Methodology of Determining the Intensity of Heat Exchange in a Polytunnel: A Case Study of Synergy Between the Polytunnel and a Stone Heat Accumulator

Abstract

This paper presents the results of laboratory tests on the intensity of mass and heat exchange in a polytunnel, with a focus on the synergy between the polytunnel and a stone accumulator. The subject of study was a standard polytunnel made of double polythene sheathing. In the process of selecting the appropriate working conditions for such a polytunnel, the characteristic operating parameters were modeled and verified. They were related to the process of mass and energy exchange, which takes place in regular controlled-environment agriculture (CEA). Then, experimental tests of a heat accumulator on a fixed stone bed were carried out. The experiments were carried out for various accumulator surfaces ranging from 18.7 m² to 74.8 m², which was measured perpendicularly to the heat medium. To standardize the results obtained, the analysis included the unit area of the accumulator and the unit time of the experiment. In this way, 835 heat and mass exchange events were analyzed, including 437 accumulator charging processes and 398 discharging processes from April to October, which is a standard period of polytunnel use in the Polish climate. During the tests, internal and external parameters of the process were recorded, such as temperature, relative humidity, solar radiation, wind speed and air flow speed in the accumulator system. Based on the parameters, a set of empirical relationships was developed using mathematical modeling. This provided the foundation for calculating heat gains as a result of its storage in a stone accumulator and its discharging process. The research results, including the developed dependencies, not only fill the scientific gap in the field of heat storage, but can also be used in engineering design of polytunnels supported by a heat storage system on a stone bed. In addition, the proposed methodology can be used in the study of other heat accumulators, not only in plant production facilities.

1. Introduction

Agricultural production involves high energy consumption, especially in protected cultivation like polytunnels. This requires technological innovations to reduce energy use in greenhouses as well as in processes like drying [1]. In Poland, the average heat demand for greenhouse tomato production is about 1400 MJ/m²/year [2], potentially making up 75% of direct costs [3]. In southern Europe, heating accounts for about 30% of operating costs [4]. In the Netherlands, a leader in greenhouse farming, 63% of energy use goes to heating and 23% to electricity [5]. Modern solutions such as diffusion glass, air drying, and LED lighting have helped cut natural gas use from 45 to 27.5 m³/m² per year. The authors also pointed out further ways to reduce heat consumption and CO₂ emissions.

Contemporary research on the energy performance of horticultural facilities is increasingly focused on comprehensive thermal modeling that takes into account both the material properties of tunnel claddings and integrated heat storage systems.

Mesmoudi et al. [6] conducted an analysis of greenhouse heating demand using coverings with various spectral and insulation parameters. Using simulation modeling, they demonstrated the significant impact of the covering material type on shaping the internal microclimate of the facility.

In the context of improving energy efficiency, growing attention is being paid to solutions based on the accumulation of excess heat and its subsequent recovery. The concept of using thermal energy storage systems (TESs) integrated with greenhouse heating systems—based on natural heat exchange processes—represents an innovative approach with significant application potential. Due to the high variability in atmospheric conditions over time (both daily and seasonal), as well as the dynamic nature of energy and mass flows in covering systems, the issue of effective heat storage in greenhouse conditions is the subject of intensive research in scientific centers around the world.

Experiments conducted under Mediterranean climate conditions have shown that the use of a rock bed with natural convection as a heat accumulator improved microclimatic parameters—such as an increase in air temperature and a decrease in relative humidity—compared to a reference object [7]. Sarbu and Sebarchievici [8] conducted a detailed review of available thermal energy storage technologies in the context of reducing energy consumption in heated and cooled buildings, including materials for both sensible and latent heat storage (PCM—phase change materials).

Studies by Bouadili et al. [9] confirm that the use of phase change materials for daytime heat storage and its recovery at night can significantly improve thermal conditions inside greenhouses while reducing primary energy consumption and CO₂ emissions. Similar conclusions were drawn by Singh et al. [10], who demonstrated that storage systems using rock beds are more efficient than PCM systems under experimental conditions.

Numerous research works have also analyzed the impact of heat storage on drying processes (Ayyappan et al. [11]), minimum internal temperatures (Kürklü et al. [12]), and the efficiency of plant production—for example, a 30% yield increase was observed by Bazgaou et al. [13].

In studies by Gourdo et al. [14,15], it was confirmed that the use of water-based heat accumulators not only improves microclimatic conditions but also reduces pest populations in tomato cultivation.

Pintaldi et al. [16] and Tian & Zhao [17] conducted an extensive review of heat storage designs, material properties, and TES system design methods. Chen et al. [18], in turn, focused on the efficiency of PCM material placement in a solar greenhouse, demonstrating significant benefits in terms of nighttime temperature stabilization.

The use of paraffin as a PCM material in greenhouses was analyzed by Öztürk [19], Kooli et al. [20], and Ghosal et al. [21,22], who evaluated the thermal, exergetic, and energetic efficiency of the tested systems. Bouhdjar et al. [23] identified the relationship between the amount of energy stored and recovered, which is crucial for assessing TES system efficiency under variable climate conditions.

Kurpaska & Latała [24] and Kurpaska et al. [25,26] conducted experimental research in plastic tunnels located in temperate climate conditions (Poland), comparing different types of accumulators (soil-based, water-based, PCM). These studies showed, among other things, that it is possible to meet the daily thermal demand of the facility under appropriate boundary conditions using rock- or water-filled accumulators.

Experiments by Bruch et al. [27], Dhifaoui et al. [28], and Vadiee & Martin [29,30] highlight the importance of precise modeling of convective airflows through heat storage beds for both vertical systems and those integrated with the ground. Moreover, it was shown that the use of thermal curtains, double glazing, and closed systems can reduce heat losses by up to 80%.

An integrated approach to greenhouse energy management, taking into account multi-source heating systems, automation, and control, was presented by Iddio et al. [31] and Taki et al. [32], emphasizing the important role of thermal energy storage in low-emission strategies and optimization systems supporting horticultural production.

The presented literature review indicates that the issue of heat storage for greenhouse production is widespread in scientific discourse, including the use of solid-bed storage systems. However, most of these reports refer to greenhouses with heating systems in various climate zones. The authors, however, identify a gap in the literature regarding the comprehensive modeling of thermal processes involving solid-bed storage systems in unheated plastic tunnels in temperate climates (e.g., Poland). Hence, the aim of the present study is a comprehensive analysis of thermal processes in a plastic tunnel. The analysis of such a system, considering the driving forces of heat exchange and the energy benefits from utilizing the heat stored in a rock-bed accumulator to ensure recommended environmental conditions for plant cultivation in an unheated production facility, constitutes the main subject of this work.

2. Materials and Methods

2.1. Experiment Set-Up

A series of tests (without cultivated plants) was carried out in a standard polytunnel, covered with a double 9 × 16 m PE foil to:

(a)

Determine thermal effects during the process of heat storage and discharge in a stone accumulator.

(b)

Determine heat losses through the polytunnel cover.

(c)

Determine heat gains from solar radiation.

(d)

Determine the intensity of air exchange in the ventilation process.

Ad (a) A stone accumulator for heat storage was installed in the polytunnel (Figure 1).

In the upper part of the tunnel, there is a perforated duct through which the air from this space was sucked by a fan (F) and pumped to a four-section stone accumulator. In an accumulator, the size of each section (separated by a layer of insulation) is 11 × 1.7 m (length and width, respectively) and the bed layer is 0.7 m high. The bed of the stone accumulator consists of porphyte granules (30–60 mm). In the lower part of each section, there are perforated ducts (two pipes per section) and four hoses collecting air from the accumulator, through which air is pumped into the tunnel. The experiments were carried out specifically for different accumulator surfaces ranging from 18.7 m² to 74.8 m². To standardize the results, the analysis was carried out for the accumulator area unit and the time unit of the experiment. The research was carried out in a polytunnel for 8 months (March–October), which constitutes a three-season climate research experiment (according to Köppen climate classification). Due to climate changes and statistical requirements, a total of 835 heat and mass exchange processes were analyzed, including 437 charging processes and 398 discharging processes. The selection of the charging or discharging process was made based on continuous measurements, and only those charging and discharging processes the duration of which was continuous and not shorter than 600 s were selected for further analysis. Hence, the minimum energy transferred to the battery or taken from the battery was at the level of 0.16 MJ. During the experiments, the following parameters were measured: air flow velocity (in a test section with a diameter of 300 mm, with a Mini Air64 air movement meter), air temperature (with PT 1000 resistance temperature meters) and relative air humidity (with a HD4917T meter). The accuracy of the measurement ranged from ±0.35 to ±0.55 K for temperature and ±1.5% for humidity and air velocity. All measurements were monitored and archived in a proprietary measurement system with a sampling time of 60 s. The sensors were placed at the air inlet and outlet of the stone accumulator.

To control the air flow through the accumulator bed, a proprietary algorithm was used. It takes into account air temperatures in the upper and middle section of the polytunnel, (t_top) and (t_ins), respectively, and the average temperature from five measurement points from individual sections of the accumulator. The algorithm controlled the operation of the fan and the position of the electric gate valves (EGV). The accumulator charging process was carried out when the temperature difference between the upper part of the tunnel and the average temperature of air in the accumulator bed was higher than 2 K. The discharge process was carried out when there was a difference between the temperature of the air in the accumulator and temperature in the center of the tunnel, also above 2 K. More details on the applied algorithm are included in reference [26]. If none of the conditions were met, the fan was switched off. During the experiments, a frequency converter was used in the fan motor power supply system to adjust the fan speed and to inject a diversified stream of air into the accumulator (q_air).

Table 1. Input parameters used in the experiments [33].

Specification, Unit			Values
Accumulator dimensions	X(m)		1.7
	Y(m)		0.7
	Z(m)		11
	number of sections		1 ÷ 4
Physical parameters of the bed	λs (W·m−1·K−1)		1.67
	ρs (kg·m−3)		2.55
	cacc (J·kg−1·K−1)		880
	bulk density (kg·m−3)		1.45
	porosity		0.43
Process parameters	air temperature at the top of the tunnel:	ttop (°C)	3.3 ÷ 48.7
	temperature inside the facility:	tins (°C)	3.6 ÷ 45.9
	relative humidity of air inside the facility:	RHins (-)	0.15 ÷ 0.9
	Charging process	tin_acc	11.1 ÷ 49.5
		RHin_acc	11.6 ÷ 67.2
	Discharging process	tin_acc	5.4 ÷ 21.5
		RHin_acc	15.7 ÷ 93.2
	Va (m·s−1)		1.34 ÷ 4.7
	τ (s)		557 ÷ 59,241

shows the input parameters of the experiments carried out during the accumulator charging and discharging processes together with the characteristics of the accumulator bed.

Ad (b) The process of heat loss through the sheathing was analyzed at the closed tunnel vent. This was carried out by monitoring the wind speed and direction outside the tunnel as well as the temperature inside the polytunnel and the ambient temperature. The WindLog™ meter was used to measure air movement.

Ad (c) Heat gains, which resulted from the conversion of solar radiation and increased the temperature inside the polytunnel, were also measured at the closed vent. A PT 1000 temperature meter and a WindLog™ meter were used for the measurement.

Ad (d) During the experiments, regardless of the degree of opening of the ventilation opening, the parameters of the external and internal climate of the tunnel were measured. The external climate parameters such as wind speed and direction, air humidity, and its temperature were measured using the WindLog™ weather station, while the LP PYRA 03 pyranometer was used to measure the solar radiation flux. These sensors were placed on a common mast, with wind and solar radiation measurements taken at a height of 10 m, and the remaining ones at a height of 2 m above ground level. The mast was installed 4 m in front of the greenhouse towards the west. The climate parameters inside the greenhouse, temperature and humidity, were measured with the integrated HD4917T sensor at a height of 1.5 m above the cultivation gutter at a distance of 4 m from the wall on which the vent was not installed.

For the purposes of thermal analysis, all parameters of the system (stone accumulator, interior of the polytunnel, parameters of the ambient climate) were monitored and collected by a digital measurement system. Its standard recording frequency was 120 s, and any fluctuations and changes were recorded additionally. To generalize the obtained results, average values of measured air parameters were calculated (W_avg). based on their instantaneous values, according to the following formula:

$${\mathit{W}}_{\mathit{a}\mathit{v}\mathit{g}}={\displaystyle \frac{1}{\mathit{\tau }}}{\int }_{{\mathit{\tau }}_1}^{{\mathit{\tau }}_2}\mathit{w}\left(\mathit{\tau }\right)\mathit{d}\mathit{\tau }$$

(1)

The time interval covers the duration of the experiment (start—τ₁, end—τ₂).

2.2. Theoretical Analysis

Heat balance for differential time $\mathit{d}\mathit{\tau }$ in macroscopic terms can be expressed in a general form as follows:

$$V\cdot \rho \cdot c_w{\displaystyle \frac{d{t}_{ins}}{d\tau }}=\pm q_{acc}+q_{rad}-q_{loss}-q_{vent}$$

(2)

where V—air volume in the polytunnel, m³; ρ—air density, kg·m^−3; c_w—specific heat of air, J·kg⁻¹·K⁻¹; t_ins—air temperature, °C; τ—time, s; q_rad—air stream supplied into the polytunnel by radiation, W; q_acc—heat flux from the stone accumulator, W; q_loss—heat loss flux, W; q_vent—heat flux exchanged by ventilation, W.

In the discussed system, the heat flux stored in the accumulator bed (q_{acc_charge}) and supplied to the inside of the tunnel as a result of its discharge (q_acc-disch) is described by the following formula:

$$d{q}_{acc}=V_a\cdot F\cdot \rho {\displaystyle \underset{\tau }{\overset{\tau +\Delta \tau }{\int }}\left(h_{in\_acc}-h_{out\_acc}\right)}d\tau$$

(3)

where V_a—air velocity in the measurement section, m·s⁻¹; F—cross-sectional area of the polytunnel’s test section, m²; h_{in_acc} and h_{out_acc}—enthalpy of the supplied air (h_{in_acc}) and air expelled from the accumulator (h_{out_acc}), respectively, J·kg. Air enthalpies were calculated using standard psychrometric relationships by first calculating the water vapor saturation pressure and next the current vapor pressure and water vapor concentration in the air.

The average air velocity in the duct along the measurement section was determined based on the relationship derived by Nikuradse [34].

$${\displaystyle \frac{V_a}{V_{max}}}=f\left(R_e\right)$$

(3a)

where Vmax—maximum air velocity measured in the center of the duct, m·s⁻¹; Re—Reynolds number.

Based on the maximum velocity measured with the MiniAir64 anemometer, the Reynolds number was initially calculated. For the measured flow, it varied in the range of (2.53 × 10⁴ ÷ 9.35 × 10⁴), which, after logarithmization, corresponds to the range (4.40 ÷ 4.97). Based on the calculations and the logarithmic curve developed by Nikuradse [34], the variability in f(Re) was determined, which was narrowed down to the range (0.81 ÷ 0.84). Therefore, for further calculations, f(Re) = 0.825 was adopted. Thus, the final calculations already take into account the average air velocity V_a determined in accordance with Equation (3a).

Heat flux entering through the tunnel’s sheathing (q_loss) was calculated based on the balance sheet (Equation (2)) for conditions when other streams (q_cc = q_vent = q_rad = 0), as described by the following equation:

$$V\cdot \rho \cdot c_w{\displaystyle \frac{d{t}_{ins}}{d\tau }}=U_{\mathrm{cov}}\cdot \left(t_{ins}-t_{out}\right)\cdot F_{\mathrm{cov}}$$

(4)

where U_cov—equivalent of the coefficient of heat transfer through the tunnel’s sheathing, W·m⁻²·K⁻¹; ${\mathbf{t}}_{\mathbf{i}\mathbf{n}\mathbf{s}}^{\mathsf{\tau }+∆\mathsf{\tau }}$ is the air temperature inside the tunnel at the end of the discussed time interval ($∆\mathsf{\tau }$), °C; F_cov—area of the tunnel sheathing, m²; ∆τ—time interval, s; ${\overline{\mathbf{t}}}_{\mathit{i}\mathit{n}\mathit{s}}$, ${\overline{\mathbf{t}}}_{\mathit{o}\mathit{u}\mathit{t}}$—average values of the air temperature inside (${\overline{\mathbf{t}}}_{\mathit{i}\mathit{n}\mathit{s}}$) and outside (${\overline{\mathbf{t}}}_{\mathit{o}\mathit{u}\mathit{t}}$) the tunnel, °C.

To calculate heat gains resulting from solar radiation (q_rad), a balance equation was used, which included the solar radiation conversion factor (f). The equation is as follows:

$$V\cdot \rho \cdot c_w{\displaystyle \frac{d{t}_{ins}}{d\tau }}=f\cdot \overline{R}\cdot F_{tun}+U_{\mathrm{cov}}\cdot \left({\overline{t}}_{ins}-{\overline{t}}_{out}\right)\cdot F_{\mathrm{cov}}$$

(5)

After transformation, it takes the form of an ordinary differential equation, as follows:

$$f={\displaystyle \frac{V\cdot \rho \cdot c_w\left(t_{ins}^{\tau +\Delta \tau }-t_{ins}^{\tau }\right)-U_{\mathrm{cov}}\cdot \left({\overline{t}}_{ins}-{\overline{t}}_{out}\right)\cdot F_{\mathrm{cov}}\cdot \Delta \tau }{\overline{R}\cdot F_{tun}\cdot \Delta \tau }}$$

(5a)

where F_tun—tunnel area, m²; $\overline{\mathit{R}}$ is the average value of solar radiation, W·m⁻².

The tests were carried out with the vents closed. Symbols ($¯$) above the parameters stand for their arithmetic mean values in the adopted time period $∆\mathit{\tau }$.

Ventilation stream (s_vent) was determined using the following balance equation:

$$V\cdot \rho \cdot c_w{\displaystyle \frac{d{t}_{ins}}{d\tau }}=s_{vent}\cdot \rho \cdot \left(h_{ins}-h_{out}\right)+U_{\mathrm{cov}}\cdot F_{\mathrm{cov}}\left({\overline{t}}_{ins}-{\overline{t}}_{out}\right)$$

(6)

which, after transformation, takes the form of an ordinary differential equation:

$$s_{vent}={\displaystyle \frac{V\cdot \rho \cdot c_w\left(t_{ins}^{\tau +\Delta \tau }-t_{ins}^{\tau }\right)-U_{\mathrm{cov}}\cdot F_{\mathrm{cov}}\cdot \Delta \tau }{\left({\overline{h}}_{ins}-{\overline{h}}_{out}\right)\cdot \rho \cdot \Delta \tau }}$$

(6a)

where ${\overline{\mathit{h}}}_{\mathbf{i}\mathbf{n}\mathbf{s}}$, ${\overline{\mathbf{h}}}_{\mathbf{o}\mathbf{u}\mathbf{t}}$—average value of air enthalpy inside the polytunnel (${\overline{\mathit{h}}}_{\mathbf{i}\mathbf{n}\mathbf{s}}$) and of ambient air ($\overline{\mathbf{h}}$_out), J·kg⁻¹. The tests were carried out at night with different opening angles of the vents. Figure 2 shows the modeled empirical system.

For the calculated values of these parameters, an equation was generated that included the relation between the dependent and independent variables. Using the non-linear estimation procedure with the quasi-Newton method, at the convergence coefficient level of 0.001, the power model was established based on the largest value of the determination coefficient.

The mean root square error was used to compare measured and calculated values:

$$\mathsf{\sigma }={\left[{\sum }_{i=1}^n{\displaystyle \frac{{\left(u_{calc,i}-u_{meas,i}\right)}^2}{n}}\right]}^{0.5}$$

(7)

where u_calc, u_meas—the values from calculations and measurements, respectively, and n—number of comparisons.

2.3. Heat Demand Simulation

Based on the determined heat fluxes, the total heat demand (dQ_h) for any tunnel, at night, can be expressed as follows:

$$\mathit{d}{\mathit{Q}}_{\mathit{h}}=\left(\mathit{d}{\mathit{q}}_{\mathit{a}\mathit{c}\mathit{c}}-\mathit{d}{\mathit{q}}_{\mathit{v}\mathit{e}\mathit{n}\mathit{t}}-\mathit{d}{\mathit{q}}_{\mathit{l}\mathit{o}\mathit{s}\mathit{s}}\right)\mathit{d}\mathit{\tau }$$

(8)

In the simulation calculations, the adopted time interval (d$\mathsf{\tau }$) is 1 h. The parameters of the ambient air consisted of long-term values of temperature, air humidity and wind speed (Figure 3).

The presented research algorithm indicates that the entire analysis was conducted in multiple stages. First, heat losses through the enclosure were determined by calculating the actual U_cov value. Then, using the obtained heat transfer coefficient (U_cov), solar energy gains—the greenhouse effect—were determined by calculating the coefficient f. These studies and analyses were conducted with the vents closed. Once the parameters (U_cov) and the conversion coefficient of solar radiation into heat (f) were determined, the ventilation intensity was calculated for different air inlet surface areas.

2.4. Verification of the Developed Dependencies in a Facility with Cultivated Plants

In a real facility where plants are cultivated, the formal record of the heat balance will change—dependency (2)—and the mass balance should be taken into account.

For the description of the thermal issues in question, based on reference [35], the following was written:

$${\displaystyle \frac{d{t}_{ins}}{d\tau }}=\frac{1}{V·\rho ·c_w}(\pm q_{acc}+q_{rad}-q_{loss}-q_{vent}-q_{TR}-q_{vent}-q_{FIR}-q_{cool})$$

(9)

If the heat gains from the plant lighting system (q_lamp) are omitted from the balance and heat losses due to the transformation of sensible heat into latent heat in the process of transpiration (λ·q_TR) and evaporation of water from the building’s cover (q_con) are taken into account, the heat losses in the near infrared range (q_FIR) and from cooling devices (q_cool) will be negligible; then, this equation is simplified to the following form:

$${\displaystyle \frac{d{t}_{ins}}{d\tau }}={\displaystyle \frac{1}{V·\rho ·c_w}}(\pm q_{acc}+q_{rad}-q_{loss}-q_{vent}-q_{TR})$$

(9a)

To calculate the transpiration flux, the model developed by Stanghellini [36] was used, which is often used by other researchers due to the physical parameters of the microclimate included in it. Hence, transpiration was described by the relationship in the following form:

$$q_{TR}={\displaystyle \frac{s\left(R-G\right)+\left(\rho \cdot c_w\frac{2\cdot LAI\cdot \left(p_s-p_a\right)}{r_r}\right)}{s+\gamma \cdot \left(1+\frac{r_i}{r_a}\right)}}$$

(9a1)

where the aerodynamic resistance to mass transfer (r_a) depends on the air velocity inside the object (u) and is defined by the following empirical formula:

$$r_a={\displaystyle \frac{245}{\left(0.54\cdot u\right)+1}}$$

(9a2)

The internal resistance to mass transfer (r_i) depends on the stomatal resistance (r_l) and was calculated using the following formula:

$$r_i=r_l\cdot \left(1+{\displaystyle \frac{1}{\exp \left(0.05\cdot \left(R-50\right)\right)}}\right)$$

(9a3)

The symbol r_r in Equation (9a2) denotes the radiative resistance to mass transfer, which is described by the following relationship:

$$r_r={\displaystyle \frac{\rho \cdot c_w}{4\cdot \sigma \cdot {\left(T+273.15\right)}^3}}$$

(9a4)

where s—slope of saturated vapor pressure curve, Pa·K⁻¹; p_s, p_a—vapor pressure for saturated (p_s) and actual (p_a) temperature, Pa; R—net radiation at the crop surface. W·m⁻²; G—soil heat flux density, W·m⁻² (this flow was omitted in the calculations); LAI—leaf area index, m²·m⁻²; γ—thermodynamic psychrometric constant, W·m⁻²; u—wind speed, m·s⁻¹; σ is the Stefan–Boltzmann constant, σ = 5.67·10⁻⁸ W·m⁻²·K⁻⁴.

Finally, the individual components of the heat balance are described by the following relationships:

-

heat from accumulator: $d{q}_{acc}=V_a\cdot F\cdot \rho {\displaystyle \underset{\tau }{\overset{\tau +\Delta \tau }{\int }}\left(h_{in\_acc}-h_{out\_acc}\right)}d\tau$

-

heat from solar radiation: $d{q}_{rad}={\int }_{\tau }^{\tau +\mathsf{\Delta }\tau }f·\mathrm{R}·F_{tun}d\tau$

-

heat exchanged through the object’s cover: $d{q}_{loss}={\int }_{\tau }^{\tau +\mathsf{\Delta }\tau }{U}_{cov}·F_{cov}·\left(t_{ins}-t_{out}\right)d\tau$

-

heat exchanged through ventilation: $d{q}_{vent}={\int }_{\tau }^{\tau +\mathsf{\Delta }\tau }{m}_{vent}·\rho ·\left(h_{ins}-h_{out}\right)d\tau$

-

heat used in the transpiration process: $d{q}_{TR}={\int }_{\tau }^{\tau +\Delta \tau }{q}_{TR}d\tau$

In turn, based on the study [33], the mass balance takes the following form:

$${\displaystyle \frac{d{m}_{ins}}{d\tau }}=\pm m_{acc}+m_{TR}+m_{EVP}+m_{cool}-m_{vent}-m_{cond}-m_{dehum}$$

(10)

After introducing simplifications in the form of omitting the mass of water in the process of evaporation (m_EVP) from the top layer of the substrate (cultivation on bales tightly covered with foil), a lack of devices for active temperature reduction as a result of using cooling mats (m_cool), omitting the mass of water in the process of condensation on the facility cover (m_cond), and not taking into account the air drying devices in the facility (m_dehum), this equation is finally simplified to the following form:

$${\displaystyle \frac{d{m}_{ins}}{d\tau }}=\pm m_{acc}+m_{TR}-m_{vent}$$

(10a)

Mass exchange during air flow through the accumulator (m_acc) is a result of the processes of evaporation/condensation of the water vapor contained in this air on the surface of the accumulator bed. If the water vapor concentration in the inlet air (C_{in_acc}) is higher than the water vapor content in the outlet air (C_{out_acc}), then its condensation in the bed will occur; otherwise, the evaporation process will occur. The mass flows can be calculated from the following relationship:

$$d{m}_{acc}=V_a\cdot F{\displaystyle \underset{{\tau }_1}{\overset{{\tau }_1+\Delta \tau }{\int }}\left(C_{in\_acc}-C_{out\_acc}\right)}d\tau$$

(10a1)

The mass flux released by the cultivated plants in the form of water vapor (m_TR) is calculated from Equation (7) by taking into account the heat of phase change (L), as follows:

$$m_{TR}={\displaystyle \frac{q_{TR}}{L}}$$

(10a2)

where L = 2.45·10⁶ J·kg⁻¹.

The mass flow rate discharged in the ventilation process is described by the relationship:

$$d{q}_{vent}={\int }_{\tau }^{\tau +\Delta \tau }{s}_{vent}·\rho ·\left(C_{ins}-C_{out}\right)d\tau$$

(10a3)

where C is the water vapor concentration in the tunnel air, g·m⁻³.

Standard psychrometric relationships are used to calculate the air parameters: slope of saturated vapor pressure curve, vapor pressure for saturated and actual temperature, and water vapor concentration in the tunnel.

3. Results and Discussion

This research was carried out in Krakow, at 19°57′ E and 50°03′ N. Figure 4 shows unit amounts of stored heat in the accumulator as a function of decision variables. The calculations were carried out for the unit of charging time (i.e., 1 h) and the unit area of the accumulator. The decision variables were the stream of supplied air and thermal conditions during the charging process.

As observed, the amount of stored heat tends to increase for the analyzed independent variables. In the experiments conducted, the range of heat was from 0.03 to 0.8 MJ·h⁻¹·m². Figure 5 shows the amount of heat supplied into the polytunnel when the accumulator is discharging.

The amount of heat supplied to the interior of the tunnel during the accumulator discharge process ranges from 0.04 to 0.86 MJ·h⁻¹·m², showing an upward trend in the function of the independent variables adopted for the analysis.

Using the non-linear estimation procedure, equations were developed to determine the unit amount of heat stored in the accumulator (i.e., the charging process) and the unit amount of heat delivered to the tunnel in the process of its discharging. The following were assumed as independent variables: hourly amount of supplied air (Q_and), the initial temperature of the accumulator (t₀), and the temperature of the supplied air (t_{in_acc}). The equations are as follows:

(a)

For the charging process:

$${\mathrm{Q}}_{\mathrm{a}\mathrm{c}\mathrm{c}\_\mathrm{c}\mathrm{h}}=0.0018{\mathrm{Q}}_{\mathrm{a}}^{1.44}+15.89{\mathrm{t}}_0^{-0.024}-16.5{\mathrm{t}}_{{\mathrm{i}\mathrm{n}}_{\mathrm{a}\mathrm{c}\mathrm{c}}}^{-0.0321}+0.13;{\mathrm{R}}^2=0.92$$

(11a)

For application: 0.76 ≤ Q_a ≤ 73.8·m⁻²·h⁻¹, 2 ≤ t₀ ≤ 31.6 K; 11.1 ≤ t_{in_acc} ≤ 44 °C.

(b)

For the discharging process:

$${\mathrm{Q}}_{\mathrm{a}\mathrm{c}\mathrm{c}\_\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{h}}=\left|-0.91{\mathrm{Q}}_{\mathrm{a}}^{-0.89}+0.27{\mathrm{t}}_0^{0.67}-0.155{\mathrm{t}}_{\mathrm{i}\mathrm{n}\_\mathrm{a}\mathrm{c}\mathrm{c}}+0.065\right|;{\mathrm{R}}^2=0.92$$

(11b)

For application: 8.36 ≤ Qa ≤ 42.16 l·m⁻²·h⁻¹, 8.3 ≤ t₀ ≤ 37.1 °C; 5.4 ≤ t_{in_acc} ≤ 21.5 °C.

In determining the remaining quantities (U_cov, f and s_vent), a procedure for assessing measurement errors was carried out. The maximum errors were calculated using the total differential method, assuming the additivity of errors from individual measuring devices. The values of these errors (δ) are provided accordingly in the analysis of these quantities.

Figure 6 shows comparisons between the calculated and measured heat values along with the calculated approximation error (σ).

The tests involving the determination of the heat loss coefficient through the building sheathing were carried out with closed vents. Figure 7 shows the variability in the coefficient U_cov as a function of the variables of the conducted experiments.

The graphs of the coefficient U_cov, in the form of a function of the wind direction (Figure 7a), demonstrate that this influence is imperceptible. The analysis conducted showed that at the significance coefficient α=0.05, there are no grounds for rejecting the hypothesis of a statistically significant differentiation in the value of the analyzed coefficient depending on the direction of the wind. In turn, the analysis (Figure 7b) shows that with the increase in wind speed and temperature difference, the value of the coefficient U_cov increases. In the examined range of independent variables, the value of the coefficient U_cov ranges from 0.034 to 3.9 W·m^−2·K⁻¹.

Based on the obtained results, a correlation equation was proposed to calculate the value of the analyzed coefficient, which takes the following form:

$${\mathrm{U}}_{\mathrm{c}\mathrm{o}\mathrm{v}}=0.36{\mathrm{V}}_{\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{d}}+6.3\mathsf{\Delta }{\mathrm{t}}^{0.17}-5.36{\mathrm{R}}^2=0.85;{\mathsf{\delta }}_{\mathrm{Ucov}}=0.137\mathrm{W}·{\mathrm{m}}^{-2}·{\mathrm{K}}^{-1}$$

(12)

For application: 0 ≤ V_wind ≤ 3.3 m·s⁻¹; 0.5 ≤$\mathsf{\Delta }$t ≤ 4.85 K.

Figure 8 presents a comparison between the calculated and measured value of the coefficient U_cov along with the calculated error (σ).

The solar-radiation-to-heat conversion coefficient as a function of ambient temperature and radiation intensity is shown in Figure 9a. Using a similar procedure, a correlation equation was determined, and the results of comparing the measured and approximated values by the proposed model are shown in Figure 9b.

The conversion factor f ranges from 0.27 to 1.06, with an average value of 0.62. The correlation equation, which allows us to calculate the value of the analyzed coefficient, takes the following form:

$$f=0.03t_{out}+0.73{\mathrm{R}}_{out}^{-0.17}+0.06{\mathrm{R}}^2=0.9;{\mathsf{\delta }}_{\mathit{f}}=0.00713\left[\text{-}\right]$$

(13)

For application: −2.3 ≤ t_out ≤ 23.5 °C; 5 ≤ R_out ≤ 785 W·m⁻².

The intensity of the ventilation process was analyzed at a variable gap area. Figure 10 shows the intensity of the external air flow at the extreme values of the vents (1.6 and 14.4 m²).

As can be seen, with increasing wind speed and air temperature difference, the intensity of air exchange increases, regardless of the size of the vent (F_w).

Based on the obtained results, a correlation equation was proposed to calculate the amount of external air inflow, which takes the following form:

$$s_{vent}=0.17F_w^{0.,5}+0.15V_{wind}^{1.,5}+0.03\mathsf{\Delta }{t}^{0.55}-0.18{\mathrm{R}}^2=0.89;{\mathsf{\delta }}_{\mathit{mvent}}=0.0422{\mathrm{m}}^3·{\mathrm{s}}^{-1}$$

(14)

For application: 1.6 ≤ F_w ≤ 14.4 m²; 0.006 ≤ V_wind ≤ 2.53 m·s⁻¹; 0.3 ≤ $\Delta$t ≤ 26.9 K.

A graphical comparison of air streams obtained from calculations and measurements, including the calculated error (σ) is shown in Figure 11.

3.1. Verification Air Parameters in the Production Object

Verification tests of the object with a 78 m² accumulator were carried out in a 135 m² polytunnel housing a tomato cultivation. During the tests, the plant density was 1.7 plants m⁻², and the LAI value was 1.6 m² m⁻². The same measuring equipment was used in the tunnel (as described in the Experiment Set-Up subsection). One day (25.06) was selected for the verification procedure, during which the average values of the surrounding climate were equal: air temperature of 17.7 °C, relative air humidity of 73.9%, solar radiation intensity of 284 W m⁻² (the sum of solar radiation energy was 5.1 kWhr), and wind speed of 3.45 m s⁻¹. In turn, inside the object, the average values of the measured parameters were as follows: temperature inside the object of 20.5 °C, and relative humidity of 73.5%. The total discharge time of the stone accumulator was 13.2 h, and the charging time was 8.35 h.

The calculated values (calculated per unit area of the tunnel) were as follows: transpiration of 6.46 kg, heat supplied from the accumulator of 0.26 MJ, and heat stored in the accumulator of 0.18 MJ.

Figure 12 shows the daily course of the measured and calculated (from Equation (11a)) temperature inside the object (Figure 12a) together with the determination of their compliance (Figure 12b). It can be seen that the parameters determined in the experimental part (U_cov, f and s_vent) together with the specified values (q_acc) satisfactorily describe the temperature changes in the analyzed object.

Similar time courses and ranges of changes in water vapor content in the air inside the tested object are shown in Figure 13.

The water vapor concentration in the greenhouse air was used for comparison because this value, in addition to the vapor pressure deficit, is used in controlling the ventilation process. It can be seen that (similarly to temperature) this comparison is characterized by high compliance.

Summing up these comparisons, it can be clearly stated that the functional dependencies obtained in the laboratory part, which allow for calculating the parameters of the heat and mass exchange process in the analyzed object (Equations (11a)–(14)), correctly describe these issues. The forms of these equations (based on easily measurable parameters) can therefore also be transferred to other objects in which plants are grown.

This work is a complementary study extending the knowledge of heat and mass exchange processes in a foil tunnel. The fundamental work of Bot [37] concerns multi-bay production greenhouses, which also comprehensively addresses the issues of heat and mass exchange based on the basic laws of physics. The relationships developed in the submitted work can be used in both short-term and long-term models of energy strategy management in search of their optimal solutions.

3.2. Forecast of Heat Consumption by the Production Object

Comparative studies were carried out in two objects of the same area, with and without a stone accumulator. More details can be found in [38]. Figure 14 shows the heat consumption in the actual test object and in an object, for which a forecast of energy consumption savings was presented, at the climatic conditions occurring during the tests.

The presented data show that for the entire cultivation season, the total savings in heat consumption, calculated based on the conducted tests, amount to 168.8 MJ·m², but 171.6 MJ·m² in the forecasts. A MJ·m² comparison of daily values shows that the value of the estimation error is s = 0.27 MJ·m²·day⁻¹.

To simulate heat savings, long-term (over 20 years) hourly average values of air temperature and humidity, as well as wind speed, were used for the object located in Krakow (19°57′ E and 50°03′ N). A polytunnel with a commercial usable area of 270 m² with a double sheathing was adopted as the test object. The area of the sheath is 580 m², the volume of air inside the object is 1016 m³, and the accumulator area is 150 m². The stream of supplied air is 30 l·m³·h. Furthermore, it was assumed that the initial temperature of the accumulator would be 19 °C, and that the accumulator would be discharged only during the night. The calculations take into account that the hourly ventilation intensity during the night would be 0.5 cubic volume. It was also assumed that the vapor pressure deficit inside the facility would be at the recommended value of 0.8 kPa [39].

It was assumed that the night air temperature would be 16 °C, and daily 21 °C, i.e., the values recommended for tomato cultivation [40]. The analysis was carried out for April–October. Due to limitations in the volume of work, only the average monthly values of the adopted parameters are listed in

Table 2. Average daily values of the parameters of the ambient climate.

Specification	Month
	April	May	June	Jule	August	September	October
Temperature, °C	8.2	13.4	18.2	17.5	17.5	13.7	9.3
RH, [-]	74.3	70.8	75.6	74.9	77.6	81.3	83.4
Wind, [m·s−1]	2.4	2.2	2.4	2	1.8	2.2	2.3

.

Figure 15 graphically shows the daily amounts of heat in the considered period, including the use of the heat accumulator in the system.

The presented data (Figure 15a) show that the highest value of the heat demand is related to the transfer of heat by penetration through the sheathing of the object. The daily heat demand ranges from 1.6 (June) to 12.8 (April) MJ·m⁻²·day⁻¹. In turn, the average heat losses through ventilation in April, May, September, and October account for approximately 1.5% of transmission losses. On the other hand, at the adopted parameters of the inside air and the long-term averages outside the building, in June, July and August, a low heat gain of heat exchanged through ventilation was observed.

With such assumptions, in the case of heat storage in a stone accumulator, monthly savings in heat demand (Figure 15b) range from 0.7 (June) to approximately MJ·m⁻²·day⁻¹ (October). Taking into account the period in which the polytunnel was used (April–October), owing to the installation of a stone accumulator, the total savings in heat demand amount to approximately MJ·m⁻².

4. Conclusions

This study focused on the analysis of heat gains from a rock heat storage system and its discharge in a plastic greenhouse tunnel. Heat fluxes were determined and temperature changes were described using differential equations. The heat transfer coefficient through the tunnel covering was established, with findings showing that wind direction does not significantly affect its value. The variability in the radiation-to-heat conversion coefficient and the ventilation intensity depending on the size of the ventilation opening were also investigated. Based on the results obtained in a 144 m² foil tunnel, equations were formulated, enabling calculations under different climatic conditions. Based on the conducted research, correlation equations that can be used to estimate heat fluxes in greenhouse facilities were developed. By including standard parameters related to the surrounding climate and construction and operating parameters of the facilities, the knowledge of the analyzed issues was enriched. A comparison of the results obtained from the simulations and measurements showed high agreement, thus confirming the agreement of the adopted methods. Based on the conducted analyses, experimental data, and simulations, the following detailed conclusions can be drawn:

• The amount of heat stored in the accumulator during charging ranged from 0.03 to 0.8 MJ·h⁻¹·m⁻², and during discharging, it ranged from 0.04 to 0.86 MJ·h⁻¹·m⁻². The developed correlation models (R² = 0.92) allow for accurate predictions based on temperature and airflow volume.
• The heat loss coefficient (U_cov) varied between 0.034 and 3.9 W·m⁻²·K⁻¹, increasing with wind speed and temperature difference. The correlation equation achieved a fit of R² = 0.85.
• The radiation conversion coefficient (f) ranged from 0.27 to 1.06 (average 0.62), with a mathematical model fit of R² = 0.90 and error δ_f = 0.00713.
• The ventilation intensity (s_vent) for openings between 1.6 and 14.4 m² ranged from 0.03 to approximately 0.6 m³·s⁻¹. The model describing ventilation achieved R² = 0.89, with an error of δ = 0.0422 m³·s⁻¹.
• The seasonal heat savings amounted to 168.8 MJ·m², compared to a projected value of 171.6 MJ·m².

The results are universal, and the derived equations can be applied in various locations and conditions. No previous studies have considered this type of heat accumulator. Future research should focus on optimizing thermal supply parameters while considering energy costs in an autonomous system.