An application of artificial neural network (ANN) for comparative performance assessment of solar chimney (SC) plant for green energy production

This study aims to optimize the power generation of a conventional Manzanares solar chimney (SC) plant through strategic modifications to the collector inlet height, chimney diameter, and chimney divergence. Employing a finite volume-based solver for numerical analysis, we systematically scrutinize influential geometric parameters, including collector height (hi = 1.85 to 0.1 m), chimney inlet diameter (dch = 10.16 to 55.88 m), and chimney outlet diameter (do = 10.16 to 30.48 m). Our findings demonstrate that reducing the collector inlet height consistently leads to increased power output. The optimal collector inlet height of hi = 0.2 m results in a significant power increase from 51 to 117.42 kW (~ 2.3 times) without additional installation costs, accompanied by an efficiency of 0.25%. Conversely, enlarging the chimney diameter decreases the chimney base velocity and suction pressure. However, as turbine-driven power generation rises, the flow becomes stagnant beyond a chimney diameter of 45.72 m. At this point, power generation reaches 209 kW, nearly four times greater than the Manzanares plant, with an efficiency of 0.44%. Nevertheless, the cost of expanding the chimney diameter is substantial. Furthermore, the impact of chimney divergence is evident, with power generation, collector efficiency, overall efficiency, and collector inlet velocity all peaking at an outer chimney diameter of 15.24 m (corresponding to an area ratio of 2.25). At this configuration, power generation increases to 75.91 kW, approximately 1.5 times more than the initial design. Remarkably, at a low collector inlet height of 0.2 m, combining it with a chimney diameter of 4.5 times the chimney inlet diameter (4.5dch) results in an impressive power output of 635.02 kW, signifying a substantial 12.45-fold increase. To model the performance under these diverse conditions, an artificial neural network (ANN) is effectively utilized.

A solar chimney (SC) power plant is a device designed for harnessing solar energy to generate power.It consists of three primary components: the collector absorber plate, a transparent cover, and a chimney.Energy conversion takes place from thermal to kinetic energy and finally into electricity [1][2][3][4] .Despite the simplicity in construction and principle, several issues exist, mostly the low efficiency of the plant.Apart from the material of the transparent cover and absorber plate, geometric parameters like collector diameter, collector inlet, chimney height, chimney diameter, and divergence influence the performance 5 .So far, the first plant is reported in Spain (Manazaranes) by Haaf et al. 6,7 in 1982 for generating ~ 50 kW power.The study on the variation of collector inlet (Table 1), variation of chimney diameter (Table 2), and divergence (Table 3) are listed in detail.
The above study of collector inlet variation envisages that the flow velocity rises but the flow volume drops for the reduction of inlet height.Several studies have mentioned the rise in power for a small inlet.However, a very small height weakens the flow and, consequently power generation.Very few of the studies have addressed optimum inlet height; this height is different for different sizes of plants.From the effect of chimney diameter, it can be claimed by a few studies that velocity rises for small chimney diameters and power rises, as power is Table 1.Impact of collector inlet heights (h i ) on the performance of SC plant.

Studied parameters Observations Remarks
Kasaeian et al. 8 Numerical and analytical study on the effect of collector inlets of 6, 8, and 12 cm on velocity and temperature 6 cm collector through showed maximum velocity and temperature No optimum entrance value is studied Patel et al. 9 Collector inlet height varies from 0.05 and 0.2 m in the numerical study The opening of 0.05 m is the best choice No optimum value is studied Vieira et al. 10 Numerical work to investigate the effect of collector inlet No significant effect has been noted No question of optimum value Ghalamchi et al. 11 Collector entrance length varied for 0.04, 0.06, 0.1 and 0.14 m, experimentally 0.06 m shows the maximum velocity The optimum condition is not studied Toghraie et al. 12 Numerical study varying collector inlet from 1 to 10 m on efficiency and power Power and efficiency are high at low inlet height No optimum height is investigated Yapici et al. 13 Inlet height is varied from 4 to 10 cm, numerical work The optimal dimension for peak performance is 6 cm -Golzardi et al. 14 Experimental assessment by varying collector entrances In one geometry, decrease in collector entrance gains exit velocity (17%) and energy transfer (62%).Other geometry by dropping 16 to 8 cm results 13% and 37% improvement No optimum value is investigated Mandal et al. 2 Numerical as well as experimental investigation for performance Effective at some value of inlet height Optimum value is noted Table 2. Impact of chimney diameter (d ch ) on the performance of SC plant.

Author Studied parameters Observations Remarks
Ming et al. 15 Numerical study to find best chimney shape Cylindrical chimney is the best choice.Optimum slenderness ratio ranges from 6 to 8 -Kasaeian et al. 8 Numerical and analytical study, velocity measurement for chimney diameters of 10, 20 and 30 cm he use of diffuser-type chimney over cylindrical one is studied numerically and experimentally both 4° shows the optimum value, and the maximum updraft -Hu et al. 20 Effect of Chimney divergence by area ratio is studied upto 32 Area ratio ~ 10 showed the optimum choice of power generation -Hassan et al. 21umerical study by Manzaranes plant, effect of divergent chimney, angle 1to 3º Optimum divergence is 1°, enhancement in velocity 27%, estimated power rise 108% -Nasraoui et al. 22 Conical chimney angle for 0, 3, 6 and 9° is studied numerically Performance drops after 3°-Yapici et al. 13 Numerical study for different shaped chimney Divergent chimney is the best for performance No optimum divergence is studied Das and Chandramohan 23 Numerical study to estimate the flow and performance parameters for divergent chimney for divergent of 1 to 5º Optimum divergence is 2°, enhancement in velocity 59%, power 280% and efficiency by 242% -Das and Chandramohan 24 Performance characteristics for different divergent chimneys numerically, divergent angle varies 1 to 5º Optimum divergence is 2°, enhancement in velocity 59%, pressure 158%, power 311% and efficiency 287% -Cuce et al. 25 Simulation on Manzaranes plant to find the effect of convergent and divergent chimney, area ratio varies from 0 to 10 Optimum area ratio is 4, efficiency rises from 0.29% to 0.83%, and power rises 54.3 to 168.5 kW - Numerous researchers have dedicated their efforts to advancing cleaner energy production, with a particular focus on enhancing the performance of solar chimney (SC) plants.While extensive work has been conducted within the field, the review of existing literature reveals a noticeable gap in the optimization of collector inlet heights.Few researchers have explored this avenue, despite its potential impact on SC plant performance.On the other hand, investigations regarding the optimization of chimney diameter and divergence have received greater attention, primarily to maximize power generation 4,14,[29][30][31] .However, when it comes to the classical Manzanares SC plant, the examination of optimal collector inlet height and chimney diameter remains notably limited.This uncharted territory within the realm of thermo-fluid flow phenomena within the SCPP, including collector inlet height and chimney diameter, has provided the impetus for our novel approach, taking on the challenge of improving SC plant performance.Our study is aimed at optimizing the standard Manzanares SC plant by altering the collector inlet (h i ), chimney diameter (d ch ), and chimney divergence based on the exit diameter (d o ).Furthermore, it involves a comparative thermal assessment of the Manzanares SC plant under various geometric constraints, spanning key parameters such as collector inlet (h i = 1.85 to 0.1 m), chimney inlet diameter (10.16 to 55.88 m or 1.0d ch to 5.5d ch ), and chimney outlet diameter (10.16 to 30.48 m or 1.0d ch to 3d ch ).The outcomes of our analysis are presented in the form of fluid flow and temperature distributions, encompassing pressure, velocity, temperature, and mass flow, as well as performance parameters including power and efficiencies.Additionally, we have endeavored to construct a performance model using an artificial neural network (ANN), a valuable tool that can greatly assist designers in the creation of prototypes.

Formation of physical domain
In the present analysis, the well-known Manzanares unit 6 is selected as the basic model as depicted in Fig. 1.For the investigation of the SC model, it is supposed that the ambient air is incompressible and surrounding ambient conditions have no change with time.Here, the density variation at low temperatures is taken care of by following the Boussinesq method.Furthermore, the solar radiation of 10 3 W/m 2 is supposed to be constant during each process 53 .

Governing equations and modeling
The transport equations for the flow model of the SC plant consist of mass, momentum, and energy conservation equations, which are solved numerically by applying the proper boundary conditions.These equations 54,55 are presented (in the tensor form) as: (1) where the indices i and j indicate 1, 2, and 3, respectively), Pr and σ t are the Prandtl number and turbulent Prandtl number.The buoyant force is expressed as ρβg i ΔT in Eq. ( 2).The symbols μ and μ t in Eqs. ( 2) and (3) correspond to the laminar and turbulent viscosity, respectively.The q rad stands for radiative heat flux obtained by solving the radiative transfer equation (RTE) given later in Eq. ( 6).The Kronecker delta is expressed by the symbol δ (wherever δ = 1, while i = j and else δ = 0).
In the present study, the fluid flow remains in a turbulent flow regime.The RNG k − ε turbulence model equations 33,[56][57][58][59] are given by Different terms in RNG k − ε turbulence model equations are shown in Table 4.In the study, the ANSYS-Fluent 18.1 solver, in combination with the validated Discrete Ordinates (DO) model for radiation transport 41,43,58 , was employed.The DO model discretizes angular space into discrete angles, facilitating directional radiation analysis and numerical determination of radiation intensity (I).The radiative transfer equation (RTE), as presented in Eq. ( 6), mathematically describes solar radiation transport within the medium, taking into account phenomena such as absorption, emission, scattering, and phase functions.The RTE predicts the distribution of radiation intensity and the resulting radiative heat flux within the computational domain.This flux serves as a critical source term (q rad ) in the energy equation, described in Eq. ( 3), which considers energy transport mechanisms including convection, conduction, and radiation.ANSYS-Fluent seamlessly integrates the RTE, DO model, and their coupling with the energy equation, ensuring accurate radiation simulations within the Solar Chimney Power Plant (SCPP).
The radiative transfer equation is expressed as where � s ′ and σ s denote scattering vector and coefficient, a absorption coefficient, m refractive index, ϕ phase function, ′ solid angle, and I radiation intensity (in W/m 2 ).The impact of different geometric parameters on the effectiveness of the SC plant is examined through power production, collector efficiency, and overall efficiency 4 .Here, the actual power generation is estimated as In the above equation, the symbols η t correspond to the turbine efficiency, which is taken as 0.8 41 , p is the pressure drop in the turbine, which is calculated as the average pressure at the chimney base (CB) × the pressure drop ratio (the general value is 2/3) 60 .
Airflow at CB, Q = chimney area × air velocity at CB.

Terms Correlations
Energy production due to mean velocity gradient, ( The dissipation rate in compressible turbulence, ( Y M ) Vol:.( 1234567890)

Solution methodology
In this study, the involved transport equations are solved numerically using the finite volume-based ANSYS-Fluent 19.2 solver 59 .The model was generated following the axisymmetric CFD model.Finally, part of the complete model (vertical slice of 15° model) is taken for the computations to reduce the total cost of computations.A similar approach has also been followed by many researchers.For instance, Hassan et al. 21have considered the 180° model, whereas the 90° and 5° model has been considered by Cuce et al. 25 , and Koonsrisuk and Chitsomboon 17 respectively.The chosen computational model along with the appropriate boundary conditions are solved in an iterative process applying the SIMPLE algorithm, which couples the pressure and velocity.Thereafter, the secondorder upwind scheme is applied to discretize the pressure, momentum, and energy equations.The computational domain is divided into smaller grids.The grids are distributed nonuniformly to capture the correct boundary layers.The maximum Y + value in the first cell is taken as 30 38 and the standard wall function is used in the mesh generation.In general, the finer meshes are distributed adjacent to the solid walls.In order to obtain the converged solution, an iterative process is continued till the reduction of the residuals until 10 -6 for all the governing equations [61][62][63] .The corresponding solution procedure is demonstrated in a flow chart, which is shown in Fig. 2. The adopted boundary conditions for the present study geometry are listed in Table 5.After the mesh independence study, a mesh of 73,862 for the element size 0.80 is taken into consideration for the present analysis.Mesh study is accomplished through different element sizes by comparing the various local parameters such as maximum velocity, temperature, chimney base velocity and temperature, and mass flow rate considering the Manzaranes plant.The considered element sizes taken as 0.90, 0.85, 0.   ) is finalized for the extensive study without further increasing computational time for the study.In fact, for obtaining the converged and stable solution under all the parametric variations, a minimum convergence criterion of 10 -6 for the maximum residuals and the mass defect is chosen for the computation.

Validation of present study
The validation study of the present solver is examined by comparing the CB velocity and power (in kW), as shown in Fig. 3, which confirms the accuracy level of the present solver.

Discussion of results
This work attempts to enhance the power production of the classical Manzanares SC plant by modifying the collector inlet height, chimney diameter, and its divergence to obtain the best design of the plant.Comparative thermal performance assessment of Manzaranes solar chimney (SC) power plant by different geometries is the main aim of this study.

Impact of collector inlet heights (h i )
The Manzaranes plant uses a collector inlet height of 1.85 m (model, M-1).However, the present study represents the impact of collector inlet height (h i ) reduction from 1.85 to 0.1 m systematically, which is defined by the different models, M-1 to M-11 (in Table 6).This analysis computes inlet velocity, chimney base velocity, pressure, temperature, mass flow rate, power produced, collector efficiency, and overall efficiency.

Flow parameters assessment
The induced buoyancy force in-between the collector and absorber plate begins the fluid flow velocity at the collector inlet, the study of the inlet velocity assists the thermo-fluid flow analysis for the plant.The impact of the reduction of collector inlet height (h i ) is illustrated in Fig. www.nature.com/scientificreports/ the collector inlet.Therefore, there should be some relationship between collector diameter and collector inlet height, here the value is about 1220 as observed in Table 6.
The impact of a reduction in collector inlet height (h i ) on the CB flow and temperature distribution is illustrated in Fig. 5.The chimney base is taken as a reference to present the flow parameters as the air turbine is installed near the CB.Chimney base velocity drops as the collector inlet height (h i ) decreases (in Fig. 5a).The drop in chimney base velocity is sharp after model, M-9.As the chimney area is constant, the volume flow rate through the chimney drops.This drop in the working flow volume in the chimney reduces the collector inlet height (h i ).The drop in the flow volume is high after the reduction in the collector inlet for the model, M-9, which is due to the inlet flow velocity drop.The reduction of flow volume gains more thermal energy from the ground plate which results in high temperature as the collector inlet drops.This has been noted in Fig. 5a.With the reduction in flow volume with collector inlet reduction, the suction pressure at the chimney base is noted to rise in Fig. 5b.The temperature of air rises, which in turn reduces the density of air with the reduction of collector inlet.Same time volume flow rate drops, therefore mass flow rate drops as the collector inlet drops (Fig. 5b).The flow contour study by velocity, pressure, and temperature near the CB are illustrated in Fig. 6 for the different models, M-1(h i = 1.85 m) and M-11 (h i = 0.1 m).It is evident in the figure, that the chimney base velocity and pressure drop and chimney base temperature rise due to the reduction in the collector inlet (h i ).

Performance assessment
Under the various geometric parameter variations, the performance study is examined by calculating the power generation and efficiencies (P act , η c , η o ), which are shown in Fig. 7.The produced kinetic energy in the air produces turbine power.This turbine work depends on the flow velocity of impinging and pressure drops through the turbine.Previous Sect."Flow parameters assessment" shows the dropping in the velocity as collector inlet height (h i ) drops tends to decrease the power generation by an air turbine.However, the increase in the suction pressure for reduction in the inlet area assists more power generation (P act ).The combined effect of velocity and pressure on power generation by the turbine shows the rising tendency of power and the power reaches 117.42 kW at h i = 0.2 m.This power generation is 2.3 times of the classical Manzaranes plant.Further reduction of h i lessens the power generated by the turbine (as in Fig. 7a).This can be attributed to that increased power is dominated by the sudden velocity drop in spite of a better pressure drop.Moreover, it can be noted that the power generation by the turbine after h i = 0.2 m is having greater value than the Manzaranes plant.The collector efficiency (η c ) drops with decreasing inlet flow area due to the reduction of mass flow rate despite of increasing temperature of air (as in Fig. 7b).The overall efficiency (η o ) shows a similar trend of curve like power (as in Fig. 7b).The efficiency increases to 0.25%.It is also 2.3 times more compared to the Manzaranes plant.Therefore, this exercise clearly shows that collector inlet height (h i ) reduction has a positive role in enhanced power generation compared to the classical Manzaranes plant.The above study summarizes that power developed by the turbine is always higher relative to the Manzaranes plant at less h i .Maximum power lies at h i = 0.2 m, P act = 117.42kW.The maximum efficiency occurs at the same location, which is 0.25%.Therefore, there is no hike in the cost but power rises from 51 to 117.42 kW.As h i reduces, the chimney base velocity drops, suction pressure increases, the temperature rises, and mass flow drops.

Impact of chimney diameter (d ch )
The impact of chimney diameter (d ch ) is carried out by raising the chimney diameter (d ch ) of the Manzaranes plant up to 55.18 m (5.5d ch ), details of different models are listed in Table 7 from M-12 to M-20.Like earlier, flow and performance are computed and illustrated subsequently.

Flow parameters assessment
The increase in the chimney diameter (d ch ) shows (as in Fig. 8) the enhancement of inlet collector flow velocity, this means the increase in the flow volume, may be due to the rising in the chimney draft.It is interesting to note that the collector inlet velocity does not rise more after 4.5d ch (M-18).Further rise in diameter stagnates the flow volume.
Figure 9 depicts the drop in the CB velocity (in Fig. 9a) as the chimney diameter enlarges.Here, in spite of the increase in the flow volume of air, velocity lessens, which is due to the rise in the chimney flow area.This high flow volume with rising chimney diameter reduces the temperature of the air.This, in turn, decreases the CB pressure as depicted in Fig. 9b.In spite of a drop in the CB velocity, the mass flow rises because of higher air density and higher flow chimney area.The rise in mass flow rate is not too much after the chimney diameter of 45.72 m (4.5d ch , M-18).The pressure, velocity, and temperature contours at the chimney base (for d ch of 10.16 m and 55.88 m) in Fig. 10 illustrate a clear understanding of flow features that velocity and temperature drops and suction pressure at the CB decrease for the model, M-20.

Performance assessment
The power generation and efficiencies variation for the different chimney inlet diameter (d ch ) is presented in Fig. 11.Earlier, it is depicted that the CB base velocity, and suction pressure both drop as d ch rises.Both the phenomenon is against the rise in flow energy by the turbine.But the power produced (P act ) by the turbine rises as observed from Fig. 11a, which is due to the dominancy of the chimney area, this, in turn, heightens the huge flow volume of air.This enhancement of flow becomes stagnant after the chimney diameter of d ch = 45.72 m (model, M-18), it results in no significant rise in the power generation further with the chimney diameter.The maximum power generation is P act = 209 kW which is ~ 4 times than the Manzaranes plant.However, it should be mentioned that some extra cost is required to obtain this increasing diameter of the chimney.The collector

Flow parameter assessment
From the collector inlet velocity variation (as in Fig. 12), it is observed that the increase in outer diameter,

Performance assessment
This study reveals the power generation and efficiencies variation for the different chimney inlet diameters (d ch ), all having an optimum value at 15.24 m (Area ratio: 2.25) outer chimney diameter (as shown in Fig. 15).Earlier studies reveal the area ratio of 10.0 20 , 4.0 25 .Here, the power generation rises to 75.91 kW, which is ~ 1.5 times more than the Manzaranes plant, corresponding efficiency is 0.16%.Comparing the effect of h i and d ch , the impact of divergence is not effective and it requires some additional cost.

Combined effect of collector inlet (h i ) and chimney diameter (d ch )
The aforesaid comparative assessment for a reduction in collector inlet height (h i ), rise in chimney diameter (d ch ), and chimney divergence (d o ) shows the performance improvement of the solar chimney power plant.The lowering of chimney inlet height (h i ) does not require any hike in the initial investment cost, whereas the chimney diameter (d ch ) and divergence (d o ) need a rise in the initial installation cost.The power generation rise is more for a reduction in h i and rises in the chimney diameter d ch .Therefore few models are chosen for the study at h i of 0.2 m and for different chimney diameters, 5.08 m to 45.72 m (0.5d ch to 4.5d ch ), as shown in Table 9 to study the superior performance of the SCCP.The remarkable power rise has been noted as illustrated in Fig. 16.At a lower value of h i of 0.2 m, the power generation with a chimney diameter of 1.

Artificial neural network (ANN) for performance analyses
Today's, artificial neural networks (ANN) play an important role in predicting the outputs.An application of artificial neural networks has been noted in solar heating systems 66,67 .The performance of the Solar chimney power plant is predicted by Amirkhani et al. 68 , Fadaei et al. 69 .A significant number of inputs and outputs are

Forward and backward propagation
The forward propagation method works on the input layer to the hidden layer and the hidden layer to the output layer.This method is used to calculate the output of each neuron in the hidden layer and output layer 70,71 .It starts operation by initializing a random weight and bias.Then calculates the output of each neuron in the hidden and output layers using the activation function.There are different types of activation functions such as Thresholds,   where z = n i=1 I i W i + B , with I is the input values, W is the transpose of the weight between input and hidden layers and B is the bias vector in a different layer.Using the above equation we can predict output from the output layer.Now at each output, the error will be calculated using the following formula: If the error is large then we use the backward propagation method to correct the weights.The backward propagation method works in the output layer to the hidden layer and the hidden layer to the input layer.In different layers weights are updated using the following equation: where λ is the learning rate of the design, which signifies how quickly the design predicts the target output.W is the weight vector in different layers.So using the above equation weights will be updated by the backward propagation method.

Determination of the number of neurons in the hidden layer
To design an efficient neural network, it is necessary to know the number of neurons in the hidden layer.The following flowchart (as shown in Fig. 18) is utilized to determine the number of neurons in the hidden layer.
To determine the number of neurons in the hidden layer, we have calculated mean square error (MSE), root mean square error (RMSE), relative square error (RSE), and correlation coefficient (R 2 ).The following formulas are used for the calculation of various error parameters.where n is the total number of data sets, y true is the actual (target) output, y true is the mean of the actual (target) output, and y train is the ANN output (predicted) after training the model.We have plotted the MSE, RMSE, RSE, and R 2 with different values of the number of neurons in the hidden layer as shown in Fig. 19.Our goal is to find the number of neurons for which MSE, RMSE, and RSE are minimum and correlation coefficient close to 1.To ascertain the optimal number of neurons for our hidden layer, we relied on a flowchart (Fig. 18) and computed various error metrics, including MSE, RMSE, RSE, and R 2 (Eqs.13-16).We aimed to identify the neuron count that simultaneously minimized MSE, RMSE, and RSE while maximizing R 2 .As depicted in Fig. 19, the hidden     It can be seen from the aforementioned Fig. 22 that the 43 neurons constructed neural network accurately predicts the output values.Therefore, we may anticipate output values for an unknown data set using this trained neural network.
To predict the developed power (P act ) of SCPP as a function of collector inlet height (h i ), chimney inlet diameter (d ch ), and chimney outlet diameter (d o ) for the constant collector diameter and same irradiation,

Conclusion
This work attempts for enhancing the power generation of the classical Manzanares solar chimney (SC) plant by modifying the collector inlet height, chimney diameter, and its divergence to obtain the best design of the plant.The performance assessment is evaluated for the different combinations of specified geometric parameters like collector inlet (h i ), chimney diameter (d ch ), and chimney divergence by exit diameter (d o ).The outcomes of the analysis are presented through the local distributions (pressure, velocity, temperature, mass flow) and performance parameters (power generation and efficiencies).Furthermore, an attempt has been made to utilize an artificial neural network (ANN) for developing the performance model, which could be very helpful for the designer of the prototype design.
This study reveals that the power developed by the turbine rises always at the lowering collector inlet height of the classical Manzaranes plant.The maximum power lies at h i = 0.2 m, which is P act = 117.42kW.The maximum efficiency occurs at the same location, which is 0.25%.Therefore, the power generation rises from 51 kW to 117.42 kW (~ 2.3 times) without increasing any additional installation cost.As h i reduces, the chimney base velocity drops, suction pressure increases, the temperature rises, and mass flow drops.The optimum collector inlet velocity occurs at h i = 0.2 m.
A rise in the chimney diameter (d ch ) lowers the chimney base velocity, and suction pressure, both.In spite of these, the power generation by the turbine rises, and the flow becomes stagnant after the chimney diameter of 45.72 m (corresponding slenderness ratio: 4.5).No significant alteration in the collector inlet velocity is noted after this diameter.The maximum power generation at this chimney diameter is 209 kW, which is ~ 4 times than the Manzaranes plant, corresponding efficiency is 0.44%.However, this rise in the chimney diameter requires a higher initial investment cost.An artificial neural network (ANN) is utilized for developing performance modeling to predict the output parameters.

Figure 1 .
Figure 1.Schematic of the SC model for cylindrical chimney.

Figure 2 .
Figure 2. Solution algorithm of the solver in ANSYS fluent.

Figure 5 .Figure 7 .
Figure 5. Variations of CB (a) velocity and temperature, (b) pressure and mass flow at different collector inlet heights (h i ).

Figure 8 .
Figure 8. Variation of collector inlet flow velocity for different chimney diameters (d ch ).

d o of 15 .
24 m (1.5 d ch ) from the base diameter of 10.16 m enhances the flow velocity.This may lead to a rise in the suction pressure by the chimney, which in turn increases the mass flow rate.This collector inlet velocity drops further rise in the chimney outlet diameter, which may be due to the higher pressure loss in the chimney.Therefore, the area ratio of 2.25 is the optimum value to obtain the maximum inlet flow velocity.The study of CB velocity, pressure, temperature, and mass flow is illustrated for the considered chimney divergence models M-21 to 24 as shown in Fig. 13.As h i and d ch are fixed, the same pattern of flow velocity is observed.The magnitude of velocity is different for different areas.Due to the nozzle action, suction pressure rises, mass flow rises and the corresponding air temperature drops at d o = 1.5d ch.Further, increase in the divergence, suction pressure drops less which lessens the mass flow rate, however, the air temperature rises as depicted.The flow contours are illustrated for d o of 10.16 m, 15.24 m, and 30.48 m (1.0d ch , 1.5 d ch and 3.0d ch ) in Fig. 14.It reveals the optimum velocity, pressure, and temperature at chimney outlet diameter of 15.24 m.

Figure 9 .
Figure 9. Variations of CB (a) velocity and temperature, (b) pressure and mass flow at different chimney diameters (d ch ).
. The 1st layer is the input layer, the 2nd layer is the hidden layer, and 3rd layer i.e. the last layer is the output layer.Each neuron in the hidden layer is assigned weight (W Ij ) and bias (B j ).Each neuron in the output layer is assigned weight (W jO ) and bias (B O ).In our design, the input layer consists of three neurons namely, I 1 (h i ), I 2 (d ch ), and I 3 (d o ), and the output layer consists of eight neurons namely, O 1 (V i ), O 2 (V CB ), O 3 (T CB ), O 4 (p CB ), O 5 (P act ), O 6 (m a ), O 7 (η c ), and O 8 (η o ).

Figure 10 .
Figure 10.Local pressure, velocity, and temperature contour plots for the variations of chimney diameters.

Figure 11 .
Figure 11.Variation of (a) power generation and, (b) collector and overall efficiency at different chimney outlet diameters (d ch ).

Figure 12 .
Figure 12.Variation of collector inlet velocity at different chimney outlet diameters.

Figure 13 .
Figure 13.Variations of CB (a) velocity and temperature, (b) pressure, and mass flow at different chimney outlet diameters (d o ).

Figure 14 .
Figure 14.Local pressure, velocity, and temperature contour plots for the variations of chimney outlet diameters.

Figure 15 .
Figure 15.Variation of (a) power production and (b) collector efficiency (η c ) and overall efficiency (η o ) for the different chimney outlet diameters (d o ).

Figure 16 .Figure 17 .
Figure 16.Variation of power generation (P act ) at different model plants.

Figure 21 .
Figure 21.Variation of error for various data sets of (a) training data set, (b) validation data set, (c) test data set, and (d) overall data set.

Table 3 .
Impact of chimney divergence (d o ) on the performance of SC plant.

Table 5 .
Adopted boundary conditions for present study geometry.53,883(M1), 62,021 (M2), 73,862 (M3), and 85,551 (M4), respectively.From the comparison, it is observed that the cumulative error level for the different local variables in the consecutive grids decreases with the decreasing element size.Furthermore, the change between element sizes 0.8 and 0.75 is much less.With this comparison, 0.8 element size (M3 Vol.:(0123456789) Scientific Reports | (2024) 14:979 | https://doi.org/10.1038/s41598-023-46505-1www.nature.com/scientificreports/ The specified geometries are collector inlet heights (h i ), chimney diameter (d ch ), and chimney divergence by chimney outlet diameter (d o ).The performance assessments are conducted for the range of key controlling geometric parameters like collector height (1.85 to 0.1 m), chimney diameter (10.16 to 55.88 m i.e. 1.0d ch to 5.5d ch ), and chimney outlet diameter (10.16 to 30.48 m i. e., 1.0d ch to 3.0d ch ).The outcomes of the analysis are presented through the fluid flow and temperature distributions (namely pressure, velocity, temperature, and mass flow) and performance parameters (power, collector, and overall efficiency).Furthermore, an attempt has been made to use an artificial neural network (ANN) for developing the generalized performance model, which could be very helpful for the designer for any prototype design.

Table 6 .
Models for variations of collector inlet heights (h i ).

Table 7 .
Models in variations of chimney diameters.

i ) Chimney diameter (d ch ) m Chimney height (h ch ), m Collector diameter (d g , m Slenderness ratio (h ch /d ch ) Chimney diametric ratio d ch /h i
5 d ch is 216.18 kW (which is 4.23 times of the Manzaranes plant), 2.0 d ch is 331.22 kW (which is 6.49 times of the Manzaranes plant), 2.5 d ch is 444.40 kW (which is 8.71 times of the Manzaranes plant), 3.0 d ch is 562.89 kW (which is 11.03 times of the Manzaranes plant), 3.5 d ch is 608.99 kW (which is 11.94 times of the Manzaranes plant), 4.0 d ch is 627.7 kW (which is 12.3 times of the Manzaranes plant), 4.5 d ch is 635.02 kW (which 12.45 times of the Manzaranes plant).Therefore, it summarizes that this combined model could be an alternative design for performance improvement,

Table 8 .
Models for the variations of chimney outlet diameters (d o ).

Chimney outlet diameter (d o ) Chimney diameter (d ch ) m Chimney height (h ch ), m Collector diameter (d g ), m Area ratio (A o /A ch ) Chimney outlet diametric ratio (d o /d ch )
Network design and predicted resultsNow we have designed the artificial neural network with 43 neurons in the hidden layer.To mitigate overfitting, we partitioned our dataset into three segments: the test set, the training set, and the validation set.The validation set played a pivotal role in monitoring our model's performance during training.If the model performs well on the training set but poorly on the validation set, it may be overfitting.We have trained the neural network using the training dataset.To reduce error, the network's weight function is corrected by the back propagation method.The validation dataset is used to calculate network generalizations, and we stopped training when generalizations did not improve.With the use of the testing dataset, the network's performance is assessed both during and after training.In this approach, the training process uses 85% of the whole data set, testing uses 10% of the data set, and assessing network performance uses the remaining dataset.The change of MSE for training, validation, and testing data sets with the number of epochs is depicted in Fig.20.Figure20shows that the best validation performance, 2.999 × 10 -5 at epoch 73, was achieved.We stopped the training process after epoch 73 because the number of iterations increased and the errors increased, which may indicate overfitting.Figure21displays the error for various data sets (training, validation, testing, and overall).The results show that the error factor is closer to 1, demonstrating the neural network's ability to fully associate the input data set with the model data set.The values error in the training data set, validation data set, test dataset, and overall data set are 0.99983, 0.999451, 0.99934, and 0.99962 respectively.This shows that the results are reasonably good for the designed neural network.With this trained network, we have now calculated expected output values for several data sets.Figure 22 displays the distribution of goals and anticipated values at the outputs O 1 , to O 8 for several data sets.

Table 9 .
Models in variations of chimney diameters (d ch ).