Analytical and Experimental Investigation of the Solar Chimney System

In this, paper the authors propose a new simplified method of solving the problem of air flow through a solar chimney system using a classical system of equations for the principles of conservation (momentum, mass, and energy), as well as a general solution to research the problem using similarity theory. The method presented in this paper allows one to design a solar chimney. The theoretical analysis was compared with experimental studies on existing solar towers. The experimental and theoretical studies were satisfactorily consistent. For clarity, the phenomenon of heat flow in the solar chimney was described using dimensionless numbers, such as the Reynolds, Grashof, Galileo, Biot, and Prandtl numbers. In the equations for the dimensionless geometric parameters, the ratios of the collector radius to the thickness gap, height, and chimney radius were used. The method used to test the system of equations allows us to analyse various solar collectors easily. In the scientific literature, there is a lack of a simple calculation method to use in engineering practice, suitable for each type of solar chimney independent of dimensions and construction parameters.

A power plant consisting of a solar collector and a chimney can work as a solar thermal power plant [6,7,[18][19][20][21][22][23][24][25], which first converts solar energy into thermal energy in the solar collector to further convert it into kinetic energy in the chimney, with final electricity generation by applying a wind turbine and generator [14,16,26]. The construction of such a solar chimney may depend on the shape and dimensions of both the collector and chimneys and their canopy profiles [10,27].
Most researchers focus on solving the problem of heat flow in solar chimneys using numerical simulations instead of analytical solutions. Compared to the analytical method, fewer assumptions are used in numerical simulations. However, it is possible to obtain more detailed descriptions of the temperature and flow field. The number of studies on numerical methods adopting computational is a continuation of the previous work [15]. Some of the theoretical considerations have been repeated and refined. The theoretical model has been strengthened by taking into account the local resistance of the air flow between the collector and the chimney. The work also included experimental research conducted on the collector-chimney model, and the results of theoretical and experimental research were compared. Experimental and theoretical studies have shown satisfactory consistency. For greater clarity, the phenomenon studied was described using dimensionless numbers, such as the numbers of Reynolds, Grashof, Galileo, Biot, and Prandtl. In the equations for dimensionless geometrical parameters of the solar chimney, the ratio of the radius of the collector to its height, as well as the height and radius of the chimney, were applied. Based on the original transformations and simplifying assumptions, a system of equations was obtained that solved the problem analytically. A universal procedure for solving a complex problem has been developed for use in engineering, for the design of a solar chimney. The proposed method allows for an easy analysis of various solar collectors.

Theoretical Model of the Solar Chimney System
Theoretical model of the solar chimney system is shown in Figure 1. between the collector and the chimney. The work also included experimental research conducted on the collector-chimney model, and the results of theoretical and experimental research were compared. Experimental and theoretical studies have shown satisfactory consistency. For greater clarity, the phenomenon studied was described using dimensionless numbers, such as the numbers of Reynolds, Grashof, Galileo, Biot, and Prandtl. In the equations for dimensionless geometrical parameters of the solar chimney, the ratio of the radius of the collector to its height, as well as the height and radius of the chimney, were applied. Based on the original transformations and simplifying assumptions, a system of equations was obtained that solved the problem analytically. A universal procedure for solving a complex problem has been developed for use in engineering, for the design of a solar chimney. The proposed method allows for an easy analysis of various solar collectors.

Theoretical Model of the Solar Chimney System
Theoretical model of the solar chimney system is shown in Figure 1.  respectively, represented by u in and p b , where the velocity of the air in the chimney is constant and equal to u c . The distributed temperature of the air inside the collector is represented by T.

The Analysis of the Air Flow in the Solar Collector
The incompressible flow of air (with a constant viscosity) inside the collector (see Figure 1), the stationary Navier-Stokes equation and principle of the conversation of mass flux crossing cylindrical surfaces have the forms [15]: where equation u r r = const results from the equation of conservation of mass.
After ignoring the small high-order parameters, the energy balance becomes where . q, u r , T, T ot , ρ, α, c p , and h represent the heat flux density and average radial velocity in the gap of the collector for the laminar flow, density of air, temperature of air, ambient temperature, and convective heat transfer coefficient between the ambient air and plate of the collector, the specific heat of air, and the depth of the collector, respectively.
By applying the boundary conditions at the collector inlet, we get The solutions to Equations (1) and (2) (in dimensionless form) in the collector can be expressed as the dimensionless pressure drop and dimensionless distribution of temperature, respectively [15]: where the following dimensionless variables were introduced Equation (4) was obtained from the solution of Equations (1) and (2) for laminar flow between the collector plates. θ q = 3, Bi = 0.005, Pr = 0.712, λ = 2200, for n = 0.04.
As seen from the theoretical analysis (see Figure 2), the pressure drop and air temperature depend on the radial coordinates of the collector. As the Reynolds number increases, the air temperature in the collector increases more slowly. However, the influence of the Reynolds number on the pressure drop in the collector is negligible.

The Analysis of the Air Flow in the Solar Collector
For the chimney, the momentum equation of the flow of air inside the chimney (see Figure 1) in the cylindrical coordinate system ( z r, ), with a vertical axis ( z ) directed upwards, can be written in In Equation (5), we neglect the free convection in the solar collector. The density in the chimney describes the equation, and represents the density of air in the chimney, which depends on the increase in temperature ( ).

ot T T c −
We assume linear dependence of density on temperature. The temperature c T in the chimney is constant, the coefficient of the thermal expansion of air is represented by β , g represents the gravitational acceleration, andν represents the kinematic viscosity of air. c p and c u represent the pressure and velocity inside the chimney, respectively. We apply the boundary conditions where the following correlations are used in the above equations: In the theoretical model, the local pressure ( p Δ ) in the flowing air between the solar collector and chimney is expressed as where ξ represents the factor for the local loss in pressure in the air flow from the collector to the chimney, which results from the change of air flow direction and change of the channel cross-section between collector and chimney.
The velocity and average velocity in the chimney from the solution to the momentum equation (Equation (5)) are equal to ( ),

The Analysis of the Air Flow in the Solar Collector
For the chimney, the momentum equation of the flow of air inside the chimney (see Figure 1) in the cylindrical coordinate system (r, z), with a vertical axis (z) directed upwards, can be written in the form In Equation (5), we neglect the free convection in the solar collector. The density in the chimney describes the equation, and ρ c = ρ[1 − β(T c − T ot )] represents the density of air in the chimney, which depends on the increase in temperature (T c − T ot ). We assume linear dependence of density on temperature. The temperature T c in the chimney is constant, the coefficient of the thermal expansion of air is represented by β, g represents the gravitational acceleration, and ν represents the kinematic viscosity of air. p c and u c represent the pressure and velocity inside the chimney, respectively.
We apply the boundary conditions where the following correlations are used in the above equations: In the theoretical model, the local pressure (∆p) in the flowing air between the solar collector and chimney is expressed as where ξ represents the factor for the local loss in pressure in the air flow from the collector to the chimney, which results from the change of air flow direction and change of the channel cross-section between collector and chimney. The velocity and average velocity in the chimney from the solution to the momentum equation (Equation (5)) are equal to Energies 2019, 12, 2060 6 of 13 The dimensionless pressure drop, Π CO , at the beginning of the chimney is equal to where Π CO represents the pressure drop calculated in the collector for r = r c . The velocity in the chimney, pressure drop at the beginning of the chimney, Reynolds number, Galileo number, Grashof number, and geometric parameter for the solar chimney can be presented in their dimensionless forms, respectively, as: The condition for the air flow in the chimney is calculated with Equation (9): From Equations (4) and (9), we achieve the coupled system below in Equation (11).
Using the functions in the coupled system (Equation (11)), θ c (Re), and Π CO (Re) enable us to calculate the Reynolds number (Re) from the following equation: The Reynolds number (Re) obtained from Equation (12)  For example, in Figure 3, the calculation of the Reynolds number was performed for the sample solar chimney system. Subsequently, the dimensionless parameters, Πco and θ c , were calculated.   Table 1 and Figure 4 display the air velocity c u and the air temperature C T in the chimney.   The theoretical analysis on the turbulent flow of air is more challenging. In this case, the velocities of the air flows in the collector channel and chimney may be different. The theoretical model of air flow should be modified. Figure 4 presents the relationships of the height of the chimney with both the temperature (calculated according to Equation (4)) and velocity of the air inside the chimney  Table 1 and Figure 4 display the air velocity u c and the air temperature T C in the chimney.   Table 1 and Figure 4 display the air velocity c u and the air temperature C T in the chimney.
The theoretical analysis on the turbulent flow of air is more challenging. In this case, the velocities of the air flows in the collector channel and chimney may be different. The theoretical model of air flow should be modified. Figure 4 presents the relationships of the height of the chimney with both the temperature (calculated according to Equation (4) The theoretical analysis on the turbulent flow of air is more challenging. In this case, the velocities of the air flows in the collector channel and chimney may be different. The theoretical model of air flow should be modified. Figure 4 presents the relationships of the height of the chimney with both the temperature (calculated according to Equation (4)) and velocity of the air inside the chimney (calculated according to Equation (9)). An increase in the chimney height causes an increase in air velocity and a decrease in air temperature. The increase in heat flux density increases both the temperature and air velocity in the chimney (see [15]).  Figure 5 shows the dependence of the energy efficiency of the solar chimney to the chimney height. Energy efficiency is the ratio of energy received to the energy supplied to the system. The energy obtained is the sum of the kinetic energy of the air and the increase in the enthalpy of the air. As shown in Figure 5, the energy efficiency of the solar chimney increases as the chimney height increases. (calculated according to Equation (9)). An increase in the chimney height causes an increase in air velocity and a decrease in air temperature. The increase in heat flux density increases both the temperature and air velocity in the chimney (see [15]). Figure 5 shows the dependence of the energy efficiency of the solar chimney to the chimney height. Energy efficiency is the ratio of energy received to the energy supplied to the system. The energy obtained is the sum of the kinetic energy of the air and the increase in the enthalpy of the air. As shown in Figure 5, the energy efficiency of the solar chimney increases as the chimney height increases. Designing the solar chimney based on the proposed theoretical flow characteristics allows one to easily determine the energy efficiency of the chimney. Particularly noteworthy is the possibility of optimizing the chimney height with the goal of maximizing its efficiency, while maintaining construction costs at an acceptable level. In this way, the profitability of the installation plant can be significantly improved. Figure 6 shows the experimental model of the solar chimney, which consists of two low cylindrical collectors and a vertical, thermally insulated pipe placed in the middle of the plate.

Experimental Investigation
The solar collector consists of two horizontal, flat disks forming an air gap spaced from each other. The diameter of the upper metal plate is 2000 mm, and the width of the air gap is 80 mm. In the middle of the board there is a vertical, thermally insulated pipe with an inner diameter of 100 mm and a height of 2500 mm.
Collector heating combined with the effect of a thermal chimney forces the airflow in the chimney. The velocity was measured using a paddle anemometer and Pitot tube. The Testo 417 anemometer was used for the measurements, which allows quick and precise measurements of air velocity from the chimney. The accuracy of the measurements was 0.1m/s ± .
The results of the air velocity were measured with the paddle anemometer, which was mounted above the chimney. Then, the results of measurements were compared with the measurements using a Pitot tube, which was mounted on 2/3 of the chimney height, as shown in Figure 6. Moreover, the pressure drop at the collector's connection to the vertical pipe was measured using a differential inclined-tube manometer. The solar radiation flux was measured using a pyranometer located at a nearby meteorological station. The results of the measurements are presented in Table 2 and Figure  7. Designing the solar chimney based on the proposed theoretical flow characteristics allows one to easily determine the energy efficiency of the chimney. Particularly noteworthy is the possibility of optimizing the chimney height with the goal of maximizing its efficiency, while maintaining construction costs at an acceptable level. In this way, the profitability of the installation plant can be significantly improved. Figure 6 shows the experimental model of the solar chimney, which consists of two low cylindrical collectors and a vertical, thermally insulated pipe placed in the middle of the plate.

Experimental Investigation
The solar collector consists of two horizontal, flat disks forming an air gap spaced from each other. The diameter of the upper metal plate is 2000 mm, and the width of the air gap is 80 mm. In the middle of the board there is a vertical, thermally insulated pipe with an inner diameter of 100 mm and a height of 2500 mm.
Collector heating combined with the effect of a thermal chimney forces the airflow in the chimney. The velocity was measured using a paddle anemometer and Pitot tube. The Testo 417 anemometer was used for the measurements, which allows quick and precise measurements of air velocity from the chimney. The accuracy of the measurements was ±0.1 m/s. The results of the air velocity were measured with the paddle anemometer, which was mounted above the chimney. Then, the results of measurements were compared with the measurements using a Pitot tube, which was mounted on 2/3 of the chimney height, as shown in Figure 6. Moreover, the pressure drop at the collector's connection to the vertical pipe was measured using a differential inclined-tube manometer. The solar radiation flux was measured using a pyranometer located at a nearby meteorological station. The results of the measurements are presented in Table 2 and Figure 7. The results obtained in the analytical and experimental studies of the air velocity in the chimney are consistent and can be considered satisfactory. The scattering of results was mainly caused by disturbances caused by wind action between the collector plates on its perimeter. In this zone, the air flow velocities were very small (approximately s m 02 . 0 ), which was consistent with the theoretical model, and was, therefore, very sensitive to external gusts, which affected the operation of the solar chimney system.

Conclusions
In this work, the obtained results of the experimental investigation and the theoretical considerations are satisfactory. The presented method is particularly suitable for laminar flow conditions in a collector (for relatively small systems, not for electricity generation). Under turbulent  The results obtained in the analytical and experimental studies of the air velocity in the chimney are consistent and can be considered satisfactory. The scattering of results was mainly caused by disturbances caused by wind action between the collector plates on its perimeter. In this zone, the air flow velocities were very small (approximately 0.02 m/s), which was consistent with the theoretical model, and was, therefore, very sensitive to external gusts, which affected the operation of the solar chimney system.

Conclusions
In this work, the obtained results of the experimental investigation and the theoretical considerations are satisfactory. The presented method is particularly suitable for laminar flow conditions in a collector (for relatively small systems, not for electricity generation). Under turbulent flow conditions (for larger systems with higher internal air velocities, which can be used for small-scale electricity generation), the results are less accurate, although their accuracy still seems to be acceptable at first. The approximated results are sufficient for the purpose of a design evaluation of solar collector-chimney systems. The considerations observe both the first and second principles of thermodynamics.
The paper presents the dependence of the energy efficiency of the solar collector system to the height of the chimney. It was shown that the energy efficiency of the solar collector system increases as the chimney height increases.
The presented modelling allows one to mimic very small systems, which is a foundation for future work to look into larger systems, up to the size of power plants.