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Counter-rotating turbines for solar chimney power plants
ARTICLE IN PRESS
Renewable Energy 31 (2006) 1873–1891 www.elsevier.com/locate/renene
Counter-rotating turbines for solar chimney power plants F. Denantes1, E. Bilgen Ecole Polytechnique, C.P. 6079 Centre Ville Montreal Qc, Canada H3C 3A7 Received 5 May 2005; accepted 13 September 2005 Available online 20 October 2005
Abstract An efficiency model at design performance for counter-rotating turbines is developed and validated. Based on the efficiency equations, an off-design performance model for counter-rotating turbines is developed. Combined with a thermodynamic model for a solar chimney system and a solar radiation model, annual energy output of solar chimney systems is determined. Two counterrotating turbines, one with inlet guide vanes, the other without, are compared to a single-runner system. The design and off-design performances are weighed against in three different solar chimney plant sizes. It is shown that the counter-rotating turbines without guide vanes have lower design efficiency and a higher off-design performance than a single-runner turbine. Based on the output torque versus power for various turbine layouts, advantageous operational conditions of counterrotating turbines are demonstrated. r 2005 Elsevier Ltd. All rights reserved. Keywords: Solar chimney; Counter-rotating turbine
1. Introduction A solar chimney is a simple system to convert energy provided by solar radiation into electrical energy. It consists of three main subsystems, which are the collector, the chimney Corresponding author. Tel.: +1 514 2340 4711x4579.
E-mail address: [email protected] (E. Bilgen). Present address: Fakulta¨t fu¨r Maschinenwesen, Technische Universita¨t Mu¨nchen, BoltzmannstraXe 15, Garching, Germany. 1
0960-1481/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2005.09.018
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Nomenclature Ac c C CRT CRTa CD CL Cz G g Hc Dh K K0 _ m P p q R0a R0b SRT T U ub w W
chimney cross-section area dimensionless absolute velocity (c ¼ C=U a ) absolute flow velocity counter-rotating turbine with stator and two rotors counter-rotating turbine without stator and two rotors drag coefficient lift coefficient axial flow velocity solar radiation intensity terrestrial gravity chimney height specific stagnation enthalpy difference head coefficient parameter of turbine operation point mass flow rate power pressure heat per unit mass geometric factor for runner a ðR0a ¼ 1 ð1=2Þðcu2 þ cu1 Þ ¼ ð1=2Þðwu2a þ wu1a ÞÞ geometric factor for runner b ðR0b ¼ ub ð1=2Þðcu3 þ cu2 Þ ¼ ð1=2Þðwu3b þ wu2b ÞÞ single-runner turbine with stator and rotor temperature peripheral velocity dimensionless velocity of runner b ðub ¼ U b =U a Þ, ub is negative dimensionless relative flow velocity relative flow velocity
Greek letters a b g e zs zra zrb zd f c ca cb o
absolute outlet flow angle, collector heat absorptance relative outlet flow angle, collector total loss coefficient correction factor lift-to-drag ratio of turbine blade (g ¼ CL/CD) flow angle change through blade stator pressure loss coefficient ðzs ¼ DPsLoss =ð1=2ÞrC 21 Þ pressure loss coefficient for rotor a ðzra ¼ DPraLoss =ð1=2ÞrW 22a Þ pressure loss coefficient for rotor b ðzrb ¼ DPrbLoss =ð1=2ÞrW 23b Þ diffuser pressure loss coefficient ðzd ¼ DPdLoss =ð1=2ÞrC 23 Þ flow coefficient ðf ¼ C z =U a Þ load coefficient ðc ¼ Dh=U 2a ¼ ðU a ðC u1 C u2 Þ þ U b ðC u2 C u3 ÞÞ=U 2a ¼ ca þ cb Þ load coefficient for runner a ðca ¼ cu1 cu2 Þ load coefficient for runner b ðcb ¼ ub ðcu2 cu3 ÞÞ rotation speed
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Subscripts a b coll D L ops dLoss raLoss rbLoss s sLoss turb ts tt u 1 2 3
runner a runner b collector drag lift operating point loss diffuser loss rotor a loss rotor b stator (or guide vane) loss at stator turbine total-static total-total projection of the considered entity on the peripheral axis position (or gas state) between the stator and runner a blades position between runners a and b position at the exit of runner b (at diffuser inlet)
and the turbine (see, e.g., [1]). Air is heated by solar radiation under a circular transparent roof open at its periphery. Suction from the chimney draws hot air from the collector, and cold air comes in from the outer perimeter. The energy contained in the air flow is converted into mechanical energy by air turbines at the base of the chimney, and into electrical energy by conventional generators. This system can also operate overnight using heat storage systems on the ground, thus annual operating hours can be as high as 8423–8723 for large power plants [1]. In large power plants, an improvement of overall conversion efficiency results in considerable energy gains as well as economic benefits. For example, following the solar chimney power plant of 200 MW conceived in Mildura, Australia [2], 1% efficiency gain means 2 MW nominal power increase or for a plant operating 3000 h/year, 6 106 kW h extra electric energy. Therefore, it would be interesting to study major subsystems of the solar chimney plant and determine potential efficiency improvement possibilities. One such subsystem is the turbine. The small prototype in Manzanares, which had a 50 kW nominal power, had a singlevertical-axis turbine with four blades made of fiber-glass hard-foam sandwich shells. Recently, the air turbine for solar chimney power plants was studied in detail [3,4]. The single-vertical-axis air turbines in these studies consist of (i) inlet blades that give the air flow a swirl, (ii) the turbine, and (iii) a diffuser. A turbine efficiency equation was developed with the dimensionless parameters, which are reaction degree R, flow coefficient f, load coefficient c, and the pressure loss coefficient zi for each of the three turbine subsections. This efficiency equation was then used to find an optimum design point. Von Backstrom and Gannon [4] used as an example a solar chimney plant of 200 MW nominal
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power and showed the following: (a) The optimum reaction degree Ropt is expressed, showing that, because the static pressure at turbine diffuser exit is important, a turbine with low or no exit whirl will have a better efficiency. This explains the necessity of introducing inlet swirl with the stator. (b) Trying to optimize a turbine for a pre-designed chimney leads to a relatively high efficiency, which implies a high c and a low f. As a result, the optimized turbine would have a slow rotation speed, implying high torque values, accompanied by lower air flow velocities than in the design point of the chimney. In turn, this means an increased turbine cross-section area. A constraint on a minimum f value is proposed. (c) The optimization of the turbine for fixed values of R and f shows that total-static efficiencies of 85% are possible, assuming that the pressure loss coefficients are low at 0.03 due to low flow angle change through turbine blades e. To improve turbine efficiency in these plants, a counter-rotating turbine system may be utilized. Conception and analysis of counter-rotating hydraulic machines show that the efficiency of counter-rotating turbines may be better than single-stage turbines in various operating conditions [5–7] and ducted fans [8]. Indeed, in the aviation industry, the concept of counter-rotating fan blades was analyzed and a prototype of an un-ducted fan engine (GE-36) was built and tested. Similarly, Antonov used counter-rotating propfan engines on aircraft An-70 and An-22. Tupolev also successfully used this type of engines on his TU-95 and TU-24 [9]. The main advantage in these applications was to reduce the propeller size and generated swirl. A counter-rotating turbine consists of several (at least two) co-axial counter-rotating blade rows and/or a stator row in front or behind the counter-rotating blade rows. Radial flow counter-rotating turbines do exist, but for the solar chimney only axial flow turbines will be considered. The main advantages of a counter-rotating turbine are the relatively low rotation speeds for high relative velocities, and thus large flow angle changes [10]. In addition, stator vanes can be eliminated, hence eliminating the resulting pressure losses. Cai et al. [11] presented different types of counter-rotating turbines, with different inlet and outflow conditions and impulsive or reactive rotating blade row. They considered in a first step a general counter-rotating turbine stage, explaining that there are five independent variables that describe a stage with two counterrotating rows. However, for better comprehension, only 11 particular configurations were analyzed more thoroughly. These configurations were compared with usual single-rotating stages. The single-runner turbine efficiency used by Von Backstrom and Gannon [4] is 0 for a load coefficient of c 0:8 at Zts 0:85 The load coefficient h¯ used by Cai 0 0 et al. [11] is defined such that h¯ ¼ c=2, thus h¯ 0:4: The calculated efficiency and load factor are not completely in accordance with their Figs. 5 and 6, but for a similar or better efficiency, the load factor in the counter-rotating turbine will be higher, which reflects the low rotation speed of counter-rotating stages. A counterrotating turbine with axial inlet and outlet flow can have a total-static efficiency of around 85%. The literature review shows that depending on the application, counter-rotating turbines are relatively more efficient conversion devices when compared to classical turbines. Therefore, our aim is to study the counter-rotating turbines in view of an application in solar chimney systems, to determine preliminary design parameters, and their operating conditions.
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2. Counter-rotating turbines 2.1. Mathematical model Performance equations are developed for counter-rotating turbines, see Fig. 1. The turbine efficiency is Zt ¼
Dh 1 . ¼ Dh þ DhLoss 1 þ DhLoss =Dh
(1)
Using the definition of enthalpy dh ¼ T ds þ ð1=rÞ dp and for an isentropic process, we obtain dh ¼ ð1=rÞdp or DhLoss ¼ DpLoss =r. Thus, the efficiency becomes 1 1 ðDpsLoss þ DpraLoss þ DprbLoss þ DpdLoss Þ . (2) Zt ¼ 1 þ rDh Using the pressure loss coefficients defined in the nomenclature, we obtain Zt ¼
1 U2 1 þ a ðzs C 21 =U 2a þ zra W 22a =U 2a þ zrb W 23b =U 2a þ zd C 23 =U 2a Þ . 2Dh
(3)
Using the variables defined in the nomenclature and after some algebra, we first obtain the dimensionless velocity expressions (see Fig. 1) and then the expressions for the two load coefficients: 1 c ca ¼ ðc 2ub ð1 R0a ub þ R0b ÞÞ; cb ¼ 2ub 1 R0a a ub þ R0b , 1 ub 2 (4) where c ¼ ca þ cb .
Diffuser d
3 Rotor
b Ua
2 Rotor
a
W1a
1 Stator Cc
Ub
Ub C1 W1b
C3 W3
Ua W1
s Cc
Fig. 1. Velocity triangle and nomenclature of a counter-rotating turbine.
C1
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Thus we have an expression for Zt which depends on the dimensionless parameters f, c, R0a , R0b , and ub . To obtain the total-total efficiency from Eq. (3), the diffuser pressure loss coefficient zd should be set to zero. It is possible to reconsider the above analysis for counter-rotating turbine without inlet guide vanes, by setting zs ¼ 0 and c1u¼0 . Hence, the counter-rotating turbine without guide vanes has only four independent variables. Both turbine total-total and total-static efficiencies can be calculated with the above method. For the total-total efficiency, the diffuser loss coefficient zd should be set to zero, while to obtain the total-static efficiency it should be set to 1. 2.2. Profile loss model The efficiency equation developed above requires knowledge of the pressure loss coefficients zr of the various turbine subsections. We will consider the available techniques, which are constant profile loss, profile loss by Lewis [12] and lift/drag ratio profile loss by Ozgur and Nathan [7]. 2.2.1. Constant profile loss model The method is to derive an expression with respect to the considered parameter. A good approximation of the profile losses is zi ¼ 0:05, which corresponds, as will be shown later by Eq. (5), to a 901 flow angle change through the blade. This value represents a safe conservative approximation. Since in our analysis the diffuser design is not considered, the pressure loss coefficient will be kept constant and equal to zd ¼ 0:75, which represents a realistic value for diffusers in solar chimneys (e.g., [4]). 2.2.2. Profile loss model by Lewis Lewis [12] gives a simple expression for profile losses for single-runner turbines, which depends on the flow angle change through a selected blade. It is very accurate given its simplicity: e 2 z ¼ 0:025 1 þ , (5) 90 where e is the flow angle change through the blade, which is es ¼ a1 for the stator, ea ¼ b1 b2a for the runner a and eb ¼ b2b b3 for the runner b. The turbine efficiency calculated using Eq. (5) is slightly higher than the one computed using the constant profile loss, which is expected since the latter was chosen using a conservative value. We note also that although this model was developed for single runners, there is no restriction, in principle, for its usage in other cases, such as counterrotating turbines. 2.2.3. Profile loss model using lift/drag ratio Ozgur and Nathan [7] analyzed an axial outlet counter-rotating turbine without guide vanes and with identical rotation speeds and specific work output for both runners. It is noticed that they are completely different from those by Lewis and they are based on the
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lift/drag ratio g of turbine blades: xs ¼
2cu1 ðc2u1 =4Þ þ f2 c21 gf ðcu1 =2Þ
and xr ¼
2c ðR2 þ f2 Þ . w2exit gf þ R
(6)
These relations modify slightly the attainable efficiencies and their operating point. Also, they are not easy to handle because of their relative complexities. 2.3. Validation and comparison of the loss models For single-runner turbines, the data provided in [4,7] will be used while for the counterrotating turbines, the velocity expressions in [11] will be verified and the plots presented in [7] will be used for comparison in the single-runner turbine case. To compare with their results, which make use of the parameters of flow coefficient f and head coefficient K, the following expression is used: c ¼ 2Kf2 . 2.3.1. Single-runner turbine The model used for the present analysis is built using equations presented in [4], thus the results presented in their Figs. 3 and 4 were exactly reproduced (not shown here). To compare results with the plots provided in [7] for a single-runner turbine with an axial outflow velocity implies R ¼ 1 c=2. The parameter K was introduced as explained above and the total-total efficiency versus f with increasing head coefficient K was plotted. The results with three models, i.e., constant loss coefficients, Lewis loss coefficients, and lift/ drag ratio loss coefficients models were plotted (not shown in figures). We observed that using the lift/drag ratio coefficients, Eqs. (1) and (6) resulted in exactly the same as in Fig. 4 of [7]. This is also expected because the same theory is used for the same case. The other profile loss models give also satisfactory simulation. The constant profile loss model resulted in slightly lower efficiencies whereas the profile loss model by Lewis produced slightly higher efficiencies than those of the reference case. The maximum efficiency with the constant and Lewis models decreases faster with increasing head coefficient than with the profile loss model using lift/drag ratio. However, since the solar chimney turbine operation point corresponds to low heads, this deficiency will not affect the overall performance results in this study. 2.3.2. Counter-rotating turbine Introducing the conditions chosen by Cai et al. [11] leads to the same velocity expressions presented in their article. The only difference is the systematically reversed sign of ub , which is expected since ub is considered positive in their work. As mentioned earlier, Ozgur and Nathan [7] considered a particular counter-rotating turbine stage: it has axial inlet and outlet velocities, equal rotation speeds, equal specific work in each blade row, and does not have a stator. These conditions lead to the following expressions: R0a ¼ 1 ub þ R0b ¼ where U b ¼ 1.
c þ 1 and 4
R0b ¼
1 c þ 2ub 2u2b c ¼ 1, 2 4 ub 1
(7)
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ηtt
0.93 0.92 0.91 0.9 K FROM 2 to 12 0.89 0.2
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(a)
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ηtt
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ηtt
0.93 0.92 0.91 0.9 K FROM 1 to 12 0.89 0.2
(c)
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φ
Fig. 2. Counter-rotating turbine efficiencies with different profile loss models: (a) constant profile loss model, (b) Lewis profile loss model, and (c) lift/drag profile loss model.
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There are now two independent variables left, which are f and K. We plotted the totaltotal efficiency as a function of the flow coefficient f with K variable from 1 to 12 as a parameter and presented in Fig. 2. The results of Ozgur and Nathan [7] in their Fig. 3 were exactly reproduced in Fig. 2(c) when the lift/drag ratio model was used. The constant loss model produced results slightly lower and that by Lewis slightly higher as it can be seen in Figs. 2(a) and (b). When compared with the results obtained for the single-runner case, we noticed that the general appearance of Fig. 2 was similar to that for the singlerunner case but the evolution of maximum efficiencies relative to K was reversed. Our observation was that the higher attainable efficiency of the counter-rotating turbines was at different operating conditions, i.e., at low K with respect to high K of the single-runner turbines. Using constant or Lewis loss coefficients does not give, at first sight, good results in agreement with those by using lift/drag ratio as can be seen in Fig. 2. Comparing with the single-runner turbine case showed, however, that the general trend was nevertheless acceptable. The efficiency remains higher with counter-rotating turbines than with singlerunner turbines when the flow coefficients increase. We note also that the results in Fig. 2 show that it is important to compare efficiencies produced with the same profile loss model. 2.3.3. Comparison of the three profile loss models We plotted the total-static efficiency Zts as a function of flow coefficient f with load coefficient c as a parameter for the same conditions but with different loss coefficient models for single (not presented in figures) and counter-rotating turbine systems, presented in Fig. 3. It is seen that the influence of the different loss coefficient models is the same on the overall performance from the simulation model. However, the lift/drag loss model changes visibly the maximum efficiency operating point more than the others. Given the available data, it is not possible to conclude which loss model gives the most accurate representation of the reality. However, the similarities pointed out show that in respect to the comparison between counter-rotating and single-runner turbines, they are more or less equivalent. This is why the presented results are computed using Lewis profile loss model, as it is widely used and it gives a bigger nominal efficiency advantage to the single-runner compared to the counter-rotating turbines, which allows a conservative approach in our comparison. 2.4. Design performance model The base for the off-design turbine efficiency remains the same as in the model with Eqs. (1)–(4), but instead of using the dimensionless parameters, the flow angle for the design point is processed and used for the off-design operating points, as suggested by Von Backstrom and Gannon [4]. The outflow angles of a blade are considered as constant, while the other angles are variable over the operating range. The constant angles are 9 a1 ¼ a1ops ¼ arctanðcu1 =fÞ ¼ arctanðð1 R0a þ ca =2Þ=fÞ > = b2a ¼ b2a ops ¼ arctanððR0a ca =2Þ=fÞ . (8) > ; b3 ¼ b3ops ¼ arctanððR0b cb =2ub Þ=fÞ
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0.8 0.7
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0.4 0.3 0.2 0.2 0
0.1 0
(c)
0.2
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1
Ψ
Fig. 3. Influence of different profile loss models on the counter-rotating turbine efficiency: R0a ¼ 1, R0b ¼ 0, ub ¼ 0:5. (a) Constant profile model, (b) Lewis profile loss model, and (c) lift/drag profile loss model.
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Dhops is the turbine specific enthalpy drop andC z ops is the axial flow velocity at the operating point considered and obtained from the thermodynamic model of the solar chimney. To have the best efficiency, the turbine load, i.e. the rotation speed, will be adapted. U ops is variable for all turbine types, i.e., single-runner and counter-rotating turbines, and depends on the current operation point of the chimney. However, in comparison to the single-runner turbine, the counter-rotating turbines offer an additional degree of freedom, which is ub . This allows for an additional optimization of the counterrotating turbine efficiency in all design points. The unknown ub ops is calculated from the following equation for U ops : Dhops C z ops ¼ ðtanða1 Þ þ ðub ops 1Þ tanðb2a Þ U ops U 2ops ub ops tanðb3 ÞÞ 1 þ ub ops u2b ops .
ð9Þ
The outlet flow angle and the turbine total-static efficiency at the considered operating point are 9 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > a3ops ¼ a cos C z ops = C 2z ops þ ðC z ops tanðb3 Þ þ U ops ub ops Þ2 ; = 1 . 2 C z ops > > ðzs cos2 ða1 Þ þ zra cos2 ðb2a Þ þ zrb cos2 ðb3 Þ þ zd cos2 ða3ops ÞÞ Zts ¼ 1 þ 2Dh ; ops (10) To optimize the efficiency at the current operating point, we choose ub ops for optimization. For the counter-rotating turbine type without guide vanes, the conditions applied to the above equations are cu1 ¼ 1 R0a þ ca =2 ¼ 0 and zs ¼ 0. 3. Solar chimney model 3.1. Thermodynamics To evaluate the off-design performance of the turbine, the operational characteristic of the solar chimney is needed. This is accomplished by using a reversible thermodynamic model on a solar chimney–gas turbine cycle used in [3]. The model gives satisfactory results when compared to those in [1], except the nominal output power is higher. This is expected since no losses have been considered in the reversible thermodynamic model. To obtain accurate values in our simulation, a proportionality coefficient is introduced to match the design operation point in [1]. The mass flow rate and power at different values of solar irradiation G for a 5 MW solar chimney are calculated and presented in Fig. 4. These results are obtained by using the proportionality coefficients: the turbine load is adjusted to _ the current operating point, so that the specified maximum efficiency pressure drop and m is matched. This technique produced exactly the same results as those for the three power plants in [1]. We note that the thermodynamic model used does not include any heat storage systems in the solar collector, but it is possible to see its influence by modifying the initial values of G
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5
10
4
8
2
P (MW)
dm/dt (kg/s)
3 6
4 1 2
0
0 0
200
400
600 G (W/m2)
800
1000
Fig. 4. Mass flow and power output in a 5 MW solar chimney.
4. Solar radiation The operating characteristic of the plant is determined by the solar radiation G, which may be calculated using the radiation model by Bird and Hulstrom [13]. The simulation was done for the location Mildura, Australia (341S, 1411E), using following equation: Z
24
Pannual ¼ 365
Pplant ðGðtÞÞZts ðGðtÞÞdt:
(11)
1
Pplant ðGÞ is a function of the thermodynamic model of the solar chimney plant and Zts ðGÞ is the off-design performance function of the turbine efficiency model.
5. Results and discussion A computer program has been developed on Matlab platform using the models presented in the previous sections. Three toolboxes have been developed, and implemented in one GUI: the first is on solar chimney model and computing air flow data for the turbine system; the second is on choosing the profile loss model, choosing the operation point for the different turbine types, counter-rotating and single-runner, and optionally using optimization algorithms for best nominal efficiency; and finally, the third is on off-design performance analysis of the chosen turbine type for the selected solar chimney plant. To study different turbine systems, a parameter K 0 is introduced, which characterizes the turbine operation point at a certain output power, with a chosen mass flow rate and axial
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Table 1 Characteristics of solar chimney plants and their nominal values of K0 Plant
5 MW
30 MW
100 MW
Chimney height (m) Collector diameter (m) Chimney diameter (m) Nominal velocity (m/s) K0
445 1110 54 9.1 2.6111
750 2200 84 12.6 2.4607
950 3600 115 15.8 2.2413
flow velocity. It is K0 ¼
c P Dh ¼ ¼ . f2 C 3z Ar C 2z
5.1. Scenarios Following [1], three different solar chimneys with nominal powers of 5, 30, and 100 MW will be considered. K 0 values corresponding to these plants are given in Table 1. Three turbine types will be considered as detailed below:
the reference single-runner turbine, with a stator and a rotor (SRT), the counter-rotating turbine with a stator and two rotors (CRT), and the counter-rotating turbine with two rotors and without a stator (CRTa).
5.2. Mechanical considerations Three layouts will be considered as shown in Fig. 5. The base configuration is the singlevertical-axis turbine. The first layout is the counter-rotating turbine installed as in a singlevertical-axis turbine case. Two other designs are six-vertical-axis turbines, and 36horizontal-axis turbines installed at the periphery of the chimney. The specific enthalpy drop at the turbine is the same in all cases and overall mass flow and density are also independent from the turbine layout. The axial velocity can be expressed as _ C z ¼ ðm=rAÞ / ð1=AÞ, where A is the total flow through area. Using trigonometric relations to calculate the flow area for the six-turbine case shows that total flow through area is 66% of the chimney cross-section area, while for the classical single-turbine case, the value is 80% since a diffuser is used. The axial velocity for the six-turbine design increases by 20% relative to the SRT. It is more difficult to evaluate the total flow area for the 36-turbine design, as it strongly depends on the lower chimney and collector outflow geometrical designs. To perform a comparison, an example of the 36-turbine configuration with a total flow through area of 78% is used. Consequently, we have K 06-turb ¼ 1:44 K 01-turb and K 036-turb ¼ 1:04 K 01-turb . For a constant K 0 , the same nondimensional turbines will have different rotation speeds. For design purposes, it is preferable to choose U, the mean peripheral velocity, as a constant design value, i.e. f ¼ constant, but o ¼ U=2pr will be higher for smaller turbines
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Fig. 5. Proposed turbine layouts. Left: single counter-rotating vertical axis turbine. Center: 36-horizontal-axis counter-rotating turbine. Right: six-vertical-axis counter-rotating turbine.
where r is the mean radius of the turbine blade. The torque on the turbine is expressed as _ which will also vary. For a constant flow area through the collector and M ¼ cU mr, chimney base, and a single turbine occupying 80% of the chimney cross section, the influence of the turbine layout on the rotation speed and torque are computed and we found that ox-turb =o1-turb ¼ 1; 2:68 and 4, and M x-turb =M 1-turb ¼ 1; 0:062 and 0:00694, where the subscript x is for single-, 6- and 36-turbine layouts, respectively. It is noticed that for the same optimal design point, the mechanical loads will decrease considerably. However, this will apply to all turbine types considered, so that it is not a factor in the efficiency comparison.
5.3. Torque comparison with SRT, CRT, and CRTa turbines To lower the output torque, a higher rotation speed may be chosen at the optimum efficiency point. As discussed earlier, at the optimal design point, the torque on the output shaft is very high. This is mainly caused by the solar chimney characteristics, namely an operation at low air flow speeds. To choose a more appropriate turbine design point, possibilities to reduce the high torque should be studied. One possibility would be to add a condition on the load factor c . The torque M on an axis delivering the power P at a rotation speed o is M ¼ P=2po. Hence, M¼
pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi _ _ _ zc rP rmDh U rmcC rmC z _ . _ z K 0 c ¼ r Pmc ¼ ¼ pffiffiffiffiffiffiffiffiffiffiffi0 ¼ rmC Cz ¼ 2 U f U Cz c=K
(12)
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The power, turbine mean radius, and mass flow rate are constant as they depend on the solar chimney characteristics. The torque is dependent on the square root of the load factor c. Hence, the possibility to influence the torque is low by modifying the load factor. This is especially true for SRTs. As will be shown later, the optimal load factor for SRTs is about 0.6 whereas for the counter-rotating ones, it is 1.6. Choosing the same load factor for the counter-rotating turbine as for the single-runner ones penalizes the nominal efficiency and torque repartition between two runners. In fact, using the data for the 5 MW plant, torques are computed for three types of turbines and the results showed that this was indeed the case. As expected, the torque reduction was minor, although it was more significant for the counter-rotating turbine without inlet guide vanes. Also, the torque on the second runner of the counter-rotating turbine dropped significantly, while the torque of the first runner approached values of a single-counter-rotating turbine. This is expected since choosing the same load factor causes the optimal design for the first runner to be similar to the single turbine runner. Hence, the second runner provides very little power. However, these observations do not apply to the counter-rotating turbine without guide vanes. It is noted that as seen before, the torque reduction with a multiple-turbine layout is much more significant than by choosing a specific turbine design point. Choosing a multiple-turbine layout allows us to design all turbine types at their optimal nominal efficiency.
5.4. Nominal turbine efficiency We obtained the attainable maximum turbine efficiency and configuration in each of the considered plant sizes. The results were obtained using the profile loss coefficients by Lewis. It was seen that with decreasing K 0 , the optimum SRT load factor c decreased as the load factor for both counter-rotating turbine increased. The results obtained using the lift/drag profile loss model showed that the optimal load factor decreased for all turbine types with decreasing K 0 and the nominal efficiency of the CRT was higher than the SRT. In this case, the CRTa showed the lowest efficiency. Generally, the calculated efficiencies were lower than those calculated with the profile loss relations by Lewis. With the lift/drag profile loss equations, the CRT has improved design efficiency by 0.09% compared to the SRT. On the other hand, if the profile loss equations by Lewis are considered as more accurate, the counter-rotating turbines do not have a design efficiency advantage over the SRTs. It also appears, as expected, that the optimum efficiency of the counter-rotating turbines is at a higher load factor than that of the singlerunner type.
5.5. Off-design performance As the results from the off-design performance model showed, the counter-rotating turbines are expected to perform better than the single-runner turbines. Therefore, to evaluate if the CRT and CRTa are able to operate with a better performance, the most disadvantageous profile loss model, which is by Lewis, will be used in the following computations.
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5.6. Efficiency Three different turbines are being considered, the single runner turbine (SRT), the counter-rotating turbine with inlet guide vanes (CRT), and the counter-rotating turbine without inlet guide vanes (CRTa). At an optimum turbine configuration, the total-static maximum efficiencies, Zts;max , are computed for SRT, CRT, and CRTa at 5, 30, and 100 MW plants. For 5 MW case, K 0 ¼ 2:6111, they are Zts;max ¼ 84:22%, 84.01%, and 84.10%; for 30 MW case, K 0 ¼ 2:4607, Zts;max ¼ 83:56%, 83.33%, and 83.44%; and for 100 MW case, K 0 ¼ 2:2413, Zts;max ¼ 82:47%, 82.20% and 82.35%. Counter-rotating turbine efficiencies over the operating range for the three plants are presented in Figs. 6–8. It can be observed that the nominal efficiency of the counterrotating turbines is lower than for the SRT, but as the load decreases, the efficiency drop is slower for both counter-rotating turbines, the axial inlet version being a little more efficient. With half load at G ¼ 500 W=m2 , the efficiency advantage of the counter-rotating turbine is already 1%. The reason for this better performance is the adaptable rotation speed of the second turbine runner as discussed earlier. Indeed, at an off-design operating point, the absolute exit velocity is not purely axial as in the design point of the turbines. With the SRT, this additional energy is lost while with the counter-rotating turbines, a part of it can be recovered by choosing an adequate ub ops . In fact, tracing the absolute exit velocity angle as a function of solar radiation showed that the absolute exit velocity angle for the SRT case decreased almost linearly from a high
0.85 CRT CRTa SRT
0.84
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0.82
0.81
200
400
600
800
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G (W/m2) Fig. 6. Turbine efficiencies over operating range for 5 MW nominal power.
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0.84
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CRT CRTa SRT
η
0.83
0.82
0.81
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200
400
600 G (W/m2)
800
1000
Fig. 7. Turbine efficiencies over operating range for 30 MW nominal power.
0.83 CRT CRTa SRT
0.82
η
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0.8
0.79
200
400
600
800
1000
G (W/m2) Fig. 8. Turbine efficiencies over operating range for 100 MW nominal power.
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Table 2 Annual energy output using Lewis profile loss relations Pnom (MW)
Pannual SRT (GW h)
Pannual CRT (GW h)
(PSRT/PCRT)1 (%)
Pannual CRTa (GW h)
(PSRT/PCRT)1 (%)
5 30 100
14.91 88.56 290.5
14.91 88.54 290.3
0.02545 0.0202 0.0810
14.93 88.66 290.8
0.1386 0.1191 0.1018
value at G ¼ 100 W=m2 to zero at G ¼ 1000 W=m2 while for the counter-rotating turbines, CRT and CRTa, it was almost zero at the same range. This confirms that the counterrotating turbines may perform better at off-design conditions, except at nominal design point at G ¼ 1000 W=m2 . It is noted that in all three different nominal power configurations, the turbine efficiencies are similar.
5.7. Annual energy output Combining all mathematical models presented earlier allows us to compute the annual energy output for each turbine type, which provides data for evaluating the real advantage or disadvantage of every turbine type considered. Table 2 shows results for all three plant sizes and turbine types. We note that the annual energy production prediction corresponds accurately to those published in [1]. The relative difference is 1% for the 5 and 30 MW plants, whereas it is 5% for the 100 MW plant. This is expected since the introduced proportionality factor is based on the published results. However, it shows the accuracy of the combination of both the solar chimney and the solar radiation model as the proportionality factor accounts only for the nominal design point. An increase by 0.14% on the annual energy production with the axial inlet counterrotating turbine is achieved. However, the advantage is lower when the plant size increases due to two reasons:
the initial nominal efficiency advantage of SRT on CRT is higher for bigger solar chimney plants and simulation by the thermodynamic model of different plant sizes using the same constant parameters is not exactly satisfactory. A detailed simulation code would be a better tool to reveal the advantages of CRT in larger power plants.
For comparison, the annual energy output advantage using the lift/drag profile loss relations is computed for 5 MW nominal power plant: the total-static nominal efficiency is 83.73% for the SRT, 83.81% for CRT, and 83.46% for CRTa; the annual energy produced by CRT is 0.78% higher than that by SRT and that by CRTa is 0.36% higher than by SRT. We see that the nominal efficiency of CRT is higher than that of SRT and, also in this case, the CRT offers a better performance than the CRTa.
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6. Conclusions As expected, it appears that the counter-rotating turbines offer their best efficiency at higher load factors than the single-runner turbines. On the other hand, the simple nominal efficiency model, containing many uncertainties, namely the profile loss model, does not allow any conclusion on the nominal operation advantage of counter-rotating turbines over the single-runner type. The main advantage of this turbine type in the solar chimney systems is its off-design performance. By considering that solar chimney power plants will be operated most of the time at a solar intensity of less than 800 W/m2, the counterrotating turbine systems will be advantageous with respect to single-runner systems from the efficiency point of view as well as annual electric energy production. One other advantage is the reduced torque on each axis compared to the single-runner turbine. An operational improvement has also been shown. Acknowledgement Financial support by Natural Sciences and Engineering Research Council, Canada, is acknowledged. References [1] Schlaich J. The solar chimney: electricity from the sun. Stuttgart: Axel Menges; 1995. [2] Enviromission, Australia, 2004. [3] Gannon AJ, von Backstrom TW. Solar chimney cycle analysis with system loss and solar collector performance. J Solar Energy Eng 2003;122(3):133–7. [4] Von Backstrom TW, Gannon AJ. Solar chimney turbine characteristics. Solar Energy 2004;76(1–3):235–41. [5] Numaki F. On the two-stage propeller pumps. Technical report no. 3, vol. 9. Tohoku Imperial University; 1930. [6] Kieves H. Untersuchungen an zwei Kaplan-Tu¨rbinen fu¨r grosse Fallho¨hen. Dissertation, Karlsruche Technische Hochshule; 1966. [7] Ozgur C, Nathan GK. Study of contrarotating turbines based on design efficiency. J Basic Eng 1971; 93:395–404. [8] Peterson GN. Ducted fans, high efficiency with contra-rotation. Technical report ACA-10, Australian Council for Aeronautics; 1944. [9] Airliners.net, www.airliners.net; 2004. [10] Louis JF. Axial flow contra-rotating turbines. In: Proceedings of the 30th international gas turbine conference and exhibit. New York: ASME; 1985. [11] Cai R, Wu W, Fang G. Basic analysis of counter-rotating turbines. In: Proceedings of the international gas turbine and aeroengine. New York: ASME; 1990. [12] Lewis RI. Turbomachinery performance analysis. Arnold; 1996. [13] Bird RE, Hulstrom RL. A simplified clear sky model for direct and diffuse insolation on horizontal surfaces. SERI technical report no. SERI/TR-642–761, Solar Energy Research Institute, Golden, CO; 1991.
Renewable Energy 31 (2006) 1873–1891 www.elsevier.com/locate/renene
Counter-rotating turbines for solar chimney power plants F. Denantes1, E. Bilgen Ecole Polytechnique, C.P. 6079 Centre Ville Montreal Qc, Canada H3C 3A7 Received 5 May 2005; accepted 13 September 2005 Available online 20 October 2005
Abstract An efficiency model at design performance for counter-rotating turbines is developed and validated. Based on the efficiency equations, an off-design performance model for counter-rotating turbines is developed. Combined with a thermodynamic model for a solar chimney system and a solar radiation model, annual energy output of solar chimney systems is determined. Two counterrotating turbines, one with inlet guide vanes, the other without, are compared to a single-runner system. The design and off-design performances are weighed against in three different solar chimney plant sizes. It is shown that the counter-rotating turbines without guide vanes have lower design efficiency and a higher off-design performance than a single-runner turbine. Based on the output torque versus power for various turbine layouts, advantageous operational conditions of counterrotating turbines are demonstrated. r 2005 Elsevier Ltd. All rights reserved. Keywords: Solar chimney; Counter-rotating turbine
1. Introduction A solar chimney is a simple system to convert energy provided by solar radiation into electrical energy. It consists of three main subsystems, which are the collector, the chimney Corresponding author. Tel.: +1 514 2340 4711x4579.
E-mail address: [email protected] (E. Bilgen). Present address: Fakulta¨t fu¨r Maschinenwesen, Technische Universita¨t Mu¨nchen, BoltzmannstraXe 15, Garching, Germany. 1
0960-1481/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2005.09.018
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Nomenclature Ac c C CRT CRTa CD CL Cz G g Hc Dh K K0 _ m P p q R0a R0b SRT T U ub w W
chimney cross-section area dimensionless absolute velocity (c ¼ C=U a ) absolute flow velocity counter-rotating turbine with stator and two rotors counter-rotating turbine without stator and two rotors drag coefficient lift coefficient axial flow velocity solar radiation intensity terrestrial gravity chimney height specific stagnation enthalpy difference head coefficient parameter of turbine operation point mass flow rate power pressure heat per unit mass geometric factor for runner a ðR0a ¼ 1 ð1=2Þðcu2 þ cu1 Þ ¼ ð1=2Þðwu2a þ wu1a ÞÞ geometric factor for runner b ðR0b ¼ ub ð1=2Þðcu3 þ cu2 Þ ¼ ð1=2Þðwu3b þ wu2b ÞÞ single-runner turbine with stator and rotor temperature peripheral velocity dimensionless velocity of runner b ðub ¼ U b =U a Þ, ub is negative dimensionless relative flow velocity relative flow velocity
Greek letters a b g e zs zra zrb zd f c ca cb o
absolute outlet flow angle, collector heat absorptance relative outlet flow angle, collector total loss coefficient correction factor lift-to-drag ratio of turbine blade (g ¼ CL/CD) flow angle change through blade stator pressure loss coefficient ðzs ¼ DPsLoss =ð1=2ÞrC 21 Þ pressure loss coefficient for rotor a ðzra ¼ DPraLoss =ð1=2ÞrW 22a Þ pressure loss coefficient for rotor b ðzrb ¼ DPrbLoss =ð1=2ÞrW 23b Þ diffuser pressure loss coefficient ðzd ¼ DPdLoss =ð1=2ÞrC 23 Þ flow coefficient ðf ¼ C z =U a Þ load coefficient ðc ¼ Dh=U 2a ¼ ðU a ðC u1 C u2 Þ þ U b ðC u2 C u3 ÞÞ=U 2a ¼ ca þ cb Þ load coefficient for runner a ðca ¼ cu1 cu2 Þ load coefficient for runner b ðcb ¼ ub ðcu2 cu3 ÞÞ rotation speed
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Subscripts a b coll D L ops dLoss raLoss rbLoss s sLoss turb ts tt u 1 2 3
runner a runner b collector drag lift operating point loss diffuser loss rotor a loss rotor b stator (or guide vane) loss at stator turbine total-static total-total projection of the considered entity on the peripheral axis position (or gas state) between the stator and runner a blades position between runners a and b position at the exit of runner b (at diffuser inlet)
and the turbine (see, e.g., [1]). Air is heated by solar radiation under a circular transparent roof open at its periphery. Suction from the chimney draws hot air from the collector, and cold air comes in from the outer perimeter. The energy contained in the air flow is converted into mechanical energy by air turbines at the base of the chimney, and into electrical energy by conventional generators. This system can also operate overnight using heat storage systems on the ground, thus annual operating hours can be as high as 8423–8723 for large power plants [1]. In large power plants, an improvement of overall conversion efficiency results in considerable energy gains as well as economic benefits. For example, following the solar chimney power plant of 200 MW conceived in Mildura, Australia [2], 1% efficiency gain means 2 MW nominal power increase or for a plant operating 3000 h/year, 6 106 kW h extra electric energy. Therefore, it would be interesting to study major subsystems of the solar chimney plant and determine potential efficiency improvement possibilities. One such subsystem is the turbine. The small prototype in Manzanares, which had a 50 kW nominal power, had a singlevertical-axis turbine with four blades made of fiber-glass hard-foam sandwich shells. Recently, the air turbine for solar chimney power plants was studied in detail [3,4]. The single-vertical-axis air turbines in these studies consist of (i) inlet blades that give the air flow a swirl, (ii) the turbine, and (iii) a diffuser. A turbine efficiency equation was developed with the dimensionless parameters, which are reaction degree R, flow coefficient f, load coefficient c, and the pressure loss coefficient zi for each of the three turbine subsections. This efficiency equation was then used to find an optimum design point. Von Backstrom and Gannon [4] used as an example a solar chimney plant of 200 MW nominal
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power and showed the following: (a) The optimum reaction degree Ropt is expressed, showing that, because the static pressure at turbine diffuser exit is important, a turbine with low or no exit whirl will have a better efficiency. This explains the necessity of introducing inlet swirl with the stator. (b) Trying to optimize a turbine for a pre-designed chimney leads to a relatively high efficiency, which implies a high c and a low f. As a result, the optimized turbine would have a slow rotation speed, implying high torque values, accompanied by lower air flow velocities than in the design point of the chimney. In turn, this means an increased turbine cross-section area. A constraint on a minimum f value is proposed. (c) The optimization of the turbine for fixed values of R and f shows that total-static efficiencies of 85% are possible, assuming that the pressure loss coefficients are low at 0.03 due to low flow angle change through turbine blades e. To improve turbine efficiency in these plants, a counter-rotating turbine system may be utilized. Conception and analysis of counter-rotating hydraulic machines show that the efficiency of counter-rotating turbines may be better than single-stage turbines in various operating conditions [5–7] and ducted fans [8]. Indeed, in the aviation industry, the concept of counter-rotating fan blades was analyzed and a prototype of an un-ducted fan engine (GE-36) was built and tested. Similarly, Antonov used counter-rotating propfan engines on aircraft An-70 and An-22. Tupolev also successfully used this type of engines on his TU-95 and TU-24 [9]. The main advantage in these applications was to reduce the propeller size and generated swirl. A counter-rotating turbine consists of several (at least two) co-axial counter-rotating blade rows and/or a stator row in front or behind the counter-rotating blade rows. Radial flow counter-rotating turbines do exist, but for the solar chimney only axial flow turbines will be considered. The main advantages of a counter-rotating turbine are the relatively low rotation speeds for high relative velocities, and thus large flow angle changes [10]. In addition, stator vanes can be eliminated, hence eliminating the resulting pressure losses. Cai et al. [11] presented different types of counter-rotating turbines, with different inlet and outflow conditions and impulsive or reactive rotating blade row. They considered in a first step a general counter-rotating turbine stage, explaining that there are five independent variables that describe a stage with two counterrotating rows. However, for better comprehension, only 11 particular configurations were analyzed more thoroughly. These configurations were compared with usual single-rotating stages. The single-runner turbine efficiency used by Von Backstrom and Gannon [4] is 0 for a load coefficient of c 0:8 at Zts 0:85 The load coefficient h¯ used by Cai 0 0 et al. [11] is defined such that h¯ ¼ c=2, thus h¯ 0:4: The calculated efficiency and load factor are not completely in accordance with their Figs. 5 and 6, but for a similar or better efficiency, the load factor in the counter-rotating turbine will be higher, which reflects the low rotation speed of counter-rotating stages. A counterrotating turbine with axial inlet and outlet flow can have a total-static efficiency of around 85%. The literature review shows that depending on the application, counter-rotating turbines are relatively more efficient conversion devices when compared to classical turbines. Therefore, our aim is to study the counter-rotating turbines in view of an application in solar chimney systems, to determine preliminary design parameters, and their operating conditions.
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2. Counter-rotating turbines 2.1. Mathematical model Performance equations are developed for counter-rotating turbines, see Fig. 1. The turbine efficiency is Zt ¼
Dh 1 . ¼ Dh þ DhLoss 1 þ DhLoss =Dh
(1)
Using the definition of enthalpy dh ¼ T ds þ ð1=rÞ dp and for an isentropic process, we obtain dh ¼ ð1=rÞdp or DhLoss ¼ DpLoss =r. Thus, the efficiency becomes 1 1 ðDpsLoss þ DpraLoss þ DprbLoss þ DpdLoss Þ . (2) Zt ¼ 1 þ rDh Using the pressure loss coefficients defined in the nomenclature, we obtain Zt ¼
1 U2 1 þ a ðzs C 21 =U 2a þ zra W 22a =U 2a þ zrb W 23b =U 2a þ zd C 23 =U 2a Þ . 2Dh
(3)
Using the variables defined in the nomenclature and after some algebra, we first obtain the dimensionless velocity expressions (see Fig. 1) and then the expressions for the two load coefficients: 1 c ca ¼ ðc 2ub ð1 R0a ub þ R0b ÞÞ; cb ¼ 2ub 1 R0a a ub þ R0b , 1 ub 2 (4) where c ¼ ca þ cb .
Diffuser d
3 Rotor
b Ua
2 Rotor
a
W1a
1 Stator Cc
Ub
Ub C1 W1b
C3 W3
Ua W1
s Cc
Fig. 1. Velocity triangle and nomenclature of a counter-rotating turbine.
C1
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Thus we have an expression for Zt which depends on the dimensionless parameters f, c, R0a , R0b , and ub . To obtain the total-total efficiency from Eq. (3), the diffuser pressure loss coefficient zd should be set to zero. It is possible to reconsider the above analysis for counter-rotating turbine without inlet guide vanes, by setting zs ¼ 0 and c1u¼0 . Hence, the counter-rotating turbine without guide vanes has only four independent variables. Both turbine total-total and total-static efficiencies can be calculated with the above method. For the total-total efficiency, the diffuser loss coefficient zd should be set to zero, while to obtain the total-static efficiency it should be set to 1. 2.2. Profile loss model The efficiency equation developed above requires knowledge of the pressure loss coefficients zr of the various turbine subsections. We will consider the available techniques, which are constant profile loss, profile loss by Lewis [12] and lift/drag ratio profile loss by Ozgur and Nathan [7]. 2.2.1. Constant profile loss model The method is to derive an expression with respect to the considered parameter. A good approximation of the profile losses is zi ¼ 0:05, which corresponds, as will be shown later by Eq. (5), to a 901 flow angle change through the blade. This value represents a safe conservative approximation. Since in our analysis the diffuser design is not considered, the pressure loss coefficient will be kept constant and equal to zd ¼ 0:75, which represents a realistic value for diffusers in solar chimneys (e.g., [4]). 2.2.2. Profile loss model by Lewis Lewis [12] gives a simple expression for profile losses for single-runner turbines, which depends on the flow angle change through a selected blade. It is very accurate given its simplicity: e 2 z ¼ 0:025 1 þ , (5) 90 where e is the flow angle change through the blade, which is es ¼ a1 for the stator, ea ¼ b1 b2a for the runner a and eb ¼ b2b b3 for the runner b. The turbine efficiency calculated using Eq. (5) is slightly higher than the one computed using the constant profile loss, which is expected since the latter was chosen using a conservative value. We note also that although this model was developed for single runners, there is no restriction, in principle, for its usage in other cases, such as counterrotating turbines. 2.2.3. Profile loss model using lift/drag ratio Ozgur and Nathan [7] analyzed an axial outlet counter-rotating turbine without guide vanes and with identical rotation speeds and specific work output for both runners. It is noticed that they are completely different from those by Lewis and they are based on the
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lift/drag ratio g of turbine blades: xs ¼
2cu1 ðc2u1 =4Þ þ f2 c21 gf ðcu1 =2Þ
and xr ¼
2c ðR2 þ f2 Þ . w2exit gf þ R
(6)
These relations modify slightly the attainable efficiencies and their operating point. Also, they are not easy to handle because of their relative complexities. 2.3. Validation and comparison of the loss models For single-runner turbines, the data provided in [4,7] will be used while for the counterrotating turbines, the velocity expressions in [11] will be verified and the plots presented in [7] will be used for comparison in the single-runner turbine case. To compare with their results, which make use of the parameters of flow coefficient f and head coefficient K, the following expression is used: c ¼ 2Kf2 . 2.3.1. Single-runner turbine The model used for the present analysis is built using equations presented in [4], thus the results presented in their Figs. 3 and 4 were exactly reproduced (not shown here). To compare results with the plots provided in [7] for a single-runner turbine with an axial outflow velocity implies R ¼ 1 c=2. The parameter K was introduced as explained above and the total-total efficiency versus f with increasing head coefficient K was plotted. The results with three models, i.e., constant loss coefficients, Lewis loss coefficients, and lift/ drag ratio loss coefficients models were plotted (not shown in figures). We observed that using the lift/drag ratio coefficients, Eqs. (1) and (6) resulted in exactly the same as in Fig. 4 of [7]. This is also expected because the same theory is used for the same case. The other profile loss models give also satisfactory simulation. The constant profile loss model resulted in slightly lower efficiencies whereas the profile loss model by Lewis produced slightly higher efficiencies than those of the reference case. The maximum efficiency with the constant and Lewis models decreases faster with increasing head coefficient than with the profile loss model using lift/drag ratio. However, since the solar chimney turbine operation point corresponds to low heads, this deficiency will not affect the overall performance results in this study. 2.3.2. Counter-rotating turbine Introducing the conditions chosen by Cai et al. [11] leads to the same velocity expressions presented in their article. The only difference is the systematically reversed sign of ub , which is expected since ub is considered positive in their work. As mentioned earlier, Ozgur and Nathan [7] considered a particular counter-rotating turbine stage: it has axial inlet and outlet velocities, equal rotation speeds, equal specific work in each blade row, and does not have a stator. These conditions lead to the following expressions: R0a ¼ 1 ub þ R0b ¼ where U b ¼ 1.
c þ 1 and 4
R0b ¼
1 c þ 2ub 2u2b c ¼ 1, 2 4 ub 1
(7)
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Fig. 2. Counter-rotating turbine efficiencies with different profile loss models: (a) constant profile loss model, (b) Lewis profile loss model, and (c) lift/drag profile loss model.
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There are now two independent variables left, which are f and K. We plotted the totaltotal efficiency as a function of the flow coefficient f with K variable from 1 to 12 as a parameter and presented in Fig. 2. The results of Ozgur and Nathan [7] in their Fig. 3 were exactly reproduced in Fig. 2(c) when the lift/drag ratio model was used. The constant loss model produced results slightly lower and that by Lewis slightly higher as it can be seen in Figs. 2(a) and (b). When compared with the results obtained for the single-runner case, we noticed that the general appearance of Fig. 2 was similar to that for the singlerunner case but the evolution of maximum efficiencies relative to K was reversed. Our observation was that the higher attainable efficiency of the counter-rotating turbines was at different operating conditions, i.e., at low K with respect to high K of the single-runner turbines. Using constant or Lewis loss coefficients does not give, at first sight, good results in agreement with those by using lift/drag ratio as can be seen in Fig. 2. Comparing with the single-runner turbine case showed, however, that the general trend was nevertheless acceptable. The efficiency remains higher with counter-rotating turbines than with singlerunner turbines when the flow coefficients increase. We note also that the results in Fig. 2 show that it is important to compare efficiencies produced with the same profile loss model. 2.3.3. Comparison of the three profile loss models We plotted the total-static efficiency Zts as a function of flow coefficient f with load coefficient c as a parameter for the same conditions but with different loss coefficient models for single (not presented in figures) and counter-rotating turbine systems, presented in Fig. 3. It is seen that the influence of the different loss coefficient models is the same on the overall performance from the simulation model. However, the lift/drag loss model changes visibly the maximum efficiency operating point more than the others. Given the available data, it is not possible to conclude which loss model gives the most accurate representation of the reality. However, the similarities pointed out show that in respect to the comparison between counter-rotating and single-runner turbines, they are more or less equivalent. This is why the presented results are computed using Lewis profile loss model, as it is widely used and it gives a bigger nominal efficiency advantage to the single-runner compared to the counter-rotating turbines, which allows a conservative approach in our comparison. 2.4. Design performance model The base for the off-design turbine efficiency remains the same as in the model with Eqs. (1)–(4), but instead of using the dimensionless parameters, the flow angle for the design point is processed and used for the off-design operating points, as suggested by Von Backstrom and Gannon [4]. The outflow angles of a blade are considered as constant, while the other angles are variable over the operating range. The constant angles are 9 a1 ¼ a1ops ¼ arctanðcu1 =fÞ ¼ arctanðð1 R0a þ ca =2Þ=fÞ > = b2a ¼ b2a ops ¼ arctanððR0a ca =2Þ=fÞ . (8) > ; b3 ¼ b3ops ¼ arctanððR0b cb =2ub Þ=fÞ
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Ψ
Fig. 3. Influence of different profile loss models on the counter-rotating turbine efficiency: R0a ¼ 1, R0b ¼ 0, ub ¼ 0:5. (a) Constant profile model, (b) Lewis profile loss model, and (c) lift/drag profile loss model.
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Dhops is the turbine specific enthalpy drop andC z ops is the axial flow velocity at the operating point considered and obtained from the thermodynamic model of the solar chimney. To have the best efficiency, the turbine load, i.e. the rotation speed, will be adapted. U ops is variable for all turbine types, i.e., single-runner and counter-rotating turbines, and depends on the current operation point of the chimney. However, in comparison to the single-runner turbine, the counter-rotating turbines offer an additional degree of freedom, which is ub . This allows for an additional optimization of the counterrotating turbine efficiency in all design points. The unknown ub ops is calculated from the following equation for U ops : Dhops C z ops ¼ ðtanða1 Þ þ ðub ops 1Þ tanðb2a Þ U ops U 2ops ub ops tanðb3 ÞÞ 1 þ ub ops u2b ops .
ð9Þ
The outlet flow angle and the turbine total-static efficiency at the considered operating point are 9 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > a3ops ¼ a cos C z ops = C 2z ops þ ðC z ops tanðb3 Þ þ U ops ub ops Þ2 ; = 1 . 2 C z ops > > ðzs cos2 ða1 Þ þ zra cos2 ðb2a Þ þ zrb cos2 ðb3 Þ þ zd cos2 ða3ops ÞÞ Zts ¼ 1 þ 2Dh ; ops (10) To optimize the efficiency at the current operating point, we choose ub ops for optimization. For the counter-rotating turbine type without guide vanes, the conditions applied to the above equations are cu1 ¼ 1 R0a þ ca =2 ¼ 0 and zs ¼ 0. 3. Solar chimney model 3.1. Thermodynamics To evaluate the off-design performance of the turbine, the operational characteristic of the solar chimney is needed. This is accomplished by using a reversible thermodynamic model on a solar chimney–gas turbine cycle used in [3]. The model gives satisfactory results when compared to those in [1], except the nominal output power is higher. This is expected since no losses have been considered in the reversible thermodynamic model. To obtain accurate values in our simulation, a proportionality coefficient is introduced to match the design operation point in [1]. The mass flow rate and power at different values of solar irradiation G for a 5 MW solar chimney are calculated and presented in Fig. 4. These results are obtained by using the proportionality coefficients: the turbine load is adjusted to _ the current operating point, so that the specified maximum efficiency pressure drop and m is matched. This technique produced exactly the same results as those for the three power plants in [1]. We note that the thermodynamic model used does not include any heat storage systems in the solar collector, but it is possible to see its influence by modifying the initial values of G
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5
10
4
8
2
P (MW)
dm/dt (kg/s)
3 6
4 1 2
0
0 0
200
400
600 G (W/m2)
800
1000
Fig. 4. Mass flow and power output in a 5 MW solar chimney.
4. Solar radiation The operating characteristic of the plant is determined by the solar radiation G, which may be calculated using the radiation model by Bird and Hulstrom [13]. The simulation was done for the location Mildura, Australia (341S, 1411E), using following equation: Z
24
Pannual ¼ 365
Pplant ðGðtÞÞZts ðGðtÞÞdt:
(11)
1
Pplant ðGÞ is a function of the thermodynamic model of the solar chimney plant and Zts ðGÞ is the off-design performance function of the turbine efficiency model.
5. Results and discussion A computer program has been developed on Matlab platform using the models presented in the previous sections. Three toolboxes have been developed, and implemented in one GUI: the first is on solar chimney model and computing air flow data for the turbine system; the second is on choosing the profile loss model, choosing the operation point for the different turbine types, counter-rotating and single-runner, and optionally using optimization algorithms for best nominal efficiency; and finally, the third is on off-design performance analysis of the chosen turbine type for the selected solar chimney plant. To study different turbine systems, a parameter K 0 is introduced, which characterizes the turbine operation point at a certain output power, with a chosen mass flow rate and axial
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Table 1 Characteristics of solar chimney plants and their nominal values of K0 Plant
5 MW
30 MW
100 MW
Chimney height (m) Collector diameter (m) Chimney diameter (m) Nominal velocity (m/s) K0
445 1110 54 9.1 2.6111
750 2200 84 12.6 2.4607
950 3600 115 15.8 2.2413
flow velocity. It is K0 ¼
c P Dh ¼ ¼ . f2 C 3z Ar C 2z
5.1. Scenarios Following [1], three different solar chimneys with nominal powers of 5, 30, and 100 MW will be considered. K 0 values corresponding to these plants are given in Table 1. Three turbine types will be considered as detailed below:
the reference single-runner turbine, with a stator and a rotor (SRT), the counter-rotating turbine with a stator and two rotors (CRT), and the counter-rotating turbine with two rotors and without a stator (CRTa).
5.2. Mechanical considerations Three layouts will be considered as shown in Fig. 5. The base configuration is the singlevertical-axis turbine. The first layout is the counter-rotating turbine installed as in a singlevertical-axis turbine case. Two other designs are six-vertical-axis turbines, and 36horizontal-axis turbines installed at the periphery of the chimney. The specific enthalpy drop at the turbine is the same in all cases and overall mass flow and density are also independent from the turbine layout. The axial velocity can be expressed as _ C z ¼ ðm=rAÞ / ð1=AÞ, where A is the total flow through area. Using trigonometric relations to calculate the flow area for the six-turbine case shows that total flow through area is 66% of the chimney cross-section area, while for the classical single-turbine case, the value is 80% since a diffuser is used. The axial velocity for the six-turbine design increases by 20% relative to the SRT. It is more difficult to evaluate the total flow area for the 36-turbine design, as it strongly depends on the lower chimney and collector outflow geometrical designs. To perform a comparison, an example of the 36-turbine configuration with a total flow through area of 78% is used. Consequently, we have K 06-turb ¼ 1:44 K 01-turb and K 036-turb ¼ 1:04 K 01-turb . For a constant K 0 , the same nondimensional turbines will have different rotation speeds. For design purposes, it is preferable to choose U, the mean peripheral velocity, as a constant design value, i.e. f ¼ constant, but o ¼ U=2pr will be higher for smaller turbines
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Fig. 5. Proposed turbine layouts. Left: single counter-rotating vertical axis turbine. Center: 36-horizontal-axis counter-rotating turbine. Right: six-vertical-axis counter-rotating turbine.
where r is the mean radius of the turbine blade. The torque on the turbine is expressed as _ which will also vary. For a constant flow area through the collector and M ¼ cU mr, chimney base, and a single turbine occupying 80% of the chimney cross section, the influence of the turbine layout on the rotation speed and torque are computed and we found that ox-turb =o1-turb ¼ 1; 2:68 and 4, and M x-turb =M 1-turb ¼ 1; 0:062 and 0:00694, where the subscript x is for single-, 6- and 36-turbine layouts, respectively. It is noticed that for the same optimal design point, the mechanical loads will decrease considerably. However, this will apply to all turbine types considered, so that it is not a factor in the efficiency comparison.
5.3. Torque comparison with SRT, CRT, and CRTa turbines To lower the output torque, a higher rotation speed may be chosen at the optimum efficiency point. As discussed earlier, at the optimal design point, the torque on the output shaft is very high. This is mainly caused by the solar chimney characteristics, namely an operation at low air flow speeds. To choose a more appropriate turbine design point, possibilities to reduce the high torque should be studied. One possibility would be to add a condition on the load factor c . The torque M on an axis delivering the power P at a rotation speed o is M ¼ P=2po. Hence, M¼
pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi _ _ _ zc rP rmDh U rmcC rmC z _ . _ z K 0 c ¼ r Pmc ¼ ¼ pffiffiffiffiffiffiffiffiffiffiffi0 ¼ rmC Cz ¼ 2 U f U Cz c=K
(12)
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The power, turbine mean radius, and mass flow rate are constant as they depend on the solar chimney characteristics. The torque is dependent on the square root of the load factor c. Hence, the possibility to influence the torque is low by modifying the load factor. This is especially true for SRTs. As will be shown later, the optimal load factor for SRTs is about 0.6 whereas for the counter-rotating ones, it is 1.6. Choosing the same load factor for the counter-rotating turbine as for the single-runner ones penalizes the nominal efficiency and torque repartition between two runners. In fact, using the data for the 5 MW plant, torques are computed for three types of turbines and the results showed that this was indeed the case. As expected, the torque reduction was minor, although it was more significant for the counter-rotating turbine without inlet guide vanes. Also, the torque on the second runner of the counter-rotating turbine dropped significantly, while the torque of the first runner approached values of a single-counter-rotating turbine. This is expected since choosing the same load factor causes the optimal design for the first runner to be similar to the single turbine runner. Hence, the second runner provides very little power. However, these observations do not apply to the counter-rotating turbine without guide vanes. It is noted that as seen before, the torque reduction with a multiple-turbine layout is much more significant than by choosing a specific turbine design point. Choosing a multiple-turbine layout allows us to design all turbine types at their optimal nominal efficiency.
5.4. Nominal turbine efficiency We obtained the attainable maximum turbine efficiency and configuration in each of the considered plant sizes. The results were obtained using the profile loss coefficients by Lewis. It was seen that with decreasing K 0 , the optimum SRT load factor c decreased as the load factor for both counter-rotating turbine increased. The results obtained using the lift/drag profile loss model showed that the optimal load factor decreased for all turbine types with decreasing K 0 and the nominal efficiency of the CRT was higher than the SRT. In this case, the CRTa showed the lowest efficiency. Generally, the calculated efficiencies were lower than those calculated with the profile loss relations by Lewis. With the lift/drag profile loss equations, the CRT has improved design efficiency by 0.09% compared to the SRT. On the other hand, if the profile loss equations by Lewis are considered as more accurate, the counter-rotating turbines do not have a design efficiency advantage over the SRTs. It also appears, as expected, that the optimum efficiency of the counter-rotating turbines is at a higher load factor than that of the singlerunner type.
5.5. Off-design performance As the results from the off-design performance model showed, the counter-rotating turbines are expected to perform better than the single-runner turbines. Therefore, to evaluate if the CRT and CRTa are able to operate with a better performance, the most disadvantageous profile loss model, which is by Lewis, will be used in the following computations.
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5.6. Efficiency Three different turbines are being considered, the single runner turbine (SRT), the counter-rotating turbine with inlet guide vanes (CRT), and the counter-rotating turbine without inlet guide vanes (CRTa). At an optimum turbine configuration, the total-static maximum efficiencies, Zts;max , are computed for SRT, CRT, and CRTa at 5, 30, and 100 MW plants. For 5 MW case, K 0 ¼ 2:6111, they are Zts;max ¼ 84:22%, 84.01%, and 84.10%; for 30 MW case, K 0 ¼ 2:4607, Zts;max ¼ 83:56%, 83.33%, and 83.44%; and for 100 MW case, K 0 ¼ 2:2413, Zts;max ¼ 82:47%, 82.20% and 82.35%. Counter-rotating turbine efficiencies over the operating range for the three plants are presented in Figs. 6–8. It can be observed that the nominal efficiency of the counterrotating turbines is lower than for the SRT, but as the load decreases, the efficiency drop is slower for both counter-rotating turbines, the axial inlet version being a little more efficient. With half load at G ¼ 500 W=m2 , the efficiency advantage of the counter-rotating turbine is already 1%. The reason for this better performance is the adaptable rotation speed of the second turbine runner as discussed earlier. Indeed, at an off-design operating point, the absolute exit velocity is not purely axial as in the design point of the turbines. With the SRT, this additional energy is lost while with the counter-rotating turbines, a part of it can be recovered by choosing an adequate ub ops . In fact, tracing the absolute exit velocity angle as a function of solar radiation showed that the absolute exit velocity angle for the SRT case decreased almost linearly from a high
0.85 CRT CRTa SRT
0.84
η
0.83
0.82
0.81
200
400
600
800
1000
G (W/m2) Fig. 6. Turbine efficiencies over operating range for 5 MW nominal power.
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CRT CRTa SRT
η
0.83
0.82
0.81
0.8
200
400
600 G (W/m2)
800
1000
Fig. 7. Turbine efficiencies over operating range for 30 MW nominal power.
0.83 CRT CRTa SRT
0.82
η
0.81
0.8
0.79
200
400
600
800
1000
G (W/m2) Fig. 8. Turbine efficiencies over operating range for 100 MW nominal power.
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Table 2 Annual energy output using Lewis profile loss relations Pnom (MW)
Pannual SRT (GW h)
Pannual CRT (GW h)
(PSRT/PCRT)1 (%)
Pannual CRTa (GW h)
(PSRT/PCRT)1 (%)
5 30 100
14.91 88.56 290.5
14.91 88.54 290.3
0.02545 0.0202 0.0810
14.93 88.66 290.8
0.1386 0.1191 0.1018
value at G ¼ 100 W=m2 to zero at G ¼ 1000 W=m2 while for the counter-rotating turbines, CRT and CRTa, it was almost zero at the same range. This confirms that the counterrotating turbines may perform better at off-design conditions, except at nominal design point at G ¼ 1000 W=m2 . It is noted that in all three different nominal power configurations, the turbine efficiencies are similar.
5.7. Annual energy output Combining all mathematical models presented earlier allows us to compute the annual energy output for each turbine type, which provides data for evaluating the real advantage or disadvantage of every turbine type considered. Table 2 shows results for all three plant sizes and turbine types. We note that the annual energy production prediction corresponds accurately to those published in [1]. The relative difference is 1% for the 5 and 30 MW plants, whereas it is 5% for the 100 MW plant. This is expected since the introduced proportionality factor is based on the published results. However, it shows the accuracy of the combination of both the solar chimney and the solar radiation model as the proportionality factor accounts only for the nominal design point. An increase by 0.14% on the annual energy production with the axial inlet counterrotating turbine is achieved. However, the advantage is lower when the plant size increases due to two reasons:
the initial nominal efficiency advantage of SRT on CRT is higher for bigger solar chimney plants and simulation by the thermodynamic model of different plant sizes using the same constant parameters is not exactly satisfactory. A detailed simulation code would be a better tool to reveal the advantages of CRT in larger power plants.
For comparison, the annual energy output advantage using the lift/drag profile loss relations is computed for 5 MW nominal power plant: the total-static nominal efficiency is 83.73% for the SRT, 83.81% for CRT, and 83.46% for CRTa; the annual energy produced by CRT is 0.78% higher than that by SRT and that by CRTa is 0.36% higher than by SRT. We see that the nominal efficiency of CRT is higher than that of SRT and, also in this case, the CRT offers a better performance than the CRTa.
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6. Conclusions As expected, it appears that the counter-rotating turbines offer their best efficiency at higher load factors than the single-runner turbines. On the other hand, the simple nominal efficiency model, containing many uncertainties, namely the profile loss model, does not allow any conclusion on the nominal operation advantage of counter-rotating turbines over the single-runner type. The main advantage of this turbine type in the solar chimney systems is its off-design performance. By considering that solar chimney power plants will be operated most of the time at a solar intensity of less than 800 W/m2, the counterrotating turbine systems will be advantageous with respect to single-runner systems from the efficiency point of view as well as annual electric energy production. One other advantage is the reduced torque on each axis compared to the single-runner turbine. An operational improvement has also been shown. Acknowledgement Financial support by Natural Sciences and Engineering Research Council, Canada, is acknowledged. References [1] Schlaich J. The solar chimney: electricity from the sun. Stuttgart: Axel Menges; 1995. [2] Enviromission, Australia, 2004. [3] Gannon AJ, von Backstrom TW. Solar chimney cycle analysis with system loss and solar collector performance. J Solar Energy Eng 2003;122(3):133–7. [4] Von Backstrom TW, Gannon AJ. Solar chimney turbine characteristics. Solar Energy 2004;76(1–3):235–41. [5] Numaki F. On the two-stage propeller pumps. Technical report no. 3, vol. 9. Tohoku Imperial University; 1930. [6] Kieves H. Untersuchungen an zwei Kaplan-Tu¨rbinen fu¨r grosse Fallho¨hen. Dissertation, Karlsruche Technische Hochshule; 1966. [7] Ozgur C, Nathan GK. Study of contrarotating turbines based on design efficiency. J Basic Eng 1971; 93:395–404. [8] Peterson GN. Ducted fans, high efficiency with contra-rotation. Technical report ACA-10, Australian Council for Aeronautics; 1944. [9] Airliners.net, www.airliners.net; 2004. [10] Louis JF. Axial flow contra-rotating turbines. In: Proceedings of the 30th international gas turbine conference and exhibit. New York: ASME; 1985. [11] Cai R, Wu W, Fang G. Basic analysis of counter-rotating turbines. In: Proceedings of the international gas turbine and aeroengine. New York: ASME; 1990. [12] Lewis RI. Turbomachinery performance analysis. Arnold; 1996. [13] Bird RE, Hulstrom RL. A simplified clear sky model for direct and diffuse insolation on horizontal surfaces. SERI technical report no. SERI/TR-642–761, Solar Energy Research Institute, Golden, CO; 1991.