# Numerical model of intermittent solar hot water production

Julien Gambade, Hervé Noël, Patrick Glouannec, Anthony Magueresse

► To cite this version:

Julien Gambade, Hervé Noël, Patrick Glouannec, Anthony Magueresse. Numerical model of intermittent solar hot water production. Renewable Energy, 2023, 218, pp.119368. 10.1016/j.renene.2023.119368. hal-05340061

HAL Id: hal-05340061

https://hal.univ-brest.fr/hal-05340061v1

Submitted on 31 Oct 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Numerical model of intermittent solar hot water production

Julien Gambade¹, Hervé Noël¹*, Patrick Glouannec¹, Anthony Magueresse¹

¹Univ. Bretagne Sud, UMR CNRS 6027, IRDL, F-56100 Lorient, France

*(corresponding author: herve.noel@univ-ubs.fr)

## Abstract:

A dynamic numerical model of a solar thermal installation is developed. The aim of the installation is to supply hot water (over 80°C) twice a day to a veal calf farm. It consists of 16 evacuated tubes solar water heaters associated to an unpressurised tank using the thermosiphon effect. The particularity of the installation is its layout, which alternates between solar collectors connected in series and those connected in parallel. This arrangement allows both pre-heating and storage of hot water. The solar installation contributed by 60% to the energy demand during a production cycle.

The model was developed on TRNSYS 18 based on existing types adapted to the configuration of the system. The modelling of the solar installation was achieved in order to simulate the energy supplied. The calibration and validation of the model are presented in this paper. The first step is to carry out a sensitivity analysis using the Morris screening method to identify the most influential model parameters. The second step is an optimization method used to minimize the difference between the simulated and measured energy to determine the influent parameters values.

Finally, the model is compared with measurements during two periods with different meteorological conditions.

## Keywords :

Solar thermal installation; Hot water production; Dynamic numerical model; Sensitivity analysis; Model calibration.
# Numerical model of intermittent solar hot water production

Julien Gambade¹, Hervé Noël¹,*, Patrick Glouannec¹, Anthony Magueresse¹

¹Univ. Bretagne Sud, UMR CNRS 6027, IRDL, F-56100 Lorient, France

*(corresponding author: herve.noel@univ-ubs.fr)

## 1. Introduction

The evolution of mankind has made it necessary to control energy, and this continues to be a critical factor in society's impact on sustainable socio-economic development. Global energy consumption has increased steadily, and this trend is anticipated to continue for at least the coming decades.

Agriculture is a sector that depends greatly on sources of non-renewable energy, especially fossil fuels, which are used for operating machinery and equipment, as well as heating or cooling buildings, providing lighting on farms [1] and heating water.

To meet this need, we should consider the use of renewable energy and more particularly solar thermal energy. Several technologies have been studied to meet the requirements of various industrial and agricultural activities [2]. Wallerand et al. attempted to determine the optimal design and implementation of solar technologies that could contribute to the needs of dairy processes. In addition, hybrid PV-thermal systems have been considered to supply heat and power to dairy farms [3]. Quijiera et al. also studied the integration of a solar thermal system to support a dairy farm [4]. These authors recommend evacuated tube collectors for use under unfavourable climatic conditions, since this type of system is more efficient when receiving predominantly diffuse solar radiation.

A thermo-economic analysis was carried out by Sokhansefat et al. to compare a flat plate collector and an evacuated tube collector under cold climate conditions [5]. For each type of solar water heater, three collectors were connected in series and modelled using Trnsys software to study the output temperatures and energy production. The results show that evacuated tube collectors perform better than flat plate collectors, while the efficiencies of flat plate collectors decrease more significantly under adverse weather conditions. In fact, under the studied cold climate conditions, tube collectors provide 30% more energy than flat plate collectors. Both models are economically viable under such a climate, with the evacuated tube system showing an advantage with a payback period of 6 years.

In this context, the European Union aims to promote the use of solar thermal energy in the agricultural sector in North West Europe where the global annual solar radiation is rather low (GHI<4287.6 MJ/m²) [6]. The solar technology considered in this study consists of evacuated tubes connected to a storage tank located directly above them. Each tube consists of two concentric tubes welded together at their upper ends.

The inner tube is coated with an absorbing layer to collect the solar radiation and transfer it to the water circulating inside. The outer tube is for insulation and limitation of long-wave radiation losses from the absorber. A vacuum is created in the annular space, drastically reducing the conduction and convection losses. Due to the thermosiphon effect, water moves from the storage tank to the glass tubes. The tube captures solar energy, converting it into heat which is then transferred to the water. As the water is heated, its density decreases, which allows it to rise to the top of the storage tank. These solar water heaters are simple in design and less expensive [7]. Their performance has been examined and many studies have aimed to improve their thermal efficiency. Bracamonte et al. showed that the collector tilt-angle has a significant influence on the daily solar energy gain, as well as on the natural circulation flow rate and stratification [8]. Kumar and Mylsamy are
48 currently investigating the enhancement of the thermal storage capacity of solar water heaters
49 using nano-embedded phase change materials [9].

50 This study deals with the modelling of a solar thermal installation for a veal farm with
51 high energy consumption due to the need to produce large volumes of water at over 80°C
52 twice a day. The main particularity of this installation is the serial and parallel arrangement of
53 water-in-glass evacuated tube solar water heaters associated to a storage tank. The numerical
54 model developed here is based on the dynamic thermal simulation software Trnsys, which is
55 then calibrated to estimate the hot water production as accurately as possible. The aims are to
56 help the sizing and to contribute to the spread of similar solar installations across Europe. In
57 this study, we adopt a three-step process to improve the response of the numerical model used
58 to simulate the solar pilot plant.

59 The first step is to use sensitivity analysis to determine the parameters that have the
60 greatest impact on calculating the energy supplied by the solar installation. The sensitivity
61 study allows us to select those parameters having an impact, while eliminating others with a
62 negligible influence on the model. In a second step, based on the conclusions of the
63 sensitivity study, we identify the parameters with the greatest impact using an optimisation
64 algorithm. The third step consists of model validation for different periods and conditions
65 using in-situ experimental measurements. Finally, we study the effects of various parameters
66 on solar water heater performance.

## 2. Materials and methods

### 2.1. *Experimental installation and instrumentation*

69 The solar water heaters modelled in this study are located in France (Lat. 48.2° N and
70 Long. 0.50° W). Sixteen solar collectors facing south are divided into two rows, supplying
71 hot water for a veal farm by means of a pump (Figure 1). Each solar water heater is made up
72 of 30 evacuated tubes titled at 45° and associated to a 241-litre storage tank open to
73 atmospheric pressure. A thermosiphon flow rate occurs between the tubes and the tank when
74 they are exposed to solar radiation. The first row consists of solar water heaters arranged in
75 series, while in the second row they are arranged in parallel. Unlike the in-series collectors,
76 the same pipe is used for the withdrawn and the subsequent filling of the parallel solar water
77 heaters. Hydraulic conflicts are avoided by the very slow filling rate (1 to 14 times the
78 withdrawal rate). The collectors in series are always filled with water and thermal
79 stratification can be observed, while, in the case of the collectors in parallel, the tanks are
80 partially emptied at each withdrawal. The number of parallel solar water heaters is chosen
81 according to the maximum daily water demand of the process and the designer recommends
82 the same number of collectors in series. A previous case study was investigated on a similar
83 solar installation [10].

84 To improve the energetic performance of the installation and to limit the risk of material
85 degradation due to high temperatures, a ground dissipation system was set up consisting of a
86 160 m circuit of serpentine cross-linked polyethylene pipes (internal and external diameters
87 of 23 and 25mm) buried in sand at two levels, 60 and 80 cm deep. The polyethylene ducts
88 used for the ground discharge can withstand a continuous operating temperature of 82°C and
89 95°C for short periods. Circulation through this dissipation system is triggered in the event of
90 overheating in the parallel solar water heaters. This occurs during prolonged periods of strong
91 sunlight when demand is low or even zero during fallowing. The stagnation temperature that
92 triggers the use of the ground discharge system is set at 85°C during operation and at 65°C
93 during the sanitary vacuum period (between two breeding cycles). This system aims to extend
94 the life of the installation. Water from the mains is introduced during the filling phases,
running in the same coil as the collectors, and is thus heated owing to the energy accumulated in the ground.

The instrumentation consists of flow meters to measure the flow at inlet ($\dot{m}_{in}$), at outlet ($\dot{m}_{out}$) and through the soil ($\dot{m}_{soil}$). Platinum probes integrated into the flow meters provide the temperature of the circulating water. Two temperature sensors were inserted through pockets at the top of the first tank in series and the last tank in parallel to measure their temperature ($T_p$ and $T_d$, respectively).

A meteorological station was set up on site to determine global horizontal irradiance (GHI), ambient temperature, relative humidity, wind speed and direction. All the climatic data is available online with a time step of 5 minutes. The instrumentation was selected for its portability and cost so that it could be reused in other similar installations. This type of instrumentation is provided by about one hundred weather stations in France, including the one installed at the pilot site. Figure 1 shows the instrumentation implemented in this study.

<figure><img src="image_2.png" /><figcaption>Figure 1: Solar installation instrumentation</figcaption></figure>

The contribution of the solar system to the daily energy demand is estimated with the flow meter located at the outlet and the temperature sensors. The daily energy supplied by the solar collectors $Q_{sup}$ is calculated by:

$$Q_{sup} = \int_{0}^{24h} \dot{m}_{out} \cdot c_p \cdot (T_{out} - T_{in}) dt \quad (1)$$

Daily global irradiation is calculated with equation (2):

$$H = \int_{0}^{24h} G \cdot dt \quad (2)$$

Table 1 reports the uncertainties of the measuring instruments associated with each physical quantity:

<table><caption>Table 1: Accuracy of the sensors</caption><thead><tr><th></th><th>Flow rate</th><th>Temperature</th><th>GHI</th><th>Wind speed</th><th>Wind direction</th></tr></thead><tbody><tr><th>Reference</th><td>US ECHO II</td><td>Sonde FKP6</td><td></td><td>Station Vantage</td><td></td></tr></tbody></table>
<table><thead><tr><th></th><th colspan="5">Pro2 Plus</th></tr></thead><tbody><tr><td>Accuracy</td><td>±2 %</td><td>±0.5 °C</td><td>±5 %</td><td>±3 km/h</td><td>±3°</td></tr></tbody></table>

## 2.2. Numerical model

### 2.2.1. Solar water heater model

The solar water heater model presented here is based on previous work by Budihardjo et al [11] [12] and is a combination of three different components referred to as: "Evacuated tube solar collector", "Horizontal cylindrical tank" and "Equation component".

The fraction of the irradiance converted into energy by the evacuated tubes is determined using the Type 71 The tank is divided into several isothermal nodes to model the thermal stratification within the solar water heaters in series. However, the specificity of parallel collectors requires the use of a modified Type 39, with a variable volume and homogeneous temperature tank model. Finally, the equation component is used to calculate the thermosiphon flow rate between the tubes and the tank according to equation (3):

$$ \dot{m} = \frac{b_0 \left[ \frac{Nu_d \cdot Gr_d}{Pr} \cos\beta \left(\frac{L}{d}\right)^{1.2} \right]^{b_1}}{4} \pi dv \rho \quad (3) $$

With:

- $b_0, b_1$ thermosiphon flow correlation constants.

- Nusselt ($Nu_d$), Grashof ($Gr_d$) and Prandtl ($Pr$) numbers.

- the tilt angle $\beta$ [°].

- $L$ and $d$ length and diameter of the tubes [m].

- the kinematic viscosity $\nu$ [$m^2 \cdot s^{-1}$]

- the density $\rho$ [kg \cdot m$^{-3}$].

Figure 2 illustrates the interactions of the flow between the evacuated tubes and the tank. The temperature at the inlet to the collector is measured at the node at the level of the tube opening. The efficiency equation is used along with the irradiance and ambient temperature to calculate the useful energy produced by the collector. The efficiency equation is a function of the average temperature of the collector. The natural circulation flow rate through the collector is calculated using correlations between the Reynolds, Nusselt and Grashof numbers and the energy gain per tube. The temperature rise through the collector is calculated using the predicted flow rate and the useful energy gain. The outlet temperature is then compared to the estimated value and iterations are performed until an energy balance is achieved in the collector. At the end of each time step, the temperature distribution in the tank is updated.

Figure 2 also highlights the characteristic coefficients of the studied solar collector, which are the subject of the sensitivity analysis presented in this study.
<figure><img src="image_3.png" /><figcaption>Figure 2: Flow chart of the numerical model highlighting key parameters</figcaption></figure>

With:

- $\eta_0$ [-], $a_l$ [kJ.hr$^{-1}$.m$^{-2}$.K$^{-1}$] and $a_2$ [kJ.hr$^{-1}$.m$^{-2}$.K$^{-2}$] the efficiency coefficients.

- the average temperature of the solar collector $T_m$ [°C].

- the gross area A [m$^2$].

- the global tilted irradiance $G_t$ [kJ.hr$^{-1}$.m$^{-2}$].

- the thermal losses $Q_{loss}$ [kJ.hr$^{-1}$].

- the external surface $S_{ext}$ [m$^2$].

- the heat loss coefficient $U_{edge}$ [kJ.hr$^{-1}$.m$^{-2}$.K$^{-1}$].

### 2.2.2. Solar water heater installation

The environmental constraints and operational requirements are the inputs to the model, taking the following assumptions into account:

* Meteorological conditions

The meteorological input data is obtained from the measurements taken at the installed weather station. The Type 16c Solar Radiation Processor is used to calculate the irradiance received on the collector plane using the global horizontal irradiance given by the pyranometer.

* Operation of the installation

The inlet and outlet flow rates of the installation are those measured directly on the site using the flow meters. The withdrawal profile therefore corresponds to the actual operation of the installation.

* Soil dissipation

To take into account heat dissipation in the ground, we used the Type 952 "Horizontal Ground Heat Exchanger" of the TESS library [13]. This subroutine models the functioning of an underground heat exchanger. The thermal conductivities of the pipe materials and the soil must be provided. The soil temperature is calculated as a function of the depth below the surface and the time of year using the Kusuda correlation [14]. The model considers the length of the buried circuit, the shape of the coil and calculates the exchange coefficients between the fluid and the soil.

* Water circuit

The in-situ flows are adequately modelled by connecting the appropriate components between themselves. The mains water circulates through the soil dissipation system before reaching the serial collectors. The water flows from one in series solar storage tank to another. The outlet of the ninth collector is connected to a manifold that redistributes the
186

incoming flow to the last seven parallel collectors. During a withdrawal phase, the extracted

187

hot water flows into the same manifold to be used for preparing the animal feed.

188

Figure 3 presents the solar installation model and the main modules are presented on

189

Table 2.

<figure><img src="image_4.png" /></figure>

190

<caption>Figure 3 : Flow chart of the solar installation model</caption>

191

192

<table><caption>Table 2 : Main modules used in the numerical model</caption><thead><tr><td>Name of the module</td><td>Serial number</td><td>Function</td></tr></thead><tbody><tr><td>Serial collectors</td><td>Type71, Type533</td><td>Collect solar energy and calculate energy conversion</td></tr><tr><td>Parallel collectors</td><td>Type71, Type39 modified</td><td>Collect solar energy and calculate energy conversion</td></tr><tr><td>Input data</td><td>Type9a</td><td>Input data</td></tr><tr><td>Radiation processor</td><td>Type16</td><td>Calculate tilted radiation</td></tr><tr><td>Soil dissipation</td><td>Type952</td><td>Model the energy transfer from a liquid-filled pipe to the soil surrounding it</td></tr><tr><td>Daily energy integration</td><td>Type24</td><td>Integrate the energy supplied</td></tr><tr><td>Outputs printer</td><td>Type25</td><td>Collect outputs</td></tr></tbody></table>

193

194

## 2.3. Calibration

195

### 2.3.1. Sensitivity analysis

196

The first step of the calibration process is to determine those parameters having the

197

most impact. The JEA software [15] is used to carry out a sensitivity analysis on 7 parameters

198

to assess their influence on the energy provided $Q_{sup}$. The values of the selected parameters

199

(highlighted in Figure 2) must be determined to calculate as accurately as possible the heat

200

production of the solar installation.

201

In the present study, we chose to apply the Morris One-Step-At-a-Time screening

202

method [16]. As the simulation takes place over several days with a 5-minute time step, this

203

method is particularly suitable. The number N of simulations required to obtain the sensitivity

204

indices (4) is lower than other methods and it is a particularly effective approach to exclude

205

non-influential parameters [17].

$$N = k \cdot (n + 1) \qquad (4)$$
where k is the number of parameters and n the number of replications.

Each iteration i (i∈[1;k]) allows the calculation of an elementary effect $E_j^{(i)}$. Two indices are determined for every parameter j (j=1...k) : the absolute mean $\mu_j^*$ [17] and the standard deviation $\sigma_j$. The first index evaluates the contribution of input j to the model outcome and is calculated with the following equation:

$$ \mu_j^* = \frac{1}{n} \sum_{i=1}^{n} |E_j^{(i)}| \qquad (5) $$

The standard deviation is a measure of the spread in the model outcome due to changes in a specific parameter [18] and is calculated with equation (6):

$$ \sigma_j = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (E_j^{(i)} - \mu_j)^2} \qquad (6) $$

The distance $d^*$ from the origin allows us to rank parameters according to their influence and is obtained from the following equation:

$$ d^* = \sqrt{\mu_j^{*2} + \sigma_j^2} \qquad (7) $$

Ten analysis of ten replications were performed to obtain an acceptable number of indices and to determine the standard deviation $s$. The results are presented within an error interval calculated in the same way as the uncertainty due to measurement repeatability (type A):

$$ u_A = \frac{s}{\sqrt{10}} \qquad (8) $$

According to Garcia-Sanchez et al. [19], it is useful to study the ratio $\sigma_j/\mu_j^*$ to characterize the linearity or non-linearity of the model response. When $\sigma_j/\mu_j^*<0.1$, this means the parameter j has an almost linear effect on the model. If this ratio lies between 0.1 and 0.5, the majority of the elementary effects have the same sign and the model response can be qualified as monotonic.

The Morris method is frequently applied to different kind of models used in hydrology [20], building [21] and agriculture [22].

### 2.3.2. Parameter identification

GENOPT software [23] was used to identify the most influential parameters. The particle swarm optimisation (PSO) algorithm with constriction is suitable for the purposes of the present study. Introduced by Clerc and Kennedy [24], it is effective in identifying several variables with continuous and independent values. At each iteration, several particles are generated which then interact with each other to reach their objective. Each particle is characterised by its speed and position in the search space. The constriction factor allows us "to improve the convergence of the algorithm, to prevent the explosion of the swarm and to control the speed of the particles in order to avoid the problem of the divergence of the swarm which causes the premature convergence of the algorithm" [25].

Almeida et al. presented an identification algorithm as an alternative method [26] for dynamic testing of solar thermal collectors. Using this approach, these authors (op. cit.) determined the efficiency coefficients of four solar collectors. A relative error of 5-7% was observed when comparing the simulations with the experimental data

This same approach was selected for our study, along with the configurations presented by Almeida et al [26] (Table 3):

<table><caption>Table 3: Configuration of the particle swarm optimisation algorithm with constriction coefficient</caption><thead><tr><td>Parameters</td><td>Value</td></tr></thead></table>
<table><tr><td>Number of particles</td><td>10</td></tr><tr><td>Cognitive acceleration</td><td>0.5</td></tr><tr><td>Social acceleration</td><td>0.5</td></tr></table>

The cost function corresponds to the PMAE (Percentage Mean Absolute Error) and is calculated with the following equation:

$$PMAE = \frac{\sum_{i=1}^{N} \frac{|Q_{sup,m} - Q_{sup,s}|}{Q_{sup,m}}}{N} \quad (9)$$

where N is the number of days simulated.

Ha et al. used a similar approach for the calibration of a TRNSYS model for building an integrated photovoltaic system [27]. The PSO algorithm required 2000 iterations to reach the optimal values, but a clear convergence is visible after 600 iterations. The calibration allows us to halve the average of the absolute differences between the model and the *in-situ* measurements.

### 2.3.3. List of parameters

For each parameter, lower and upper bounds are set around a reference value using data from the literature and previous model results (Table 4). The first parameters selected are the coefficients used in the solar collector efficiency equation. Their values were determined in previous studies and are recalculated here to take into account the changes made to the present model. The coefficients $b_0$ and $b_1$ need to be investigated as regards their influence on the energy calculation. The initial values and the ranges of variation are taken from Budihardjo and Morrison [12]. The reference value of the heat loss coefficient of the tank is determined by taking into account the equivalent thermal resistance of the materials (60 mm of polyurethane between two stainless steel plates). The reference value of the albedo is obtained from on-site measurements.

<table><caption>Table 4: Parameters studied</caption><thead><tr><th></th><th>Reference value</th><th>Lower boundary</th><th>Upper boundary</th><th>Units</th></tr></thead><tbody><tr><td>η<sub>0</sub></td><td>0.6</td><td>0.2</td><td>1.0</td><td>-</td></tr><tr><td>a<sub>1</sub></td><td>5</td><td>2</td><td>15</td><td>kJ.hr<sup>-1</sup>.m<sup>-2</sup>.K<sup>-1</sup></td></tr><tr><td>a<sub>2</sub></td><td>0.001</td><td>0.0005</td><td>0.01</td><td>kJ.hr<sup>-1</sup>.m<sup>-2</sup>.K<sup>-2</sup></td></tr><tr><td>b<sub>0</sub></td><td>0.1914</td><td>0.18</td><td>0.21</td><td>-</td></tr><tr><td>b<sub>1</sub></td><td>0.4084</td><td>0.40</td><td>0.42</td><td>-</td></tr><tr><td>U<sub>edge</sub></td><td>3.7</td><td>2</td><td>15</td><td>kJ.hr<sup>-1</sup>.K<sup>-1</sup></td></tr><tr><td>ρ</td><td>0.5</td><td>0</td><td>1.0</td><td>-</td></tr></tbody></table>

### 2.4. Experimental sequences

The period used for the sensitivity analysis was chosen for its varied weather conditions. In addition, the period must be long enough to ensure that internal energy of the tank has only a small influence on the calculation of $Q_{sup}$.

The selected time-sequence runs from midnight on 1 September to midnight on 12 September. The start and end times were chosen to correspond to periods with no irradiance or flow rate, either for withdrawal or filling. The weather conditions during the studied period are shown in Figure 4 and Figure 5.
272
273
274

The intermittent production of the solar installation can be seen in Figure 6 where the filling and withdrawal flow rates are plotted. The withdrawal profile is more precisely shown on Figure 7 over two consecutive days.

<figure><img src="image_5.png" /><figcaption>Figure 4: Global horizontal irradiance and relative humidity (during 11 days in September)</figcaption></figure>

275
276

<figure><img src="image_6.png" /><figcaption>Figure 5: Ambient temperature and wind speed (during 11 days in September)</figcaption></figure>

277
278
<figure><img src="image_7.png" /><figcaption>Figure 6: Inlet and outlet flow rates (during 11 days in September)</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 7: Withdrawal profile during two consecutive days with corresponding volume</figcaption></figure>

The selected 11-day period includes alternating sunny and cloudy days. The maximum irradiance varies between 450 W/m² and 1000 W/m², with an average irradiance around 279 W/m². The daily global horizontal irradiations range from 8.6 MJ/m² to 17.7 MJ/m². The ambient temperature is between 11°C and 28°C (Figure 5). The average wind speed is moderate and reaches a maximum value of 30 km/h.

To check whether the model is capable of correctly simulating the solar installation, the results were confronted with in-situ measurements during different periods. The first period tested was from 14 to 27 August, showing the following weather conditions:
<figure><img src="image_9.png" /><figcaption>Figure 8 : Global horizontal irradiance and relative humidity (August)</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 9 : Ambient temperature and wind speed (August)</figcaption></figure>

It is appropriate to check whether the numerical model works with less favourable weather conditions. The model yields a satisfactory response for the summer using the identified coefficients. To ensure that its response remains acceptable throughout the year, the results also have to be tested under winter conditions.

In-situ measurements were taken during a period from 16/02/2022 at 6 am to 6/03/2022 at 6 am and include days with varying conditions (Figure 10). The Global Horizontal Irradiance (GHI) is much lower compared to the other periods studied, with a maximum value of around 650 W/m². Daily global radiation received horizontally varies between 2.5 MJ/m² and 12.1 MJ/m². The surrounding air is also colder with a temperature between -2°C and 13°C (Figure 11).

291
292

293
294

295
296
297
298
299
300
301
302
303
304
<figure><img src="image_11.png" /><figcaption>Figure 10 : Global horizontal irradiance and relative humidity (Winter)</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 11 : Ambient temperature and wind speed (Winter)</figcaption></figure>

To validate the performance of the model over the long term, we examine a sequence of 5 months. The simulation covers three different seasons, from 29/07 to 21/12. Finally, we investigate whether varying the collector tilt angle (β) for three different orientations will affect the energy supplied by the solar water heater.

## 3. Results and discussion

### 3.1. Sensitivity analysis

Figure 12 shows the sensitivity indices obtained from the Morris method. The greater the distance of a point from the origin, the greater the impact of the associated coefficient. In order of increasing impact, $U_{edge}$, $a_1$ and $\eta_0$ are the three most influential parameters for the numerical model of the pilot site. The other four parameters have a smaller influence on the energy supplied. Albedo is the fourth most influential parameter. The ratio $\sigma_j/\mu_j^*$ indicates

305
306

307

308

309

310

311

312

313

314

315

316

317

318

319
that the model response to the most influential parameter $\eta_0$ can be qualified as monotonic. The average of the elementary effects induced by the variation of the optical efficiency is positive. Therefore, if the optical efficiency increases, the energy supplied increases. The coordinates $(\mu_j^*; \sigma_j)$ of $a_1$ and $U_{edge}$ are on the borderline between “almost monotonic” and “monotonic”, which indicates that they have a non-linear influence on $Q_{sup}$, and/or there is an interaction between parameters. It is therefore not possible to determine how the energy supplied might vary as a function of these two parameters alone.

<figure><img src="image_13.png" /><figcaption>Figure 12: Sensitivity indices from the Morris method</figcaption></figure>

In conclusion, the sensitivity analysis identifies $\eta_0$, $a_1$ and $U_{edge}$ as the most influential parameters. Consequently, it is essential to take these parameters into account to improve the comparison between measurement and simulation. Furthermore, the effects of $b_0$, $b_1$, and $a_2$ are not relevant. Since albedo is less influential than the first three parameters and has been measured experimentally, it is not included in the optimisation process. Integrating the albedo into the identified parameters would add a dimension to the optimisation problem and therefore increase the computation time. For the non-influential parameters, the values used in the simulations are the reference values presented in Table 4.

## 3.2. Coefficient identification

The optimisation process results in a cost function (PMAE) of 2.2%. The values of the identified coefficients are presented in Table 5 and the speed of convergence of each parameter can be seen on Figure 13:

<table><caption>Table 5: Identified parameters</caption><thead><tr><th></th><th>Identified value</th><th>Initial value</th><th>Units</th></tr></thead><tbody><tr><td>$\eta_0$</td><td>0.75</td><td>0.6</td><td>-</td></tr><tr><td>$a_1$</td><td>4.64</td><td>5</td><td>kJ.hr<sup>-1</sup>.m<sup>-2</sup>.K<sup>-1</sup></td></tr></tbody></table>
$U_{edge}$ 3.16 3.7 kJ.hr⁻¹.K⁻¹

342

343

The triplet of coefficients differs from the usual values for evacuated tube collectors.

344

The values are representative of the performance of a specific collector arrangement in
operation and not in a normative test. In addition, they are related to the Global Horizontal

345

346

Irradiance measured on site.

<figure><img src="image_14.png" /><figcaption>Figure 13 : Results of the identification</figcaption></figure>

347

348

Nevertheless, the values are consistent with similar collectors. $U_{edge}$ is close to the initial value obtained by taking into account the equivalent thermal resistance of the materials.

349

350

351

352

### 3.3. Model validation

353

#### 3.3.1. September

354

The results from the September simulation are shown on Figure 14 :
<figure><img src="image_15.png" /><figcaption>Figure 14 : Comparison between computed and measured energies (September)</figcaption></figure>

The model is highly consistent with the experimental measurements. The relative error over the whole period is 0.5%.

**3.3.2. August**

The simulation for August gives the results presented in Figure 15 :

<figure><img src="image_16.png" /><figcaption>Figure 15 : Comparison between computed and measured energy supplied (August)</figcaption></figure>
363

364

365

366

367

368

369

With a PMAE of 2.9% and a relative error of 0.2%, the results are promising. Despite different meteorological conditions, the model response is very satisfactory with a maximum error obtained on $Q_{sup}$ not exceeding 7.1%.

The simulations allow us to evaluate the temperatures in the tanks. The modified *Type 533* used to model the in-line collector tank gives access to the water temperature distribution at different levels. Figure 16 shows the top and bottom temperatures of the first and last solar collector in series.

<figure><img src="image_17.png" /></figure>

370

371

372

<caption>Figure 16: Simulated temperatures in the first and last solar water heater in series and the inlet flow rate</caption>

373

374

375

376

377

378

A marked thermal stratification can be observed in both tanks, and is particularly noticeable in the eighth solar water heater in the periods after filling. This corresponds to the time when the tank is filled with cold water in the lower part while the hotter water remains in the upper part. During this period, the temperature in the first tank appears to be more homogeneous because the water at its inlet is has a temperature close to that of the mains water.

379

### 3.3.3. February/March

380

381

Figure 17 shows a comparison between the results obtained from model simulations and measurements:
<figure><img src="image_18.png" /><figcaption>Figure 17: Comparison between computed and measured energies supplied (February/March)</figcaption></figure>

The results are satisfactory. The model slightly overestimates the energy supplied over the whole period by 2.8%. The average deviation between measurements and simulation is 5.8%. The absolute value of the error in winter is of the same order of magnitude as for the other two periods. For this time-sequence, the lower $Q_{sup}$ values imply a higher daily relative error but this does not exceed 15%. In comparison with the previous periods, it appears that the energy supplied is better estimated under clear sky conditions than when it is cloudy. Similar observations have been made for a model with flat plate collectors arranged in series [28].

An analysis of the inlet and outlet temperatures illustrates the contribution of this installation to energy needs (Figure 18). It is noteworthy that the model validates the operating assumptions of the system. The solar water heaters arranged in series have the main function of preheating the water, while those in parallel also heat but mainly fulfil the storage function. The temperature at the outlet of the system is very close to the temperature measured between the last collector in series and the first one in parallel.
<figure><img src="image_19.png" /><figcaption>Figure 18: Temperatures at the input, at the output of the serial arrangement and at the output of the parallel sensors</figcaption></figure>

### 3.3.4. Farm production cycle

Having validated the model over periods of several days, the simulation could be carried out over the entire farm production cycle (Figure 19). The model evaluates the energy supplied satisfactorily and does not diverge after more than a hundred days. Alternating weather conditions do not affect the model performance, and it works equally well in summer and winter. The relative error over the cycle is -0.2% and the average daily absolute error is 16 MJ. The PMAE is 8.5%. Errors of the same order of magnitude are obtained by Bava and Furbo for the Trnsys model of a large array of solar collectors [29]. However, these authors (op. cit.) only simulated the energy supplied over a period of 5 days.

<figure><img src="image_20.png" /><figcaption>Figure 19: Comparison between computed and measured energy supplied (production cycle)</figcaption></figure>
412
413
414
415

In-situ measurements and the simulation show the solar installation contribution to heat water at 80°C is equal to 61.2% to the energy required (83.6 GJ during the production cycle). Over 40700 litres of hot water circulated through the dissipation system, equivalent to 4.3 GJ of energy stored in the ground.

416

**3.4. Impacts of tilt-angle and orientation on the energy supplied**

417
418
419
420
421
422
423
424

The cumulative energy delivered during the 5-month period was calculated for 3 tilt angles ($\beta$=15°, 45° and 75°) and 3 orientations (south, southwest and southeast). Figure 20 shows the total irradiation received on the plane of the collectors and the cumulative energy supplied by the solar water heater. The highest amount of energy received by the collectors is achieved with a 45° tilt angle for all three orientations. In addition, the maximum cumulative tilted irradiation is received on a south-facing plane with a 45° inclination ($1.8 \times 10^6$ MJ). Having a 75° slope increases the collection of solar radiation in winter, hence a greater cumulative energy is supplied with the south and southeast orientations.

425
426
427
428
429
430
431
432

For all three orientations, the maximum energy delivered is attained when $\beta$=45°. This observation is consistent with previous studies of individual collectors [8] [30]. In order to enhance the energy supplied, the solar radiation collection needs to be maximized. This explains why the maximum amount of heat is delivered with a south-facing installation ($4.5 \times 10^4$ MJ). Evacuated tube collectors are nevertheless still efficient with diffuse irradiation, which is predominant in north-western Europe during the winter. In this way, the solar water heater system can maintain a constant contribution to hot water production throughout the veal farm production cycle.

433
434
435
436

In cases where the collectors cannot be oriented towards the south, our results show that the energy supplied with a 45° angle decreases by 13% and 6% for the south-west and south-east, respectively. In both situations, the use of fossil energy to heat water is reduced in favour of renewable energy.

<figure><img src="image_21.png" /></figure>

437
438

<caption>Figure 20: Cumulative energy a) received on the collector plane and b) supplied by the solar installation</caption>

439

**4. Conclusion**

440
441
442
443
444
445

This study presents the approach used to validate the numerical model of a solar installation designed to produce hot water twice a day in an agricultural application.

First, a sensitivity analysis is carried out to eliminate the parameters with a negligible influence on the estimation of the energy delivered by the solar collectors. The most influential parameters are the optical efficiency $\eta_0$, the first-order efficiency coefficient $a_1$ and the storage tank thermal loss coefficient $U_{edge}$.
446

447

448

449

450

451

452

453

454

These three identified parameters are used to minimize the error between the measured and simulated energy supplied by the installation. The values obtained for η₀, a₁ and U edge are 0.75, 4.64 kJ.hr⁻¹.m⁻².K⁻¹ and 3.16 kJ.hr⁻¹.K⁻¹, respectively, which are representative of the operating performance of the solar water heaters. These three identified coefficients are used to validate the model during different experimental sequences. Relative errors of 0.2% and 2.8% are obtained for selected periods during the summer and winter, respectively. A good agreement is also observed between measurements and the model for a long-duration simulation with varied weather conditions. The energy supplied is estimated over a 5-month period with a relative error of -0.2% and a PMAE of 8.5%.

455

456

457

458

459

460

461

462

463

A parametric study was carried out to investigate the effects of characteristic parameters such as collector tilt-angle and orientation. According to the simulations, the collector tilt-angle of the solar collectors has a substantial impact on the installation's daily solar heat gain and collected radiation. The optimal energy output is obtained with an arrangement of solar water heaters facing south with a tilt-angle of 45°, corresponding to a cumulative value close to 4.5×10⁴ MJ. This value is reduced by 13% and 6% when the solar collectors are facing south-west or south-east, respectively. The same investigation could be carried out at different locations to assess and optimize the potential for solar water heating applications.

464

465

466

467

468

469

470

471

472

The validated numerical model is a tool to assist in the dimensioning and improvement of future solar hot water installations in order to reduce fossil energy consumption in the agricultural sector. Such a model can take into account the unpredictability of weather conditions, as well as energy demand and installation inertia, making it more accurate and reliable than a static model. It is also possible to assess the influence of site constraints, in particular imposed orientation and the presence of surface shading. The present analysis is based on a single set of meteorological data that will be consolidated with complementary long-term investigations.

473

## Credit author statement

474

**Julien Gambade:** Methodology, Investigations, Formal analysis, Writing – Original

475

**Draft Hervé Noël:** Validation, Writing – Review & Editing, Supervision, Project

476

**administration Patrick Glouannec:** Validation, Writing – Review & Editing, Supervision,

477

**Project administration Anthony Magueresse:** Investigation, Resources

478

## Acknowledgements

479

480

481

482

483

This research was carried out in the framework of the ICaRE4Farms project, funded by the INTERREG North West Europe programme. Fengtech designed the solar installation and made it available for the study. Dr M.S.N. Carpenter post-edited the English style and grammar.
484 References

485 [1] B. Aydoğan, G. Vardar, Evaluating the role of renewable energy, economic growth and agriculture on CO2 emission in E7 countries, International Journal of Sustainable Energy. 39 (2020) 335–348. https://doi.org/10.1080/14786451.2019.1686380.

486
487
488 [2] K.P. Sakhare, Kiran, H. Balsoriya, J.P. Kesari, Opportunities for solar thermal systems across dairy, agricultural, hotel & automobile industries, Materials Today: Proceedings. (2022) S2214785321080925. https://doi.org/10.1016/j.matpr.2021.12.353.

491
492
493 [3] A.S. Wallerand, M. Kermani, R. Voillat, I. Kantor, F. Maréchal, Optimal design of solar-assisted industrial processes considering heat pumping: Case study of a dairy, Renewable Energy. 128 (2018) 565–585. https://doi.org/10.1016/j.renene.2017.07.027.

494
495
496 [4] J.A. Quijera, M.G. Alriols, J. Labidi, Integration of a solar thermal system in a dairy process, Renewable Energy. 36 (2011) 1843–1853.
https://doi.org/10.1016/j.renene.2010.11.029.

497 [5] T. Sokhansefat, A. Kasaeian, K. Rahmani, A.H. Heidari, F. Aghakhani, O. Mahian, Thermoeconomic and environmental analysis of solar flat plate and evacuated tube collectors in cold climatic conditions, Renewable Energy. 115 (2018) 501–508.
https://doi.org/10.1016/j.renene.2017.08.057.

501 [6] R. Prăvălie, C. Patriche, G. Bandoc, Spatial assessment of solar energy potential at global scale. A geographical approach, Journal of Cleaner Production. 209 (2019) 692–721. https://doi.org/10.1016/j.jclepro.2018.10.239.

504
505
506
507 [7] A. Uniyal, Y.K. Prajapati, L. Ranakoti, P. Bhandari, T. Singh, B. Gangil, S. Sharma, V.V. Upadhyay, S.M. Eldin, Recent Advancements in Evacuated Tube Solar Water Heaters: A Critical Review of the Integration of Phase Change Materials and Nanofluids with ETCs, Energies. 15 (2022). https://doi.org/10.3390/en15238999.

508
509
510
511 [8] J. Bracamonte, J. Parada, J. Dimas, M. Baritto, Effect of the collector tilt angle on thermal efficiency and stratification of passive water in glass evacuated tube solar water heater, Applied Energy. 155 (2015) 648–659.
https://doi.org/10.1016/j.apenergy.2015.06.008.

512 [9] P.M. Kumar, K. Mylsamy, A comprehensive study on thermal storage characteristics of nano-CeO2 embedded phase change material and its influence on the performance of evacuated tube solar water heater, Renewable Energy. 162 (2020) 662–676.
https://doi.org/10.1016/j.renene.2020.08.122.

516
517
518 [10] J. Gambade, H. Noël, P. Glouannec, A. Magueresse, In-situ assessment of a solar vacuum tube collectors installation dedicated to hot water production, Energy Reports. 8 (2022) 605–615. https://doi.org/10.1016/j.egyr.2022.10.160.

519
520
521 [11] G.L. Morrison, I. Budihardjo, M. Behnia, Measurement and simulation of flow rate in a water-in-glass evacuated tube solar water heater, Solar Energy. 78 (2005) 257–267.
https://doi.org/10.1016/j.solener.2004.09.005.

522
523 [12] I. Budihardjo, G.L. Morrison, Performance of water-in-glass evacuated tube solar water heaters, Solar Energy. 83 (2009) 49–56. https://doi.org/10.1016/j.solener.2008.06.010.

524 [13] TESS Component Library Package - TESS Libraries | TRNSYS: Transient System Simulation Tool, (n.d.). https://www.trnsys.com/tess-libraries/index.html (accessed October 14, 2022).

527
528 [14] T. Kusuda, P.R. Achenbach, Earth temperature and thermal diffusivity at selected stations in the United States, National Bureau of Standards Gaithersburg MD, 1965.

529
530 [15] Y. Zhang, L. Jankovic, JEA, An Interactive Optimisation Engine for Building Energy Performance Simulation, (2017) 10.

531
532 [16] M.D. Morris, Factorial Sampling Plans for Preliminary Computational Experiments, Null. 33 (1991) 161–174. https://doi.org/10.1080/00401706.1991.10484804.
533

534

535

[17] F. Campolongo, J. Cariboni, A. Saltelli, An effective screening design for sensitivity analysis of large models, Environmental Modelling & Software. 22 (2007) 1509–1518.
https://doi.org/10.1016/j.envsoft.2006.10.004.

536

537

538

[18] K. Menberg, Y. Heo, R. Choudhary, Sensitivity analysis methods for building energy models: Comparing computational costs and extractable information, Energy and Buildings. 133 (2016) 433–445.
https://doi.org/10.1016/j.enbuild.2016.10.005.

539

540

541

542

[19] D. Garcia Sanchez, B. Lacarrière, M. Musy, B. Bourges, Application of sensitivity analysis in building energy simulations: Combining first- and second-order elementary effects methods, Energy and Buildings. 68 (2014) 741–750.
https://doi.org/10.1016/j.enbuild.2012.08.048.

543

544

545

[20] D.M. King, B.J.C. Perera, Morris method of sensitivity analysis applied to assess the importance of input variables on urban water supply yield – A case study, Journal of Hydrology. 477 (2013) 17–32.
https://doi.org/10.1016/j.jhydrol.2012.10.017.

546

547

548

[21] M. Labat, K. Attonaty, Numerical estimation and sensitivity analysis of the energy demand for six industrial buildings in France, Journal of Building Performance Simulation. 11 (2018) 223–240.
https://doi.org/10.1080/19401493.2017.1322637.

549

550

551

[22] H. Ben Touhami, R. Lardy, V. Barra, G. Bellocchi, Screening parameters in the Pasture Simulation model using the Morris method, Ecological Modelling. 266 (2013) 42–57.
https://doi.org/10.1016/j.ecolmodel.2013.07.005.

552

553

554

555

[23] M. Wetter, GenOpt® -- A Generic Optimization Program, (n.d.) 9.

[24] M. Clerc, J. Kennedy, The particle swarm - explosion, stability, and convergence in a multidimensional complex space, IEEE Transactions on Evolutionary Computation. 6 (2002) 58–73.
https://doi.org/10.1109/4235.985692.

554

555

556

557

558

[25] M. Zemzami, N. Elhami, A. Makhloufi, M. Itmi, N. Hmina, Application d'un modèle parallèle de la méthode PSO au problème de transport d'électricité, IncertFia. 17 (2017).
https://doi.org/10.21494/ISTE.OP.2017.0127.

559

560

561

[26] P. Almeida, M.J. Carvalho, R. Amorim, J.F. Mendes, V. Lopes, Dynamic testing of systems – Use of TRNSYS as an approach for parameter identification, Solar Energy. 104 (2014) 60–70.
https://doi.org/10.1016/j.solener.2014.02.010.

562

563

564

[27] S.-W. Ha, S.-H. Park, J.-Y. Eom, M.-S. Oh, G.-Y. Cho, E.-J. Kim, Parameter Calibration for a TRNSYS BIPV Model Using In Situ Test Data, Energies. 13 (2020) 4935.
https://doi.org/10.3390/en13184935.

565

566

567

568

[28] Z. Tian, B. Perers, S. Furbo, J. Fan, Analysis and validation of a quasi-dynamic model for a solar collector field with flat plate collectors and parabolic trough collectors in series for district heating, Energy. 142 (2018) 130–138.
https://doi.org/10.1016/j.energy.2017.09.135.

569

570

571

[29] F. Bava, S. Furbo, Development and validation of a detailed TRNSYS-Matlab model for large solar collector fields for district heating applications, Energy. 135 (2017) 698–708.
https://doi.org/10.1016/j.energy.2017.06.146.

572

573

574

575

[30] R. Tang, Y. Yang, W. Gao, Comparative studies on thermal performance of water-in-glass evacuated tube solar water heaters with different collector tilt-angles, Solar Energy. 85 (2011) 1381–1389.
https://doi.org/10.1016/j.solener.2011.03.019.