A modified model for parabolic trough solar receiver

Transcription

1 American Journal o Engineering Research (AJER) e-issn : p-issn : Volume-02, Issue-05, pp Research Paper Open Access A modiied model or parabolic trough solar receiver M-C. EL JAI, F-Z. CHALQI (Lab. IMAGES. University o Perpignan. France) Abstract: The aim o this paper is to give an original mathematical model that describes the heat exchange between the main components o a thermal solar collector in an Integrated Solar Combined Cycle (ISCC) plant. The obtained model is used to perorm easier simulations o the studied system and gives the temperature evolutions o the heat transer luid and o the metal tube receiver. The model could also be used to optimize the solar collector design according to desired objectives. Keywords: Distributed parameter systems, heat transer luid, modelling, solar parabolic trough collector. I. INTRODUCTION The ight against the problem o climate change caused by pollution o air and water is increasing. It becomes overwhelmingly urgent. This is mainly due to the continued exploitation o ossil uels resources. It is thereore essential to ind a solution allowing production o CO 2 -ree energy to meet our daily and industrial needs. The solar energy is one o the renewable energies. It is ree and especially clean, and can perectly help to solve this problem. The exploitation o this energy would be useul and more advantageous in solar plants by concentrating the sunlight. This energy can be stored as heat energy or 2 hours by using as heat transer luid the molten salt, the stone or the phase change materials. The process o concentrating solar energy can be achieved by a system based on concentration o lenses, or relective mirrors such that the sunrays converge onto a target o a smaller size and located at the ocal plan o this surace. Generally, there are two main methods used to perorm the concentration o solar energy (see Fig. ): Line-ocusing systems: linear concentration. Point-ocusing systems: concentration point. In the irst class, there are two types o solar plant: solar power tower (Big Solar Furnace in Odeillo, France, MW), and solar power o parabolic Dish-Stirling (Dish Stirling prototype plants o 0 kw each in Almeria, Spain). In the second class, we ind two types o solar plant: solar plant o parabolic trough (Nevada Solar One Power Plant, 64MW and Ain Beni-Mathar solar plant northeast Morocco, 472 MW), and solar plant o Fresnel linear collectors ( PE solar plant in Murcia, Spain,.4 MW). Fig. : Dierent methods or solar concentration w w w. a j e r. u s Page 200

2 The thermodynamic solar power plants (known concentration in Fig. 2) use a lot o mirrors that make direct the solar rays to a heat transer luid to be heated at high temperatures. For this reason, the relecting mirrors have to ollow the sun s movement to collect and concentrate a maximum rate o solar radiation throughout the solar cycle used. The heat produced by the heat transer luid will be used to generate electricity using steam turbines or gas. Fig. 2: Dierent types o solar concentration plants [] The technology o thermodynamic solar has current applications as power generation, solar power booster, and generation o steam or industrial processes. In this paper, we are interested in solar plant o parabolic trough and speciically in solar plant which operating with the Integrated Solar Combined Cycle (ISCC). Here we concentrate on the case o cylindroparabolic plants as that developed in Ain Beni-Mathar, located northeast o Morocco. The studied plant uses an integrated solar system combined with a system running a natural gas to produce electricity continuously even in the absence o the sun. This solar plant consists o a solar ield, a solar heat exchanger, two gas turbines, a steam turbine and an air condenser. The solar plant principle can be described as ollows (Fig. 3). The extracted gases rom the turbines are injected in two boilers. The solar energy collected by the trough parabolic mirrors, allows increasing the low o vapor produced in the recovery boilers. An amount o water rom the condenser enters the boiler. When it has been heated to the evaporation point, a part o the water will be led to the solar heat exchanger where it will be heated to the boiling point, evaporated and overheated to return then to the steam generator. It will be re-overheated beore being introducing into the steam turbine o three levels (high, medium and low pressure) [2]. Fig. 3: Principle o Ain Beni-Mathar solar thermal plant [2]. However, the annual eiciency o these solar plants is aected by the instantaneous variations o the weather. The movement o the sun, the clouds, and the wind speed deines these variations. They are observed essentially at the parabolic trough o solar ield. Thereore it is necessary to model the operating o these solar collectors or improving the eiciency o the solar ield and to contribute to a better perormance o the entire solar plant. w w w. a j e r. u s Page 20

3 The paper is organized as ollows. The next section describes the physical interactions o the solar plant components. Based on the energy balance, a general model is established. The various parameters depend strongly on the temperature o the main elements o the collector. However this model which looks simple is very complex or numerical implementation. The next section is devoted to a dierent writing o the model which leads to a look-like logistic model. The new version can be easily implemented and used or design or control problems. It is illustrated by simulation results. II. THE PHYSICAL SYSTEM 2. DESCRIPTION OF THE SOLAR FIELD The solar plant is an Integrated Combined Cycle Thermo-Solar Power plant, located in northeast o Morocco. It consists o 256 parabolic trough solar collectors. These collectors are classiied in 64 parallel loops; each loop is 68 meters long, see Fig. 4. The receiver tubes are located at the ocal axis o the parabolic trough solar collectors. They contain a heat transer luid which temperature can reach 393 C. Fig. 4: The receiver loop [2] The collector used in this solar thermal plant consists o parabolic relectors (a series o mirrors), a metallic structure, a solar tracking system, and receiver tube. This type o parabolic trough solar collectors may have a concentration ratio o about 80%. The mirror is made o borosilicate glass, whose transmittance is approximately 98%. This glass is covered with a layer o silver in its lower part, with a special coating and protection. The best relector can relect 97% o incident radiation. The role o the solar tracking mechanism is adapted to maintain the incident solar radiation perpendicular to the relector. The radiation is relected to the ocal line o the parabola where a receiver tube contains the heat transer luid. The tube receiver or heat collection element (HCE) is o Schott PTR 70 type [3]. It is composed o two concentric tubes. The stainless-steel absorber tube, surrounded by a partially evacuated glass envelope to minimize heat losses, see Fig. 5. The receiver tube contains a heat transer luid which is a synthetic oil (Therminol VP-, [4]). Fig. 5: The solar receiver tube [3] 2.2 FIRST MODELLING APPROACH Modelling parabolic trough solar collectors has been explored by many authors [5, 6, 7, 8, 9, 0]. The modelling principle is based on energy balance between the essential elements o the heat collection element (HCE) which are the receiver tube, and the heat transer luid. The Fig. 6 shows a transversal section o the dierent thermal exchanges [5, 6, 8] between the receiver tube, and the heat transer luid, and their environment. w w w. a j e r. u s Page 202

4 Fig. 6: Scheme o the dierent thermal exchanges o the HCE The main purpose o this modelling is to predict the equilibrium temperature o the luid at the output o the solar ield generated by the low rate at the entrance o the receiver tube. The ollowing hypotheses are considered:. The properties o the luid depend on the temperature. 2. In each section o the tube, the luid low is assumed to be uniormly distributed and equal to a mean 3. The solar radiation and the luid low vary on time and are the same or the whole receiver tube 4. The luid is assumed to be incompressible. The state variables we consider are the temperature o the luid T, the absorber tube T m and the glass envelope T gl. The energy balance or the heat collection element leads to three partial dierential equations o three temperatures. The irst equation describes the luid temperature T. It depends o time t and on space x. The second and third equations describe the absorber tube temperature T m and the glass envelope temperature T gl. The obtained system o equations is then given by: In this system o equations, the amounts o energy were considered and deined rom thermodynamics. These energies are oten heat exchange between the dierent components o the HCE, either by convection or conduction, or radiation, taking into account the impact o the environment [5, 6, 7, 8, 0]. The heat transer luid, lowing inside the metal tube, receives by convection an amount o heat that depends mainly to the characteristics o luid such as the density, the speciic heat, the kinematic viscosity, the thermal conductivity [2] etc. These characteristics are related to the luid temperature. The heat received is given by: Q h A ( T T ) (2) r = m, sur, m, in m The amount o the absorbed solar energy, Q sol,abs, by the parabolic trough solar collector depends on the weather and the cleanliness o the collectors and is deined by: Q = I F(cos ) D sol, abs s e opt (3) The two concentric tubes (metal and glass) o the receiver tube, are exchanging the heat by conduction and by radiation [5, 6, 7, 8]. This heat energy depends on the dierent characteristics o the stainless steel and the glass, and is deined by: where Q in Qin, rad Qin, cond = (4) w w w. a j e r. u s Page 203

5 Q A = sur, m, out in, rad gl m gl 4 Tm T d 2 ke, air ( Tm Tgl ) Q, = (6) in cond d gl, in ln( ) dm, out An amount o thermal energy is exchanged, by convection and radiation [5, 6, 7, 8], between the glass envelope and the environment. This thermal energy is deined by the characteristics o the glass and those deining the ambient air, as ollows: = Q Q (7) where Q d Q out out, rad out, conv 4 gl m, out gl, in 4 4 = A ( T T ) (8) out, rad gl sur, gl, out gl amb out, conv = hgl, amb Asur, gl, out ( Tgl Tamb Q ) (9) Ater replacing in equations () and dividing by ρca or each, we obtain a system o partial dierential equations (PDE) which can be rewritten in the ollowing simple orm (5) where the coeicients a i, b i and c i are given in the annex. The model o the solar trough collector allows the knowledge o the luid temperature evolution only at the output o the receiver tube. Furthermore it could help to choose the parameters o the plant in such a way that the temperature can be maintained at a desired equilibrium value, despite instantaneous variations o the weather. This could be regulated by a convenient low rate at the entrance o the receiver tube. However numerical simulation o the complete obtained model above (0), leads to various diiculties, due to the complexity o the model coeicients which all depend nonlinearly on the temperature. So it becomes necessary to modiy the obtained model or easier simulations. The irst assumption is to assume that a perect vacuum exists between the two concentric metal and glass tubes. In this case we can neglect the glass behavior and thus we obtain a system coupling luid and metal temperatures dynamics. This assumption allows to study the heat exchange between the luid and the metal tube. The irst exchange describes the amount Q g o energy deined in equation (2), while the second exchange is deined by Q abs,sol - Q g - Q m,amb. The amount o energy Q m,amb describes the heat exchange between the metal tube and its environment. This thermal energy is given by : w w w. a j e r. u s Page 204

6 Where, the coeicients βi are dierent rom the bi s and are given in the annex. One can notice that the various coeicients depend on the temperature o the luid or the metal tube. III. MODIFIED MODEL Firstly the model (2) can be easily studied rom mathematical point o view. For that purpose we rewrite it in a matrix orm, by considering the vector z deined by z = easily represented in the ollowing vector orm. Thus the above system can be z = A z B (3) Where A is a matrix o order (2 * 2) and B is a matrix o order (2 * ), deined by a0 a a2 0 x A = and B = (4) b 2 This ormulation shows that the considered system is well posed rom mathematical point o view and, under the condition that the coeicients are bounded, the system admits a unique solution. In what ollows we consider the model (2) and we explore some coeicients which aect dramatically the resolution. In spite o the simplicity o the model (2), it is not consistant with the nature o the system. Furthermore its numerical implementation leads to surprising numerical results. It is diicult to ind in the literature models which can lead to realistic simulations. Most o the models describe very complex physics but lead to non possible simulations. A model can be considered as a ine model i it is simple and can be implemented easily or simulations. That is why we consider a modiied model which could be used by solar plant modelers without complex physical considerations. 3. MODELLING APPROACH The main diiculties in the above model are due to the temperatures sensitivity with respect to the model coeicients. However we have noticed that some o the coeicients depend nonlinearly o the temperature. Various simulations show that the solution evolves dramatically or small variations o certain coeicients. A numerical study o the coeicients leads to a second assumption which consists to neglect the coeicients which value are lower than. On an other hand the coeicients and are slightly linear with respect to the luid temperature as shown in Fig. 7. Fig. 7: Evolution o the coeicients and This suggests to consider a linear description or these coeicients and to reormulate the system under the orm w w w. a j e r. u s Page 205

7 This ormulation can be seen as equations o an equilibrium model where the temperature will be stabilized around a certain value. One can consider other approaches or the coeicients modelization but they do not lead to any improvement o the model. 3.2 SIMULATIONS The obtained model consists in a system o two nonlinear partial dierential equations. It evolves in time t and depends on one-dimensional space x. Formally it derives rom the system (2) and thereore it is well posed rom mathematical point o view. In this section we give a numerical approach or the resolution o (7). The discretization principle or solving boundary-value problems consist o replacing each o the derivatives in the dierential equation by an approximate dierence quotient approximation. The dierence quotient is generally chosen so that a certain approximation order error is maintained. Other methods or solving (inite elements) these equations could be considered. The model (7) is explicitly nonlinear. In this system, we know the mean values or all the remaining coeicients. The range o variation o the temperature can lead to an explicit approximation o the coeicients. Thus we ind the ollowing mean values or the considered coeicients: For the discretization let L be the length o a loop o the HCE and x a step size. Denote N x the number o space paths o length x (see Fig. 8), then. In the considered plant we have meters. For the time horizon, we consider the time step denoted and N t the number o time steps. I we denote t period the time length between the sunrise and the sunset (with a maximum solar lux), then period. Fig. 8: Discretization o the heat collection element w w w. a j e r. u s Page 206

8 The resolution o the reormulated model requires initial and boundary conditions. We assume that the initial known temperature o the luid. We also denote, the initial temperature o the metal tube, assumed to be known. In practical applications we consider TIME EVOLUTION TEMPERATURES The model previously modiied is given in (7) where is the luid temperature and is the metal tube temperature. For the time evolution o the luid temperature at any point o the receiver tube, it is not necessary to consider the space evolution. So we consider irstly that the partial derivative in space vanishes, thus the irst equation o (7) becomes T t x, t a K T T a2 Tm = (8) Fig. 9: Evolution o the luid temperature. In this case we neglect the space variable and denote T n T ( nt) and we apply a modiied Euler method (more accurate) with the initial conditions. Then we obtain the evolution graphs given in Fig. 9 or the luid temperature and in Fig. 0 or the metal tube temperature. Fig. 0: Time evolution o the metal tube temperature. We notice that the metal tube temperature is always higher than that o the luid. For the luid and the receiver tube the igures show that the temperature evolves until an equilibrium level which maintains the luid temperature at about 400 Celsius. These results are consistant with the measurements obtained by the output, under ideal conditions. Remark 3. In the applications depending on the considered plant one can introduce a weighting term ξ to adapt the evolution o the luid temperature (which may depend on physical parameters and day insulation). For that purpose we can rewrite the term considering w w w. a j e r. u s Page 207

9 T a = K T a a The igure () shows the inluence o the coeicient ξ on the time evolution o the luid temperature. Fig. : Evolution o the luid temperature T in time in the cases where the weighting term ξ = 0.9, and ξ = 0.6 The Fig. shows that at the beginning o the time interval, and or a weighting term equal to 0.9, the luid temperature T increases with time to a maximum value ranging rom 352 C to 362 C. Then the luid temperature T decreases to a value o about 324 C. However when the weighting term ξ is equal to 0.6, the luid temperature T increases with time to reach a maximum value o about C and remains maintained at this value. The weighting term ξ can be used in the discretized equation or an adaptation o the evolution to any situation TIME AND SPACE EVOLUTION OF THE TEMPERATURES Introduce now the mesh points o coordinates, or and and let be an approximated value o the temperature (T or the luid and T m or the metal tube) denoted T n j T( jx, nt ) Fig. 2: Element o the distributed parameter model With these notations we illustrate the eiciency o the given model considering the ollowing inite dierence irst order approximations o the derivatives T ( n j jx, nt) = ( T t T n j )/ t and T ( j x, nt) = ( T x n j n Tj )/ x (9) The resolution o the reormulated model requires initial and boundary conditions. The initial conditions have been stated in previous section. Additionally we consider boundary condition at the entrance o the receiver tube given by and assumed to be known. In this case, we have to discretize the principal model in equation (7). Denote and apply the inite dierence discretization in time and space given in (9) (see []), we obtain w w w. a j e r. u s Page 208

10 Fig. 3: Global (time-space) temperature evolution o the luid. For the illustrative simulation, we have considered a time step equal to second and a space step equal to 0. meter. The Fig. 3 presents the obtained results or the luid temperature throughout the receiver tube at dierent times, and Fig. 4 that o the metal tube. Fig. 4: Global (time-space) temperature evolution o the metal tube. The Fig. 3 and 4 show that the temperatures increase rom initial temperature at the entrance o the tube to reach a maximal value o about 400 C or the luid and 600 C or the metal tube. The metal tube temperature reaches its equilibrium value very quickly all along the tube, because all the tube is in the ocal line o the solar receiver and is directly exposed to the sunlight. The nomenclature and the annex given ater would help to achieve the illustrative simulations. Remark 3.2 The modiied model introduced in the previous section has reduced the model to a system o two dierential equations (7). We can notice that the receiver tube is all located in the ocal line o the parabolic trough. Consequently we can consider that the whole receiver tube is excited by the same amount o energy and thus its temperature is equal to a mean value. This is obviously illustrated by the Fig. 4. Additionally the users could be interested only by the luid temperature evolution and not that o the metal tube. This suggests that the luid receives the same amount o energy all along the tube which can be considered as a passive control on the luid. Thereore we can simpliy the model by neglecting the receiver tube equation and considering that the luid is excited by its contact with the metal tube, by a certain amount to be identiied. This allows to consider the ollowing simpliied partial dierential equation which state is the luid temperature T T x, t T x, t a = t 0 x a ~ K T T a T (2) where is assumed to be known. The system is augmented by initial and boundary conditions. We can assume that the mean value is equal to that given by measurements o the metal tube temperature. In this case the various coeicients o the model do not it with the physical values o the previous model. Thus the model can be improved using an identiication o the other coeicients. The ollowing igures show that the results are very signiicant and that the model can be drastically reduced. 2 w w w. a j e r. u s Page 209

11 Fig. 5: Fluid temperature evolution (at a ixed point o the tube) without considering the space impact. The numerical simulations have been achieved considering the ollowing values:,, and. The Fig. 5 shows that the result is quite similar to that o the complex model considered in the irst section. Fig. 6: Fluid temperature evolution, at dierent times, all along the receiver tube. The Fig. 5 and 6 show that the temperature level at the end o the tube is about 375. This temperature level can be adjusted and regulated at any convenient desired level, depending o the real conditions o the trough solar receiver. IV. CONCLUSION In this paper we give an original modiied model o the heat collector element within an Integrated Combined Cycle Thermo-Solar Power Plant. The given approach is more accurate and easier to implement. This model has necessitated the study o heat exchange between the three components o the heat collection element. The reormulation o the model was based by considering a perect vacuum between the two concentric tubes (metal tube inside the glass envelope) o the heat collection element (HCE). The model established consists in two irst order partial dierential equations depending on time and one dimension space variable, and may be reduced to one partial dierential equation. The dierent simulation results show that both the luid temperature T and the metal tube temperature Tm evolve until reaching a certain equilibrium value. The obtained results are consistent with the plant values. The proposed model can be improved by considering an identiication o some o the coeicients. This identiication depends on the considered solar plant and will be explored in a uture work. V. ACKNOWLEDGEMENTS We would like to thank ) Proessor A. EL JAI (University o Perpignan, France) or his advice throughout this work; as well or modelling aspects as or numerical approach o the studied problem. 2) Lorenzo Castro Gómez-Valadés, Process Engineer (ISCC Ain Beni Mathar Morocco), or his kind help and the data o the ISCC plant.. w w w. a j e r. u s Page 20

12 REFERENCES []. S. Andrieux, Le solaire thermodynamique à concentration, 202, -4. [2]. F-Z. CHALQI, Bilan thermique de la central thermo-solaire d Ain Béni-Mathar (ABM), Université de Perpignan Via Domitia, Août 202. [3]. Schott Solar Csp Gmbh, Schott PTR 70 Receivers, Mainz (Germany), [4]. Solutia, Therminol VP-, Louvain-la-Neuve (France), [5]. Thorsten A. Stuetzle, Automatic Control o the 30 MWe SEGS VI Parabolic Trough Plant, University o Wisconsin-Madison, Madison, [6]. R. Forristall, Heat transer analysis and modeling o a parabolic trough solar receiver implemented in engineering equation solver, Technical report, National Renewable Energy laboratory, [7]. N. Hamani, A. Moummi, N. Moummi, A. Saadi and Z. Mokhtari, Simulation de la température de sortie de l eau dans un capteur solaire cylindro-parabolique dans le site de Biskra, Revue des Energies renouvelables, Vol. 0. N 2, 2007, [8]. F. Burkhholder and C. Kulscher, Heat loss testing o Schott s 2008 PTR70 parabolic trough receiver, Technical report (NREL/TP May 2009). [9]. B. Kelly and D. Kearney, Parabolic Trough Solar System Piping Model, Final Report, National Renewable Energy laboratory, [0]. Ming Qu, D. H. Archer and S. V. Masson, A linear Parabolic trough Solar Collector Perormance Model, Renewable Energy Resources and a Greener Future Vol.VIII-3-3, Proceedings o the Sixth International Conerence or Enhanced Building Operations, Shenzhen, China, November 6-9, []. A. El Jai, Eléments d analyse numérique (PUP, 2nd edition, 200). ANNEX : HTF density : Metal (Stainless steel) density : Glass density : HTF speciic heat : Metal (Stainless steel) speciic heat : Glass speciic heat : Inside diameter o the metal tube : Outside diameter o the metal tube : Inside diameter o the glass : Outside diameter o the glass : Noted also. Cross-sectional area inside the metal tube : Cross-sectional area o the metal tube : Cross-sectional area o the glass : Inner surace area per length o the metal tube : Outer surace area per length o the metal tube : Outer surace area per length o the glass envelope : Solar Direct irradiance : Incidence modiier unction θ : Angle o incidence, degree : Eective width o the collector : Optical eicienc: Factor depending o dirt on the mirrors : Eective thermal air conductivity : Ambient temperature : Overall HTF volume low rate : Total number o collectors : Metal (Stainless steel) emissivity : Glass emissivity : Stean-Boltzmann constant : Convection heat transer coeicient between the metal tube and the HTF : Convection heat transer coeicient between the glass and the ambient : Convection heat transer coeicient between the metal tube and the ambient. w w w. a j e r. u s Page 2