Improving Thermal Performance of Solar Heating Systems

Abstract

The solar energy reaching the immediate surroundings of a single-family house throughout the year is sufficient to repeatedly and fully cover its heating needs during the heating season in a temperate climate. Nevertheless, modern technology is not yet able to fully solve the problem of thermal self-sufficiency in single-family houses. It is therefore advisable to seek solutions that improve the thermal efficiency of domestic solar installations. Efficient use of solar radiation heat accumulated during the summer months for heating requires the use of high-volume storage tanks. Another option is to discharge excess heat outside the system during the summer. This publication focuses on the latter solution. A model of the solar heating system for a residential building and pool with a storage tank powered by solar energy has been developed. Simulation calculations were performed, showing that the removal of excess heat is a beneficial solution, especially when this energy can be used to heat water in the pool. The calculations concerned the heating of a single-family house in a temperate climate. Lowering the temperature of the water in the storage tank reduces heat losses from the tank to the environment (ground), while supplying the solar collectors with lower-temperature fluid increases the driving force of the heat transfer process.

1. Introduction

The use of solar energy is a natural, economical, and environmentally friendly method to maintain adequate thermal comfort in buildings, provide heat for domestic hot water and in some cases support for industrial or commercial processes. It is relatively mature technology, with high efficiency, and its compatibility with existing building services make it an attractive option in the transition toward low-carbon energy solutions [1].

A variety of solar energy systems for a wide range of processes and numerous types of solar collectors can be applied. Methods of storing solar energy for later use, particularly during periods of low solar radiation, are also extensively discussed in the article [2]. Solar heating systems can perform one or more functions simultaneously. The simplest are single-function systems, such as domestic hot water heating systems. A dual-function system, on the other hand, is a system that provides both space heating and instantaneous domestic hot water, like showers and sinks. A simulation of a domestic water heating system with SCs in Portugal is presented in [3]. The article focuses mainly on the impact of solar collectors’ area A_c and storage tank volume V on heating efficiency. The authors also examined the modeling of long-term solar heating processes. They concluded that overly complex models are unnecessary, as they do not significantly improve accuracy. This limitation stems from the inherent uncertainty in the weather data used for simulations.

The use of solar collectors (SCs) for heating has become widespread in many countries around the world, including those with cool climates. The simplest method is storing thermal energy in its sensible form, i.e., through fluid temperature changes in the storage. In this case, water is most often used as the storage medium. Heat can also be stored by exploiting latent heat, which occurs during medium phase changes in the storage. In this case, paraffins or salt hydrates are the most frequently applied. Heat can also be stored by utilizing reversible chemical reactions and sorption processes. Solid materials can also serve as storage media, e.g., sand. Sweet and McLeskey presented a study of solar heating of a house with such a heat accumulator in their paper [4]. In contrast, metal was proposed as a heat storage material in the study of Soni et al. [5]. They presented a simulation of the process in which thermal energy was stored in an insulated steel block embedded in the ground.

Most research to date on solar space heating focuses on the efficiency indicators that can be achieved with specific SCs installation parameters in different climatic conditions. Redpath et al. [6] analyzed solar heating of houses in different zones of Europe. Their results showed that in the Mediterranean zone, seasonal thermal energy storage can fully cover the thermal energy demand of a typical house. However, for the Atlantic and continental climate zones, full coverage is achievable only in passive houses, i.e., those with an annual energy demand of less than 15 kWh/m². Clarke et al. in [7] investigated the influence of A_c and V on system efficiency. The simulation was carried out for a passive house in Ireland. The usable floor area of the house was 215 m², A_c was 10.6 m², and V was 23 m³. It was shown that increasing A_c to 20 m² raised the solar fraction SF from 0.47 to 0.63. The results of solar heating performance tests under various conditions are also presented in [8,9,10,11]. In a related study [12], the authors of this paper conducted simulation analyses of solar space heating in temperate climate conditions. The influence of A_c, V, and excess heat removal outside the installation on the SF was investigated. The influence of solar collectors’ inclination angle on the efficiency of solar space heating was presented in [13].

The TRNSYS simulation tool is widely used for the analysis and design of various energy systems, supporting both optimization and performance evaluation. In [14], a dynamic simulation model of a tank was developed using TRNSYS 18 software. The model was successfully validated against experimental data for a vertical cylindrical tank with a volume of 0.3 m³. In [15], a hybrid heating model for a multi-family building in Sweden was presented. The system combined district heating as well as heat pumps, and electric heating.

One of the challenges with solar space heating is the occurrence of excess heat during the summer months. To avoid these surpluses, the surface area of the installed collectors should be limited or large-volume heat storage tanks should be used. The first approach, however, reduces the amount of heat that can be used during the heating season. The large volume of the tanks is also disadvantageous in several respects. In addition to the obvious reasons related to investment costs and location, large tanks have a high external surface area, which leads to increased heat losses.

In temperate climates, indoor spaces do not require heating during the summer, unlike pools, where heating is often recommended and sometimes even necessary. This is primarily due to water evaporation into the environment, which involves a high thermal effect associated with phase change. In addition, the comfortable water temperature in a pool is considerably higher than the air temperature indoors or outdoors. The large mass of water, combined with its high heat capacity, also plays a role, as it results in slow direct heating of the pool. The solution to the problem of excess heat in storage tanks and heat deficiency in outdoor swimming pools is the transfer of excess heat from the storage tank to the pool water during the summer.

Numerous studies in the literature address pool heating with renewable sources. In [16], a method for evaluating energy savings achieved by flat plate SCs assisted heat pumps for heating swimming pool water was presented. Chow et al. [17] analyzed the performance of a solar-assisted heat pump system for indoor swimming pool water and space heating under the subtropical climatic condition of Hong Kong. Ilgaz and Yumrutaş [18] presented a case study on the heating performance of a system combining a solar-assisted heat pump with an underground thermal energy storage tank for swimming pool applications under real climatic conditions in Gaziantep Province, Turkey. The developed analytical model was used to evaluate several performance parameters of the system. A comprehensive state-of-the-art review of swimming pool advanced heating technologies has been presented by Li et al. in [19]. This review covered various systems, including conventional boilers, heat pumps, solar thermal systems, passive and active technologies and hybrid solutions, evaluating their energy efficiency, environmental impact, and economic performance. Modeling of water heating in swimming pools using solar radiation is also described in papers [20,21,22,23,24].

Analysis of a swimming pool with solar heating using the utilizability method was discussed in the article [25] by Gonçalves et al. This approach allows for evaluating the solar energy contribution under varying climatic and operational conditions. The study demonstrates that the utilizability method is an effective tool for optimizing solar thermal system design and predicting energy savings in pool heating applications. The simulations were carried out for various locations in Brazil. SF was determined under the following conditions: pool surface area $A_p$ = 250 m², A_c equal to 150 m², and water temperature 30 °C. For the warmest location among those studied, Fortaleza (3.76° S), SF = 0.69 was determined, while for the coldest, Porto Alegre (30.02° S), SF was equal to 0.21. The article [26] reviewed recent developments in solar water heaters and collectors, with particular relevance to applications such as swimming pool heating.

The literature review shows that solar systems rarely cover the total energy demand under peak conditions. Achieving full solar coverage is impractical, especially in temperate climates, and oversizing of equipment is not recommended.

A novel contribution of this study lies in the integration of the concept of solar utilizability with functional approximations of physical and geometric variables that fluctuate seasonally. This approach yields smoothed courses, excluding short-term weather extremes, but provides a robust basis for assessing the impact of parameters on long-term system performance. Another original aspect of this work is the inclusion of excess heat discharge in the process model, a consideration largely overlooked in earlier studies. This strategy enhances summer energy usage and improves overall system efficiency.

This paper considers solar heating of a single-family house in a temperate climate with accompanying excess heat discharge (e.g., to a pool) during the summer. The heat flows between the individual components of the system were analyzed. Particular attention was paid to the impact of excess heat discharge to the pool on the SF of the building heating system. A model of the solar heating process for space and pool heating from a storage tank loaded by SCs was developed.

2. Solar System Model

2.1. Principles and Basic Relationships

A simplified diagram of the considered solar system consisting of SCs, a hot water storage tank, a heat exchanger for space heating, and a heat exchanger for pool heating is shown in Figure 1. The diagram does not include pumps, valves, or an additional heater in the building. There are two types of solar gains in the installation: direct gains, absorbed by the water in the pool, and indirect gains, absorbed by the water in the tank through the SCs. The heat exchanger in the building operates only during the heating season, while the pool heat exchanger operates during the summer.

Figure 1. Solar system diagram.

The visualization of solar radiation energy flowing through the considered system is presented in the Sankey diagram (Figure 2).

Figure 2. Sankey diagram for solar heating system.

The amount of useful energy generated in the collector during a 24 h period, $Q_u$, and delivered to the tank is described by the following equation:

$$Q_u=A_c{F}_R\left(\tau \alpha \right)H_T\varphi$$

(1)

where $F_R$ is the collector heat removal factor, $\left(\tau \alpha \right)$ is the effective transmittance–absorptance product, $H_T$ is the daily radiation on a tilted collector surface, and $\varphi$ is the daily utilizability. For determining $Q_u$ according to Equation (1), it was necessary to specify $\varphi$. The method for that was described in detail in [27,28], among others. The basic relationships used in calculating solar radiation were presented in Appendix A.

The model presented in this paper (and in [12]) is founded on the concept of utilizability. The monthly averaged daily utilizability represents the ratio of usable energy obtained during a given month to the amount of solar energy incident on the SCs over the same time. This parameter takes into account that part of the solar energy is unused during periods when the radiant flux is less than the heat losses from the SCs to the environment. Utilizability depends, among other things, on the critical solar irradiance. This is the radiation level where the absorbed solar radiation and SCs thermal losses are equal. In this article, the authors used daily utilizability values. Since the basis of the calculations were monthly average values, it was necessary to use functions approximating the relationship between utilizability and time.

Heat losses from the tank to the surroundings $Q_L$ are:

$$Q_L=U_t{A}_t\left(T_t-T_g\right)·∆t$$

(2)

where $U_t$ is the overall heat transfer coefficient of the tank, $A_t$ is the tank wall surface area, $T_t$ is the water temperature in tank, $T_g$ is tank surroundings temperature, and $∆t$ is a time step.

To determine daily heat transferred from the tank to the radiator in the building $Q_s$, two quantities should be considered: the amount of heat that can potentially be transferred to the radiator $Q_0$ during the day, and the amount of heat that should be transferred to the radiator $Q_b$ during the day to maintain the desired room temperature in the building $T_r$ (under winter conditions). The heat transfer equation for $Q_0$ is as follows:

$$Q_0=\dot{m}c\epsilon \left(T_{in}-T_r\right)·∆t$$

(3)

where $\dot{m}$ is the water flow rate in the building circuit, $c$ is the water heat capacity, $\epsilon$ is the effectiveness of the heat exchanger, and $T_{in}$ is the inlet water temperature.

The daily heat required to heat a room $Q_b$ is calculated as the multiplication of the overall heat transfer coefficient of the whole building $U_b$, the building envelope area A_w, and monthly heating degree days ${DD}_m$:

$$Q_b=U_b{A}_w{\displaystyle \frac{{DD}_m}{30.4}}·∆t$$

(4)

For heating purposes, heat is transferred from the tank in the amount of $Q_s$, and, if necessary, supplemented by an additional source $Q_{aux}$, depending on the current water temperature in the tank. The critical temperature is denoted as $T_0$. If the water in the tank exceeds $T_0$, it is mixed with the water returning from the radiator before being supplied to the radiator. This ensures that the temperature of the supplied water will be equal to $T_0$. If the water in the tank is at a temperature below $T_0$ but remains above the minimum temperature $T_{min}$, it is supplied directly to the radiator.

Heat $Q_s$ is equal to the smaller value of $Q_0$ and $Q_b$ and is transferred to the building only during the heating season [12,29]:

$$Q_s=\mathrm{m}\mathrm{i}\mathrm{n}\left(Q_0,Q_b\right)$$

(5)

The limits of the heating period can be determined based on the air temperature. The daily amount of excess heat transferred to the pool $Q_p$ is limited by the pool heat exchanger and is determined from:

$$Q_p=\dot{m_p}c{\epsilon }_p\left(T_t-T_p\right)·∆t$$

(6)

where $\dot{m_p}$ is the water flow rate in the pool circuit, ${\epsilon }_p$ is the effectiveness of the pool heat exchanger, and $T_p$ is the pool water temperature.

Heat accumulated in the walls of the building, denoted as $Q_{acc}$, is formulated as follows:

$$Q_{acc}=m_w{c}_w\left(T_w-T_{w0}\right)$$

(7)

where $m_w$ is the mass of building, $c_w$ is the heat capacity of walls, and $T_{w0}$ is the wall temperature in the previous time interval.

2.2. Calculation Algorithm

The flowchart of the calculation algorithm is shown in Figure 3. In brief, the algorithm calculates the energy balance of the solar heating system through iterative daily computations of meteorological and operational parameters. It is based on the heat balance of the tank, which is supplied with heat in the amount of $Q_u$. The individual components of heat losses from the tank are $Q_s$, $Q_p$, $Q_L$, and $Q_{acc}$.

Figure 3. Flowchart of the calculation algorithm.

The thermal balance of the tank in the solar system is as follows:

$$Q_u-\left(Q_s+Q_L+Q_p+Q_{acc}\right)=V\rho c·∆T$$

(8)

where $∆T$ is the daily water temperature variation in the tank.

Calculations were performed for successive time intervals (days) throughout the year. The water temperature in the reservoir at the beginning of the year $T_{init}$ was taken as the initial approximation. For each day, the calculation involved determining the final water temperature, which then served as the starting temperature for the next day. The calculated final water temperature on the last day of the year was expected to match the assumed initial temperature. Convergence of the algorithm, defined by the condition $\left|T_{\mathrm{init}}-T_{\mathrm{init}0}\right|<0.1 °\mathrm{C}$, was generally achieved in the second iteration.

Compared to the algorithm described in our previous publication [12] it was assumed that only excess heat would be used for pool heating (Figure 3). Therefore, simultaneous heating of the pool water and the building rooms will not occur. Another distinction introduced in the present calculations concerns the control of water flow between the receivers, which depends on the relationship between air temperature $T_a$ and the limiting temperature of 14 °C. Based on $T_a$, the heating season was defined as the period extending from day $t_2$ = 260 to day $t_1$ = 134 of the following year. There was also a significant difference in terms of reaching the maximum $T_t$. In [12], only the amount of heat to be removed from the system was specified. In this article, the interaction between the water in the heated pool and the environment was included in the model. This made it possible to determine the temporal changes of $T_p$.

The calculation algorithm was validated by comparison with the results obtained from the TRNSYS application, which was described in detail in the authors’ publication [12]. Good consistency of results obtained by different methods has been proved.

The basic indicators determined are SF and solar efficiency SE. SF is the ratio of solar energy delivered to the radiator ${\Sigma Q}_s$ to the energy required to heat the building $\Sigma Q_b$:

$$SF={\displaystyle \frac{{\Sigma Q}_s}{\Sigma Q_b}}$$

(9)

where by the summation is performed for individual days of the year. On the other hand, solar efficiency SE is the ratio of the sum ${\Sigma Q}_s$ and ${\Sigma Q}_p$ to the amount of solar energy incident on the collectors:

$$SE={\displaystyle \frac{{\Sigma Q}_s+{\Sigma Q}_p}{{A_c·\Sigma H}_T}}$$

(10)

3. Heating Water in an Outdoor Pool

The water in an outdoor pool is exposed to thermal interaction with the atmosphere, which often makes it necessary to heat it in order to maintain comfortable temperatures in the range of 25–30 °C. Figure 4 presents the heat transfer between the water in the pool and the environment. Heat transfer occurs via solar radiation flux, long-wave radiation flux, convective flux, and evaporation flux. These “natural” fluxes are supplemented by heat transfer from the heating system. The assumption was made that the water in the pool $T_p$ has a uniform temperature (i.e., it is completely mixed).

Figure 4. Heat fluxes during pool heating.

The relationships between the individual heat fluxes are presented below. The solar radiation flux is the ratio of daily insolation on a horizontal surface H and the time interval Δt. Long-wave radiation LW is emitted in the opposite direction to solar radiation, i.e., from the water surface to the sky. The LW flux between the water surface and the sky is calculated using the Stefan–Boltzmann equation:

$$LW=0.95·\sigma \left(T_p^4-T_{sky}^4\right)$$

(11)

where $\sigma$ is the Stefan–Boltzmann constant and $T_{sky}$ is the temperature of the sky.

$T_{sky}$ depends on the air temperature $T_a$ and humidity RH. According to ASHRAE [28], it can be calculated using the following formula:

$$T_{sky}={ T}_a-\left(1105.8-7.562{·T}_a+0.01333{·T}_a^2-31.292·RH+14.58{·RH}^2\right)$$

(12)

where in this equation $T_a$ is in K.

Convective flux $H_{conv}$ results from the difference between $T_p$ and $T_a$ and, unlike other fluxes, can be absorbed or released by water. The heat transfer equation determines $H_{conv}$:

$$H_{conv}=h\left(T_p-T_a\right)$$

(13)

The heat transfer coefficient $h$ between the environment and the water surface depends on the hydrodynamic conditions at the water surface. For a known wind speed $u$, $h$ can be calculated using the following equation:

$$h=5.7+3.8·u$$

(14)

In addition to convective and radiative fluxes, the pool loses heat due to water evaporation into the environment. The partial pressure of water vapor at the water surface is generally greater than the partial pressure of vapor in atmospheric air. This difference is the driving force behind the transfer of water from its surface to the environment. The evaporative heat flux EV is given by the following relationship:

$$EV=A\left(p_{sat}-p\right)·h$$

(15)

This relationship, together with the definition of the constant A, is explained in Appendix B.

The heat balance of the water in the pool is as follows. The amount of heat supplied to the water in the pool during the day $Q_{in}$ includes reaching solar radiation and $Q_p$:

$$Q_{in}=Q_p+H{A}_p$$

(16)

where $H$ is daily insolation on the horizontal surface. On the other hand, the water in the pool loses heat flux $Q_{out}$ through LW, EV and $H_{conv}$. The value of $Q_{out}$ depends on the surface area of the water in the pool and the time interval Δt under consideration:

$$Q_{out}=\left(LW+EV+H_{conv}\right)·A_p·∆t$$

(17)

The daily change in the internal energy of the pool water ${\mathsf{\Delta }U}_p$ causes a change in water temperature $∆T_p$:

$$∆U_p=V_p\rho c·∆T_p$$

(18)

According to the law of conservation of energy, the following applies:

$$Q_{in}-Q_{out}=∆U_p$$

(19)

4. Results

The appropriate calculations were performed for different values of A_c, A_p, V and $\dot{m_p}$. The latter value directly determines $Q_p$. The main purpose of the calculations was to determine whether redirecting heat outside the space heating system affects its overall efficiency. In addition, the relationships between heat flows and temperatures in the solar system are presented. Due to the high number of parameters in the simulated process, some were fixed at constant values. An overview of these is provided in Table 1.

Table 1. Data used in the simulation.

Parameter	Value	Parameter	Value
DDa	3016 K·day	UL	3.0 W/(m2K)
$F_R$	0.88	Ut	0.16 W/(m2K) [30]
$h_p$	1.35 m	Tr	21 °C
ΣHT	4126.9 MJ/m2	Tg	9 °C
$m_w$	120 × 103 kg	(τα)	0.81
$\dot{m}$	3.0 kg/min	ε	0.6 [31]
u	0.5 m/s	εp	0.7 [31]
(Ub·Aw)	150 W/(m2K)	ψ	50°

4.1. Climate Parameters

For the location considered in this study (Krakow, Poland), Figure 5 shows the average $T_a$ for individual days of the year, generated from data taken from the Photovoltaic Geographical Information System (PVGIS) [32]. This relationship was approximated by a periodic function (a). After converting the daily average data to monthly averages, a function was created to approximate these data as well. Its mathematical form (b) and graphical representation are almost identical to those for the daily average data.

Figure 5. Approximation of the temporal temperature course.

For the calculation of H, hourly insolation data obtained from the PVGIS database [32] were used. These data were converted to daily, and then to monthly average daily values, and an approximation function (c), presented in Figure 6, was created based on them. This figure shows the values H for individual days of the year for the location under consideration, i.e., Krakow. In Figure 6, the course of the approximating function is represented by a continuous red line.

Figure 6. Approximation of daily solar radiation on a horizontal surface.

The time courses of the daily clearness index $K_T$ for the location considered in this study were calculated according to the following approximation function [12]:

$$K_T=0.42-0.08·\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t+0.62\right)$$

(20)

The daily insolation on the inclined surface $H_T$ depends on the value of $H$, β, $K_T$, and the diffuse reflectance ${\rho }_g$. The time courses of insolation under various conditions are shown in Figure 7 and Figure 8. These figures are based on the course of daily solar radiation on a horizontal surface shown in Figure 6.

Figure 7. The influence of $K_T$ on $H_T$.

Figure 8. The influence of β and ${\rho }_g$ on $H_T$.

The course of $H_T$ changes depending on $K_T$ is shown in Figure 7. Individual curves refer to $K_T$ values that remain constant throughout the year and $K_T$ values determined from Equation (20). As can be seen, $H_T$ values increase with a greater $K_T$ parameter.

Figure 8 shows the influence of β and ${\rho }_g$ on $H_T$. With β growth, $H_T$ decreases, but the shape of the curve characterizing this effect also changes. For small β (30°), the curve is unimodal, while for a vertical surface (90°) there are two distinct maxima, especially for ${\rho }_g$ equal to 0. The ${\rho }_g$ significantly affects $H_T$ for large β values.

Figure 9 presents examples of temporal curves of solar energy transferred during a day, referred to A_c equal to 1 m². The calculations were performed for ψ = 50°. The highest energy values refer to daily insolation outside the atmosphere $H_0$, which is not attenuated by any factor. The maximum $H_0$ values in the summer slightly exceed 40 MJ/m². The amount of solar energy falling on the Earth’s surface is significantly lower than $H_0$, and the maximum value does not exceed $H$ = 20 MJ/m². The $H$ values were determined for the $K_T$ coefficient from Equation (20) and ${\rho }_g$ = 0. The maximum of the curve characterizing $H_T$ is flattened compared to the maxima for $H_0$ and $H$.

Figure 9. Time courses of solar radiation in various conditions.

4.2. Courses of Heat Fluxes and Temperatures

Figure 10 shows example courses of daily heats in a solar system. $Q_u$ is the only item in the heat balance supplied to the system, while $Q_s$, $Q_L$, and $Q_p$ characterize the daily heats leaving the system. The time course of $Q_u$ is shown as negative values (−$Q_u$) to highlight the algebraic sum of the individual values.

Figure 10. Daily heats in a solar system.

The time course of $Q_s$ can be divided into three periods (Figure 10). In the first period, $Q_s$ values initially increase due to increased solar radiation, and then decrease as a result of reduced heat demand for space heating. In the middle period, $Q_s$ = 0 because there is no space heating operation. In the third period, $Q_s$ values increase at the beginning of the heating season due to increased demand for heat, and then decrease as a result of the depletion of heat stored in the tank.

The value $∆U$ characterizes changes in the internal energy of the system (water in the tank). In the first and third quarters of the year, these changes are close to zero (Figure 11), as discussed below. Since the heat balance of the system must be closed over the year, the average annual value of $\sum ∆U=0$. $∆U$ changes are related to $T_t$, because the derivative d$T_t$/dt is proportional to $∆U$. This is shown in Figure 11. The temperature changes in the second and fourth quarters are significant, and therefore the temperature derivatives (and $∆U$) are non-zero, unlike the values for the other quarters of the year.

Figure 11. Time courses of $T_t$ and $∆U$.

Figure 11 also illustrates the daily heat $Q_{acc}$ changes. The time course of $Q_{acc}$ consists of two symmetrical periods and is time-correlated with calendar half-years. In the first half of the year, the cold building walls absorb heat from inside the rooms. In the second half of the year, however, walls transfer heat in the opposite direction, returning it to the system. Therefore, the average annual value of $Q_{acc}$ is equal to zero.

The time courses of heat fluxes and temperatures during pool heating are shown in Figure 12 and Figure 13. For comparison, Figure 12 also shows $T_t$. The dashed vertical lines represent the time range of pool heating. During this period, $T_t$ initially drops sharply, while $T_p$ rises abruptly. This is due to large temperature differences. Subsequently, $T_t$ and $T_p$ stabilize until the end of the pool heating period. After switching from pool heating to room heating, $T_t$ rises sharply due to the low heat demand for room heating during this period. Then, the heat demand for heating increases and $T_t$ decreases. Figure 12 also shows $T_a$ and $T_{sky}$. $T_p$ throughout the heating period is about 5–6 K higher than the $T_a$, and the maximum $T_p$ is about 22 °C.

Figure 12. Temperature courses in solar system.

Figure 13. Heat rates in solar system.

Figure 13 shows the heat fluxes transferred in the pool. Heat losses are shown as negative values, which allows us to estimate the algebraic sum of all fluxes. It is close to zero, so $T_t$ during the pool heating period is relatively stable (except for the initial period). The largest contribution to water heating comes from heat absorbed as a result of solar radiation. The excess heat from the tank through the heat exchanger is smaller, and changes in its thermal power are not very variable throughout the period. The greatest heat losses are due to water evaporation, slightly smaller losses occur to the sky, and the smallest are convective losses.

4.3. The Impact of Climatic Parameters on $T_p$

Climatic conditions have a strong influence on $T_p$. $RH$ is also particularly important. As $RH$ increases, the rate of water evaporation decreases, resulting in reduced $EV$. This can be seen in Figure 14. An increase in $RH$ from 0.3 to 0.9 causes an increase in the maximum value of $T_p$ by 6 K.

Figure 14. Time courses of $T_p$ at different $RH$ values.

At the water surface,$u$ also has a significant impact. Figure 15 shows the results of calculations of time courses of $T_p$ for $u$ ranging from 0.5 to 2 m/s. As $u$ increases, the intensity of convective heat and mass transfer increases (Figure 15). This results in an increase in $H_{conv}$ and $EV$, in accordance with Equations (13) and (15), resulting from the analogy of heat and mass transfer. The values in Figure 15 refer to $RH$ = 0.7. At $u$ of 2 m/s, the maximum $T_p$ reaches approximately 20 °C, while at $u$ = 0.5 m/s, the temperature can reach 23 °C.

Figure 15. Time courses of $T_p$ at different wind speed values.

4.4. Removing Excess Heat from the Tank—Impact on $SE$  and $SF$

The analysis of the impact of process parameters on the total (annual) amount of heat transferred and the maximum $T_p$ and $T_t$ was presented. The impact of the following parameters was considered: $A_c$, $A_p$, $V$ and $\dot{m_p}$.

Figure 16 illustrates the impact of the flow rate of water in the pool heat exchanger $\dot{m_p}$ on the amounts of heats transferred in the system over the year.

Figure 16. Influence of $\dot{m_p}$ on ${\sum Q}_u$, $\sum Q_s$, ${\sum Q}_p$, ${\sum Q}_L$ and the maximum $T_t$.

The dependencies of $\sum Q_u$ and ${\sum Q}_p$ with $\dot{m_p}$ have similar shapes (Figure 16). For any value of $\dot{m_p}$, the difference between $\sum Q_u$ and ${\sum Q}_p$ is almost the same. Therefore, transferring more heat from the tank to the pool ${\sum Q}_p$ causes almost the same increase in the amount of heat supplied to the tank from the collector $\sum Q_u$. This pattern can be explained by analyzing the instantaneous values of the transferred heats. High ${\sum Q}_p$ values result in a significant reduction in the temperature of the water in the tank. Therefore, SCs are supplied with water at a lower temperature, which means that the driving force for heat transfer in the SCs and also the utilizability $\varphi$ are significant. According to Equation (1), $Q_u$ values are proportional to $\varphi$, which explains their high values.

For large $\dot{m_p}$ values, i.e., at lower temperatures in the tank, heat losses ${\sum Q}_L$ are lower, as shown in Figure 16. On the other hand, the constancy of ${\sum Q}_s$ for different $\dot{m_p}$ values results from Equations (3) and (5). The inlet water temperature of the radiator in the room $T_{in}$ remains constant throughout the heating season, regardless of the water temperature in the tank. The relationship between the instantaneous values of $Q_p$, $Q_u$, and $Q_L$ is particularly straightforward outside the heating season, especially when the heat accumulated in the walls, $Q_{acc}$, is neglected. In summer, $Q_s$ = 0. In addition, excess heat is intensively transferred into the pool, which causes slight changes in $T_t$, and therefore, from Equation (8), it follows that $Q_u=Q_p+Q_L$.

The basic parameter affecting the solar heating system is $A_c$. The impact of $A_c$ on the $SE$ and $SF$ indicators is shown in Figure 17. With $A_c$ increase, $SF$ rises monotonically, with a strong rise for small $A_c$ values and a gradually smaller increase for larger values. This relationship results from the variability of $Q_s$, to which $SF$ is proportional. $A_c$ has a different effect on $SE$. In this case, as $A_c$ increases, both the numerator and denominator of expression in Equation (10) change. This causes the $SE$-$A_c$ curve to have a maximum. The maximum value of $SE$ = 0.61 achieved for $A_c$ = 5 m² is not noteworthy due to the corresponding low $SF$ = 0.15. However, in the range of $A_c$ between 15 and 20 m², the $SE$ values decrease only slightly, while the $SF$ values increase, which suggests searching for the optimum $A_c$ in this range.

Figure 17. Influence of $A_c$ on $SE$ and $SF$.

Figure 18 shows the effect of $V$ on the heat transfer values in a solar system. Tank volume has virtually no effect on $\sum Q_s$, ${\sum Q}_p$, and ${\sum Q}_u$, while the impact on $\sum Q_L$ is negligible. However, $V$ has a strong impact on the maximum $T_t$. In tanks with a $V$ of less than 2.5 m³, the maximum temperature exceeds 100 °C, which makes it impossible to use such tanks with $A_c$ = 20 m².

Figure 18. Influence of $V$ on ${\sum Q}_u$, ${\sum Q}_s$, ${\sum Q}_p$, $\sum Q_L$ and the maximum $T_t$.

An important parameter affecting $T_t$ and $T_p$ is the pool surface area $A_p$ (Figure 19). The fixed value of $A_p$ determines $V_p$ based on the assumed pool depth of 1.35 m. For $A_p$ > 10 m², maximum $T_p$ is approximately 21 °C, and maximum $T_t$ is about 61 °C. However, significant changes occur for small $A_p$; for example, for $A_p$ < 2 m², the maximum $T_p$ exceeds 50 °C.

Figure 19. Influence of $A_p$ on the maximum $T_t$ and $T_p$.

5. Conclusions

This article presents a model of the process of heating a residential building and a pool with solar energy obtained by solar collectors and then stored in a heat storage tank. The analyses presented, based on the concept of solar utilizability, concerned the heating of a single-family house in a temperate climate. They demonstrated that the removal of excess heat is a beneficial solution, especially when it can be used to heat water in a pool.

The main conclusions are as follows:

(1)

The method proposed by the authors combines the concept of solar utilizability with a functional approximation of physical and geometric quantities that undergo cyclical changes throughout the year. It generates smooth curves, disregarding short-term fluctuations in weather conditions relative to the average values. The developed numerical model enables quick and comprehensive analysis of the process and assessment of heat fluxes and temperatures occurring in the solar heating system for buildings and pools.

(2)

For the input data used in simulation calculations, the thermal effects of heating the pool are insufficient. In general, the water in pools is heated to 25–30 °C. However, the purpose of this article was to examine the effect of using excess heat generated for space heating during the heating season; heating the water in the pool was only a result of this process. For the considered input data, pool water temperature throughout the heating period was approximately 5–6 K higher than the air temperature, and the water temperature in the tank was relatively stable. The largest contribution to pool water heating comes from heat absorbed directly from solar radiation. The greatest heat losses in the pool are due to evaporative heat flux.

(3)

The impact of the solar collectors’ area on the solar efficiency and solar fraction indicators was analyzed. Achieving $SF$ values close to unity in temperate climates is quite difficult. It requires the use of large collector areas and large tank volumes. However, since buildings are always equipped with additional heat sources, it is advisable to forego high $SF$ values in favor of increasing $SE$. This can be achieved by heating the water in the pool with excess heat from the solar installation.

(4)

Climatic conditions have a strong influence on pool water temperature. A change in air humidity from 0.3 to 0.9 causes an increase in the maximum value of pool water temperature by 6 K. An increase in wind speed of 1.5 m/s can cause a reduction of 3 K in the maximum pool water temperature. Mass losses caused by water evaporation from pools are significant. For air at a temperature of 20 °C and relative humidity of 0.7, and water temperature of 23 °C, the difference in partial water vapor pressure above the water surface is 1170 Pa. Assuming a heat transfer coefficient (for u = 0.5 m/s) of h = 7.6 W/(m²K), the evaporating water stream is equal to 0.22 kg/(m²∙h). Accordingly, the daily evaporation from a pool with a surface area of 18 m² amounts to approximately 0.09 m³ of water.

(5)

Transferring excess heat from the storage tank does not reduce the potential heating capabilities of the space heating system. Therefore, this action is equivalent to improving the thermal performance of solar heating systems.