Study of Long-Term Thermal Performance of Solar Pool Heating Systems at Selected Locations in Europe

Abstract

Heating water in outdoor pools is common, particularly in regions with cool or temperate climates. Several factors, including solar radiation, ambient temperature, wind speed, and humidity, influence the pool water temperature. A key design challenge is to determine the collector surface area required to achieve the desired pool water temperature. In this study, a mathematical model was developed that accounts for the aforementioned factors. Under various operating conditions, thermal performance calculations were carried out. Climatic conditions at three locations across Europe, representing different climate regimes, were analyzed. The model was compared with results from the POLYSUN simulation software. Most of the calculations were performed for a pool surface area of 24 m². The calculations showed that wind speed above the pool water surface has a significant impact on heat losses. Locating the pool in a sheltered area results in a consistent reduction in heat losses. It was determined that, under the climatic conditions of Kraków, the installation of solar collectors with a surface area equal to 50% of the pool surface enables the maintenance of daytime water temperatures above 21 °C for approximately 100 days. In the absence of solar collectors, achieving such temperatures is not feasible.

1. Introduction

Solar energy constitutes a sustainable and economically viable solution for low-temperature heat applications, including swimming pool heating. Given the relatively moderate temperature levels required and the strong coincidence between solar availability and pool usage periods, solar thermal systems are particularly well-suited for this purpose [1]. In summer, solar energy can be used to heat the pool water, thereby mitigating the problem of surplus thermal energy produced by solar collectors. Swimming pools experience significant and persistent heat losses driven by evaporation, convective heat transfer, and long-wave radiation flux, which leads to continuous heating demand. As a result, their specific energy consumption is often considerably high, making them a critical target for energy efficiency improvements and the integration of renewable energy technologies [2,3]. In this context, the decarbonization of pool heating systems is essential not only to reduce greenhouse gas emissions but also to limit long-term operational costs.

Solar thermal heating systems are mature, energy-efficient technology for swimming pool applications, particularly in outdoor and semi-open configurations. Their straightforward integration with existing hydraulic and filtration systems, combined with their ability to significantly reduce auxiliary energy demand, makes them a key strategy for lowering fossil fuel dependency. A comprehensive state-of-the-art review by Li et al. [4] indicates that unglazed solar collectors are frequently employed in such applications due to their high efficiency at low temperature differences, low investment cost, and simple operation. Nevertheless, the actual performance of solar pool heating systems is strongly influenced by climatic conditions, system configuration, collector type, and control strategies. Parameters such as solar irradiance, ambient temperature, wind speed, and humidity directly affect both heat gains and losses, thereby determining the achievable solar fraction and seasonal efficiency.

Several experimental and numerical studies have demonstrated the technical feasibility and energy benefits of solar pool heating systems. Ruiz and Martínez [5] developed a TRNSYS model validated with experimental data for an outdoor swimming pool and showed that properly sized solar systems can supply a large share of the annual heating demand. Similarly, Lugo et al. [6] combined numerical simulations with experimental validation for warm climates, highlighting the strong influence of collector area, mass flow rate, and control strategies on system efficiency. Article [7] presents a dynamic Modelica-based model of an outdoor swimming pool equipped with thermal cover, integrated with solar thermal collectors and auxiliary heating. The results show that evaporation is the dominant heat-loss mechanism, leading to high energy demand, while solar collectors cover only a limited share of this demand. Ghabour et al. [8] investigated the feasibility of using solar thermal energy for heating swimming pools in a Central European climate, using Hungary as a case study. Simulations were performed using T*SOL software version 2018. The results indicate that the most cost-effective solution is a solar system without auxiliary heating or a heat exchanger, employing flat-plate collectors, and applying a pool cover and a windshield. Dongellini et al. [9] developed in the Matlab/Simulink R2014a environment, a dynamic model of a solar heating system composed of horizontal solar flat collectors coupled to an outdoor swimming pool. The model allows predicting the system’s thermal energy performance on an hourly basis. The results demonstrate that unglazed collectors are suitable for this application.

Hydraulic parameters and control strategies have been identified as key factors in optimizing system performance by Zhao et al. [10]. They experimentally showed that reducing flow rate and pump speed improves collector efficiency while lowering electricity consumption. These results were confirmed by Cunio and Sproul [11], who reported enhanced energy savings and thermal efficiency for unglazed solar collectors operating under reduced flow conditions.

Analysis of solar heating of swimming pools using the utilizability method was presented in the article [2] by Gonçalves et al. This approach allows the evaluation of 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. Simulations were performed for several locations in Brazil. The solar fraction values were determined under the following conditions: a pool surface area of 250 m², a collector surface area of 150 m², and a water temperature of 30 °C. In [12], the possibilities of solar heating of residential buildings were investigated. The benefits of removing excess heat from the storage tank and transferring it to the swimming pool water during the summer were demonstrated.

Beyond conventional solar thermal systems, hybrid solutions have received increasing attention. Solar-assisted heat pump systems have been investigated by Starke et al. [13] and Li et al. [14], who demonstrated that the integration of solar collectors with heat pumps and thermal energy storage significantly improves seasonal performance. Similar conclusions were drawn by Ren et al. [15], who showed that hybrid thermal energy storage enhances operational stability and reduces auxiliary energy use. Sezen and Gungor [16] further confirmed through a comparative review that solar-assisted heat pump systems outperform standalone systems in terms of efficiency.

More recently, integrated photovoltaic/thermal systems have been proposed for pool heating applications. Luo et al. [17] analyzed a solar PV/T pool system experimentally and theoretically and demonstrated that simultaneous heat and electricity generation can significantly improve overall system efficiency. At a real scale, Katsaprakakis [18] reported the successful implementation of a solar-combi system for swimming pool heating and domestic hot water production in Greece, achieving substantial fossil fuel savings.

From an economic and sustainability perspective, Mardi El et al. [19] conducted a life-cycle cost analysis and showed that solar thermal collectors and PV-assisted heat pump systems represent some of the most cost-effective and environmentally sustainable solutions for indoor swimming pools. However, their performance and economic viability are highly sensitive to local climatic conditions and energy prices.

Water in open reservoirs generally has a lower temperature than the surrounding air due to heat losses caused, among other factors, by evaporation. In high-latitude countries, solar energy alone is often insufficient to maintain a comfortable water temperature in outdoor pools, which necessitates additional heating. Water heating is also commonly used in warm-climate countries to extend the swimming season. Due to the high cost of heating with conventional energy sources, solar energy is widely used for heating outdoor pools. The use of heat pumps is also a beneficial solution. However, this does not necessarily mean abandoning solar energy, which can significantly support the operation of heat pumps. Solar heating of outdoor pools is a very cost-effective method, particularly when simple unglazed solar collectors are used, directly connected to the pool heating system.

Pool water temperature $T_p$ is governed by multiple interacting factors, such as solar radiation, ambient temperature $T_a$, wind speed u, relative air humidity $RH$, and the surface area of installed solar collectors $A_c$. A novel contribution of this study lies in a mathematical model which incorporates these variables by dividing the daily cycle into two distinct periods, daytime and nighttime, defined by sunrise and sunset. When integrated with the concept of utilizability, in combination with functional approximation of physical and geometric quantities that vary cyclically throughout the year, this approach results in a relatively simple model that adequately represents the real process. The model generates smoothed profiles by filtering out short-term weather fluctuations, while still providing a robust framework for evaluating the influence of key parameters on long-term system performance. Another novel aspect of this study is the evaluation of the system’s thermal performance across a range of operating conditions. Furthermore, achievable water temperatures under different climatic conditions in three European cities were analyzed: Thessaloniki, Kraków, and Stockholm. These cities are located near the 20° E meridian and at latitudes of 40°, 50°, and 60° N, respectively. Simulations were conducted to assess the maximum achievable $T_p$ at these locations assuming identical collector and pool surface areas. The calculations were performed using the proposed model and the POLYSUN simulation software version 2026.2.

2. Mathematical Model

2.1. Solar Pool Heating System Configuration

A simplified schematic of a solar water-heating system for a pool is shown in Figure 1. In the system presented in Figure 1a, solar heat is supplied directly to the water flowing in the pool-pump-collector loop (filtering and control devices are not depicted). Assuming perfect mixing of the water in the pool and ignoring heat losses in the pipes, the temperature at the collector inlet is equal to $T_p$.

Figure 1b presents an installation with a coil placed in a pool where the $T_p$ remains constant regardless of position. The temperature at the outlet of the heat exchanger $T_p^{\prime }$ (equal to the temperature at the inlet to the collector) is higher than $T_p$ in this case. The following relationship is valid [20]:

$${\displaystyle \frac{T_c-T_p^{\prime }}{T_c-T_p}}=1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-NTU\right)$$

(1)

where $T_c$ is the outlet water temperature in the solar collector. The number of transfer units $NTU$ is defined as follows:

$$NTU={\displaystyle \frac{{UA}_{HEX}}{{\dot{m}}_{HEX}c}}$$

(2)

where $U$ is the overall heat transfer coefficient, $A_{HEX}$ is the heat exchanger area, ${\dot{m}}_{HEX}$ is the water flow rate in the heat exchanger, and $c$ is the water heat capacity.

2.2. Modeling Daily Changes in Insolation and Temperature

One way to model processes involving solar radiation is to use instantaneous data (e.g., hourly), which accurately reflects actual weather changes. However, a disadvantage of this calculation method is that the generated curves of process parameter variations correspond only to the specific time period represented by the weather data used. Therefore, they cannot be treated as generalized curves.

Another option is to use data that has been averaged according to certain rules. For seasonal or annual calculations, data for a typical meteorological year can be used. Such data can be processed numerically by developing functions that approximate the values of individual variables over time (e.g., ambient temperature, solar radiation, clearness index). The disadvantage of this method of calculation is that such approximations generally do not apply to hourly values, as the approximation functions would then be too complex. Using average daily weather data can lead to significant errors, as it assumes, among other things, that the intensity of solar radiation is the same during the day and at night.

To avoid the disadvantages of both methods, this study uses a compromise model based on dividing each day into two periods, conventionally referred to as day (d) and night (n), separated by sunrise and sunset. The method used to calculate daytime and nighttime durations for individual days and nights, along with $T_a$, is outlined below. The input data for individual days of the year are $T_a$ and the daily insolation on the collector surface $H_T$. The length of the day ${∆t}_d$ (in hours) is calculated according to the following relationship [21]:

$${∆t}_d={\displaystyle \frac{24}{\pi }}·\mathrm{a}\mathrm{b}\mathrm{s}\left({\omega }_s\right)$$

(3)

where sunset (or sunrise) hour angle ${\omega }_s$ is calculated using the following formula [21]:

$${\omega }_s=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\left(-\mathrm{t}\mathrm{a}\mathrm{n}\left(\psi \right)\right)·\mathrm{t}\mathrm{a}\mathrm{n}\left(\delta \right)$$

(4)

where declination $\delta$ was determined from

$$\delta =23.45·{\displaystyle \frac{\pi }{180}}·\mathrm{s}\mathrm{i}\mathrm{n}\left({\displaystyle \frac{2\pi }{365}}\left(284+n\right)\right)$$

(5)

Knowing the length of the day, it is easy to determine the beginning and end time of the daily period, $t_1$ and $t_2$, respectively:

$$t_1=12-{∆t}_d/2$$

(6a)

$$t_2=12+{∆t}_d/2$$

(6b)

For a given $H_T$ value, the average daily insolation relative to the length of the day period $H_{Td}$ is equal to the following:

$$H_{Td}={\displaystyle \frac{24}{{∆t}_d}}{·H}_T$$

(7)

whereas at night $H_{Tn}=0$, which is graphically presented in Figure 2.

To determine the ambient temperature for the day $T_d$ and for the night $T_n$ period, the daily temperature time courses must be estimated. The daily ambient temperature time courses were approximated by the following relationship:

$$T=T_a-B_d\mathrm{c}\mathrm{o}\mathrm{s}\left({\omega }_{d}t\right)$$

(8)

where $B_d$ is the daily temperature amplitude, and ${\omega }_d$ is the frequency (=2π/24).

$T_d$ and $T_n$ were calculated as the integral averages of the function (8) within the limits marked in Figure 3. The following results were obtained:

$$T_d=T_a-{\displaystyle \frac{B_d}{{\omega }_d·{∆t}_d}}\left[\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega }_d{t}_2\right)-\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega }_d{t}_1\right)\right]$$

(9a)

$$T_n=T_a-{\displaystyle \frac{B_d}{{\omega }_d·\left(24-{∆t}_d\right)}}\left[\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega }_d\left(t_1+24\right)\right)-\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega }_d{t}_2\right)\right]$$

(9b)

The interpretation of average temperature and solar radiation values for the day and night periods is presented in Figure 2 and Figure 3. The time courses correspond to the 135th day of the year (15 May, Kraków). A value of $B_d$ = 4 K was assumed. According to weather data for that day, the calculated values are as follows: $H_T$ = 12.7 MJ/m² and $T_a$ = 13.3 °C. ${∆t}_d$ is equal to 15.3 h, which results in $t_1$ = 4.3 h and $t_2$ = 19.3 h. The average day period temperature, i.e., between 4.3 h and 19.3 h, calculated from Equation (9a), is $T_d$ = 15.4 °C. The average night period temperature, i.e., between 19.3 h and 4.3 (+24) h, according to Equation (9b), is $T_n$ = 10.1 °C.

Figure 4 presents a detailed comparison between $T_a$ time courses obtained from the proposed model and the reference data derived from PVGIS [22] for Kraków during the late spring period. The model reproduces the general upward trend in temperature observed over time, as well as the characteristic diurnal fluctuations. The phase agreement between the two datasets remains satisfactory, indicating that the model correctly captures the timing of temperature changes.

2.3. Climate Parameters

Climatic conditions at three European locations representing different climate regimes, specifically Thessaloniki, Kraków, and Stockholm, were analyzed. The weather data for these locations were obtained from the Photovoltaic Geographical Information System (PVGIS) version 5.3 [22]. Typical meteorological year data for the period PVGIS-ERA5: 2005–2023 were used. The following parameters were considered: daily radiation on the horizontal surface $H$, $T_a$, $RH$, and $u$. Variations in $RH$ and $u$ are irregular throughout the year; therefore, average values were used. These values are presented in

Table 1. Average annual values of $RH$ and $u$.

	Thessaloniki	Kraków	Stockholm
$RH$	0.64	0.77	0.75
$u$ [m/s]	4.4	2.3	3.1

.

The clearness index is defined as the following ratio:

$$K_T={\displaystyle \frac{H}{H_0}}$$

(10)

As can be seen in Figure 5 and Figure 6, the time courses of $H$ and $T_a,$ at each of the locations considered, show regularity. The following relationship was used to approximate the data:

$$Y\left(t\right)=A-B·\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t-P_s\right)$$

(11)

where $A$, $B$, $P_s$ are constants, and ω = 2π/365.

Table 2. Coefficients for determining approximation relationships for $H$, $T_a$, and $K_T$.

		Thessaloniki	Kraków	Stockholm
H [MJ/m2]	$A$	12.3	8.3	7.0
	$B$	7.8	6.7	7.2
	$P_s$	−0.03	−0.03	−0.22
$T_a$ [°C]	$A$	16.8	9.6	8.2
	$B$	9.9	10.2	8.8
	$P_s$	0.49	0.34	0.54
KT	$A$	0.418	0.321	0.626
	$B$	0.077	0.069	−0.146
	$P_s$	−0.62	−0.74	−0.05

presents constant values, determined using the least squares method, applied in Equation (11) to calculate $H$, $T_a$, and $K_T$ for individual locations. The approximate values of $H$ and $T_a$ were taken from [22] or calculated using Equation (10).

$H_T$ is calculated using the following formula:

$$H_T=R·H$$

(12)

In this equation, $R$ is the coefficient of daily heat transfer from horizontal to inclined surface. The value of $R$ refers to the total radiation and is calculated from the following equation:

$$R=\left(1-{\displaystyle \frac{H_d}{H}}\right)R_b+\left({\displaystyle \frac{H_d}{H}}\right)\left({\displaystyle \frac{1+\cos \beta }{2}}\right)+{\rho }_g\left({\displaystyle \frac{1-\cos \beta }{2}}\right)$$

(13)

The mean deviation of the modeled and experimental $T_a$ time courses shown in Figure 4 from the daily average temperature curve determined by Equation (11), using the coefficients $A$, $B$, and $P_s$ presented in Table 2, was calculated. These values are 6.9 K for the experimental data and 4.6 K for the model.

The individual components in Equation (13) denote direct radiation, diffuse radiation, and reflected radiation, respectively. The share of diffuse radiation ${\displaystyle \frac{H_d}{H}}$ can be determined based on empirical relationships as a function of the clearness index $K_T$. $R_b$ is the ratio of daily beam radiation on a tilted surface to that on a horizontal surface and is a function of $\psi$ and $\beta$. In the analysis conducted, it was assumed that the collector’s slope $\beta$ was equal to the latitude $\psi$ of the considered installation location. The solar collectors in the installations were oriented towards the south.

Figure 7 shows the time courses of the $R$, $R_n$, $RH$ and $r_{t,n}$ under the climatic conditions of Kraków. ${\rho }_g$ = 0.2 was assumed. Time courses shown were approximated using the periodic function given in Equation (11). The determined coefficients $A$, $B$, and $P_s$ are presented in

Table 3. Approximation of $R$ coefficients.

	Thessaloniki, ψ = 40°	Kraków, ψ = 50°	Stockholm, ψ = 60°
A	1.175	1.233	1.501
B	−0.365	−0.455	−0.762
Ps	−0.08	0	−0.17

.

2.4. Utilizability

The model uses the concept of daily solar utilizability $\varphi$. It is defined as the ratio of usable energy obtained during the day to the total solar energy incident on the solar collector during the same period. $\varphi$ considers the fact that some of the solar energy is not used when the radiation flux is lower than the heat flux lost from the collector to the environment. The critical solar radiation intensity I_Tc plays a key role, as it is the radiation intensity below which collectors do not supply heat to the system. From the definition of I_Tc, the following relationship follows:

$$I_{Tc}={\displaystyle \frac{U_L\left(T_p-T_a\right)}{\left(\tau \alpha \right)}}$$

(14)

The value of $\varphi$ therefore depends on I_Tc and indirectly on the inlet temperature of the collector. $\varphi$ should be calculated according to the following equation [21]:

$$\varphi =\mathrm{e}\mathrm{x}\mathrm{p}\left\{\left[a+b\left({\displaystyle \frac{R_n}{R}}\right)\right]\left(X_c+c{X}_c^2\right)\right\}$$

(15)

where a, b, and c are constants dependent on $K_T$. Thus, the utilizability method is based on the correlation of $\varphi$ with $K_T$, the ratio $R_n/R$, and the dimensionless critical radiation $X_c$ defined as follows:

$$X_c={\displaystyle \frac{I_{Tc}}{r_{t,n}{R}_{n}H}}$$

(16)

The monthly average ratio of total radiation on the tilted plane to that on the horizontal plane $R_n$, and the ratio of total radiation in an hour to total in a day (noon) $r_{t,n}$, appearing in Equations (15) and (16) are functions of time and angles:

$$R_n=f_1\left(t,\psi ,\beta \right)$$

(17)

$$r_{t,n}=f_2\left(t,\psi \right)$$

(18)

The use of the utilizability method requires that the critical radiation level remains approximately constant. In practice, this occurs in systems with very large storage tanks or in systems without storage, where the medium entering the collector comes from a source with a constant temperature. Therefore, the concept of utilizability can be used in modeling solar water heating in a pool.

Figure 7 shows examples of the time courses of the $R_n$ and $r_{t,n}$. The shape of these profiles provides the basis for their approximation by a relationship given in Equation (11). The constants obtained are presented in

Table 4. Approximation of $R_n$ and $r_{t,n}$ coefficients using the utilizability method.

	Thessaloniki			Kraków			Stockholm
	$A$	$B$	$P_s$	$A$	$B$	$P_s$	$A$	$B$	$P_s$
$R_n$	1.045	−0.108	0.06	1.025	−0.120	0.16	1.494	−0.553	−0.120
$r_{t,n}$	0.145	−0.029	−0.16	0.149	0.044	−0.16	0.163	−0.079	−0.16

.

2.5. Heat Transfer at the Water Surface

Heat is transferred between the surface of the water in the pool and the surrounding environment because of convection, radiation, and evaporation. The convective heat flow $H_{conv}$ is described by the following equation:

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

(19)

The heat transfer coefficient between the water surface and the environment $h$ depends primarily on $u$ above the water surface. There are numerous empirical equations for determining the value of $h$. The following relationship was used as a starting point to estimate the value of the coefficient h [23]:

$$h=3.1+2.1·u$$

(20)

In the installation discussed, there are two different radiation fluxes between the surface of the water in the pool and the surroundings: solar radiation (shortwave) flux and longwave radiation flux $LW$. The solar radiation flux was discussed in Section 2.3. $LW$ is directed opposite to the solar radiation flux: from the water surface to the sky. It can be described by the Stefan–Boltzmann equation:

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

(21)

where ${\epsilon }_s$ is the emissivity of the water surface equal 0.95; $\sigma$ is the Stefan–Boltzmann constant; and $T_{sky}$ is the temperature of the sky. $T_{sky}$ depends on $T_a$ and $RH$. According to ASHRAE [23], it can be determined as follows:

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

(22)

where in this equation $T_a$ is in K.

The heat flux resulting from water evaporation $EV$ is related to phase change. The difference in the partial pressures of water vapor at the water surface and in the air drives the transport of water in vapor form. With the transfer of the material component, heat is transported from the water surface to the air. Using the analogy between heat and mass transfer to calculate the rate of water vapor transport, the relationship describing the heat flux associated with evaporation can be determined as follows [23]:

$$EV=A \left(P_{sat}-p\right)·h$$

(23)

where $A$ is a constant defined in the following way:

$$A={\displaystyle \frac{L{M}_w}{P{M}_{a}c}}·{\left({\displaystyle \frac{a}{D}}\right)}^{-2/3}$$

(24)

where $L$ is the heat of vaporization of water; $M_a$ = 0.029 kg/mol and $M_w$ = 0.018 kg/mol are in sequence the molar masses of air and water; $D$ = 22.5 × 10⁻⁶ m²/s is the diffusion coefficient of water in air; $c$ = 1010 J/(kg·K) is heat capacity of air; and a = 19.4 × 10⁻⁶ m²/s is thermal diffusivity of air [24]. For a pressure $P$ = 1 × 10⁵ Pa, $A$ = 0.0168 K/Pa was obtained [25].

2.6. Calculation Algorithm

The heat balance of the water in the pool is as follows. The amount of solar heat supplied to the water in the pool during the day $Q_{in}$ was calculated from the following:

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

(25)

where daily useful heat transferred to the pool system $Q_p$ was obtained from the following:

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

(26)

where $F_R$ is the collector heat removal factor and $\left(\tau \alpha \right)$ is the effective transmittance–absorptance product.

The water in the pool loses heat $Q_{out}$ through $LW$, $EV,$ and $H_{conv}$. The value of $Q_{out}$ depends on the surface area of the water in the pool $A_p$ and the time interval $∆t$. It can be determined from the following equation:

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

(27)

The calculations were performed alternately for night and day periods of individual days. For both periods, the calculations are iterative. The calculation algorithm for a single day is as follows. For the night period, $Q_{in}$ = 0. Additionally, a tentative value of $T$ is assumed. On this basis, $H_{conv}$, $EV$, $T_{sky}$, and $LW$ are calculated, substituting ambient temperature for the night period $T_n$ in each case. Then, $Q_{out}$ and, finally, the water temperature at the end of the night period were calculated:

$$T_p=T_{p0}+{\displaystyle \frac{Q_{in}-Q_{out}}{V_p\rho c}}$$

(28)

where $T_{p0}$ is the pool water temperature at the end of the daily period on the previous day. If f = Abs ${(T}_p-T)$ is less than ε = 0.02 °C, which characterizes the accuracy of the calculations, then pool water temperature at the end of the night period $T_{pn}$ is found.

For the daily period, a tentative value of $T$ should also be assumed. Then, $H_{conv}$, $EV$, $T_{sky}$, and $LW$ are calculated, substituting $T_d$. In the next step, determine $\varphi$ and $Q_p$, followed by $Q_{in}$ and $Q_{out}$. Ultimately, the water temperature at the end of the day period is calculated using Equation (28). $T_{p0}$ should be replaced with $T_{pn}$. The temperature at the end of the day period is calculated by solving the nonlinear equation f(T) = 0, similarly to the night period.

The bisection method was employed to solve the mathematical model, ensuring stable and reliable convergence of the solution procedure. The analytical model was developed to maintain conceptual consistency with the POLYSUN software, ensuring alignment at a level sufficient for meaningful comparison of results. Further information regarding the calculation algorithm can be found in Appendix A.

3. Modeling with the POLYSUN Software

A schematic diagram of a pool solar water heating system generated in POLYSUN software version 2026.2 [26] is shown in Figure 8. The main component of the installation was an unglazed solar collector field that directly transfers the generated solar heat to an open pool, without the use of an intermediate heat exchanger. This approach minimizes additional thermal losses associated with heat exchangers and eliminates various issues related to the use of different heat transfer media. The thermal performance of the solar collectors was calculated in accordance with the internationally recognized ISO 9806:2025 standard [27]. The calculations were performed for an unglazed solar collector, whose characteristic parameters are presented in

Table 5. The process parameters values used in the calculations.

Parameter	Value
Diffuse reflectance, ${\rho }_g$	0.2
Pool surface area, $A_p$	24 m2
Pool depth, $h_p$	1.35 m
Solar collector efficiency, $F_R\left(\tau \alpha \right);$${\eta }_0$	0.9 [2]
Solar collector heat losses coefficient, $F_{R}U;$$a_1$	21 W/(m2K) [2]
Water heat capacity, $c$	4190 J/(kgK)
Water density, $\rho$	1000 kg/m3
Water flow rate, $\dot{m_p}$	30 L/min/m2

. Night-time operation of the circulation pump was not permitted. Instead, it was regulated by a differential temperature controller that activates the pump when $T_c$ exceeds $T_p$, by a predefined threshold (cut-in and cut-off temperatures of 3 °C and 1 °C, respectively), thereby ensuring efficient heat gain. A fixed water flow of 30 L/min/m² was set. The fraction of the local wind speed affecting the collector array was set to 50%. Heat loss to the ground surrounding the pool was considered in the calculations. Similar to the numerical model, no heat loss in the pipelines was assumed.

In the POLYSUN program, climate data are generated by default based on the Meteonorm climate database (version 7.2). However, for the analyzed installation, more recent climate data from the PVGIS database (PVGIS-ERA5, 2005–2023) [22] were applied to the selected locations and updated in the program’s source directory.

For the pool, an absorbance (i.e., the percentage of global radiation absorbed by the pool water) of 60% was assumed, in accordance with the data presented by Duffie and Beckman [21]. Additionally, a solar-radiation reflection of 8% at the water surface was considered. The underlying physical models and calculation methods used in POLYSUN to determine pool heat losses—as well as the key factors influencing the energy efficiency of pool heating—are described in [28].

4. Calculation Results

4.1. Model Comparison with POLYSUN Results

Time courses of $T_p$ were determined for solar-heated pools at selected locations in Europe. In each case, a ratio of $A_c$/$A_p$ = 1 was assumed. The curves were determined using the model presented in Section 2 and the POLYSUN application. The simulation period covered the time when the greatest amount of solar radiation reaches the surface of the solar collectors, corresponding to the period when the pools are typically in use. An overview of the most important process parameter values used in the calculations is presented in Table 5.

Calculations based on our own model showed a strong dependence of the resulting curves on the heat transfer coefficient between the water surface and the environment $h$. This coefficient depends mainly on $u$, a quantity that is difficult to determine. This is because the wind speed for a given location depends on the pool’s placement, the season, the time of day, and it is also largely random. The following algorithm was used to determine $T_p$. First, results were generated using the POLYSUN application. Next, a trial value of $h$ was set, and $T_p$ were calculated using the author’s model. The sum of squares of the deviations ($SS$) between the temperatures obtained from both models was calculated over the entire season. By repeating this procedure for different values of $h$, the function curves shown in Figure 9 were generated. As can be seen, in each case, the functions have a distinct minimum. This makes it possible to identify the values of $h$ for which the agreement between the predictions of both models is the closest.

For values of $h$, corresponding to the minimum $SS$ values, time curves of $T_p$ were determined and shown in Figure 10, Figure 11 and Figure 12. The curves generated by the proposed model are based on smoothed weather data; therefore, as they do not account for diurnal variations, they appear smooth, in contrast to the curves obtained from the POLYSUN application, which are based on hourly data. Taking this into account, the model reproduces the actual process trend correctly, albeit with limited accuracy.

Figure 10 shows the time courses of $T_p$ for the climatic conditions of Thessaloniki. Additionally, a profile corresponding to low wind speed (0.5 m/s)—conventionally considered a wind-sheltered location—has been plotted. The temperatures reached under these conditions are approximately 5 K higher than those obtained under typical conditions. The shape of the time courses of $T_p$ obtained using the model presented in Section 2, and the numerical values are consistent with the literature data. Ruiz and Martinez [5] presented the results of measurements conducted in a Mediterranean climate (Alicante, Spain). For $A_c/A_p$ = 1, the maximum temperature was reached at the turn of July and August, approximately 29 °C.

Figure 11 presents the results obtained for the climatic conditions of Kraków. The temperature levels are noticeably lower than those observed for Thessaloniki. For instance, the maximum daytime temperature is 5.2 K lower, while the nighttime maximum is 4.9 K lower. Additionally, Figure 11 includes temperature curves that account for variations in $RH$, whereas in the other calculations, $RH$ was assumed to remain constant throughout the year. The analysis incorporates monthly average $RH$ values for Kraków, as shown in Figure 6. Including $RH$ variability results in slight modifications of the pool water temperature time courses; however, these differences remain relatively small.

Figure 12 presents the time courses of $T_p$ for a pool located in Stockholm. On average, temperatures are 2–3 K lower than those in Kraków. The figure also includes results for a $A_c$/$A_p$ ratio of 1.5, which leads to a noticeable increase in water temperature of approximately 2.0–2.5 K. Furthermore, Figure 12 illustrates the influence of the initial condition on $T_p$. In this case, the initial temperature was set to $T_{init}=15 °\mathrm{C}$, whereas in the remaining calculations, $T_{init}=10 °\mathrm{C}$ was assumed. As observed, this modification affects $T_p$ only over a limited period—approximately a dozen days at the beginning of the season.

Table 6. Selected results of solar pool heating system simulations in various locations.

Parameter	Thessaloniki	Kraków	Stockholm
Estimated $h$ [W/(m2K)]	8.0 ± 0.5	10.2 ± 0.5	6.4 ± 0.5
Maximum $T_p$ at day [°C]	29.5	24.3	21.4
Maximum $T_p$ at night [°C]	27.1	22.2	20.2

summarizes the key parameters determined for the analyzed locations based on the model described in Section 2. The table also shows the maximum daytime and nighttime water temperatures recorded at each location.

To compare the determined values of $h$, Figure 13 provides a graphical representation of the results. The coordinates of the plotted points correspond to the annual average $u$ at the analyzed locations and the numerically determined values of $h$, treated as control parameters. The line shown in the figure represents the empirical relationship given by Equation (20). Although the agreement between the empirical curve and the numerical results is limited, the comparable magnitude of $h$ obtained using different approaches suggests that the model captures the underlying physical processes with reasonable accuracy.

4.2. Effect of $A_c$ on $T_p$

Starting from this subsection, all the drawings considered concern Kraków. The effect of collector area on solar pool water heating is best represented by the dimensionless ratio of collector area to pool area, $A_c/A_p$. This is a key parameter in the process under consideration. Figure 14 shows the temperature curves for several values of the $A_c/A_p$ ratio, as well as for the case without solar collector heating ($A_c/A_p$ = 0). The results clearly demonstrate that increasing $A_c$ leads to a systematic rise in $T_p$ over the entire analyzed period. The differences between the curves are most evident during the summer months (June–August), when solar radiation intensity is highest. In this period, larger $A_c$ enable $T_p$ to exceed 25 °C, while in the absence of solar heating, the temperature remains at a substantially lower level. This indicates that $A_c$ not only affects peak temperatures but also extends the duration of favorable thermal conditions for pool use. It was found that, under Kraków’s conditions, installing solar collectors with a surface area half that of the pool surface area makes it possible to maintain the water temperature (during the day) above 21 °C for 100 days.

4.3. Pool Water Heat Losses to the Environment

Figure 15 illustrates the temporal variation in heat fluxes transferred between the pool water surface and the surrounding environment over the analyzed period. The dominant contributions to total heat loss are associated with $EV$ and $LW$. The $LW$ component exhibits relatively smooth and gradual seasonal variation.

In contrast, $EV$ shows pronounced variability, with a clear maximum during the warmest part of the season, indicating its strong sensitivity to temperature, humidity, and wind conditions. A distinct diurnal pattern is observed for $EV$, characterized by significant fluctuations between daytime and nighttime values. On the other hand, $LW$ remains comparatively stable over the daily cycle, with only minor variations. $H_{conv}$ is smaller than previously mentioned and exhibits significant day-to-night flux variations.

The presented pool water heat losses to the environment are consistent with data from the literature. For example, Zhao et al. [10] presented the results of a simulation conducted using the TRNSYS application for conditions in the Southern Hemisphere (Sydney). They concluded that, under the conditions of the analysis, the sum of heat losses from the water surface was almost equal to solar heat gains, resulting in only slight differences between the water and ambient temperatures. The dominant heat losses were evaporation and long-wave radiation. In contrast, the convective flux played only a minor role due to the small temperature difference between the pool water and the surrounding air.

4.4. Preventing Heat Losses from the Water’s Surface

Assuming that individual heat losses to the environment were completely eliminated, $T_p$ values were analyzed. In practice, however, such losses can only be partially reduced. The purpose of these calculations was therefore to illustrate the theoretical potential for minimizing heat losses. Nevertheless, several practical measures—such as the periodic (nighttime) use of pool covers and the construction of pools in sheltered locations—can substantially reduce this heat loss.

The application of a cover to the water surface effectively limits evaporation and reduces long-wave radiative heat losses. In the model, the complete elimination of evaporation can be simulated by setting $A=0$ in Equation (23), while suppression of $LW$ is achieved by assuming a water emissivity coefficient equal to 0 in Equation (21). For both cases, the resulting pool water temperature time courses, corresponding to the elimination of these heat-flux components, are presented in Figure 16.

As noted, the dominant mechanisms of heat loss in the pool are $EV$ and $LW$. Nevertheless, the mitigation of convective heat transfer remains of considerable importance. This effect is illustrated by the uppermost time courses in Figure 16, which were obtained by substituting $u=0$ into Equation (20). A substantial reduction in overall heat loss can be achieved by locating the pool in a wind-sheltered environment. Within the framework of the model, this condition corresponds to zero wind velocity, for which $h$ assumes a significantly lower—albeit non-zero—value, characteristic of free convection.

A reduction in $u$ yields additional benefits beyond the decrease in $H_{conv}$:

• $EV$ is also reduced, due to the analogy between heat and mass transfer, whereby the heat transfer coefficient occurs on Equation (23);
• For a wind-sheltered pool, a decrease in heat loss is observed under both daytime and nighttime conditions.

4.5. Solar System with a Heat Exchanger

Figure 17 illustrates the corresponding temperature time courses for the system with a heat exchanger. The calculations were performed for a heat exchanger with a parameter ${UA}_{HEX}=1000$ W/K and for water flow rates of 0.1 and 0.3 kg/s, corresponding to $NTU$ values of 2.4 and 0.8, respectively. An increase in the water flow rate results in a lower temperature at the inlet to the pool $T_c$. At the same time, higher flow rates reduce the temperature difference between the inlet and outlet of the heat exchanger, leading to an increase in the temperature at the collector inlet $T_p^{\prime }$. This temperature, in turn, affects the collector’s efficiency.

The temperature time courses of $T_c, T_p$, and $T_p^{\prime }$ are shown in Figure 17. The water flow rate has a strong influence on $T_c$ time courses and a less pronounced effect on $T_p^{\prime }$. However, within the analyzed range and for a constant value of ${UA}_{HEX}$, the flow rate through the heat exchanger does not significantly affect $T_p$.

The behavior of the utilizability parameter used in the model was also analyzed. As shown in Figure 17, the values of $\varphi$ remain close to unity and are nearly constant over time. Slightly higher values were observed at lower water flow rates, which can be attributed to the higher collector inlet temperature under these conditions, resulting in a reduced driving force for heat transfer in the collector. The fact that $\varphi$ remains close to unity is due to two main reasons. First, in the present model, the collector operation is limited to daytime, so the absence of solar radiation at night does not influence the value of $\varphi$. The second reason is that the analysis considered only the summer period, which means that $\varphi$ is not affected by the low heat fluxes that occur in winter.

5. Conclusions

The present study investigated the performance of a solar pool water heating system using both a simplified analytical model and numerical simulations in POLYSUN. Analysis of the calculation results allows the formation of the following conclusions:

• Solar heating with a properly designed heat exchanger yields thermal performance comparable to that achieved by direct circulation of water from the collector to the pool. This has been demonstrated for heat exchangers with $NTU$ values of 0.8 and 2.4.
• Increasing the collector area leads to higher pool water temperatures; however, the applicability of this approach is limited by economic and spatial constraints.
• In high-latitude regions (e.g., 60°), solar pool water heating alone is insufficient to ensure acceptable water temperature conditions.
• Minimizing heat losses to the environment is crucial in solar pool water heating systems. Wind speed has been identified as a key factor influencing these losses. Locating the pool in a sheltered, low-wind area reduces both convective and evaporative heat losses. Moreover, such a solution is effective continuously, both during the day and at night, unlike the intermittent use of pool covers.
• A solar heating model was developed based on dividing each day into two periods—daytime and nighttime—separated by sunrise and sunset. When combined with the concept of utilizability, this approach yields a simple model that provides an adequate representation of the real process. Although it does not capture short-term fluctuations resulting from the use of smoothed weather data, it successfully reproduces the overall temperature trends. The proposed model can be used as a practical tool for preliminary design, parametric studies, and long-term performance assessment of solar pool heating systems. In addition, it is preferable in cases where limited input data are available or when computational efficiency is required, while still providing sufficiently accurate results compared to detailed hourly weather data and other simulation tools.
• The use of the proposed model to simulate solar water heating in a pool confirmed its suitability. A comparison with the results obtained from the POLYSUN application showed good agreement under various climatic conditions in Europe.
• It should be emphasized that the conclusions presented above strictly reflect the analyzed data and conditions. Since the parametric analysis was conducted specifically for Kraków, the results cannot be directly generalized to other climatic regions without further validation.