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Numerical analysis of the optimal turbine pressure drop ratio in a solar chimney power plant
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Solar Energy 98 (2013) 42–48 www.elsevier.com/locate/solener
Numerical analysis of the optimal turbine pressure drop ratio in a solar chimney power plant Penghua Guo, Jingyin Li ⇑, Yuan Wang, Yingwen Liu School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China Available online 14 June 2013 Communicated by: Associate Editor Xinping Zhou
Abstract In a solar chimney power plant, only a fraction of the available total pressure difference can be used to run the turbine to generate electrical power. The optimal ratio of the turbine pressure drop to the available total pressure difference in a solar chimney system is investigated using theoretical analysis and 3D numerical simulations. The values found in the literature for the optimal ratio vary between 2/3 and 0.97. In this study, however, the optimal ratio was found to vary with the intensity of solar radiation, and to be around 0.9 for the Spanish prototype. In addition, the optimal ratios obtained from the analytical approach are close to those from the numerical simulation and their differences are mainly caused by the neglect of aerodynamic losses associated with skin friction, flow separation, and secondary flow in the theoretical analysis. This study may be useful for the preliminary estimation of power plant performance and the power-regulating strategy option for solar chimney turbines. Ó 2013 Elsevier Ltd. All rights reserved. Keywords: Solar chimney power plant; Optimal ratio; Turbine pressure drop; Numerical analysis
1. Introduction A solar chimney power plant (SCPP) system presents interesting possibilities for the large-scale use of solar energy. The Spanish prototype, which ran continuously from mid-1986 to early 1989, proved that an SCPP is a practical technology capable of highly reliable operation (Haaf et al., 1983; Haaf, 1984). SCPPs have been considered as feasible options for electricity generating and have been the subject of applied research worldwide (Dai et al., 2003; Ketlogetswe et al., 2008; Nizetic et al., 2008; Zhou et al., 2010a; Cao et al., 2011; Hamdan, 2011). The schematic of a typical solar chimney power plant is shown in Fig. 1. The collector is the element used to heat the in-coming air by means of the greenhouse effect. Owing to the chimney effect, a total pressure difference is generated between the chimney base and the ambient, which drives ⇑ Corresponding author. Tel.: +86 29 13152181528.
E-mail address: [email protected] (J. Li). 0038-092X/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.solener.2013.03.030
the turbine installed at the chimney base to generate electrical power. Continuous operation can be guaranteed by using heat storing material beneath the ground, which is heated during the day and releases the stored heat throughout the night. A detailed review of the SCPP by Zhou et al. (2010b) provides a comprehensive and concise picture of the research and development of the SCPP technology over the past few decades. Efficient conversion of fluid energy to electrical power depends primarily on the operation of the turbine. The turbine in the Spanish prototype is a shrouded turbine which belongs to the category of pressure-based turbine. Before and after, the turbine air speeds are about the same, but the pressure changes significantly. The available total pressure difference Dptot can be divided into two parts: the pressure drop across the turbine Dpturb and the flow losses Dploss. The variation in solar radiation causes the volume flow rate through the system to change accordingly. During operation, the turbine blade pitch is then adjusted to regulate power output according to the changing airflow. This
P. Guo et al. / Solar Energy 98 (2013) 42–48
43
Nomenclature Achim Acoll cp ft fopt g G P Ta V
section area of the chimney (m2) area of the collector (m2) specific heat capacity (J kg1 K1) pressure drop ratio optimal pressure ratio gravitational acceleration (m s2) global horizontal radiation (W m2) available power for the turbine (k W) ambient temperature (K) updraft velocity (m s1)
then raises the pertinent question of how to maximize power production by adjusting the pressure drop across the turbine under a specified condition. To find the maximum power condition, the optimum combination of the turbine pressure drop Dpturb and volume flow rate Q_ must be determined. Researchers defined the ratio of the turbine pressure drop to the available total pressure difference as the pressure drop ratio ft = Dpturb/ Dptot, first mentioned by Haaf (1984) in his pioneer study of the SCPP. A turbine pressure drop amounting to 2/3 of the available total pressure difference was considered to be theoretically optimal. However, some researchers using different analytical methods showed that the assumption of 2/3 is proved to be true only when the available total pressure difference is constant (von Backstro¨m and Fluri, 2006; Koonsrisuk and Chitsomboon, 2010; Nizetic and Klarin, 2010). A range of values for the optimal ratio has been suggested in the literature and is generally considered to be higher than 2/3. Schlaich et al. (2005) reported a value of about 0.80 and indicated that the optimal ratio depended on the power plant characteristics. von Backstro¨m and Fluri (2006) developed a simple analytical method to estimate the optimal ratio, and a typical value of 0.9 was calculated using data from the Spanish prototype. Bernardes et al. (2003) mentioned an optimal value
Fig. 1. Schematic diagram of the SCPP.
Greek symbols b thermal expansion coefficient (K1) Dpturb turbine pressure drop (Pa) Dptot available total pressure difference (Pa) Dploss flow losses (Pa) DT temperature rise (K) g overall efficiency (%) gcoll collector efficiency (%) gsc chimney efficiency (%) q density of air (kg m3)
as high as 0.97, but they pointed out that this value was hard to achieve in reality and using a value between 0.8 and 0.9 was recommended. Later, the same research team conducted theoretical simulations to find the maximum power point of a reference SCPP using two heat transfer schemes. A higher optimal ratio of around 0.9 was found for the lower heat transfer coefficients, while higher heat transfer coefficients resulted in a lower optimal ratio of 0.8 (Bernardes and von Backstro¨m, 2010). Nizetic and Klarin (2010) also concluded, through a simplified analytical approach, that the optimal ratio was in the range of 0.8– 0.9. It is obvious that there is no unified accepted optimal ratio for maximum power generation and a range of values has been suggested. In this study, a simple, generally applicable formula was derived for the evaluation of the optimal turbine pressure drop ratio through an analytical approach. In addition, a comprehensive 3D numerical simulation incorporating the radiation model, solar load model and turbine model was carried out for the Spanish prototype. The simulations are employed for verifying the analytical model and for in detail investigation of the optimal turbine pressure drop ratio.
Fig. 2. Variation of the available power for a turbine with the volume flow rate.
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2. Theoretical analysis of the optimal ratio for maximum power production Fig. 2 shows the variation in the available power for a turbine with the volume flow rate. In this figure, the dashed line represents the available total pressure difference, often referred to as the driving force. The solid line denotes the flow losses in the system. The difference between the driving force and the flow losses is then the turbine pressure drop. For an incompressible flow, the available power extracted by the turbine is equal to the product of the volume flow rate Q_ and the pressure drop across the turbine Dpturb. As shown in Fig. 2, the areas of different colored rectangles denote the available power of the turbine under different volume flow rates. Zero power is gained under two extreme conditions: (a) When the pressure drop over the turbine becomes very large the flow rate and the useful work approach zero. (b) When the pressure drop over the turbine approaches zero the useful work becomes zero. Between these two extreme cases, there is an optimum turbine blade setting to maximize power generation. To investigate the optimal ratio for the turbine pressure drop, it is necessary to analyze the overall efficiency of the SCPP. Generally speaking, the overall efficiency of the SCPP can be defined as follows: g ¼ gsc gcoll ft ;
ð1Þ
where g, gsc, gcoll, ft are the overall efficiency, the chimney efficiency, the collector efficiency and the turbine pressure drop ratio. According to the theoretical analysis in a previous study (Gannon and von Backstrom, 2000; Schlaich et al., 2005), when ignoring the aerodynamic losses, the expressions for the partial efficiencies of the chimney and the collector are derived as follows: gsc ¼
Dptot gH ¼ ; qcp DT cp T 0
ð2Þ
qVAchim cp DT ; Acoll G
ð3Þ
gcoll ¼ ft ¼
Dptot Dploss qcp DT gsc 12 qV 2 ¼ : Dptot qcp DT gsc
ð4Þ
Substituting Eqs. (2)–(4) into the expression for the overall efficiency g, it becomes: g¼
gH qVAchim cp DT qcp DT gsc 12 qV 2 : cp T 0 Acoll G qcp DT gsc
ð5Þ
For convenience, the two coefficients C1 and C2 are defined as follows: C1 ¼
qAchim ; Acoll G
C2 ¼
gH : T0
ð6Þ
The overall efficiency can then be further written as: 1 g ¼ C 1 C 2 V DT C 1 V 3 : 2
ð7Þ
The overall efficiency is a function of the air velocity and temperature rise in the system. Following the research of von Backstro¨m and Fluri (2006), a simple but useful power law assumption for the relationship between the air velocity and temperature rise is: DT ¼ C 3 V m ;
ð8Þ
Substituting Eq. (8) into Eq. (7), yields: 1 g ¼ C 1 C 2 C 3 V mþ1 C 1 V 3 : 2 The maximum power available is then found when @g 3 ¼ C 1 C 2 C 3 ðm þ 1ÞV m C 1 V 2 ¼ 0: @V 2
ð9Þ @g @V
¼ 0: ð10Þ
By solving the above equation, the optimal air flow velocity is then given by: 1 2m 2 V opt ¼ C 2 C 3 ðm þ 1Þ : ð11Þ 3 Substituting Eq. (11) into the expression for the turbine pressure drop ratio in Eq. (4), yields an optimal ratio for the pressure drop across the turbine: fopt ¼ 1
V 2m mþ1 ¼1 : 3 2C 2 C 3
ð12Þ
The optimal ratio is determined by the value of m. In the special case when m = 0, which means the air temperature rise in the collector is independent of the air velocity in the system, the optimal turbine pressure drop ratio is 2/3. This corresponds to the conclusion mentioned in the introduction. However, the temperature rise in the collector is significantly affected by the system volume flow rate and can hardly remain constant in practical operation. Both the turbine pressure drop and solar radiation greatly influence the temperature rise in the collector. Hence, the variation of the value m needs to be investigated further to accommodate the variation in the daily solar radiation intensity. The Computational Fluid Dynamics (CFD) method has been confirmed as a powerful tool for detailed analyses of the SCPP. A comprehensive 3D numerical simulation was conducted for predicting the performance of the SCPP in the following sections. 3. Numerical method The simulations were all conducted for steady flow using the commercially available CFD package (ANSYS Fluent). Since this paper mainly focuses on the optimal turbine pressure drop ratio in the SCPP, the ambient domain was neglected in the simulation, which is also the commonly adopted approach in the literature for the performance prediction of the SCPP (Pastohr et al., 2004; Ming et al., 2006; Xu et al., 2011). On the other hand, the inclusion of the ambient domain in the simulation is mainly for the consideration of the effect of the ambient crosswind on the SCPP or the flow-induced vibration of the structure by crosswind.
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Table 1 Boundary conditions and model parameters. Place
Type
Value
Bottom of the ground External surface of the collector roof Surface of the chimney Inlet of the collector Outlet of the chimney Pressure drop across the turbine
Wall; temperature Wall; mixed Wall; heat flux Pressure inlet Pressure outlet Reversed fan
300K h = 8 W/(m2 K), e = 0.94 qchim = 0 W/m2 Pi = 0 Pa, Ta = 302 K Po = 0 Pa DPturb
In addition, an ambient domain is usually several times larger than the SCPP in numerical simulation. The number of the grid points would be a challenge to the compute capability of the present PC if both the internal and external flows of the SCPP were included in the simulations. The heat transfer process in the SCPP system involves all three phenomena: conduction, convection and radiation and the inclusion of all the three phenomena in numerical simulations can give better prediction than those neglecting the radiation phenomenon in the computations, according to the author’s investigation. Usually the collector in the SCPP system is regarded as a greenhouse with different types of semi-transparent covering materials such as glass or plastic. In fact, the chimney installed at the center of the collector makes it obviously different from a real greenhouse. In the SCPP, due to the chimney effect, the convection in the collector is greatly enhanced. In such a case, the process by which the collector absorbs the outgoing radiated energy in the infrared spectrum from the ground and re-emits most of it back towards earth would play a key role in the heat transfer process inside the SCPP. In previous numerical studies, however, the radiation heat transfer in the collector is rarely considered. To obtain a more accurate performance prediction of the SCPP, the discrete ordinates (DO) radiation model was adopted in this numerical simulation. Given that the flow inside the SCPP is induced because the force of gravity acting on the density variations and has proved to be turbulent flow (Ming et al., 2008; Xu et al., 2011), the RNG k–e turbulence model was selected taking into consideration the buoyancy effect. The Boussinesq approximation was adopted in the simulation which treated the air density as a constant value in all solved equations, except for the buoyancy term in the momentum equation. Thereby computational effort is saved compared to compressible simulations where the continuity equation is solved. The solar load model provided by Fluent was adopted to calculate the radiation effects from the sun’s rays entering the computational domain. The solar radiation was modeled using the sun position vector and illumination parameters. Such a model is an efficient and practical approach to applying solar loads as heat sources in the energy equations and is available for 3D simulation only. The turbine pressure drop was used as the independent control variable in the power output control strategies. In the simulation, a reversed fan model (actuator disk model)
Fig. 3. Computational grid.
Fig. 4a. Comparison of power output between simulation results and experimental data.
was used to determine the pressure drop across the turbine. The turbine was treated as an infinitely thin disk and the discontinuous pressure drop across it could be specified as a constant value or a function of the velocity through the turbine. The SIMPLEC algorithm was selected as the pressurevelocity coupling scheme. The convective terms were dis-
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with the experimental data. More precisely, the simulation results tended to slightly overestimate the power output and the updraft velocity. This was mainly because the simulation was conducted for steady flow, and thermal equilibrium was achieved in the collector for each specified condition, while in reality the soil layer had thermal inertia. 4. Results and discussion
Fig. 4b. Comparison of updraft velocity between simulation results and experimental data.
Fig. 5. Variations of updraft velocity and temperature rise with solar radiation and turbine pressure drop.
cretized with a second-order-accurate upwind scheme. The Body Force Weighted algorithm was selected as the discretization method for pressure term. Structured grid was adopted and the grid was refined adaptively near walls. The comparisons of the computational results using different strategies for grid refinement indicated that, the grids are fine enough to be used to obtain grid-independent solutions. The main boundary conditions for the SCPP are listed in Table 1. The numerical method was validated using the experimental data of the Spanish prototype obtained from the literature (Schlaich et al., 2005). The main dimensions for the Spanish prototype are: the chimney is 194.6 m high and 10.16 m in diameter; the collector is 1.85 m high and 244 m in diameter. The computational grid was shown in Fig. 3. Comparison between the numerical results and experimental data was carried out to verify the validity of the numerical method in this study. As indicated in Figs. 4a and 4b, the simulation results were quite consistent
To verify the theoretical analysis and further investigate the optimal turbine pressure drop ratio, numerical simulations were carried out for the Spanish prototype by setting the solar radiation at 200 W/m2, 400 W/m2, 600 W/m2 and 800 W/m2. The updraft velocity in the chimney was altered by adjusting the pressure drop across the turbine. Fig. 5 shows the variations in updraft velocity and temperature rise with the solar radiation and turbine pressure drop at the entrance of the chimney. Obviously, under constant solar radiation, when the turbine pressure drop increases, the temperature rise increases while the updraft velocity gradually decreases. An increase in the turbine pressure drop means more energy is extracted by the turbine resulting in a reduction of the updraft velocity. Hence, the heating time of the airflow inside the collector is prolonged and the temperature rise increases. Meanwhile, when the turbine pressure drop remains constant, both the temperature rise and updraft velocity increase significantly with solar radiation. A power law assumption was adopted in the analysis approach in Section 2 for the relationship between temperature rise and updraft velocity. To verify the assumption, the simulation results for the temperature rise and updraft velocity were curve fitted using the power law assumption in Eq. (8), and the fitting curves were plotted in Fig. 6. Obviously, the fitting curve coincides well with the simulated results, thus verifying the validity and applicability of the power law assumption used in the theoretical analysis. The root mean squared error RMSE (a value closer to 0 indicates a better fit) and the coefficient of multiple determinations R-square (a value closer to 1 indicates a better fit) were chosen as the criteria for quality evaluation of the fits. The fitting equations for different solar radiation intensities are listed in Table 2. The RSME and R-square values listed in Table 2 also indicate that the curves fit the data points obtained from the simulations quite accurately. In addition, the value of the exponent m can be obtained from the equations. Clearly, the value of m decreases with the increase in the solar radiation intensity. However, for a given solar radiation, the value of m is barely influenced by the variation in the updraft velocity in the chimney. According to Eq. (10), the optimal pressure drop ratio fopt can be conveniently calculated using the value of m. Hence it can be concluded that the optimal pressure drop ratio is mainly affected by the solar radiation for a specified SCPP. During the dynamic control of a real SCPP, the turbine pressure drop
P. Guo et al. / Solar Energy 98 (2013) 42–48
47
Table 3 Comparison of the optimal turbine pressure drop ratio between the theoretical and numerical results.
Fig. 6. Simulated results and fitting curves for temperature rise and updraft velocity.
Table 2 Fitting equations for different solar radiation intensities. Solar radiation (W/m2) 200 400 600 800
Fitting equations 0.734
DT = 12.92V DT = 37.00V0.7678 DT = 62.79V0.7808 DT = 89.34V0.7889
RSME
R-square
0.01939 0.02626 0.02634 0.02369
0.9998 0.9999 0.9999 1
Solar radiation (W/m2)
m
fopt From theoretical model
fopt From numerical simulation
200 400 600 800
0.7340 0.7678 0.7808 0.7889
0.9110 0.9226 0.9269 0.9296
0.9134 0.9051 0.8942 0.8782
results are consistent with the conclusions of Ming et al. (2010). The maximum overall efficiencies under different solar radiation intensities can be obtained from Fig. 7 and their corresponding pressure drop ratios ft are the optimal pressure drop ratios fopt. The comparison of the optimal pressure ratio between the theoretical and numerical results is listed in Table 3. Obviously, for the Spanish prototype, all these values for the optimal pressure ratio were around 0.9 and in the range suggested by previous researchers. In addition, the optimal ratios obtained from the theoretical model were close to those from the numerical simulation with a maximum relative difference of less than 6%, which is acceptable for this kind of simplified analysis. The neglect of the flow losses in the theoretical analysis is the main reason for this deviation. With the increase of solar radiation, the flow losses in the system, which are generally directly proportional to the square of the air velocity, increase accordingly. A larger fraction of the available total pressure difference was then used to overcome the flow losses and less was used to drive the turbine. This also explains why the optimal ratios from theoretical and numerical results were closest when the solar radiation was 200 W/m2, and their difference gradually increased with the level of solar radiation. 5. Conclusions
Fig. 7. Simulated results of the overall efficiency.
needs to be varied by adjusting the turbine blade pitch or other power-regulating approaches to accommodate the variation in the intensity of the daily solar radiation. The overall efficiencies of the SCPP were also calculated under different conditions using the simulation results as shown in Fig. 7. Apparently, the overall efficiencies were significantly affected by the solar radiation and the turbine pressure drop. It was found that the overall efficiencies of the Spanish prototype corresponding to changing turbine pressure drop of the prototype were all less than 0.2% under different solar radiation intensities. The simulated
In this study, the optimal ratio for the turbine pressure drop to the available total pressure difference has been investigated comprehensively using theoretical analysis and 3D numerical simulations. It was found that the optimal ratio mainly varied with a change in the intensity of solar radiation. Therefore, a turbine with adjustable blade pitch or other power-regulating approaches is recommended for the practical operation of a solar chimney system, to accommodate the variation in the intensity of the daily solar radiation. In addition, a simplified theoretical analysis formula was developed, with a power law assumption, for the relationship between the updraft velocity and air temperature rise at the entrance of the chimney. The validity and applicability of the theoretical formula was verified by comprehensive 3D numerical simulations. The comparison results show that the optimal ratios obtained from the theoretical model are close to those from the numerical simulation with a maximum relative difference
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of less than 6%, and their deviation was mainly caused by the neglect of flow losses in the theoretical analysis. Acknowledgements This work was supported by “the Fundamental Research Funds for the Central Universities” and “the National Natural Science Foundation of China” under Contract No. 51276137. References Bernardes, M.A. dos S., von Backstro¨m, T.W., 2010. Evaluation of operational control strategies applicable to solar chimney power plants. Solar Energy 84 (2), 277–288. Bernardes, M.A. dos S., Voss, A., Weinrebe, G., 2003. Thermal and technical analyses of solar chimneys. Solar Energy 75 (6), 511–524. Cao, F., Zhao, L., Guo, L.J., 2011. Simulation of a sloped solar chimney power plant in Lanzhou. Energy Conversion and Management 52 (6), 2360–2366. Dai, Y.J., Huang, H.B., Wang, R.Z., 2003. Case study of solar chimney power plants in Northwestern regions of China. Renewable Energy 28 (8), 1295–1304. Gannon, A.J., von Backstrom, T.W., 2000. Solar chimney cycle analysis with system loss and solar collector performance. Journal of Solar Energy Engineering – Transactions of the ASME 122 (3), 133–137. Haaf, W., 1984. Solar chimneys part II: preliminary test results from the Manzanares pilot plant. International Journal of Sustainable Energy 2 (2), 141–161. Haaf, W., Friedrich, K., Mayer, G., Schlaich, J., 1983. Solar chimneys part I: principle and construction of the pilot plant in Manzanares. International Journal of Sustainable Energy 2 (1), 3–20. Hamdan, M., 2011. Analysis of a solar chimney power plant in the Arabian Gulf region. Renewable Energy 36 (10), 2593–2598. Ketlogetswe, C., Fiszdon, J.K., Seabe, O.O., 2008. Solar chimney power generation project – the case for Botswana. Renewable and Sustainable Energy Reviews 12 (7), 2005–2012.
Koonsrisuk, A., Chitsomboon, T. 2010. Theoretical turbine power yield in solar chimney power plants. In: 3rd International Conference on Thermal Issues in Emerging Technologies Theory and Applications (ThETA 3), Cairo, Egypt, pp. 339–346. Ming, T.Z., Liu, W., Xu, G.L., 2006. Analytical and numerical investigation of the solar chimney power plant systems. International Journal of Energy Research 30 (11), 861–873. Ming, T.Z., Liu, W., Xu, G.L., Xiong, Y.B., Guan, X.H., Pan, Y., 2008. Numerical simulation of the solar chimney power plant systems coupled with turbine. Renewable Energy 33 (5), 897–905. Ming, T.Z., Zheng, Y., Liu, C., Liu, W., Pan, Y., 2010. Simple analysis on thermal performance of solar chimney power generation systems. Journal of the Energy Institute 83 (1), 6–11. Nizetic, S., Klarin, B., 2010. A simplified analytical approach for evaluation of the optimal ratio of pressure drop across the turbine in solar chimney power plants. Applied Energy 87 (2), 587–591. Nizetic, S., Ninic, N., Klarin, B., 2008. Analysis and feasibility of implementing solar chimney power plants in the Mediterranean region. Energy 33 (11), 1680–1690. Pastohr, H., Kornadt, O., Gurlebeck, K., 2004. Numerical and analytical calculations of the temperature and flow field in the upwind power plant. International Journal of Energy Research 28 (6), 495–510. Schlaich, J., Bergermann, R., Schiel, W., Weinrebe, G., 2005. Design of commercial solar updraft tower systems – utilization of solar induced convective flows for power generation. Journal of Solar Energy Engineering – Transactions of the ASME 127 (1), 117–124. von Backstro¨m, T.W., Fluri, T.P., 2006. Maximum fluid power condition in solar chimney power plants – an analytical approach. Solar Energy 80 (11), 1417–1423. Xu, G.L., Ming, T.Z., Pan, Y., Meng, F.L., Zhou, C., 2011. Numerical analysis on the performance of solar chimney power plant system. Energy Conversion and Management 52 (2), 876–883. Zhou, X.P., Wang, F., Fan, J., Ocieng, R.M., 2010a. Performance of solar chimney power plant in Qinghai-Tibet Plateau. Renewable and Sustainable Energy Reviews 14 (8), 2249–2255. Zhou, X.P., Wang, F., Ocieng, R.M., 2010b. A review of solar chimney power technology. Renewable and Sustainable Energy Reviews 14 (8), 2315–2338.
Solar Energy 98 (2013) 42–48 www.elsevier.com/locate/solener
Numerical analysis of the optimal turbine pressure drop ratio in a solar chimney power plant Penghua Guo, Jingyin Li ⇑, Yuan Wang, Yingwen Liu School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China Available online 14 June 2013 Communicated by: Associate Editor Xinping Zhou
Abstract In a solar chimney power plant, only a fraction of the available total pressure difference can be used to run the turbine to generate electrical power. The optimal ratio of the turbine pressure drop to the available total pressure difference in a solar chimney system is investigated using theoretical analysis and 3D numerical simulations. The values found in the literature for the optimal ratio vary between 2/3 and 0.97. In this study, however, the optimal ratio was found to vary with the intensity of solar radiation, and to be around 0.9 for the Spanish prototype. In addition, the optimal ratios obtained from the analytical approach are close to those from the numerical simulation and their differences are mainly caused by the neglect of aerodynamic losses associated with skin friction, flow separation, and secondary flow in the theoretical analysis. This study may be useful for the preliminary estimation of power plant performance and the power-regulating strategy option for solar chimney turbines. Ó 2013 Elsevier Ltd. All rights reserved. Keywords: Solar chimney power plant; Optimal ratio; Turbine pressure drop; Numerical analysis
1. Introduction A solar chimney power plant (SCPP) system presents interesting possibilities for the large-scale use of solar energy. The Spanish prototype, which ran continuously from mid-1986 to early 1989, proved that an SCPP is a practical technology capable of highly reliable operation (Haaf et al., 1983; Haaf, 1984). SCPPs have been considered as feasible options for electricity generating and have been the subject of applied research worldwide (Dai et al., 2003; Ketlogetswe et al., 2008; Nizetic et al., 2008; Zhou et al., 2010a; Cao et al., 2011; Hamdan, 2011). The schematic of a typical solar chimney power plant is shown in Fig. 1. The collector is the element used to heat the in-coming air by means of the greenhouse effect. Owing to the chimney effect, a total pressure difference is generated between the chimney base and the ambient, which drives ⇑ Corresponding author. Tel.: +86 29 13152181528.
E-mail address: [email protected] (J. Li). 0038-092X/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.solener.2013.03.030
the turbine installed at the chimney base to generate electrical power. Continuous operation can be guaranteed by using heat storing material beneath the ground, which is heated during the day and releases the stored heat throughout the night. A detailed review of the SCPP by Zhou et al. (2010b) provides a comprehensive and concise picture of the research and development of the SCPP technology over the past few decades. Efficient conversion of fluid energy to electrical power depends primarily on the operation of the turbine. The turbine in the Spanish prototype is a shrouded turbine which belongs to the category of pressure-based turbine. Before and after, the turbine air speeds are about the same, but the pressure changes significantly. The available total pressure difference Dptot can be divided into two parts: the pressure drop across the turbine Dpturb and the flow losses Dploss. The variation in solar radiation causes the volume flow rate through the system to change accordingly. During operation, the turbine blade pitch is then adjusted to regulate power output according to the changing airflow. This
P. Guo et al. / Solar Energy 98 (2013) 42–48
43
Nomenclature Achim Acoll cp ft fopt g G P Ta V
section area of the chimney (m2) area of the collector (m2) specific heat capacity (J kg1 K1) pressure drop ratio optimal pressure ratio gravitational acceleration (m s2) global horizontal radiation (W m2) available power for the turbine (k W) ambient temperature (K) updraft velocity (m s1)
then raises the pertinent question of how to maximize power production by adjusting the pressure drop across the turbine under a specified condition. To find the maximum power condition, the optimum combination of the turbine pressure drop Dpturb and volume flow rate Q_ must be determined. Researchers defined the ratio of the turbine pressure drop to the available total pressure difference as the pressure drop ratio ft = Dpturb/ Dptot, first mentioned by Haaf (1984) in his pioneer study of the SCPP. A turbine pressure drop amounting to 2/3 of the available total pressure difference was considered to be theoretically optimal. However, some researchers using different analytical methods showed that the assumption of 2/3 is proved to be true only when the available total pressure difference is constant (von Backstro¨m and Fluri, 2006; Koonsrisuk and Chitsomboon, 2010; Nizetic and Klarin, 2010). A range of values for the optimal ratio has been suggested in the literature and is generally considered to be higher than 2/3. Schlaich et al. (2005) reported a value of about 0.80 and indicated that the optimal ratio depended on the power plant characteristics. von Backstro¨m and Fluri (2006) developed a simple analytical method to estimate the optimal ratio, and a typical value of 0.9 was calculated using data from the Spanish prototype. Bernardes et al. (2003) mentioned an optimal value
Fig. 1. Schematic diagram of the SCPP.
Greek symbols b thermal expansion coefficient (K1) Dpturb turbine pressure drop (Pa) Dptot available total pressure difference (Pa) Dploss flow losses (Pa) DT temperature rise (K) g overall efficiency (%) gcoll collector efficiency (%) gsc chimney efficiency (%) q density of air (kg m3)
as high as 0.97, but they pointed out that this value was hard to achieve in reality and using a value between 0.8 and 0.9 was recommended. Later, the same research team conducted theoretical simulations to find the maximum power point of a reference SCPP using two heat transfer schemes. A higher optimal ratio of around 0.9 was found for the lower heat transfer coefficients, while higher heat transfer coefficients resulted in a lower optimal ratio of 0.8 (Bernardes and von Backstro¨m, 2010). Nizetic and Klarin (2010) also concluded, through a simplified analytical approach, that the optimal ratio was in the range of 0.8– 0.9. It is obvious that there is no unified accepted optimal ratio for maximum power generation and a range of values has been suggested. In this study, a simple, generally applicable formula was derived for the evaluation of the optimal turbine pressure drop ratio through an analytical approach. In addition, a comprehensive 3D numerical simulation incorporating the radiation model, solar load model and turbine model was carried out for the Spanish prototype. The simulations are employed for verifying the analytical model and for in detail investigation of the optimal turbine pressure drop ratio.
Fig. 2. Variation of the available power for a turbine with the volume flow rate.
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P. Guo et al. / Solar Energy 98 (2013) 42–48
2. Theoretical analysis of the optimal ratio for maximum power production Fig. 2 shows the variation in the available power for a turbine with the volume flow rate. In this figure, the dashed line represents the available total pressure difference, often referred to as the driving force. The solid line denotes the flow losses in the system. The difference between the driving force and the flow losses is then the turbine pressure drop. For an incompressible flow, the available power extracted by the turbine is equal to the product of the volume flow rate Q_ and the pressure drop across the turbine Dpturb. As shown in Fig. 2, the areas of different colored rectangles denote the available power of the turbine under different volume flow rates. Zero power is gained under two extreme conditions: (a) When the pressure drop over the turbine becomes very large the flow rate and the useful work approach zero. (b) When the pressure drop over the turbine approaches zero the useful work becomes zero. Between these two extreme cases, there is an optimum turbine blade setting to maximize power generation. To investigate the optimal ratio for the turbine pressure drop, it is necessary to analyze the overall efficiency of the SCPP. Generally speaking, the overall efficiency of the SCPP can be defined as follows: g ¼ gsc gcoll ft ;
ð1Þ
where g, gsc, gcoll, ft are the overall efficiency, the chimney efficiency, the collector efficiency and the turbine pressure drop ratio. According to the theoretical analysis in a previous study (Gannon and von Backstrom, 2000; Schlaich et al., 2005), when ignoring the aerodynamic losses, the expressions for the partial efficiencies of the chimney and the collector are derived as follows: gsc ¼
Dptot gH ¼ ; qcp DT cp T 0
ð2Þ
qVAchim cp DT ; Acoll G
ð3Þ
gcoll ¼ ft ¼
Dptot Dploss qcp DT gsc 12 qV 2 ¼ : Dptot qcp DT gsc
ð4Þ
Substituting Eqs. (2)–(4) into the expression for the overall efficiency g, it becomes: g¼
gH qVAchim cp DT qcp DT gsc 12 qV 2 : cp T 0 Acoll G qcp DT gsc
ð5Þ
For convenience, the two coefficients C1 and C2 are defined as follows: C1 ¼
qAchim ; Acoll G
C2 ¼
gH : T0
ð6Þ
The overall efficiency can then be further written as: 1 g ¼ C 1 C 2 V DT C 1 V 3 : 2
ð7Þ
The overall efficiency is a function of the air velocity and temperature rise in the system. Following the research of von Backstro¨m and Fluri (2006), a simple but useful power law assumption for the relationship between the air velocity and temperature rise is: DT ¼ C 3 V m ;
ð8Þ
Substituting Eq. (8) into Eq. (7), yields: 1 g ¼ C 1 C 2 C 3 V mþ1 C 1 V 3 : 2 The maximum power available is then found when @g 3 ¼ C 1 C 2 C 3 ðm þ 1ÞV m C 1 V 2 ¼ 0: @V 2
ð9Þ @g @V
¼ 0: ð10Þ
By solving the above equation, the optimal air flow velocity is then given by: 1 2m 2 V opt ¼ C 2 C 3 ðm þ 1Þ : ð11Þ 3 Substituting Eq. (11) into the expression for the turbine pressure drop ratio in Eq. (4), yields an optimal ratio for the pressure drop across the turbine: fopt ¼ 1
V 2m mþ1 ¼1 : 3 2C 2 C 3
ð12Þ
The optimal ratio is determined by the value of m. In the special case when m = 0, which means the air temperature rise in the collector is independent of the air velocity in the system, the optimal turbine pressure drop ratio is 2/3. This corresponds to the conclusion mentioned in the introduction. However, the temperature rise in the collector is significantly affected by the system volume flow rate and can hardly remain constant in practical operation. Both the turbine pressure drop and solar radiation greatly influence the temperature rise in the collector. Hence, the variation of the value m needs to be investigated further to accommodate the variation in the daily solar radiation intensity. The Computational Fluid Dynamics (CFD) method has been confirmed as a powerful tool for detailed analyses of the SCPP. A comprehensive 3D numerical simulation was conducted for predicting the performance of the SCPP in the following sections. 3. Numerical method The simulations were all conducted for steady flow using the commercially available CFD package (ANSYS Fluent). Since this paper mainly focuses on the optimal turbine pressure drop ratio in the SCPP, the ambient domain was neglected in the simulation, which is also the commonly adopted approach in the literature for the performance prediction of the SCPP (Pastohr et al., 2004; Ming et al., 2006; Xu et al., 2011). On the other hand, the inclusion of the ambient domain in the simulation is mainly for the consideration of the effect of the ambient crosswind on the SCPP or the flow-induced vibration of the structure by crosswind.
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Table 1 Boundary conditions and model parameters. Place
Type
Value
Bottom of the ground External surface of the collector roof Surface of the chimney Inlet of the collector Outlet of the chimney Pressure drop across the turbine
Wall; temperature Wall; mixed Wall; heat flux Pressure inlet Pressure outlet Reversed fan
300K h = 8 W/(m2 K), e = 0.94 qchim = 0 W/m2 Pi = 0 Pa, Ta = 302 K Po = 0 Pa DPturb
In addition, an ambient domain is usually several times larger than the SCPP in numerical simulation. The number of the grid points would be a challenge to the compute capability of the present PC if both the internal and external flows of the SCPP were included in the simulations. The heat transfer process in the SCPP system involves all three phenomena: conduction, convection and radiation and the inclusion of all the three phenomena in numerical simulations can give better prediction than those neglecting the radiation phenomenon in the computations, according to the author’s investigation. Usually the collector in the SCPP system is regarded as a greenhouse with different types of semi-transparent covering materials such as glass or plastic. In fact, the chimney installed at the center of the collector makes it obviously different from a real greenhouse. In the SCPP, due to the chimney effect, the convection in the collector is greatly enhanced. In such a case, the process by which the collector absorbs the outgoing radiated energy in the infrared spectrum from the ground and re-emits most of it back towards earth would play a key role in the heat transfer process inside the SCPP. In previous numerical studies, however, the radiation heat transfer in the collector is rarely considered. To obtain a more accurate performance prediction of the SCPP, the discrete ordinates (DO) radiation model was adopted in this numerical simulation. Given that the flow inside the SCPP is induced because the force of gravity acting on the density variations and has proved to be turbulent flow (Ming et al., 2008; Xu et al., 2011), the RNG k–e turbulence model was selected taking into consideration the buoyancy effect. The Boussinesq approximation was adopted in the simulation which treated the air density as a constant value in all solved equations, except for the buoyancy term in the momentum equation. Thereby computational effort is saved compared to compressible simulations where the continuity equation is solved. The solar load model provided by Fluent was adopted to calculate the radiation effects from the sun’s rays entering the computational domain. The solar radiation was modeled using the sun position vector and illumination parameters. Such a model is an efficient and practical approach to applying solar loads as heat sources in the energy equations and is available for 3D simulation only. The turbine pressure drop was used as the independent control variable in the power output control strategies. In the simulation, a reversed fan model (actuator disk model)
Fig. 3. Computational grid.
Fig. 4a. Comparison of power output between simulation results and experimental data.
was used to determine the pressure drop across the turbine. The turbine was treated as an infinitely thin disk and the discontinuous pressure drop across it could be specified as a constant value or a function of the velocity through the turbine. The SIMPLEC algorithm was selected as the pressurevelocity coupling scheme. The convective terms were dis-
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with the experimental data. More precisely, the simulation results tended to slightly overestimate the power output and the updraft velocity. This was mainly because the simulation was conducted for steady flow, and thermal equilibrium was achieved in the collector for each specified condition, while in reality the soil layer had thermal inertia. 4. Results and discussion
Fig. 4b. Comparison of updraft velocity between simulation results and experimental data.
Fig. 5. Variations of updraft velocity and temperature rise with solar radiation and turbine pressure drop.
cretized with a second-order-accurate upwind scheme. The Body Force Weighted algorithm was selected as the discretization method for pressure term. Structured grid was adopted and the grid was refined adaptively near walls. The comparisons of the computational results using different strategies for grid refinement indicated that, the grids are fine enough to be used to obtain grid-independent solutions. The main boundary conditions for the SCPP are listed in Table 1. The numerical method was validated using the experimental data of the Spanish prototype obtained from the literature (Schlaich et al., 2005). The main dimensions for the Spanish prototype are: the chimney is 194.6 m high and 10.16 m in diameter; the collector is 1.85 m high and 244 m in diameter. The computational grid was shown in Fig. 3. Comparison between the numerical results and experimental data was carried out to verify the validity of the numerical method in this study. As indicated in Figs. 4a and 4b, the simulation results were quite consistent
To verify the theoretical analysis and further investigate the optimal turbine pressure drop ratio, numerical simulations were carried out for the Spanish prototype by setting the solar radiation at 200 W/m2, 400 W/m2, 600 W/m2 and 800 W/m2. The updraft velocity in the chimney was altered by adjusting the pressure drop across the turbine. Fig. 5 shows the variations in updraft velocity and temperature rise with the solar radiation and turbine pressure drop at the entrance of the chimney. Obviously, under constant solar radiation, when the turbine pressure drop increases, the temperature rise increases while the updraft velocity gradually decreases. An increase in the turbine pressure drop means more energy is extracted by the turbine resulting in a reduction of the updraft velocity. Hence, the heating time of the airflow inside the collector is prolonged and the temperature rise increases. Meanwhile, when the turbine pressure drop remains constant, both the temperature rise and updraft velocity increase significantly with solar radiation. A power law assumption was adopted in the analysis approach in Section 2 for the relationship between temperature rise and updraft velocity. To verify the assumption, the simulation results for the temperature rise and updraft velocity were curve fitted using the power law assumption in Eq. (8), and the fitting curves were plotted in Fig. 6. Obviously, the fitting curve coincides well with the simulated results, thus verifying the validity and applicability of the power law assumption used in the theoretical analysis. The root mean squared error RMSE (a value closer to 0 indicates a better fit) and the coefficient of multiple determinations R-square (a value closer to 1 indicates a better fit) were chosen as the criteria for quality evaluation of the fits. The fitting equations for different solar radiation intensities are listed in Table 2. The RSME and R-square values listed in Table 2 also indicate that the curves fit the data points obtained from the simulations quite accurately. In addition, the value of the exponent m can be obtained from the equations. Clearly, the value of m decreases with the increase in the solar radiation intensity. However, for a given solar radiation, the value of m is barely influenced by the variation in the updraft velocity in the chimney. According to Eq. (10), the optimal pressure drop ratio fopt can be conveniently calculated using the value of m. Hence it can be concluded that the optimal pressure drop ratio is mainly affected by the solar radiation for a specified SCPP. During the dynamic control of a real SCPP, the turbine pressure drop
P. Guo et al. / Solar Energy 98 (2013) 42–48
47
Table 3 Comparison of the optimal turbine pressure drop ratio between the theoretical and numerical results.
Fig. 6. Simulated results and fitting curves for temperature rise and updraft velocity.
Table 2 Fitting equations for different solar radiation intensities. Solar radiation (W/m2) 200 400 600 800
Fitting equations 0.734
DT = 12.92V DT = 37.00V0.7678 DT = 62.79V0.7808 DT = 89.34V0.7889
RSME
R-square
0.01939 0.02626 0.02634 0.02369
0.9998 0.9999 0.9999 1
Solar radiation (W/m2)
m
fopt From theoretical model
fopt From numerical simulation
200 400 600 800
0.7340 0.7678 0.7808 0.7889
0.9110 0.9226 0.9269 0.9296
0.9134 0.9051 0.8942 0.8782
results are consistent with the conclusions of Ming et al. (2010). The maximum overall efficiencies under different solar radiation intensities can be obtained from Fig. 7 and their corresponding pressure drop ratios ft are the optimal pressure drop ratios fopt. The comparison of the optimal pressure ratio between the theoretical and numerical results is listed in Table 3. Obviously, for the Spanish prototype, all these values for the optimal pressure ratio were around 0.9 and in the range suggested by previous researchers. In addition, the optimal ratios obtained from the theoretical model were close to those from the numerical simulation with a maximum relative difference of less than 6%, which is acceptable for this kind of simplified analysis. The neglect of the flow losses in the theoretical analysis is the main reason for this deviation. With the increase of solar radiation, the flow losses in the system, which are generally directly proportional to the square of the air velocity, increase accordingly. A larger fraction of the available total pressure difference was then used to overcome the flow losses and less was used to drive the turbine. This also explains why the optimal ratios from theoretical and numerical results were closest when the solar radiation was 200 W/m2, and their difference gradually increased with the level of solar radiation. 5. Conclusions
Fig. 7. Simulated results of the overall efficiency.
needs to be varied by adjusting the turbine blade pitch or other power-regulating approaches to accommodate the variation in the intensity of the daily solar radiation. The overall efficiencies of the SCPP were also calculated under different conditions using the simulation results as shown in Fig. 7. Apparently, the overall efficiencies were significantly affected by the solar radiation and the turbine pressure drop. It was found that the overall efficiencies of the Spanish prototype corresponding to changing turbine pressure drop of the prototype were all less than 0.2% under different solar radiation intensities. The simulated
In this study, the optimal ratio for the turbine pressure drop to the available total pressure difference has been investigated comprehensively using theoretical analysis and 3D numerical simulations. It was found that the optimal ratio mainly varied with a change in the intensity of solar radiation. Therefore, a turbine with adjustable blade pitch or other power-regulating approaches is recommended for the practical operation of a solar chimney system, to accommodate the variation in the intensity of the daily solar radiation. In addition, a simplified theoretical analysis formula was developed, with a power law assumption, for the relationship between the updraft velocity and air temperature rise at the entrance of the chimney. The validity and applicability of the theoretical formula was verified by comprehensive 3D numerical simulations. The comparison results show that the optimal ratios obtained from the theoretical model are close to those from the numerical simulation with a maximum relative difference
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