Effects of pulsating flow frequency and dimensionless amplitude on the thermal performance of SEGS LS-2 parabolic trough solar collector

Abstract

This study presents a numerical investigation of the combined effects of frequency and dimensionless amplitude of sinusoidal pulsating flow on the thermal performance of the SEGS LS-2 parabolic solar collector. The analysis combines a MCRT method for non-uniform solar flux distribution with sinusoidal inlet conditions implemented through user-defined functions (UDFs) in ANSYS Fluent using the RNG k-ε turbulence model. The effect of pulsating flow parameters (frequency: 0.2–6 Hz; dimensionless amplitude: 0.3–0.9) on the heat transfer characteristics, including Nusselt number and thermal efficiency, was evaluated using Syltherm 800 oil as the heat transfer fluid (HTF) at a Reynolds number of 4761. The results showed that at a frequency of 5 Hz (Strouhal = 0.131) and a dimensionless amplitude of 0.5, the time-averaged Nusselt number reaches 5.1 and the time-averaged thermal efficiency reaches 77%, while according to previous findings, the highest thermal efficiency in steady state is reported to be 74%. Increasing frequency turbulence in the viscous sublayer causes the growth of the hydrodynamic layer and improves heat transfer. Higher frequencies (e.g. 6 Hz) cause turbulence to freeze and limit the increase in performance. These findings highlight the potential of frequency-controlled solar collectors in areas with high solar flux. The use of frequency valves, while saving additional costs, also increases the thermal efficiency by 3-4.5%.

Introduction

The global surge in energy demand, coupled with environmental concerns over fossil fuel use, has accelerated the adoption of renewable energy sources, particularly solar energy, due to its widespread availability and minimal environmental impact¹. Parabolic trough collectors (PTCs), such as the Solar Electric Generating System (SEGS LS-2), are critical for harnessing solar energy by concentrating radiation onto an absorber tube to heat a working fluid for applications like power generation and industrial heating². Enhancing the thermal efficiency of PTCs remains a key research focus, with studies exploring geometric modifications, advanced heat transfer fluids, and novel numerical methods.

Pioneering work by Dudley et al.² experimentally optmized the SEGS LS-2 collector by varying material properties, achieving improved thermal performance. Sadaghiyani et al.³ used Monte Carlo Ray Trace (MCRT) methods to model non-uniform solar flux, demonstrating that reflector geometry adjustments enhance energy absorption. Cheng et al.⁴ coupled finite volume and MCRT methods to study the impact of Syltherm 800 and vacuum conditions on PTC efficiency. Mwesigye et al.⁵ introduced perforated plates to increase heat absorption, while Wang et al.⁶ analyzed thermal stresses in PTC systems. Recent studies have explored advanced techniques, such as Vahidinia et al.⁷, who reported an 88% thermal efficiency improvement using combined finned absorber tubes, and Panja et al.⁸, who achieved an 86.9% increase in Nusselt number with nanofluids and porous inserts.

In parallel, pulsating flow studies in simpler geometries, such as horizontal pipes or channels, have shown that frequency and amplitude variations enhance turbulence and heat transfer near walls^9–14. For instance, Kurtulmuş et al.¹⁵ reported a 1.5-fold increase in heat transfer coefficient under pulsating conditions, while Papadopoulos et al.¹⁶ noted turbulence freezing at high frequencies. However, To the best of the authors’ knowledge and based on a comprehensive literature search conducted in Scopus and Web of Science, only a limited number of studies have addressed the effects of pulsating flow in tubes. One of the points that has led to the reason and nature of the variation of the heat transfer coefficient in pulsating flow is the lack of consideration of viscous sublayers near the walls. In experimental studies, no precise method has been proposed to observe the hydrodynamic changes of viscous sublayers, or the necessary and sufficient hardware facilities have not been provided for such a task. Numerical methods such as the Re-Normalization Group (RNG) k-ε turbulence model have the ability to predict the hydrodynamic changes in the boundary layer. Also, this method does not reduce the accuracy of numerical results in other spaces such as areas near the center of the tube. It should be noted that factors such as the type and dimensions of the geometry and the type of fluid affect the rate of change in thermal efficiency resulting from pulsating flow. The geometry of the LS2 type of parabolic collector has high potential for frequency sensitivity due to its large lateral surface area and small cross-sectional area. The objectives of the present study are threefold:

(a) To numerically investigate the combined influence of sinusoidal pulsation frequency (0.2–6 Hz) and dimensionless amplitude (0.3–0.9) on the time-averaged Nusselt number, heat transfer coefficient, and thermal efficiency of the commercial SEGS LS-2 parabolic trough collector using Syltherm 800 as the heat transfer fluid at Re =4761.

(b) To identify the optimum frequency–amplitude combination that maximizes thermal performance and to explain the underlying physical mechanisms, including enhancement of near-wall turbulence in the viscous sublayer and the onset of turbulence freezing at higher frequencies.

(c) To evaluate the technical feasibility and economic viability of implementing frequency-controlled actuation systems (solenoid or rotary valves) in large-scale solar fields, thereby providing a practical pathway for efficiency improvement in existing and future PTCs plants.

Research significance

By reviewing and studying reputable articles from recent years, it can be concluded that investigating the thermal efficiency of flat or PTCs using sinusoidal inlet flow with different frequencies has not received the attention of researchers to date. So that in a number of numerical articles, two-phase methods and nanoparticles have been used to increase the thermal efficiency of the collector, or in others, adding porous space has been the main topic of research. In a limited number of studies, such as the articles by Sadaghiyani and Cheng^3,4, the main discussion has been about how to model the PTC and the amount of energy absorbed from the reflector, and in this regard, the use of the MCRT algorithm (the method used in this study) has been proposed. In the past five years, a number of authors, including Kurtolmus¹⁵, Shu¹⁷, Molochnikov¹⁸ and Su¹⁹, have investigated the effect of frequency on the thermal performance of simple or finned horizontal tubes. A study of the paper by Navinkumar²⁰ also shows that researchers have gradually sought solutions (including the absorber tube rotation method) to increase the thermal efficiency of PTCs. Finally, due to the lack of comprehensive research, the very small number of reliable articles, the dispersion of results and the ambiguities in the results have led to the idea of investigating the effect of frequency and dimensionless amplitude variables on the thermal performance of the PTC. Therefore, contrary to previous studies that were limited to simple horizontal tubes and did not take a scientific and accurate look at the truth of the matter, this study investigated the effect of pulsed flow on the thermal performance of a PTC. It is expected that changing the frequency and amplitude within a certain range will lead to changes in the hydrodynamic behavior of the fluid in the boundary layer and change the mechanism of fluid motion in viscous sublayer, which is likely to result in greater absorption of solar energy by the heat transfer fluid (HTF). Numerical studies of frequency and dimensionless amplitude variables require the use of unsteady and turbulent numerical methods. Numerical studies of boundary layers near the wall, including the viscous layer and the buffer zone, are also among the most important discussions of the present research. Therefore, considering the imbalance and excessive consumption of fossil energies in the world, it is hoped that the frequency PTC can be a suitable alternative to fossil energy consuming devices. In this study, frequencies greater than 6 Hz and dimensionless amplitudes greater than 0.9 were not studied due to the occurrence of freezing and backflow phenomena, respectively.

Also, to highlight the novelty of this study, a comprehensive comparison of the present work with previous studies of pulsating flow and PTCs is presented in Table 9.

Table 9.

Comparison of pulsating flow studies and parabolic trough collector research.

Study	Flow type	Geometry	Method	Gap filled	Key findings	Key limitations	Ref
Present work	Unsteady pulsating flow coupled with MCRT algorithm	SEGS LS-2 PTC	Numerical	Combined effects of frequency and dimensionless amplitude on SEGS LS-2 thermal performance	77% thermal efficiency at Fe= 5 Hz (St=0.131), A=0.5	_	_
Dudley et al.	Steady flow	SEGS LS-2 PTC	Experimental	Baseline thermal efficiency and heat loss reduction	70% efficiency	baseline steady conditions and no efficiency improvement	2
Kurtulmus et al.	Pulsating flow	Sinusoidal channel	Experimental	Studied pulsating flow in wavy channels	1.5x increase in heat transfer coefficient at St=1.03, Re=4000	constancy of heat flux and the thermodynamic properties of HTF	15
Xu et al.	Pulsating flow + nanofluids	Microchannel heat sink	Experimental	Combined pulsating flow and nanofluids	16.5% increase in Nusselt number with square pulse at 3.5–4.5 Hz	The high price of nanofluids, uniform heat flux and Dimensionless amplitude constancy	17
Molochnikov et al.	Pulsating laminar flow	Smooth horizontal pipe	Experimental	Studied sinusoidal pulsating flow effects	Increase local Nusselt number in Stokes region at high frequencies	Using a simple horizontal pipe, laminar flow and uniform heat flux	18
Naveenkumar et al.	Steady flow	PTC	Experimental	Introduced alternating rotation of absorber tube	18% increase in outlet temperature; 39% increase in heat transfer rate	baseline steady conditions and high implementation costs	20
Cheng et al.	Steady flow	PTC	Numerical	Studied fluid type and vacuum effects on efficiency	Fluid properties and vacuum conditions impact efficiency	baseline steady conditions and no efficiency improvement	4

SEG-LS2 PTC model

PTCs are divided into two general categories: concentrating and non-concentrating. Concentrating collectors consist of parabolas that reflect sunlight onto their focal point. By placing these parabolas together and moving a fluid over their focal point, the solar radiant heat can be absorbed and used for heating or electricity generation. PTCs are of the concentrating type, and flat and vacuum collectors are of the non-concentrating type. In this study, an attempt was made to show the effect of frequency and dimensionless amplitude variables of sinusoidal inlet flow on thermodynamic variables such as heat transfer coefficient, Nusselt number, and thermal efficiency of the collector. In general, a PTC consists of the following components²¹. Table 1 presents the dimensions and material type of PTC components.

• A.Reflector
• B.Absorber Tube
• C.Evacuated Space
• D.Heat Transfer Fluid
• E.Glass Tube
• F.Solid Plug

Table 1.

Dimensions and material type of SEG-LS2 PTC components.

Component	Dimension	Material type
Reflector	A=39 m2 and rim angle= 90 degree2	Stainless steel2
Absorber tube	L=7.8 m and Dh=0.004 m21	Copper2
Evacuated space	L=7.8 m and Dh=0.039 m21	Air21
Heat transfer fluid	L=7.8 m and Dh=0.015 m21	Syltherm 80031
Glass tube	L=7.8 m and Dh=0.006 m21	Pyrex glass antireflectiv21
Solid plug	L=7.8 m and D=0.05 m21	Stainless steel2

As can be seen in Fig. 1, the reflector or reflecting plate receives sunlight and reflects it in different directions. Since energy is transferred in the form of radiation, there is no need for a material medium. Therefore, this form of energy is absorbed at different angles on the lateral surface of the tube. A very important point is the amount of radiation energy absorbed by the absorbing plate. Sunlight enters the reflector with an approximate and constant radiant energy of 1000 W/m². The reflector, keeping the energy constant, emits it in different directions. Given the low radiation absorption rate of Pyrex glass and the passage of radiant energy through a vacuum, more than 95% of the solar radiation energy is absorbed by the absorber tube. Also convection and radiation losses to the ambient were conservatively neglected due to their minor contribution (<1 W/m²K). However, the amount of absorption also depends on the angle of sunlight emission and rim angle of the reflector. Some parts of the tube have more energy and other parts have a smaller share. According to experiments conducted by Abed et al.²² on the SEG-LS2 model with Syltherm 800 operating fluid, the radiation absorption rate has been reported in different areas of the absorber tube surface. The reports based on the absorbed heat flux and the absorber tube angle are depicted in Fig. 2.

Fig. 1.

Components of a SEG-LS2 PTC²¹.

Fig. 2.

Variation of absorbed heat flux under absorber tube angle (MCRT algorithm) [21 and 22].

Computational methodology

Governing equations

All numerical methods are defined based on the fundamental equations of continuity, momentum, and energy. Equations equation (1), equation (2) and equation (3) represent the continuity, momentum, and energy equations, respectively.

1

2

3

Considering the importance of examining the amount of longitudinal and transverse turbulence resulting from changes in frequency and dimensionless amplitude variables and using the ANSYS Fluent software guide²³, the RNG type turbulence equation ƙ-Ɛ is proposed. RNG k-ε model with modifications derived from the RNG theory, provides higher accuracy in predicting eddy flows and sinusoidal oscillations in bulk and near-wall regions. With the Enhanced Wall Treatment (EWT) method and y⁺=1 meshing, this model predicts the viscous layer behavior well and provides good numerical stability in unsteady simulations with User-Define Function (UDF).

In this solution method²⁴, the terms related to the Navier-Stokes equation²⁵ are solved instantaneously. Therefore, if the velocity profile near the walls can be changed by applying frequency and changing the velocity in the viscous and buffer sublayers, it can be hoped that the heat transfer coefficient rate will also increase with increasing turbulence near the wall. Considering the importance of the boundary layer variables and viscous sublayers, the use of the EWT method is proposed to model the areas near the tube wall. In this paper, equation (4) and equation (5) have been used to define the RNG model, and equation (6) and equation (7) have been used to investigate turbulent flow near walls.

4

5

6

7

In the presented equations, G_k denotes the turbulence produced by changes in average velocity (kinetic energy). G_b represents the turbulence produced by buoyancy forces, Y_m is the ratio of the expansion changes of compressible fluids to the total loss flux, α is the inverse of the effective Pr number, u+ refers to the dimensionless velocity, and y+ represents the dimensionless distance from the wall.

Numerical approach

In order to solve the problem, it is necessary to present and simplify all numerical methods and algorithms appropriate to the type of flow. Syltherm 800 oil is an incompressible fluid, and for this reason, the pressure-based solver was used. Considering the use of the pressure-based solver, solving the governing equations requires determining the methods of separating the equations and interpolating the pressure. Considering the type of problem, the Quick Algorithm was employed to separate the equations and the Standard algorithm was used to interpolate the pressure. The Simple Algorithm was used to connect the continuity and momentum equations. In the pressure-based solver, the under relaxation factor is used to control the numerical error of the variables. Considering the nonlinearity and simultaneousness of the governing differential equations, the under relaxation factor was decreased. with using this method, the probability of divergence of the results will be much lower. In the present study, an under relaxation factor²⁶ of 0.3 was used. the convergence criterion for the flow and energy solving variables is the residual value less than 10⁻⁶. Figure 3 shows the residuals under iteration changes. Also viscous heating (0.5% I_b) is negligible compared to solar radiative input, and enabling it would introduce numerical errors and artificial overheating. Thus, it was disabled per standard solar collector CFD modeling practices to maintain result accuracy. in the present simulation using the RNG k-ε model with EWT method, the No Slip boundary condition with zero surface roughness (Roughness Height = 0) was applied to the absorber tube wall, equivalent to a smooth surface where roughness effects were not investigated.

Fig. 3.

Unsteady numerical solution residuals under iteration variations.

Geometric and grid design

The SEGS-LS2 PTC, according to the experimental data of Dudley et al.², has a hydraulic diameter of 15.2 mm and a length of 7.8 m. Moreover, the inner and outer radius of the absorber tube are reported to be 0.0254 and 0.0330 m, respectively. In the present study, Gambit software²⁷ was used for modeling, grid creation, and boundary conditions. Figure 4 shows a three-dimensional view of the designed collector along with an organized rectangular grid. The quality of the grids was examined from two perspectives: skewness and orthogonality of the grid lines, and due to the selection of an organized rectangular grid, no distortion or angle is seen between the perpendicular lines of the rectangular grids. Considering the importance of studying the heat transfer in the boundary layer, the grids of the viscous sublayer are defined in 10 main layers with a geometric growth of 1.2, so that the viscous sublayer can be modeled by considering y+ of about one.

Fig. 4.

3D view of the model with meshing of the viscous sublayer.

Grid independency

In order to determine the required number of grids, it is necessary to examine the sensitivity of the results to the number of grids. Figure 5 examines the sensitivity of the numerical results to increasing the number of grids. This Figure is drawn in the range of 500,000 to 3,200,000 grids. As shown in Fig. 5, from 500,000 to about 2,000,000 grids, the outlet temperature decreases significantly, but with further increase in the number of grids, the sensitivity decreases. In the range of 2,120,000 to 3,200,000 grids, the change in the numerical results is very small and can be ignored. Therefore, in all the present numerical analyses, the number of grids is 2,120,000. Table 2 shows the grid independence for the six types of grid.

Fig. 5.

Change in absorber tube outlet temperature under changes in grid number.

Table 2.

Grid independence test.

Grid	Radial nodes	Angular nodes	Axial nodes	Total nodes	Tout	Ɛ with exp (%)2
1	22	40	100	512000	448.12	0.374
2	30	50	150	1275000	447.83	0.309
3	34	54	180	1846800	447.76	0.293
4	38	56	200	2120000	447.68	0.279
5	42	60	220	2642000	447.63	0.275
6	46	70	240	3220000	447.61	0.266

Boundary conditions

Numerical analyses using the Finite Volume Method (FVM) require the determination of boundary conditions and the type of solution. To determine the boundary conditions, as mentioned, the C programming language was used. The outer surface of the collector receives solar radiation in a time-constant and spatially variable manner, therefore, the boundary condition used for the outer surface is defined as heat flux. The energy of sunlight is transferred to the surface of the absorber tube by a reflector with an area of ​​39 m², so that about 0.96% of the energy is absorbed by the outer surface of the absorber tube of the collector. Also, due to the low wind speed in hottest and dry regions and the large hydraulic diameter of the annulus space compared to the diameter of other PTC components, heat losses resulting from convection around the PTC have been ignored. The non-uniform solar heat flux distribution on the outer surface of the absorber tube was implemented using a UDF based on the MCRT results reported by Dudley et al.² for the real SEGS LS-2 collector under clear-sky conditions (I_b = 968.2 W/m²). This approach reproduces the experimentally measured Local Concentration Ratio (LCR) without the statistical noise inherent in direct MCRT coupling. The angular variation of the LCR is described by a third-order polynomial fitted to the experimental data. Equations (8) and (9) show the variation of the LCR in terms of (degree). the MCRT algorithm was created using numerical functions in the form of identifiable codes and entered into the ANSYS Fluent software. The written codes operate like a MCRT algorithm, as shown in Fig. 6, which shows the temperature distribution of the outer surface of the absorber tube. The coding method is also used to create the flow inlet boundary condition. The function defined through the Define menu and the UDF^28,29 submenus is compiled and defined in the software memory as a new boundary condition. The flow inlet is one of the time-varying and space-constant boundary conditions, meaning that it is constant on any surface where it is defined and changes based on time changes. The added code is presented as a mass flow rate and is called the Mass Flow Inlet boundary condition. The principles of its definition are based on equations (10), (11) and (12). It should be noted that in the present study, the dimensionless amplitude is defined as the ratio of the maximum velocity of the oscillating flow to the average velocity of the flow. In this definition, the average velocity of the oscillating flow and the non-oscillating flow are the same, and instantaneous decreases and increases in velocity occur around the average velocity.

8

9

10

11

Fig. 6.

Temperature distribution of the absorber outer wall surface in the present study.

12

Also, the pressure Outlet boundary condition is used for the flow output from the collector.

LCR validation

Figure 7 shows the changes in LCR values ​​with respect to changes in the angle of the outer surface of the absorber tube. In this Figure, the values ​​obtained by the UDF method are compared with the experimental results of the Dudley². Table 3 shows the coefficients of Equation (9) for 7 angle ranges. Also, in this Table, the numerical LCR values ​​obtained by the UDF method are compared with the experimental results of the experiment, and the average error obtained is about 3.7 %.

Fig. 7.

LCR Validation of the present results with experimental results².

Table 3.

Coefficients of LCR equation and numerical LCR error.

zone	(degree)	B0	B1	B2	B3	LCR	Ɛ with exp (%)2
1	0–75	1.112046	−8.100954E-4	−1.07117E-4	0	0.84	8.53
2	75–104	1.685403E3	−5.97439E1	6.87860E-1	2.54443E-3	25.47	1.92
3	104–171.2.2	3.275329E2	−5.280388	3.196692E-2	−6.602394E-5	37.15	0.87
4	171.2–188.8.2.8	4.957224E3	−5.488588E1	1.524597E-1	0	22.21	1.11
5	188.8–256.8	−4.403785E2	6.979938	−3.504845E-2	5.961826E-5	38.54	1.02
6	256–285	−4.480387E4	5.427366E2	−2.01952	2.493475E-3	24.47	2.34
7	285–360	−9.66886	5.688045E-2	−7.511141E-5	0	0.97	7.12

Unsteady flow simulation

Since all variables, including frequency and dimensionless amplitude, are time-dependent, the solution type will be unsteady and all time equations are implicitly solved. The time step is a variable that is strongly dependent on the grid dimensions and the Courant number. To solve all variables within the time of numerical analyses, it is necessary that the Courant number is less than one. Therefore, the Courant number is defined as equation (13).

13

In all numerical analyses³⁰, the range of dimensionless amplitude changes (A) is defined between 0.3 and 0.9. Also, the frequency is defined with the symbol Fe and was numerically investigated in the range of 0.2 to 6 Hz. Also, based on the experimental data of Dudley et al.², Re = 4761 corresponds to the typical turbulent flow regime of existing SEGS LS-2 installations at design mass flow rate. For this reason, a Reynolds number of 4761 has been used in all numerical simulations.

Time step independency

Choosing a time step appropriate to the grid dimensions is very important because in any situation the Courant number must be less than one. By applying this restriction and assuming that the dimensions of the cells are constant, the time step variable will be decisive. Considering that the frequency changes in each analysis, it is necessary to change the time step to maintain the limits of the Courant number. For the analysis with a frequency of 1 Hz, a time step of 0.08 s was used, and for the analysis with a frequency of 6 Hz, a time step of 0.007 s was used. For this reason, the time required for convergence of the results for small frequencies was about 8 hours and for larger frequencies, including the frequency of 6 Hz, more than 170 hours. Considering the unsteady solution of the equations for a period of 20 s, it is necessary to present the independence of the numerical results from the time step. Figure 8 shows the changes in outlet temperature and absorber outer wall temperature in the time range of 0 to 20 s. The left vertical axis shows the absorber outer wall temperature and the right axis shows the outlet temperature of the absorber tube. The numerical results are shown for three time steps of 0.03, 0.05 and 0.08. By observing Fig. 8, it can be seen that there is no difference in the results obtained under different time steps, so the main reason for using smaller time steps is to satisfy the condition of using the RNG model for unsteady numerical analyses. In this regard, considering the constant number of grid and the average flow velocity, the time step size according to equation (13) is chosen in such a way that the Courant number is always equal to or less than one. Table 4 shows the changes in the Courant number under changes in the time step at different frequencies. According to the data in this Table, the average Courant number of the investigated cases is 0.3.

Fig. 8.

Time step independence study at different time steps (0.03, 0.05, and 0.08 s).

Table 4.

Courant number under time step changes at different frequencies.

Fe (Hz)	Time step (s)	Cell size (m)	Vave	Courant number
1	0.08	0.0485	0.5792	0.95
2	0.025	0.0485	0.5792	0.29
3	0.016	0.0485	0.5792	0.19
4	0.0125	0.0485	0.5792	0.14
5	0.01	0.0485	0.5792	0.11
6	0.007	0.0485	0.5792	0.08

HTF properties

Syltherm 800 oil is a good choice for heat transfer due to its viscosity and low pressure drop. On the other hand, this oil has relatively good stability in the temperature range of 233 K to 673 K, which reduces system maintenance costs. The thermodynamic properties of Syltherm 800 oil are a function of temperature and are expressed as polynomials. With regard to Table 5 and the data of Dow Company³¹, the coefficients of the polynomial equations defined for thermal conductivity, viscosity, specific heat and density are presented. Also, the appropriate temperature range for the equations is between 300 and 650 K.

Table 5.

Thermodynamic properties of Syltherm 800 oil³¹.

Coefficient	Cp (J/kgK)	ρ (kg/m3)	K(W/mK)	µ (Pa.s)
a0	1.1078 E3	1.2690	0.1901	8.4866 E-2
a1	0	−1.5208	−1.8802 E-4	−5.5412 E-4
a2	0	1.7905 E-3	0	1.3882 E-6
a3	0	−1.6708 E-6	0	−1.5660 E-9
a4	0	0	0	6.6723 E-13

Validation of numerical results

Given the importance of validating and proving the accuracy and precision of numerical solution methods, the issue was examined and evaluated from two perspectives. First, given that the SEG-LS PTC has been used in the present study, it is necessary to compare the geometric dimensions, material type, boundary conditions, flow type and its suitability with the numerical methods used with the experimental and numerical results of other researchers. In this regard, the experimental studies of Dudley et al.² and the numerical results of Kaloudis et al.²¹ have been utilized. It should be noted that in all calculations, equation (14) was used to calculate the thermal efficiency²¹ and equation (15) was used to calculate the amount of computational error²⁸. On the other hand, due to the impossibility of validating the sinusoidal inlet boundary conditions and the unsteady solution for the PTC geometry, it is necessary to use another geometry with sinusoidal inlet boundary conditions. In this study, the experimental results of Kurtulmuş et al.¹⁵, which were conducted in 2020 for the wave channel geometry (with sinusoidal inlet), have been utilized. They have used sinusoidal inlet in their investigations for the wave-shaped channel. For this purpose, they used a DC motor and a variable speed rotary valve, which could serve as a model for designing and manufacturing the frequency-controlled solenoid valves required by PTC. The motor operates under the user’s command to create different frequencies in the range of Strouhal number 2.07>St>0.11, so that over time, the hydrodynamic and thermal variables of the channel gradually change.

SEG-LS PTC geometry validation

Figure 9 shows the changes in the thermal efficiency of the PTC under changes in the Reduced Temperature Parameter (RTP). In this Figure, the results of the present study have been compared with the experimental² and numerical²¹ results. As shown in Fig. 9, for cases related to RTP less than 0.2, the error rate of the present numerical results compared to the experimental results of Dudley² and the numerical results of Kaloudis²¹ is less than 5 percent. However, with increasing RTP, the error of the results increases and reaches about 10%. In general, the average error of the numerical results obtained with the Dudley² experimental results are about 7% and with the Kaloudis²¹ numerical results are about 8%. The increase in error between the present numerical results and the numerical results of Kaloudis²¹ is due to the use of the RNG k-ε turbulence model. In the Kaloudis results, the effect of the viscous sublayer of the walls has been neglected. Table 6 shows the error percentage of all simulated cases with experimental and numerical results.

Fig. 9.

SEG-LS PTC geometry validation with numerical and experimental results^2,21.

Table 6.

Comparison of present study results with experimental and numerical data^2,21 for PTC.

Case	RTP (m2K/W)	Effps	Effexp2 (%)	Effcfx21 (%)	Ɛexp (%)	Ɛnum (%)
1	0.133	74.77	71	75.4	5.45	0.83
2	0.176	74.65	70.17	71.94	6.38	3.76
3	0.247	74.32	70.25	65.10	5.79	13.98
4	0.287	73.96	67.98	67.18	8.78	10.09

Sinusoidal inlet boundary condition validation

Figure 10 shows the time-averaged Nusselt number changes at Reynolds number 7000, relative to the changes in Strouhal number. In addition to the experimental results of Kurtulmuş, this diagram also shows the numerical results of Hamzah et al.³². By comparing the results of the present study with the experimental results, an error of about 4 percent was observed. Also, the error of the present results with the numerical results of Hamzah et al.³² was obtained to be about 2 percent. Table 7 compares the percentage error of the numerical results of the present study with the experimental results of Kurtulmus¹⁵ and the numerical results of Hamzah³². These results show that the use of the UDF method for defining the sinusoidal function is valid.

14

15

Fig. 10.

Sinusoidal inlet boundary condition validation with the reports of Kurtulmuş and Hamza^15,32.

Table 7.

Comparison of present results with experimental and numerical data^15,32 for pulsating flow.

Case	St	<Nu>ps	<Nu>ex15	<Nu>num32	Ɛexp (%)	Ɛnum (%)
1	0.122	139.89	143.49	137.80	2.50	1.51
2	0.241	143.12	137.80	143.08	3.86	0.02
3	0.363	150.24	137.39	147.56	9.35	1.81
4	0.482	156.36	147.96	153.25	5.67	2.02
5	0.601	163.80	151.22	159.35	8.32	2.79
6	0.73	168.29	164.22	163.99	2.47	2.62
7	0.961	170.52	168.29	168.32	1.32	1.31
8	1.202	168.72	168.29	167.88	0.25	0.49

Uncertainty quantification

This section presents a comprehensive quantification of the numerical uncertainties associated with the simulation conducted using ANSYS Fluent. The main sources of uncertainty considered are mesh resolution, time step size, numerical schemes, and property interpolation. The evaluation methods include grid independence tests, time step sensitivity analysis, comparison of numerical discretization schemes, and assessment of polynomial-based property interpolation. The quantified uncertainties are summarized in Table 8. These analyses demonstrate that all uncertainty sources remain below 2%. The low levels of uncertainty across all evaluated sources confirm the robustness and reliability of the numerical model. This confidence is further supported by the close agreement with experimental results.

Table 8.

Sources and quantification of numerical uncertainty.

Uncertainty source	Evaluation method	Key quantity	Uncertainty (%)	Remarks
Grid resolution	Grid independence (6 levels)	Tout	1.11	2.1 million
Time step size	Time step size (3 levels)	Tw	0.89	0.08-0.007.08.007.08.007.08.007 s
Numerical scheme	2nd-order vs 1 st order upwind	Nuav	1.72	1 st order
Property interpolation	Polynomial vs linear and constant	Nuav	1.25	Polynomial

Results and discussions

Velocity changes under different frequencies with constant dimensionless amplitude

Before reporting and analyzing the obtained results, it is necessary to explain the solution of the boundary layer equations, especially the von Karman similarity solution³³. According to Equation (16) to Equation (25), the flow in the boundary layer has specific hydrodynamic characteristics. The fluid velocity in the boundary layer is a function of the free stream and wall characteristics. The boundary conditions according to Equation (17), Equation (18), Equation (19) and Equation (20) cause the similarity solution of Equation (16) and finally, by obtaining the values ​​of the constant coefficients, Equation (23), Equation (24) and Equation (25) are solved. In Equation (23), the fluid velocity in the boundary layer is predicted. So that with increasing the free stream velocity and the distance from the wall, the fluid velocity in the viscous sublayer increases and with increasing the boundary layer thickness, the fluid velocity decreases. Also, based on the similarity solution, Equation (24) and Equation (25) represent the boundary layer thickness and the wall friction coefficient, respectively. All these equations were obtained under normal flow conditions and without creating an inlet frequency. The purpose of using the frequency inlet flow was to change the velocity pattern in the boundary layer, buffer areas and in general the viscous sublayer. The solar energy source is emitted as a flux varying with location on the collector surface. If under certain frequencies, the velocity of the viscous sublayer reaches a value higher than the no-frequency case, it can be expected that the collector heat transfer rate and solar energy absorption will increase with the increase in the heat transfer coefficient. Figure 11 shows the velocity vectors at a distance of 3.5 m from the absorber tube inlet under 4 frequencies of 0, 2, 5 and 6 Hz. In all 4 cases, the dimensionless amplitude is constant and its value is considered to be 0.3. By comparing the velocity vectors, it is clear that the velocity profiles have changed with the change in frequency and the von Karman similarity solution variables are no longer constant and are changing continuously. The velocity profile at 0 Hz indicates a flat profile related to normal turbulent flow, but with increasing frequency, the velocity profiles have become subject to unusual changes. So that for the frequency of 5 Hz, these changes are much more tangible. At this frequency, the shape of the turbulent flow profile has become much more irregular. So that the areas near the walls have higher velocities, but there is not much difference between the velocity profiles related to the frequencies of 5 and 6 Hz. In Fig. 12, the velocity contours for the areas near the collector wall, including the boundary layer, buffer areas and viscous sublayer, are shown for two frequency modes of 0 and 5 Hz. By comparing these two modes, it is clear that in the frequency mode of 5 Hz, the velocity in the viscous sublayer is higher than in the frequency mode of 0 Hz, the fluids with higher inertia (areas closer to the absorber tube center) have transferred some of their energy under frequency momentum to the fluids in the viscous region and have caused longitudinal and transverse turbulence in the viscous sublayer. By referring back to Fig. 11 and examining the central regions, it is observed that at a frequency of 5 Hz, the inertia of the regions closer to the tube center has been reduced. This means that the energy transfers of the high-energy particles closer to the tube center to the particles in the viscous region has been successful. When energy is to be transferred from the solid surface to the fluid inside the tube, the best possible way to increase thermal efficiency is to create more turbulence in the regions near the wall so that more energy is transferred to the fluid inside the tube. By comparing the results, it is observed that at a frequency of 5 Hz, the average velocity outside the boundary layer (non-viscous regions) is reported to be 0.57 m/s and in the viscous regions near the wall, 0.12 m/s. Also, in the 0 Hz frequency mode, the average velocity outside the boundary layer is 0.61 m/s and in the viscous regions is 0.09. Such conditions are not possible without longitudinal and transverse collisions of fluid particles, and the only factor that can transfer this energy is the instantaneous changes in velocity in all directions. The irregularity of the instantaneous collisions created under the 5 Hz frequency has increased the velocity of the viscous regions. Also, by comparing the average velocity values ​​in the 5 and 6 Hz modes, not much difference can be seen. So that the average velocity for the 6 Hz mode outside the boundary layer is 0.56 m/s and for the 5 Hz mode, it is 0.57 m/s. The author concludes that in the 6 Hz frequency mode, less energy is transferred from high momentum particles to lower energy fluid particles (viscous sublayer and adjacent layers). It can be said that a further increase in frequency has caused turbulent freezing. So that the frequency has increased to such an extent that the fluid flow feels the changes less. Figure 13 shows the changes in velocity with respect to the change in the distance of the fluid particles from the wall. This diagram shows the velocity of Syltherm 800 oil for frequencies of 0, 2, and 5 Hz at a distance of 0 to 0.1 mm from the wall and at a distance of 3.5 m from the collector inlet (fully developed region). By comparing the velocity values ​​in the viscous sublayer, it can be shown that the velocity value for the 5 Hz frequency at a distance of 0.055 mm from the wall is about 8 percent higher than the 0 Hz frequency case and about 2.5 percent higher than the 2 Hz frequency case.

16

17

18

19

20

21

22

23

24

25

Fig. 11.

Velocity vectors for different frequencies at a distance of 3.5 m from the inlet of absorber tube with constant amplitude (A=0.3) after 20 s.

Fig. 12.

Velocity contour in viscous regions near the wall for frequencies of 0 and 5 Hz with constant amplitude (A=0.3) at Re=4761 and after 20 s.

Fig. 13.

Velocity values ​​for different frequencies in viscous sublayer at Re=4761 and after 20 s.

Temperature contours of PTC under different frequencies with constant dimensionless amplitude

Figure 14 shows the temperature contours of the absorber tube of the SEG-LS2 PTC. In this Figure, the outlet temperature contours of the absorber tube are shown for three frequencies of 0, 2 and 5 Hz and under a dimensionless amplitude of 0.3. The total duration of the numerical analysis in ANSYS Fluent software is 20 s, the time steps are 0.025 and 0.01 s, respectively. Due to the non-uniform heat flux, the temperature difference between the left and right sides of the absorber tube is significant. For more detailed studies, the contours of the left and right sides of the absorber tube are shown separately in sections (a) and (b) of Fig. 14. Section (a) shows that with increasing frequency, the boundary layer, and especially the viscous and buffer layers, are affected by instantaneous and continuous velocity changes, and as a result, more heat is transferred to the central regions of the absorber tube. On the other hand, due to the type of collector and the angle of solar energy irradiation, a part of the absorber tube has received much less energy (b). The color of the temperature contour in these areas has changed drastically under frequency changes. The area of ​​the lowest temperature areas at 0 Hz (dark blue) is much larger than the blue areas at 2 and 5 Hz. These areas have reached their minimum value at 5 Hz. Given the importance of examining the boundary layer and areas close to the wall, microscopic analysis is essential. The boundary layer is where the fluid velocity is very low due to the no-slip condition and increased wall friction, and tends to zero in areas very close to the wall. Frequency changes are a factor in disrupting these conditions and increasing the velocity in the viscous and buffer areas, so that at 5 Hz, velocity changes, molecular collisions, and ultimately temperature changes can be felt. The frequency changes caused a change in the turbulent flow boundary layer pattern and an increase in the heat transfer coefficient, and more energy is absorbed by the Syltherm 800 oil. According to the von Karman similarity equations, an increase in the average tube velocity reduces the thickness of the boundary layer and increases the heat transfer coefficient. In this numerical study, with frequency changes, the velocity of the central regions (maximum velocity) has decreased and the velocity of the viscous and near-viscous region has increased. According to the numerical results obtained, the time-averaged outlet temperature of the collector after 20 s is 446.89 K for the 0 Hz frequency mode, 447.11 K for the 2 Hz frequency mode, and 447.51 K for the 5 Hz frequency mode.

Fig. 14.

Variations in the absorber tube outlet temperature contour at different frequencies (Fe=0, 2 and 5 Hz) and constant amplitude (A=0.3) at Re=4761. (a) Temperature contour at left side of PTC. (b) Temperature contour at right side of PTC.

Thermal performance PTC at variable frequencies (Fe=0–6 Hz) and constant dimensionless amplitude (A=0.3)

Figure 15 shows the changes in the Oscillatory Nusselt number under time changes at different frequencies and under a dimensionless amplitude of 0.3. In this diagram, the left vertical axis shows the Oscillatory Nusselt number, the right vertical axis shows the Oscillatory heat transfer coefficient, and the horizontal axis shows the time changes. As can be seen, in all cases, the Nusselt number and the heat transfer coefficient decrease from the start of the analysis to 7 s and then increase until 14 s and then increase or decrease as a frequency in a certain dimensionless amplitude. With more accuracy, it can be seen that after 7 s, the changes in the Oscillatory Nusselt number and the Oscillatory heat transfer coefficient at frequencies of 5 and 6 Hz gradually become more different than other frequencies and their values ​​are higher than other frequencies. The highest values ​​of the Nusselt number and the heat transfer coefficient are obtained after 14 s from the start of the analysis and for the frequency of 5 Hz, and their values ​​are 5.1 and 36.3 W/m²K, respectively. On the other hand, with a further increase in frequency (results related to the frequency of 6 Hz), the range of changes in the Oscillatory Nusselt number and the Oscillatory heat transfer coefficient has decreased, and a further increase in frequency has no effect on the thermal performance of the parabolic collector absorber tube. From the author’s point of view and some researchers, including Molochnikov¹⁸, the layers closer to the wall are exposed to more energy absorption, but due to the greater contribution of viscous forces, they do not have sufficient ability to absorb more energy. According to the results obtained, frequency changes affect the viscous layers near the wall and, by creating successive frequency fluctuations, increase the inertial forces. The hydrodynamic and thermal boundary layers are strongly interdependent, and changes in each cause changes in the other. With this interpretation and by re-examining Fig. 16 and Fig. 17, a further increase in the velocity for a frequency of 5 Hz in the viscous sublayer justifies an increase in the Nusselt number for a frequency of 5 Hz. For better visibility and simplification, the variables of Nusselt number and heat transfer coefficient in Fig. 16 are defined as a power function (R²=0.92). These functions predict the average changes of the variables over time. Power functions predict all instantaneous changes of the variables as a function. A limited number of variable values ​​may not apply in this function, but it can be said that more than 90% of the variable values ​​apply in the power function. This function predicts that the behavior of the variables has not stabilized after 14 s from the start of the analysis and the changes continue. By referring back to Fig. 16 and observing the enlarged diagram below from the time interval 14 to 18 s, it can be inferred that the variables of Nusselt number and heat transfer coefficient are decreasing. The defined power function predicts the slope of the decrease of the Nusselt number in the no-frequency state much higher than in other cases. However, this slope is very small for frequencies of 5 and 6 Hz. According to the model of this function, the Nusselt number in the no-frequency state will reach a constant value after a short time and will not change. Also, according to this prediction, a longer period of time is needed for the Nusselt numbers related to the 5 and 6 Hz. In fact, the power function refers to the freezing of the flow. In the author’s opinion, the fluid’s habituation to changes in thermodynamic properties, boundary and inlet conditions and the lack of reaction to them is a type of flow freezing. From this point of view, all flows experience freezing, but the time required for freezing can be increased by frequency changes. By comparing the Nusselt numbers related to the 5 and 6 Hz frequency states, no particular difference is observed. In fact, the thermal variables of the flow do not sense the further increase in frequency in the flow rate of the tube inlet. This lack of reaction is also another type of freezing. By carefully examining Fig. 15, it can be seen that the increase in the frequency of the inlet flow rate has reduced the range of changes in the average Nusselt number and the heat transfer coefficient. Thermal variables have a time delay compared to hydrodynamic variables. For example, the fluid flow must first move for the temperature to change. Further increases in the frequency of the inlet flow must be transferred to the thermal variables in some way, but decreasing the dimensionless amplitude of the thermal variables prevents this from happening. In fact, decreasing the dimensionless amplitude of the Nusselt number changes acts like an anesthetic. Figure 17 shows the changes in the average temperature of the inner wall of the absorber tube under changes in time. As can be seen in the figure, with increasing the frequency of the inlet flow, the average temperature of the absorber tube walls decreases. The largest temperature decrease is related to the frequencies of 5 and 6 Hz. According to Equation (26), changes in the Nusselt number depend on variables such as the thermal conductivity of the fluid, the hydraulic diameter of the absorber tube, and also the heat transfer coefficient of the fluid. With increasing the heat transfer coefficient, the Nusselt number also increases. Also, according to Equation (27), changes in the heat transfer coefficient depend on the difference in bulk temperature and the surface temperature of the absorber tube. Referring back to Fig. 17, it is observed that at frequencies of 5 and 6 Hz, the surface temperature of the absorber tube has decreased. As a result, the difference between the bulk temperature and the surface temperature of the tube also decreases. Finally, as the temperature difference decreases and the heat transfer coefficient increases, the Nusselt number increases.

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27

Fig. 15.

Oscillatory Nusselt number and heat transfer coefficient of PTC under time variation at Re=4761 in the frequency range of 0 to 6 Hz with constant amplitude (A=0.3).

Fig. 16.

Nusselt number and heat transfer coefficient of PTC under time variation with using power functions in the frequency range of 0 to 6 Hz with constant amplitude (A=0.3).

Fig. 17.

Outer wall absorber tube temperature under time variations at Re=4761.

Thermal performance of PTC at constant frequency and variable dimensionless amplitude (0.3–0.9.3.9)

This section examines the effect of changing the inlet flow dimensionless amplitude in the constant frequency mode. Figure 18 shows the thermal performance of the absorber under time variation. In this diagram, the range of dimensionless amplitude changes is 0.3 to 0.9 and the frequency value is constant and equal to 2 Hz. As can be seen in part (a) of Figure, increasing the inlet flow dimensionless amplitude increases the range of changes in the Nusselt number. Also, part (c) of Figure shows the changes in thermal efficiency under time variations. In this part, the frequency is constant, like part (a), and its value is 2 Hz. It can be seen carefully in part (c) that with increasing the dimensionless amplitude of the fluid inlet flow, the range of changes in thermal efficiency also increases over time. For a better understanding of the subject, part (b) and (d) of Fig. 18 shows the changes in the Nusselt number and thermal efficiency, respectively, using linear functions. As shown in part (b), increasing the dimensionless amplitude causes an increase in the Nusselt number, so that for a fixed frequency of 2 Hz, the highest value of the Nusselt number is related to the dimensionless amplitude of changes of 0.9 and its value is reported as 5.3. However, in part (d) of Fig. 18, with an increase in the dimensionless amplitude of changes of the fluid inlet flow, the thermal efficiency has decreased so that the lowest value of the thermal efficiency is related to the dimensionless amplitude of 0.9 and its value is reported as 74%. The contradiction between the increase in the Nusselt number and the decrease in the thermal efficiency can be justified by expressing the variables dependent on the thermal efficiency. According to Equation (28), the thermal efficiency is a function of the mass flow rate, the specific heat coefficient, and the difference in the inlet and outlet temperatures of the fluid. When the dimensionless amplitude changes by a factor of 0.9, the flow rate increases and decreases sharply, and this increase and decrease causes a change in the outlet temperature of the absorber tube. In this case, the fluid is cooled and its outlet temperature decreases, and as a result, the thermal efficiency decreases. Increasing the dimensionless amplitude will be desirable when it does not lead to a decrease in the thermal efficiency. Carefully looking at the results of part (b) and part (d) of Fig. 18, the desired dimensionless amplitude for the frequency of 2 Hz is about 0.3 to 0.5 and its further increase is not recommended. It should be noted that Equation (28) was used to calculate the instantaneous thermal efficiency at each time step, and the method of averaging the numerical results of all time steps in the range of 13 to 20 s was used to calculate the time-averaged thermal efficiency. Considering the desired results obtained for the frequency of 5 Hz, the changes in the Nusselt number and thermal efficiency at this frequency are examined. Part (a) and (b) of Fig. 19 shows the changes in the Nusselt number and thermal efficiency with using linear functions, respectively (R²=0.95 and R²=0.93). According to the results of part (b), the thermal efficiency in the dimensionless amplitude of 0.3 or 0.5 did not decrease significantly and they can be used in frequency flows. Also, according to the results of part (a) and considering the higher values ​​of the Nusselt number for the dimensionless amplitude of 0.5, the use of this dimensionless amplitude is recommended. Figure 20 shows the variations of the local Nusselt number along the length of the PTC. In all cases, the dimensionless amplitude is constant and its value is 0.5. The local Nusselt number has decreased in the inlet region (up to a length of 0.5 m) and for all the studied cases. The decrease in the local Nusselt number in the inlet region of the absorber tube at a frequency of 0 Hz is greater than in other cases. From 0.5 m onwards, fully developed hydrodynamic and thermal boundary layer profiles are formed. An increase in the local Nusselt number has been reported along the collector and for all cases. Changes in kinetic energy, temperature changes of thermodynamic properties and non-uniform heat flux have caused an increase in the local Nusselt number. With increasing frequency, the changes in the kinetic energy of the particles have increased. The highest value of the local Nusselt number has been observed at frequencies 5 and 6 and its value has been reported to be about 7. With further increase in frequency, no significant change in the local Nusselt number has been observed.

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Fig. 18.

Thermal performance of PTC under time variation at constant frequency (Fe=2 Hz) and variable dimensionless amplitude (A=0.3–0.9.3.9). (a) Oscillatory Nu number (b), Nu number with using linear function method, (c) Oscillatory thermal efficiency, (d) thermal efficiency with using linear function method.

Fig. 19.

thermal performance of PTC at constant frequency (Fe=5 Hz) and variable dimensionless amplitude (A=0.3–0.9.3.9). (a) Nusselt number with using linear function method, (b) thermal efficiency with using linear function method.

Fig. 20.

local Nusselt number in axial direction of PTC with constant dimensionless amplitude (A=0.5) and different frequencies at t=20 s.

Thermal efficiency of optimal PTC (Fe=5 Hz, A=0.5) under solar heat flux changes

Figure 21 shows the changes in the thermal efficiency of the PTC under changes in the solar radiant flux for a frequency of 5 Hz and a dimensionless amplitude of 0.5. The range of changes in the radiant flux is between 850 and 1100 W/m². According to the numerical results, the lowest thermal efficiency occurred for a radiant flux of 860 W/m² and its value is 75.5%. In contrast, the highest thermal efficiency was for a radiant flux of 1100 W/m² and its value is 79%. Also, to examine the effect of changing the radiant flux on the thermal efficiency of the optimal frequency PTC (Fe=5 Hz, A=0.5), a graph of relative changes in efficiency is also presented. In the optimal frequency flow with radiant fluxes greater than 950 W/m², the thermal efficiency increases with a better slope. The relative changes in thermal efficiency obtained in radiant fluxes of 850 and 1100 W/m² were 3.5 and 4.5 %, respectively. The results of the present study show that by using optimal frequency PTCs (Fe=5 Hz, A=0.5) in hot and dry regions (I_b>1100 W/m²), the thermal efficiency can be increased to 80 %.

Fig. 21.

Thermal efficiency of optimal PTC under solar heat flux changes.

Thermal performance of PTC under Strouhal number changes

Time-averaged Nusselt number variation under Strouhal number changes

Figure 22 shows the Time-averaged Nusselt number under changes in Strouhal number. The Strouhal number is defined according to Equation (29) and depends on the variables of the hydraulic diameter of the absorber tube, frequency and average fluid velocity. The dimensionless Strouhal number can also be defined as the ratio of the frequency force to the inertial force. According to the results shown in this Figure, the highest increase in heat transfer occurred in the dimensionless amplitude of 0.5 and the Strouhal number was 0.131 (related to frequency 5 Hz), where the time-averaged Nusselt number reached its maximum value of 5.1.

Fig. 22.

Time-averaged Nusselt number under changes in Strouhal number at Re=4761.

Time-averaged thermal efficiency under Strouhal number changes

Figure 23 shows Time-averaged thermal efficiency under changes in Strouhal number. According to the quantitative results obtained, no specific changes in the time-averaged thermal efficiency are observed in the Strouhal number range of 0 to 0.05. But in the range of Strouhal number 0.105, the longitudinal and transverse turbulences resulting from frequency changes have reached their maximum value, so that under a frequency of 5 Hz, the time-averaged thermal efficiency has reached 77%. With a further increase in the Strouhal number and a more severe decrease in the frequency of the flow, there is not enough time to sense the thermal changes.

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Fig. 23.

Time-averaged thermal efficiency under changes in Strouhal number at Re=4761.

Also to highlight the novelty of this study, Table 9 compares the present work with previous studies on pulsating flows and PTCs, focusing on the unique investigation of the effects of frequency and dimensionless amplitude on the thermal performance of SEGS LS-2 PTC.

Economic evaluation of the use frequency-controlled solenoid valves

This section examines the benefits of using frequency valves to increase the thermal efficiency of PTCs. Based on the results, frequency-controlled solenoid valves can be designed and installed in PTCs to optimize the pulsating flow with a frequency of 5 Hz and a dimensionless amplitude of 0.5 and improve the thermal efficiency of SEGS LS-2 by 3%. The price of each frequency valve is about $110. Also, the price of each m² of the LS2 model is $200². Considering the area of ​​39 m² (width 5 m × length 7.8 m) of a complete module, the total price will be $7800. In this case, the price of the frequency valve is only 1.41% of the price of a complete module. This ratio reaches 1.25% in industrial plants with 100 modules, potentially saving millions of dollars. Given the vast expanse of hot and dry regions of the world, such systems can achieve superior thermal efficiency. Table 10 shows the changes in thermal efficiency and financial burden resulting from modifications to PTCs. Changes in geometry or the use of nanoparticles increase the thermal efficiency of the PTC, but these changes require costs. In the present study, the thermal efficiency has been increased by 3% at a cost of less than 2% of the value of a standard LS2 PTCs. By reviewing similar studies and comparing the cost and the resulting increase in thermal efficiency, we can see that using frequency valves is the best economic-practical option.

Table 10.

Comparing the cost and increase in thermal efficiency achieved.

Authors	HTF	Eff improve	Extra work	Cost increase
Dudley et al.2	Syltherm 800	0%	_	0% (base)
present Study	Syltherm 800	3%	Frequency Valve	1.25–2.25%
Mwesigye et al.5	Syltherm 800	9%	Perforated Plates	13–18%
Naveenkumar et al.20	Water	2%	Rotational Absorber	15–17 %
Vahidinia et al.7	Dowtherm A	0.88%	Fins & Turbulator	19–23 %
Abed et al.22	Therminol VP	14.6%	6 % nanoparticles	26–33 %

Conclusion

In the present study, SEG-LS2 model of PTC with a length of 7.8 meters and a hydraulic diameter of 15.2 mm was modeled using Gambit software and then analyzed under ANSYS Fluent software. The steps performed include initial modeling, organized meshing, applying boundary conditions, selecting and applying numerical equations, independence from the number of grids and time steps, validation, numerical analysis, and extracting results. The fluid used was Syltherm 800 oil, which was preferable compared to other fluids due to its viscosity and lower pressure drop. Due to the Reynolds number range (Re = 4761), the flow was turbulent and unsteady. The absorber tube of the PTC is subject to more viscous forces due to having two external and internal solid boundaries, and applying simultaneous frequency changes affects the two boundary layer structures. According to the results obtained for zero Strouhal number, the velocity profile is completely flat in other areas except for the buffer and viscous sublayer, and its thermal efficiency is about 74%.

By applying frequency changes in the range of 0 to 6 Hz, the hydrodynamic layer adjacent to the walls and consequently the transverse velocity profile has changed so that for a Strouhal number of 0.131 (Fe=5 Hz), the highest velocity values ​​have been observed in the areas near the wall and viscous sublayer. The increase in velocity in the boundary areas of the wall has increased the heat transfer coefficient and thermal efficiency of the collector. As the frequency increases in the Strouhal number range of 0.131, a larger portion of the inertial forces are transferred to the buffer regions and viscous sublayer in the form of frequency forces. So that it can be said that 10% of the total inertial forces belong to frequency fluctuations.

Further increase in frequency causes a sharp decrease in the periodicity of the mass flow rate, so that with an increase in frequency from 5 to 6 Hz, the pattern of the previous states is not repeated. It can be said that further decrease in periodicity has caused the flow and mass flow rate to freeze, so that the fluid flow no longer feels the frequency changes and no further effect is observed in the velocity profile near the wall.

Changing the dimensionless amplitude can increase or decrease the average Nusselt number and thermal efficiency. Excessive increase in dimensionless amplitude causes the fluid to cool and ultimately reduces thermal efficiency. As a result, a dimensionless amplitude of 0.5 is suggested to maintain the growth of both variables.

The following studies are recommended for further development of frequency-controlled PTCs.

1. Numerical study of the effect of frequency under step functions.

2. Experimental study of installing frequency-controlled solenoid valves in the PTC.

3. Two-phase analysis of the effect of frequency change with the presence of nanoparticles.

Limitations

Numerical limitation

The use of numerical methods in the study of turbulent and unsteady flows is time-consuming and complex. Solving a numerical problem under a short time step and a Courant number less than one requires a computer with a dedicated CPU and RAM. Resource limitations in providing the required equipment have increased the analysis time. Thus, the software runtime for the 6 Hz frequency mode, 0.3 dimensionless amplitude, and 0.007 second time step has been more than a week. In this simulation with a 2.2 million cells, performed using Ansys Fluent on an 8-core Core i7 processor, the RAM usage was approximately 32 GB and the average CPU utilization was around 85%.

Practical limitations for large-scale pulsed flow control in SEGS LS2 systems

Actuator Scaling: Single solenoid valves are not sufficient for more than 100 LS-2 modules (0.5 kg/s mass flow rate at Re=4761); requires parallel arrays or industrial rotary valves, which increases system complexity and maintenance requirements.

Field Synchronization: Phase locking in kilometer-scale PTC arrays requires centralized PLC with fiber optic timing control (±10 ms accuracy).

Proposed experimental validation roadmap

A. Laboratory Prototype: Single LS-2 module with variable speed DC motor + rotary flap¹⁵ in a well-equipped laboratory facility.

b. Field Demonstration: 5-module array testing with time-averaged Nusselt number vs. frequency in a well-equipped laboratory facility.

c. Commercial validation: Retrofitting of existing SEGS plants with the aim of verifying a thermal efficiency of 3% (77% achieved at f=5Hz, A=0.5).

These limitations can be overcome by using proven modular control systems from HVAC pacing applications.

Latin symbols

A

Dimensionless amplitude

A_ap

Reflector surface (m²)

C

Courant number

C_p

Specific heat (J/kgK)

C_f

Wall friction coefficient

D_h

Hydraulic diameter of the absorber tube (m)

dt

Time step (s)

Eff

Thermal efficiency (%)

Fe

Frequency (Hz)

g

Gravitational acceleration (m/s²)

h

Heat transfer coefficient (W/m²K)

<h>

Time-limited heat transfer coefficient (W/m²K)

h_av

Time-averaged heat transfer coefficient (W/m²K)

P

Pressure (Pa)

I_b

Solar flux (W/m²)

k

Thermal conductivity coefficient (W/mK)

L

Length of parabolic collector (m)

Nu

Nusselt number

<Nu>

Time-limited Nusselt number

Nu_av

Time-averaged Nusselt number

P

Pressure (Pa)

Re

Reynolds number

St

Strouhal number

T_w

Absorber outer wall temperature (K)

T_out

Outlet temperature (K)

T_in

Inlet temperature (K)

T_bulk

Bulk temperature (k)

U_in

Inlet velocity (m/s)

U_ave

Average velocity (m/s)

U_max

Maximum oscillation velocity (m/s)

u_x

Velocity in the boundary layer (m/s)

Greek symbols

Ɛ

Calculation error (%)

µ

Fluid viscosity (Pa.s)

ρ

Density (kg/m³)

µ

Viscosity (Pa.s)

Angle (degree)

∆h

Mesh length (m)

∆P

Pressure difference (Pa)

∆T

Temperature difference (K)

δ

Boundary layer thickness (m)

Abbreviations

EWT

Enhanced wall treatment

FVM

Finite volume method

HTF

Heat transfer fluid

PTC

Parabolic trough collector

LCR

Local concentration ratio

MCRT

Monto carlo ray tracing

RNG

Re- normalization group

SEGS

Solar electric generating system

UDF

User defined function

Author contributions

S.F. conceived the research idea, designed the numerical methodology, performed the simulations, analyzed the data, and drafted the manuscript.I.M. contributed to the development of the physical model, supervised the numerical implementation, and critically revised the manuscript.M.A. contributed to the validation strategy, interpretation of the results, and revision of the manuscript.M.K. supervised the overall project, contributed to the discussion of the findings and approved the final version of the manuscript.All authors reviewed and approved the final manuscript.

Data availability

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.