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\title{Mathematical Modeling of Temperature Lapse Rate in Solar Chimney Power Plant}
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             \author[1]{Dr. Shashank.B.  Thakre}

             \author[2]{Shashank  Thakre}

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\date{\small \em Received: 9 December 2012 Accepted: 3 January 2013 Published: 15 January 2013}

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\begin{abstract}
        


A solar chimney power plant consists of a large greenhouse-type collector surrounding a tall chimney. The air, heated within the collector, passes through an inlet guide vane (IGV) cascade and then through a transition section to a turbine that powers a generator. The transition section contains the turbine inlet guide vanes that support the whole chimney and guides the flow entering the turbine. There is a surge in the use of the solar chimney power plant in the recent years which accomplishes the task of converting solar energy into kinetic energy. The existing models are sufficient to accurately describe the mechanism and evaluate the performance of a solar chimney power plant system, in which the effects of various parameters on the relative static pressure, driving force, power output and efficiency. This paper aims at development of mathematical model for obtaining the temperature lapse rate that takes in the solar chimney.

\end{abstract}


\keywords{collector, turbine, chimney, mathematical models, review.}

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\let\tabcellsep& 	 	 		 
\section[{INTRODUCTION}]{INTRODUCTION}\par
he widespread use of solar energy, as an alternate and non depletable resource for agriculture and industry as well as other applications, is dependent on the development of solar systems which possess the reliability, performance and economic characteristics that compare favorably with the conventional systems. The solar chimney power plant system, which is composed of the solar collector, the chimney and the turbine, has been investigated all over the world since the German researcher Jorg Schliaich first made the brainchild in the 1970s. The main objective of the collector is collecting solar radiation to heat up the air inside. As the air density inside the system is less than that of the environment at the same height, natural convection affected by buoyancy which acts as driving force comes into existence. Due to the existence of the chimney, the cumulative buoyancy results in a large pressure difference between the system and the environment, then the heated air rises up into the chimney with great speed. If an axis-based turbine is placed inside the chimney where there is a large pressure drop, the potential and heat energy of the air can be converted into kinetic energy and ultimately into electric energy. 
\section[{II. DESCRIPTION OF THE SYSTEM}]{II. DESCRIPTION OF THE SYSTEM}\par
A typical solar chimney power plant consists of a solar hot air collector, a solar chimney and a turbine with generator. All three essential elements have been familiar from time immemorial. A solar chimney power plant simply combines them in a new way \hyperref[b0]{[1,}\hyperref[b1]{2,}\hyperref[b2]{3]}, as is shown in Fig. \hyperref[fig_0]{1}. Air is heated by solar radiation under a low circular transparent cover open at the periphery; this and the natural ground below it form a hot air collector. In the middle of the cover is a vertical chimney with large air inlets at its base. The joint between the cover and the chimney base is airtight. As hot air is lighter than cold air it rises up the chimney. Suction from the chimney then draws in more hot air from the collector, and cold air enters from the outer perimeter. Thus solar radiation causes a constant up-draught in the chimney. The energy that the hot air contains is converted into mechanical energy by pressure staged wind turbines at the base of the chimney, and into electrical energy conventional generators. The characteristics of this solar chimney power plant are listed below.\par
? Efficient solar radiation use. The hot air collector used in the system, can absorb both direct and diffused radiation. Thus the solar chimney can operate on both clear and overcast days. The other major large-scale solar thermal power plants, which are often driven by high temperature steam generated from solar concentrators, can only use direct radiation. Free dual functions, natural energy storage and greenhouse effect. The collector provides storage for natural energy, as the ground under the transparent cover can absorb some of the radiated energy during the day and releases it into the collector at night. Thus solar chimneys also produce a significant amount of electricity at night. The collector itself can also be used as a greenhouse, which will benefit agriculture production accordingly.\par
? Low operation cost. Unlike conventional power stations, and also other solar-thermal type power stations, solar chimneys do not need cooling water. This is a key advantage in northwestern China where there have already been problems with drinking water. ? Low construction cost. The building materials needed for solar chimneys, mainly concrete and transparent materials, are available everywhere in sufficient quantities. ? Particularly important is that no investment in a hightech manufacturing plant is needed, as both wind turbine and solar collectors are well-developed industrial products. 
\section[{III.}]{III.}\par
MATHEMATICAL MODELS DEVELOPED  {\ref Xinping Zhou et.al., (2009)} developed a theoretical model for evaluating maximum height and optimum height of chimney for maximum power output. The models proposed are as follows.\par
The electric power generated by the turbine generators, Pout, can be expressed as Collector: A simple energy balance across a section of the collector yields the fluid temperature at the outlet of that section. For example, the exit temperature at the first section can be evaluated using. . p u a avg f mc q t t + = (4)\par
The temperature avg f t . so obtained becomes the inlet temperature for the second section and so on. The temperature attained by the air at the end of the 25th section is the temperature of the air entering the chimney section.\par
Chimney: The velocity at the exit of each chimney section can be found using following equation. 
\section[{(}]{(}\par
) z Pe z ga v a v a w ch ch a i ch ch i ch ch ? ? ? ? + = + ? ? ? ? ? sec . 2 sec . 
\section[{( )}]{( )} 
\section[{A Year}]{A Year}\par
The electric power output can be related to the kinetic power by the following equation:max P P gen elec ? = (7)\par
Typical values of the generator efficiency gen ? vary from 75\% to 85\%. A value of 80\% for gen ? is used throughout this analysis. An overall efficiency can be defined for the solar chimney as the ratio of the useful electric power output and the insolation over the area of the collector:coll elec overall Sa P = ? (8)\par
Having developed the models for the analysis, the authors finally concluded that mathematical model capable of predicting the performance of solar chimney systems has been developed. The model is capable of estimating the temperature and power output of solar chimneys as well as examining the effect of various ambient conditions and structural dimensions on the power output.\par
The potential of solar chimney for application in rural areas of developing countries is discussed by Onyango, F.N and Ochieng, R.M.,  {\ref (2006)}. The authors have considered the appropriateness of a solar chimney to rural villages and highlight some features of such a power generating plant. A mathematical correlation was developed for temperature ratio? , and the instantaneous electric power produced by a single turbine P i , for square collector of side L, a circular chimney of radius R and velocity V, at which the air impinges on the rotor blades. The important expressions are as follows. 
\section[{? ? ?}]{? ? ?}? ? ? ? ? ? ? = s m H s T T T T ? (10)\par
Where, ? is the difference between the collector surface temperature and the temperature at the turbine (T s -T H ) to the difference between the air mass temperature under the roof and the collector surface temperature (T m -T s ), Where T s and T H are the temperatures at the surface covered by a selective material and at any position H in the covered area A and h is the convective transfer coefficient.\par
If it is assumed that the area swept by the rotor is the same as the cross-sectional area of the chimney, the instantaneous electric power Pi produced by a single turbine is readily derived as? ? ? ? ? ? = 3 2 2 1 27 16 V R P m i ? ? (11)\par
Where, m ? is the density of the air at temperature Tm and the factor of 16/27 is the ideal limit for the extraction of power. Further can be written as, ? ? ? ? ? ? ? ? ? = ? R H L X P i (14)\par
Thermal and technical analyses of solar chimneys were carried out by Bernardes, M.A. and Weinrebe, A.  {\ref (2003)}. In the research work, an analysis for the solar chimneys was developed, aimed particularly at a comprehensive analytical and numerical model, which describes the performance of solar chimneys. This model was developed to estimate power output of solar chimneys as well as to examine the effect of various ambient conditions and structural dimensions on the power output. Results from the mathematical model were compared with experimental results and the model was further used to predict the performance characteristics of large-scale commercial solar chimneys. The results show that the height of chimney, the factor of pressure drop at the turbine, the diameter and the optical properties of the collector are important parameters for the design of solar chimneys. \hyperref[b6]{Mullet (1987)} presented an analysis to derive the overall efficiency of the solar chimney. The author found that the overall efficiency is directly related to the height of the chimney and he demonstrated it to be about 1\% for the height of 1000m and finally concluded that the solar chimney is essentially a power generator of large scale.\par
The equations developed by the researcher are as follows. Driving force inside the chimney is given by,h K K p ? ? ? ? ? ? ? ? ? = ? 1 0 1 67 . 11 (15)\par
Where, K 0 = Ambient temperature (293 k) K 1 = Air temperature in the chimney. h = Height of the chimney. 
\section[{( ) A Year}]{( ) A Year}\par
Velocity of air flowing inside the chimney is given as,2 / 1 0 2 ? ? ? ? ? ? ? = K Kgh V (16)\par
Where, Î?"K = Temperature rise in K K 0 = Ambient temperature (293 k) Power rating: The mechanical power rating can be obtained as the rate of delivery of kinetic energy at the top of the chimney.2 / 3 0 1 2 ? ? ? ? ? ? ? = K Kgh W ? (17)\par
Where, 1 ? = Density of air at temperature K 1 .\par
W = Mechanical power rating. Overall efficiency; ? 1 is assumed to be 80\% with a reasonable rise in temperature. ? 2 = 0.033 x 10 -3 h ? 3 is assumed to be in the range of 40 to 80 \%. ? = ? 1X ? 2 x ? 3 = 1.07\%  {\ref Zhou et.al., (2007)} conducted experiments on a model having 8 PVC pipes measuring 0.35m radius and 1m height. Eight peaces of pipes were taken to get variation in height of the chimney. In the similar fashion diameter of chimney was made to vary from 1m to 5m. A multiple-blade designed on the operating principle of turbine blade was installed at the base of the chimney. The generator, commercially available, was a permanent magnetism motor with direct current. The updraft drove the turbine, which drove the generator to generate electricity. In the course of experimental measuring, the turbine-generator is under no load conditions. Platimum resistance thermometer sensors (Pt 100) were used to measure hot air temperatures; a mercury thermometer with an accuracy of ±0.5 1C was fixed outside the equipment to measure ambient temperature; a thermal anemometer with an accuracy of ±0.01 m/s was used to measure the velocity of airflow. Measurements were sampled every 10 min. Further a mathematical model based on energy balance has been developed to predict the performance of the solar chimney thermal power generating equipment for different conditions. The models are as follows. 
\section[{(}]{(}\par
)? ? ? = T T T gH V ch / 2 0 (18)\par
Where, V denotes hot air velocity at the solar collector outlet. Power output of the total system can be found as3 3 1 V A P ch w out ? ? = (19)\par
Where w ? is the turbine efficiency, usually between 50\% to 90\%.\par
On the basis of results obtained the researchers finally concluded that the simulated power outputs are basically in agreement with the results calculated with the measurements, which validates the mathematical model of the solar chimney thermal power generating system. Furthermore, based on the simulation and the specific construction costs at a specific site, the optimum combination of chimney and collector dimensions can be selected for a required electric power output.\par
Analytical and numerical investigation of the solar chimney power plant systems was carried out by  {\ref Tengzhen et.al., (2006)}. Different models were developed by the authors for determining static pressure, driving force, power output and efficiency. The authors felt that there is a surge in the use of the solar chimney power plant in the recent years which accomplishes the task of converting solar energy into kinetic energy and also the existing models are insufficient to accurately describe the mechanism. Therefore a more comprehensive model is presented by the authors to evaluate the performance of a solar chimney power plant system, in which the effects of various parameters on the relative static pressure, driving force, power output and efficiency have been further investigated.\par
The density difference may be expressed in terms of the volume coefficient of expansion as The maximum power output can be expressed as the product of driving force and the volumetric flow rate and shows that the air properties, the chimney height, the collector radius and the solar radiation have significant effect on the maximum power output. It can be anal sized from above literature that more emphasis is given in development of models for obtaining velocity of flow through the chimney, power generated by the plant and heat flow through the collector, but significant efforts for determination of adiabatic temperature lapse rate is not done, hence in this paper an attempt is made for obtaining the adiabatic temperature lapse rate with the help of simple mathematical model. 
\section[{IV.}]{IV.} 
\section[{MATHEMATICAL CORRELATION FOR ADIABATIC LAPSE RATE}]{MATHEMATICAL CORRELATION FOR ADIABATIC LAPSE RATE}\par
Adiabatic lapse rate is given by Cp g dz dT = ? ( ) ( ) A Z Z Cp g T T h h + ? = ? ? 1 1 or, ( ) ( ) A Z Z Cp g T T h h + ? = ? 1 1\par
(24) at Z = 0, T h =T 1 where, Z -is the height at which turbine is installed. Therefore,1 1 Z Cp g T T h ? = (26)\par
The temperature of air at any height (Z+h) can similarly be expressed as V. 
\section[{CONCLUSION}]{CONCLUSION}\par
It is concluded that the mathematical models developed by the different researcher are quite useful for predicting the performance of solar chimney power plant. The theoretical mathematical model developed in the present study is quite useful for obtaining the adiabatic temperature lapse rate for a long chimney where the heat flow from the walls of the chimney cannot be measured accurately.\begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Fig. 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-3.png}
\caption{\label{fig_1}}\end{figure}
      			\footnote{© 2013 Global Journals Inc. (US)} 			\footnote{Mathematical Modeling of Temperature Lapse Rate in Solar Chimney Power Plant} 		 		\backmatter  			  				\begin{bibitemlist}{1}
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\end{document}
