- Email: [email protected]
A simplified analytical approach for evaluation of the optimal ratio of pressure drop across the turbine in solar chimney power plants
Applied Energy 87 (2010) 587–591
Contents lists available at ScienceDirect
Applied Energy journal homepage: www.elsevier.com/locate/apenergy
A simplified analytical approach for evaluation of the optimal ratio of pressure drop across the turbine in solar chimney power plants S. Nizetic a,*, B. Klarin b a
University of Split, Faculty of Electrical and Mechanical Engineering and Naval Architecture, Department of Thermodynamics, Thermotechnics and Heat Engines, R.Boskovica b.b., 21000 Split, Croatia b University of Split, Faculty of Electrical and Mechanical Engineering and Naval Architecture, LAHES – Laboratory for Aero and Hybrid Energy Systems, R.Boskovica b.b., 21000 Split, Croatia
a r t i c l e
i n f o
Article history: Received 8 January 2009 Received in revised form 21 April 2009 Accepted 14 May 2009 Available online 5 June 2009 Keywords: Solar chimney Turbine pressure drop Overall efficiency
a b s t r a c t In this paper, a simplified analytical approach for evaluating the factor of turbine pressure drop in solar chimney power plants is presented. This characteristic factor (or pressure drop ratio in turbines, according to the total pressure drop in the chimney) is important because it is related to the output power. The determined factor (or ratio) values of the turbine pressure drop are found to be within a value range consistent with other studies. It was concluded that for solar chimney power plants, turbine pressure drop factors are in the range of 0.8–0.9. This simplified analytical approach is useful for preliminary analysis and fast evaluation of the potential of solar chimney power plants. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Solar chimney power plants (SC) present a renewable energy power source. The key factor in the economic feasibility of SC power plants is determined by solar irradiation at a particular geographic location. Desert areas are convenient locations, where the solar irradiation intensity averages over 2000 kW h/m2 year. In [1– 3], the possibility of SC power plant use in other geographic locations with lower solar irradiation is analyzed. The concept of SC power plants, as well as a prototype plant, was developed by a group of scientists led by Professor J. Schlaich [4–6]. The working principle of the plant is based on the buoyancy effect, as a fundamental physical principle. Relatively cold surrounding air is heated by a collector. Due to the density difference, warm air flows from the collector periphery to the chimney in the collector center. The wind turbine is placed at the bottom of the chimney because of construction (practical) reasons. Under certain conditions, the wind turbine provides rotation to an electric generator, producing electrical energy. Since the inception of SC power plants, many scientific papers have been published, providing comprehensive and concise overviews of the work done in this field [7]. A very important factor for optimal electrical power output is the pressure drop at the turbine, which corresponds to the maximum electric power output for a given condition. In [6], this factor * Corresponding author. Tel.: +385 21 3055948; fax: +385 21 463877. E-mail address: [email protected] (S. Nizetic). 0306-2619/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2009.05.019
is mentioned for the first time and is reported to have a value of 2/ 3. However, in later works [4] and [8], higher values for this factor are presented. Factor has a value of 2/3 only when there is a constant air temperature increase in the collector (or when the available total pressure drop is constant). In [7], the maximum power is achieved when the factor of the pressure drop at the turbine is equal to approximately 0.97. In the same paper, the authors indicate that this value of pressure drop is difficult to reach, and realistically, the value is expected to range from 0.8 to 0.9. Von Backström and Fluri [8] analyzed the influence of the volume flow of air on power output in different conditions of aerodynamic losses and turbine working characteristics. The value range of the pressure drop estimated from their analytical theory is consistent with values estimated by other researchers. In the same paper [8], assuming typical values from Schlaich, the calculated optimal value of the turbine pressure ratio is typically 0.9. According to other studies, the pressure drop is: Bernardes [7], about 0.83; Hedderwick [9], about 0.7; Schlaich et al. [10], about 0.8. Hence, no unified value of the factor exists, but a range of values has been suggested, as previously mentioned. Also it is obvious that all calculated values for optimal factor of pressure drop, by other different authors, converge to the value that is close to the value of 0.9. So probably that value is close to the real one for optimal factor of turbine pressure drop. The objective of this paper is to estimate reliable results for the optimal pressure drop factor (or a value range) using a simplified analytical approach. We theorize that, even for a constant solar irradiation, the pressure potential of a solar chimney power plant
588
S. Nizetic, B. Klarin / Applied Energy 87 (2010) 587–591
Nomenclature Ac Acoll cp f g G Hc k _ m p0 p2 Pcl Pe Ptc Q_ qcoll T0 T1
v1
wc w1 wcopt
cross-sectional area of solar chimney (m2) solar collector area (m2) specific heat capacity of air (kJ/kg C) ratio, (K) acceleration of gravity (m/s2) solar irradiance (W/m2) solar chimney height (m) air density reduction factor mass flow of air through the collector, (kg/s) atmospheric pressure (Pa) pressure at collector outlet (Pa) power loss due to exit kinetic energy (kW) electric output form the solar chimney power plant (kW) power of theoretical air cycle (kW) heat gain of the air in the collector (W) specific heat gain of the air in the collector (W/m2) ambient temperature (K) temperature of air in the collector outlet (K) specific volume of air at collector outlet (m3/kg) inlet air flow velocity of the chimney (m/s) optimal inlet air flow velocity of the chimney (m/s)
is not fixed but is a function of the air temperature increase in the collector.
Greek symbols effective absorption coefficient b factor proportional to convective energy loss (W/m2 K) gcoll solar collector efficiency gsc solar chimney efficiency gsp overall SC power plant efficiency gt turbine efficiency gtopt optimal pressure drop ratio (factor) gwt blade, transmission and generator efficiency q0 density of ambient air (kg/m3) q1 qc density of air at the inlet in the solar chimney (kg/m3) DT opt optimal air temperature increase through the collector, (K) DT air temperature increase between the collector inflow and outflow (°C) Dht specific enthalpy difference (J/kg) Dpt pressure drop across the turbine (Pa) Dptot Dpac total available pressure drop (pressure difference produced between the chimney base and the surroundings) (Pa)
a
chimney efficiency,
gsc ¼
2. Simplified expression for the overall efficiency of a solar chimney power plant
v 1 Dpac gHc ; ¼ cp ðT 1 T 0 Þ cp T 0
ð2Þ
collector efficiency,
In order to estimate the optimal turbine pressure drop factor, it is necessary to analyze the overall efficiency of a SC power plant. Generally, the overall efficiency of a SC power plant gsp is defined as the product of the partial efficiencies of the SC components:
gsp ¼ gsc gcoll gt gwt
technical work (J/kg)
wt
ð1Þ
where gsc, gcoll, gt are the efficiencies of the chimney, collector, turbine and gwt transmission, blades and generator efficiency, respectively. In [3], the theoretical and real air cycles in SC power plants are analyzed, according to the Fig. 1. In the same paper, based on the analysis depicted in Fig. 1, expressions for the partial efficiencies of the main components of the SC power plant are derived as follows:
gcoll ¼
q w1 Ac cp DT Q_ ¼ 1 ; Acoll G Acoll G
ð3Þ
and an alternative collector efficiency, according to Schlaich [4],
gcoll ¼ a
bDT G
ð3aÞ
where the coefficient values are a = 0.75–0.8 and b = 5–6 W/m2 K for DT = 30 K. In [3], the turbine efficiency is defined as
gt ¼
P tc Pcl w21 ¼1 ; Pcl 2cp DT gsc
ð4Þ
where w1 wc. According to the aforementioned expressions, the overall efficiency of a SC power plant should be estimated as
gsp ¼
_ p DT gHc w2c =2 mc 1 gwt : Acoll G cp DT gsc cp T 0
ð5Þ
A similar analysis of the theoretical air cycle in a SC power plant is derived in papers [11,12]. 2.1. Overall efficiency as a function of velocity at the chimney inlet The air flow velocity at the chimney inlet is an important parameter that influences the overall efficiency of a SC power plant, and its effect on overall efficiency is analyzed in subsequent sections of the paper. The relationship between the overall efficiency and the air flow velocity at the chimney inlet is described by
gsp ¼ gsp ðwc Þ:
Fig. 1. Theoretical and real air cycle in SC power plants [3].
Assuming that heating occurs at a constant pressure (aerodynamic losses are neglected), the mass flow of air through the col_ is calculated by lector m
589
S. Nizetic, B. Klarin / Applied Energy 87 (2010) 587–591
0.250
Optimal velocity of air at chimney inlet, w c(m/s)
20.0
sp (%)
0.200 0.150 0.100 0.050 0.000
0
2
4
6
8
10
12
14
16
wc (m/s)
ð6Þ
where the air density in the chimney qc is expressed as the air density at the collector inlet q0, reduced by a factor k (because the air in the chimney is lighter than the surrounding air),
qc ffi q0 k ffi q0 1 þ
DT T0
1 :
ð7Þ
Therefore, according to expression (4), the overall efficiency of a SC power plant, as a function of the air flow velocity at the chimney inlet (gsp ¼ gsp ðwc Þ), is determined by the expression,
gsp ¼ gsp ðwc Þ 2
3 Ac wc cp DT 7 gH w2 =2 c 1 c gwt : 5 cp DT gc Acoll G cp T 0
DT 6q0 1 þ T 0
¼4
18.0 17.0 16.0 15.0 14.0 13.0 12.0 11.0 10.0 9.0
Hc=200 m Hc=300 m
8.0 7.0
Hc=500 m
15
Fig. 2. Overall efficiency of a SC power plant as a function of air flow velocity at the chimney inlet.
_ ¼ qc Ac wc ; m
19.0
1
ð8Þ
In Fig. 2, the graphical interpretation of expression (8) is shown, using data similar to that of the prototype plant of Manzanares [6]. The above plot shows that an optimal air flow velocity at the chimney inlet exists and is related to the maximum overall efficiency of the SC power plant. 3. Optimal velocity of air at the chimney inlet for a constant air temperature increase in the collector In order to estimate the optimal air flow velocity, expression (8) d gsp ¼ 0; from which we can calculate should be maximized, dw c
20
25
30
Fig. 3. Dependence of optimal air flow velocity at the chimney inlet on the constant temperature increase in collector.
pffiffiffiffiffiffiffiffiffiffiffiffi Hc or the product Hc DT . Fig. 3 shows the optimal air flow velocity, for different chimney heights and different air temperature increases in the collector. From Fig. 4, it is obvious that in SC power plants, the optimal air flow velocity at the chimney inlet has a square root dependence on the air temperature increase in the collector DT for a specified chimney height Hc. In order to verify expression (9), calculations were performed using data from the Manzanares prototype [6]. According to the measured data (measured on 2 September, 1982.), the optimal air flow velocity at the chimney inlet is approximately wcopt ffi 9:8 m/s, the air temperature increase through the collector DTopt = 19 K and the outlet power Pe = 48 kW. With Hc = 195 m, expression (9) yields wcopt ffi 9:1 m/s. In this example, the numerical error is 6.8%, which is acceptable for this kind of simplified analysis. If we insert the expression for turbine efficiency, expression (9), into expression (4), then the optimal pressure drop factor becomes gtopt ¼ 2=3, which corresponds to the value in expression (4). However, it is important to note that at greater pressure drop values, the conditions change to the case in which the air temperature increase in the collector is constant. Therefore, the impact of the air temperature increase change on the optimal turbine pressure drop factor must be analyzed.
0.93
wcopt . Therefore, with the derivative of expression (8) defined as,
wcopt
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 gHc DT ; ¼ 3 T0
ð9Þ
2
d given that dw > 0. c For mean flow conditions, as in Manzanares [6], expression (9) can be simplified to
wcopt ffi 0:15
pffiffiffiffiffiffiffiffiffiffiffiffi H c DT :
ð10Þ
Results from expression (10) are only applicable to the unrealistic condition where temperature rise DT is independent of mass flow. Hence, results from Fig. 3 only have sense for a case where the air temperature increase in the collector is constant (which is hard to achieve in reality). Expression (10) shows that the most influential parameter on optimal air flow velocity at the chimney inlet is the chimney height
Factor of turbine pressure drop (pressure ratio)
d 2 gH g ¼ w2c c DT ¼ 0; dwc sp 3 T0 and an optimal air flow velocity gspmax ,
35
Air temperature increase, Δ T [K]
0.91
0.88
0.86
0.83 8.0
9.0
10.0 11.0 12.0 13.0 14.0 15.0 16.0
Velocity at chimney inlet, (m/s) Fig. 4. Impact of flow velocity change on the pressure drop factor.
S. Nizetic, B. Klarin / Applied Energy 87 (2010) 587–591
0.92
4. Optimal ratio of pressure drop across the turbine as a function of air temperature increase in the collector In order to derive an analytical expression for the optimal factor of turbine pressure drop across a turbine in the case of a varying temperature increase, the previously determined expression for the total efficiency of a SC power plant (expression (8)) should be used. Therefore, an alternative expression, expression (3a), should be used instead of the general expression for collector efficiency, expression (3),
bDT gHc w2 =2 gsp ¼ a 1 c gwt : G cp DT gc cp T 0
ð11Þ
Factor of turbine pressure drop
590
DT opt
ð12Þ
Expression (12), combined with the expression for turbine efficiency, expression (3a), yields an optimal ratio of pressure drop across the turbine,
Dpt Dptot
¼1f opt
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 a ; Gf b
ð13Þ
where Dpt is the pressure drop across the turbine, Dptot is the total : Factor f represents ratio available pressure drop and gtopt DDpptott opt
w2 T 0 f ¼ c : 2gHc As seen in the previous case, the optimal pressure drop ratio
gtopt ðDDpptott Þopt depends on the air flow velocity at the chimney inlet, as well as the coefficient ratio a/b. The impact of the flow velocity wc and the ratio a/b will be analyzed after verification of expression (12). Authors in [8] (Von Backström and Fluri) had previously come to the same conclusion but from another approach. In order to verify expression (13), typical values were taken from Manzanares data [4], and the results are presented in Table 1. from Table 1 shows that, for The determined value of DDpptott opt
certain conditions, the pressure drop factor is 0.9. This value matches well with the data provided in [7] and [8]. Namely, for initial conditions similar to those in [7], the pressure drop factor ranges from 0.83 to 0.91. The same paper indicated that the expected factor value ranged from 0.8 to 0.9, while according to same authors, the theoretical maximum value of the ratio is 0.97. On the other hand, the authors of [8] compared their results with those of [4] and [7] and concluded that the typical ratio for Manzanares conditions is 0.9. Hence, based on these previous results, it can be concluded that the value estimated here matches well with values calculated by other groups, and that the suggested analytical approach provides a useful estimation for factor of turbine pressure drop. However, according to this approach, it is obvious that the chardepends on the air flow velocity at the acteristic ratio DDppt tot
opt
chimney inlet, as well as the ratio a/b. Therefore, the impact of the air flow velocity at the chimney inlet and the ratio a/b should
Table 1 Calculation of gtopt with measured data from Manzanares. Time (h)
G (W/m2)
a
b (W/m2 K)
Hc (m)
T0 (K)
wc (m/s)
gtopt
12
890
0.8
6
194.6
299
9.0
0.9
0.88
α=0.7 α=0.8 α=0.9
0.86 0.050
From the derivative of expression (11) (for an air temperature increase DT, or dDd T ðgsp Þ ¼ 0), we can calculate DTopt, (the optimal air flow temperature increase in the collector) which corresponds to the optimal pressure drop across the turbine,
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a w2c GT 0 : ¼ b 2gHc
0.90
0.090
0.130
Ratio
0.170
0.210
/
Fig. 5. Impact of ratio a/b on the pressure drop factor.
also be analyzed. Similar analysis for total pressure difference can be found in [13] and for a case of solar chimney in [14]. 4.1. Influence of air flow velocity at the chimney inlet on pressure ratio According to results presented in [4], the expected flow velocities in SC power plants are in the range of 9.0–15.0 m/s (for normal working conditions). Thus, the characteristic ratio values should be estimated across that air velocity range. For this purpose, specified initial conditions, in accordance with the data provided in [4], are used. The results are graphically presented in Fig. 4. From the same figure, it is obvious that, within the range of velocity value considered, the pressure drop factor ranges from 0.83 to 0.91. Furthermore, as the air flow velocity at the chimney inlet decreases, the characteristic ratio increases, as expected. Finally, it can be concluded that, as the air flow velocity at the chimney inlet changes, the value of the ratio is in range Dpt ¼ 0:8 0:9. Dp tot
opt
4.2. Influence of ratio a/b on the pressure ratio The ratio a/b takes into consideration the convective and thermal losses incurred by absorption through soil. In Fig. 5, the impact of changes in the ratio a/b is shown. The value of the ratio a/b varies between 0.09 and 0.18, which corresponds to the expected conditions in the normal working regime of SC power plants (and conditions of the prototype plant, [4]). Hence, the value range is almost identical to that established by examining the impact of air flow velocity at the chimney inlet. Ananalysis of the influential parameters of the characteristic shows that: ratio DDppt tot
opt
– both impact parameters (wc and a/b) have the same range of value ratio DDppt , tot
opt
– in both cases, the determined range of values is in accordance with the values presented in other studies. 5. Conclusions In this work, a simplified analytical approach for the evaluation of the optimal pressure drop ratio in solar chimney power plants is presented. The approach is based on a simplified thermodynamic analysis of the overall SC cycle. It is estimated that the ratio (Dpt/Dptot)opt depends on two parameters: the air flow velocity at the solar chimney inlet w1 and the ratio of coefficients a/b. It is also shown that changes in either parameters (from their assumed
S. Nizetic, B. Klarin / Applied Energy 87 (2010) 587–591
values) result in similar changes to the ratio, (Dpt/Dptot)opt = 0.8– 0.92. This ratio value is in accordance with values provided by other authors. Hence, this proposed simplified analytical approach demonstrates that in solar chimney power plants, the turbine pressure drop factor (or characteristic ratio (Dpt/Dptot)opt) varies from 0.8 to 0.9. Therefore, it can be concluded that the proposed simplified approach is reliable and useful for a preliminary power estimation of solar chimney power plants for a given conditions. Acknowledgement Authors wish to thank the Ministry of Science, Education and Sports of the Republic of Croatia for support in this work. References [1] Dai YJ, Huang HB, Wang NRZ. Case study of solar chimney power plants in north-western regions of China. Renew Energy 2003;28:1295–304. [2] Bilgen E, Rheault J. Solar chimney power plants for high latitudes. Sol Energy 2005;79:449–58.
591
[3] Nizetic S, Ninic N, Klarin B. Analysis and feasibility of implementing solar chimney power plants in the Mediterranean region. Energy 2008;33:1680–90. [4] Schlaich J. The solar chimney: electricity from the sun. Geislingen: Maurer C; 1995. pp. 55. [5] Haaf W, Friedrich K, Mayr G, Sclaich J. Solar chimneys: Part I: Principle and construction of the pilot plant in Manzanares. Int J Sol Energy 1983;2(1):3–20. [6] Haaf W. Solar chimneys, Part II: Preliminary test results from the Manzanares pilot plant. Int J Sol Energy 1984;2:141–61. [7] Bernardes MAS, Voß A, Weinrebe G. Thermal and technical analyses of solar chimneys. Sol Energy 2003;75:511–24. [8] Von Backström TW, Fluri TP. Maximum fluid power condition in solar chimney power plants – an analytical approach. Sol Energy 2006;80:1417–23. [9] Hedderwick RA. Performance evaluation of a solar chimney power plant. M.Sc. Eng. Thesis, University of Stellenbosh; 2001. [10] Schlaich J, Bergermann R, Schiel W, Weinrebe G. Design of commercial solar tower system-utilization of solar induced convective flows for power generation. In: Proceedings of the international solar energy conference, Kohala Coast, United States; 2003. p. 573–81. [11] Von Backström TW, Gannon AJ. The solar chimney air standard thermodynamic cycle. SAIMechE R&D J 2000;16(1):16–24. [12] Gannon AJ, Von Backström TW. Solar chimney cycle analysis with system loss and solar collector performance. J Sol Energy Eng 2000;122:133–7. [13] Pinelli M, Bucci GTW. Numerical based desing of exhaust gas system in a cogeneration power plant. Appl Energy 2009;86:857–66. [14] Hariss DJ, Helwig N. Solar chimney and building ventilation. Appl Energy 2007;84:135–46.
Contents lists available at ScienceDirect
Applied Energy journal homepage: www.elsevier.com/locate/apenergy
A simplified analytical approach for evaluation of the optimal ratio of pressure drop across the turbine in solar chimney power plants S. Nizetic a,*, B. Klarin b a
University of Split, Faculty of Electrical and Mechanical Engineering and Naval Architecture, Department of Thermodynamics, Thermotechnics and Heat Engines, R.Boskovica b.b., 21000 Split, Croatia b University of Split, Faculty of Electrical and Mechanical Engineering and Naval Architecture, LAHES – Laboratory for Aero and Hybrid Energy Systems, R.Boskovica b.b., 21000 Split, Croatia
a r t i c l e
i n f o
Article history: Received 8 January 2009 Received in revised form 21 April 2009 Accepted 14 May 2009 Available online 5 June 2009 Keywords: Solar chimney Turbine pressure drop Overall efficiency
a b s t r a c t In this paper, a simplified analytical approach for evaluating the factor of turbine pressure drop in solar chimney power plants is presented. This characteristic factor (or pressure drop ratio in turbines, according to the total pressure drop in the chimney) is important because it is related to the output power. The determined factor (or ratio) values of the turbine pressure drop are found to be within a value range consistent with other studies. It was concluded that for solar chimney power plants, turbine pressure drop factors are in the range of 0.8–0.9. This simplified analytical approach is useful for preliminary analysis and fast evaluation of the potential of solar chimney power plants. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Solar chimney power plants (SC) present a renewable energy power source. The key factor in the economic feasibility of SC power plants is determined by solar irradiation at a particular geographic location. Desert areas are convenient locations, where the solar irradiation intensity averages over 2000 kW h/m2 year. In [1– 3], the possibility of SC power plant use in other geographic locations with lower solar irradiation is analyzed. The concept of SC power plants, as well as a prototype plant, was developed by a group of scientists led by Professor J. Schlaich [4–6]. The working principle of the plant is based on the buoyancy effect, as a fundamental physical principle. Relatively cold surrounding air is heated by a collector. Due to the density difference, warm air flows from the collector periphery to the chimney in the collector center. The wind turbine is placed at the bottom of the chimney because of construction (practical) reasons. Under certain conditions, the wind turbine provides rotation to an electric generator, producing electrical energy. Since the inception of SC power plants, many scientific papers have been published, providing comprehensive and concise overviews of the work done in this field [7]. A very important factor for optimal electrical power output is the pressure drop at the turbine, which corresponds to the maximum electric power output for a given condition. In [6], this factor * Corresponding author. Tel.: +385 21 3055948; fax: +385 21 463877. E-mail address: [email protected] (S. Nizetic). 0306-2619/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2009.05.019
is mentioned for the first time and is reported to have a value of 2/ 3. However, in later works [4] and [8], higher values for this factor are presented. Factor has a value of 2/3 only when there is a constant air temperature increase in the collector (or when the available total pressure drop is constant). In [7], the maximum power is achieved when the factor of the pressure drop at the turbine is equal to approximately 0.97. In the same paper, the authors indicate that this value of pressure drop is difficult to reach, and realistically, the value is expected to range from 0.8 to 0.9. Von Backström and Fluri [8] analyzed the influence of the volume flow of air on power output in different conditions of aerodynamic losses and turbine working characteristics. The value range of the pressure drop estimated from their analytical theory is consistent with values estimated by other researchers. In the same paper [8], assuming typical values from Schlaich, the calculated optimal value of the turbine pressure ratio is typically 0.9. According to other studies, the pressure drop is: Bernardes [7], about 0.83; Hedderwick [9], about 0.7; Schlaich et al. [10], about 0.8. Hence, no unified value of the factor exists, but a range of values has been suggested, as previously mentioned. Also it is obvious that all calculated values for optimal factor of pressure drop, by other different authors, converge to the value that is close to the value of 0.9. So probably that value is close to the real one for optimal factor of turbine pressure drop. The objective of this paper is to estimate reliable results for the optimal pressure drop factor (or a value range) using a simplified analytical approach. We theorize that, even for a constant solar irradiation, the pressure potential of a solar chimney power plant
588
S. Nizetic, B. Klarin / Applied Energy 87 (2010) 587–591
Nomenclature Ac Acoll cp f g G Hc k _ m p0 p2 Pcl Pe Ptc Q_ qcoll T0 T1
v1
wc w1 wcopt
cross-sectional area of solar chimney (m2) solar collector area (m2) specific heat capacity of air (kJ/kg C) ratio, (K) acceleration of gravity (m/s2) solar irradiance (W/m2) solar chimney height (m) air density reduction factor mass flow of air through the collector, (kg/s) atmospheric pressure (Pa) pressure at collector outlet (Pa) power loss due to exit kinetic energy (kW) electric output form the solar chimney power plant (kW) power of theoretical air cycle (kW) heat gain of the air in the collector (W) specific heat gain of the air in the collector (W/m2) ambient temperature (K) temperature of air in the collector outlet (K) specific volume of air at collector outlet (m3/kg) inlet air flow velocity of the chimney (m/s) optimal inlet air flow velocity of the chimney (m/s)
is not fixed but is a function of the air temperature increase in the collector.
Greek symbols effective absorption coefficient b factor proportional to convective energy loss (W/m2 K) gcoll solar collector efficiency gsc solar chimney efficiency gsp overall SC power plant efficiency gt turbine efficiency gtopt optimal pressure drop ratio (factor) gwt blade, transmission and generator efficiency q0 density of ambient air (kg/m3) q1 qc density of air at the inlet in the solar chimney (kg/m3) DT opt optimal air temperature increase through the collector, (K) DT air temperature increase between the collector inflow and outflow (°C) Dht specific enthalpy difference (J/kg) Dpt pressure drop across the turbine (Pa) Dptot Dpac total available pressure drop (pressure difference produced between the chimney base and the surroundings) (Pa)
a
chimney efficiency,
gsc ¼
2. Simplified expression for the overall efficiency of a solar chimney power plant
v 1 Dpac gHc ; ¼ cp ðT 1 T 0 Þ cp T 0
ð2Þ
collector efficiency,
In order to estimate the optimal turbine pressure drop factor, it is necessary to analyze the overall efficiency of a SC power plant. Generally, the overall efficiency of a SC power plant gsp is defined as the product of the partial efficiencies of the SC components:
gsp ¼ gsc gcoll gt gwt
technical work (J/kg)
wt
ð1Þ
where gsc, gcoll, gt are the efficiencies of the chimney, collector, turbine and gwt transmission, blades and generator efficiency, respectively. In [3], the theoretical and real air cycles in SC power plants are analyzed, according to the Fig. 1. In the same paper, based on the analysis depicted in Fig. 1, expressions for the partial efficiencies of the main components of the SC power plant are derived as follows:
gcoll ¼
q w1 Ac cp DT Q_ ¼ 1 ; Acoll G Acoll G
ð3Þ
and an alternative collector efficiency, according to Schlaich [4],
gcoll ¼ a
bDT G
ð3aÞ
where the coefficient values are a = 0.75–0.8 and b = 5–6 W/m2 K for DT = 30 K. In [3], the turbine efficiency is defined as
gt ¼
P tc Pcl w21 ¼1 ; Pcl 2cp DT gsc
ð4Þ
where w1 wc. According to the aforementioned expressions, the overall efficiency of a SC power plant should be estimated as
gsp ¼
_ p DT gHc w2c =2 mc 1 gwt : Acoll G cp DT gsc cp T 0
ð5Þ
A similar analysis of the theoretical air cycle in a SC power plant is derived in papers [11,12]. 2.1. Overall efficiency as a function of velocity at the chimney inlet The air flow velocity at the chimney inlet is an important parameter that influences the overall efficiency of a SC power plant, and its effect on overall efficiency is analyzed in subsequent sections of the paper. The relationship between the overall efficiency and the air flow velocity at the chimney inlet is described by
gsp ¼ gsp ðwc Þ:
Fig. 1. Theoretical and real air cycle in SC power plants [3].
Assuming that heating occurs at a constant pressure (aerodynamic losses are neglected), the mass flow of air through the col_ is calculated by lector m
589
S. Nizetic, B. Klarin / Applied Energy 87 (2010) 587–591
0.250
Optimal velocity of air at chimney inlet, w c(m/s)
20.0
sp (%)
0.200 0.150 0.100 0.050 0.000
0
2
4
6
8
10
12
14
16
wc (m/s)
ð6Þ
where the air density in the chimney qc is expressed as the air density at the collector inlet q0, reduced by a factor k (because the air in the chimney is lighter than the surrounding air),
qc ffi q0 k ffi q0 1 þ
DT T0
1 :
ð7Þ
Therefore, according to expression (4), the overall efficiency of a SC power plant, as a function of the air flow velocity at the chimney inlet (gsp ¼ gsp ðwc Þ), is determined by the expression,
gsp ¼ gsp ðwc Þ 2
3 Ac wc cp DT 7 gH w2 =2 c 1 c gwt : 5 cp DT gc Acoll G cp T 0
DT 6q0 1 þ T 0
¼4
18.0 17.0 16.0 15.0 14.0 13.0 12.0 11.0 10.0 9.0
Hc=200 m Hc=300 m
8.0 7.0
Hc=500 m
15
Fig. 2. Overall efficiency of a SC power plant as a function of air flow velocity at the chimney inlet.
_ ¼ qc Ac wc ; m
19.0
1
ð8Þ
In Fig. 2, the graphical interpretation of expression (8) is shown, using data similar to that of the prototype plant of Manzanares [6]. The above plot shows that an optimal air flow velocity at the chimney inlet exists and is related to the maximum overall efficiency of the SC power plant. 3. Optimal velocity of air at the chimney inlet for a constant air temperature increase in the collector In order to estimate the optimal air flow velocity, expression (8) d gsp ¼ 0; from which we can calculate should be maximized, dw c
20
25
30
Fig. 3. Dependence of optimal air flow velocity at the chimney inlet on the constant temperature increase in collector.
pffiffiffiffiffiffiffiffiffiffiffiffi Hc or the product Hc DT . Fig. 3 shows the optimal air flow velocity, for different chimney heights and different air temperature increases in the collector. From Fig. 4, it is obvious that in SC power plants, the optimal air flow velocity at the chimney inlet has a square root dependence on the air temperature increase in the collector DT for a specified chimney height Hc. In order to verify expression (9), calculations were performed using data from the Manzanares prototype [6]. According to the measured data (measured on 2 September, 1982.), the optimal air flow velocity at the chimney inlet is approximately wcopt ffi 9:8 m/s, the air temperature increase through the collector DTopt = 19 K and the outlet power Pe = 48 kW. With Hc = 195 m, expression (9) yields wcopt ffi 9:1 m/s. In this example, the numerical error is 6.8%, which is acceptable for this kind of simplified analysis. If we insert the expression for turbine efficiency, expression (9), into expression (4), then the optimal pressure drop factor becomes gtopt ¼ 2=3, which corresponds to the value in expression (4). However, it is important to note that at greater pressure drop values, the conditions change to the case in which the air temperature increase in the collector is constant. Therefore, the impact of the air temperature increase change on the optimal turbine pressure drop factor must be analyzed.
0.93
wcopt . Therefore, with the derivative of expression (8) defined as,
wcopt
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 gHc DT ; ¼ 3 T0
ð9Þ
2
d given that dw > 0. c For mean flow conditions, as in Manzanares [6], expression (9) can be simplified to
wcopt ffi 0:15
pffiffiffiffiffiffiffiffiffiffiffiffi H c DT :
ð10Þ
Results from expression (10) are only applicable to the unrealistic condition where temperature rise DT is independent of mass flow. Hence, results from Fig. 3 only have sense for a case where the air temperature increase in the collector is constant (which is hard to achieve in reality). Expression (10) shows that the most influential parameter on optimal air flow velocity at the chimney inlet is the chimney height
Factor of turbine pressure drop (pressure ratio)
d 2 gH g ¼ w2c c DT ¼ 0; dwc sp 3 T0 and an optimal air flow velocity gspmax ,
35
Air temperature increase, Δ T [K]
0.91
0.88
0.86
0.83 8.0
9.0
10.0 11.0 12.0 13.0 14.0 15.0 16.0
Velocity at chimney inlet, (m/s) Fig. 4. Impact of flow velocity change on the pressure drop factor.
S. Nizetic, B. Klarin / Applied Energy 87 (2010) 587–591
0.92
4. Optimal ratio of pressure drop across the turbine as a function of air temperature increase in the collector In order to derive an analytical expression for the optimal factor of turbine pressure drop across a turbine in the case of a varying temperature increase, the previously determined expression for the total efficiency of a SC power plant (expression (8)) should be used. Therefore, an alternative expression, expression (3a), should be used instead of the general expression for collector efficiency, expression (3),
bDT gHc w2 =2 gsp ¼ a 1 c gwt : G cp DT gc cp T 0
ð11Þ
Factor of turbine pressure drop
590
DT opt
ð12Þ
Expression (12), combined with the expression for turbine efficiency, expression (3a), yields an optimal ratio of pressure drop across the turbine,
Dpt Dptot
¼1f opt
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 a ; Gf b
ð13Þ
where Dpt is the pressure drop across the turbine, Dptot is the total : Factor f represents ratio available pressure drop and gtopt DDpptott opt
w2 T 0 f ¼ c : 2gHc As seen in the previous case, the optimal pressure drop ratio
gtopt ðDDpptott Þopt depends on the air flow velocity at the chimney inlet, as well as the coefficient ratio a/b. The impact of the flow velocity wc and the ratio a/b will be analyzed after verification of expression (12). Authors in [8] (Von Backström and Fluri) had previously come to the same conclusion but from another approach. In order to verify expression (13), typical values were taken from Manzanares data [4], and the results are presented in Table 1. from Table 1 shows that, for The determined value of DDpptott opt
certain conditions, the pressure drop factor is 0.9. This value matches well with the data provided in [7] and [8]. Namely, for initial conditions similar to those in [7], the pressure drop factor ranges from 0.83 to 0.91. The same paper indicated that the expected factor value ranged from 0.8 to 0.9, while according to same authors, the theoretical maximum value of the ratio is 0.97. On the other hand, the authors of [8] compared their results with those of [4] and [7] and concluded that the typical ratio for Manzanares conditions is 0.9. Hence, based on these previous results, it can be concluded that the value estimated here matches well with values calculated by other groups, and that the suggested analytical approach provides a useful estimation for factor of turbine pressure drop. However, according to this approach, it is obvious that the chardepends on the air flow velocity at the acteristic ratio DDppt tot
opt
chimney inlet, as well as the ratio a/b. Therefore, the impact of the air flow velocity at the chimney inlet and the ratio a/b should
Table 1 Calculation of gtopt with measured data from Manzanares. Time (h)
G (W/m2)
a
b (W/m2 K)
Hc (m)
T0 (K)
wc (m/s)
gtopt
12
890
0.8
6
194.6
299
9.0
0.9
0.88
α=0.7 α=0.8 α=0.9
0.86 0.050
From the derivative of expression (11) (for an air temperature increase DT, or dDd T ðgsp Þ ¼ 0), we can calculate DTopt, (the optimal air flow temperature increase in the collector) which corresponds to the optimal pressure drop across the turbine,
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a w2c GT 0 : ¼ b 2gHc
0.90
0.090
0.130
Ratio
0.170
0.210
/
Fig. 5. Impact of ratio a/b on the pressure drop factor.
also be analyzed. Similar analysis for total pressure difference can be found in [13] and for a case of solar chimney in [14]. 4.1. Influence of air flow velocity at the chimney inlet on pressure ratio According to results presented in [4], the expected flow velocities in SC power plants are in the range of 9.0–15.0 m/s (for normal working conditions). Thus, the characteristic ratio values should be estimated across that air velocity range. For this purpose, specified initial conditions, in accordance with the data provided in [4], are used. The results are graphically presented in Fig. 4. From the same figure, it is obvious that, within the range of velocity value considered, the pressure drop factor ranges from 0.83 to 0.91. Furthermore, as the air flow velocity at the chimney inlet decreases, the characteristic ratio increases, as expected. Finally, it can be concluded that, as the air flow velocity at the chimney inlet changes, the value of the ratio is in range Dpt ¼ 0:8 0:9. Dp tot
opt
4.2. Influence of ratio a/b on the pressure ratio The ratio a/b takes into consideration the convective and thermal losses incurred by absorption through soil. In Fig. 5, the impact of changes in the ratio a/b is shown. The value of the ratio a/b varies between 0.09 and 0.18, which corresponds to the expected conditions in the normal working regime of SC power plants (and conditions of the prototype plant, [4]). Hence, the value range is almost identical to that established by examining the impact of air flow velocity at the chimney inlet. Ananalysis of the influential parameters of the characteristic shows that: ratio DDppt tot
opt
– both impact parameters (wc and a/b) have the same range of value ratio DDppt , tot
opt
– in both cases, the determined range of values is in accordance with the values presented in other studies. 5. Conclusions In this work, a simplified analytical approach for the evaluation of the optimal pressure drop ratio in solar chimney power plants is presented. The approach is based on a simplified thermodynamic analysis of the overall SC cycle. It is estimated that the ratio (Dpt/Dptot)opt depends on two parameters: the air flow velocity at the solar chimney inlet w1 and the ratio of coefficients a/b. It is also shown that changes in either parameters (from their assumed
S. Nizetic, B. Klarin / Applied Energy 87 (2010) 587–591
values) result in similar changes to the ratio, (Dpt/Dptot)opt = 0.8– 0.92. This ratio value is in accordance with values provided by other authors. Hence, this proposed simplified analytical approach demonstrates that in solar chimney power plants, the turbine pressure drop factor (or characteristic ratio (Dpt/Dptot)opt) varies from 0.8 to 0.9. Therefore, it can be concluded that the proposed simplified approach is reliable and useful for a preliminary power estimation of solar chimney power plants for a given conditions. Acknowledgement Authors wish to thank the Ministry of Science, Education and Sports of the Republic of Croatia for support in this work. References [1] Dai YJ, Huang HB, Wang NRZ. Case study of solar chimney power plants in north-western regions of China. Renew Energy 2003;28:1295–304. [2] Bilgen E, Rheault J. Solar chimney power plants for high latitudes. Sol Energy 2005;79:449–58.
591
[3] Nizetic S, Ninic N, Klarin B. Analysis and feasibility of implementing solar chimney power plants in the Mediterranean region. Energy 2008;33:1680–90. [4] Schlaich J. The solar chimney: electricity from the sun. Geislingen: Maurer C; 1995. pp. 55. [5] Haaf W, Friedrich K, Mayr G, Sclaich J. Solar chimneys: Part I: Principle and construction of the pilot plant in Manzanares. Int J Sol Energy 1983;2(1):3–20. [6] Haaf W. Solar chimneys, Part II: Preliminary test results from the Manzanares pilot plant. Int J Sol Energy 1984;2:141–61. [7] Bernardes MAS, Voß A, Weinrebe G. Thermal and technical analyses of solar chimneys. Sol Energy 2003;75:511–24. [8] Von Backström TW, Fluri TP. Maximum fluid power condition in solar chimney power plants – an analytical approach. Sol Energy 2006;80:1417–23. [9] Hedderwick RA. Performance evaluation of a solar chimney power plant. M.Sc. Eng. Thesis, University of Stellenbosh; 2001. [10] Schlaich J, Bergermann R, Schiel W, Weinrebe G. Design of commercial solar tower system-utilization of solar induced convective flows for power generation. In: Proceedings of the international solar energy conference, Kohala Coast, United States; 2003. p. 573–81. [11] Von Backström TW, Gannon AJ. The solar chimney air standard thermodynamic cycle. SAIMechE R&D J 2000;16(1):16–24. [12] Gannon AJ, Von Backström TW. Solar chimney cycle analysis with system loss and solar collector performance. J Sol Energy Eng 2000;122:133–7. [13] Pinelli M, Bucci GTW. Numerical based desing of exhaust gas system in a cogeneration power plant. Appl Energy 2009;86:857–66. [14] Hariss DJ, Helwig N. Solar chimney and building ventilation. Appl Energy 2007;84:135–46.