Uncertainty analysis and experimental validation of solar parabolic trough collector simulation model

solar parabolic trough collector, uncertainty analysis, experimental models, System Advisor Model, validation

1. Introduction

In recent years, industrial activities have played a significant role in the economic growth of several countries, resulting in substantial increases in productivity and improved living standards. However, the expansion of these industrial activities has led to a notable surge in global energy consumption [1, 2], while contributing 40% of global carbon dioxide (CO₂) emissions [3]. In Kenya, the industrial sector relies significantly on imported petroleum and wood fuel for heat energy. The tea factories, in particular, depend heavily on the burning of wood fuel, which has become increasingly expensive, to generate process heat for tea drying. This reliance on wood fuel also leads to the emission of greenhouse gases (GHG).

Renewable energy offers a significant opportunity to reduce pollution and address the environmental imbalances associated with biomass in the tea sector [1]. Solar energy, in particular, holds great promise for addressing multiple challenges, including the growing demand for energy [4], global warming [5], and environmental pollution [6]. In the industrial sector, a 52% share of heat is utilized within low and medium temperature range (<400°C) [7]. This is a temperature regime that is particularly accessible for commercially available solar–thermal technologies [8]. The parabolic trough collector (PTC) has demonstrated its potential as an excellent choice for industrial applications, establishing itself as the most promising and well-developed concentrated solar–thermal (CST) technology [9]. PTC technology currently dominates the market, with an 81% share [10], and can be a better technological alternative to the traditional wood-fired steam boilers in the tea industry in Kenya. Even though it has considerable potential for industrial heat, the development of the PTC field is still very limited, particularly in regions with high solar potential such as Africa.

Modeling is a powerful tool at the level of technical improvements that provides an efficient means for characterizing the performance of PTC systems by providing an approximation of their behavior [1, 11]. Models or dynamic simulation tools, such as the System Advisor Model (SAM), which was developed by the National Renewable Energy Laboratory (NREL) in the USA, can be used for assessing the performances of PTC systems under variable conditions. Other simulation software in solar–thermal applications include the Green Energy System Analysis Tool (Greenius) that was developed by the Institute of Solar Research (DLR) in Germany. This software has been used in several studies; for instance, one study [12] used it to simulate the performances of different heat-transfer fluids (HTFs) for a 1-MWe parabolic trough in Oujda (Eastern Morocco) and another study [13] used it to simulate the performances of a parabolic trough in Maan (Jordan). Transient System (TRNSYS) is another popular simulation tool that is used to simulate the behavior of transient systems, including CST technologies. For instance, one study [14] validated a TRNSYS model for a thermodynamic plant with parabolic trough solar–thermal power in Ain-Témouchent (Algeria), while another study [15] used is to analyse the performance of a 30-MWe SEGS VI parabolic trough plant in a Californian desert (USA). Nevertheless, for solar–thermal systems, SAM is the preferred option as it provides a wide range of performance metrics that are specific to solar–thermal systems, enabling users to gain insights into various aspects of system performance, including thermal output, efficiency, and more. SAM also allows a high degree of customization in defining system configurations, making it adaptable to different technologies and project requirements [16, 17].

There is no doubt that an accurate model for solar prediction allows improved designs of energy systems [18, 19]. In the context of the current rapid development of large-scale solar energy projects, modeling allows the saving of time and effort while providing a significant amount of information. Yet, the accuracy of the modeled radiation datasets is of utmost importance in providing a high degree of confidence to solar project developers while demonstrating the feasibility of industrial thermal PTC projects in countries such as Kenya, where CST activities are still very nascent. This calls for careful validation, normally against high-quality measurements [20]. Validation can be defined as the process of building an acceptable level of confidence that an inference about a simulated process is correct or valid inference for the actual process [21]. It is a critical step in establishing the credibility of a simulation model among its users [22]. It is therefore necessary to subject the models to experimental facilities or model prototypes to permit the validation of the results.

However, few studies have been published in the literature regarding the development of outdoor validation tests [23] or even prototypes for the purposes of PTC models validation. Some of the studies include, for instance, the work by Schiricke et al. [24], who made one of the first attempts at the experimental validation of CST systems by analysing the optical modeling of PTCs that was developed by using commercial Monte Carlo ray-tracing software. The experimental measurements confirmed the reliability of the model. Alfellag [25] presented an experimental investigation of a PTC that was developed in MATLAB^® software by using a test rack containing a circulating pump, a storage tank, thermocouples, a heat exchanger, and a flow meter. The results showed acceptable agreement even though there were some variances. Mateo et al. [26] conducted the experimental validation of a mathematical model of a PTC for a solar adsorption refrigerator using experimental data gathered from similar collectors in two different locations. The model was satisfactorily validated by experimental data. Similarly, Soudani et al. [27] experimentally validated a mathematical model of a PTC and registered close results between theoretical and experimental data, with the difference in thermal efficiency recorded as being <6%. Mouaky et al. [2] studied a PTC analytical model and validated it by using experimental data from a PTC test loop, with the result that the model used for PTC simulation was accurate and the daily average deviation was ~4.8% under clear-sky conditions. In 2020, Boretti et al. [28] conducted the first validation of SAM for both PTC and solar tower models by using high-frequency data collected on existing facilities and concluded that computational results were relatively close to measured values for the PTC but very far away for solar tower systems.

It is evident from the review of the literature that most CST validation assessments that employ the use of experimental results rarely account for errors from the most influential factors in the PTC field measurement parameters in order to increase the accuracy of the measured data. This leads to less reliable field data output results. Extensive analysis of the uncertainties associated with experimental data—also referred to as acceptance tests—helps to address and quantify the contributions of errors that are inherent in the collection of input data. CST performance is subject to a number of uncontrollable effects, including solar position, level of direct normal irradiation (DNI), wind velocity, and ambient temperature. While the effect of some variables such as the DNI is unquestionably more prominent than those of others, each significant effect needs to be quantified in order to obtain a sufficiently precise prediction of solar field performance. Because no measurement technique can perfectly determine the quantity it measures (every measurement device has inherent uncertainty associated with the measurement values), the goal of uncertainty analysis is to determine whether the performance of the designed plant model in question will meet or exceed the target, given a particular uncertainty and required confidence level [29]. This is because a realistic assessment of model suitability is not possible without uncertainty analysis [30].

This paper seeks to demonstrate a novel approach to the validation of a solar PTC model developed in SAM for industrial process heat application in tea drying in Kenya by using experimental data from a PTC prototype. The novel validation approach of the SAM model begins with quantification of the uncertainties in the experimental data and further establishes thermal energy output and efficiency model equations that are then utilized for validation of the PTC simulation model. The validation equations are derived from a data-fitting technique through linear regression based on the least squares method. Experimental data are derived from performance measurements of the PTC model prototype that was designed, fabricated, and tested in Kajiado, Kenya. This is the first study in Kenya, to the best of the authors’ knowledge, to have used this approach to perform the validation of a PTC plant simulation model.

2. System description and experimental test set-up

A parabolic trough plant simulation model for industrial process heat application was initially developed in SAM by using meteorological and process heat demand data from the Toror tea factory in Kericho, Kenya [1]. A small-scale PTC prototype was built for the purpose of validating the predictions of this model—specifically the solar field component. The system was installed at a site in Kimuka Kajiado, Kenya (latitude 1.418426° S and longitude 36.59749° E) for experimental testing under outdoor weather conditions, using water as the system HTF.

A collection of governing equations is employed in determining the design parameters for the prototype. The initial parameter under consideration is the curve length S of the reflective surface, which is identical to the width of the aluminum reflector mirror, which is 1150 mm. By using this information, we can calculate the latus rectum of the parabola, which, in turn, will assist in determining the remaining parameters. The formula for calculating the curve length is derived from the work of Kalogirou and it is expressed as follows [31]:

\begin{array}{l} S = \frac{H_{p}}{2} {[\sec (\frac{φ_{r}}{2}) \tan (\frac{φ_{r}}{2})] + \ln [\sec (\frac{φ_{r}}{2}) + \tan (\frac{φ_{r}}{2})]} \end{array}

(1)

where H_p represents the latus rectum of the parabola, which corresponds to the width of the parabolic shape at its focal point, and φ_r denotes the rim angle, which is the angle formed between the center line of the collector and the line connecting the outer rim of the reflector to the focal point [32].

It is possible to obtain different rim angles for the same aperture; for the different rim angles, the focus-to-aperture ratio, which defines the curvature of the parabola, changes. The optimum rim angle is between 70° and 110° [33]. Values of <70° reduce the aperture width and the collector working temperature while values of >110° will increase the reflective area without increasing the effective area [32]. Kalogirou also demonstrated that, for a 90° rim angle, the mean focus-to-reflector distance, and hence the reflected beam spread, is minimized so that the slope and tracking errors are also minimized, leading to higher optical efficiency [31]. For these reasons, this study adopts a rim angle of 90°. Therefore, H_p was established to be ~1000 mm by using Equation (1).

Another important parameter related to the rim angle φ_r is the aperture of the parabola W_a, which is given by the following equation [34]:

\begin{array}{l} W_{a} = 4 f \tan (\frac{φ_{r}}{2}) \end{array}

(2)

where f is the focal length of the parabola, derived from Equation (3):

\begin{array}{l} f = \frac{W_{a}}{4 \tan (\frac{φ_{r}}{2})} \end{array}

(3)

The parabola latus rectum H_p is equal to W_a when φ_r = 90° [31]. Therefore, the value of W_a is equal to 1000 mm and, from Equation (3), our focal length f is 250 mm, as illustrated in Fig. 1.

Figure 1.

The characteristic dimensions of the fabricated PTC.

Another important parameter is the collector geometrical concentration ratio C_g, defined as the ratio of the effective area of the aperture A_a to the area of the receiver A_r, given by [35]:

\begin{array}{l} C_{g} = \frac{A_{a}}{A_{r}} = \frac{(W_{a} - D_{o u t}) L_{c}}{0.5 π d_{o u t} L_{r}} = \frac{W_{a} - D_{o u t}}{0.5 π d_{o u t}} \end{array}

(4)

where D_out is the external diameter of the glass envelope, d_out is the external diameter of the absorber tube, and L_c and L_r are the lengths of the collector and the receiver, respectively.

In this research, the external diameter of the glass envelope is measured as 95 mm, while the external diameter of the absorber tube is 40 mm. The length of the absorber is consistent with that of the collector mirror, which is 2000 mm. Consequently, the concentration ratio, calculated as the ratio of the mirror width to the absorber diameter, was calculated to be 14.41. This value is considered desirable, as it exceeds 10, as recommended by [36].

The manufacturer provides several receiver parameters, including the absorber tube transmittance, absorptance, and glass envelope surface absorptance. For other parameters, such as the glass envelope surface emittance [37] and the absorber tube surface emittance [38], material composition data from the manufacturer were used to obtain values from existing literature. Table 1 lists both the selected and computed parameters for the collector mirror and absorber tube, while Fig. 2 shows the fabricated PTC prototype.

Table 1.

Design parameters and considerations.

Parameter	Value
Collector
Reflective aperture area, A_a	1.81 m²
Length of collector, L_c	2.00 m
Width of collector (curve length), S	1.15 m
Aperture of parabola, W_a	1.00 m
Focal length, f	0.25 m
Latus rectum, H_p	1.00 m
Rim angle, φ_r	90°
Mirror reflectance, 𝜌_m	0.95
Geometric concentration ratio, C_g	14.41
Receiver tube
Length of receiver tube, L_r	2.0 m
Absorber tube internal diameter, d_in	0.037 m
Absorber tube external diameter, d_out	0.040 m
Glass envelope internal diameter, D_in	0.091 m
Glass envelope external diameter, D_out	0.095 m
Absorber tube surface emittance, ℇ_abs	0.32
Absorber tube surface absorptance, α_abs	0.96
Glass envelope surface emittance, ℇ_env	0.88
Glass envelope surface absorptance, α_env	0.02
Glass envelope transmittance, 𝜏_env	0.92

Parameter	Value
Collector
Reflective aperture area, A_a	1.81 m²
Length of collector, L_c	2.00 m
Width of collector (curve length), S	1.15 m
Aperture of parabola, W_a	1.00 m
Focal length, f	0.25 m
Latus rectum, H_p	1.00 m
Rim angle, φ_r	90°
Mirror reflectance, 𝜌_m	0.95
Geometric concentration ratio, C_g	14.41
Receiver tube
Length of receiver tube, L_r	2.0 m
Absorber tube internal diameter, d_in	0.037 m
Absorber tube external diameter, d_out	0.040 m
Glass envelope internal diameter, D_in	0.091 m
Glass envelope external diameter, D_out	0.095 m
Absorber tube surface emittance, ℇ_abs	0.32
Absorber tube surface absorptance, α_abs	0.96
Glass envelope surface emittance, ℇ_env	0.88
Glass envelope surface absorptance, α_env	0.02
Glass envelope transmittance, 𝜏_env	0.92

Table 1.

Design parameters and considerations.

Parameter	Value
Collector
Reflective aperture area, A_a	1.81 m²
Length of collector, L_c	2.00 m
Width of collector (curve length), S	1.15 m
Aperture of parabola, W_a	1.00 m
Focal length, f	0.25 m
Latus rectum, H_p	1.00 m
Rim angle, φ_r	90°
Mirror reflectance, 𝜌_m	0.95
Geometric concentration ratio, C_g	14.41
Receiver tube
Length of receiver tube, L_r	2.0 m
Absorber tube internal diameter, d_in	0.037 m
Absorber tube external diameter, d_out	0.040 m
Glass envelope internal diameter, D_in	0.091 m
Glass envelope external diameter, D_out	0.095 m
Absorber tube surface emittance, ℇ_abs	0.32
Absorber tube surface absorptance, α_abs	0.96
Glass envelope surface emittance, ℇ_env	0.88
Glass envelope surface absorptance, α_env	0.02
Glass envelope transmittance, 𝜏_env	0.92

Parameter	Value
Collector
Reflective aperture area, A_a	1.81 m²
Length of collector, L_c	2.00 m
Width of collector (curve length), S	1.15 m
Aperture of parabola, W_a	1.00 m
Focal length, f	0.25 m
Latus rectum, H_p	1.00 m
Rim angle, φ_r	90°
Mirror reflectance, 𝜌_m	0.95
Geometric concentration ratio, C_g	14.41
Receiver tube
Length of receiver tube, L_r	2.0 m
Absorber tube internal diameter, d_in	0.037 m
Absorber tube external diameter, d_out	0.040 m
Glass envelope internal diameter, D_in	0.091 m
Glass envelope external diameter, D_out	0.095 m
Absorber tube surface emittance, ℇ_abs	0.32
Absorber tube surface absorptance, α_abs	0.96
Glass envelope surface emittance, ℇ_env	0.88
Glass envelope surface absorptance, α_env	0.02
Glass envelope transmittance, 𝜏_env	0.92

Figure 2.

Solar PTC prototype.

3. Experimental methodology and validation approach

This study adopts four significant steps that are necessary for successful model validation by using experimental data, namely:

analysis and quantification of the uncertainties associated with experimental data;
development of experimental model functions with regression coefficients determined by using the least squares method;
evaluation of the quality of fit of the model functions;
validation of the PTC model by using thermal output and efficiency experimental model functions.

3.1 Approach to uncertainty analysis

The parabolic trough uncertainty analysis, or performance acceptance test, aims to determine whether the measured performance metrics meet the guaranteed results. To achieve this, accurate data measurement tools are crucial. However, any real measurement system is prone to both random and systematic errors, which can be attributed to various sources such as calibration errors, instrument inaccuracies, and methodological limitations [29].

Both NREL and the American Society of Mechanical Engineers (ASME) have developed guidelines for uncertainty analysis in parabolic trough systems. These guidelines provide recommendations for procedures that can yield accurate results, consistently with good engineering practices. The guidelines are specifically designed for PTC systems with high-temperature synthetic oil but the principles are applicable to other CST systems [39]. Additionally, the international testing standard (ISO 9806:2013) provides guidance on uncertainty analysis in low-temperature collectors.

This study adopts the approach presented by Kearney [39] and Wagner et al. [29] to carry out the uncertainty analysis. This approach adopts both NREL and ASME procedures. Generally, the following steps are observed:

(i) Define the uncertainty problem by identifying the specific uncertainty aspect through the parameters to be analysed. In this study, uncertainties in the solar field thermal output and thermal efficiency are selected for uncertainty analysis, as the same have been selected as the validation parameters.
(ii) Identify and quantify various sources of uncertainty, such as measurement errors and parameter variations.
(iii) Formulate mathematical models of validation parameters to propagate the uncertainties through the system, accounting for correlations between variables.
(iv) Measure and evaluate the performance metrics (e.g. direct normal irradiance, reduced temperature difference, mass flowrates) for each mathematical model.
(v) Analyse and interpret the results to understand the impact of each source of uncertainty on the overall system performance.

3.2 Measurement parameters and instruments

The measurements required for the uncertainty tests are the HTF inlet and outlet temperatures, the HTF flow rate, and real-time weather data, including ambient temperature, wind velocity, and direct normal irradiance. The real-time weather data are recorded on-site by using a digital weather station equipped with a color light-emitting diode display. This station includes a solar panel, transmission module, wind vane, high-speed anemometer, thermo-hygrometer, and a rain funnel. It measures the ambient temperature (–40°C to 60°C), wind direction and speed (0–45 m/s), atmospheric pressure (600–1100 hPa), humidity (1%–95%), and rainfall levels (0–9999 mm).

For SAM, the key input data include the ambient temperature, wind speed, wind direction, relative humidity, atmospheric pressure, and solar irradiance. For the PTC prototype, essential measurements are the inlet and outlet temperatures of the collector, as well as the mass flow rate. The mass flow rate and inlet temperature were monitored by using a digital liquid crystal display on a water flow sensor meter, which operates within a range of 1–25 liters/min (0.017–0.42 kg/s). The collector outlet temperature was measured by using a GM320 infrared digital thermometer that was capable of measuring temperatures from –50°C to 400°C. Global irradiance was recorded by using an EKO MS-602, while diffuse radiation was measured by using another MS-602 pyranometer with a shading ball to block direct sunlight. Consequently, the direct normal irradiance was calculated by using the relation [40]:

\begin{array}{l} G_{g} = G_{d} + G_{b} \cos θ_{z} \end{array}

(5)

where G_g, G_d, and G_b are the global, diffuse, and direct normal radiation, respectively, while θ_z is the zenith angle of the Sun.

Table 2 lists the types of parameters measured and the accuracy of each instrument used in the solar field measurements.

Table 2.

Parameters and instrument uncertainties.

Parameter	Instrument uncertainty
HTF temperature	±1.5°C
Flow rate	±0.1%
Direct normal irradiance	±2.0%
Ambient temperature	±0.1°C
Relative humidity	±1.0%
Wind speed	±0.5 m/s
Atmospheric pressure	±3 hPa

Parameter	Instrument uncertainty
HTF temperature	±1.5°C
Flow rate	±0.1%
Direct normal irradiance	±2.0%
Ambient temperature	±0.1°C
Relative humidity	±1.0%
Wind speed	±0.5 m/s
Atmospheric pressure	±3 hPa

Table 2.

Parameters and instrument uncertainties.

Parameter	Instrument uncertainty
HTF temperature	±1.5°C
Flow rate	±0.1%
Direct normal irradiance	±2.0%
Ambient temperature	±0.1°C
Relative humidity	±1.0%
Wind speed	±0.5 m/s
Atmospheric pressure	±3 hPa

Parameter	Instrument uncertainty
HTF temperature	±1.5°C
Flow rate	±0.1%
Direct normal irradiance	±2.0%
Ambient temperature	±0.1°C
Relative humidity	±1.0%
Wind speed	±0.5 m/s
Atmospheric pressure	±3 hPa

3.3 Test conditions

A reliable uncertainty analysis requires testing under steady-state conditions and thermal equilibrium. However, the constantly changing energy input from the Sun and other factors in the solar field make it difficult to achieve this goal. Failing to achieve thermal equilibrium or steady-state test conditions can contribute to uncertainty in the test results. To minimize this uncertainty, it is essential to keep variations in key test parameters low so that they do not significantly impact the uncertainty band in the results. This can be achieved by ensuring that the solar system is under stable test conditions and in a thermal equilibrium state before testing begins [41].

Taking into account the impact of variations in test parameters on the overall uncertainty of the test results, a set of stabilization criteria and test conditions has been established to ensure a steady state for PTC energy tests, as presented below [29, 39].

Stabilization criteria are applicable to four parameters with allowable variability over the test period provided in brackets below:

HTF volumetric flow rate (≤0.5%)
Direct normal irradiance (≤0.5%)
HTF inlet temperature (≤0.2%)
HTF outlet temperature (≤0.2%)

The steady-state test, on the other hand, is defined by the following conditions:

Tests should be carried out between 9 a.m. and 4 p.m.
Duration of test run should be ~15–30 minutes with test-run data points collected every 10 seconds on average. Therefore, the number of test points in each test run should be between 90 and 180
Maximum wind speed should be ≤13 m/s
DNI on the solar field area has to be >500 W/m² with DNI variations smaller than ±20 W/m²

In the experimental validation of PTCs, steady-state conditions are established to ensure that the system reaches stable and uniform operating conditions, allowing accurate measurement and analysis of the performance of the collector. The steady-state conditions during experimental validation ensures that results accurately represent the long-term performance of the collector under stable operating conditions. This is crucial for the design and optimization of PTCs for practical applications.

The given stabilization criteria and test conditions have been chosen in such a way that the variations in the key test parameters are low enough to contribute in only a minor way to the uncertainty band in the results. At the same time, the set of stabilization criteria parameters were settled upon because their variability has a strong influence on the total uncertainty of the test results. The tests were conducted by using a short duration of 10-second time steps to capture the impact of transient effects resulting from a sudden change in the solar field inlet temperature, otherwise the final stabilization criteria for a specific project will be strongly influenced by the design of the solar system and associated instrumentation.

3.4 Calculation of measurement uncertainty

Due to the natural variability of resources and the imperfections in control systems, variations in all measured parameters are unavoidable. Uncertainties are typically classified into two types: systematic errors and random errors. While random errors tend to average out when multiple measurements are taken over a period of time, systematic errors persist regardless of the number of measurements. Systematic errors can arise from various sources, including the calibration process, instrument malfunctions, transducer errors, and fixed errors in the measurement method. Similarly, random errors can be based on the manufacturer’s specifications. For repeated measurements, the random standard uncertainty can be defined by:

\begin{array}{l} S_{\bar{X}} = σ_{X} / \sqrt{N} \end{array}

(6)

where 𝜎_X is the standard deviation of a series of sampled data and N is the number of data points collected over the test interval.

Calculated results, such as the delivered thermal energy and the solar–thermal efficiency, are not typically measured directly but rather are based on parameters that are measured during the course of one or multiple acceptance tests. Equation (7) describes how the uncertainties in each of the measured variables X propagate into the value of a calculated resulting quantity R. In this case, the result R is a function of the individual or average values of these independent parameters:

\begin{array}{l} R = f ({\bar{X}}_{1}, {\bar{X}}_{2}, \dots, {\bar{X}}_{i}) \end{array}

(7)

where the subscript i describes the number of parameters used in the calculation of the result and $\bar{X}$ is either the value of a single measurement of the parameter or the average value of the parameter based on a number of N repeated measurements.

The expression for the combined standard measurement uncertainty of a calculated result based on multiple error sources can in many cases be calculated from the root-sum-square of the total uncertainty of the individual random and systematic error sources, as shown in Equation (8):

\begin{array}{l} u_{R} = \sqrt{{(S_{R})}^{2} + {(b_{R})}^{2}} \end{array}

(8)

where S_R is the standard random uncertainty component of a result and b_R is the systematic standard uncertainty component of a result, calculated as follows:

\begin{array}{l} S_{R} = {[{\sum^{​}}_{I}^{i = 1} {(\frac{\partial R}{\partial {\bar{X}}_{i}} S_{{\bar{X}}_{i}})}^{2}]}^{1 /_{2}} \end{array}

(9)

\begin{array}{l} b_{R} = {[\sum_{i = 1}^{I} {(\frac{\partial R}{\partial {\bar{X}}_{i}} b_{{\bar{X}}_{i}})}^{2}]}^{^{1} /_{2}} \end{array}

(10)

where $S_{{\bar{X}}_{i}}$ is the random standard uncertainty of the mean of N measurements (Equation (6)) and $b_{{\bar{X}}_{i}}$ is defined as the systematic standard uncertainty component of a parameter, calculated as follows:

\begin{array}{l} b_{{\bar{X}}_{i}} = \frac{α}{\sqrt{3}} \end{array}

(11)

where 𝛼 is the estimated accuracy of the measurement or the instrument uncertainty. When the uncertainty is presented as a percentage, the systematic standard uncertainty is the product of the measurement uncertainty (as a percentage) and the nominal measurement [28].

The combined standard uncertainty in Equation (8) implies that the calculated result will capture the true result within a 68% confidence level (or 1 standard deviation). Typically, a confidence level of 95% (or 2 standard deviations) is desirable. Therefore, the expanded uncertainty in the result is given by:

\begin{array}{l} U_{R, 95} = 2 u_{R} \end{array}

(12)

3.5 Thermal energy output and efficiency calculations

Once the thermal equilibrium and test condition stability have been established, the criteria for thermal energy output and efficiency measurements are primarily based on the level of uncertainty in the test results that are calculated by using the equations discussed under section 3.4 on ‘Calculation of measurement uncertainty’.

The solar field thermal output, or useful heat gain, of the prototype collector is determined from the measured inlet and outlet temperatures, mass flow rate, and specific heat of water (HTF) by using the following equation:

\begin{array}{l} Q_{g a i n} = \dot{m} C_{p} (T_{o u t} - T_{i n}) \end{array}

(13)

where T_in and T_out are the inlet and the outlet temperatures, respectively, $\dot{m}$ is the mass flow rate, and C_p is the specific heat of the water.

The partial derivative terms symbolically represent the manner in which the result R varies with the changing independent variable. For example, the response of the thermal energy output to the mass flow rate given Equation (13) is:

\begin{array}{l} \frac{\partial Q_{g a i n}}{\partial \dot{m}} = C_{p} (T_{o u t} - T_{i n}) \end{array}

(14)

By extending the principle shown in Equation (14) to the entire systematic uncertainty calculation for solar field thermal energy output, the absolute standard random uncertainty are given in Equations (15) and (16):

\begin{array}{l} \begin{array}{l} S_{R}^{2} = & {(\frac{\partial R}{\partial \dot{m}} S_{\dot{m}})}^{2} + {(\frac{\partial R}{\partial C_{p}} S_{C_{p}})}^{2} + {(\frac{\partial R}{\partial T_{i n}} S_{T_{i n}})}^{2} & + {(\frac{\partial R}{\partial T_{o u t}} S_{T_{o u t}})}^{2} \end{array} \end{array}

(15)

\begin{array}{l} \begin{array}{l} S_{R}^{2} = & {(C_{p} (T_{o u t} - T_{i n}))}^{2} {S_{\dot{m}}}^{2} + {(\dot{m} (T_{o u t} - T_{i n}))}^{2} {S_{C_{p}}}^{2} & + {(\dot{m} C_{p})}^{2} S_{T_{o u t}}^{2} + {(\dot{m} C_{p})}^{2} S_{T_{i n}}^{2} \end{array} \end{array}

(16)

Similarly, the absolute standard systematic uncertainty can be derived as shown in Equation (16).

\begin{array}{l} \begin{array}{l} b_{R}^{2} = & {(C_{p} (T_{o u t} - T_{i n}))}^{2} {b_{\dot{m}}}^{2} + {(\dot{m} (T_{o u t} - T_{i n}))}^{2} {b_{C_{p}}}^{2} & + {(\dot{m} C_{p})}^{2} b_{T_{o u t}}^{2} + {(\dot{m} C_{p})}^{2} b_{T_{i n}}^{2} \end{array} \end{array}

(17)

The solar field thermal efficiency is evaluated experimentally by using the equation below [33]:

\begin{array}{l} η_{g} = \frac{\dot{m} C_{p} (T_{o u t} - T_{i n})}{A_{a} G_{b}} \end{array}

(18)

where A_a is the effective aperture area of the collector and G_b is the direct solar irradiance component in the aperture plane of the collector.

The methodology for obtaining Equations (16) and (17) can be applied identically to the solar field efficiency in Equation (18) to arrive at the estimated uncertainties. The resulting equations for the absolute standard random systematic uncertainty and absolute standard systematic uncertainty associated with the solar field efficiency are presented in Equations (19) and (20), respectively:

\begin{array}{l} \begin{array}{l} S_{R}^{2} = & {(\frac{C_{p}}{A_{a} G_{b}} (T_{o u t} - T_{i n}))}^{2} {S_{\dot{m}}}^{2} + {(\frac{\dot{m}}{A_{a} G_{b}} (T_{o u t} - T_{i n}))}^{2} {S_{C_{p}}}^{2} & + {(\frac{\dot{m} C_{p}}{A_{a} G_{b}})}^{2} S_{T_{o u t}}^{2} + {(\frac{\dot{m} C_{p}}{A_{a} G_{b}})}^{2} S_{T_{i n}}^{2} + {(\frac{\dot{m} C_{p} (T_{o u t} - T_{i n})}{A_{a}^{2} G_{b}})}^{2} S_{A_{a}}^{2} & + {(\frac{\dot{m} C_{p} (T_{o u t} - T_{i n})}{A_{a} G_{b}^{2}})}^{2} S_{G_{b}}^{2} \end{array} \end{array}

(19)

\begin{array}{l} \begin{array}{l} b_{R}^{2} = & {(\frac{C_{p}}{A_{a} G_{b}} (T_{o u t} - T_{i n}))}^{2} {b_{\dot{m}}}^{2} + {(\frac{\dot{m}}{A_{a} G_{b}} (T_{o u t} - T_{i n}))}^{2} {b_{C_{p}}}^{2} & + {(\frac{\dot{m} C_{p}}{A_{a} G_{b}})}^{2} b_{T_{o u t}}^{2} + {(\frac{\dot{m} C_{p}}{A_{a} G_{b}})}^{2} b_{T_{i n}}^{2} + {(\frac{\dot{m} C_{p} (T_{o u t} - T_{i n})}{A_{a}^{2} G_{b}})}^{2} b_{A_{a}}^{2} & + {(\frac{\dot{m} C_{p} (T_{o u t} - T_{i n})}{A_{a} G_{b}^{2}})}^{2} b_{G_{b}}^{2} \end{array} \end{array}

(20)

3.6 Experimental model functions and data fitting

Data fitting is the process of utilizing the uncertainty analysis to develop experimental model functions with parameters that represent observed data with the greatest accuracy. A model function depicts the underlying functional relationship between the outcome of the measurement and the corresponding conditions, whether known or assumed. Therefore, the goal of data fitting is to find those parameters of the experimental model function that best describe the connection between observations and conditions [42].

The least squares method is a widely used approach in statistics and numerical analysis to find the best-fitting parameters of experimental model functions to a given set of data points. The method aims to minimize the sum of the squared differences between the observed data points and the corresponding values that are predicted by the experimental model [43]. In other words, any least squares curve- or line-fitting algorithm optimizes the constants of a fitting equation by minimizing the sum of the squares of the deviations of the actual (data) values from the values that are predicted by the equation [44].

In this study, we develop two experimental model functions that are later used for validation of the SAM PTC simulation model. These are the ‘solar–thermal output model function’ and the ‘thermal efficiency model function’.

3.7 Solar–thermal output model function

The model functions are derived by using the following procedure:

(i) The parameters for the steady-state model equation for the solar–thermal output of the collector prototype is established. The solar–thermal output model is a linear three-parameter function, adapted from [45–47]:

\begin{array}{l} Q_{g a i n} = a_{1} + a_{2} G_{b} + a_{3} (Δ T) \end{array}

(21)

where Q_gain is the solar–thermal output or dependent variable, and is a function of the direct normal irradiance G_b and the difference between the average collector temperature T_m and the ambient temperature T_a, denominated as ΔT (in this simplified model, the average collector temperature T_m is calculated by using (T_in + T_out)/2). The parameter 𝛼₁ is the universal constant or y-intercept, otherwise known as the general mean [48], while 𝛼₂ and 𝛼₃ are the regression coefficients measuring the average effect of the respective meteorological (independent) variables, in this case G_b and ΔT(T_m – T_a) respectively.

Equation (21) is the fitted trend line and can be used to forecast the future solar energy output.

(ii) The next step is to find the linear regression coefficients 𝛼₁, 𝛼₂, and 𝛼₃ together with their standard uncertainties by computing a least squares solution of Equation (21) as explained by [49]. That is, Equation (21) is treated as a system of equations in the unknowns 𝛼₁, 𝛼₂, and 𝛼₃. In matrix form, this can be written as Ax = b, where:

\begin{array}{l} \begin{array}{l} \begin{array}{l} A = [\begin{array}{l} \begin{array}{l} Δ T_{1} & G_{b, 1} Δ T_{2} & G_{b, 2} \end{array} & \begin{array}{l} 1 1 \end{array} \begin{array}{l} ⋮ & ⋮ \begin{array}{l} ⋮ Δ T_{n} \end{array} & \begin{array}{l} ⋮ G_{b, n} \end{array} \end{array} & \begin{array}{l} ⋮ \begin{array}{l} ⋮ 1 \end{array} \end{array} \end{array}] & x = [\begin{array}{l} a_{3} a_{2} a_{1} \end{array}] \end{array} & b = [\begin{array}{l} Q_{g a i n, 1} Q_{g a i n, 2} \begin{array}{l} ⋮ \begin{array}{l} ⋮ Q_{g a i n, n} \end{array} \end{array} \end{array}] \end{array} \end{array}

As a least squares solution, $\hat{x}$ provides the best-fit parameters of Equation (21) and is determined by multiplying both sides of Ax = b by the transpose to form an augmented matrix (A^TA. A^Tb), which is then row reduced to obtain $\hat{x}$ ⁠:

\begin{array}{l} \hat{x} = [\begin{array}{l} a_{3} a_{2} a_{1} \end{array}] \end{array}

The standard uncertainties of the parameters are obtained from the matrix (A^TA)^–1 where the diagonal elements are the squared uncertainties of the regression parameters and the off-diagonal elements are the co-variances between them.

(iii) Once the regression parameters are established, it is important to test the goodness-of-fit of the solar energy output model function. A satisfactory measure of the “goodness-of-fit” of the model is crucial to deem the combined standard uncertainties in model parameters trustworthy and to validate the suitability of employing the model [29]. A number of statistical indicators are available for evaluating the model functions that govern the experimental behavior of measured data. In this study, the coefficient of determination, or regression, R² is used for evaluating the goodness-of-fit.

R² represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s). It is therefore a measure of how well the regression line approximates and fits the real data points. Hence, a coefficient of determination with a value of 1 indicates that the regression line perfectly fits the data [50]. It can be estimated by using Equation (22) [51]:

\begin{array}{l} R^{2} = {[\frac{n (\sum_{i = 1}^{n} X_{i} Y_{i}) - (\sum_{i = 1}^{n} X_{i}) (\sum_{i = 1}^{n} Y_{i})}{\sqrt{[n \sum_{i = 1}^{n} X_{i}^{2} - {(\sum_{i = 1}^{n} X_{i})}^{2}] [n \sum_{i = 1}^{n} Y_{i}^{2} - {(\sum_{i = 1}^{n} Y_{i})}^{2}]}}]}^{2} \end{array}

(22)

where n is the number of data points, and X_i and Y_i are the model and measured values, respectively, at time step i.

3.8 Thermal efficiency model function

The thermal efficiency model is a three-parameter quadratic function adapted from [52]:

\begin{array}{l} η_{g} = U_{o} + U_{1} T^{*} + U_{2} G_{b} {T^{*}}^{2} \end{array}

(23)

where the quantity T^* is referred to as the reduced temperature difference and is expressed as (T_m – T_a)/G_b.

The parameters U₀, U₁ and U₂ that best fit the collector efficiency model function are the solutions of a multiple linear regression problem [53]. The model function is derived by using a similar procedure to that for solving the solar–thermal output model function, discussed above.

4. Results and discussion

4.1 Test conditions

Tests were conducted on a clear-sky day (31 March 2024) and the period that was selected for this analysis falls between 12 p.m. and 12:15 p.m. (East African Time). This time period was selected because it corresponds to a relatively steady solar resource level and the incident conditions of almost solar noon (at 12:00 solar time). This also allows the tests to be performed under almost the same incidence angle values. The data readings were 10 seconds apart for a total of 90 readings over the 15-minute test duration. The test duration of 15 minutes was selected based on the data-reading time step of 10 seconds and the number of test readings that enables the acceptable uncertainty test conditions that are described under Section 3.1.2 to be met. The irradiance during the tests was in the range of between 890 and 908 W/m², resulting in DNI variations of only 18 W/m². The maximum wind speed and relative humidity recorded were 2.6 m/s and 22%, respectively. The mass flow rate was kept constant at 0.09 kg/s. Table 3 shows the comparison between the allowable steady-state variability of the four critical test parameters and the results from this study. The study has met the stabilization criteria for a steady-state performance test.

Table 3.

Stabilization criteria for short-duration steady-state tests.

Parameter	Allowable variability over test period $S_{\bar{X}} / \bar{X} (%)$	Actual test variability over test period $S_{\bar{X}} / \bar{X} (%)$
HTF flow rate, kg/s	0.5%	0.00%
Direct normal irradiance, W/m²	0.5%	0.05%
Solar field inlet temperature, °C	0.2%	0.14%
Solar field outlet temperature, °C	0.2%	0.18%

Parameter	Allowable variability over test period $S_{\bar{X}} / \bar{X} (%)$	Actual test variability over test period $S_{\bar{X}} / \bar{X} (%)$
HTF flow rate, kg/s	0.5%	0.00%
Direct normal irradiance, W/m²	0.5%	0.05%
Solar field inlet temperature, °C	0.2%	0.14%
Solar field outlet temperature, °C	0.2%	0.18%

Table 3.

Stabilization criteria for short-duration steady-state tests.

Parameter	Allowable variability over test period $S_{\bar{X}} / \bar{X} (%)$	Actual test variability over test period $S_{\bar{X}} / \bar{X} (%)$
HTF flow rate, kg/s	0.5%	0.00%
Direct normal irradiance, W/m²	0.5%	0.05%
Solar field inlet temperature, °C	0.2%	0.14%
Solar field outlet temperature, °C	0.2%	0.18%

Parameter	Allowable variability over test period $S_{\bar{X}} / \bar{X} (%)$	Actual test variability over test period $S_{\bar{X}} / \bar{X} (%)$
HTF flow rate, kg/s	0.5%	0.00%
Direct normal irradiance, W/m²	0.5%	0.05%
Solar field inlet temperature, °C	0.2%	0.14%
Solar field outlet temperature, °C	0.2%	0.18%

The prototype collector system was set up and tested approximately 1 hour before the performance test began at 12 p.m. This precaution was taken to ensure that heat loss by the system remained relatively consistent throughout the 15-minute experimental period. Otherwise, if the solar field is not in equilibrium—for instance, the temperatures of the solar field HTF and the piping are still changing over time—the delivered energy will be less than the absorbed power because some of the excess energy is used to heat the solar field HTF and piping. The largest HTF inlet and outlet temperature difference observed during the 15-minute test was 3.3°C, with the highest recorded inlet and outlet temperatures being 43.8°C and 47.1°C, respectively. The system was operated at lower temperatures to minimize heat-transfer fluid losses and achieve optimal efficiency.

4.2 Uncertainty evaluation

Tables 4–7 summarize the uncertainty data and results that were derived from the steady-state tests based on the methodology and equations discussed earlier as applied to the solar field energy output and efficiency calculations, respectively. These results provide the uncertainty contributions of the various measured data parameters over the course of the acceptance test time period. The tables are presented in the standard ASME PTC format for reporting uncertainty. The primary sources of uncertainty in the experimental results are attributed to the measurement of the solar irradiance and temperature difference. To improve the accuracy of the data, it is essential to focus on precise measurements of these variables, as they are critical factors in determining the thermal output and efficiency of the solar PTC.

Table 4.

Solar field energy output uncertainty for measured data.

Parameter information (in parameter units)									Uncertainty contribution of parameters to the result (in units squared)
Symbol	Description	Units	Nominal value	Standard deviation $σ_{i}$	Number of measurements $N_{i}$	Absolute systematic standard uncertainty ${b_{\bar{X}}}_{_{i}}$	Absolute random standard uncertainty ${S_{\bar{X}}}_{_{i}}$	Absolute sensitivity $\frac{\partial R}{\partial {\bar{X}}_{i}}$	Absolute systematic standard uncertainty contribution ${(\frac{\partial R}{\partial {\bar{X}}_{i}} {b_{\bar{X}}}_{_{i}})}^{2}$	Absolute random standard uncertainty contribution ${(\frac{\partial R}{\partial {\bar{X}}_{i}} {S_{\bar{X}}}_{_{i}})}^{2}$
$\dot{m}$	Mass flow rate	kg/s	0.09	0.0000	90	0.0009	0.0000	12.60	1.29 × 10–4	########
Cp	HTF specific heat	kJ/kg °C	4.20	0.0000	90	0.0000	0.0000	0.270	########	########
Tout	HTF outlet temperature	°C	46.1	0.4189	90	0.0580	0.0442	0.378	4.81 × 10–4	2.79 × 10–4
Tin	HTF inlet temperature	°C	43.1	0.2280	90	0.0580	0.0240	0.378	4.81 × 10–4	8.25 × 10–5

Parameter information (in parameter units)									Uncertainty contribution of parameters to the result (in units squared)
Symbol	Description	Units	Nominal value	Standard deviation $σ_{i}$	Number of measurements $N_{i}$	Absolute systematic standard uncertainty ${b_{\bar{X}}}_{_{i}}$	Absolute random standard uncertainty ${S_{\bar{X}}}_{_{i}}$	Absolute sensitivity $\frac{\partial R}{\partial {\bar{X}}_{i}}$	Absolute systematic standard uncertainty contribution ${(\frac{\partial R}{\partial {\bar{X}}_{i}} {b_{\bar{X}}}_{_{i}})}^{2}$	Absolute random standard uncertainty contribution ${(\frac{\partial R}{\partial {\bar{X}}_{i}} {S_{\bar{X}}}_{_{i}})}^{2}$
$\dot{m}$	Mass flow rate	kg/s	0.09	0.0000	90	0.0009	0.0000	12.60	1.29 × 10–4	########
Cp	HTF specific heat	kJ/kg °C	4.20	0.0000	90	0.0000	0.0000	0.270	########	########
Tout	HTF outlet temperature	°C	46.1	0.4189	90	0.0580	0.0442	0.378	4.81 × 10–4	2.79 × 10–4
Tin	HTF inlet temperature	°C	43.1	0.2280	90	0.0580	0.0240	0.378	4.81 × 10–4	8.25 × 10–5

Table 4.

Solar field energy output uncertainty for measured data.

Parameter information (in parameter units)									Uncertainty contribution of parameters to the result (in units squared)
Symbol	Description	Units	Nominal value	Standard deviation $σ_{i}$	Number of measurements $N_{i}$	Absolute systematic standard uncertainty ${b_{\bar{X}}}_{_{i}}$	Absolute random standard uncertainty ${S_{\bar{X}}}_{_{i}}$	Absolute sensitivity $\frac{\partial R}{\partial {\bar{X}}_{i}}$	Absolute systematic standard uncertainty contribution ${(\frac{\partial R}{\partial {\bar{X}}_{i}} {b_{\bar{X}}}_{_{i}})}^{2}$	Absolute random standard uncertainty contribution ${(\frac{\partial R}{\partial {\bar{X}}_{i}} {S_{\bar{X}}}_{_{i}})}^{2}$
$\dot{m}$	Mass flow rate	kg/s	0.09	0.0000	90	0.0009	0.0000	12.60	1.29 × 10–4	########
Cp	HTF specific heat	kJ/kg °C	4.20	0.0000	90	0.0000	0.0000	0.270	########	########
Tout	HTF outlet temperature	°C	46.1	0.4189	90	0.0580	0.0442	0.378	4.81 × 10–4	2.79 × 10–4
Tin	HTF inlet temperature	°C	43.1	0.2280	90	0.0580	0.0240	0.378	4.81 × 10–4	8.25 × 10–5

Parameter information (in parameter units)									Uncertainty contribution of parameters to the result (in units squared)
Symbol	Description	Units	Nominal value	Standard deviation $σ_{i}$	Number of measurements $N_{i}$	Absolute systematic standard uncertainty ${b_{\bar{X}}}_{_{i}}$	Absolute random standard uncertainty ${S_{\bar{X}}}_{_{i}}$	Absolute sensitivity $\frac{\partial R}{\partial {\bar{X}}_{i}}$	Absolute systematic standard uncertainty contribution ${(\frac{\partial R}{\partial {\bar{X}}_{i}} {b_{\bar{X}}}_{_{i}})}^{2}$	Absolute random standard uncertainty contribution ${(\frac{\partial R}{\partial {\bar{X}}_{i}} {S_{\bar{X}}}_{_{i}})}^{2}$
$\dot{m}$	Mass flow rate	kg/s	0.09	0.0000	90	0.0009	0.0000	12.60	1.29 × 10–4	########
Cp	HTF specific heat	kJ/kg °C	4.20	0.0000	90	0.0000	0.0000	0.270	########	########
Tout	HTF outlet temperature	°C	46.1	0.4189	90	0.0580	0.0442	0.378	4.81 × 10–4	2.79 × 10–4
Tin	HTF inlet temperature	°C	43.1	0.2280	90	0.0580	0.0240	0.378	4.81 × 10–4	8.25 × 10–5

Table 5.

Total solar field energy output uncertainty.

Symbol	Description	Units	Calculated mean value R	Absolute systematic standard uncertainty of result b_R	Absolute random standard uncertainty of result S_R	Combined standard uncertainty u_R	Expanded uncertainty of the result U_R,95%	Expanded uncertainty of the result U_R,95% (%)
Q_gain	Solar field energy output	kW_thh	1.146	0.000	0.000	0.000	0.000	0.00%

Symbol	Description	Units	Calculated mean value R	Absolute systematic standard uncertainty of result b_R	Absolute random standard uncertainty of result S_R	Combined standard uncertainty u_R	Expanded uncertainty of the result U_R,95%	Expanded uncertainty of the result U_R,95% (%)
Q_gain	Solar field energy output	kW_thh	1.146	0.000	0.000	0.000	0.000	0.00%

Table 5.

Total solar field energy output uncertainty.

Symbol	Description	Units	Calculated mean value R	Absolute systematic standard uncertainty of result b_R	Absolute random standard uncertainty of result S_R	Combined standard uncertainty u_R	Expanded uncertainty of the result U_R,95%	Expanded uncertainty of the result U_R,95% (%)
Q_gain	Solar field energy output	kW_thh	1.146	0.000	0.000	0.000	0.000	0.00%

Symbol	Description	Units	Calculated mean value R	Absolute systematic standard uncertainty of result b_R	Absolute random standard uncertainty of result S_R	Combined standard uncertainty u_R	Expanded uncertainty of the result U_R,95%	Expanded uncertainty of the result U_R,95% (%)
Q_gain	Solar field energy output	kW_thh	1.146	0.000	0.000	0.000	0.000	0.00%

Table 6.

Solar field thermal efficiency uncertainty for measured data.

Parameter information (in parameter units)									Uncertainty contribution of parameters to the result (in units squared)
Symbol	Description	Units	Nominal value	Standard deviation $σ_{i}$	Number of measurements $N_{i}$	Absolute systematic standard uncertainty ${b_{\bar{X}}}_{_{i}}$	Absolute random standard uncertainty ${S_{\bar{X}}}_{_{i}}$	Absolute sensitivity $\frac{\partial R}{\partial {\bar{X}}_{i}}$	Absolute systematic standard uncertainty contribution ${(\frac{\partial R}{\partial {\bar{X}}_{i}} {b_{\bar{X}}}_{_{i}})}^{2}$	Absolute random standard uncertainty contribution ${(\frac{\partial R}{\partial {\bar{X}}_{i}} {S_{\bar{X}}}_{_{i}})}^{2}$
$\dot{m}$	Mass flow rate	kg/s	0.09	0.0000	90	0.000 90	0.0000	0.007 743 41	4.84 × 10^–11	0.000 000 0
C_p	HTF specific heat	kJ/kg °C	4.20	0.0000	90	0.0000	0.0000	0.000 165 93	0.000 000 0	0.000 000 0
T_out	HTF outlet temperature	°C	46.1	0.4189	90	0.0580	0.0442	0.000 232 30	1.81 × 10^–10	1.05 × 10^–10
T_in	HTF inlet temperature	°C	43.1	0.2280	90	0.0580	0.0240	0.000 232 30	1.81 × 10^–10	3.10 × 10^–11
G_b	Direct normal irradiance	W/m²	899	4.8162	90	18.020	0.5077	0.000 696 91	1.58 × 10^–4	1.25 × 10^–7
A_a	Effective aperture area	m²	1.81	0.0000	90	0.0000	0.0000	0.000 696 91	0.000 000 0	0.000 000 0

Parameter information (in parameter units)									Uncertainty contribution of parameters to the result (in units squared)
Symbol	Description	Units	Nominal value	Standard deviation $σ_{i}$	Number of measurements $N_{i}$	Absolute systematic standard uncertainty ${b_{\bar{X}}}_{_{i}}$	Absolute random standard uncertainty ${S_{\bar{X}}}_{_{i}}$	Absolute sensitivity $\frac{\partial R}{\partial {\bar{X}}_{i}}$	Absolute systematic standard uncertainty contribution ${(\frac{\partial R}{\partial {\bar{X}}_{i}} {b_{\bar{X}}}_{_{i}})}^{2}$	Absolute random standard uncertainty contribution ${(\frac{\partial R}{\partial {\bar{X}}_{i}} {S_{\bar{X}}}_{_{i}})}^{2}$
$\dot{m}$	Mass flow rate	kg/s	0.09	0.0000	90	0.000 90	0.0000	0.007 743 41	4.84 × 10^–11	0.000 000 0
C_p	HTF specific heat	kJ/kg °C	4.20	0.0000	90	0.0000	0.0000	0.000 165 93	0.000 000 0	0.000 000 0
T_out	HTF outlet temperature	°C	46.1	0.4189	90	0.0580	0.0442	0.000 232 30	1.81 × 10^–10	1.05 × 10^–10
T_in	HTF inlet temperature	°C	43.1	0.2280	90	0.0580	0.0240	0.000 232 30	1.81 × 10^–10	3.10 × 10^–11
G_b	Direct normal irradiance	W/m²	899	4.8162	90	18.020	0.5077	0.000 696 91	1.58 × 10^–4	1.25 × 10^–7
A_a	Effective aperture area	m²	1.81	0.0000	90	0.0000	0.0000	0.000 696 91	0.000 000 0	0.000 000 0

Table 6.

Solar field thermal efficiency uncertainty for measured data.

Parameter information (in parameter units)									Uncertainty contribution of parameters to the result (in units squared)
Symbol	Description	Units	Nominal value	Standard deviation $σ_{i}$	Number of measurements $N_{i}$	Absolute systematic standard uncertainty ${b_{\bar{X}}}_{_{i}}$	Absolute random standard uncertainty ${S_{\bar{X}}}_{_{i}}$	Absolute sensitivity $\frac{\partial R}{\partial {\bar{X}}_{i}}$	Absolute systematic standard uncertainty contribution ${(\frac{\partial R}{\partial {\bar{X}}_{i}} {b_{\bar{X}}}_{_{i}})}^{2}$	Absolute random standard uncertainty contribution ${(\frac{\partial R}{\partial {\bar{X}}_{i}} {S_{\bar{X}}}_{_{i}})}^{2}$
$\dot{m}$	Mass flow rate	kg/s	0.09	0.0000	90	0.000 90	0.0000	0.007 743 41	4.84 × 10^–11	0.000 000 0
C_p	HTF specific heat	kJ/kg °C	4.20	0.0000	90	0.0000	0.0000	0.000 165 93	0.000 000 0	0.000 000 0
T_out	HTF outlet temperature	°C	46.1	0.4189	90	0.0580	0.0442	0.000 232 30	1.81 × 10^–10	1.05 × 10^–10
T_in	HTF inlet temperature	°C	43.1	0.2280	90	0.0580	0.0240	0.000 232 30	1.81 × 10^–10	3.10 × 10^–11
G_b	Direct normal irradiance	W/m²	899	4.8162	90	18.020	0.5077	0.000 696 91	1.58 × 10^–4	1.25 × 10^–7
A_a	Effective aperture area	m²	1.81	0.0000	90	0.0000	0.0000	0.000 696 91	0.000 000 0	0.000 000 0

Parameter information (in parameter units)									Uncertainty contribution of parameters to the result (in units squared)
Symbol	Description	Units	Nominal value	Standard deviation $σ_{i}$	Number of measurements $N_{i}$	Absolute systematic standard uncertainty ${b_{\bar{X}}}_{_{i}}$	Absolute random standard uncertainty ${S_{\bar{X}}}_{_{i}}$	Absolute sensitivity $\frac{\partial R}{\partial {\bar{X}}_{i}}$	Absolute systematic standard uncertainty contribution ${(\frac{\partial R}{\partial {\bar{X}}_{i}} {b_{\bar{X}}}_{_{i}})}^{2}$	Absolute random standard uncertainty contribution ${(\frac{\partial R}{\partial {\bar{X}}_{i}} {S_{\bar{X}}}_{_{i}})}^{2}$
$\dot{m}$	Mass flow rate	kg/s	0.09	0.0000	90	0.000 90	0.0000	0.007 743 41	4.84 × 10^–11	0.000 000 0
C_p	HTF specific heat	kJ/kg °C	4.20	0.0000	90	0.0000	0.0000	0.000 165 93	0.000 000 0	0.000 000 0
T_out	HTF outlet temperature	°C	46.1	0.4189	90	0.0580	0.0442	0.000 232 30	1.81 × 10^–10	1.05 × 10^–10
T_in	HTF inlet temperature	°C	43.1	0.2280	90	0.0580	0.0240	0.000 232 30	1.81 × 10^–10	3.10 × 10^–11
G_b	Direct normal irradiance	W/m²	899	4.8162	90	18.020	0.5077	0.000 696 91	1.58 × 10^–4	1.25 × 10^–7
A_a	Effective aperture area	m²	1.81	0.0000	90	0.0000	0.0000	0.000 696 91	0.000 000 0	0.000 000 0

Table 7.

Total solar field thermal efficiency uncertainty.

Symbol	Description	Units	Calculated mean value R	Absolute systematic standard uncertainty of result b_R	Absolute random standard uncertainty of result S_R	Combined standard uncertainty u_R	Expanded uncertainty of the result U_R,95%	Expanded uncertainty of the result U_R,95% (%)
η_g	Solar field efficiency	–	0.706	0.0126	0.0004	0.0129	0.0258	3.66%

Symbol	Description	Units	Calculated mean value R	Absolute systematic standard uncertainty of result b_R	Absolute random standard uncertainty of result S_R	Combined standard uncertainty u_R	Expanded uncertainty of the result U_R,95%	Expanded uncertainty of the result U_R,95% (%)
η_g	Solar field efficiency	–	0.706	0.0126	0.0004	0.0129	0.0258	3.66%

Table 7.

Total solar field thermal efficiency uncertainty.

Symbol	Description	Units	Calculated mean value R	Absolute systematic standard uncertainty of result b_R	Absolute random standard uncertainty of result S_R	Combined standard uncertainty u_R	Expanded uncertainty of the result U_R,95%	Expanded uncertainty of the result U_R,95% (%)
η_g	Solar field efficiency	–	0.706	0.0126	0.0004	0.0129	0.0258	3.66%

Symbol	Description	Units	Calculated mean value R	Absolute systematic standard uncertainty of result b_R	Absolute random standard uncertainty of result S_R	Combined standard uncertainty u_R	Expanded uncertainty of the result U_R,95%	Expanded uncertainty of the result U_R,95% (%)
η_g	Solar field efficiency	–	0.706	0.0126	0.0004	0.0129	0.0258	3.66%

The calculated total uncertainty, expressed as the expanded uncertainty, of the mean observed solar–thermal output during the performance test at the 95% confidence level is 9.05%, which is equivalent to 0.104 kWh_th. The total expanded uncertainty of the solar field thermal efficiency is in the order of 3.66% (equivalent to 0.0258) at the same confidence level of 95%. The mean collector efficiency that was determined from experimental tests is 70.6%. The maximum efficiency of the solar collector is ~76%, which is attributed to the good-quality components with the properties listed in Table 1 and the operation of the system at lower temperatures. The uncertainties in the thermal output and efficiency are taken into account when analysing the data, allowing a more accurate assessment of the experimental models and validation of the SAM PTC model, in the following sections.

The experimental results indicate that the main source of uncertainty in calculating the solar field thermal output and efficiency is due to errors in measuring the solar irradiance and temperature difference. These errors can be minimized by employing high-quality sensors with very low uncertainty margins, thereby enhancing measurement reliability. Additionally, a more robust design of the solar PTC prototype can significantly improve the overall measurement accuracy, optimizing the experimental set-up to mitigate potential sources of error such as vibration and misalignment. Furthermore, maintaining a controlled experimental environment can help to reduce external influences on the performance measurements. In this study, factors such as wind and the presence of dust or soiling on the collector may have negatively impacted the solar irradiance measurements and hence the overall energy output and efficiency, particularly given the significant wind conditions at the measurement site.

4.3 Analysis of experimental models

Table 8 is a summary of the results of measurement data from the short-duration performance acceptance test that was used in developing the thermal output and efficiency experimental models by using the methodology and equations that were discussed in Section 3.2. The regression coefficients for the two models are presented in Table 9.

Table 8.

Measurement data for experimental analysis.

Count no.	G_b (W/m²)	T_m (°C)	ΔT (°C)	T^* (°C/W m²)	Count no.	G_b (W/m²)	T_m (°C m²/W)	ΔT (W_thh)	T^* (°C/W m²)
1	893	43.73	16.1	0.016 46	46	895	44.45	15.5	0.015 75
2	896	43.78	16.6	0.016 85	47	897	44.60	15.8	0.016 05
3	894	43.70	15.9	0.016 22	48	899	44.65	16.3	0.016 46
4	892	43.83	16.2	0.016 59	49	902	44.70	16.8	0.016 96
5	892	43.83	16.1	0.016 48	50	900	44.70	16.7	0.016 89
6	895	43.80	16.2	0.016 54	51	902	44.73	17.0	0.017 18
7	893	43.88	16.6	0.016 91	52	901	44.70	16.7	0.016 87
8	894	43.80	15.9	0.016 22	53	897	44.73	17.0	0.017 28
9	894	43.80	15.9	0.016 22	54	898	44.70	16.8	0.017 04
10	894	43.85	16.4	0.016 67	55	898	44.70	16.9	0.017 15
11	891	43.78	15.7	0.016 05	56	897	44.83	17.0	0.017 28
12	894	43.93	16.2	0.016 55	57	900	44.83	16.9	0.017 11
13	895	44.05	16.4	0.016 65	58	903	44.88	17.5	0.017 61
14	890	44.08	17.0	0.017 42	59	904	44.85	17.2	0.017 31
15	891	44.13	17.7	0.018 18	60	904	44.85	17.3	0.017 37
16	893	44.08	16.7	0.017 02	61	902	44.78	16.5	0.016 63
17	897	44.10	16.9	0.017 17	62	901	44.93	17.0	0.017 20
18	896	44.13	17.2	0.017 52	63	899	44.90	16.7	0.016 91
19	896	44.23	17.2	0.017 52	64	898	44.93	17.1	0.017 32
20	897	44.23	17.1	0.017 39	65	897	44.93	17.0	0.017 28
21	898	44.18	16.5	0.016 70	66	902	45.05	17.2	0.017 35
22	896	44.13	16.1	0.016 41	67	902	45.08	17.5	0.017 63
23	890	44.20	17.2	0.017 64	68	903	45.10	17.6	0.017 72
24	892	44.23	17.2	0.017 60	69	903	45.05	17.2	0.017 28
25	897	44.33	17.0	0.017 28	70	904	45.10	17.6	0.017 70
26	897	44.35	17.4	0.017 61	71	905	45.20	17.6	0.017 68
27	898	44.33	17.0	0.017 26	72	905	45.23	17.9	0.018 01
28	898	44.48	17.5	0.017 71	73	904	45.23	17.9	0.018 03
29	899	44.40	16.8	0.017 02	74	903	45.20	17.6	0.017 72
30	896	44.30	16.0	0.016 29	75	904	45.20	17.7	0.017 81
31	898	44.43	17.1	0.017 37	76	905	45.23	17.8	0.017 90
32	899	44.43	17.0	0.017 24	77	907	45.20	17.7	0.017 75
33	900	44.40	16.8	0.017 00	78	907	45.20	17.6	0.017 64
34	899	44.40	16.7	0.016 91	79	906	45.18	17.4	0.017 44
35	895	44.45	17.3	0.017 54	80	906	45.33	17.9	0.017 99
36	896	44.48	16.6	0.016 85	81	907	45.30	17.5	0.017 53
37	899	44.45	16.3	0.016 46	82	904	45.30	17.5	0.017 59
38	900	44.50	16.7	0.016 89	83	906	45.33	17.9	0.017 99
39	897	44.53	17.0	0.017 28	84	908	45.30	17.6	0.017 57
40	896	44.58	17.6	0.017 86	85	908	45.33	17.8	0.017 84
41	894	44.53	17.0	0.017 34	86	907	45.30	17.6	0.017 64
42	900	44.68	17.5	0.017 67	87	908	45.33	17.7	0.017 68
43	900	44.60	16.8	0.017 00	88	907	45.35	18.1	0.018 08
44	901	44.60	16.7	0.016 87	89	908	45.43	17.9	0.017 95
45	899	44.55	16.4	0.016 57	90	908	45.43	17.7	0.017 73

Count no.	G_b (W/m²)	T_m (°C)	ΔT (°C)	T^* (°C/W m²)	Count no.	G_b (W/m²)	T_m (°C m²/W)	ΔT (W_thh)	T^* (°C/W m²)
1	893	43.73	16.1	0.016 46	46	895	44.45	15.5	0.015 75
2	896	43.78	16.6	0.016 85	47	897	44.60	15.8	0.016 05
3	894	43.70	15.9	0.016 22	48	899	44.65	16.3	0.016 46
4	892	43.83	16.2	0.016 59	49	902	44.70	16.8	0.016 96
5	892	43.83	16.1	0.016 48	50	900	44.70	16.7	0.016 89
6	895	43.80	16.2	0.016 54	51	902	44.73	17.0	0.017 18
7	893	43.88	16.6	0.016 91	52	901	44.70	16.7	0.016 87
8	894	43.80	15.9	0.016 22	53	897	44.73	17.0	0.017 28
9	894	43.80	15.9	0.016 22	54	898	44.70	16.8	0.017 04
10	894	43.85	16.4	0.016 67	55	898	44.70	16.9	0.017 15
11	891	43.78	15.7	0.016 05	56	897	44.83	17.0	0.017 28
12	894	43.93	16.2	0.016 55	57	900	44.83	16.9	0.017 11
13	895	44.05	16.4	0.016 65	58	903	44.88	17.5	0.017 61
14	890	44.08	17.0	0.017 42	59	904	44.85	17.2	0.017 31
15	891	44.13	17.7	0.018 18	60	904	44.85	17.3	0.017 37
16	893	44.08	16.7	0.017 02	61	902	44.78	16.5	0.016 63
17	897	44.10	16.9	0.017 17	62	901	44.93	17.0	0.017 20
18	896	44.13	17.2	0.017 52	63	899	44.90	16.7	0.016 91
19	896	44.23	17.2	0.017 52	64	898	44.93	17.1	0.017 32
20	897	44.23	17.1	0.017 39	65	897	44.93	17.0	0.017 28
21	898	44.18	16.5	0.016 70	66	902	45.05	17.2	0.017 35
22	896	44.13	16.1	0.016 41	67	902	45.08	17.5	0.017 63
23	890	44.20	17.2	0.017 64	68	903	45.10	17.6	0.017 72
24	892	44.23	17.2	0.017 60	69	903	45.05	17.2	0.017 28
25	897	44.33	17.0	0.017 28	70	904	45.10	17.6	0.017 70
26	897	44.35	17.4	0.017 61	71	905	45.20	17.6	0.017 68
27	898	44.33	17.0	0.017 26	72	905	45.23	17.9	0.018 01
28	898	44.48	17.5	0.017 71	73	904	45.23	17.9	0.018 03
29	899	44.40	16.8	0.017 02	74	903	45.20	17.6	0.017 72
30	896	44.30	16.0	0.016 29	75	904	45.20	17.7	0.017 81
31	898	44.43	17.1	0.017 37	76	905	45.23	17.8	0.017 90
32	899	44.43	17.0	0.017 24	77	907	45.20	17.7	0.017 75
33	900	44.40	16.8	0.017 00	78	907	45.20	17.6	0.017 64
34	899	44.40	16.7	0.016 91	79	906	45.18	17.4	0.017 44
35	895	44.45	17.3	0.017 54	80	906	45.33	17.9	0.017 99
36	896	44.48	16.6	0.016 85	81	907	45.30	17.5	0.017 53
37	899	44.45	16.3	0.016 46	82	904	45.30	17.5	0.017 59
38	900	44.50	16.7	0.016 89	83	906	45.33	17.9	0.017 99
39	897	44.53	17.0	0.017 28	84	908	45.30	17.6	0.017 57
40	896	44.58	17.6	0.017 86	85	908	45.33	17.8	0.017 84
41	894	44.53	17.0	0.017 34	86	907	45.30	17.6	0.017 64
42	900	44.68	17.5	0.017 67	87	908	45.33	17.7	0.017 68
43	900	44.60	16.8	0.017 00	88	907	45.35	18.1	0.018 08
44	901	44.60	16.7	0.016 87	89	908	45.43	17.9	0.017 95
45	899	44.55	16.4	0.016 57	90	908	45.43	17.7	0.017 73

Table 8.

Measurement data for experimental analysis.

Count no.	G_b (W/m²)	T_m (°C)	ΔT (°C)	T^* (°C/W m²)	Count no.	G_b (W/m²)	T_m (°C m²/W)	ΔT (W_thh)	T^* (°C/W m²)
1	893	43.73	16.1	0.016 46	46	895	44.45	15.5	0.015 75
2	896	43.78	16.6	0.016 85	47	897	44.60	15.8	0.016 05
3	894	43.70	15.9	0.016 22	48	899	44.65	16.3	0.016 46
4	892	43.83	16.2	0.016 59	49	902	44.70	16.8	0.016 96
5	892	43.83	16.1	0.016 48	50	900	44.70	16.7	0.016 89
6	895	43.80	16.2	0.016 54	51	902	44.73	17.0	0.017 18
7	893	43.88	16.6	0.016 91	52	901	44.70	16.7	0.016 87
8	894	43.80	15.9	0.016 22	53	897	44.73	17.0	0.017 28
9	894	43.80	15.9	0.016 22	54	898	44.70	16.8	0.017 04
10	894	43.85	16.4	0.016 67	55	898	44.70	16.9	0.017 15
11	891	43.78	15.7	0.016 05	56	897	44.83	17.0	0.017 28
12	894	43.93	16.2	0.016 55	57	900	44.83	16.9	0.017 11
13	895	44.05	16.4	0.016 65	58	903	44.88	17.5	0.017 61
14	890	44.08	17.0	0.017 42	59	904	44.85	17.2	0.017 31
15	891	44.13	17.7	0.018 18	60	904	44.85	17.3	0.017 37
16	893	44.08	16.7	0.017 02	61	902	44.78	16.5	0.016 63
17	897	44.10	16.9	0.017 17	62	901	44.93	17.0	0.017 20
18	896	44.13	17.2	0.017 52	63	899	44.90	16.7	0.016 91
19	896	44.23	17.2	0.017 52	64	898	44.93	17.1	0.017 32
20	897	44.23	17.1	0.017 39	65	897	44.93	17.0	0.017 28
21	898	44.18	16.5	0.016 70	66	902	45.05	17.2	0.017 35
22	896	44.13	16.1	0.016 41	67	902	45.08	17.5	0.017 63
23	890	44.20	17.2	0.017 64	68	903	45.10	17.6	0.017 72
24	892	44.23	17.2	0.017 60	69	903	45.05	17.2	0.017 28
25	897	44.33	17.0	0.017 28	70	904	45.10	17.6	0.017 70
26	897	44.35	17.4	0.017 61	71	905	45.20	17.6	0.017 68
27	898	44.33	17.0	0.017 26	72	905	45.23	17.9	0.018 01
28	898	44.48	17.5	0.017 71	73	904	45.23	17.9	0.018 03
29	899	44.40	16.8	0.017 02	74	903	45.20	17.6	0.017 72
30	896	44.30	16.0	0.016 29	75	904	45.20	17.7	0.017 81
31	898	44.43	17.1	0.017 37	76	905	45.23	17.8	0.017 90
32	899	44.43	17.0	0.017 24	77	907	45.20	17.7	0.017 75
33	900	44.40	16.8	0.017 00	78	907	45.20	17.6	0.017 64
34	899	44.40	16.7	0.016 91	79	906	45.18	17.4	0.017 44
35	895	44.45	17.3	0.017 54	80	906	45.33	17.9	0.017 99
36	896	44.48	16.6	0.016 85	81	907	45.30	17.5	0.017 53
37	899	44.45	16.3	0.016 46	82	904	45.30	17.5	0.017 59
38	900	44.50	16.7	0.016 89	83	906	45.33	17.9	0.017 99
39	897	44.53	17.0	0.017 28	84	908	45.30	17.6	0.017 57
40	896	44.58	17.6	0.017 86	85	908	45.33	17.8	0.017 84
41	894	44.53	17.0	0.017 34	86	907	45.30	17.6	0.017 64
42	900	44.68	17.5	0.017 67	87	908	45.33	17.7	0.017 68
43	900	44.60	16.8	0.017 00	88	907	45.35	18.1	0.018 08
44	901	44.60	16.7	0.016 87	89	908	45.43	17.9	0.017 95
45	899	44.55	16.4	0.016 57	90	908	45.43	17.7	0.017 73

Count no.	G_b (W/m²)	T_m (°C)	ΔT (°C)	T^* (°C/W m²)	Count no.	G_b (W/m²)	T_m (°C m²/W)	ΔT (W_thh)	T^* (°C/W m²)
1	893	43.73	16.1	0.016 46	46	895	44.45	15.5	0.015 75
2	896	43.78	16.6	0.016 85	47	897	44.60	15.8	0.016 05
3	894	43.70	15.9	0.016 22	48	899	44.65	16.3	0.016 46
4	892	43.83	16.2	0.016 59	49	902	44.70	16.8	0.016 96
5	892	43.83	16.1	0.016 48	50	900	44.70	16.7	0.016 89
6	895	43.80	16.2	0.016 54	51	902	44.73	17.0	0.017 18
7	893	43.88	16.6	0.016 91	52	901	44.70	16.7	0.016 87
8	894	43.80	15.9	0.016 22	53	897	44.73	17.0	0.017 28
9	894	43.80	15.9	0.016 22	54	898	44.70	16.8	0.017 04
10	894	43.85	16.4	0.016 67	55	898	44.70	16.9	0.017 15
11	891	43.78	15.7	0.016 05	56	897	44.83	17.0	0.017 28
12	894	43.93	16.2	0.016 55	57	900	44.83	16.9	0.017 11
13	895	44.05	16.4	0.016 65	58	903	44.88	17.5	0.017 61
14	890	44.08	17.0	0.017 42	59	904	44.85	17.2	0.017 31
15	891	44.13	17.7	0.018 18	60	904	44.85	17.3	0.017 37
16	893	44.08	16.7	0.017 02	61	902	44.78	16.5	0.016 63
17	897	44.10	16.9	0.017 17	62	901	44.93	17.0	0.017 20
18	896	44.13	17.2	0.017 52	63	899	44.90	16.7	0.016 91
19	896	44.23	17.2	0.017 52	64	898	44.93	17.1	0.017 32
20	897	44.23	17.1	0.017 39	65	897	44.93	17.0	0.017 28
21	898	44.18	16.5	0.016 70	66	902	45.05	17.2	0.017 35
22	896	44.13	16.1	0.016 41	67	902	45.08	17.5	0.017 63
23	890	44.20	17.2	0.017 64	68	903	45.10	17.6	0.017 72
24	892	44.23	17.2	0.017 60	69	903	45.05	17.2	0.017 28
25	897	44.33	17.0	0.017 28	70	904	45.10	17.6	0.017 70
26	897	44.35	17.4	0.017 61	71	905	45.20	17.6	0.017 68
27	898	44.33	17.0	0.017 26	72	905	45.23	17.9	0.018 01
28	898	44.48	17.5	0.017 71	73	904	45.23	17.9	0.018 03
29	899	44.40	16.8	0.017 02	74	903	45.20	17.6	0.017 72
30	896	44.30	16.0	0.016 29	75	904	45.20	17.7	0.017 81
31	898	44.43	17.1	0.017 37	76	905	45.23	17.8	0.017 90
32	899	44.43	17.0	0.017 24	77	907	45.20	17.7	0.017 75
33	900	44.40	16.8	0.017 00	78	907	45.20	17.6	0.017 64
34	899	44.40	16.7	0.016 91	79	906	45.18	17.4	0.017 44
35	895	44.45	17.3	0.017 54	80	906	45.33	17.9	0.017 99
36	896	44.48	16.6	0.016 85	81	907	45.30	17.5	0.017 53
37	899	44.45	16.3	0.016 46	82	904	45.30	17.5	0.017 59
38	900	44.50	16.7	0.016 89	83	906	45.33	17.9	0.017 99
39	897	44.53	17.0	0.017 28	84	908	45.30	17.6	0.017 57
40	896	44.58	17.6	0.017 86	85	908	45.33	17.8	0.017 84
41	894	44.53	17.0	0.017 34	86	907	45.30	17.6	0.017 64
42	900	44.68	17.5	0.017 67	87	908	45.33	17.7	0.017 68
43	900	44.60	16.8	0.017 00	88	907	45.35	18.1	0.018 08
44	901	44.60	16.7	0.016 87	89	908	45.43	17.9	0.017 95
45	899	44.55	16.4	0.016 57	90	908	45.43	17.7	0.017 73

Table 9.

Regression coefficients.

Thermal energy output				Thermal efficiency
	$a_{1}$	$a_{2}$	$a_{3}$	U₀	$U_{1}$ (W/m²/°C)	$U_{2}$ (W/m²/°C)
Parameter value	–6.76	0.0077	0.056	–1.53	131.80	–38.34

Thermal energy output				Thermal efficiency
	$a_{1}$	$a_{2}$	$a_{3}$	U₀	$U_{1}$ (W/m²/°C)	$U_{2}$ (W/m²/°C)
Parameter value	–6.76	0.0077	0.056	–1.53	131.80	–38.34

Table 9.

Regression coefficients.

Thermal energy output				Thermal efficiency
	$a_{1}$	$a_{2}$	$a_{3}$	U₀	$U_{1}$ (W/m²/°C)	$U_{2}$ (W/m²/°C)
Parameter value	–6.76	0.0077	0.056	–1.53	131.80	–38.34

Thermal energy output				Thermal efficiency
	$a_{1}$	$a_{2}$	$a_{3}$	U₀	$U_{1}$ (W/m²/°C)	$U_{2}$ (W/m²/°C)
Parameter value	–6.76	0.0077	0.056	–1.53	131.80	–38.34

4.4 Thermal output experimental model

The thermal output experimental model equation is presented below:

\begin{array}{l} Q_{g a i n} = - 6.76 + 0.0077 G_{b} + 0.056 (Δ T) \end{array}

(24)

To determine whether the system has met the acceptance criteria, we need to compare the actual thermal output of the system (including its associated uncertainty) with the predicted thermal output that was calculated by using Equation (24). In statistical terms, we are testing the “null hypothesis” that the observed thermal output does not exceed the predicted thermal output. The purpose of the acceptance test is to verify that the thermal output of the system meets or exceeds a predetermined level so, if our test shows that the null hypothesis is false, then it indicates that the system has successfully passed the acceptance test [27, 39].

From Table 5, the mean thermal output observed during the test is 1.150 kWh_th, with an uncertainty of 9.05% (or 0.104 kWh_th) at the 95% confidence level. In order to prove the null hypothesis, the predicted thermal output value must be greater than the observed value plus its uncertainty, which is 1.150 + 0.104 = 1.254 kWh_th. This means that, if the modeled thermal output is >1.254 kWh_th, then we can be 95% confident that the solar field is not performing at the guaranteed level that is predicted by the experimental model. Fig. 3 shows the observed and modeled thermal output values over time, as well as the average values (dotted lines) for the duration of the acceptance test.

Figure 3.

Comparison of observed and predicted thermal output.

The mean experimental model thermal output value for the test period is 1.117 kWh_th, which is significantly lower than the 1.254 kWh_th required to prove the null hypothesis. As a result, the null hypothesis is disproved, as the predicted thermal output does not exceed the observed level. Therefore, the system has successfully passed the acceptance test based on its thermal output performance. The coefficient of determination, or regression, R² is used to assess the ability of the experimental model to accurately predict the thermal output. From calculations, the R² value is 0.972, which is statistically significant and acceptable. This suggests that the experimental model is a reliable tool for predicting the thermal output of the collector.

4.5 Thermal efficiency experimental model

Based on the regression coefficient parameters that are presented in Table 9, the thermal efficiency curve of the experimental model is presented below:

\begin{array}{l} η_{g} = - 1.53 + 131.80 (\frac{T_{m} - T_{a}}{G_{b}}) - 38.34 {(\frac{T_{m} - T_{a}}{G_{b}})}^{2} \end{array}

(25)

The predicted thermal efficiency of the model is 72.5%, which is lower than the observed value of 73.2% (0.706 + 0.0258) obtained from the data (see Table 7). This also disproves the null hypothesis, as it suggests that the predicted thermal efficiency does not exceed the observed level. As a result, the system has passed the acceptance test based on its thermal efficiency performance.

Additionally, the R² value of 0.989 indicates that the model provides a good fit to the observed data, suggesting that it is reliable for predicting thermal efficiency by using measured data. The plot in Fig. 4 shows the comparison between the observed and predicted thermal efficiency values, along with the average values (dotted lines), for the duration of the acceptance test.

Figure 4.

Comparison of observed and predicted thermal efficiency.

4.6 Validation of the SAM PTC model

Following the verification of the ability of the experimental models to accurately predict thermal output and efficiency of the collector, the models are used to validate the performance of the parabolic trough collector as predicted by using the SAM software. The SAM model is tested by simulating various scenarios using the same boundary conditions as the experimental data (collector properties, solar irradiance, mass flow, inlet, and outlet temperatures, ambient temperature, wind speed, and relative humidity). The data for this analysis were collected on a separate day between 9 a.m. and 6 p.m., at an interval of 1 hour.

The simulations were carried out hourly, replicating operational conditions that were similar to those in the experimental set-up. Each simulation time step incorporated specific input conditions that were derived from the experimental set-up, as follows:

Meteorological data—the solar irradiance, ambient temperature, wind speed, wind direction, atmospheric pressure, and relative humidity.
Design point parameters—mainly inlet and outlet temperatures of the HTF and the HTF flow rate.
Collector and receiver parameters as detailed in Table 1.

Since SAM usually requires extensive annual data for robust simulations, the short-term experimental data are integrated within a broader annual dataset that includes the selected experimental day along with other relevant weather and performance data from the entire year at the location of the experiment. These annual data were sourced from a nearby weather station. However, for validation purposes, the specific SAM simulation results for the days of the experiment were required.

The experimental results are then compared to the simulated outcomes in order to validate the accuracy of the SAM model. The error bars of the experimental results were utilized to validate the SAM model by assessing how closely the simulation results aligned with these error margins. Additional validation was conducted by using three statistical parameters: the root mean square error (RMSE), the mean bias error (MBE), and the t-statistics method (t_sta). These performance indices are calculated by using the following equations [54, 55]:

\begin{array}{l} R M S E = \sqrt{\frac{1}{n} \underset{i = 1}{\sum^{n}} {(I_{m e a s ., i} - I_{m o d ., i})}^{2}} \end{array}

(26)

\begin{array}{l} M B E = \frac{1}{n} \underset{i = 1}{\sum^{n}} (I_{m e a s ., i} - I_{m o d ., i}) \end{array}

(27)

\begin{array}{l} t_{s t a} = \sqrt{[\frac{(n - 1) M B E^{2}}{R M S E^{2} - M B E^{2}}]} \end{array}

(28)

where I_mod.,i is the value of the simulation model result at time step i, I_meas,i is the value of the experimental result at time step i and n is the number of values under investigation.

The comparison between the experimental and simulated data, as shown in Figs 5 and 6, reveals that the predictions of the SAM model fall within the error bars of the experimental results. This implies that the SAM model accurately predicts the thermal output and efficiency of the PTC throughout the day, as reflected in Table 10.

Table 10.

Experimental and simulation values for validation of SAM.

T_a (°C)	G_b (W/m²)	Q_gain expt (kW_thh)	Q_gain SAM (kW_thh)	η_g expt	η_g SAM
	Thermal energy output			Thermal efficiency
22.6	328	(0.331 ± 0.104)	0.301	(0.39 ± 0.026)	0.37
23.8	433	(0.499 ± 0.104)	0.438	(0.51 ± 0.026)	0.5
25.9	564	(0.646 ± 0.104)	0.591	(0.53 ± 0.026)	0.53
26.3	701	(0.977 ± 0.104)	0.901	(0.61 ± 0.026)	0.62
27.6	746	(1.054 ± 0.104)	1.001	(0.68 ± 0.026)	0.69
28.1	859	(1.108 ± 0.104)	1.097	0.72 ± 0.026)	0.7
27.8	788	(1.007 ± 0.104)	1.093	(0.70 ± 0.026)	0.71
26.3	744	(0.933 ± 0.104)	0.964	0.68 ± 0.026)	0.69
23.8	484	(0.688 ± 0.104)	0.632	(0.67 ± 0.026)	0.69
21.1	302	(0.494 ± 0.104)	0.446	(0.54 ± 0.026)	0.58

T_a (°C)	G_b (W/m²)	Q_gain expt (kW_thh)	Q_gain SAM (kW_thh)	η_g expt	η_g SAM
	Thermal energy output			Thermal efficiency
22.6	328	(0.331 ± 0.104)	0.301	(0.39 ± 0.026)	0.37
23.8	433	(0.499 ± 0.104)	0.438	(0.51 ± 0.026)	0.5
25.9	564	(0.646 ± 0.104)	0.591	(0.53 ± 0.026)	0.53
26.3	701	(0.977 ± 0.104)	0.901	(0.61 ± 0.026)	0.62
27.6	746	(1.054 ± 0.104)	1.001	(0.68 ± 0.026)	0.69
28.1	859	(1.108 ± 0.104)	1.097	0.72 ± 0.026)	0.7
27.8	788	(1.007 ± 0.104)	1.093	(0.70 ± 0.026)	0.71
26.3	744	(0.933 ± 0.104)	0.964	0.68 ± 0.026)	0.69
23.8	484	(0.688 ± 0.104)	0.632	(0.67 ± 0.026)	0.69
21.1	302	(0.494 ± 0.104)	0.446	(0.54 ± 0.026)	0.58

Table 10.

Experimental and simulation values for validation of SAM.

T_a (°C)	G_b (W/m²)	Q_gain expt (kW_thh)	Q_gain SAM (kW_thh)	η_g expt	η_g SAM
	Thermal energy output			Thermal efficiency
22.6	328	(0.331 ± 0.104)	0.301	(0.39 ± 0.026)	0.37
23.8	433	(0.499 ± 0.104)	0.438	(0.51 ± 0.026)	0.5
25.9	564	(0.646 ± 0.104)	0.591	(0.53 ± 0.026)	0.53
26.3	701	(0.977 ± 0.104)	0.901	(0.61 ± 0.026)	0.62
27.6	746	(1.054 ± 0.104)	1.001	(0.68 ± 0.026)	0.69
28.1	859	(1.108 ± 0.104)	1.097	0.72 ± 0.026)	0.7
27.8	788	(1.007 ± 0.104)	1.093	(0.70 ± 0.026)	0.71
26.3	744	(0.933 ± 0.104)	0.964	0.68 ± 0.026)	0.69
23.8	484	(0.688 ± 0.104)	0.632	(0.67 ± 0.026)	0.69
21.1	302	(0.494 ± 0.104)	0.446	(0.54 ± 0.026)	0.58

T_a (°C)	G_b (W/m²)	Q_gain expt (kW_thh)	Q_gain SAM (kW_thh)	η_g expt	η_g SAM
	Thermal energy output			Thermal efficiency
22.6	328	(0.331 ± 0.104)	0.301	(0.39 ± 0.026)	0.37
23.8	433	(0.499 ± 0.104)	0.438	(0.51 ± 0.026)	0.5
25.9	564	(0.646 ± 0.104)	0.591	(0.53 ± 0.026)	0.53
26.3	701	(0.977 ± 0.104)	0.901	(0.61 ± 0.026)	0.62
27.6	746	(1.054 ± 0.104)	1.001	(0.68 ± 0.026)	0.69
28.1	859	(1.108 ± 0.104)	1.097	0.72 ± 0.026)	0.7
27.8	788	(1.007 ± 0.104)	1.093	(0.70 ± 0.026)	0.71
26.3	744	(0.933 ± 0.104)	0.964	0.68 ± 0.026)	0.69
23.8	484	(0.688 ± 0.104)	0.632	(0.67 ± 0.026)	0.69
21.1	302	(0.494 ± 0.104)	0.446	(0.54 ± 0.026)	0.58

Figure 5.

Comparison of predicted and simulated thermal output results.

Figure 6.

Comparison of predicted and simulated efficiency results.

10.1186/s40807-023-00077-w

Moreover, the analysis that employs the three statistical metrics that were derived from the experimental and simulation values presented in Table 10 demonstrates statistically significant results. For the thermal energy output, the calculated values for the RMSE, MBE, and t-statistics were 0.057, 0.033, and 2.135, respectively. Similarly, for the thermal efficiency, the corresponding values for the RMSE, MBE, and t-statistics were 0.018, 0.010, and 1.993, respectively. As a result, the SAM PTC model has been successfully validated, demonstrating its reliability in simulating the performance of the collector.

It is essential to recognize that the experimental testing of the solar PTC has two significant limitations, and those are soiling and dust deposition, which can hinder the performance measurement of the parabolic trough by blocking the solar radiation from reaching the surface of the collector. This accumulation reduces the effective aperture area, leading to lower solar irradiance and thermal energy absorption, and thus diminishing the efficiency. Moreover, it complicates the interpretation of the performance data, as the output variations may be wrongly attributed to the system design rather than environmental conditions. Regular cleaning and maintenance are necessary for accurate assessments, but they can also introduce additional variability in the results. Despite this challenge, experimental testing is deemed essential for validating simulation models and ensuring trust in their applications.

5. Conclusion

This study outlines a detailed acceptance test process that integrates uncertainty analysis to validate experimental data in predicting the performance of a PTC prototype. The findings are used to create experimental performance models, which help to validate a simulation model that is developed in SAM for industrial heating applications. The acceptance test focuses on analysing the thermal energy output and efficiency of the solar field under clear-sky conditions during a steady state, reducing the risk of modeling errors. The results from the experimental models and their uncertainties are compared with the performance projections from the SAM model for these parameters.

The uncertainty analysis provided a realistic evaluation of the thermal performance of the prototype during testing, resulting in expanded uncertainties of 9.05% (0.104 kWh_th) for thermal output and 3.66% (0.0258) for thermal efficiency. The predictions of the experimental models showed strong correlations with the observed values, indicated by R² regression coefficients of 0.972 for thermal output and 0.989 for thermal efficiency. Comparison of the experimental results with the simulation outcomes confirmed the validity of the simulation model, as the results fell within the experimental error margins. Additionally, statistical analysis yielded significant values: RMSE of 0.057, MBE of 0.033, and t-statistics of 2.135 for thermal energy output, and RMSE of 0.018, MBE of 0.010, and t-statistics of 1.993 for thermal efficiency, underscoring the accuracy and reliability of the simulation model. Consequently, the simulation model has been validated and can effectively predict the thermal output and efficiency of parabolic trough collectors, and it can be utilized to forecast their thermal behavior under various operating conditions in Kenya, even without experimental data.

The four outlined simulation model validation steps—uncertainty analysis, regression model development, quality evaluation of experimental models, and simulation model validation—have produced several key research outputs that can enhance the design and implementation of solar PTC systems in industrial applications. First, the uncertainty analysis tables provide a reference for future studies, helping to refine experimental set-ups by identifying sources of variability. Second, the regression models offer a robust mathematical framework for predictive analytics, linking input variables such as solar irradiance and HTF temperatures to performance metrics such as thermal efficiency. Third, the quality-of-fit assessment using metrics such as R-squared values enhances the credibility of the models, allowing researchers to ascertain their representation of experimental data. Finally, validated performance predictions ensure that simulations accurately reflect real-world performance. Together, these outputs generate critical insights and can facilitate the development of more efficient, reliable, and scalable solar–thermal solutions for industrial applications.

Author contributions

Philip Akello (Conceptualization [Lead], Formal analysis [Supporting], Resources [Equal], Software [Equal], Writing—original draft [Lead]), Churchill Saoke (Formal analysis [Equal], Software [Supporting], Visualization [Lead], Writing—review & editing [Supporting]), Joseph Kamau (Formal analysis [Supporting], Methodology [Equal], Validation [Equal], Writing—review & editing [Equal]), and Jared Ndeda (Formal analysis [Supporting], Methodology [Equal], Validation [Equal], Writing—review & editing [Equal])

Conflict of interest statement

None declared.

Funding

None declared.

Data availability

The main data underlying this article are available in the article. Any remaining data will be shared on reasonable request to the corresponding author.

References

[1]

Akello

POO

Saoke

Kamau

et al.

Modeling and performance analysis of solar parabolic trough collectors for hybrid process heat application in Kenya’s tea industry using system advisor model

Sustain Energy Res

2023

;

. https://doi.org/

10.1016/j.solener.2019.11.040

[2]

Mouaky

Merrouni

Laadel

NE.

Simulation and experimental validation of a parabolic trough plant for solar thermal applications under the semi-arid climate conditions

Sol Energy

2019

;

194

969

–

. https://doi.org/

. https://iea.blob.core.windows.net/assets/a846e5cf-ca7d-4a1f-a81b-ba1499f2cc07/Renewables_2019.pdf

[3]

IEA

International Energy Agency

2019

(30 September 2023, date last accessed)

[4]

Akbarzadeh

Valipour

MS.

Heat transfer enhancement in parabolic trough collectors: a comprehensive review

Renew Sustain Energy Rev

2018

;

198

–

218

. https://doi.org/

10.1016/j.rser.2018.04.093

10.1016/j.rser.2016.09.071

[5]

Conrado

Rodriguez-Pulido

Calderón

Thermal performance of parabolic trough solar collectors

Renew Sustain Energy Rev

2017

;

1345

–

. https://doi.org/

10.1080/15567036.2020.1731018

[6]

Zhang

Shang

Optimal sizing of a grid-connected hybrid renewable energy systems considering hydroelectric storage

Energy Sources Part A

2020

;

5043

–

. https://doi.org/

10.1533/9780857096173.3.602

[7]

IEA, International Energy Agency

World Energy Statistics 2016, Online Tables

. www.iea.org/statistics/

(6 December 2018, date last accessed)

[8]

Häberle

Concentrating solar technologies for industrial process heat and cooling

. In:

Lovegrove

Stein

(eds),

Concentrating Solar Power Technology

, 2nd edn.

Cambridge, UK

Woodhead Publishing

2021

602

–

. https://doi.org/

[9]

Hussain

Norton

Duffy

Technological assessment of different solar-biomass systems for hybrid power generation in Europe

Renew Sustain Energy Rev

2017

;

1115

–

. https://doi.org/

10.1016/j.rser.2016.08.016

. https://pubdocs.worldbank.org/en/849341611761898393/WorldBank-CSP-Report-Concentrating-Solar-Power-Clean-Power-on-Demand-24-7-FINAL.pdf

[10]

World Bank

Concentrating solar power: Clean Solar Power 24/7

Washington, DC

2021

(21 June 2023, date last accessed)

[11]

Quezada–García

Sánchez–Mora

Polo–Labarrios

et al.

Modeling and simulation to determine the thermal efficiency of a parabolic solar trough collector system

Case Stud Thermal Eng

2019

;

100523

. https://doi.org/

10.1016/j.csite.2019.100523

10.1109/EITech.2015.7162978

[12]

Ouali

HAL

Merrouni

Moussaoui

et al.

Electricity yield analysis of a 50 MW solar tower plant under Moroccan climate

. In: 2015 International Conference on Electrical and Information Technologies (ICEIT), Marrakech, Morocco,

25–27 March

New York

IEEE

2015

252

–

. https://doi.org/

[13]

Liqreina

Qoaider

Dry cooling of concentrating solar power (CSP) plants, an economic competitive option for the desert regions of the MENA region

Sol Energy

2014

;

103

417

–

. https://doi.org/

10.1016/j.solener.2014.02.039

[14]

Remlaoui

Benyoucef

Assi

et al.

A TRNSYS dynamic simulation model for a parabolic trough solar thermal power plant

Int J Energetica

2020

;

–

. https://doi.org/

10.47238/ijeca.v4i2.106

[15]

Jones

Pitz-Paal

Schwarzboezl

et al.

TRNSYS modeling of the SEGS VI parabolic trough solar electric generating system

. In: International Solar Energy Conference, Washington, DC, USA,

21–25 April 2001

, Vol.

16702

New York

American Society of Mechanical Engineers

2001

405

–

. https://doi.org/

10.1115/SED2001-152

[16]

Blair

DiOrio

Freeman

et al.

System Advisor Model (SAM) General Description (Version 2017.9. 5) (No. NREL/TP-6A20-70414)

Golden, CO

National Renewable Energy Laboratory (NREL)

2018

. https://doi.org/

10.2172/1440404

10.1016/j.rser.2014.07.117

[17]

Naaim

Ouhammou

Aggour

et al.

Multi-utility solar thermal systems: harnessing parabolic trough concentrator using SAM software for diverse industrial and residential applications

Energies

2024

;

3685

. https://doi.org/

[18]

de Simón-Martín

Díez-Mediavilla

Alonso-Tristán

Modelling solar data: reasons, main methods and applications

. In: International Conference on Renewable Energies and Power Quality (ICREPQ’13) Proceedings, Bilbao, Spain,

20–22 March 2013

, Vol.

, pp.

–

2013

. https://doi.org/

10.24084/repqj11.440

[19]

Barhmi

Heynen

Golroodbari

van Sark

A review of solar forecasting techniques and the role of artificial intelligence

Solar

2024

;

–

135

. https://doi.org/

[20]

Gueymard

CA.

A review of validation methodologies and statistical performance indicators for modeled solar radiation data: towards a better bankability of solar projects

Renew Sustain Energy Rev

2014

;

1024

–

. https://doi.org/

10.1016/0166-3615(94)00045-r

[21]

Jagdev

Browne

Jordan

Verification and validation issues in manufacturing models

Comput Ind

1995

;

331

–

. https://doi.org/

10.1016/j.rser.2010.03.012

[22]

Collins

Koehler

Lynch

Methods that support the validation of agent-based models: an overview and discussion

J Artif Soc Soc Simul

2024

;

Article 11

. https://doi.org/

[23]

Fernández-García

Zarza

Valenzuela

et al.

Parabolic-trough solar collectors and their applications

Renew Sustain Energy Rev

2010

;

1695

–

721

. https://doi.org/

. https://commons.erau.edu/edt/11

[24]

Schiricke

Pitz-Paal

Lüpfert

et al.

(2009). Experimental verification of optical modeling of parabolic trough collectors by flux measurement

J Sol Energy Eng

0110

;

131

. https://doi.org/

[25]

Alfellag

MAA.

Modeling and experimental investigation of parabolic trough solar collector

Master’s thesis

2014

Florida

Embry-Riddle Aeronautical University

[26]

Mateo

Echarri

Samsón

Thermal analysis and experimental validation of parabolic trough collector for solar adsorption refrigerator

Energy Power Eng

2017

;

687

–

702

. https://doi.org/

10.4236/epe.2017.911044

10.1007/s12206-017-0747-3

[27]

Soudani

Aiadi

Bechki

et al.

Experimental and theoretical study of parabolic trough collector (PTC) with a flat glass cover in the region of Algerian Sahara (Ouargla)

J Mech Sci Technol

2017

;

4003

–

. https://doi.org/

10.1016/s0038-092x(99)00034-1

[28]

Boretti

Nayfeh

Al-Kouz

Validation of SAM modeling of concentrated solar power plants

Energies

2020

;

1949

. https://doi.org/

[29]

Wagner

Mehos

Kearney

et al.

Modeling of a parabolic trough solar field for acceptance testing: a case study

. In: ASME 2011 5th International Conference on Energy Sustainability, Washington, DC, USA,

7–10 August 2011

, 54686.

2011

595

–

603

. https://doi.org/

10.1115/ES2011-54245

[30]

Mathioulakis

Voropoulos

Belessiotis

Assessment of uncertainty in solar collector modeling and testing

Sol Energy

1999

;

337

–

. https://doi.org/

[31]

Kalogirou

SA.

Solar Energy Engineering: Processes and Systems

Amsterdam

Elsevier

2023

[32]

Akhbari

Bidi

Bakhtiari

et al.

Design and fabrication of a parabolic trough collector and experimental investigation of wind impact on direct steam production in Tehran

Int J Mech Ind Eng

2019

;

140

–

. https://www.researchgate.net/publication/289941948_Parabolic_Trough_Technology

[33]

Guenther

Michael

Csambor

et al.

Parabolic trough technology

. In:

Advanced CSP Teaching Materials

Cologne

Deutsches Zentrums für Luft- und Raumfahrt

2011

–

106

10.1016/j.renene.2013.04.019

[34]

Jaramillo

Venegas-Reyes

Aguilar

et al.

Parabolic trough concentrators for low enthalpy processes

Renew Energy

2013

;

529

–

. https://doi.org/

[35]

El-Husseini

Mostafa

El-Gindy

et al.

Solar heating system using parabolic collector for thermal optimum conditions of biogas production in winter

Arab Univ J Agric Sci

2019

;

123

–

. https://doi.org/

10.21608/ajs.2019.43073

. https://www.researchgate.net/publication/343988690

[36]

Prasad

Basha

Janjanam

Design, fabrication and evaluation of thermal performance of parabolic trough collector with elliptical absorber using Zn/H₂O nanofluid

Int J Adv Sci Technol

2020

;

7805

–

10.1016/j.jallcom.2016.08.073

[37]

Shao

Cui

et al.

High emissivity MoSi₂–TaSi₂–borosilicate glass porous coating for fibrous ZrO₂ ceramic by a rapid sintering method

J Alloys Compd

2016

;

690

–

. https://doi.org/

.https://inis.iaea.org/search/search.aspx?orig_q=RN:33040336

[38]

Nam

Lee

Kim

et al.

Experimental study on the emissivity of stainless steel

. In: Proceedings of the Korean Nuclear Society Spring Meeting, Cheju, Republic of Korea,

24–25 May 2001

, Vol.

, No.

2001

[39]

Kearney

Utility-Scale Parabolic Trough Solar Systems: Performance Acceptance Test Guidelines, April 2009-December 2010 (No. NREL/SR-5500-48895)

Golden, CO

National Renewable Energy Laboratory (NREL)

2011

. https://doi.org/

10.2172/1015504

. https://link.springer.com/book/9783658114558

[40]

Pashiardis

Pelengaris

Kalogirou

SA.

Geographical distribution of global radiation and sunshine duration over the Island of Cyprus

Appl Sci

2023

;

5422

. https://doi.org/

[41]

Elmaihy

El Weteedy

Short-duration performance acceptance test of parabolic trough solar field: a case study

J Sol Energy Eng

20170510

;

139

. https://doi.org/

[42]

Strutz

Data Fitting and Uncertainty: A Practical Introduction to Weighted Least Squares and Beyond

, Vol.

Wiesbaden, Germany

Vieweg+ Teubner

2011

[43]

Pandey

AK.

Least Squares Curve Fitting: Original Data, Fitted Curve, and Residuals

2023

. https://arunp77.medium.com/least-squares-curve-fitting-original-data-fitted-curve-and-residuals-5349f1b1c892

(30 March 2024, date last accessed)

. https://cbe.udel.edu/wp-content/uploads/2019/03/FittingData.pdf

[44]

Shine

AD.

Fitting Experimental Data to Straight Lines (Including Error Analysis)

University of Delaware

2006

(16 September 2023, date last accessed)

10.1016/j.solener.2012.07.025

[45]

Mathioulakis

Panaras

Belessiotis

Uncertainty in estimating the performance of solar thermal systems

Sol Energy

2012

;

3450

–

. https://doi.org/

10.1016/j.solener.2009.10.024

[46]

Belessiotis

Mathioulakis

Papanicolaou

Theoretical formulation and experimental validation of the input–output modeling approach for large solar thermal systems

Sol Energy

2010

;

245

–

. https://doi.org/

. https://lepten.ufsc.br/publicacoes/solar/eventos/2006/EUROSUN/Manfred.pdf. (

[47]

Kratzenberg

Beyer

Colle

Analytic Determination of the Expanded Uncertainty for Steady State and Quasi Dynamic Collector Tests Under Outdoor Conditions.

2023

28 November 2024

, date last accessed).

[48]

Tasie

Sigalo

Omubo-Pepple

et al.

Modelling of solar power production in dry and rainy seasons using some selected meteorological parameters

Energy Power Eng

2022

;

274

–

. https://doi.org/

10.4236/epe.2022.147015

. https://textbooks.math.gatech.edu/ila/ila.pdf

[49]

Margalit

Rabinoff

Interactive Linear Algebra

2019

(15 April 2024, date last accessed)

[50]

Zaversky

García-Barberena

Sanchez

et al.

Probabilistic modeling of a parabolic trough collector power plant: an uncertainty and sensitivity analysis

Sol Energy

2012

;

2128

–

. https://doi.org/

10.1016/j.solener.2012.04.015

10.1016/j.solener.2006.08.007

[51]

Kim

Jenkins

Rumsey

et al.

Simulation and model validation of a horizontal shallow basin solar concentrator

Sol Energy

2007

;

463

–

. https://doi.org/

10.1016/s0196-8904(01)00180-7

[52]

Sabatelli

Marano

Braccio

et al.

Efficiency test of solar collectors: uncertainty in the estimation of regression parameters and sensitivity analysis

Energy Convers Manage

2002

;

2287

–

. https://doi.org/

. https://www.researchgate.net/publication/265107397_Numerical_Recipes

[53]

Press

Teukolsky

Vetterling

et al.

Numerical Recipes

Cambridge, UK

Cambridge University Press,

1992

(30 May 2024, date last accessed)

10.1007/s40095-014-0151-z

[54]

Mecibah

Boukelia

Benyahia

NE.

Management and exploitation of direct normal irradiance resources for concentrating

Int J Energy Environ Eng

2015

;

–

. https://doi.org/

10.1016/j.apenergy.2023.122144

[55]

Kosmopoulos

Dhake

Melita

et al.

Multi-layer cloud motion vector forecasting for solar energy applications

Appl Energy

2024

;

353

122144

. https://doi.org/