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Solar Energy 98 (2013) 42–48

Numerical analysis of the optimal turbine pressure drop ratioin a solar chimney power plant

Penghua Guo, Jingyin Li ⇑, Yuan Wang, Yingwen Liu

School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China

Available online 14 June 2013

Communicated by: Associate Editor Xinping Zhou

Abstract

In a solar chimney power plant, only a fraction of the available total pressure difference can be used to run the turbine to generateelectrical power. The optimal ratio of the turbine pressure drop to the available total pressure difference in a solar chimney system isinvestigated using theoretical analysis and 3D numerical simulations. The values found in the literature for the optimal ratio varybetween 2/3 and 0.97. In this study, however, the optimal ratio was found to vary with the intensity of solar radiation, and to be around0.9 for the Spanish prototype. In addition, the optimal ratios obtained from the analytical approach are close to those from the numericalsimulation and their differences are mainly caused by the neglect of aerodynamic losses associated with skin friction, flow separation, andsecondary flow in the theoretical analysis. This study may be useful for the preliminary estimation of power plant performance and thepower-regulating strategy option for solar chimney turbines.� 2013 Elsevier Ltd. All rights reserved.

Keywords: Solar chimney power plant; Optimal ratio; Turbine pressure drop; Numerical analysis

1. Introduction

A solar chimney power plant (SCPP) system presentsinteresting possibilities for the large-scale use of solarenergy. The Spanish prototype, which ran continuouslyfrom mid-1986 to early 1989, proved that an SCPP is apractical technology capable of highly reliable operation(Haaf et al., 1983; Haaf, 1984). SCPPs have been consid-ered as feasible options for electricity generating and havebeen the subject of applied research worldwide (Dai et al.,2003; Ketlogetswe et al., 2008; Nizetic et al., 2008; Zhouet al., 2010a; Cao et al., 2011; Hamdan, 2011). The sche-matic of a typical solar chimney power plant is shown inFig. 1. The collector is the element used to heat the in-com-ing air by means of the greenhouse effect. Owing to thechimney effect, a total pressure difference is generatedbetween the chimney base and the ambient, which drives

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http://dx.doi.org/10.1016/j.solener.2013.03.030

⇑ Corresponding author. Tel.: +86 29 13152181528.E-mail address: jyli@mail.xjtu.edu.cn (J. Li).

the turbine installed at the chimney base to generate electri-cal power. Continuous operation can be guaranteed byusing heat storing material beneath the ground, which isheated during the day and releases the stored heat through-out the night. A detailed review of the SCPP by Zhou et al.(2010b) provides a comprehensive and concise picture ofthe research and development of the SCPP technology overthe past few decades.

Efficient conversion of fluid energy to electrical powerdepends primarily on the operation of the turbine. The tur-bine in the Spanish prototype is a shrouded turbine whichbelongs to the category of pressure-based turbine. Beforeand after, the turbine air speeds are about the same, butthe pressure changes significantly. The available total pres-sure difference Dptot can be divided into two parts: the pres-sure drop across the turbine Dpturb and the flow lossesDploss. The variation in solar radiation causes the volumeflow rate through the system to change accordingly. Duringoperation, the turbine blade pitch is then adjusted to regu-late power output according to the changing airflow. This

Nomenclature

Achim section area of the chimney (m2)Acoll area of the collector (m2)cp specific heat capacity (J kg�1 K�1)ft pressure drop ratiofopt optimal pressure ratiog gravitational acceleration (m s�2)G global horizontal radiation (W m�2)P available power for the turbine (k W)Ta ambient temperature (K)V updraft velocity (m s�1)

Greek symbols

b thermal expansion coefficient (K�1)Dpturb turbine pressure drop (Pa)Dptot available total pressure difference (Pa)Dploss flow losses (Pa)DT temperature rise (K)g overall efficiency (%)gcoll collector efficiency (%)gsc chimney efficiency (%)q density of air (kg m�3)

P. Guo et al. / Solar Energy 98 (2013) 42–48 43

then raises the pertinent question of how to maximizepower production by adjusting the pressure drop acrossthe turbine under a specified condition.

To find the maximum power condition, the optimumcombination of the turbine pressure drop Dpturb and vol-ume flow rate _Q must be determined. Researchers definedthe ratio of the turbine pressure drop to the available totalpressure difference as the pressure drop ratio ft = Dpturb/Dptot, first mentioned by Haaf (1984) in his pioneer studyof the SCPP. A turbine pressure drop amounting to 2/3of the available total pressure difference was consideredto be theoretically optimal. However, some researchersusing different analytical methods showed that the assump-tion of 2/3 is proved to be true only when the availabletotal pressure difference is constant (von Backstrom andFluri, 2006; Koonsrisuk and Chitsomboon, 2010; Nizeticand Klarin, 2010). A range of values for the optimal ratiohas been suggested in the literature and is generally consid-ered to be higher than 2/3. Schlaich et al. (2005) reported avalue of about 0.80 and indicated that the optimal ratiodepended on the power plant characteristics. von Backs-trom and Fluri (2006) developed a simple analyticalmethod to estimate the optimal ratio, and a typical valueof 0.9 was calculated using data from the Spanish proto-type. Bernardes et al. (2003) mentioned an optimal value

Fig. 1. Schematic diagram of the SCPP.

as high as 0.97, but they pointed out that this value washard to achieve in reality and using a value between 0.8and 0.9 was recommended. Later, the same research teamconducted theoretical simulations to find the maximumpower point of a reference SCPP using two heat transferschemes. A higher optimal ratio of around 0.9 was foundfor the lower heat transfer coefficients, while higher heattransfer coefficients resulted in a lower optimal ratio of0.8 (Bernardes and von Backstrom, 2010). Nizetic and Kla-rin (2010) also concluded, through a simplified analyticalapproach, that the optimal ratio was in the range of 0.8–0.9.

It is obvious that there is no unified accepted optimalratio for maximum power generation and a range of valueshas been suggested. In this study, a simple, generally appli-cable formula was derived for the evaluation of the optimalturbine pressure drop ratio through an analyticalapproach. In addition, a comprehensive 3D numerical sim-ulation incorporating the radiation model, solar loadmodel and turbine model was carried out for the Spanishprototype. The simulations are employed for verifying theanalytical model and for in detail investigation of the opti-mal turbine pressure drop ratio.

Fig. 2. Variation of the available power for a turbine with the volume flowrate.

44 P. Guo et al. / Solar Energy 98 (2013) 42–48

2. Theoretical analysis of the optimal ratio for maximum

power production

Fig. 2 shows the variation in the available power for aturbine with the volume flow rate. In this figure, the dashedline represents the available total pressure difference, oftenreferred to as the driving force. The solid line denotes theflow losses in the system. The difference between the driv-ing force and the flow losses is then the turbine pressuredrop. For an incompressible flow, the available powerextracted by the turbine is equal to the product of the vol-ume flow rate _Q and the pressure drop across the turbineDpturb. As shown in Fig. 2, the areas of different coloredrectangles denote the available power of the turbine underdifferent volume flow rates.

Zero power is gained under two extreme conditions: (a)When the pressure drop over the turbine becomes verylarge the flow rate and the useful work approach zero.(b) When the pressure drop over the turbine approacheszero the useful work becomes zero. Between these twoextreme cases, there is an optimum turbine blade settingto maximize power generation.

To investigate the optimal ratio for the turbine pressuredrop, it is necessary to analyze the overall efficiency of theSCPP. Generally speaking, the overall efficiency of theSCPP can be defined as follows:

g ¼ gsc � gcoll � ft; ð1Þ

where g, gsc, gcoll, ft are the overall efficiency, the chimneyefficiency, the collector efficiency and the turbine pressuredrop ratio. According to the theoretical analysis in a previ-ous study (Gannon and von Backstrom, 2000; Schlaichet al., 2005), when ignoring the aerodynamic losses, theexpressions for the partial efficiencies of the chimney andthe collector are derived as follows:

gsc ¼Dptot

qcpDT¼ gH

cpT 0

; ð2Þ

gcoll ¼qVAchimcpDT

AcollG; ð3Þ

ft ¼Dptot � Dploss

Dptot

¼qcpDT gsc � 1

2qV 2

qcpDT gsc: ð4Þ

Substituting Eqs. (2)–(4) into the expression for the overallefficiency g, it becomes:

g ¼ gHcpT 0

� qVAchimcpDTAcollG

�qcpDTgsc � 1

2qV 2

qcpDT gsc: ð5Þ

For convenience, the two coefficients C1 and C2 are definedas follows:

C1 ¼qAchim

AcollG; C2 ¼

gHT 0

: ð6Þ

The overall efficiency can then be further written as:

g ¼ C1C2V DT � 1

2C1V 3: ð7Þ

The overall efficiency is a function of the air velocity andtemperature rise in the system. Following the research ofvon Backstrom and Fluri (2006), a simple but useful powerlaw assumption for the relationship between the air veloc-ity and temperature rise is:

DT ¼ C3V m; ð8Þ

Substituting Eq. (8) into Eq. (7), yields:

g ¼ C1C2C3V mþ1 � 1

2C1V 3: ð9Þ

The maximum power available is then found when @g@V ¼ 0:

@g@V¼ C1C2C3ðmþ 1ÞV m � 3

2C1V 2 ¼ 0: ð10Þ

By solving the above equation, the optimal air flowvelocity is then given by:

V opt ¼2

3C2C3ðmþ 1Þ

� � 12�m

: ð11Þ

Substituting Eq. (11) into the expression for the turbinepressure drop ratio in Eq. (4), yields an optimal ratio forthe pressure drop across the turbine:

fopt ¼ 1� V 2�m

2C2C3

¼ 1� mþ 1

3: ð12Þ

The optimal ratio is determined by the value of m. In thespecial case when m = 0, which means the air temperaturerise in the collector is independent of the air velocity in thesystem, the optimal turbine pressure drop ratio is 2/3. Thiscorresponds to the conclusion mentioned in the introduc-tion. However, the temperature rise in the collector is sig-nificantly affected by the system volume flow rate andcan hardly remain constant in practical operation. Boththe turbine pressure drop and solar radiation greatly influ-ence the temperature rise in the collector. Hence, the vari-ation of the value m needs to be investigated further toaccommodate the variation in the daily solar radiationintensity. The Computational Fluid Dynamics (CFD)method has been confirmed as a powerful tool for detailedanalyses of the SCPP. A comprehensive 3D numerical sim-ulation was conducted for predicting the performance ofthe SCPP in the following sections.

3. Numerical method

The simulations were all conducted for steady flow usingthe commercially available CFD package (ANSYS Fluent).

Since this paper mainly focuses on the optimal turbinepressure drop ratio in the SCPP, the ambient domain wasneglected in the simulation, which is also the commonlyadopted approach in the literature for the performance pre-diction of the SCPP (Pastohr et al., 2004; Ming et al., 2006;Xu et al., 2011). On the other hand, the inclusion of theambient domain in the simulation is mainly for the consid-eration of the effect of the ambient crosswind on the SCPPor the flow-induced vibration of the structure by crosswind.

Table 1Boundary conditions and model parameters.

Place Type Value

Bottom of the ground Wall; temperature 300KExternal surface of the collector roof Wall; mixed h = 8 W/(m2 K), e = 0.94Surface of the chimney Wall; heat flux qchim = 0 W/m2

Inlet of the collector Pressure inlet Pi = 0 Pa, Ta = 302 KOutlet of the chimney Pressure outlet Po = 0 PaPressure drop across the turbine Reversed fan DPturb

Fig. 3. Computational grid.

Fig. 4a. Comparison of power output between simulation results andexperimental data.

P. Guo et al. / Solar Energy 98 (2013) 42–48 45

In addition, an ambient domain is usually several times lar-ger than the SCPP in numerical simulation. The number ofthe grid points would be a challenge to the compute capa-bility of the present PC if both the internal and externalflows of the SCPP were included in the simulations.

The heat transfer process in the SCPP system involvesall three phenomena: conduction, convection and radiationand the inclusion of all the three phenomena in numericalsimulations can give better prediction than those neglectingthe radiation phenomenon in the computations, accordingto the author’s investigation. Usually the collector in theSCPP system is regarded as a greenhouse with differenttypes of semi-transparent covering materials such as glassor plastic. In fact, the chimney installed at the center ofthe collector makes it obviously different from a real green-house. In the SCPP, due to the chimney effect, the convec-tion in the collector is greatly enhanced. In such a case, theprocess by which the collector absorbs the outgoing radi-ated energy in the infrared spectrum from the groundand re-emits most of it back towards earth would play akey role in the heat transfer process inside the SCPP. Inprevious numerical studies, however, the radiation heattransfer in the collector is rarely considered. To obtain amore accurate performance prediction of the SCPP, thediscrete ordinates (DO) radiation model was adopted inthis numerical simulation.

Given that the flow inside the SCPP is induced becausethe force of gravity acting on the density variations and hasproved to be turbulent flow (Ming et al., 2008; Xu et al.,2011), the RNG k–e turbulence model was selected takinginto consideration the buoyancy effect. The Boussinesqapproximation was adopted in the simulation which trea-ted the air density as a constant value in all solved equa-tions, except for the buoyancy term in the momentumequation. Thereby computational effort is saved comparedto compressible simulations where the continuity equationis solved.

The solar load model provided by Fluent was adoptedto calculate the radiation effects from the sun’s rays enter-ing the computational domain. The solar radiation wasmodeled using the sun position vector and illuminationparameters. Such a model is an efficient and practicalapproach to applying solar loads as heat sources in theenergy equations and is available for 3D simulation only.

The turbine pressure drop was used as the independentcontrol variable in the power output control strategies. Inthe simulation, a reversed fan model (actuator disk model)

was used to determine the pressure drop across the turbine.The turbine was treated as an infinitely thin disk and thediscontinuous pressure drop across it could be specifiedas a constant value or a function of the velocity throughthe turbine.

The SIMPLEC algorithm was selected as the pressure-velocity coupling scheme. The convective terms were dis-

Fig. 4b. Comparison of updraft velocity between simulation results andexperimental data.

Fig. 5. Variations of updraft velocity and temperature rise with solarradiation and turbine pressure drop.

46 P. Guo et al. / Solar Energy 98 (2013) 42–48

cretized with a second-order-accurate upwind scheme. TheBody Force Weighted algorithm was selected as the discret-ization method for pressure term. Structured grid wasadopted and the grid was refined adaptively near walls.The comparisons of the computational results using differ-ent strategies for grid refinement indicated that, the gridsare fine enough to be used to obtain grid-independent solu-tions. The main boundary conditions for the SCPP arelisted in Table 1.

The numerical method was validated using the experi-mental data of the Spanish prototype obtained from the lit-erature (Schlaich et al., 2005). The main dimensions for theSpanish prototype are: the chimney is 194.6 m high and10.16 m in diameter; the collector is 1.85 m high and244 m in diameter. The computational grid was shown inFig. 3. Comparison between the numerical results andexperimental data was carried out to verify the validity ofthe numerical method in this study. As indicated inFigs. 4a and 4b, the simulation results were quite consistent

with the experimental data. More precisely, the simulationresults tended to slightly overestimate the power outputand the updraft velocity. This was mainly because the sim-ulation was conducted for steady flow, and thermal equilib-rium was achieved in the collector for each specifiedcondition, while in reality the soil layer had thermal inertia.

4. Results and discussion

To verify the theoretical analysis and further investigatethe optimal turbine pressure drop ratio, numerical simula-tions were carried out for the Spanish prototype by settingthe solar radiation at 200 W/m2, 400 W/m2, 600 W/m2 and800 W/m2. The updraft velocity in the chimney was alteredby adjusting the pressure drop across the turbine.

Fig. 5 shows the variations in updraft velocity and tem-perature rise with the solar radiation and turbine pressuredrop at the entrance of the chimney. Obviously, under con-stant solar radiation, when the turbine pressure dropincreases, the temperature rise increases while the updraftvelocity gradually decreases. An increase in the turbinepressure drop means more energy is extracted by the tur-bine resulting in a reduction of the updraft velocity. Hence,the heating time of the airflow inside the collector is pro-longed and the temperature rise increases. Meanwhile,when the turbine pressure drop remains constant, boththe temperature rise and updraft velocity increase signifi-cantly with solar radiation.

A power law assumption was adopted in the analysisapproach in Section 2 for the relationship between temper-ature rise and updraft velocity. To verify the assumption,the simulation results for the temperature rise and updraftvelocity were curve fitted using the power law assumptionin Eq. (8), and the fitting curves were plotted in Fig. 6.Obviously, the fitting curve coincides well with the simu-lated results, thus verifying the validity and applicabilityof the power law assumption used in the theoreticalanalysis.

The root mean squared error RMSE (a value closer to 0indicates a better fit) and the coefficient of multiple deter-minations R-square (a value closer to 1 indicates a betterfit) were chosen as the criteria for quality evaluation ofthe fits. The fitting equations for different solar radiationintensities are listed in Table 2.

The RSME and R-square values listed in Table 2 alsoindicate that the curves fit the data points obtained fromthe simulations quite accurately. In addition, the value ofthe exponent m can be obtained from the equations.Clearly, the value of m decreases with the increase in thesolar radiation intensity. However, for a given solar radia-tion, the value of m is barely influenced by the variation inthe updraft velocity in the chimney. According to Eq. (10),the optimal pressure drop ratio fopt can be conveniently cal-culated using the value of m. Hence it can be concludedthat the optimal pressure drop ratio is mainly affected bythe solar radiation for a specified SCPP. During thedynamic control of a real SCPP, the turbine pressure drop

Fig. 6. Simulated results and fitting curves for temperature rise andupdraft velocity.

Table 2Fitting equations for different solar radiation intensities.

Solar radiation (W/m2) Fitting equations RSME R-square

200 DT = 12.92V�0.734 0.01939 0.9998400 DT = 37.00V�0.7678 0.02626 0.9999600 DT = 62.79V�0.7808 0.02634 0.9999800 DT = 89.34V�0.7889 0.02369 1

Fig. 7. Simulated results of the overall efficiency.

Table 3Comparison of the optimal turbine pressure drop ratio between thetheoretical and numerical results.

Solar radiation(W/m2)

m fopt From theoreticalmodel

fopt Fromnumericalsimulation

200 �0.7340 0.9110 0.9134400 �0.7678 0.9226 0.9051600 �0.7808 0.9269 0.8942800 �0.7889 0.9296 0.8782

P. Guo et al. / Solar Energy 98 (2013) 42–48 47

needs to be varied by adjusting the turbine blade pitch orother power-regulating approaches to accommodate thevariation in the intensity of the daily solar radiation.

The overall efficiencies of the SCPP were also calculatedunder different conditions using the simulation results asshown in Fig. 7. Apparently, the overall efficiencies weresignificantly affected by the solar radiation and the turbinepressure drop. It was found that the overall efficiencies ofthe Spanish prototype corresponding to changing turbinepressure drop of the prototype were all less than 0.2%under different solar radiation intensities. The simulated

results are consistent with the conclusions of Ming et al.(2010).

The maximum overall efficiencies under different solarradiation intensities can be obtained from Fig. 7 and theircorresponding pressure drop ratios ft are the optimal pres-sure drop ratios fopt. The comparison of the optimal pres-sure ratio between the theoretical and numerical results islisted in Table 3. Obviously, for the Spanish prototype,all these values for the optimal pressure ratio were around0.9 and in the range suggested by previous researchers. Inaddition, the optimal ratios obtained from the theoreticalmodel were close to those from the numerical simulationwith a maximum relative difference of less than 6%, whichis acceptable for this kind of simplified analysis. Theneglect of the flow losses in the theoretical analysis is themain reason for this deviation. With the increase of solarradiation, the flow losses in the system, which are generallydirectly proportional to the square of the air velocity,increase accordingly. A larger fraction of the available totalpressure difference was then used to overcome the flowlosses and less was used to drive the turbine. This alsoexplains why the optimal ratios from theoretical andnumerical results were closest when the solar radiationwas 200 W/m2, and their difference gradually increasedwith the level of solar radiation.

5. Conclusions

In this study, the optimal ratio for the turbine pressuredrop to the available total pressure difference has beeninvestigated comprehensively using theoretical analysisand 3D numerical simulations. It was found that the opti-mal ratio mainly varied with a change in the intensity ofsolar radiation. Therefore, a turbine with adjustable bladepitch or other power-regulating approaches is recom-mended for the practical operation of a solar chimney sys-tem, to accommodate the variation in the intensity of thedaily solar radiation. In addition, a simplified theoreticalanalysis formula was developed, with a power law assump-tion, for the relationship between the updraft velocity andair temperature rise at the entrance of the chimney. Thevalidity and applicability of the theoretical formula wasverified by comprehensive 3D numerical simulations. Thecomparison results show that the optimal ratios obtainedfrom the theoretical model are close to those from thenumerical simulation with a maximum relative difference

48 P. Guo et al. / Solar Energy 98 (2013) 42–48

of less than 6%, and their deviation was mainly caused bythe neglect of flow losses in the theoretical analysis.

Acknowledgements

This work was supported by “the Fundamental Re-search Funds for the Central Universities” and “the Na-tional Natural Science Foundation of China” underContract No. 51276137.

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