Simple Empirical Relation for an Evacuated-Tube Solar Collector Performance Prediction from Solar Intensity

Abstract

In this paper, the effect of solar intensity on the heat pipe tip temperature in a heat pipe type—evacuated-tube solar collector (HP-ETSC) was investigated. A simple relation was proposed, relating the solar intensity to the heat pipe tip temperature generated from the experimental data. This simple empirical relation was applied in a set of heat transfer equations derived to predict the heating medium temperature at the manifold outlet of the evacuated-tube solar collector. The calculated results corresponding to two types of heating medium, i.e., palm oil and water, were compared with experimental results from the literature. The results show that the average error was 6.41% for the case of palm oil and 4.66% for the case of water. Based on the case of water as a heating medium fluid, it was found that the flow rate of the heating medium fluid affected the accuracy of prediction, as the percentage error increased with the heating medium flow rate. The maximum percentage error increased from only 1.83% for a water inlet flowing at a Reynolds number of about 2.4 × 10³ to 15.23% for a water flow rate at a Reynolds number of about 2.6 × 10⁴. The correction factor was added into the correlation to predict the heat transfer coefficients of heating medium fluids. With this correction factor, the maximum error could be reduced from 11.78% to 7.29% for the palm oil case and from 15.23% to 5.57% for the water case. The average errors corresponding to palm oil and water cases could be reduced to 0.74% and 1.26%, respectively.

1. Introduction

Nowadays, awareness of global warming and climate change is such that many countries, including Thailand, have a common approach of wanting to reduce the amount of greenhouse gas emissions. Therefore, renewable energy has been developed to replace the use of fossil fuels by focusing on clean and sustainable energy, such as wind power, hydropower, biomass, biogas and solar power. Solar thermal energy is one of the renewable energy sources that never runs out.

The solar thermal collector is a device that converts solar radiation into thermal energy. There are three main types of solar collectors currently in use: the focusing solar collector, the flat plate solar collector (FPSC) and the evacuated-tube solar collector (ETSC). In this work, the ETSC was investigated. An evacuated tube is composed of two concentric glass tubes in which the inner one’s surface is coated with a radiation-absorbing substance and a vacuum is created between the two concentric tubes. The gap between them is evacuated to achieve excellent thermal vacuum insulation, leading to higher efficiency compared with other types of solar collectors [1,2,3,4]. The main feature is reducing the convection and conduction loss between the inner glass and ambient air. Several researchers have attempted to improve and develop the ETSC systems, aiming to increase efficiency and reduce investment costs. Li et al. [5] investigated the thermal performance and internal flow of horizontal double-row water-in-glass evacuated-tube solar collectors with different inclination angles. The three-dimensional simulation results were validated with an experimental result that measured temperature inside the tubes of the horizontal double-row ETSC. They found that the declination angle had a significant effect on flow patterns, energy conversion efficiency and stratification inside evacuated tubes. Papadimitratos et al. [6] modified ETSCs using solar water heaters by filling the tubes with some phase change materials, like tritriacontane paraffin and erythritol, aiming to store more thermal energy in the form of phase change (latent) heat in the system. They found that the dual PCM system had a longer cooling time than normal and kept the water at a higher temperature compared to the standard system during the evening and at night, when there was no solar radiation. Reducing thermal resistance through ESTC internal restructuring is another way to improve efficiency. Heyhat et al. [7] studied the effects of using copper metal foam, CuO/water nanofluids and the combinations of both on thermal efficiency. They found that the combination of copper metal foam and CuO/water nanofluids provided the maximum thermal efficiency, higher than that of purified water with copper metal foam and CuO/water nanofluid. Abd-Elhady et al. inserted thermal oil and foamed copper into the evacuation tubes to improve the heat transfer rate. They examined the cases of a normal evacuated tube, an ETSC filled with oil and an ETSC filled with oil and foamed copper instead of a finned surface. They found that the thermal efficiency of the ESTC increased in the case of replacing the finned surface with copper foam and filling it with oil. Supankanok et al. [8] modified the evacuated tube by inserting a cheap material like stainless steel scrubber in the inner tube gap to promote heat transfer and found that it should increase its efficiency by up to about 35%. The other way to help us understand the ETSC and to improve the performance of the ETSC is to perform process modelling and optimization. Kotb et al. [9] set up the mathematical model to estimate the thermal efficiencies of ETSCs with different arrangements and found that the optimum number of tubes and their arrangement depended on climate and operating conditions. They reported the maximum error of the predicted exit water temperature as 2%. The optimum arrangement could reduce the number of tubes by up to 41%. Mahdi et al. [10] reviewed and studied the hybrid heat transfer optimization techniques in thermal energy storage (TES) systems. They found that the best performance gain was achieved by using hybrid heat pipes with fins or metal foam and that using hybrid nanoparticles with metal foam or fins was more effective than using only nanoparticles. Nokhosteen and Sobhansarbandi [11] developed a modified resistance network (RN) model which accounted for ambient conditions and angular distribution of solar radiation on the periphery of the HP-ETSC to predict the temperature distribution and average fin temperature of the collector, reporting the maximum error of the model as 10%. Elsheniti et al. [12] developed a theoretical model to predict the performance of the ETSC. They found that, with general heat transfer equations, the maximum relative error between calculated and experimental results was 12.5%. However, when they added the term of thermal mass into the thermal energy balance equation, the maximum relative error could be reduced to 4.4%. They also proposed a new generalized correlation for ETSC efficiency that linked the mass flow and outlet temperature for high inlet temperature applications. Schumann et al. [13] simulated the system of an evacuated-tube solar collector with a low-cost diffuse reflector and found that the performance of the collector with a cheap diffuse reflector could be increased by up to 33% compared to that of the conventional one.

One of the greatest difficulties of simulating the solar thermal collecting system accurately is unreliable solar irradiance during the day, which directly relates to the amount of energy absorbed and transferred through the evacuated tube. Many researchers have developed theoretical models to predict the solar collector performance by estimating the heat absorbed from solar irradiance, which sometimes unpredictably fluctuates based on thermal radiation and heat transfer via heat pipe based on heat transfer principles [9,11,12]. In this manner, many complicated heat transfer equations with complicated thermal resistance networks inside the HP-ETSC must be set up and solved. However, the thermal performance of the HP-ETSC in view of its ability to heat up the heat pipe tip, where the thermal energy transfers directly to the heating fluid, has never been reported.

In this work, the heat pipe tip temperature change with solar irradiance was directly observed. From previous study by our research group, the efficiencies of the HP-ESTC and the modified HP-ESTC with the stainless steel scrubber were estimated by assuming that the tip temperature of the heat pipe stored in the evacuated tube was constant at 140 °C, which was the average tip temperature during the experimental period of 9:00 a.m.–4:00 p.m. [8]. However, the heat pipe tip temperature can vary, particularly during days with inclement weather. The main objective of this work was to investigate the effect of solar intensity on the temperature of the heat pipe tip, which is the area where heat directly transfers to the heat transfer fluid flowing along the manifold. Instead of calculating the heat absorbed from a set of heat transfer equations, the heat pipe tip temperatures at different solar intensities were collected. An empirical relation was generated to relate the solar intensity to the heat pipe tip temperature. The heat pipe tip temperature predicted from this simple relation was applied to estimate the temperature of the heat transfer fluid using MATLAB R2021b and compared with the experimental results. Correction factors were applied to compensate for the deviation due to the assumptions set up to simplify the system and calculation.

2. Experimental

The experimental set shown in Figure 1a was set up to measure the temperatures of the heat pipe tips. The experiment was set up at the School of Engineering, King Mongkut’s Institute of Technology Ladkrabang, Bangkok (latitude 13°43′35″ N and longitude 100°46′20″ E), while the experimental set in Figure 1b was set up for the heat pipe evacuated-tube solar collector performance.

As the climate cannot be controlled or simulated, the experiment cannot be directly repeated. To make the measured data reliable, three single evacuated tubes were installed under sunlight as shown in the figure. In this way, three tip temperatures were collected at the same time with the same solar intensity. A single evacuated-tube solar collector consists of a copper heat pipe that contains 10 mol% of ethylene-glycol (b.p. 197.3 °C) in water functioning as a working fluid. According to the vapor–liquid equilibrium of this binary mixture, the liquid mixture is completely vaporized at 138.7 °C [8], which is not too high for this mixture to absorb the solar thermal energy. It completely vaporizes in a short time and allows us to provide a higher heat pipe tip temperature than when using pure water. K-type thermocouples with an accuracy of ±1 °C (Comcube Co., Ltd., Bangkok, Thailand) were connected to the heat pipe tips to measure their temperatures. The data were recorded by a data logger. Solar intensity was measured every 30 min from 9:00 a.m. to 4:00 p.m. by using a solar power meter (TM-206, Tenmars Electronics Co., Ltd., Taipei, Taiwan) with an accuracy of less than ±5% error.

To observe the evacuated-tube solar collector performance, the experimental set shown in Figure 1b and set up in our previous work [8] was applied. A 20 ETSC set tilted 45° from the horizontal was connected to a storage tank, where palm oil with six storage capacities varying from 50 to 160 L was stored. The palm oil was circulated from the storage outlet through the manifold and flowed back to the tank with a constant mass flow rate of 1.92 kg/min. The manifold, pipes and palm oil storage tank were insulated with the thermal insulator (Aeroflex^®), as seen in the figure.

The evacuated tubes used in this work were supplied by Zhejiang Chinatide Solar & Luminous Energy, Co., Ltd., Haining, China. The ETSC system characteristics, including dimensions and materials of the evacuated tube, heat pipe, storage tank, aluminum fin and manifold, are shown in

Table 1. ETSC system characteristics [8].

Evacuated Tube			Heat Pipe
Inner diameter	0.047	m	Heat pipe material	Copper
Outer diameter	0.058	m	Inner diameter	0.012	m
Tube length	1.8	m	Outer diameter	0.014	m
Tube material	Borosilicate glass		Thermal conductivity	401	W/m.K
Absorptive coating	SS-Cu-AlN		Aluminum fin
Number of tubes	20	tubes	Fin thickness	0.5	m
Storage tank			Thermal conductivity	237	W/m.K
Heating medium	Palm oil		Manifold
Inner diameter	0.58	m	Inner diameter	0.019	m
Insulator thickness	0.05	m	Outer diameter	0.022	m
Tank height	0.76	m	Length	1.55	m

.

To perform the experiment, palm oil was circulated in the system. Solar intensity, manifold inlet temperature, manifold outlet temperature, ambient temperature and palm oil temperature inside the storage tank were measured and recorded every hour from 9:00 a.m. to 4:00 p.m.

3. Methodology

3.1. System Configuration

Figure 2 represents the components inside a heat pipe evacuated-tube solar collector. Figure 2a shows the schematic cross section sketched from the evacuated tube in Figure 2b, while Figure 2c shows the geometry and dimensions of an evacuated-tube solar collector.

A space between concentric glass tubes was vacuumed to reduce thermal energy loss from conduction and convection. The copper heat pipe was wrapped with an aluminum sheet to promote the heat conduction. The phenomenon occurring inside the heat pipe when absorbing heat from the sunlight is illustrated via Figure 3.

When the absorber coating on the surface of the evacuated tube absorbs energy from the sunlight (the yellow arrows) and transfers to the heat pipe, the working fluid is heated up until some or all of it vaporizes and flows up to the heat pipe tip due to natural convection as shown by the red arrow. Since the heat pipe tip is placed inside the manifold where the heat transfer fluid passes through, the latent heat of vaporization from the working fluid is released to the heat transfer fluid, causing the working fluid to condense and flow down the bottom of the heat pipe as shown by the blue arrow to re-store heat from the sunlight in a cycle if the solar intensity is high enough to boil the working fluid.

3.2. Heat Transfer Equations

Temperatures in the evacuated-tube solar collector system configurated in Figure 1b were predicted based on the heat transfer principle under some reasonable assumptions, listed below.

• The major heat loss is considered as the losses via the manifold and tank walls. The heat loss from the pipes connecting the manifold with the tank outlet is negligible. Therefore, the outlet temperature of the palm oil from the tank (T_tank) is assumed to be equal to the manifold inlet temperature (T_in,man).
• The palm oil in the tank is well mixed so that its temperature is uniform and equals the outlet temperature.

3.2.1. Manifold Domain

As previously described, thermal energy, mainly in the form of latent heat, transfers through the copper heat pipe tips to the palm oil flowing through the manifold. The heat absorbed along the manifold (${\dot{Q}}_{man}$) causes the palm oil temperature to rise from the manifold inlet temperature ($T_{in,man}$) to the outlet temperature ($T_{out,man}$), and hence, it can be calculated using the following equation.

$${\dot{Q}}_{man}=\dot{m}{c}_p(T_{out,man}-T_{in,man})$$

(1)

According to the assumption that the heat loss from the pipes linking the palm oil well-mixed tank with the manifold is negligible, the manifold inlet temperature equals to the temperature of palm oil in the tank ($T_{in,man}=T_{tank}$). The ${\dot{Q}}_{man}$ can also be calculated in terms of heat transfer from the surface of N heat pipe tips to the palm oil flowing along the manifold as follows.

$${\dot{Q}}_{man}=\dot{m}{c}_p(T_{out,man}-T_{tank})={hNA}_{tip}\Delta T_{lm}$$

(2)

where h is the heat transfer coefficient, $A_{tip}$ is the outer surface area of a heat pipe tip and $\Delta T_{lm}$ is the logarithmic mean temperature difference between the heat pipe tip ($T_{tip}$) and the palm oil heating medium given by

$$\Delta T_{lm}=\frac{(T_{tank}-T_{out,man})}{\ln [(T_{tip}-T_{out,man})/(T_{tip}-T_{tank})]}$$

(3)

Combining Equation (2) with Equation (3) gives the equation to estimate the temperature of palm oil leaving the manifold as

$$T_{man,out}=T_{tip}-(T_{tip}-T_{tank})\exp \left(\frac{-N{A}_{tip}h}{\dot{m}{c}_p}\right)$$

(4)

The heat transfer coefficient (h) can be determined through the Nusselt number, which is the ratio of convective to conductive heat transfer at a boundary in a fluid. In this work, the heat pipe tips were placed inside the manifold where the heating medium fluid flowed across. The pattern of the flow corresponded with the flow of fluid through one-line in-line tube banks, and hence, the Nusselt number correlation proposed by Zukauskas et al. [14,15] and shown in Equation (5) was modified and applied.

$$Nu=\frac{hD}{k}=fC{\mathrm{Re}}^m{\mathrm{Pr}}^n{(\mathrm{Pr}/{\mathrm{Pr}}_s)}^{0.25}$$

(5)

where C, m and n are constants depending on the range of Reynolds number (Re) of the fluid flowing through the tube banks, expressed in

Table 2. Constants in the Nusselt number correlation of Zukauskas et al. [14,15].

Range of Re	Constant
	C	m	n
0–100	0.900	0.40	0.36
100–1000	0.520	0.50	0.36
1000–200,000	0.270	0.63	0.36

, and f is a correction factor for one-line in-line tube banks which is equal to 1 for Re < 1000 and equal to 0.7 for Re > 1000. Re in this equation is calculated based on the fluid properties at the manifold inlet temperature and at the average velocity, while $\mathrm{Pr}$ and ${\mathrm{Pr}}_s$ are the Prandtl numbers calculated based on properties at the manifold inlet temperature and heat pipe tip temperature, respectively.

As seen in Table 2, the constants C and m depend on the range of Reynolds numbers. Since Reynolds number varies with linear velocity and temperature-depending properties like kinematic viscosity of fluid, its change should also affect the heat transfer coefficient value. Therefore, the correlation was modified before being applied to this work by fitting the median of each range of Reynolds numbers with the constants C and m to give these constants at a particular Reynolds number, as in Equations (6) and (7).

$$m=0.0294\ln (\mathrm{Re})+0.297$$

(6)

$$C=1.52{\mathrm{Re}}^{-0.153}$$

(7)

Heat loss from the manifold is estimated by considering the thermal resistance (the reciprocal of thermal conductivity) of each component arranged in series along the manifold cross-section, as shown in Figure 4. In each loop of calculation, we approximated the difference between the palm oil manifold inlet temperature and the ambient temperature as a driving force to calculate heat loss term.

The total thermal resistance of the manifold can be calculated as follows:

$$R_{man}=R_{palm}+R_{Cu}+R_{ins1}+R_{Al}+R_{ins2}+R_{air}$$

(8)

By considering the overall heat transfer coefficient timing area (UA) in the form of inversed total thermal resistance, the equation to estimate heat loss from the manifold surface is expressed as the following.

$${\dot{Q}}_{man,loss}=\frac{T_{tank}-T_{amb}}{R_{man}}$$

(9)

3.2.2. Storage Tank Domain

As the palm oil in the tank is circulated through the manifold to absorb heat from the sunlight, with the heat loss through the tank wall as ${\dot{Q}}_{loss}$, heat accumulated in the palm oil tank ($Q_{tank}$) can be calculated as follows.

$$\frac{d{Q}_{tank}}{dt}=m{C}_p\frac{d{T}_{tank}}{dt}={\dot{Q}}_{man}-{\dot{Q}}_{loss}$$

(10)

Considering the heat loss term ${\dot{Q}}_{loss}$, expressed in Figure 5a and represented in the form of an equivalent thermal resistance circuit in Figure 5b, the thermal resistances of the storage tank wall ($R_{wall,tank}$) and the bottom ($R_{botton,tank}$) can be calculated from Equations (11) and (12), respectively.

$$R_{wall,tank}=\frac{1}{h_{palm oil}{A}_{in,tank}}+\frac{\ln (r_{out,tank}/r_{in,tank})}{2\pi k_{tank}L}+\frac{\ln (r_{out,ins}/r_{in,ins})}{2\pi k_{ins}L}+\frac{1}{h_{air}{A}_{out,ins}}$$

(11)

$$R_{bottom,tank}=\frac{1}{h_{palm oil}{A}_{in,tank}}+\frac{r_{out,tank}-r_{in,tank}}{k_{tank}{A}_{out,tank}}+\frac{1}{h_{air}{A}_{bottom,ins}}$$

(12)

R_wall,tank is the thermal resistance of the wall of the storage tank, including the convective loss of palm oil to the radius of the inner storage tank wall ($r_{\mathrm{in},tank}$), the conductive loss through the thickness of the storage tank wall ($r_{out,tank}-r_{in,tank}$), the conductive loss through the thickness of the insulator ($r_{out,ins}-r_{in,ins}$), and the convective loss from the outer surface of the insulator to the ambient air. Meanwhile, R_bottom,tank is the thermal resistance of the bottom of storage tank. In Equations (11) and (12), $h_{palm oil}$ and $h_{air}$ are the heat transfer coefficients of palm oil and air, respectively, while $k_{tank}$ and $k_{ins}$ are the thermal conductivities of the tank wall and insulator, respectively. Combining the thermal resistances of the tank wall and bottom expressed by the equivalent thermal resistances network in Figure 5b, the total equivalent resistance of the storage tank ($R_{tank}$) can be determined as the following.

$$R_{tank}=\frac{1}{R_{top}}+\frac{1}{R_{bottom}}+\frac{1}{R_{wall}}$$

(13)

Therefore, the term ${\dot{Q}}_{loss}$ in Equation (10) can be calculated as a function of the total equivalent resistance as follows:

$${\dot{Q}}_{\mathit{loss}} {=UA(T}_{tank}{-T}_{\mathit{amb}})=\frac{{(T}_{tank}{-T}_{\mathit{amb}})}{R_{tank}}$$

(14)

where $T_{amb}$ is the ambient temperature.

Substituting ${\dot{Q}}_{man}$ from Equation (2) and ${\dot{Q}}_{loss}$ from Equation (14) into Equation (10) gives

$$\frac{d{T}_{tank}}{dt}=\frac{1}{{(m{C}_p)}_{tank}}[\dot{m}{C}_p(T_{man,out}-T_{tank})-\frac{(T_{tank}-T_{amb})}{R_{tank}}]$$

(15)

The physical and thermal properties of palm oil and water as a function of temperature (in °C) are expressed in

Table 3. Properties of palm oil and water.

Property	Palm Oil	Water
Density (kg/m3)	$\rho =0.004T^2-0.5279T+900.5$	$\rho =-0.0012T^2-0.0761T+1001$
Dynamic viscosity (kg/m·s)	$\mu =1.3284\times {10}^7{T}^{-1.63}$	$\mu =2\times {10}^{-7}{T}^2-3\times {10}^{-5}T+0.0016$
Heat capacity (J/kg·K)	$C_p=0.003T^2+2.5T+1796.6$	$C_p=-1.6768T+4211.4$
Thermal conductivity (W/m·K)	$k=1\times {10}^{-7}{T}^2-1\times {10}^{-4}T+0.1745$	$k=0.0012T+0.5643$

.

3.3. Calculation Flow Chart

The above system of equations shown in the previous section was solved by using MATLAB R2021b with the calculation steps shown in Figure 6.

Before running the program, the ambient temperature, manifold inlet palm oil temperature measured at the initial time, measured solar intensity, mass flow rate of the palm oil, time step and end time were input into the program. In the program, the ETSC starting time was set as t = 0. The time step and end time were set as 1 s and 7 h (9:00–16:00), respectively. Then, when the program was run, the heat pipe tip was calculated. Next, the properties of palm oil at the manifold inlet and the palm oil temperature were calculated following Reynolds number, Prandtl numbers and heat transfer coefficient calculation. Finally, the manifold inlet palm oil temperature and the palm oil temperature in the storage tank were calculated. The palm oil temperature in the storage tank was set as the manifold inlet palm oil temperature for the next loop. The program was terminated when the operating time was longer than the end time. The palm oil manifold inlet and outlet temperatures were exported and compared with the experimental results.

4. Results and Discussion

The main objective of this study was to generate the relation to estimate the heat pipe tip temperature, which changed with solar intensity during the day, and to use this relation for ETSC performance prediction. The results are presented as the following.

4.1. Correlation for Tip Temperature Prediction from Solar Intensity

To observe the change of heat pipe tip temperature with the variation of solar intensity, the heat pipe tip temperature was measured, and solar intensity was recorded from 9:00 a.m. to 4:00 p.m. for several days during September 2020, using the experimental set in Figure 1a. The solar intensities and heat pipe tip temperatures were plotted to investigate the trend of tip temperature change with the solar intensity as shown in Figure 7. It must be noted that commercial-grade evacuated tubes may not be identical. Some few differences, such as the thickness and uniformity of absorptive coating, degree of vacuum, amount and fraction of working fluid and arrangement of aluminum fin, can lead to significant overall difference in the performance of each tube. The error bars shown in this plot include mainly the unidentical performance of the evacuated tubes, while the bullet data represent their average performance at different solar intensity.

The relation between tip temperature ($T_{tip}$) and solar intensity ($I_G$) fitted with the plot in Figure 7 was obtained as an exponential function with R² = 0.7371. By adding the correction factor obtained from the average error into the equation, the empirical relation was provided as in Equation (16).

$$T_{tip}=11.5+236.9\left[\exp \left(\frac{-496.6}{I_G}\right)\right]$$

(16)

The simple empirical relation in Equation (16) was applied with the other set of solar intensities collected in June 2021 to predict the heat pipe tip temperatures and compare them with the measured ones. The results are shown in Figure 8.

It is seen that the calculated tip temperatures follow the trend of the measured tip temperatures that change with the solar intensity during the day, particularly at high solar intensity. As observed in the red dashed box, the predicted heat pipe tip temperature deviates more when solar intensity immediately drops because, though the solar intensity drops at a particular time, the heat absorbed still remains in the working fluid inside the heat pipe in the form of latent heat of condensation and sensible heat (Elsheniti et al. [12] defined this term as “the heat from thermal mass”) where the vaporization–condensation cycle occurs. It is also observed that the effect of this remaining heat declines within a short time, since the amount of working fluid stored in the heat pipe is very little (about 5 mL [8]). The predicted values are closer to the measured ones again when the solar intensity stays at a similar level. Overall, the average deviation of calculated results and experimental results was 9.8 °C, or about 7.10% deviation from the average measured temperature. It should be noted that the massive error compensated for in the all-day total error was at 16:00, when the solar intensity drastically dropped. Excluding this datum, the average error could be reduced to 6.0 °C (about 4.34%). Furthermore, the error is expected to decrease when increasing the frequency of solar intensity measurement.

Considering the error due to the measurement devices, the percentage error is defined as the following equation.

$$\% error=\frac{T_{measured}-T_{calculated}}{T_{measured}}\times 100$$

(17)

Since the accuracy of the solar intensity measurement device is ±5%, it can make the predicted tip temperature deviate in the range of ±3.5%. For the accuracy of the thermal couple specified as ±1 °C from the previous result, the average measured tip temperature was 138.5 °C, meaning that, for the accuracy in the range of ±1 °C, the error was in the range of −0.73 to 0.72%.

It must be noted that Equation (16) was generated from the collected data under the condition that the heat pipe tips were surrounded by ambient air. Therefore, the bare tips had a somewhat different heat transfer rate from those installed inside the manifold where heating medium fluid passed through. The direct application of this relation to predict the ETSC performance could cause some error due to this difference.

4.2. Verification and Validation of Mathematical Model

The palm oil outlet temperatures from the evacuated-tube solar collectors arranged in series were examined using heat transfer calculation and experiments, as described in the previous section. The initial temperatures of the inlet palm oil measured at the ambient level from the six experiments (six storage capacities) ranged from 32 °C to 34 °C [8], with a mass flow rate of 0.032 kg/s (Re ≈ 50). These were input in the MATLAB program. As previously mentioned, the experiment cannot be directly repeated with the same condition, as the solar intensity cannot be controlled; the calculated results compared with six sets of experimental data should make the validation of the mathematical model more reliable. The level of underestimation is shown through the plot in Figure 9. The diagonal line in the figure stands for 0% deviation from the experimental results. The average percentage deviations (A-PD) for different storage capacities lie in the range of 4.52–7.75%, while the overall absolute percentage deviation (ABS-PD) and maximum deviation (MAX-PD) are 6.41% and 11.78%, respectively.

The empirical relation proposed in Equation (16) was also applied with the experimental results reported by Elsheniti et al. [12], who investigated the thermal performance of the ETSC using water as a heating medium under different inlet water temperatures, numbers of evacuated tubes, water mass flow rates and solar intensities. In their work, they aimed to produce 70–90 °C hot water from a series of three sets of 15-tube ETSCs. The experiments were performed in Alexandria, Egypt. The information used for calculation is shown in

Table 4. Physical characteristics based on one set of ETSCs. Elsheniti et al. [12].

Evacuated Tube			Manifold
Outer diameter	0.058	m	Inner diameter	0.0158	m
Tube length	1.8	m	Outer diameter	0.0162	m
Number of tubes	15	tubes	Length	1.0	m
Heat pipe material	copper		Overall heat transfer	0.06	W/m2·K
			coefficient (Uloss)

.

Apart from the type of heating medium fluid, the difference between the experiment of Elsheniti et al. and our experiment is that in their experiment, the temperature of the water entering the ETSC manifold in each cycle was controlled by a fan cooling coil and a heater, while in our experiment, the palm oil was directly circulated in the system. In their experiments, the flow rates of water were controlled at 0.03, 0.065, 0.11 and 0.3 kg/s, with water inlet temperature ranges of 32–34 °C, 70–85 °C, 53–76 °C and 36–40 °C, which corresponded to Reynolds numbers calculated based on the water manifold inlet temperatures at about 2.4 × 10³, 9.0 × 10³, 1.5 × 10⁴ and 2.6 × 10⁴, respectively. As a different heating medium was used, the order of magnitude of Reynolds numbers with respect to Elsheniti et al.’s work was two to three orders higher.

The calculation steps followed the procedure in Figure 6 from “Start” to “Calculate $T_{man,out}$”. Since in Elsheniti et al.’s work, the water inlet temperature was controlled by a fan cooling coil and a heater, the water outlet temperature was calculated based on the reported water inlet temperature at a particular time. The degree of deviation can be considered via the plot in Figure 10.

From Figure 10, the maximum error increased with increasing water mass flow rate, i.e., 1.83, 4.50, 6.14 and 15.23% at the water flow rates of 0.03, 0.065, 0.11 and 0.3 kg/s, respectively, while the overall absolute percentage deviation was 4.76%. This means that the greater the Reynolds number, the higher the error. This might be because of the deviation in the heat transfer coefficient predicted by the correlation in Equation (5), which was fitted with the data from multiple-line tube bank systems. Although it was suggested that we add the correction factor of f = 0.7 into the correlation for cases of fluid flow with Reynolds numbers greater than 1000, this correction factor may not be a constant for the whole wide range but may vary with the Reynolds number. The other possible reason is that the Reynolds numbers substituted into the correlation in this work are the ones calculated from average velocities, but Zukauskas et al. [15], who proposed the correlation, defined Reynolds number as a function of the maximum velocity (V_max), which was calculated based on the geometry of the tube banks as follows.

$$V_{\max }=\frac{S_T}{S_T-D}V$$

(18)

where V, S_T and D are the average velocity, distance from the center of the tube to the center of the tube in the adjacent line and diameter of the tube, respectively. Therefore, the maximum velocity defined previously cannot be applied to a one-line tube bank system, which may lead to the deviation of the predicted heat transfer coefficient. Considering the correlation, the greater the Reynolds number deviation, the more the heat transfer coefficient increased.

Elsheniti et al. [12] reported that, compared with the experimental result for the water mass flow rate of 0.03 kg/s, the maximum error of the case without adding thermal mass into the energy balance equation was as high as 12.5%, while the maximum error of the case with added thermal mass term reduced to 4.4%. Compared with this work, the maximum error for the water mass flow rate was only 1.83%. For high water mass flow rates, the theoretical model proposed by Elsheniti et al. provided the results with a maximum error of 3.6%.

From the previous results, the average deviations of the predicted ETSC outlet temperatures from the measured ones for both palm oil and water cases at different Reynolds numbers of heating medium fluids are summarized in

Table 5. Shows the tendency of the average errors of the ETSC performance prediction as a function of Reynolds number.

Re	Average Error (%)	Heating Medium Fluid	Source of Experimental Data
50	6.41	Palm oil	This work
2400	−0.40	Water	[12]
9000	−3.55	Water	[12]
15,000	−5.14	Water	[12]
26,000	−10.92	Water	[12]

.

It is seen that the deviation of the prediction changes from under-prediction at a low Reynolds number into over-prediction when the Reynolds number increases, and the degree of over-prediction increases when increasing the Reynolds number. As previously described, the relation used to calculate the heat pipe tips was set up from the measured heat tips installed in ambient air. Under this condition, heat from the heat pipe tips transferred to almost stagnant air. The heat release rate from the heat pipe tips and, therefore, heat absorbed from the sunlight should be significantly lower than those occurring in the manifold, where the heating medium fluid was circulated. Meanwhile, the higher the Reynolds number of the fluid, the higher the heat transfer rate obtained.

With the previous observation, the correction factor α was generated and added into Equation (5) to compensate for the deviation, giving

$$Nu=\frac{hD}{k}=\alpha fC{\mathrm{Re}}^m{\mathrm{Pr}}^n{(\mathrm{Pr}/{\mathrm{Pr}}_s)}^{0.25}$$

(19)

$$\mathrm{where} \hspace{1em}\hspace{1em}\hspace{1em}\alpha =\frac{2400}{\mathrm{Re}}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{for} \mathrm{Re} < 2500$$

(20)

$$\mathrm{and} \hspace{1em}\hspace{1em}\hspace{1em}\alpha =\frac{5500}{\mathrm{Re}} \hspace{1em}\hspace{1em}\hspace{1em}\mathrm{for} \mathrm{Re} > 2500$$

(21)

Figure 11 shows the comparisons of measured and predicted inlet and outlet palm oil temperatures at various palm oil capacities without and with application of the correction factor. The dashed lines stand for the calculated results without applying the correction factor into the heat transfer coefficient correlation, while the solid lines are the calculated results after applying the correction factor in the correlation.

It is seen that the calculated results show good agreement with the experimental results. The predicted outlet and inlet temperatures are close to the measured ones, particularly during the initial hour, and deviate from the measured values when operating longer. This should be because the heat transfer coefficient calculated from Equation (5) was based on the manifold inlet temperature. However, when the palm oil passed the heat pipe tip, its temperature rose, resulting in an increased Reynolds number and hence enhancing the heat transfer coefficient along the manifold. As a result, small errors accumulated in every palm oil circulating cycle, causing the calculated results to be underestimated more when operating longer. For every storage capacity, the calculated results are slightly underestimated.

For the case of water, the calculated results in comparison with the experimental results are shown in Figure 12a–d. In terms of the water inlet temperatures, their experiments can be divided into two cases, which are (1) the low inlet temperature case (32–40 °C) as in Figure 12a,b and (2) the high inlet temperature case (53–85 °C) as in Figure 12c,d.

It is seen that the solar intensity change follows a similar trend for every experiment, i.e., it increases in the morning, reaches the highest level at about 12:00, stays at a similar level for 2–3 h and then decreases. No extreme drop of the solar intensity is observed for any experiment. The solar intensity level seems to be higher for the experiments with the water mass flow rates of 0.03 and 0.3 kg/s (Figure 12a,b). Moreover, it can be observed that the difference between inlet and outlet temperatures of water decreases when decreasing the solar intensity.

It is observed that without applying the correction factor α, the predicted outlet temperatures are close to the experimental results for the entire period of operation, particularly at low mass flow rates. The deviation of calculated results from the measured values is maintained at the same level for the entire operation. This should be because no error accumulation occurs from the heating medium circulation without controlling the inlet temperature, as was discussed earlier in the case of palm oil experiments. As described earlier, the calculated values are more overestimated when the flow rate of water is increased for both inlet temperature ranges, which should be because of more deviation from the system, which the relation to predict the heat pipe tip temperature was derived from while the Reynolds number of the fluid was increased. This deviation was compensated for by adding the correction factor term into the heat transfer coefficient correlation. As observed in Figure 12, the accuracies of the predictions are significantly improved.

Figure 13 shows the percentage errors of the predicted heating medium outlet temperatures after adding the correction factor α into the correlation to estimate the heat transfer coefficient. It is seen that the prediction is obviously more accurate for both palm oil and water cases. The maximum percentage error of the palm oil case decreases from 11.78% to 7.29%, while the maximum percentage error of water case decreases from 15.23% to 5.57%.

The empirical relation generated in this work should be general enough to be applied to predict the performance of a similar type and configuration of HP-ETSCs operated at different locations with different heating mediums and for different mass flow rates in the range of Reynolds numbers up to 3.0 × 10⁴. However, it should be noted that there are other parameters, such as the period of time during the year and inclination angle of the ETSC, which affect the heat pipe tip temperature and therefore, influence the accuracy of the ETSC performance prediction [16]. To evaluate these effects, further investigations should be carried out in the future.

5. Conclusions

The empirical relation relating the heat pipe tip temperature of the heat-pipe-type evacuated-tube solar collector was generated and applied to calculate the heating medium temperature at the manifold outlet. The calculated results were compared with the experimental results in which palm oil and water were used as heating medium fluids. The conclusions are as follows.

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The calculated results show good agreement with the experimental results for both heating medium types. The overall average errors for palm oil and water cases were 6.41 and 4.66%, respectively.

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For the case of palm oil as a heating medium, there was no significant effect of palm oil capacity on the accuracy of the model. The average percentage errors were in the range of 4.52–7.75% for the palm oil storage volume of 50–160 L. The maximum percentage error was 11.78%.

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For the case of water as a heating medium, the prediction was very accurate at a low water flow rate. The percentage error increased with the flow rate of water. The average and maximum percentage deviations were in the range of 0.80–10.52% and 1.83–15.23%, respectively, for a water flow rate range of 0.03–0.3 kg/s.

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Adding the correction factor α into the heat transfer coefficient correlation could significantly reduce the overall maximum errors from 11.78% to 7.29% for the palm oil case and from 15.23% to 5.57% for the water case, while the overall average errors were 0.74 and 1.26%, respectively.

In summary, compared with prediction by theoretical heat transfer models of HP-ETSC, the prediction using the simple empirical relation provided comparable relative errors. The simplicity of the relation could provide some advantages in some applications, such as setting up the transfer function in a control system or a cursory estimation when lacking some HP-ETSC characteristics and properties.