exergy efficiency and exergy destruction of the system is far greater than the influence of \({T}_{a}.\) By increasing \({I}_{T}\) from \(350\) W/m2 to \(1200\) W/m2 in constant \({T}_{a}\), the system's exergy efficiency decreases from 22% to 16.2% and the system's exergy destruction increases s from 1.25 kW to 2.5 kW.

Fig. 21
figure 21

Influence of \({I}_{T}\) on the exergy efficiency and exergy destruction of the IDX-SAHP

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4.4.3 Influence of \({{\varvec{A}}}_{{\varvec{c}}{\varvec{l}}}\) on \({{\varvec{\eta}}}_{{\varvec{e}}{\varvec{x}}-{\varvec{s}}{\varvec{y}}{\varvec{s}}}\) and \({\dot{{\varvec{E}}{\varvec{\chi}}}}_{{\varvec{d}}{\varvec{e}}{\varvec{s}}-{\varvec{s}}{\varvec{y}}{\varvec{s}}}\)

Figure 22 depicts the influence of \({A}_{cl}\) on exergy efficiency and exergy destruction of the system. With the increase in \({A}_{cl}\) in constant \({T}_{a}\), the amount of \({\dot{E\chi }}_{rad}\) is enhanced as per. Accordingly, the exergy efficiency decreases and the exergy destruction increases. By increasing \({A}_{cl}\) from \(1.5\) m2 to \(6.5\) m2 in constant \({T}_{a}\), the system's exergy efficiency decreases from 18 to 14% and the system's exergy destruction increases from 1.4 kW to 4 kW.

Fig. 22
figure 22

Influence of \({A}_{cl}\) on the exergy efficiency and exergy destruction of the IDX-SAHP

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4.4.4 Influence of \({\varvec{\omega}}\) on \({{\varvec{\eta}}}_{{\varvec{e}}{\varvec{x}}-{\varvec{s}}{\varvec{y}}{\varvec{s}}}\) and \({\dot{{\varvec{E}}{\varvec{\chi}}}}_{{\varvec{d}}{\varvec{e}}{\varvec{s}}-{\varvec{s}}{\varvec{y}}{\varvec{s}}}\)

Figure 23 depicts the influence of \(\omega\) on exergy efficiency and exergy destruction of the system. By increasing the \(\omega\) in constant \({T}_{a}\) and \({I}_{T}\), the temperature of R-134a at the compressor outlet enhances which leads to increase in \({\dot{Q}}_{\mathrm{cond}}\). Therefore, the exergy efficiency increases and the exergy destruction in the condenser decreases. It should also be noted that as the \(\omega\) increases, the compressor does more work and as a result, the rate of exergy destruction in the compressor increases; however, increasing the \(\omega\) reduces the exergy destruction in the condenser. Considering the sum of all the irreversibilities of the system components, increasing the \(\omega\) lead to decrease in the exergy destruction of the whole system. Accordingly, the influence of \(\omega\) on the exergy destruction of condenser is larger than compressor. By increasing \(\omega\) from \(1200 (rpm)\) m2 to \(2800 (rpm)\) in constant \({T}_{a}\) and \({I}_{T}\), the system's exergy efficiency increases from 14 to 18% and the system's exergy destruction decreases from 2.4 kW to 1.5 kW.

Fig. 23
figure 23

Influence of \(\omega\) on the exergy efficiency and exergy destruction of the IDX-SAHP

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4.5 Optimization Results

This part of the research presents the results of the maximized IDX-SAHP. It should be noted that December month is opted to optimize the IDX-SAHP because of having the least amounts of \({I}_{T}\) (\(307.8684\) \(W/{m}^{2}\)), MSHs \((161 \mathrm{h})\), and \({T}_{a}\) (\(5\) °C) along with the highest amount of working hour \((142 \mathrm{h})\) throughout the year. Besides, the maximized IDX-SAHP is fully compared to the initial model.

4.5.1 SOO and MOO Results

Table 6 presents the SOOs and MOO results of the IDX-SAHP. In the COP-based maximization, the highest amount of \({\mathrm{A}}_{cl}\) (\(6.5\) m2) and the least amount of \(\omega\) (\(1205\) rpm) were achieved. However, CE-based maximization, the least amount of \({A}_{cl} (1.5 {{\mathrm{{m}}}^{2}})\) and the highest amount of \(\upomega\) \((2780 {\mathrm{ rpm}})\) were achieved. Additionally, the \({{L}_{\mathrm{cond}}}\) of \(50 {\mathrm{(m)}}\), \({D}_{cl}\) almost \(13.0 {\mathrm{(mm)}}\), \({k}_{p}\) almost \(400 {\mathrm{ (W/m.K)}}\), and the \(\delta\) of about \(1.0 {\mathrm{(mm)}}\) were achieved through the SOOs for COP and CE.

The optimum parameters achieved by the COP-based maximization (provided in Table 6) were implemented in the initial IDX-SAHP model. The total working hour was \(1101 {\mathrm{h}}\) throughout the year, resulting in \(218 {\mathrm{hours}}\) reduction compared to the initial IDX-SAHP. Besides, the optimum parameters achieved by the CE-based maximization (provided in Table 6) were implemented to the initial IDX-SAHP model. The total working hour was \(1258 {\mathrm{h}}\) throughout the year, resulting in \(61 {\mathrm{h}}\) reduction compared to the initial IDX-SAHP. Accordingly, the design of IDX-SAHP based on the SOO of COP gave a better performance compared to the SOO of CE.

Similar to the achieved conclusion from the SA, the SOOs also showed there is an optimum \({A}_{cl}\) and \(\omega\) to achieve the highest performance for IDX-SAHP.

To scrutinize the COP and CE interactions, the MOO may be performed using the NSGA-II algorithm. Figure 24 presents the Pareto curve obtained by the MOO process which depicts the COP and CE interactions. As observed, as COP lessens, CE is enhanced and vice versa, namely, it is unfeasible to enhance both COP and CE, simultaneously.

Fig. 24
figure 24

Pareto curve obtained by the MOO process

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As presented, the least COP is at point B with a value of \(1.5\) while CE has the largest amount with a value of \(91\%\). Besides, the least CE is at point A with a value of \(42\%\) whereas COP has the largest amount with a value of \(8.7\). Accordingly, the optimum solutions obtained by the MOO process lay in \(1.5\le COP\le 8.7\) and \(\%42\le CE\le \%91\).

To achieve the final optimum layout of the IDX-SAHP, the AHP MCDA method was integrated with the MOO process. Accordingly, an maximized IDX-SAHP with \({\mathrm{A}}_{\mathrm{cl}}\) of \(6.35 {m}^{2}\), \(\omega\) of \(1430 \mathrm{(rpm)}\), \({L}_{\mathrm{{cond}}}\) of \(48 \mathrm{(m)}\), \({D}_{\mathrm{{cl}}}\) almost \(13.0\mathrm{ (mm)}\), \({k}_{p}\) of \(390 \mathrm{(W/m K)}\), and \(\delta\) of about \(8.7 \mathrm{(mm)}\) were yielded. In Fig. 24, point D and point C, respectively, represent the values of COP and CE of the maximized IDX-SAHP system and the initial one.

As presented in Table 6, in the SOO of COP, the COP is enhanced greatly from \(3\) to \(8.7\) whereas the CE lessened a little from \(52\%\) to \(42\%\). Besides, in the SOO of CE, the COP lessened slightly from \(3\) to \(1.5\) whereas the CE is greatly enhanced from \(52\) to \(91\%\). It is deduced that increasing the COP reduces the CE, and accordingly, both fitness functions cannot be improved at the same time. Furthermore, the AHP MCDA method integrated with the MOO process yielded that COP is largely enhanced from \(3\) to \(7\) whereas the CE reduced a little from \(52\%\) to \(46.4\%\).

4.5.2 Maximized IDX-SAHP Evaluation

In this part, the IDX-SAHP system as per the MOO results is assessed, in-depth. In this context, the optimum parameters achieved by the MOO (provided in Table 6) were implemented to the initial IDX-SAHP and the maximized system was examined. As provided in Table 7, the largest COP was \(8.4\) in with the highest values for \({T}_{a}\) (\(31\) °\(C\)), \({I}_{T}\) (\(570.552 \mathrm{W/{m}}^{2}\)), and MSHs (\(343\) hours) throughout the year. Additionally, the smallest COP is \(4.6\) in December with the least values for \({I}_{T}\) (307.8684 \(\mathrm{W/{m}}^{2}\)), MSHs (\(161\) hours), and \({T}_{a}\) (5 \(^\circ \mathrm{C}\)) throughout the year. On the contrary, the largest CE is \(77\%\) in September with a small value for of \({I}_{T}\) (\(524.7188\) \(W/{m}^{2}\)), and almost high values for \({T}_{a}\) (\(26\) \(^\circ C\)) and MSHs (\(320\) hours) throughout the year. Furthermore, the least CE is \(43\%\) in February with a high value for \({I}_{T}\) (\(391.942\) \(\mathrm{W/{m}}^{2}\)), and almost small values for \({T}_{a}\) (\(6\) \(^\circ C\)) and MSHs (\(189\) hours) throughout the year. To have \({T}_{W,o}\) of \(5\) 0 \(^\circ\) C, the total working hour of the IDX-SAHP is \(1037\) hours throughout the year and ranges from \(51\) hours in July to \(119\) hours in December. In addition, \({\dot{Q}}_{\mathrm{cond}}\) ranges from \(100\mathrm{ kWh}\) in July to \(323 kWh\) in December. In the warm months, the working hour and \({\dot{Q}}_{\mathrm{cond}}\) is smaller because \({T}_{W,i}\), \({T}_{a}\), \({I}_{T}\), and MSHs are larger and vice versa.

Table 7 Comparison of the maximized and initial IDX-SAHP
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Consequently, although the CE of the maximized IDX-SAHP drops slightly, its COP is highly enhanced, resulting in a reduction in the total working hour up to \(282\) hours compared to the initial IDX-SAHP. In other words, the overall power use of the maximized IDX-SAHP largely lessens compared to the initial IDX-SAHP. Accordingly, the MOO presented a more optimal IDX-SAHP system compared to the SOOs.

5 Conclusions

This investigation proposed the numerical modeling of the indirect-expansion solar-assisted heat pump (IDX-SAHP), and the exergy-energy analysis and multiple-objective optimization (MOO) of the developed system were carried out. The performance of the proposed IDX-SAHP was completely assessed under the temperate climate of Iran. Next, the influences of design IDX-SAHP parameters such as \({T}_{a}\), \({I}_{T}\), \({T}_{W,o}\), \(\omega\), \({A}_{\mathrm{cl}}\), \({k}_{p}\), \({N}_{\mathrm{gl}}\), \({D}_{\mathrm{cl}}\), \({L}_{\mathrm{cond}}\), and \(\delta\) were analyzed on the COP and CE using OPAT technique. Based on the results of OPAT, the IDX-SAHP performance was affected by the design parameters considerably so that the performance would be largely enhanced by selecting the appropriate IDX-SAHP components. Afterward, the exergy analysis of the IDX-SAHP system was conducted and the influence of \({T}_{a}\), \({I}_{T}\), \({A}_{cl}\) and \(\omega\) on exergy efficiency (\({\eta }_ {\mathrm{ex-sys}}\)) and exergy destruction \(({\dot{E\chi }}_{\mathrm{des-sys}})\) of the system were investigated. The exergy analysis showed that increasing the \({T}_{a}\) leads to increase in \({\eta }_{\mathrm{ex-sys}}\) and reduction in \({\dot{E\chi }}_{\mathrm{des-sys}}\) of the system. However, \({I}_{T}\) had a more pronounced effect on exergy of the system compared to \({T}_{a}\) and greatly reduced the \({\eta }_{\mathrm{ex-sys}}\) and increased the \({\dot{E\chi }}_{\mathrm{des-sys}}\). Besides, increasing the \({A}_{\mathrm{cl}}\) reduced the \({\eta }_{\mathrm{ex-sys}}\) and increased \({\dot{E\chi }}_{\mathrm{des-sys}}\) while increasing the \(\omega\) increased the \({\eta }_{\mathrm{ex-sys}}\) and reduced the \({\dot{E\chi }}_{\mathrm{des-sys}}\). Moreover, SOOs and MOO processes were implemented on the developed IDX-SAHP. The SOOs results indicated that the design of IDX-SAHP according to the COP-based optimization recommended better efficiency compared to the CE-based optimization. Also, the Pareto curve obtained by the MOO process depicted the COP and CE interactions. To achieve the final optimum layout of the IDX-SAHP, AHP MCDA method was integrated with MOO process. The achieved results demonstrated that the MOO approach yielded better comprehensive performance compared to the SOOs in such a way that although the CE of the maximized IDX-SAHP lessened a little from \(52\) to \(47\%\), its COP is greatly enhanced from \(3\) to \(7\), resulting in reducing the total working hour up to \(282\) hours compared to the initial IDX-SAHP, and therefore, the overall power consumption of the maximized system largely lessened. This study declared the importance of SA, SOO, MOO, and MADA during the design of the IDX-SAHP to improve the performance and lessen the power use of the system via opting the best design parameters of the system. Our next research focuses on the environmental-economic evaluations of the maximized IDX-SAHP, construction of the maximized IDX-SAHP and performance analysis in different climate conditions of Iran, development and performance analysis of IDX-SAHP applying PV/T.