OPTIMUM SMALL-SCALE OPEN AND DIRECT SOLAR THERMAL BRAYTON CYCLE FOR PRETORIA, SOUTH AFRICA. W G Le Roux T Bello-Ochende J P Meyer

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1 Proceedings of the ASME th International Conference on Energy Sustainability ES July 3-,, San Diego, CA, USA ES-935 OPTIMUM SMALL-SCALE OPEN AND DIRECT SOLAR THERMAL BRAYTON CYCLE FOR PRETORIA, SOUTH AFRICA W G Le Roux T Bello-Ochende J P Meyer Department of Mechanical and Aeronautical Engineering, University of Pretoria, Pretoria, Private Bag x, Hatfield, 8, South Africa ABSTRACT The open and direct solar thermal Brayton cycle is exposed to various weather conditions like changing solar irradiation, wind and surrounding temperature. The geometries of the receiver and recuperator and the turbine operating point as parameter can be optimised in such a way that they accommodate these weather changes and allow for high power output throughout a typical year. In this paper, a method of obtaining these parameters based on total entropy generation minimisation is presented. A parabolic dish concentrator with a diameter of.8 m is used as well as an off-the-shelf turbomachine. Results show that the absorbed power at the receiver and maximum allowed receiver surface temperature play important roles in determining the optimum operating point and maximum power output.. INTRODUCTION In the solar thermal Brayton cycle, rotational power is generated with a turbine using compressed air heated in a solar cavity receiver. In the solar cavity receiver, heat obtained from concentrating the sun s rays is transferred to the air. The air is used as working fluid in the open Brayton cycle. The air as working fluid makes this cycle very attractive for the waterscarce Southern Africa. The irreversibilities of the recuperative solar thermal Brayton cycle are mainly due to heat transfer across a finite temperature difference and fluid friction. For the solar thermal Brayton cycle with micro-turbine, a modified cavity receiver and a counterflow plate-type recuperator are used as suggested from the literature [, ]. The geometries of these components should be optimised by minimising the total irreversibility rate so that the system can produce maximum power output. In nature, the best organ is not the infinitely large organ, as recommended by the thermodynamics of the organ alone. The best organ is imperfect since it destroys exergy, but the imperfect organ makes the whole animal the least imperfect that it can be [3]. The open and direct solar thermal Brayton cycle s components and operating point can be optimised in such a way that they make the whole system the least imperfect when operating at steady state in any condition. In recent work, a geometry optimisation method based on total entropy generation minimisation was developed and was applied to establish the maximum power output of a smallscale open and direct solar thermal Brayton cycle with cavity receiver and recuperator at any steady-state condition []. This method allows for the global performance of the system to be optimised, by minimising the total irreversibility rate by sizing the receiver and recuperator accordingly, as suggested by various authors [5]. The absorbed heat rate at the receiver was determined as a function of the receiver aperture diameter with the use of a receiver sizing algorithm []. The effects of wind, receiver inclination, rim angle, atmospheric temperature and pressure, recuperator height, solar irradiance and concentration ratio on the optimum geometries and performance of the open and direct solar thermal Brayton cycle were also investigated by considering a single system in more detail [7]. In recent work it was found that the effects of changing solar radiation and wind are the most important and significant [8]. The change in solar irradiance changes the maximum power output of the system as well as the optimum geometries (of receiver and recuperator) and optimum operating point of the micro-turbine. A change in temperature, pressure and inclination, however, has similar consequences but the optimum operating point stays almost constant. Copyright by ASME

2 The open and direct solar thermal Brayton cycle should thus be optimised to operate over a period of time, since the environment in which it is situated and its heat source are functions of time. This can be done by taking into consideration the probability of certain environmental conditions being present throughout the year. The best database for solar irradiance would be the long-term measured data at the site of the proposed solar system [9]. Measured solar radiation is available for Pretoria, South Africa []. In this work different possibilities of handling the changing weather throughout the year are investigated by changing the system geometries or operating point of the microturbine which are optimised with the method of total entropy generation minimisation. Due to the area ratio constraint, the receiver diameter is written as a function of the receiver aperture diameter, D = 3d () The receiver aperture diameter can be calculated using Eq. (3) since A w = D rec L rec. d = D L rec rec π (3). PHYSICAL AND MATHEMATICAL MODEL The open and direct solar thermal Brayton cycle with recuperator is shown in Fig.. A.8 m diameter parabolic dish concentrates the solar heat for the cavity receiver. The specular reflectivity of the concentrator is assumed to be 85%. The rate of intercepted heat by the cavity receiver, & *, depends on the cavity receiver aperture (which depends on the geometries of the cavity receiver). W & is the power output of the system. Air out Compressor 3 Air in Recuperator 5 9 Receiver & * 7 8 Turbine W & = W& W& Load Figure. The open and direct solar thermal Brayton cycle with recuperator... Solar receiver model The modified cavity receiver [] is shown in Fig.. A closely wound copper tube with diameter, D rec, and with length, L rec, through which the working fluid travels, constructs the cavity receiver walls and cavity receiver aperture. The convection heat loss takes place through the receiver aperture with diameter, d. An area ratio of A w A a = 8 is recommended [] as it was found to be the ratio that gives the minimum heat loss from the cavity receiver. Since the surface area of a sphere is πd, the diameter of the spherical receiver can be calculated as ( A w A ) 3π D = + () a t c Figure. Modified cavity receiver. According to [], the total rate of heat loss due to convection and radiation, can be approximated [] as &.9.98 loss.39grd ( + cos β ) ( Tw T ).5 ( d D) ( ka D)( T T ) a w.37.. Determination of absorbed power The absorbed power is the amount of power transferred to the working fluid by the receiver after intercepting the power of the solar radiation and losing power through its aperture and via conduction. For the concentrator with constant diameter, focal length and rim angle, the rate of heat absorbed by the working fluid is a function of receiver aperture diameter, d. At the focal point of a solar concentrator, the reflected rays do not form a point but an image of finite size centred about the focal point. This is due to the sun s rays not being truly parallel and due to concentrator errors. The larger the receiver aperture diameter, the larger the rate of heat intercepted by the receiver, & *. Also, the larger the aperture diameter, the larger the heat loss rate, &, due to convection loss and radiation in Eq. (). The rate of absorbed heat, &, is the intercepted heat rate minus the total heat loss rate. The sizing algorithm [] is applied to determine & * for a specific aperture diameter. The sizing algorithm takes into consideration the concentrator area, rim angle, specular () Copyright by ASME

3 reflectance, inclination, irradiance, parabolic concentrator error and wind and heat loss rate to determine the absorbed heat rate as a function of the receiver aperture diameter. The shadow of the receiver and its insulation is also accounted for when calculating the intercepted heat rate. The absorbed heat rate as a function of receiver aperture diameter, from the sizing algorithm, can be numerically approximated with Eq. (5) using the discrete least squares approximation method [3], where y i is a set of constants used to describe the function. = i = i & y id (5).3. Recuperator model A counterflow plate-type recuperator is used as shown in Fig. 3. The channels with hydraulic diameter, D h,, length, L, and aspect ratio, ab are shown. The number of flow channels in the recuperator, n, depends on the recuperator height, H, channel height, b, and thickness of the channel separating surface, t, and can be written as a function of the channel aspect ratio, H n = H ( t + b) = () t + D b t a ( a b) ) ( a b) h, + ( ) L Figure 3. Counterflow plate-type recuperator. Equation (7) gives the mass flow rate per channel. m& k = m& n (7) The surface area, A s, for a channel as a function of the channel aspect ratio is ( a + b) L = D L ( a b) ( + ) ( + ( a b) ) A (8) s = h, The thickness of the material between the hot and cold stream, t, is mm and the recuperator height is m. The Reynolds number for a flow channel is H ( a b) a Re = m& (9) k Dh, µ Using the definition of the hydraulic diameter and Eq. (9), the Reynolds number can be calculated with Re = µ D ( a b) ( a b) m& ( + ) h, k () Heat exchanger irreversibilities can be reduced by slowing down the movement of fluid through a heat exchanger []. Small Reynolds numbers can thus be expected for the optimised recuperator channels and the Gnielinski equation [5] can be used to determine the Nusselt number, Nu D ( ( )( )).5 ( ) 3 Pr Re f f 8 Pr = () The Petukhov equation [] is used to calculate the friction factor, (.79ln Re.) f = () With the use of the friction factor, Reynolds number and the definition of the pressure drop [7], the pressure drop through the recuperator can be written in terms of the geometric variables as m& k P =.79ln µ Dh, 8m& ρ k ( a b) ( a b + ) ( a b) ( a b + ) 5 ( L D ) h,. (3).. Compressor and turbine properties When considering geometric optimisation, the pressure ratios and mass flow rates can be chosen as parameters [8]. A standard off-the-shelf micro-turbine is used in the analysis [9]. The turbine operating point (turbine corrected mass flow rate and turbine pressure ratio) is chosen as a parameter. Note that the turbine corrected mass flow rate is a function of the turbine pressure ratio as shown on the turbine map. The compressor isentropic efficiency, compressor corrected mass flow rate and compressor pressure ratio are intrinsically coupled to each other and are available from the compressor map. The compressor should operate within its compressor map range, otherwise flow surge or choking can occur. The turbine isentropic efficiency depends on the load []. 3. RESEARCH METHODOLOGY 3.. Variables 3 Copyright by ASME

4 There are five geometric variables to be optimised: The cavity receiver tube diameter, D rec, the tube length of the cavity receiver, L rec, the hydraulic diameter of the recuperator channels, D h,, the length of the recuperator channels, L, and aspect ratio of the recuperator channels, ab. The objective function ( power output of the system) in terms of the scaled geometry variables, parameters and constants is maximised using the dynamic trajectory optimisation method [] in MATLAB. 3.. Selection of most common weather conditions The hourly solar inclination for Pretoria was determined for each day of the year between 7: and 7: []. The average inclination angle was found accordingly. The hourly solar beam irradiance for Pretoria was found with []. The average beam irradiance was found to be 58 Wm. After speculation and considering the maximum solar radiation during summer and winter, it was assumed that the solar inclination is a function of the beam irradiance, I, as shown in Eq. (): ( 9) β = sin I () The weather conditions assumed most likely to occur are shown in Table. Weather condition B thus describes the weather condition most likely to occur since it consists of the average solar inclination and beam irradiance. Weather condition D describes the condition which would cause the maximum heat rate on the receiver. Weather condition A and C are the conditions most likely to occur between Weather condition B and D. Table. The proposed four most common weather conditions for Pretoria. Weather condition Solar inclination, β Solar beam irradiation, I (Wm ) Temperature ( C) A B 3 58 C 8 D Modelling and the objective function The objective function is the function which is maximised by the optimisation of variables. The power output of the system is written in terms of the total entropy generation rate in the system. The entropy generation mechanisms are identified and the objective function is constructed. The optimisation of the geometry variables is done over a range of operating points of the micro-turbine Temperatures and pressures in terms of geometry variables: The modeling of the temperatures and pressures is required since the objective function requires the values of the temperatures and pressures at each point in Fig.. It is assumed that P = P = P = 8 kpa (see Fig. ). The temperatures and pressures in all the ducts are calculated with an assumed temperature drop or pressure drop which is small. When considering experimental results for turbines and their mass flow rates [3], it is assumed that the turbine corrected mass flow rate and turbine pressure ratio can be modeled with the use of the turbine map. The turbine map shows the turbine corrected flow as a function of the turbine pressure ratio. After choosing the turbine operating point, T 7 is chosen for the first iteration. For the second iteration, the system mass flow rate is guessed, where after dp 9-, P 9, P 8 and P 7 are calculated so that the mass flow rate can be calculated using Eq. (5), m& tcf P7.7 m& = (5) t ( T + ) 59 7 where P 7 is in psi and T 7 in degrees Fahrenheit respectively [9]. The corrected compressor mass flow rate can then be calculated with Eq. () since the mass flow rate through the compressor is equal to the mass flow rate through the turbine. Note that P and T are in psi and degrees Fahrenheit respectively [9]. ( T ) m + 55 m& ccf = & () P 3.95 For the third iteration, the compressor pressure ratio is guessed so that dp 3-, P, P 5, P, P 3 and P can be calculated. This allows for the compressor pressure ratio to be obtained with iteration. The compressor isentropic efficiency is available from the compressor map and can be interpolated using the corrected compressor mass flow rate and compressor pressure ratio. A turbine isentropic efficiency of.8 is assumed since it is a function of the load and the load is not known. The recuperator efficiency ( η ) is calculated using the ε-ntu method with the fouling factor for air given as R f =. [7], where after the temperatures can be found so that T 7 can be iterated Construction of the objective function: The finite heat transfers and pressure drops in the compressor, turbine, recuperator, receiver and ducts are identified as entropy generation mechanisms. When doing an exergy analysis for the system and assuming V = V and Z = Z, the objective function can be assembled as given in Eq. (7). The function to be maximised (the objective function), is W & (the power Copyright by ASME

5 output). Equation (8) shows the total entropy generation rate in terms of the temperatures and pressures (with reference to Fig. ). The entropy generation rate for each component is added and is shown in block brackets. For the analysis in this work, T* is assumed to be 7 K [] and at a point between the concentrator and receiver. W& = gen + (7) + mc & where p T S& ( T T ) mt & c ln( T T ) T & * T * p [ m p ] compressor [ & T + mc & p ln( T3 T ) mr & ln( P3 P )] Duct 3 S & = & c ln ( T T ) + mr & ln( P P ) + gen + mc & + & T T T P P ( k ) k p ln T T P P & T recuperator [ + mc & p ln( T5 T ) mr & ln( P5 P )] Duct 5 & * & ( T T ) mr & ln( P P ) loss mc p ln 5 5 T * T + & + & [ T + mc & p ln( T7 T ) mr & ln( P7 P )] Duct7 [ m& cp ln ( T7 T8 ) + mr & ln( P7 P8 )] turbine & T + mc & ln( T T ) mr & ln( P P ) [ p ] Duct89 receiver + (8) Note that & * & loss = & Constraints: The recuperator aspect ratio is constrained to a maximum of. The concentration ratio between concentrator area and receiver aperture area, CR is constrained to CR min = as shown in Eq. (9). Dh Lrec 8 As, conc CRmin (9), rec Equation () prevents the receiver from losing its cavity shape, by only allowing a minimum of two diameters in the distance between the aperture edge and the edge of the receiver. ( 3 ) ) D L π h, rec h, rec rec D () The cavity receiver tubes are constructed from copper. The maximum surface temperature of the receiver tubes should stay well below its melting temperature. A maximum receiver surface temperature of T s,max = K is identified for the analysis [, 9]. The surface area of a tube and the Dittus- Boelter equation [] help to construct Eq. (), which is the allowable maximum surface temperature of the receiver.. (.3π L k Pr ( m& ( D )) ).8 T = & µπ () s, max T + rec h, rec The longer the recuperator the more beneficial it is to the system. There needs to be a constraint on its length. To make sure the system stays compact, the recuperator s length cannot exceed the length of the radius of the dish, L () D conc. RESULTS AND DISCUSSION The open and direct solar thermal Brayton cycle should be able to handle various weather conditions throughout a typical year in Pretoria. One solution is to identify four weather conditions which are most likely to occur and optimise the geometries of the receiver and recuperator and find the optimum operating point of the micro-turbine for each weather condition. These geometries can then be available for the system on demand. (W) (m) Figure. Net absorbed heat rate at cavity receiver depending on cavity receiver aperture diameter. Figure shows the relation between & and the receiver aperture diameter for each of the proposed most common weather conditions in Table. The method of obtaining & was discussed in Section.. Note that Fig. was generated by using a parabolic concentrator error of.7 rad and rim angle of 5 as suggested from the literature []. For each of the proposed weather conditions in Table, the objective function was maximised at different operating points of the turbo-machine s turbine, as shown on its turbine map [9]. These results are shown in Fig. 5 as a function of compressor pressure ratio. Each data point in Fig. 5 represents an optimised set of geometries as obtained from the maximization of the objective function. The results for Weather D B A C 5 Copyright by ASME

6 condition A is not shown since a positive power output could not be obtained by the micro-turbine for the low power input. Note that in Fig. 5 there exists an operating point for each weather condition, which provides the highest maximum power output. This point is the optimum operating point. The optimum operating points with the highest maximum power outputs and their optimum geometries are shown in Table for each weather condition. These optimum operating points are also shown in Fig. on the compressor map. Table 3 shows the absorbed heat rate, power output and maximum receiver surface temperature for each of the maxima in Table. It is shown that at each of these cases the receiver surface temperature is at its allowable maximum temperature of K (see Section ) or very close to it. operation in various weather conditions any one of these three geometries sets with its optimised operating points can be chosen according to how close the weather condition is to the condition the geometry set was optimised for. To establish whether the three geometry sets with their operating points (Table ) are able to handle the various weather conditions throughout a typical year, a few other weather conditions are considered. In Table, for data set D* (see Table ), different absorbed heat rates are implemented at positions and 3 as shown in Fig D C B (kw) Figure 5. Maximum power output at different operating points of the micro-turbine at the different weather conditions. Table. Operating points with highest maximum power output for each of the four weather conditions. r m& r t m& ab D rec L rec D h, L ccf tcf (kgs) (lbmin) (cm) (m) (mm) (m) D* C* B* Table 3. Operating points with highest maximum power output for each of the four weather conditions. T s,max (K) & (W) W & (W) D* C* B* The geometries shown in Table are thus optimised for each weather condition most likely to occur. During system Figure. Optimum operating points shown on the compressor map of the micro-turbine. In Fig. 7 it is shown that the decrease in solar beam irradiance has a large effect on the power output of the system. When the solar beam irradiance falls below 9 Wm, the system is unable to generate a positive power output. The data set D* can thus only be used when the solar beam irradiance is between 9 Wm and Wm. The maximum surface temperatures of points to 3 are shown in Table. Note that the maximum receiver surface temperature drops as the solar beam irradiance decreases. The maximum receiver surface temperature would thus also increase when the solar beam irradiance increases. This can result in tube burnout. The result is that the system with any optimised data set in Table, can only handle absorbed heat rates which are smaller than the value the data set was optimised for. When such a system handles the smaller absorbed heat rates, it Copyright by ASME

7 would also have much lower power output. The cost and the impracticality of having several components with different geometries available for the system are also noted. Table. The effect of changing absorbed heat input when geometry and turbine operating point stays constant. Number r m& W & I w β T T (Wm ccf s,max ) (K) (K) (kgs) (W) & (W) In Fig. 5, note that for each different weather condition, the point of highest maximum power output lies on a very specific line. Note that, according to Table 3 and Table 5, when the power output lies on this line, the maximum receiver surface temperature is K, which is the allowable maximum receiver surface temperature. Table shows that the geometry set optimised for Weather condition D can be used to get the same power output and maximum receiver surface temperature as was obtained with a geometry set optimised for Weather conditions B and C respectively in Table (kw) 5 (kw) Figure 7. Effect of changing solar beam irradiance on data set optimised for Weather condition D. Another option is to select a single geometry set and control the mass flow rate through the system as the absorbed heat rate changes. The optimum geometry found in Table for Weather condition D ( & = 5.9 kw) can be chosen to stay constant for example. Instead of keeping the turbine operating point fixed as in Fig. 7, when the solar beam irradiance drops to 9 Wm (or & =.7 kw), the mass flow rate can be decreased as shown in Fig. 8. Similarly when the absorbed heat rate drops to. kw (solar beam irradiance of 8 Wm and a slight wind), the mass flow rate can be decreased so that the power output can be increased. With a good reaction time, this would prevent the receiver surface temperature from becoming too hot or too cold as the absorbed heat rate changes. Each of the data points shown in Fig. 8 are shown in Table 5. Note that as the mass flow rate (compressor pressure ration in Table 5) decreases, the maximum receiver surface temperature increases. The maximum receiver surface temperature cannot exceed K, so that the maximum power output and optimum operating point is at the point where this maximum receiver surface temperature is reached. Figure 8. Using the mass flow rate to increase power output when the maximum receiver surface temperature is low. Table 5. The effect of changing mass flow rate (operating point) when geometry stays constant. Symbol X & (W) W & (W) T s,max (K) r & (W) W & (W) T s,max (K) r Table. Comparing results of data set D with data sets B and C. Weather condition B Weather condition C As obtained with optimum geometry set in Table As obtained with geometry set of Weather condition D & (W) W & (W) T s,max (K) 7 r.7.9 & (W) W & (W) 3 5 T s,max (K) 73 r..9 7 Copyright by ASME

8 Similarly, as shown in Table 7, the optimum geometry found for Weather condition B can be used to get the same results as was obtained with the optimised geometry sets of Weather conditions C and D respectively, shown in Table. Note that in Table and Table 7, the maximum receiver surface temperature is not always exactly the same when using the different data sets. This is because the data points in Table were not necessarily at the most optimum operating points, but were only very close to it. Other data sets, even those reasonably close to the highest maximum power output in Fig. 5, could not be used at other weather conditions to give comparable results. Table 7. Comparing results of data set B with data sets D and C. Weather condition D Weather condition C As obtained with optimum geometry set in Table As obtained with geometry set of Weather condition B & (W) W & (W) T s,max (K) 7 r..9 & (W) W & (W) T s,max (K) 73 r.9.93 Note that from Eq. (), the maximum receiver surface temperature is a function of T, L rec, & m& and D h,rec and that T = f η, η, η, r, r, T, dt, dt, dt, dt, ). ( c t t T When considering Table and Table 7, it is shown that for maximum power output, a specific optimum set of geometries and operating point for a specific weather condition does not necessarily exist, but rather, there is a correct combination of a set of geometries and & m& so that the allowable maximum receiver surface temperature can be obtained. The results showed that when using total entropy generation minimisation to optimise the geometries of the receiver and recuperator, the point of highest maximum power output (with optimum operating point) lies on the constraint of the allowable maximum receiver surface temperature of K. The results further showed that a geometry set optimised for maximum power output at the optimum operating point for a specific absorbed heat rate (as shown in Table ), can be used to give maximum power output at any other absorbed heat rate since its maximum receiver surface temperature will also be at K. Figure 9 shows that the power output of the small-scale open and direct solar thermal Brayton cycle with concentrically wound tube cavity receiver (.8 m diameter dish) depends on the maximum receiver surface temperature, absorbed heat rate and micro-turbine operating point. Figure 9 shows lines of constant absorbed heat rate and constant maximum receiver surface temperature. Thus for any absorbed heat rate available at the cavity receiver, the maximum power output and optimum operating point can be determined from Fig. 9 for a specific allowable maximum receiver surface temperature. The maximum power output is found at the intersection of the line of absorbed heat rate with the line of allowable maximum receiver surface temperature. Any one of the geometry sets in Table can be used for such a system when an allowable maximum receiver surface temperature of K is considered. Fig. 9 is specific to the micro-turbine used. (kw) 8 =.8 kw, T = 38 K, = 8.3 kw, T = 79 K, 3 = 9.7 kw, T 3 = 5 K, =. kw, T = 9 K 5 =.7 kw, =. kw, 7 = 5.9 kw T T Figure 9. Performance map for maximum power output and optimum operating point. 5. CONCLUSION The absorbed heat rate at the cavity receiver of the open and direct solar thermal Brayton cycle, changes throughout a day, month and year. The system should be able to accommodate these changes. In this work, the method of total entropy generation minimisation was applied to show that the mass flow rate through the system can be changed as the absorbed heat rate changes so that maximum power output can be attained. Results showed that the highest maximum power output or optimum operating point has maximum receiver surface temperature of K (the allowable maximum temperature). Results showed that the maximum power output and optimum operating point can be determined for any absorbed heat rate, and the optimum geometry set to be used can be a geometry set optimised for any absorbed heat rate where the maximum receiver surface temperature is T 3 T 8 Copyright by ASME

9 K. One geometry set optimised correctly can thus be used together with mass flow rate control to accommodate the changing weather conditions throughput a typical year in Pretoria. NOMENCLATURE a Longer side length of recuperator channel (m) A Area (m ) A w Receiver inner wall area (m ) b Shorter side length of recuperator channel (m) c p Zero pressure constant pressure specific heat (JkgK) c p Constant pressure specific heat (JkgK) c v Constant volume specific heat (JkgK) CR Concentration ratio (D conc d) d Receiver aperture diameter (m) D Receiver diameter (m) D conc Parabolic dish concentrator diameter (m) D rec Receiver tube diameter (m) D h, Recuperator channel hydraulic diameter (m) f Friction factor Gr Grashof number H Recuperator height (m) I Solar beam irradiance (Wm ) k Thermal conductivity of a fluid (WmK) k Gas constant (c p c v ) L Length (m) m& System mass flow rate (kgs) m& k Recuperator channel mass flow rate (kgs) n Number of flow channels NTU Number of transfer units Nu Nusselt number P Pressure (Pa) Pr Prandtl number Constant used in Fig. 9 for absorbed heat rate (W) & * Rate of intercepted heat at receiver cavity (W) & Rate of heat loss from component (W) & Rate of heat loss from the cavity receiver (W) loss & Net rate of absorbed heat (W) r Compressor pressure ratio r t Turbine pressure ratio R Gas constant (JkgK) Re Reynolds number R f Fouling factor S & Entropy generation rate (WK) gen t Plate thickness between recuperator flow channels (m) T Temperature (K) T* Apparent sun temperature as an exergy source (K) T Environment temperature (K) V Velocity (ms) w Wind factor W & Power (W) W & Net power output of system (W) y Z Numerical approximation constant Height (m) β Inclination of receiver Change in ε Effectiveness (in the ε-ntu method) ρ Density (kgm 3 ) µ Dynamic viscosity (kgms) η Efficiency Subscripts Environmentloss,,3.. Refer to Fig. c Compressor a Receiver aperture CF Corrected flow conc Concentrator D Based on receiver diameter D Based on recuperator channel diameter h Hydraulic k Channel max Maximum min Minimum Net rec Receiver Recuperator s Surface t Turbine w Receiver inner wall Superscript * At weather condition ACKNOWLEDGEMENTS The authors acknowledge the Centre for Renewable and Sustainable Energy Studies (CRSES) (University of Stellenbosch, South Africa) and prof. E. van Dyk (Nelson Mandela Metropolitan University, South Africa) for their contributions. REFERENCES [] Sendhil Kumar, N. and Reddy, K. S., 8, Comparison of Receivers for Solar Dish Collector System, Energy Conversion and Management, 9, pp [] Shah, R. K., 5, Micro Gas Turbines,, Compact Heat Exchangers for Microturbines, Neuilly-sur-Seine, France: RTO, pp Available from: [3] Bejan, A. and Lorente, S.,, The Constructal Law of Design and Evolution in Nature, Philosophical Transactions of the Royal Society Biological Sciences, 35, pp [] Le Roux, W. G., Bello-Ochende, T. and Meyer, J.P.,, Thermodynamic Optimisation of an Integrated Design of a 9 Copyright by ASME

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