# Modeling the Thermal Performance for Different Types of Solar Chimney Power Plants

Authors: Ghassan F. Smaisim; Azher M. Abed; Ali Shamel
Journal: Complexity
 (2022)

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## Abstract

Nowadays, due to restrictions on fossil fuels, the use of renewable energies is increasing day by day. Among renewable energies, solar energy has received more attraction due to its availability in all places. Among solar energy technologies, the solar tower has been welcomed due to its high power generation of electrical energy. For accurate modeling of the studied system, each component of the system has been evaluated and modeling has been done. Therefore, in this research, solar tower modeling has been conducted to achieve high electrical energy production, and to better compare the production rate, 5 cities in Iran with different weather conditions have been considered. According to the results, it can be mentioned that the highest production power by the studied system is related to Shiraz city with an average production of 20 kW/m2, and among the cities studied, the lowest rate is related to Mashhad with a production power of 15 kW/m2.


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## Body

## 1. Introduction

Nowadays, the importance of using clean energy and replacing it with fossil fuels is not covered as a basic solution for protecting the environment. Also, the advantages of consuming solar energy as a clean energy source, especially in Iran, are clear [1, 2]. A simple technology that uses solar energy to generate electricity is the solar chimney power plant. In addition to using a simple technology, this type of power plant has special advantages such as no need for water and the possibility of generating electricity during the day and night [3, 4].Figure1 shows a solar chimney power plant in the form of a schema. The power plant consists of a transparent collector and a cylinder chimney at its center. One or more turbines are also installed near the chimney base, which is connected to a generator. The function of this power plant is that the sun’s radiation after passing through the ceiling of the collector (which is a transparent layer like a glass) mainly absorbs the surface of the earth, leading to the warming of the earth’s surface and, consequently, the air adjacent to the earth, the higher temperature of the air inside the chimney column than the outside air, causing the effect of the chimney, which leads to the suction of the ambient air into the collector and then the chimney. Inside the chimney, it moves turbines and generates electricity [5–7].Figure 1
Schema of a solar chimney power plant.The beginning of the most important studies on this subject dates back to 1931 when the basic principles and description of the solar chimney power plant were reported by Günter [8]. The original example of the solar chimney power plant with an output power of 50 kilowatts was designed by Cuce in 2020, 150 kilometers south of Madrid, Manzanares, Spain [2, 3]. Cuce et al. presented a brief discussion of energy balance, design rules, production power, and power plant cost analysis [9]. In the next study, Cuce reported the preliminary results of testing the power plant built in Spain [10]. In 2021, Malta provided an analysis to determine the total efficiency of the power plant. He found that even though a solar chimney has little efficiency, building a large-scale power plant is the only economically viable option [11]. Belkhode et al. presented an evaluation of the experimental data to determine the performance of a solar chimney power plant [12].Khidhir considered the airflow inside the collector as an extended radial flow between the two parallel disks in order to obtain the heat transfer coefficient of the solar chimney collector. In this study, the required analytical equations were presented to estimate the coefficients of heat transfer and pressure drop obtained from friction [13]. Galia studied the performance of the turbine used in the solar chimney power plant [14].To predict the performance of solar chimney power plants, many researchers have used computational fluid dynamics (CFD). As an illustration, Torabi et al. presented a two-dimensional numerical model to estimate the air temperature distribution within the power plant collector [15]. In the following sections, Arzpeyma et al. simulated the solar chimney power plant and its correlation with the ambient wind effect [16]. In Reference [17], the authors presented the developed mathematical model CFD analysis for solar chimney efficiency evaluation with height variation.Habibollahzade et al. developed a comprehensive theoretical model considering hourly changes in solar radiation and investigated the effect of different geometric parameters of the power plant on the output power [18]. Moreover, Aligholami et al. studied the feasibility of implementing solar chimney power plants in the Mediterranean regions [19].Many researchers investigated the performance of solar chimney power plants and the quantity of electrical energy production in Iran [20, 21]. However, in most studies, transmission phenomena in the power plant have been modeled in a reliable state [22–24]. While the constraints of accumulation, especially the accumulation of heat in the surface layers of the earth and the chimney wall, can cause a significant error in the simulation results [25].In Reference [4], the authors presented the effects of geometric parameters on the performance of solar chimney power plants. Torabi et al. studied the investigation the performance of the solar chimney power plant for improving the efficiency and increasing the outlet power of turbines using computational fluid dynamics [15]. In Reference [26], the authors presented the solar chimney power plants—dimension matching for optimum performance. Cuce et al. studied the performance assessment of solar chimney power plants with the impacts of divergent and convergent chimney geometry [27].In this study, the modeling of all parts of the power plant has been done seamlessly and in unstable conditions. The use of the discontinuous model allows, along with considering the changes in the ambient temperature and the amount of sunlight during the day and night, heat reserves in different power plant environments, including in the surface layers of the earth and the chimney wall, are included in the calculations. In addition, the instantaneous solar radiation for Iranian cities is calculated using associate professors’ equations, which have been developed based on Iran’s climatic conditions. In addition, the information about ambient temperature during the day and night has been extracted from meteorological data and used.After verifying the accuracy of the proposed model by comparing its results with the existing empirical data, a simulation of the power plant with dimensions and same-sex power plants constructed in Manzanares, Spain (capacity of 50 kW) has been carried out in 5 different cities of Iran (Shiraz, Bushehr, Kerman, Tehran, and Mashhad) and its average year-long power has been calculated. The results of the modeling, while emphasizing on higher efficiency of this power plant in Iran compared to Spain, indicate a significant difference in the performance of solar chimney power plants in different cities of Iran. Accordingly, the construction of power plants in Shiraz will have the highest efficiency among the five selected cities.


## 2. Materials and Methods

In this section, the governing equations of transmission phenomena are presented in different parts of the power plant which include the collector, turbine, and chimney tower. In order to be more accurate in calculations, in modeling all parts of the collector and chimney, accumulation expression is considered for continuity, motion, and energy equations.


### 2.1. Collector

The governing equations in the collector, assuming that the radial current is fully developed, are as follows.


#### 2.1.1. Equation of Continuity

(1)−1r∂∂rrρu=∂ρ∂t,where ρ and u better is the density and speed of airflow within the collector, r is the radius of the collector, and t is the time.


#### 2.1.2. Equation of Motion

This equation is derived from the momentum balance as follows:(2)−∂P∂r−ρu∂u∂r−τrHr−τgHr=ρ∂u∂t.Here,P and Hr are better indications of the air pressure inside the collector and the height of the collector’s cover from the ground up. Also, τr and τg are better shear stress in the ceiling of the collector and the ground surface.By condoning the roughness of the collector’s ceiling, during the fully developed turbulence, the valueτr can be obtained from the following equation:(3)τr=0.023ρu2Re0.2.The Reynolds number is defined and calculated as follows:(4)Re=ρudhμ,where μ is the dynamic viscosity and dh is the hydraulic diameter. This quantity for flowing between parallel pages is twice the distance between pages. Shear stress caused by the radial flow on the earth’s surface can be calculated by the Kruger–Bayes equation (6).(5)τg=12fρu2,where f is the friction coefficient and can be calculated as follows:(6)f=0.02975εgdh0.2541.75μρuεg0.51+1.Here,εg is earth roughness. The combination of the above two equations can be expressed as follows:(7)τg=0.014875ρur2εg2Hr0.2541.75μρurεg0.51+1.


#### 2.1.3. The Equation of Air Energy inside the Collector

Energy balance on a fluid element that is in contact from below with the ground and from above is in contact with the collector, as expressed in the following form:(8)−ρCpHrur∂Tf∂r+hgfTg−Tf+hrfTr−Tf=ρCpHr∂Tf∂t.In this equation,Cp is special thermal capacity, and Tg, Tf, and Tr are ground surface temperature, airflow temperature inside the collector, and ceiling temperature of collector cover, respectively.Due to low airspeed, the displacement mechanism is a combination of free and compulsory displacement. In this case, the displacement coefficient between the airf and the ground (hgf) can be calculated from the following relationship [8]:(9)hgf=Nuforcekf2Hr4+NufreekfHr41/4.Thatkf is the thermal conductivity of the air inside the collector. Also, Nusselt no-dimension numbers are obtained based on the following equations:(10)Nufree=0.15GrPr1/3,Gr=gβTg,1−TfH3υ2,Nuforce=0.024Re0.8Pr0.4,where Gr and Pr are better than numbers without the Grashof dimension and Prandtl, g is the acceleration of earth’s gravity, β volumetric expansion coefficient, υ cinematic viscosity, and Tg,1 ground surface temperature.(11)Re=2uHrυ.The heat transfer coefficient of the displacement between the inner airflow and the cover of the collector (hrf) is similar to the method of calculating the heat transfer coefficient of the displacement between the airflow and the ground, of course, with this method.The difference in the Grashof equation should be the temperature of the collector’s surface instead of the temperature of the earth’s surface.


#### 2.1.4. Equation of Collector Ceiling Energy

The energy balance on the roof of the collector is as follows:(12)Ibαb+Idαd+hgrTg−Tr−hraTr−Ta−hrsTr−Ts+hrsTr−Ts=ρrCprδr∂Tr∂t.In this equation,Ib and Id are direct solar radiation intensity and scattered solar radiation intensity, respectively, which are calculated by associate professors’ equations (17). αb is the direct effective absorption coefficient, αd is the effective absorption coefficient of dispersion, and Ta is the ambient air temperature. Also, ρr, Cpr, and δr are better density, special thermal capacity, and thickness of collector coating. The heat transfer coefficient of the displacement between the cover surface of the collector and the outer air (hra), which is affected by the wind on the flat surface, is calculated as follows [18]:(13)hra=2.8+3uw.Here,uw is the ambient wind speed. Also, hgr and hrs, which are the radiation heat transfer coefficient between the earth’s two surfaces and the coating, respectively, as well as the radiation heat transfer coefficient between the earth and the sky, can be calculated from the following equations (19):(14)hgr=σ1/εg+1/εr−1Tg2+Tr2Tg+Tr,hrs=σεrTs2+Tr2Ts+Tr,where σ, εg and εr are Stephen Boltzmann’s constant, land propagation coefficient, and collector cover dike coefficient, respectively, and Ts is sky temperature, the amount of which is estimated in terms of Kelvin as follows [20]:(15)Ts=0.0552Ta1.5.


#### 2.1.5. Equation of Earth’s Energy

In this section, the Earth can be considered as a semiunlimited body. Therefore, the Earth’s temperature equation is as follows:(16)∂2Tg∂Z2=ρgcpgkg∂Tg∂t,where ρg, cpg, and kg are density, special thermal capacity, and thermal conductivity coefficient of the earth, respectively.The first boundary condition of this equation is obtained using the energy balance at the ground level as follows:(17)Ibτrαgb+Idτrαgd=hgrTg−Tr+hgfTg−Tf−kg∂Tg∂Z.Due to the inseparability of the Earth’s radius and its low thermal conductivity, the second boundary condition is as follows.(18)Z⟶∞:∂Tg∂Z=0.


### 2.2. Turbine

Turbine pressure drop can be calculated using the Betz power rule [8]:(19)Δpt=827ρut2,where ut is the airspeed at the turbine’s output. The power output power of the power plant is also calculated as follows [12]:(20)Power=mηtΔptρ,where ηt is the turbine efficiency, considering the experimental results obtained in the Spanish laboratories, this parameter is considered to be 0.83. [2].


### 2.3. Chimney

By condoning the changes in air velocity in line with the radius of the governing equations in the chimney, it is expressed as follows.


#### 2.3.1. Equation of Continuity

(21)−∂∂Zρuz=∂ρ∂t.


#### 2.3.2. Equation of Motion

(22)−∂P∂Z−ρuz∂uz∂Z−2τchRch−ρg=ρ∂uz∂t.Shear stress on the chimney wall can be calculated from the following equation:(23)τch=18fρuz2.Due to the magnitude of the chimney diameter, the roughness effect of its surface is negligent and the friction coefficient of the (f) chimney wall is obtained from the following equation (22):(24)f=1.82logRe−1.64−2.


#### 2.3.3. The Equation of Air Energy inside the Chimney

By condoning the changes in air temperature in the direction of radius and in addition to giving up thermal conductivity in comparison with displacement, the equation of air temperature distribution inside the chimney is written as follows.(25)−ρuzCp∂Tz∂Z−2hfRchTz−Twi=ρCp∂Tz∂t.The required boundary condition of this part is obtained from equal to the temperature of the entrance to the chimney with the air temperature of the output from the collector. The heat transfer coefficient between the air inside the tower and the inner wall of the tower is calculated from the following equations:(26)hf=Nufkairdch,Nuf=0.023Re0.8Pr1/3,Re=ρudchμ.


#### 2.3.4. Equation of Chimney Wall Energy

Since the thickness of the chimney is insignificant compared to its height, heat conductivity can be assumed to be one-dimensional and unstable.(27)1r∂∂rrkw∂Tw∂r=∂∂tρwCρwTw.Boundary conditions at the internal and external levels of energy balance on those surfaces are obtained as follows:The equation for the internal surface is as follows:(28)kw∂Tw∂r|r=Ri=hfTw|r=Ri−T.The equation for the external surface is as follows:(29)gT−kw∂Tw∂r|r=Rout+haTar=Rout−T=0.In this regard,qt is the total direct and solar intrusive radiation absorbed by the external surface of the chimney.The heat transfer coefficient between the ambient air and the external wall of the tower can also be calculated from the following equation (19):(30)ha=11.4+1.4uwind.


### 2.4. Method of Performance Calculations

To simulate the performance of a solar chimney power plant, it is necessary to solve all the governing equations seamlessly. In the prepared simulation, these equations are solved using the finite difference numerical method.Simulation calculations are performed in two circles of trial and error. In the external circle, the early temperature values of the fluid and power plant components (including the air inside the collector and the chimney, the surface layers of the earth, the cover surface, and the temperature of the chimney wall) are guessed at zero time; all of these temperatures must be equal to the computational values for the temperature at the end of the day at 24 o’clock.In the inner circle at each time stage, the mass discharge of the air inside the power plant is guessed. The correct condition of this guess is the equal value calculated for the air pressure reached at the end of the chimney with the outside air pressure at that altitude. Figure2 shows the design of the calculations in the prepared simulator.Figure 2
Layout of calculations.


## 2.1. Collector

The governing equations in the collector, assuming that the radial current is fully developed, are as follows.


### 2.1.1. Equation of Continuity

(1)−1r∂∂rrρu=∂ρ∂t,where ρ and u better is the density and speed of airflow within the collector, r is the radius of the collector, and t is the time.


### 2.1.2. Equation of Motion

This equation is derived from the momentum balance as follows:(2)−∂P∂r−ρu∂u∂r−τrHr−τgHr=ρ∂u∂t.Here,P and Hr are better indications of the air pressure inside the collector and the height of the collector’s cover from the ground up. Also, τr and τg are better shear stress in the ceiling of the collector and the ground surface.By condoning the roughness of the collector’s ceiling, during the fully developed turbulence, the valueτr can be obtained from the following equation:(3)τr=0.023ρu2Re0.2.The Reynolds number is defined and calculated as follows:(4)Re=ρudhμ,where μ is the dynamic viscosity and dh is the hydraulic diameter. This quantity for flowing between parallel pages is twice the distance between pages. Shear stress caused by the radial flow on the earth’s surface can be calculated by the Kruger–Bayes equation (6).(5)τg=12fρu2,where f is the friction coefficient and can be calculated as follows:(6)f=0.02975εgdh0.2541.75μρuεg0.51+1.Here,εg is earth roughness. The combination of the above two equations can be expressed as follows:(7)τg=0.014875ρur2εg2Hr0.2541.75μρurεg0.51+1.


### 2.1.3. The Equation of Air Energy inside the Collector

Energy balance on a fluid element that is in contact from below with the ground and from above is in contact with the collector, as expressed in the following form:(8)−ρCpHrur∂Tf∂r+hgfTg−Tf+hrfTr−Tf=ρCpHr∂Tf∂t.In this equation,Cp is special thermal capacity, and Tg, Tf, and Tr are ground surface temperature, airflow temperature inside the collector, and ceiling temperature of collector cover, respectively.Due to low airspeed, the displacement mechanism is a combination of free and compulsory displacement. In this case, the displacement coefficient between the airf and the ground (hgf) can be calculated from the following relationship [8]:(9)hgf=Nuforcekf2Hr4+NufreekfHr41/4.Thatkf is the thermal conductivity of the air inside the collector. Also, Nusselt no-dimension numbers are obtained based on the following equations:(10)Nufree=0.15GrPr1/3,Gr=gβTg,1−TfH3υ2,Nuforce=0.024Re0.8Pr0.4,where Gr and Pr are better than numbers without the Grashof dimension and Prandtl, g is the acceleration of earth’s gravity, β volumetric expansion coefficient, υ cinematic viscosity, and Tg,1 ground surface temperature.(11)Re=2uHrυ.The heat transfer coefficient of the displacement between the inner airflow and the cover of the collector (hrf) is similar to the method of calculating the heat transfer coefficient of the displacement between the airflow and the ground, of course, with this method.The difference in the Grashof equation should be the temperature of the collector’s surface instead of the temperature of the earth’s surface.


### 2.1.4. Equation of Collector Ceiling Energy

The energy balance on the roof of the collector is as follows:(12)Ibαb+Idαd+hgrTg−Tr−hraTr−Ta−hrsTr−Ts+hrsTr−Ts=ρrCprδr∂Tr∂t.In this equation,Ib and Id are direct solar radiation intensity and scattered solar radiation intensity, respectively, which are calculated by associate professors’ equations (17). αb is the direct effective absorption coefficient, αd is the effective absorption coefficient of dispersion, and Ta is the ambient air temperature. Also, ρr, Cpr, and δr are better density, special thermal capacity, and thickness of collector coating. The heat transfer coefficient of the displacement between the cover surface of the collector and the outer air (hra), which is affected by the wind on the flat surface, is calculated as follows [18]:(13)hra=2.8+3uw.Here,uw is the ambient wind speed. Also, hgr and hrs, which are the radiation heat transfer coefficient between the earth’s two surfaces and the coating, respectively, as well as the radiation heat transfer coefficient between the earth and the sky, can be calculated from the following equations (19):(14)hgr=σ1/εg+1/εr−1Tg2+Tr2Tg+Tr,hrs=σεrTs2+Tr2Ts+Tr,where σ, εg and εr are Stephen Boltzmann’s constant, land propagation coefficient, and collector cover dike coefficient, respectively, and Ts is sky temperature, the amount of which is estimated in terms of Kelvin as follows [20]:(15)Ts=0.0552Ta1.5.


### 2.1.5. Equation of Earth’s Energy

In this section, the Earth can be considered as a semiunlimited body. Therefore, the Earth’s temperature equation is as follows:(16)∂2Tg∂Z2=ρgcpgkg∂Tg∂t,where ρg, cpg, and kg are density, special thermal capacity, and thermal conductivity coefficient of the earth, respectively.The first boundary condition of this equation is obtained using the energy balance at the ground level as follows:(17)Ibτrαgb+Idτrαgd=hgrTg−Tr+hgfTg−Tf−kg∂Tg∂Z.Due to the inseparability of the Earth’s radius and its low thermal conductivity, the second boundary condition is as follows.(18)Z⟶∞:∂Tg∂Z=0.


## 2.1.1. Equation of Continuity

(1)−1r∂∂rrρu=∂ρ∂t,where ρ and u better is the density and speed of airflow within the collector, r is the radius of the collector, and t is the time.


## 2.1.2. Equation of Motion

This equation is derived from the momentum balance as follows:(2)−∂P∂r−ρu∂u∂r−τrHr−τgHr=ρ∂u∂t.Here,P and Hr are better indications of the air pressure inside the collector and the height of the collector’s cover from the ground up. Also, τr and τg are better shear stress in the ceiling of the collector and the ground surface.By condoning the roughness of the collector’s ceiling, during the fully developed turbulence, the valueτr can be obtained from the following equation:(3)τr=0.023ρu2Re0.2.The Reynolds number is defined and calculated as follows:(4)Re=ρudhμ,where μ is the dynamic viscosity and dh is the hydraulic diameter. This quantity for flowing between parallel pages is twice the distance between pages. Shear stress caused by the radial flow on the earth’s surface can be calculated by the Kruger–Bayes equation (6).(5)τg=12fρu2,where f is the friction coefficient and can be calculated as follows:(6)f=0.02975εgdh0.2541.75μρuεg0.51+1.Here,εg is earth roughness. The combination of the above two equations can be expressed as follows:(7)τg=0.014875ρur2εg2Hr0.2541.75μρurεg0.51+1.


## 2.1.3. The Equation of Air Energy inside the Collector

Energy balance on a fluid element that is in contact from below with the ground and from above is in contact with the collector, as expressed in the following form:(8)−ρCpHrur∂Tf∂r+hgfTg−Tf+hrfTr−Tf=ρCpHr∂Tf∂t.In this equation,Cp is special thermal capacity, and Tg, Tf, and Tr are ground surface temperature, airflow temperature inside the collector, and ceiling temperature of collector cover, respectively.Due to low airspeed, the displacement mechanism is a combination of free and compulsory displacement. In this case, the displacement coefficient between the airf and the ground (hgf) can be calculated from the following relationship [8]:(9)hgf=Nuforcekf2Hr4+NufreekfHr41/4.Thatkf is the thermal conductivity of the air inside the collector. Also, Nusselt no-dimension numbers are obtained based on the following equations:(10)Nufree=0.15GrPr1/3,Gr=gβTg,1−TfH3υ2,Nuforce=0.024Re0.8Pr0.4,where Gr and Pr are better than numbers without the Grashof dimension and Prandtl, g is the acceleration of earth’s gravity, β volumetric expansion coefficient, υ cinematic viscosity, and Tg,1 ground surface temperature.(11)Re=2uHrυ.The heat transfer coefficient of the displacement between the inner airflow and the cover of the collector (hrf) is similar to the method of calculating the heat transfer coefficient of the displacement between the airflow and the ground, of course, with this method.The difference in the Grashof equation should be the temperature of the collector’s surface instead of the temperature of the earth’s surface.


## 2.1.4. Equation of Collector Ceiling Energy

The energy balance on the roof of the collector is as follows:(12)Ibαb+Idαd+hgrTg−Tr−hraTr−Ta−hrsTr−Ts+hrsTr−Ts=ρrCprδr∂Tr∂t.In this equation,Ib and Id are direct solar radiation intensity and scattered solar radiation intensity, respectively, which are calculated by associate professors’ equations (17). αb is the direct effective absorption coefficient, αd is the effective absorption coefficient of dispersion, and Ta is the ambient air temperature. Also, ρr, Cpr, and δr are better density, special thermal capacity, and thickness of collector coating. The heat transfer coefficient of the displacement between the cover surface of the collector and the outer air (hra), which is affected by the wind on the flat surface, is calculated as follows [18]:(13)hra=2.8+3uw.Here,uw is the ambient wind speed. Also, hgr and hrs, which are the radiation heat transfer coefficient between the earth’s two surfaces and the coating, respectively, as well as the radiation heat transfer coefficient between the earth and the sky, can be calculated from the following equations (19):(14)hgr=σ1/εg+1/εr−1Tg2+Tr2Tg+Tr,hrs=σεrTs2+Tr2Ts+Tr,where σ, εg and εr are Stephen Boltzmann’s constant, land propagation coefficient, and collector cover dike coefficient, respectively, and Ts is sky temperature, the amount of which is estimated in terms of Kelvin as follows [20]:(15)Ts=0.0552Ta1.5.


## 2.1.5. Equation of Earth’s Energy

In this section, the Earth can be considered as a semiunlimited body. Therefore, the Earth’s temperature equation is as follows:(16)∂2Tg∂Z2=ρgcpgkg∂Tg∂t,where ρg, cpg, and kg are density, special thermal capacity, and thermal conductivity coefficient of the earth, respectively.The first boundary condition of this equation is obtained using the energy balance at the ground level as follows:(17)Ibτrαgb+Idτrαgd=hgrTg−Tr+hgfTg−Tf−kg∂Tg∂Z.Due to the inseparability of the Earth’s radius and its low thermal conductivity, the second boundary condition is as follows.(18)Z⟶∞:∂Tg∂Z=0.


## 2.2. Turbine

Turbine pressure drop can be calculated using the Betz power rule [8]:(19)Δpt=827ρut2,where ut is the airspeed at the turbine’s output. The power output power of the power plant is also calculated as follows [12]:(20)Power=mηtΔptρ,where ηt is the turbine efficiency, considering the experimental results obtained in the Spanish laboratories, this parameter is considered to be 0.83. [2].


## 2.3. Chimney

By condoning the changes in air velocity in line with the radius of the governing equations in the chimney, it is expressed as follows.


### 2.3.1. Equation of Continuity

(21)−∂∂Zρuz=∂ρ∂t.


### 2.3.2. Equation of Motion

(22)−∂P∂Z−ρuz∂uz∂Z−2τchRch−ρg=ρ∂uz∂t.Shear stress on the chimney wall can be calculated from the following equation:(23)τch=18fρuz2.Due to the magnitude of the chimney diameter, the roughness effect of its surface is negligent and the friction coefficient of the (f) chimney wall is obtained from the following equation (22):(24)f=1.82logRe−1.64−2.


### 2.3.3. The Equation of Air Energy inside the Chimney

By condoning the changes in air temperature in the direction of radius and in addition to giving up thermal conductivity in comparison with displacement, the equation of air temperature distribution inside the chimney is written as follows.(25)−ρuzCp∂Tz∂Z−2hfRchTz−Twi=ρCp∂Tz∂t.The required boundary condition of this part is obtained from equal to the temperature of the entrance to the chimney with the air temperature of the output from the collector. The heat transfer coefficient between the air inside the tower and the inner wall of the tower is calculated from the following equations:(26)hf=Nufkairdch,Nuf=0.023Re0.8Pr1/3,Re=ρudchμ.


### 2.3.4. Equation of Chimney Wall Energy

Since the thickness of the chimney is insignificant compared to its height, heat conductivity can be assumed to be one-dimensional and unstable.(27)1r∂∂rrkw∂Tw∂r=∂∂tρwCρwTw.Boundary conditions at the internal and external levels of energy balance on those surfaces are obtained as follows:The equation for the internal surface is as follows:(28)kw∂Tw∂r|r=Ri=hfTw|r=Ri−T.The equation for the external surface is as follows:(29)gT−kw∂Tw∂r|r=Rout+haTar=Rout−T=0.In this regard,qt is the total direct and solar intrusive radiation absorbed by the external surface of the chimney.The heat transfer coefficient between the ambient air and the external wall of the tower can also be calculated from the following equation (19):(30)ha=11.4+1.4uwind.


## 2.3.1. Equation of Continuity

(21)−∂∂Zρuz=∂ρ∂t.


## 2.3.2. Equation of Motion

(22)−∂P∂Z−ρuz∂uz∂Z−2τchRch−ρg=ρ∂uz∂t.Shear stress on the chimney wall can be calculated from the following equation:(23)τch=18fρuz2.Due to the magnitude of the chimney diameter, the roughness effect of its surface is negligent and the friction coefficient of the (f) chimney wall is obtained from the following equation (22):(24)f=1.82logRe−1.64−2.


## 2.3.3. The Equation of Air Energy inside the Chimney

By condoning the changes in air temperature in the direction of radius and in addition to giving up thermal conductivity in comparison with displacement, the equation of air temperature distribution inside the chimney is written as follows.(25)−ρuzCp∂Tz∂Z−2hfRchTz−Twi=ρCp∂Tz∂t.The required boundary condition of this part is obtained from equal to the temperature of the entrance to the chimney with the air temperature of the output from the collector. The heat transfer coefficient between the air inside the tower and the inner wall of the tower is calculated from the following equations:(26)hf=Nufkairdch,Nuf=0.023Re0.8Pr1/3,Re=ρudchμ.


## 2.3.4. Equation of Chimney Wall Energy

Since the thickness of the chimney is insignificant compared to its height, heat conductivity can be assumed to be one-dimensional and unstable.(27)1r∂∂rrkw∂Tw∂r=∂∂tρwCρwTw.Boundary conditions at the internal and external levels of energy balance on those surfaces are obtained as follows:The equation for the internal surface is as follows:(28)kw∂Tw∂r|r=Ri=hfTw|r=Ri−T.The equation for the external surface is as follows:(29)gT−kw∂Tw∂r|r=Rout+haTar=Rout−T=0.In this regard,qt is the total direct and solar intrusive radiation absorbed by the external surface of the chimney.The heat transfer coefficient between the ambient air and the external wall of the tower can also be calculated from the following equation (19):(30)ha=11.4+1.4uwind.


## 2.4. Method of Performance Calculations

To simulate the performance of a solar chimney power plant, it is necessary to solve all the governing equations seamlessly. In the prepared simulation, these equations are solved using the finite difference numerical method.Simulation calculations are performed in two circles of trial and error. In the external circle, the early temperature values of the fluid and power plant components (including the air inside the collector and the chimney, the surface layers of the earth, the cover surface, and the temperature of the chimney wall) are guessed at zero time; all of these temperatures must be equal to the computational values for the temperature at the end of the day at 24 o’clock.In the inner circle at each time stage, the mass discharge of the air inside the power plant is guessed. The correct condition of this guess is the equal value calculated for the air pressure reached at the end of the chimney with the outside air pressure at that altitude. Figure2 shows the design of the calculations in the prepared simulator.Figure 2
Layout of calculations.


## 3. Results and Discussion

To validate the prepared model and numerical method of solving the simulation results equations are compared with the experimental data of The Manzanares Solar Chimney Power Plant in Spain. The power plant is built from a collector with a diameter of 224 meters and a tower with a diameter of 10 meters and a height of 200 meters. Also, the average height of the collector from the ground level is 85.1 meters. Other thermophysics dimensions and properties of different parts of the power plant are listed in Table1 [28].Table 1
Thermophysics properties and power plant dimensions.
The ceiling of the collector (glass)Glass emission coefficient0.85The roughness of glass0Thickness of glass0/004External radius112Internal radius12Average elevation from the ground level1/85Earth (soil)Emission coefficient0/9Absorb coefficient0/91Density1900 Kg/m3Special thermal capacity840 Jules per kilogram degree celsiusRoughness0/05TurbineOutput0/83Chimney (cement)Height200Inner diameter10Thickness0/1Roughness on the surface0/002Publication coefficient0/84Figure3, the output results of the Manzanars power plant can be compared with the results of this research. As can be seen, the simulation results are in very good agreement with the experimental data, which confirms the accuracy of the calculations. Table 2 shows the average daily power and energy generated on September 2 for two power plants. As can be seen, this power plant in Shiraz has approximately 92% more production capacity than the Spanish pilot. This indicates the importance of investigating and the possibility of building such power plants in Iran, especially in Shiraz.Figure 3
The output of Manzanares power plant in Spain and its comparison with experimental data.Table 2
Comparison of production power and energy production in September.
Daily generated energy (MJ)Average production (kW/day)Shiraz2671/4930/92Spain1390/1816/09Figure4 shows changes in ambient temperature and solar radiation in Shiraz and Manzanares cities. A comparison of the power output of the Shiraz power plant compared to the Spanish pilot is shown in Figure 5.Figure 4
Comparison of (a) ambient temperature and (b) solar radiation between Shiraz and Manzanares on September 2.
(a)(b)Figure 5
Comparison of power production in Shiraz and Manzanares cities on September 2.In this research, to investigate the effect of geographical location and environmental factors on the performance of solar chimney power plants and to optimize their construction location in Iran, 5 cities in Iran (Shiraz, Bushehr, Kerman, Tehran, and Mashhad) with different climatic and environmental conditions and a high amount of solar radiation, have been selected. In Table3, the climate and climatic conditions of the selected cities can be seen [17].Table 3
Climate and climatic conditions of selected cities.
CityLatitudeAverage annual relative humidity (%)The average yearly temperature differenceAnnual solar radiation (MJ/m)Shiraz29/5328/414/587675/51Bushehr28/2962/78/497337/18Kerman30/2512/518/557071/19Tehran35/6823/48/546678/48Mashhad36/2624/311/026571/66Figure6 shows the solar energy radiated per unit area of the horizon on the middle of the moon in 5 selected cities, fulfilled using the associate’s equations (17). As can be seen, Shiraz has the highest MJ/m2/year (51.7675) and Mashhad has the lowest solar/year MJ/m (66.6571), and the average monthly temperature in the 5 selected cities is shown in Figure 7.Figure 6
Average monthly total solar radiation per day in different months of the Gregorian year in selected cities.Figure 7
Monthly average temperature changes in different months of the year for selected cities in Iran.Figure8 shows the average monthly production power of the modeled power plant in 5 cities in Iran (Shiraz, Bushehr, Kerman, Tehran, and Mashhad).Figure 8
Average changes in power plant production power in different months of the year for selected cities.As can be seen in Figure8, among these five selected cities, the construction of power plants in Shiraz has the highest efficiency and is better than Mashhad, Tehran, Kerman, and Bushehr has better 40%, 13%, 3%, and 16% more production capacity than Mashhad and Tehran, comparing average year power and average year radiation for 5 cities of Iran (Shiraz, Bushehr, Kerman, Tehran). In Figure 9, we can see the results of Mashhad city, which includes the average production power.Figure 9
Comparison of average year-on-year power production in different cities of Iran.Bushehr, despite having the stronger radiation, has less efficiency than Shiraz; the humidity of the air as well as the amplitude of temperature changes during the day.


## 4. Conclusion

In this paper, the governing equations of the solar chimney simulator were presented and it was illustrated that the prepared household simulator has good accuracy. Using a simulator of a solar power plant in the dimensions of a Spain power plant, for five different cities of Iran (Shiraz, Bushehr, Kerman, Tehran, and Mashhad), the model and the following results were observed.(1)
If this power plant is constructed in Shiraz, the energy produced per month in September is 92% higher than the same amount in Spain.(2)
Among the 5 selected cities, Shiraz has the highest efficiency, and power plants in Shiraz are 7.14% more than the average power in four cities (Bushehr, Kerman, Tehran, and Mashhad) estimated.(3)
Among the five cities studied, after Shiraz, the priority of constructing power plants is in Kerman, Tehran, Bushehr, and Mashhad, respectively.(4)
The reason for the lower efficiency of the power plant in the coastal city of Bushehr seems to be the high humidity of the city’s air, as well as the lower scope of the daily temperature changes in this city.



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*Source: 3656482-2022-07-18.xml*