An Installed Hybrid Direct Expansion Solar Assisted Heat Pump Water Heater to Monitor and Modeled the Energy Factor of a University Students’ Accommodation

Abstract

This paper focused on the performance monitoring and modeling of a 6.0 kW, 2000 L hybrid direct expansion solar assisted heat pump (DX-SAHP) water heater used for the production of hot water in a university students’ accommodation with 123 females. The data of total electrical energy consumed, volume of hot water consumed, ambient temperature, relative humidity, and solar irradiance were obtained from the data acquisition systems and analyzed in conjunction with the energy factor (EF) of the system. A multiple linear regression model was developed to predict the EF. The EF of the hybrid DX-SAHP water heater was determined from the summation of the coefficient of performance (COP) of the heat pump unit and the solar fraction (SF) of the solar collectors. The operations of the hybrid energy system were analyzed based on three phases (first phase from 00:00–08:00, second phase from 08:30–18:30, and third phase from 19:00–23:30) over 24 h for the entire monitoring period. The average EF of the hybrid energy system per day during the second phase of operation was 4.38, while the SF and COP were 0.697 and 3.683, respectively. The developed multiple linear regression model for the hybrid DX-SAHP water heater accurately predicted the determined EF.

1. Introduction

Solar assisted heat pump (SAHP) technologies operate primarily on conventional vapor compression refrigeration cycles (VCRCs), enhanced by solar energy. The SAHP are commonly known as hybrid solar assisted air source heat pumps (ASHPs) and are composed of a solar water heater and a conventional air source heat pump (ASHP) water heater. These technologies are recommended for sanitary hot water production due to their excellent energy consuming efficiency and significant reduction in carbon footprint. Enormous research has been conducted to determine the performance of SAHP water heaters through the experimental and modeling methods and in the residential sector. This study presents a state of the art hybrid solar assisted heat pump (SAHP) water heater installed in a students’ residence of a university campus with the goal of monitoring and modeling its performance.

Research has been conducted on model development and validation of the thermal performance of a direct expansion solar air source heat pump (DX-SAHP) water heater and exploitation of the impact of different factors on the system performance [1]. The study treated the determined solar collector efficiency and the coefficient of performance of the DX-SAHP water heaters as two fitness functions. An experimental study has been conducted on two designed direct expansion solar assisted heat pumps (DX-SAHPs) using the refrigerants R134a and R290 to perform a comparative analysis of the thermal performance of the systems under various winter weather conditions [2]. Research conducted on the operational parameters of a photovoltaic solar assisted heat pump system proved that upon optimizing the power consumption of the system, a reduction of 55.7% can be achieved relative to the power consumed from a conventional electric water heater of similar tank capacity used for hot water heating [3]. A robust simulation model of the heating performance of the direct expansion solar assisted heat pump (DX-SAHP) was established based on experimental data under frost conditions from a prototype system and the modeled results mimic the experimental values [4]. Research conducted on a single- and double-glazed solar assisted heat pump water heater revealed that the COP of the double-glazed system was 3.37, while that of the single-glazed system was 2.69 with a similar heat gain rate [5]. The results of the thermal performance depicted that the double-glazed system performed better than the single-glazed system. An experimental study conducted on solar assisted heat pump water heaters confirmed potential space heating increased by 1–3% and had a cooling load reduction of between 7 and 15%, whereas conventional heat pump water heaters increased the space heating by 8–17% and decreased spaced cooling in the range 7–15% [6]. A review of the heating mode of different SAHPs under various ambient conditions demonstrated that the heat loss from the bare collector–evaporator starts with high solar irradiance and that frost had a negligible disruptive effect in the flat air type evaporator as opposed to the finned type [7]. A mixed integer linear programming method has been developed as an optimization model for the optimal management of direct expansion solar assisted photovoltaic/thermal heat pumps and an energetic and economic analysis was conducted on an annual basis on the system [8]. A study of a hybrid thermosolar system conducted in the northwest of Greece revealed that the concentrating solar power system increases the operational costs as compared to fossil fuel thermal systems [9]. The hybrid thermosolar system could serve as an accurate technology to energy engineers and researchers as well as for critical managerial decisions regarding electrical energy economics. Enormous studies have been conducted on residential hybrid solar assisted heat pump (HSAHP) water heaters, especially in South Africa, but this is the first of its kind to be implemented at a university campus [10].

Mindful of the daunting cost of electricity due to hot water heating in the students’ accommodations as a result of inefficient heating by electric boilers, the implementation of the DX-SAHP water heaters can lead to both permanent reduction in the electrical energy consumed and in the cost of the electricity bill. This study focused on the implementation of an efficient DX-SAHP water heater in one of the female accommodations with 123 students at the university of Fort Hare, Alice campus in the Eastern Cape province of South Africa. Furthermore, a detailed evaluation of the energy performance was conducted exploring the experimental determined electrical, thermal, and weather parameters associated with the operation of the installed DX-SAHP water heater. Lastly, a multiple linear regression model was developed to predict the energy factor of the system.

2. Aim and Objectives of the Study

2.1. Aim of the Study

The aim of this study is to design an efficient hybrid DX-SAHP water heater and model the energy factor (EF) using a multiple linear regression model with electrical energy consumed, ambient temperature, relative humidity, and wind speed as the predictors.

2.2. Objectives of the Study

The formulated objectives of the study include:

i.

To design and install an efficient hybrid DX-SAHP water heater.

ii.

To design and build a DAS to monitor the performance of the hybrid DX-SAHP water heater.

iii.

To determine the solar fraction (SF) of the solar collectors of the hybrid DX-SAHP water heater during operation throughout the monitoring period.

iv.

To determine the COP of the air source heat pump (ASHP) of the hybrid DX-SAHP water heater during operation of the system over the monitoring period.

v.

To determine the energy factor (EF) of the hybrid DX-SAHP water heater during operation for the entire monitoring period.

vi.

To develop and build a multiple linear regression model to predict the EF of the hybrid DX-SAHP water heater with electrical energy consumed, ambient temperature, relative humidity, and wind speed as the predictors.

3. Limitation and Assumptions for the Study

3.1. Limitation Associated with the Study

The research is limited to one location (female accommodation at the University of Fort Hare), with one type of solar collector (‘its solar’ evacuated tubes collectors), and an ASHP unit from a specific manufacturer (‘its solar’ heat pump unit) due to the high capital cost involved in purchasing the water heating devices and the sensors. This can be justified as the total expenditure (both the cost of equipment and installation) amounted to ZAR 650,000 (approximately USD 37 456.37).

3.2. Assumptions Employed in the Study

The following assumptions were taken into consideration in achieving the findings from the study:

i.

The volume of the hot water consumed by the occupants in the residence is equal to the volume of hot water measured by the flow meter connected to the hot water copper pipe supplying hot water into the accommodation from the hybrid DX-SAHP water heater;

ii.

The electrical energy consumed by the water circulation pump installed on the side of the solar collectors of the hybrid DX-SAHP water heater is negligible;

iii.

The operational cycles of the hybrid DX-SAHP water heater were divided into three phases (phase one from 00:00–08:00, phase two from 08:30–18:30, and phase three from 19:00–23:30) over 24 h, and the values of the EF were determined during the operation of the system in phase two;

iv.

The ASHP unit of the hybrid DX-SAHP water heater did not practically operate in phase one and phase three throughout the monitoring period.

4. Fundamentals of SAHP Water Heater

An SAHP water heater comprises an integrated solar thermal collector and an ASHP water heater. The main drive that characterized the exceptional interest in the utilization of the heat pump water heating system was the ability to harness a substantial free low-grade heat source within the vicinity of the evaporator into gained thermal energy by refrigerant during VCRC. The heat pump operates between two temperature reservoirs and the heat pump assists in pumping heat from a low temperature reservoir source to a high temperature reservoir sink. The electrical energy required to pump the heat is proportional to the temperature lift between the two reservoirs [11]. Hence, a higher evaporation temperature leads to an increase in COP, due to the increase in free solar or aerothermal energy that the heat pump is capable of extracting from the evaporator unit. In hot and humid countries, an ideal technology for domestic hot water heating is the SAHP water heater [12]. This is undeniable as there is frequent and available solar radiation is this region. Despite the higher setup cost of this technology when compared to standalone ASHP and electric heating, the operational cost is very low and favors its utilization. A potential setback of SAHP is associated with the installation of the system in taller buildings, and this is adversely affecting the market penetration of the technology [13]. The SAHP is an excellent technology for producing hot water in the low temperature range (40 °C to 70 °C) [14]. In addition, in tropical countries where space heating is not a concern, hot water in this temperature range is preferable for domestic hot water heating. Like the conventional ASHP water heater, the research on SAHP water heaters has gained interest from researchers around the globe, as far back as 1950 [15]. Furthermore, most of the research has focused on the enhancement and improvement of the energetic and exergetic efficiency, the operational optimization, and the potential reduction of the daunting capital cost in the residential sector [16,17]. This study is based on the utilization of an efficient DX-SAHP water heater and the prediction of its performance in a female accommodation in the university campus.

4.1. Basic Components of SAHP and Investigated Parameters

The ultimate performance of an SAHP system depends on the size and configuration of the sub-components that must match the demands of the load and the availability of the daily solar radiation [18]. An SAHP system comprises four basic sub-components, including the collector–evaporator (which may act as single or combined components depending on the type of SAHP), compressor, thermal expansion valves, and the storage–heat exchanging tank. Studies conducted on SAHP systems show that the collector (also the evaporator for direct series type SAHP systems) and the compressor are the most impacted components with respect to the amount of heat incidence on the system from the heat source [19]. Additionally, both the compressor and water circulation pumps are the power consuming devices in an SAHP system.

4.1.1. Collector–Evaporator

Glazed and unglazed flat plate (FP) solar collectors are the two main collector types widely used in SAHP system applications based on past research conducted with this technology. Both the unglazed and glazed type FP solar collectors have been extensively used in research conducted on the SAHP [20]. It is imperative to mention that the unglazed FP collector is commonly employed and experimented among the two types of FP solar collectors [21]. This is due to the better collector efficiencies achieved by the unglazed FP solar collector to the counterpart glazed type. The heat loss is significantly minimized in the unglazed FP collector, as the available heat is practically absorbed by the unglazed flat plate collector. Additionally, the unglazed FP collector in turn transmits the absorbed heat to the low temperature flowing fluid. This main benefit of the uncovered FP solar collectors has been highlighted in enormous research studies of SAHP for both hot water heating and space heating [17]. Other types of solar collectors incorporated to an ASHP water heater that have been investigated include the photovoltaic/thermal and evacuated solar water technologies [22]. These types of solar collectors can yield high temperature differences and are worth combining both photovoltaic/thermal as collector at the heat source side of SAPHs. The experiment by Ji et al. [23] on the performance of the photovoltaic panels with heat pump units under the outdoor weather conditions of Hefei in Central China determined a COP of closed to 10.4 for the overall system. The minimization of heat losses through various designs of the SAHPs has constantly been implemented in the evaluation of the system performance. The critical factors that determine the system performance include the size and material of the collectors, the length and diameter of the pipes, fluid properties, wind velocity, ambient conditions, and solar radiation [24]. More so, these factors have been analyzed with reference to the SAHP unit. Based on these parameters, the ambient temperature and solar radiation are the primary factors that influence the performance of the SAHP water heater [24].

4.1.2. Compressor

Efficiency and speed are the two main factors of interest for compressor units employed in SAHP systems [25]. These factors are strongly correlated with the refrigerant pressure and the rise in heating temperature. However, selecting an under-size compressor is a call for concern, if the required output temperature achieved is too high. This can lead to a mismatch between the compressor speed and instability of the available solar radiation. Research has shown that these factors have strong influence on the performance of the compressor [26]. Studies from existing literature revealed that a better compressor performance can be achieved by employing variable speed compressors. Yousefi and Moradali [27] obtained a COP between 2.6 and 3.3 with a configuration of an SAHP that employed a variable speed compressor. A different study was conducted and the results showed that using a variable frequency compressor in an SAHP system gave higher COPs [28]. This has an extra advantage that was associated with high solar radiation rate yield at the collector–evaporator end and, subsequently, enhanced heat gain in the condenser. It is postulated that setting the compressor speed at a low rate will increase the COP and also the lifespan of the compressor. Xu et al. [29], in their experimental study on the performance of SAHP water heaters, demonstrated that the COP decreased from 2.82 to 1.86 as the compressor speed increased. Additionally, Xu et al. [30] depicted from their research conducted that decreasing compressor speed from 5000 r/min to 1500 r/min resulted in an exceptional increase in system COP from 5.89 to 7.36 in Nanjing and from 5.59 to 7.09 in Hong Kong. In addition to the improvement of the COP, the setting of the compressor at a low speed provides a potential possibility of decreasing the compressor power and energy consumed.

4.1.3. Condenser

It is worth mentioning that for the purpose of low temperature water heating applications by SAHP, the condensers function both as heat exchangers and heat storage tanks. The copper tube piping has been widely used for refrigerant flow and heat exchange between the vapor compressor refrigerant fluid and the cold water in the tank [12]. Hawlader et al. [31] conducted an experiment on a fiber-glass-made water condensing tank. The condenser performance depends on the reduction of heat loss through radiation. Therefore, the design of the condenser plays a crucial part in a bid to improve the performance characteristics. A number of studies have focused on the design parameters of the condenser in relation to other sub-components of the SAHP system. Wang et al. [32] exploited the numerical modeling of an SAHP water heating system and confirmed that the compressor speed can strongly affect the amount of heat gained in the condenser. Furthermore, the condensing refrigerant properties such as the heat transfer coefficient, water, and ambient temperature are the factors that should be taken into consideration in the design and performance of a condenser. Anderson et al. [33] investigated the available area of the condenser in connection to the water tank as a strategy to enhance the overall performance of the SAHP water heater.

4.1.4. Expansion Valve

An expansion valve is a device that controls the pressure and flow rate of the refrigerant in a heat pump cycle. There exist various types of expansion valves and they are named according to their functions. It has been demonstrated that the expansion valve can play a key role in the effective performance of compressor and mass flow rate of the refrigerant in any SAHP system. The common expansion valves that are widely used include the thermostatic expansion valve and the electronic type, and both are used in SAHP systems [34]. The electronic expansion valve with a controller and a variable speed compressor are very suitable to be used in an SAHP system [35]. This type of expansion valve can guarantee a high system performance and a balance of components between solar collector and compressor units.

4.1.5. Types of Refrigerants Used in SAHP Systems

An effective working fluid or refrigerant for SAHP systems should have a high thermal conductivity, critical temperature, and evaporation enthalpy to achieve a high heat transfer rate and COP [36]. The refrigerant must possess other excellent qualities that range from a low freezing point, low specific volume capacity, and low viscosity to comply with the heat pump operation and reduce power consumption. Ozone layer depletion and global warming potential are two major refrigerant-associated problems that must be a factor in the selection criteria for a refrigerant in a heat pump system. In SAHP systems, there are two types of refrigerants commonly utilized: pure and blend refrigerants. More so, the blend refrigerants have been widely used of late for the SAHP systems as an environmentally benign and energy efficient refrigerant. Nouri et al. [37] replaced R12 with R134A and conducted an experimental comparison of the system performance. They depicted a degradation of between 2 and 4% of the system performance with the collector temperature range from 0–20 °C as a result of replacing the R12 with R134A. Additionally, a comparison between other refrigerants was conducted and the findings showed that R12 yielded the best COP followed by R22, R134A, R410A, and R407C/R404A, respectively. Mohanraj et al. [38] carried out a comparison study between R22 and a mixture of R407C for an SAHP system. The energy performance ratio of R407C was depicted to be 2–5% lower compared to R22. However, the total equivalent warming impact of R407C was reported to be lower with reference to R22 under leakage conditions. Although the performance of the blend type refrigerants was observed to be lower than that of conventional pure refrigerants, the promising results revealed the likely potential of blend refrigerant’s performance to outperform that of pure refrigerant in the future.

4.2. Classification of SAHP Water Heaters

SAHP is a type of heat pump that combines with solar thermal collectors which supply heat directly or indirectly to the evaporator [39]. When the SAHP configuration is such that heat is supplied directly to the evaporator, it is called a direct expansion solar assisted heat pump (DX-SAHP). On the contrary, when the heat is indirectly supplied to the evaporator it is called an indirect expansion solar assisted heat pump (IX-SAHP). In the parallel system, a conventional heat pump is a typical ASHP utilized in parallel with solar collectors. In this configuration, the solar collector supplies direct heat whenever the available solar radiation is above the critical threshold radiation level of the collectors. This can be assisted by a heat pump, if the solar heat absorbed is inadequate to raise the water temperature to the set point temperature. The fourth category employed both series and parallel configuration of the SAHP. The main concept of SAHP is the utilization of solar heat to increase the evaporation temperature or reduce the condenser load of the heat pumps. As a consequence, the COP of the SAHP is higher relative to a standalone ASHP of the same size operating under same prevailing weather conditions.

4.2.1. Series Configuration SAHP Water Heater

The series system may be divided into two types: direct expansion (DX-SAHP) and indirect expansion (IX-SAHP) [40,41]. The IX-SAHP utilizes a water loop heat pump system where the evaporator side of the heat pump unit is connected to the hot water produced by the solar collector. The heat harnessed by the solar collectors is transported indirectly to the evaporator of the heat pump as shown in Figure 1.

In the series connection for the DX-SAHP configuration, a two phase solar collector—evaporator is used as the evaporator of the heat pump unit. The refrigerant is forced directly from the compressor into the solar collector—evaporator end and thus eliminates the requirement of an additional water loop circuit to transport heat from the collector to the evaporator as shown in Figure 2.

4.2.2. Parallel Configuration SAHP Water Heater

In the parallel configuration both the solar collector and the heat pump unit supply heat directly to the domestic hot water tank, but with preference allocated to the solar collector. Figure 3 shows a schematic diagram of a parallel SAHP water heater.

4.2.3. Dual Configuration SAHP Water Heater

In the dual configuration mode, the SAHP system can operate in series, parallel, or both configurations simultaneously in a bid to enhance the overall performance. The dual configuration SAHP water heater can trigger a specific heating mode depending on the domestic hot water load and the prevailing solar radiation, as well as the ambient weather conditions. Figure 4 presents a schematic diagram of a dual SAHP water heater. Nonetheless, SAHP offers the potential for significant improvement in the COP over ASHP [42]. This can be accounted for by the existence of a higher evaporation temperature from direct exposure to solar radiation relative to ambient temperature.

4.3. Energetic and Exergetic Efficiency of SAHP

Heat pump performance depends on both the heat sink and heat source temperatures. The higher the heat source temperature, the better the performance of the SAHP water heater [21]. The main parameters associated with the performance of the SAHP are the COP and the collector efficiencies and are primarily related to the collector operating temperature. As the collector temperature increases, the collector efficiency decreases while the efficiency of the heat pump unit rises.

Exergy is the amount of maximum useful work that can be done by a system under complete interaction with the enclosed environment at the same pressure and temperature [43]. The analysis of exergy dynamics in a solar-driven system may result in the identification of inefficient components within the system and optimization of the operating conditions [44]. The goal of an exergy analysis is to obtain an optimized system design by maximizing the exergetic efficiency at all components levels. Extensive research has been conducted that revealed significant findings of the performance of DX-SAHP based on both energy and exergy analysis [45]. The COP of SAHP changes with variation in the evaporator and condenser temperatures and is sensitive to available solar radiation and hot water output temperature [46]. The higher the difference between the condenser and evaporation pressures, the higher the power consumed by the compressor. In the domestic hot water application, hot water outlet temperature is typically set at a single temperature known as the set point temperature. The evaporation pressure/temperature at the solar collector—evaporator side can be adjusted to decrease the system COP. A higher evaporation pressure/temperature can result in lower compressor pressure lift and a decrease in the corresponding input power. However, when evaporation pressure/temperature is higher than the ambient temperature, the solar collector—evaporator loss is pronounced and leads to an overall reduction in the system COP. There exists a maximum evaporation pressure/temperature that will provide the highest overall COP for a DX-SAHP. The choice of the optimum evaporating temperature is based on the design of the solar collector—evaporator of the system and the weather condition. Vaishak et al. [47] depicted that a variation of the evaporator temperature from 0 °C to 10 °C under favorable solar conditions can ensure a high collector efficiency. Riffat and Cuce [48] confirmed in their research that a maximum exergy efficiency can be achieved by an SAHP system when the evaporation temperature ranges between 5 °C and 8 °C above ambient temperature. Miglioli et al. [18] proposed that the evaporation temperature shall be maintained at 5–10 °C above ambient temperature through a well-matching heat pumping (evaporative) capacity of the compressor and the collector capacity. When the operating evaporator is within this temperature range, this often leads to lower collector heat loss and the heat pump performance increases. The utilization of the variable speed compressor to maintain the temperature level can reasonably increase the COP of a DX-SAHP [49]. However, according to Hu et al. [50], operating at higher evaporation temperature may result in a higher compressor discharge temperature beyond its operating range and lead to a lower lifespan of the compressor. They suggested that the operating evaporator at temperatures lower than ambient temperature may cause lower compressor discharge temperature and can enable the collector to absorb heat both from the ambient and the solar radiation with minimum collector heat loss. Exergetic analysis carried out by numerous researchers proved that the highest exergy loss occurred at the compressor while the least took place at the expansion valve [51]. According to Torio et al. [52], very important outcomes have been achieved with the improvement of SAHP system performance through exergy analysis. They showed that the system performance depends on the wind speed, collector size, compressor speed, and duration of operation. A number of researchers have been able to report on the exergy efficiency of the overall SAHP system [53] while others have evaluated the exergy efficiency of individual components of an SAHP system [54]. An important benefit of the exergy analysis is that efficiency enhancement and reduction of thermodynamic heat losses associated with sub-components or the overall SAHP can be identified [55].

4.4. Performance and Modeling of SAHP Water Heaters

The persistent hikes of energy cost and environmental impacts are the core drivers of all research in the energy related field. The improvement in energy conversion efficiency and the alleviation of economic challenges is of primary concern. The experiment by Badiei et al. [56] focused on a heat pipe integrated solar-assisted heat pump water heater that operated with dual heat sources and utilized both the performance of the conventional heat pump and the solar heat pipe collector. The outdoor test for the heat pipe SAHP showed that the COP for the hybrid-mode operation can reach 3.32, with an increase of 28.7% as compared to the COP for the heat pump mode (2.58). Shi et al. [35] analyzed the thermal performance of a DX-SAHP for various refrigerants using two collector configurations, namely an unglazed collector and a glazed collector. They showed that R12 gives the highest value of COP, followed by R22 and R134A. Furthermore, they showed that the performance of their systems using blend refrigerant R410A was more efficient than either R407C or R404A. However, the R410A had decreased in efficiency by 15–20% when compared to the efficiency obtained with R134A. Chaturvedi et al. [57] conducted research on the long-term performance of a multifunctional DX-SAHP system for residential application. They confirmed that the daily average COP of a heat pump for space heating ranged from 2.6 to 3.3, while for domestic hot water heating at 50 °C, the COP of the system was between 2.1 to 3.5. Additionally, using the same system, but focusing on space cooling at night, the COP of the heat pump was determined to be 2.9. It was depicted that the multi-functional DX-SAHP system could provide a long-term operation under different weather conditions and with a low operating cost for the entire year.

Li et al. [37] investigated DX-SAHP systems using DX-SAHP with a fixed speed rotary compressor, immerse condenser coil, and an aluminum plate collector—evaporator. The achieved COP of the DX-SAHP water heater was up to 6.61 during daytime and 3.11 at a rainy night. The seasonal average value of the COP and the collector efficiency was determined as 5.25 and 1.08, respectively. As heat sink temperature increases above 60 °C, the COP of a single stage SAHP begins to decrease. Kamel et al. [58] proposed two stage DX-SAHP systems with R134A as refrigerant and one glaze collector for meeting load demand with high temperature in the range of 60 °C to 90 °C. He proved that a two-stage DX system had better performance than did a single-stage DX system at high condensing temperature. A mathematical model of the system was developed to predict its operating performance under specified weather conditions. They suggested that the system performance is governed by the variation of the circulation flow rate, solar collector area, and initial water temperature in the preheating solar tank. Ji et al. [59] examined the potential application of a DX-SAHP using a variable speed reciprocating compressor, immersed condenser coil, tube in plate heat exchanger, and capillary tube. The system achieved an annual COP of 6.46 and was more efficient than the conventional heat pump system performance. Xu et al. [60] developed novel low-concentrating solar photovoltaic/thermal combined heat pump system with the output as electricity and thermal energy generation. Fixed truncated parabolic concentrators was used to reflect the incident sunlight onto the surface of PV cells and also served as the evaporator of the heat pump system. Wang et al. [61] proposed a novel IX-SAHP system which integrates a domestic heat pump with a solar water heater. Çağlar and Yamalı [62] investigated the performance of a solar-assisted heat pump with an evacuated tubes collector based on theoretical and experimental methods. The maximum value of the COP of the SAHP from the experimental method was 6.38 and the deviation in the determined COP from the experimental and theoretical methods was negligible.

Kim et al. [48] presented the modeling and optimization of an SAHP using ice slurry. Solar collectors were utilized as the main source of thermal energy, with two distinct loops allowing the collectors to function in series with an ice tank or a warm water tank. The proposed heat pump with ice slurry achieved an 86% reduction over an electrical resistance heating system and a 5% decrease relative to SAHP with sensible storage only.

4.5. Mathematical Models Employed in the Study

A mathematical model is a mathematical equation or computational program that describes the dynamic behavior of a physical system or process [63,64]. A mathematical model is beneficial in control systems applications, optimization of physical systems or processes, and prediction during the “what if scenarios” of a system or process. A mathematical model can be presented by three components: namely, the input(s), mathematical equation(s), and output(s) [65]. The mathematical model used in the study is multiple linear regression derived by the least square regression method [65]. The multiple linear regression model is used to predict the energy factor of the designed and monitored DX-SAHP water heater connected in parallel configuration and installed in a students’ residence at a university campus. The input parameters for the multiple linear regression model include ambient weather factors, thermodynamics, and electrical factors of the hybrid SAHP water heater.

5. Materials and Methods

It comprises the materials and methods utilized in the realization of this study. The list of materials used in the experimental setup consists of devices that make up the hybrid DX-SAHP water heater, the sensors, and data loggers required for the construction of the data acquisition system (DAS).

The methods involved both quantitative and qualitative analyses. The methods were broken down into three deliverables as follows:

i.

The design and installation of the hybrid DX-SAHP water heater and the construction of the DAS to monitor the operating performance of the efficient hot water heating system.

ii.

The recording and storing of data from the measuring sensors into the data loggers. Furthermore, the exportation of the stored data for analysis using MATLAB, to reveal the quantitative operating performance of the hybrid DX-SAHP water heater.

iii.

The building and development of an accurate multiple linear regression model to predict the EF of the designed and installed hybrid DX-SAHP water heater.

Table 1. Equipment, sensors, and data loggers used in the setup.

Item	Material	Quantity
1	Power meter with inbuilt data logger (3 phase quality track analyzer, category A power meter from Advanced Monitoring Solution, Cape Town, SA)	1
2	Flow meters (T-MINOL-130-NL flow meter from Hobo corporation, Lakeville, MN, USA)	3
3	Temperature sensors (copper pipe thermocouple sensor TMC50-HD water and soil temperature sensors from Hobo corporation, Lakeville, MN, USA)	12
4	Ambient temperature and relative humidity sensor (TEMP/RH sensor, 12 bit, 2 m-S-THB-M002, from Hobo corporation, Lakeville, MN, USA)	1
5	24 kW, 2000 L (Kwikhot electric boiler from Kwikhot PTY, SA)	1
6	6.0 kW input ASHP unit (air source heat pump unit from ‘its solar’, Johannesburg, SA)	1
7	Evacuated tubes collectors (solar collectors from ‘its solar’ SA)	8
8	Controller systems (heat pump controller unit and solar collector controller unit, from ‘its solar’, Johannesburg, SA)	2
9	Pyranometer (silicon pyranometer 3m cable—S-LIB-M003 from Hobo cooperation, Lakeville, MN, USA)	1
10	Anenometer (wind speed sensor-S-WSB-M003 from Hobo cooperation, Lakeville, MN, USA)	1
11	Waterproof and weatherproof enclosures	2
12	Wind Vane (wind direction sensor-S-WDA-M003 from Hobo corporation, Lakeville, MN, USA)	1
13	Data logger (Hobo 4 channel pulse input logger -UX120-017M from Hobo corporation, Lakeville, MN, USA)	1
14	Data loggers (Hobo 4 channel analog logger -UX120-006M from Hobo corporation, Lakeville, MN, USA)	3
15	Data logger (U30 NRC logger weather station-U30-NRC-000-10-S from Hobo cooperation, Lakeville, MN, USA)	1
16	5 W solar panel (solar panel 5 Watt-Solar-5W from Hobo corporation, Lakeville, MN, USA)	1
17	4.5 V DC battery (U30 battery-HRB-U30-S100 from Hobo corporation, Lakeville, MN, USA)	1
18	Solar radiation shield (RS3-B Radiation shield-ODD72 from Hobo corporation, Lakeville, MN, USA)	1
19	USB-Cable (USB interface cable-cable-USBMB)	1
20	Hoboware pro software (version 3.7.25) (Hobo cooperation, Lakeville, MN, USA)	1
21	MATLAB software (version 2021a)	1

shows the devices, sensors, and data loggers used in the experimental setup.

Experimental Setup

Figure 5 shows the 2D schematic diagram of the installed efficient DX-SAHP water heater. It comprises a 2000 L, 24 kW electric boiler retrofitted by solar collectors (8 evacuated tubes collectors arranged as a balance parallel system with each branch containing four series-connected evacuated tubes collectors) and a 6.0 kW input ASHP unit. The 24 kW heating element in the electric boiler was disabled and retrofitted with both the solar collectors and the 6.0 kW input ASHP unit. The major components of the ASHP unit are the evaporator, compressor, condenser, and expansion valve. Makeup cold water from the incoming mains water was used to fill up the storage tank during hot water consumption as a result of the demand for hot water by the students in the accommodation. A copper pipe was connected between the storage tank and the inlet of the solar collectors while a three-phase 370 W water circulation pump (PU2) and a flow meter (F2) were installed on the copper pipeline. A copper pipe was connected between the outlet of the solar collectors and the storage tank. The 370 W water circulation pump (PU2) helped the water to circulate between the storage tank and the solar collectors during the heating cycles. The water temperatures at the inlet and outlet of the solar collectors were measured using temperature sensors (T6 and T7). The flow meter (F2) measured the volume of water heated by the solar collectors during the heating cycles and the water was stored in the storage tank. A copper pipe was connected between the storage tank and the inlet of the ASHP unit while a single phase 100 W water circulation pump (PU1) and a flow meter (F1) were installed on the copper pipeline. A copper pipe was connected between the outlet of the ASHP unit and the storage tank. The 100 W water circulation pump (PU1) assisted in the circulation of water between the storage tank and the ASHP unit. The flow meter (F1) measured the volume of water heated by the ASHP unit during the vapor compression refrigeration cycle (VCRC). The water temperatures at the inlet and the outlet of the ASHP unit were measured using temperature sensors (T5 and T4). Temperature sensors (T1, T2, and T3) were installed at the bottom, middle, and top of the storage tank and measured the water temperatures at the respective positions in the storage tank. A copper pipe was connected to the storage tank and supplied hot water to the destinations in the building, while a temperature sensor (T8) and a flow meter (F3) were installed on the copper pipe and in the proximity of the outlet of the storage tank. The flow meter (F3) measured the volume of hot water consumed by the occupants in the building while the temperature sensor (T8) measured the outlet water temperature from the storage tank into the building. Temperature sensors (T9 and T10) were installed at the inlet and outlet of the evaporator of the ASHP unit. These temperature sensors (T9 and T10) measured the refrigerant temperatures at the inlet and outlet of the evaporator. Temperature sensors (T11 and T12) were installed at the inlet and outlet of the condenser of the ASHP unit. These temperature sensors (T11 and T12) measured the refrigerant temperatures at the inlet and outlet of the condenser. Ambient temperature and relative humidity sensor, pyranometer, anemometer, and wind vane were installed in the vicinity of the solar collectors. The ambient temperature and relative humidity sensor measured the ambient temperature and relative humidity. The pyranometer measured the global solar irradiance incidence on the solar collectors. The anemometer measured the wind speed while the wind vane measured the wind direction. A quality track analyzer (three-phase power meter with inbuilt data logging capability) was installed on the power supply line powering the DX-SAHP water heater. All the measuring sensors were connected to data loggers to store the recorded data. All the data loggers were configured to log in 1 min intervals throughout the monitoring period using hoboware pro software. The stored data from the data loggers were downloaded to a laptop via USB cable for further processing using MATLAB. The installed DX-SAHP water heater was automatically controlled with the incorporation of the solar controller and the controller for the ASHP unit.

6. The Design and Configuration of the DAS

The twelve temperature sensors were copper pipe thermocouples and were connected to three, four-channel UX120 hobo analog data loggers. The temperature sensors (T1, T2, T3, and T8) were connected to one of the UX120 hobo analog data loggers (UX120-1, as shown in Figure 6). The temperature sensors (T4, T5, T6, and T7) were connected to the second UX120 hobo analog data logger (UX120-2, as shown in Figure 6) while the rest of the temperature sensors (T9, T10, T11, and T12) were connected to the third UX120 hobo analog data logger (UX120-3, as shown in Figure 6). These UX120 hobo analog data loggers (UX120-006M) were powered by two AA 1.5 V batteries. The flow meters (F1, F2, and F3) were connected to a four-external-channel input pulse data logger (UX120-017M) and were powered by two AA 1.5 V batteries. The ambient temperature and relative humidity sensors, pyranometer, anemometer, and wind vane were connected to a no-remote communication data logger (NRC-U30 data logger). The NRC-U30 data logger was powered by a 4.5 V battery which was charged from a 5 W solar PV (photovoltaic) panel, and is shown in Figure 6. The electrical power consumed by the DX-SAHP water heater was measured using a three-phase power track analyzer with an inbuilt data logger. The installed temperature sensors were insulated with insulation tapes and the ambient temperature and relative humidity sensors were protected by the solar radiation shield. The installed flow meters were enclosed by protective weatherproof enclosures. All the data loggers were configured to log every 1-min interval throughout the monitoring period (3 month period, from January to March 2022) using hoboware pro software.

7. Theory and Calculations

The COP of the ASHP of the hybrid DX-SAHP water heater is the ratio of the thermal energy gained by stored water due to the ASHP unit to the electrical energy consumed during the vapor compression refrigeration cycle and is given by Equation (1) [66].

$$COP=\frac{Q_{tank}}{E_{ASHP}}=\frac{m_{F1}c\left(T_4-T_5\right)}{p_{ASHP}t}$$

(1)

where $COP=$ coefficient of performance, $Q_{tank}=$ thermal energy gained by water in the storage tank due to ASHP, $E_{ASHP}=$ electrical energy consumed by ASHP, $m_{F1}=$ mass of the water heated by ASHP, $c=$ specific heat capacity of water, $T_4=$ water temperature at outlet of ASHP, $T_5=$ water temperature at inlet of ASHP, and $t=$ time taken.

The solar fraction of the solar collectors of the hybrid DX-SAHP water heater is the ratio of the thermal energy gained by water due to the solar collectors to the solar energy harnessed by the solar collectors and is given by Equation (2) [67]:

$$SF=\frac{Q_{SC}}{I{A}_{SC}t}=\frac{m_{F2}c\left(T_7-T_6\right)}{I{A}_{SC}t}$$

(2)

where $SF=$ solar fraction, $I=$ global solar irradiance incidence on collectors, $t=$ time taken, $c=$ specific heat capacity of water, $Q_{SC}=$ thermal energy gained by solar collectors, $A_{SC}=$ area of solar collectors and $A_{SC}=nLD$, n = number of evacuated tubes, L = length of an evacuated tube, D = diameter of an evacuated tube, T₇ = water temperature at outlet of the solar collectors, and T₆ = water temperature at inlet of the solar collectors.

The energy factor of the hybrid DX-SAHP water heater is the sum of the COP of the ASHP and the SF of the solar collectors and is given by Equation (3) [67].

$$EF=COP+SF$$

(3)

where $EF=$ energy factor of the hybrid DX-SAHP water heater, $COP=$ coefficient of performance of the ASHP, and $SF=$ solar fraction of the solar collectors.

The derived multiple linear regression model of the energy factor of the hybrid DX-SAHP water heater with the chosen predictors is given by Equation 4. The derived and built regression model can be expressed as a function of EF in terms of $E_t,T_{Amb},RH,and WS$ and is expressed as $EF=\left(E_t,T_{Amb},RH,WS\right)$, as represented in Equation (4).

$$EF={\mathsf{\gamma }}_0+{\mathsf{\gamma }}_1{E}_t+{\mathsf{\gamma }}_2{T}_{Amb}+{\mathsf{\gamma }}_{3}RH+{\mathsf{\gamma }}_{4}WS$$

(4)

where ${\mathsf{\gamma }}_0=$ forcing constant, ${\mathsf{\gamma }}_1\dots {\mathsf{\gamma }}_4=$ scaling constants, $EF=$ energy factor, $E_t=$ electrical energy consumed by DX-SAHP water heater, $T_{Amb}=$ ambient temperature, $RH=$ relative humidity, and $WS=$ wind speed.

The dataset over the monitoring period (January–March 2022) were divided into the test dataset (dataset for January to February 2022) and the evaluation dataset (dataset for the month of March 2022). The multiple linear regression model exploited the least square method using the test dataset to determine the scaling constants attributed to each of the predictors. The determination coefficient between the calculated EF and the modeled EF was determined as well as the root mean square error. The determination coefficient and root mean square error were used to test the accuracy of the model. Finally, the evaluation dataset for the predictors were used in conjunction with the derived multiple linear regression model to validate the model.

7.1. Measurement Accuracies and Uncertainties

The uncertainties derived from the calculated parameters based on the result of the error measurements from the set of independent variables is given by Equation (5) [68].

$$w_r={\left[{\left[w_1\frac{\partial R}{\partial X_1}\right]}^2+{\left[w_2\frac{\partial R}{\partial X_2}\right]}^2+\dots \dots \dots \dots +{\left[w_n\frac{\partial R}{\partial X_n}\right]}^2\right]}^{\frac{1}{2}}$$

(5)

where $R$ = the given function; $w_r$ = total uncertainty; $X_1, X_2, \dots \dots .X_n$ = independent variables; and $w_1, w_2, \dots \dots .w_n$ = uncertainty in the independent variables.

The type A uncertainties were based on the statistical means and standard deviations from the recorded measurements [69] and are shown in

Table 2. Uncertainties of the measured and derived quantities.

Quantity	Type A Uncertainty	Type B Uncertainty	Combined Uncertainty
Ambient temperature (°C)	±0.200	±0.120	±0.233
Relative humidity (%)	±0.250	±0.140	±0.286
Water flow rates measurements (L/min)	±0.010	±0.006	±0.012
Power consumed by split type ASHP system (kW)	±0.130	±0.003	±0.130
Water and refrigerant temperatures measurements (°C)	±0.250	±0.120	±0.277
Global solar radiation measurements (W/m2)	±10.00	±4.350	±10.905
Electrical energy consumed (kWh)	±0.130	±0.025	±0.132
Thermal energy gained (kWh)	±0.190	±0.042	±0.195
Coefficient of performance (COP)	±0.260	±0.203	±0.330
Solar fraction (SF)	±0.060	±0.020	±0.080
Energy factor (EF)	±0.320	±0.223	±0.390

. In addition, the type B uncertainties were determined on the basis of the accuracy of the sensors or by the use of Equation (5) to obtain the derived uncertainty of the desired quantities (SF, COP, and EF). The combined uncertainties of the prescribed quantities are provided in Table 2.

7.2. Data Analysis

The stored data in the data loggers were downloaded using hoboware pro software installed on a laptop. The data were exported into MATLAB and the raw data were processed into a 30 min interval over a 24 h period for the entire monitoring of the performance of the DX-SAHP water heater. The critical input parameters (electrical energy consumed, ambient temperature, relative humidity, and wind speed) were determined and the desired output (energy factor) calculated during phase two (08:30–18:30) of the operation of the DX-SAHP water heater. Phase two was the period of operation of the DX-SAHP water heater whereby both the solar collectors and the ASHP unit were both operating. The desired output (EF = SF + COP) of the DX-SAHP water heater was determined. The derived EF values were vital in the development of the multiple linear regression model.

8. Results and Discussion

8.1. Profiles of Parameters to Assess Performance of the Hybrid DX-SAHP Water Heater

The performance of the hybrid DX-SAHP water heater for an average weekday (7–11 February 2022) was analyzed based on the profiles of the electrical power consumed (electrical power consumed by the ASHP and the water circulation pump activating the solar collectors), global solar irradiance, ambient temperature, and relative humidity.

8.1.1. Average Weekday Profiles of the Electrical Power Consumed and Global Irradiance

Figure 7 shows the average weekday profiles of the electrical power consumed by the hybrid DX-SAHP water heater on the left y-axis and the global solar irradiance on the right y-axis in half-hourly intervals over a 24 h period from the 7–11 February 2022. The profiles were divided into three phases with the first phase from 00:00–08:00, the second phase from 08:30–18:30, and the third phase from 19:00–23:30. It was observed that during the first phase, the electrical power consumed varied between 0 and 0.472 kW while the average was 0.082 kW. The solar irradiance profile for the first phase shows that the minimum, maximum, and mean global irradiance were 0.60, 56.475, and 11.98 W/m², respectively. During the third phase, the minimum and maximum electrical power consumed were 0 and 0.007 kW while the mean was 0.004 kW. The average of the global solar irradiance during the third phase was 0.60 W/m². It was deduced that during the third phase of operation of the hybrid DX-SAHP water heater, the ASHP unit never switched on. Alternatively, during the second phase of operation, the profile of the electrical power consumed revealed five peaks at times 09:00, 10:00, 11:00, 13:30, and 15:00, and corresponded to peaks of 0.81, 1.15, 1.24, 1.58, and 1.52 kW. The associated peak power consumed coincided with the global solar irradiance of 150, 788, 900, 752, and 595 W/m², respectively. The peak electrical power consumed was due to the moment of operation of the heat pump unit of the DX-SAHP water heater during the monitoring period. During the second phase, the minimum and maximum electrical power consumed were 0 and 1.58 kW with a mean of 0.62 kW. The solar irradiance in the second phase ranged from 2.07–913 W/m² with a mean of 530.58 W/m². The results show that in the first phase of operation, a very negligible contribution of the EF of the system was achieved because of the lesser frequency of operation of the heat pump unit and the lower COP values achieved [70]. The third phase shows that the EF was practically 0 as neither the heat pump unit nor the solar collectors were active. Hence, the bulk of the contribution of the EF of the hybrid DX-SAHP water heater occurred in the second phase of operation.

8.1.2. Average Weekday Profiles of Power Consumed and Ambient Temperature

Figure 8 depicts that during the first phase of operation, the electrical power consumed ranged from 0 to 0.474 kW with a mean of 0.082 kW. The profile of the ambient temperature during the first phase of operation ranged from 12.32 to 22.76 °C and the mean was 12.32 °C. The third phase of operation showed that the electrical power consumed was between 0 and 0.01 kW while the mean was 0.002 kW. The profile of the ambient temperature during the third phase of operation showed a variation between 17.98 and 23.96 °C with an average of 20.94 °C. It was determined that during the second phase of operation, the peaks of the electrical power consumed were 0.81, 1.15, 1.24, 1.58, and 1.52 kW and occurred at times 09:00, 10:00, 11:30, 13:30, and 15:00. The corresponding ambient temperatures associated with the peak power consumed were 25.52, 26.58, 28.61, 30.49, and 30.87 °C, respectively. It was depicted that the ambient temperatures recorded in the second phase of operation were higher than those determined from the other two phases. During the second phase of operation the ambient temperatures fluctuated between 24.39 and 31.00 °C with an average of 28.51 °C and were favorable for both performance of the heat pump unit and the solar collectors [71].

8.1.3. Average Weekday Profiles of Power Consumed and Relative Humidity

Figure 9 shows that during the first phase of operation, the electrical power consumed varied between 0 and 0.474 kW with a mean of 0.059 kW, while the profile of the relative humidity fluctuated between 31.86 and 56.16% with a mean of 59.19%. The third phase of operation showed that the electrical power consumed was between 0 and 0.01 kW with a mean of 0.002 kW. The profile of the relative humidity during the third phase of operation revealed a variation between 26.16 and 48.67% and the average was 37.69%. The profile of the electrical power consumed in the second phase of operation showed five peaks of 0.81, 1.15, 1.24, 1.58, and 1.52 kW, which correspond to five relative humidities at times 09:00, 10:00, 11:30, 13:30, and 15:00, respectively. The peak electrical power consumed, and the associated relative humidity, were 0.81 kW and 24.40%; 1.15 kW and 23.03%; 1.24 kW and 21.42%; 1.58 kW and 20.54%; and 1.52 kW and 19.99%, respectively. It is of absolute relevance to affirm that the relative humidity may influence the electrical power consumed [72].

8.1.4. Average Weekday Profiles of Power Consumed and Wind Speed

Figure 10 depicts that during the first phase of operation, the electrical power consumed ranged from 0 to 0.472 kW with a mean of 0.082 kW, while the profile of the wind speed fluctuated between 0.288 and 0.302 m/s with an average of 0.295 m/s. The third phase of operation showed that the electrical power consumed was between 0 and 0.07 kW with a mean of 0.004 kW. The profile of the wind speed showed a variation between 0.146 and 0.325 m/s with a mean of 0.276 m/s. There were five peak of electrical power consumed which corresponded to five associated relative humidities (0.81 kW and 0.304 m/s; 1.15 kW and 0.305 m/s; 1.24 kW and 0.308 m/s; 1.57 kW and 0.311 kW; and 1.52 kW and 0.314 m/s).

8.2. Performance of the ASHP Unit of the Hybrid DX-SAHP Water Heater

Table 3. Performance of heat pump of the system during the second phase of its operation.

SD	V-Ph2 (L)	Tin-HP (°C)	Tout-HP (°C)	Q-HP (kWh)	E-HP (kWh)	COP
1	2971	55.35	59.93	6.669	1.980	3.368
2	3770	52.74	57.04	22.97	4.340	5.294
3	4020	52.86	55.93	5.210	2.070	2.517
4	3993	52.45	57.15	20.28	4.580	4.428
5	5027	52.02	56.89	10.65	2.990	3.562
6	4095	53.53	57.91	8.05	2.380	3.384
7	3804	53.16	57.67	5.33	1.790	2.979
8	2452	47.46	51.95	25.57	5.450	4.693
9	4387	53.02	56.80	15.67	3.64	4.305
10	4353	52.23	56.54	8.936	2.60	3.437
11	4353	52.27	55.57	7.782	2.57	3.028
12	3989	52.74	57.15	15.267	3.430	4.451
13	2971	51.99	55.06	4.759	1.520	3.131
14	5125	53.33	56.89	4.166	1.740	2.394
15	3028	49.85	52.94	16.574	3.920	4.228

SD = sample days, V-Ph2 = volume of hot water consumed during the second phase, FR-HP = flow counts through the heat pump, Tin-HP = water temperature at inlet of ASHP unit, Tout-HP = water temperature at outlet of ASHP unit, Q-HP = thermal energy gained by water from the ASHP unit, E-HP = electrical energy consumed by ASHP unit, COP = coefficient of performance.

shows sample days and the critical parameters (for the months of January to March 2022) in the second phase of the operation of the hybrid DX-SAHP water heater. The sample days are very good representations of the determined data for the days in the specified months. The volume of hot water consumed per day by the students occupying the building during the second phase of operation varied between 2452 and 5125 L with an average of 3889.2 L. The flow count of the water flowing into the inlet of the ASHP unit per day during the second phase of operation ranged from 38.0 to 131.0 counts with a mean of 82.3 counts, and 1 count was equivalent to 3.758 L/min. The thermal energy gained due to the ASHP heating cycle per day during the second phase of operation was between 4.17 and 25.58 kWh and the average was 11.86 kWh. The electrical energy consumed by the ASHP as a result of the heating cycles per day during the second phase of operation was between 1.52 and 5.45 kWh with a mean of 3.00 kWh. The COP of the ASHP unit of the hybrid DX-SAHP water heater per day during the second phase of operation was between 2.39 and 5.29 with a mean of 3.68. The achieved COP of the ASHP unit of the hybrid DX-SAHP water heater was better than the COP obtained from a typical standalone ASHP water heater [73].

8.3. Performance of the Collectors of the Hybrid DX-SAHP Water Heater

Table 4 shows the sample of the days and the critical parameters (for the months of January to March 2022) in the second phase of the operation of the system. The sample days accurately represent the determined data for the days in the specified months. The flow count of the water flowing into the inlet of the solar collectors per day during the second phase of operation was between 1212 and 2616 counts with a mean of 2283 counts (1 count = 3.758 L/min). The thermal energy gained due to the heating cycle of the solar collectors per day during the second phase of operation was between 19.36 and 36.90 kWh and the mean was 28.91 kWh. The solar energy harnessed by the solar collectors and used in the heating of water per day during the second phase of operation was between 22.33 and 48.39 kWh and the average was 42.23 kWh. The SF of the solar collectors of the hybrid DX-SAHP water heater per day during the second phase of operation ranged from 0.562 to 0.867 with an average of 0.697.

The results depicted that the SF of the solar collectors of the hybrid DX-SAHP water heater was higher when compared to the majority of standalone solar water heaters. The average electrical energy consumed by the hybrid DX-SAHP water heater per day during the second phase of operation was 5.77 kWh while the average thermal energy gained by both the ASHP unit and the solar collectors per day was 40.77 kWh. The average EF of the hybrid DX-SAHP water heater per day during the second phase of operation was 4.38. The excellent EF values determined may lead to a conclusion that the hybrid DX-SAHP water heater operates with a better efficiency relative to the EF of other hybrid energy systems used for domestic hot water heating [5].

Table 4. Performance of solar collectors in the hybrid DX-SAHP water heater during the second phase of its operation.

SD	V-Ph2 (L)	Tin-SC (°C)	Tout-SC (°C)	Q-SC (kWh)	E-Sol (kWh)	SF
1	2971	57.37	57.85	22.26	25.826	0.862
2	3770	54.32	54.70	29.66	46.201	0.642
3	4020	54.34	54.64	28.24	43.581	0.648
4	3993	53.95	54.17	30.44	42.110	0.723
5	5027	53.51	53.82	29.22	46.090	0.634
6	4095	54.87	55.16	19.36	22.330	0.867
7	3804	54.91	55.32	26.46	47.080	0.562
8	2452	53.33	53.75	26.71	46.780	0.571
9	4387	54.54	54.79	30.41	45.060	0.675
10	4353	53.71	54.07	27.11	44.961	0.603
11	4353	53.79	54.07	27.11	44.961	0.603
12	3989	54.33	54.56	27.16	35.098	0.774
13	2971	55.24	55.37	36.63	48.391	0.757
14	5125	54.67	54.81	35.90	48.319	0.743
15	3028	54.46	54.58	36.90	46.713	0.790

SD = sample days, V-Ph2 = volume of hot water consumed during the second phase, Q-SC = thermal energy gained by water from the solar collectors, E-Sol = solar energy harness from the sun, SEF = solar efficiency.

Table 4. Performance of solar collectors in the hybrid DX-SAHP water heater during the second phase of its operation.

SD	V-Ph2 (L)	Tin-SC (°C)	Tout-SC (°C)	Q-SC (kWh)	E-Sol (kWh)	SF
1	2971	57.37	57.85	22.26	25.826	0.862
2	3770	54.32	54.70	29.66	46.201	0.642
3	4020	54.34	54.64	28.24	43.581	0.648
4	3993	53.95	54.17	30.44	42.110	0.723
5	5027	53.51	53.82	29.22	46.090	0.634
6	4095	54.87	55.16	19.36	22.330	0.867
7	3804	54.91	55.32	26.46	47.080	0.562
8	2452	53.33	53.75	26.71	46.780	0.571
9	4387	54.54	54.79	30.41	45.060	0.675
10	4353	53.71	54.07	27.11	44.961	0.603
11	4353	53.79	54.07	27.11	44.961	0.603
12	3989	54.33	54.56	27.16	35.098	0.774
13	2971	55.24	55.37	36.63	48.391	0.757
14	5125	54.67	54.81	35.90	48.319	0.743
15	3028	54.46	54.58	36.90	46.713	0.790

SD = sample days, V-Ph2 = volume of hot water consumed during the second phase, Q-SC = thermal energy gained by water from the solar collectors, E-Sol = solar energy harness from the sun, SEF = solar efficiency.

8.4. Sample Days and Dataset Used in Developing the Model

The sample days and dataset represented in

Table 5. Test dataset of predictors and the desired output of the hybrid DX-SAHP water heater.

SD	Period (hh:mm)	Et (kWh)	Tamb (°C)	RH (%)	WS (m/s)	EF
1	08:30–18:30	3.878	16.70	57.70	0.622	4.329
2	08:30–18:30	7.450	23.08	29.36	0.946	5.937
3	08:30–18:30	5.255	31.12	20.24	0.167	3.166
4	08:30–18:30	7.607	27.90	18.63	0.123	5.152
5	08:30–18:30	5.403	29.34	20.51	0.152	4.197
6	08:30–18:30	4.246	20.68	69.56	0.181	4.361
7	08:30–18:30	4.187	26.69	30.28	0.210	3.542
8	08:30–18:30	7.745	30.02	21.64	0.224	5.265
9	08:30–18:30	6.828	29.45	25.32	0.253	4.981
10	08:30–18:30	5.714	30.33	23.89	0.267	4.041
11	08:30–18:30	5.714	30.33	23.89	0.267	3.633
12	08:30–18:30	6.418	31.33	23.06	0.311	5.225
13	08:30–18:30	4.697	25.48	27.12	0.325	3.888
14	08:30–18:30	4.874	26.97	26.05	0.354	3.137
15	08:30–18:30	6.465	25.39	29.94	0.339	5.019

SD = sample days, Et = total electrical energy consumed, Tamb = ambient temperature, RH = relative humidity, WS = wind speed.

span the second phase of the operation of the hybrid DX-SAHP water heater over a two-month period (January–February 2022). The dataset for the entire two-month period were used to build and develop the mathematical model of the EF of the hybrid DX-SAHP water heater. It is depicted in Table 5 that the total electrical energy consumed per day ranged from 3.878 to 7.745 kWh with an average of 5.765 kWh. The ambient temperature was between 16.70 and 31.33 °C with an average of 26.99 °C, while the relative humidity ranged from 18.63 to 69.56% with an average of 29.81%. The wind speed over the period considered and used in building the model ranged from 0.123 to 0.946 m/s with an average of 0.316 m/s. In addition, the EF of the hybrid DX-SAHP water heater was determined and varied between 3.137 and 5.937 with a mean of 4.392.

8.5. Ranking of the Predictors According to the Weight of Importance

The reliefF test was used to rank the predictors (total electrical energy consumed (Et), ambient temperature (Tamb), relative humidity (RH), and wind speed (WS)) by weights of importance to the energy factor (EF) of the hybrid DX-SAHP water heater. The reliefF test is a statistical test that ranks predictors by weights of importance of the output using regression algorithm analysis [74]. The weights of contribution of each predictor to the EF is between −1 and 1, and a predictor with a weight of one revealed that it was a primary factor and showed a strong positive correlation to the desired output [75]. A predictor under the reliefF test with a weight of −1 insinuates that it is a secondary factor and shows a strong negative correlation with the desired output. The weights of the ranked predictors according to the reliefF test depict that the total electrical energy consumed, ambient temperature, relative humidity, and wind speed were 0.604, −0.0644, −0.2750, and −0.056, respectively, as shown in Figure 11. The results demonstrated that the total electrical energy consumed (Et) was a primary factor and was contributing the most to the EF. The remaining predictors (wind speed (WS), ambient temperature (Tamb), and relative humidity (RH)) were secondary factors, with relative humidity contributing the least to the EF. Lastly, the contribution by weights of importance to the EF of the hybrid DX-SAHP water heater due to wind speed was greater than that of the ambient temperature.

8.6. Building and Developing of the Mathematical Model

The multiple linear regression model of the EF of the hybrid DX-SAHP water heater using the total electrical energy consumed, ambient temperature, relative humidity, and wind speed as input parameters was developed by applying the least square method [76].

Table 6. Forcing and scaling values for the developed multiple linear regression model.

Input Parameter	Symbols	Constants	Symbols	Scaling Values	Output
		Forcing	${\mathsf{\gamma }}_0$	0.5822	Energyfactor (EF)
Total electrical energy consumed	Et	Scaling	${\mathsf{\gamma }}_1$	0.6863
Ambient temperature	Tamb	Scaling	${\mathsf{\gamma }}_2$	−0.0368
Relative humidity	RH	Scaling	${\mathsf{\gamma }}_3$	0.0234
Wind speed	WS	Scaling	${\mathsf{\gamma }}_4$	0.4708

shows the scaling and forcing constants of the predictors used to develop the mathematical model. The forcing constant was 0.5822 and was equivalent to the constant associated with the lump factors (other factors outside of the chosen four predictors) that contributed to the desired output. The scaling factor associated with the total electrical energy consumed was 0.6863, and any increase in the total electrical energy consumed can result in an increase in the EF, provided the rest of the factors remain unchanged. This can be attributed to the fact that the scaling constant associated with the total electrical energy consumed was positive. The scaling constant associated with the ambient temperature was −0.0368, and a potential rise in the ambient temperature may lead to a decrease in the EF. Finally, the scaling factors of the relative humidity and the wind speed were 0.0234 and 0.4708, respectively. An increase in either of these predictors (relative humidity or wind speed) can result in an increase in the EF of the hybrid DX-SAHP water heater, provided the other factors remain constant.

Figure 12 shows the calculated and modeled EF of the hybrid DX-SAHP water heater based on the test dataset of the input and output parameters. The modeled EF accurately represented the calculated EF with very good determination coefficient (r² = 0.902), root mean square error (RMSE = 0.809), and covariance (0.597), and are acceptable for the developed model [77]. The modeled indicator parameters were very good over the 95% confidence level and insinuate that the developed EF model would mimic the test EF with a high accuracy. The value of the root mean square error was smaller than the least value of the calculated EF. The modeled EF mimicked the calculated EF and the determination coefficient was very close to one, while the graph of the calculated and modeled EF showed no outliers. The covariance between the test EF and the modeled EF showed a strong positive correlation, as both quantities can increased concurrently.

8.7. Validation of the Developed Model

Table 7. Validation dataset of predictors and desired output of the hybrid DX-SAHP system.

SD	Period	Et (kWh)	Tamb (°C)	RH (%)	WS(m/s)	EF
1	08:30–18:30	6.425	36.45	49.82	0.572	4.644
2	08:30–18:30	6.187	34.17	42.43	0.319	4.532
3	08:30–18:30	7.401	27.30	41.78	0.560	5.495
4	08:30–18:30	6.969	27.12	47.47	0.315	5.506
5	08:30–18:30	6.260	33.67	54.39	0.610	4.711
6	08:30–18:30	6.994	33.93	45.10	0.360	4.897
7	08:30–18:30	4.690	32.51	48.77	0.295	3.451
8	08:30–18:30	3.692	20.87	43.35	0.318	3.740
9	08:30–18:30	3.916	21.67	51.26	0.475	4.000
10	08:30–18:30	3.213	21.82	49.00	0.370	3.524

SD = sample days, Et = total electrical energy consumed, Tamb = ambient temperature, RH = relative humidity, WS = wind speed.

shows the sample days and dataset (in March 2022) of the selected predictors and the output used in the validation of the model based on the second phase of operation of the hybrid DX-SAHP water heater. The variation of the total electrical energy consumed per day during the second phase of operation was between 3.213 and 7.401 kWh with an average of 5.575 kWh. The ambient temperature ranged from 20.87 to 36.45 °C with a mean of 28.95 °C. The relative humidity was between 41.78 and 54.39% with a mean of 47.34%. The wind speed fluctuated between 0.295 and 0.610 m/s, with an average of 0.419 m/s. Lasty, the EF was determined and ranged from 3.351 to 5.506 with a mean of 4.450.

Figure 13 shows the graph of the evaluated EF and the modeled EF of the hybrid DX-SAHP water heater. It can be observed that both graphs accurately represent each other. The determination coefficient for the validated EF and the modeled EF was 0.970, while the root mean square error and the covariance were 0.779 and 0.598, respectively. The determination coefficient was much closer to one and the root mean square error was smaller than the least value of the EF. Therefore, without any loss of generality, it can be concluded that the modeled EF accurately predicted the validated EF.

8.8. Comparing the Calculated and Modeled Energy Factor Using One-Way ANOVA

Figure 14a,b show the one-way analysis of variance (ANOVA) plots of the calculated and modeled EF for the test dataset and the evaluation (validation) dataset. It can be observed from both ANOVA plots that the EF was uniformly distributed, and no outliers existed on either the calculated or the modeled EF with reference to the test and validation dataset. The red horizontal line on the ANOVA plot shows the mean of the EF, while the black lower and upper horizontal lines represent the lower and upper limit values of the EF at the lower and upper 95% confidence level. It is very important to know that if the p-value is larger than the significance level between the calculated and modeled EF it is an indication that no significant different exists between the groups of the calculated and modeled EF. The p-value of the one-way ANOVA conducted between the calculated and modeled EF with 95% confidence level with the test dataset was 0.998, while the p-value with the validation dataset was 0.996 [77]. Both p-values for the calculated and modeled EF with the test dataset and the validation dataset are greater than the significance level of 0.05. Therefore, there exist no significant differences between the calculated and modeled EF for the test dataset and for the validation dataset.

9. Conclusions

An efficient 6.0 kW input hybrid DX-SAHP water heater was designed and installed as a replacement of an inefficient 24 kW electric boiler in a university students’ accommodation. The results of the performance of the hybrid DX-SAHP water heater proved that the system was operating efficiently with an average weekday EF of 4.38, while the average SF and COP were 0.697 and 3.683, respectively. The average weekday volume of hot water consumed during the second phase of operation contributed 56.74% of the total volume of hot water consumed over the three phases of operations. The average volume of hot water consumed per weekday per student was 56.7 L. The total electrical energy consumed and the thermal energy gained per weekday by water due to the hybrid DX-SAHP water heater were 5.77 kWh and 40.77 kWh, respectively, during the second phase of operation of the system. The average electrical power consumed by the hybrid DX-SAHP water heater during the second phase of operation contributed to 80.5% of the electrical power consumed over the three phases of operations (first phase, second phase, and third phase) of the hybrid DX-SAHP water heater. The ranking by weights of importance confirmed that the total electrical energy consumed was a primary factor, while the ambient temperature, relative humidity, and wind speed were secondary factors. The predictors’ contributions by weights of importance revealed that the total electrical energy consumed (0.604) was most significant, while the least was relative humidity (−0.2750). The derived scaling factors of the developed model justified that increase in the total electrical energy consumed, wind speed, and relative humidity can result in an increase in EF, if the other factors are invariant. Furthermore, an increase in the ambient temperature could lead to a decrease in the EF, provided the rest of the factors remained constant. The model of the EF demonstrated a very high accuracy as the determination coefficient was 0.912 while the root mean square error between the modeled and the test EF was 0.3241.

10. Recommendation

A rollout in the field of energy efficiency and in the domain of hot water heating should be implemented in the students’ accommodations of the university by replacing the inefficient electric boilers with DX-SAHP water heaters. The developed multiple linear regression model of the energy factor should be utilized in predicting the efficiency of the proposed efficient DX-SAHP water heaters upon implementation at the university campus. A techno-economic analysis of the design and installed DX-SAHP water heater should be conducted to ascertain the viability of the technology.