Solar-driven Joule cycle reciprocating Ericsson engines for small scale applications. From improper operation to high performance

Energy Conversion and Management 135 (2017) 101–116

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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

Solar-driven Joule cycle reciprocating Ericsson engines for small scale applications. From improper operation to high performance Dorin Stanciu a,⇑, Viorel Ba˘descu b a b

Department of Engineering Thermodynamics, University POLITEHNICA of Bucharest, Splaiul Independentei, 313, 060042 Bucharest, Romania Candida Oancea Institute, University POLITEHNICA of Bucharest, Splaiul Independentei, 313, 060042 Bucharest, Romania

a r t i c l e

i n f o

Article history: Received 14 September 2016 Received in revised form 21 November 2016 Accepted 24 December 2016

Keywords: Parabolic trough collector Ericsson engine Proper design Operation strategy

a b s t r a c t The paper focuses on a Joule cycle reciprocating Ericsson engine (JCREE) coupled with a solar parabolic trough collector (PTC). A small scale application located at mid Northern Hemisphere latitude (44°2500 N) is considered. A new dynamic (time-dependent) model is developed and used to design the geometry and estimate the performance of the PTC-JCREE system under the most favorable weather conditions (i.e. summer day and clear sky). The paper brings two main contributions. First, specific constraints on the design parameters have been identified in order to avoid improper JCREE operation, such as gas under-compression in the compressor cylinder and gas over-compression and/or overexpansion in the expander cylinder. Second, increasing the work generated per day requires using a proper strategy to switch between different rotation speeds. Specific results are as follows. For the (reference) constant engine rotation speed 480 rpm, the output work per day is 39,270 kJ and the overall efficiency is 0.134. The output work decreases by increasing the rotation speed, since the operation interval during a day diminishes. A better operation strategy is to switch among three rotation speed values, namely 480, 540 and 600 rpm. In this case the output work is 40,322 kJ and the overall efficiency is 0.137. The performance improvement is quite small and the reference constant rotation speed 480 rpm may be a suitable choice, easier to use in practice. For both the constant and variable rotation speed strategies, the overall efficiency is almost constant along the effective operation time interval, which is from 8:46 to 15:15. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Low cost engines with reasonably good thermal efficiency and compatible with many types of energy sources is expected to cover the entire output power range required by small scale domestic applications. Several solutions are presently available, including Organic Rankine Cycles (ORC) based engines, Stirling engines and Micro-gas turbine engines [1–4]. They have advantages and disadvantages, which depend not only on the operating cycle, but also on the targeting power range. When choosing among available solutions, acquisition and operating costs as well as thermal efficiency represent the main criteria. From this point of view, the Ericsson engine could be one of the best solutions in the power range up to 5 kW [3]. This engine belongs to the class of hot air engines and, beyond its name, it operates under the Joule cycle. Unlike the well-known gas turbine engines working on Joule cycle, the Ericsson engine is externally heated and performs the com⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (D. Stanciu). http://dx.doi.org/10.1016/j.enconman.2016.12.070 0196-8904/Ó 2016 Elsevier Ltd. All rights reserved.

pression and expansion processes in reciprocating compressors and expanders, respectively. Several names are associated with this kind of engine, such as Reciprocating Joule-cycle engine [2], volumetric hot air Joule cycle engine [8], Joule cycle reciprocating Ericsson engine (JCREE) [7,8]. The last name is adopted here. Small power applications involve low mass flow rates; this make JCREEs a better option than micro-gas turbines, despite peculiar irreversibilities are associated with the operation of these engines, like those due to mechanical friction and flow through valves. Several JCREE applications have been already reported. Some of them were treated using steady state models, based on the classical Joule cycle, further improved by taking into account the irreversible processes specific to piston engines operation. These models were used to estimate the JCREE performance in the low and middle power range (up to 5 kW and from 5 to 30 kW, respectively). For example, Moss et al. [2] developed a steady state design procedure for a JCREE used for domestic micro CHP (combined heat and power), producing 5 kW electrical power and 8 kW of heat. The goal was to achieve a higher efficiency than that of a gas-turbine plant of comparable size. The study predicted a thermal efficiency of 35% for

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Nomenclature cp cv CD D E e EVC EVO f GB H IVC IVO JCREE h k Kh L Lv M _ m Nu Nr P p Pr Q_ qs R Ra Re T V VS VC w

specific heat at constant pressure [J/(kg K)] specific heat at constant volume [J/(kg K)] discharge coefficient [–] diameter [m] energy [J] rib height [m] closing advance of expander exhaust valve [deg] opening delay of compressor exhaust valve [deg] friction factor beam (direct) solar energy flux [W/m2] width [m] closing advance of expander inlet valve [deg] opening delay of compressor inlet valve [deg] Joule cycle reciprocating Ericsson engine enthalpy [J/kg], convection heat transfer coefficient [W/ (m2 K)] adiabatic exponent incident angle modifier [–] length [m] valve lift [m] air mass [kg] mass flow rate [kg/s] Nusselt number [–] engine rotation speed [rpm] pressure [Pa] pitch [m] Prandtl number [–] heat flux [W] surface heat flux [W/m2] air specific constant [J/(kg K)] Rayleigh number Reynolds number temperature [K] volume [m3] swept volume [m3] clearance volume [m3] velocity [m/s]

the pressure ratio 7.5 and 1000 rpm. The authors concluded that this engine has considerable advantages compared with other prime movers in terms of efficiency, emissions and multi-fuel capability. Bonnet et al. [3] carried out energy, exergy and exergoeconomic analyses for a JCREE driven by external natural gas combustion. Their study focused on the real engine by considering several irreversibilities due to low mechanical efficiency, valve pressure losses and heat rejected at the chimney. The authors concluded that the JCREE may be a suitable and profitable solution for thermal energy conversion in the low power range (1–5 kW). The same kind of global approach was performed by Creyx et al. [4] to explore the operation of an Ericsson engine working upon the Joule or Ericsson cycles and designed for micro CHP purposes. A sensitive analysis was carried out in order to find the optimal engine operation conditions. The authors found that the engine performances are maximum for an optimum pressure ranging between 5 and 8 bar, when the heat exchanger operates at the highest possible temperature that it can reach. Touré and Stouffs [5] analyzed a JCREE driven by recuperated heat and established relationships between the geometrical characteristics and operation parameters of the engine and its thermodynamic performances. Other applications were analyzed by using dynamic models, which take into account the time-dependent movement of the piston and cylinder valves. Losses in valves, heat exchangers and pipes pressure were treated in details, while other mechanical

_ W

power [W]

Greek symbols absorptivity [–], helix angle [deg] b pressure ratio [–], tilt angle [deg], expansion coefficient [1/K] e emissivity [–] g efficiency [–] k ratio of connecting road to crank radius [–] m kinematic viscosity [m2/s] s time [s], transmissivity [–]

a

Subscript a c e eng env high gls m ptc sky usf in ex out (–)

air in absorber tube compressor, cycle expander engine environment, envelope high side glass spatial average parabolic trough collector sky useful inner, inflow outer outflow time averaged per cycle

Superscript (a) absorption (c) convection (r) radiation (rb) ribbed surface (sm) smooth surface

losses were described by using empirical coefficients, in a way which is similar to steady state models. For instance, by using a dynamic model, Wojewoda and Kazimierski [6] investigated the performances of a closed JCREE designed to supply moderate levels of power (up to 30 kW). The engine-heat exchanger assembly was highly pressurized and greater values of engine rotation speed were considered. Additionally, two recirculation blowers were used to ensure the air circulation in the heat exchangers when the engine valves are closed. The output power and the efficiency of the engine were evaluated for a large range of rotation speeds (i.e. 500–3500 rpm) by taking into account the recirculation mass flow rate provided by the blowers. Results showed output powers up to 25 kW and thermal efficiency of about 25% for engine rotation speed of 3000 rpm and exhaust compressor pressure around 90 bar. Lontsi et al. [7,8] developed a dynamic model for predicting the performances of low-speed open JCREE. The air is compressed inside the compressor cylinder until a pressure of about 4 bar is obtained and the heat is supplied by a shell- and-tube heat exchanger in which the working fluid flows inside the tubes, which have a constant wall temperature of 873 K. Numerical simulations predicted rapid hot start of the engine (about 5 s) and stable engine operation with output power of 1716 W at efficiency around 23%, as well as good transient response to the perturbation induced by a sudden pressure drop occurring in the compressor intake manifold. A dynamic model for JCREE operation was proposed by

D. Stanciu, V. Ba˘descu / Energy Conversion and Management 135 (2017) 101–116

Oudkerk and Lemort [9]. Results revealed engine efficiencies around 24% and output powers about 1.6 kW, for an expander supply temperature of 800 °C. The authors concluded that, when micro CHP applications are considered, the JCREE engine deserves the same attention as the Stirling engine or the power plant based on Organic Rankine cycle. Creyx et al. [10] proposed a dynamic model for the operation of an open JCREE integrated into a biomass-fueled micro-CHP system. The expansion cylinder has been designed in such a way that the engine performance is maximized. Due to its availability, solar energy is the main renewable energy source. For this reason, solar based heat power technologies at large and small scales have attracted increased interest in the last decades. Recently, Modi et al. [11] reviewed results obtained for concentration solar thermal systems, with special emphasis on the small scale applications based on Stirling and Rankine cycles. Other small scale solar power generation systems based on dish/Stirling engine assembly [12,13], PTC/steam or Organic Rankine cycles configurations [14,15] and evacuated flat-plate collectors/Organic Rankine Cycle [16] have been also investigated and optimized by using various thermodynamic methods and optimization procedures. All these works prove that the solar driven heat engines designed for small scale applications may play a major role in the attempt to remove the dependency on fossil fuels and to regain the clearness of the environment. Few works refer to solar energy applications of Ericsson engines. For example, Alaphilippe et al. [17] conducted a theoretical investigation on the coupling of solar parabolic trough collectors (PTC) with Ericsson air engines. The study relies on a global mathematical model, which assumes a fixed solar irradiance of 1000 W/m2 concentrated by a parabolic trough collector whose aperture area is of 6.3 m2. The results proved the potential of using the JCREE in the field of micro-power generation based on solar energy conversion. The goal of this paper is to investigate in more details the potential of solar driven JCREEs. A time dependent model is built for a PTC-JCREE system. The JCREE model improves the previous works of Lontsi et al. [7,8], while the time dependent PTC model develops the approaches by Forristall [20] and Kalogirou [21]. Some of the new contributions are as follows. Steady-state models were pro-

103

posed in [20,21], while a transient model is developed here. The basic assumption of constant incident solar radiation has been adopted in [17], whereas a time dependent beam solar irradiance model is employed in this work. This allows treating more realistic clear-sky operation regimes instead of the academic cases treated previously. Further, the geometrical parameters of the PTC-JCREE system are properly designed in this paper, in order to avoid under-compression or over-expansion regimes due to the variation of air temperature inside the PTC absorber tube during the day. The existence of potentially improper operation regimes has not been specifically emphasized in literature. Here we propose procedures to avoid these undesirable situations. Finally, in order to increase the work supplied during a day, simulations are performed for different constant values of the engine rotation speed, as well as by adopting a strategy consisting of switching between different rotation speeds. The structure of the paper is as follows. The system is described in Section 2. The new model is presented in Section 3, while Section 4 shows the numerical solution methods. Section 5 contains the system set-up. The strategy to improve the performance of the PTC-JCREE system is proposed in Section 6, while conclusions are drawn in Section 7. 2. System description Fig. 1 shows the system considered here. It consists of an Ericsson engine coupled to a Parabolic Trough Collector. The PTC is aligned along the E-W axis. Solar energy is concentrated on the absorber tube of the PTC and heats up the air flow inside. The hot air constitutes the high temperature reservoir for the Ericsson engine. Basically, the Ericsson engine consists of a reciprocating compressor and expander and some heat exchangers. Depending on heat exchangers arrangements, the engine can perform a close or an open cycle, with or without regeneration. The compression and expansion processes are adiabatic, and the heat exchange process is isobaric. Therefore, in this work the Ericsson engine operates on the Joule cycle instead of the Ericsson cycle. The thermodynamic system is the air enclosed inside the PTCJCREE system. During a full rotation of engine cranks shaft, the thermodynamic processes of admission, compression, expansion

Fig. 1. PTC-JCREE configuration (left) and mean section in the tubular receiver of the PTC (right).

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and exhaust, respectively, take place inside the cylinders. Their sequence depends on which of the compressor or expansion cylinder is considered. The duration of these processes depends on the timing of the inlet and exhaust valves. When both valves are closed, the compression or expansion processes occur. When one of the valves is open, the admission or the exhaust process takes place. Let us focus on a certain air mass quantity which follows the Joule cycle. Before entering the engine, it has the temperature T0 and pressure P0 (Fig. 1a). During the compressor admission process, the inlet valve is opened and the air quantity crosses the admission manifold and enters the compressor cylinder. Now, assume that the temperature and pressure do not change during the flow of air through the intake manifold; thus the parameters of the air coming into the compressor cylinder are T1 = T0 and P1 = P0. After closing the inlet valve, the air is compressed inside the compressor cylinder until the exhaust valve is opened. This is the moment when the evacuation process begins. During the evacuation process, the air mass leaves the compressor and enters the PTC absorber pipe, where it is heated by the concentrated solar energy. Theoretically, the heating process takes place at constant pressure; however, due to friction, a pressure loss occurs. At the end of the heating process, the air temperature and pressure get the values T3 and P3, respectively. During the time the expander inlet valve is opened, the PTC evacuated air passes inside the expander cylinder and the expansion process begins. After the inlet valve is closed, the expansion process continues until the expander exhaust valve is unlocked. Now, the amount of air leaves the expander cylinder, flows through the exhaust manifold and meets the atmospheric air. The PTC-JCREE system is located at 44.25°N latitude (Bucharest, Romania, South-Eastern Europe). The day considered for performance evaluation is June 21st for which clear sky conditions are assumed. During this day, the PTC is kept at a fixed tilt angle. The main design parameters of the PTC-JCREE system are as follows: (a) Ericsson engine: swept and clearance volumes; ratio of connecting road length to crank radius; timing of the valves for both compressor and expander; engine rotation speed. (b) Parabolic trough collector: collector aperture (width and length); inner and outer diameters of both absorber tube and glass envelope, tilt angle (kept constant during the considered day). Several configurations of Ericsson engines are presented in [4]. All of them are driven by internal or external combustion, so that the temperature of the gas entering the expander is constant for all operation time. An overview of the PTCs existing on the market, as well as of the prototypes currently under development, can be found in [18]. In the present case, the amount of input heat depends on the aperture size of the PTC. Also, that heat flux is time dependent, since the incident solar flux is variable during the day. Therefore, the temperature of the hot air entering the expander is continuously varying, which may affect the proper operation of the engine. As a result, the main design and operation parameters of the Ericsson engine should be set up in accordance with the PTC size and the time variation of the input heat flux. The set-up procedure and the design parameter values finally adopted are presented in Section 5.

3. Model The model consists of two parts. The first part deals with the operation of the JCREE while the second part describes the opera-

tion of the coupled system consisting of PTC and JCREE. Both parts rely on a dynamic approach enabling to catch the main transient operation features of the JCREE-PTC system. 3.1. Models of engine components 3.1.1. Hypotheses and equations The models of the JCREE components rely on mass and energy balance equations for control volumes defined for compressor, expander and absorber tube, respectively. The assumptions are: (1) The working fluid is air, which behaves like a perfect gas. Note that the temperature variation of the physical properties of the air (specific heat, viscosity, thermal conductivity) is properly taken into account. (2) The air properties are uniformly distributed inside the engine components. (3) The compressor and expander cylinders are adiabatically insulated. (4) The gas leakages through the piston rings of both cylinders are neglected. Also, the pressure losses occurring inside the intake and exhaust manifolds are neglected. With these assumptions, the formal mass and energy balance equations for the fluid inside a specific component of the engine are as follows:

dM _ in  m _ out ¼m ds
 dcv dT dM _ _ in hin  m _ out hout þ Q_  W þ cv T ¼m Mcv þ MT ds dT ds

ð1Þ ð2Þ

where s is time, M, T, cp and cv are the mass, temperature and specific heats at constant pressure and constant volume, respectively, _ in;out stands for inlet or outlet mass flow rates, respectively, hin,out m is the specific enthalpy of inflow or outflow streams, respectively, _ represent heat flux and mechanical power, while Q_ and W respectively. The instantaneous pressure inside the engine component is given by the state equation of perfect gas:

P¼

MRT V

ð3Þ

where R represents the gas constant. Using the formal Eqs. (1)–(3) requires taking into account the peculiarities of each engine component. These peculiarities are presented in Sections 3.1.2–3.1.4. 3.1.2. Compressor When the compressor is considered, the unknowns are the air mass enclosed in the cylinder, Mc(s), its temperature, Tc(s) and pressure, Pc(s). The time s [s] is linked with the crank angle h by the relation h ¼ 2pN r s=60 where Nr [rpm] represents the crank shaft rotation speed. The compressor is equipped with reciprocating valves. The opening of inlet and exhaust valves is delayed by the IVO (inlet valve opening) angle and EVO (exhaust valve opening) angle, respectively, while their closing occurs at bottom-dead center and top-dead-center, respectively. This means that the expansion takes place for 0 < h < IVO and the compression occurs for p < h < EVO. In both cases the thermodynamic system is closed, _ c;out ¼ 0. The admission process and the exhaust pro_ c;in ; m so that m cess take place for IVO < h < p and p + EVO < h < 2p, respectively. During admission and exhaust, the thermodynamic system is open and the exchanged mass flow rates are given by [19]:

D. Stanciu, V. Ba˘descu / Energy Conversion and Management 135 (2017) 101–116

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 kþ1 2 Phigh;j 2k _ j ¼ Aj C Dj pffiffiffiffiffiffiffiffiffiffiffiffiffiffi bjk  bj k m RT high;j k  1

ð4Þ

In the above relation, the subscript j denotes the inlet valve (in) or the exhaust valve (ex), Aj ¼ pLv ;j Dv ;j represents the valve curtain area, CDj stands for discharge coefficient and k is the specific heats ratio. The pressure ratio, bj is evaluated by bj ¼ Plow;j =P high;j if the k

flow through the valve j is subsonic or by bj ¼ ½2=ðk þ 1Þk1 otherwise. The inlet and exhaust valve lift, Lv,j are modeled by the relations [7,8]:








Lv c;max h  IVO 1  cos 2p 2 p  IVO
  Lv c;max h  EVO  p ¼ 1  cos 2p p  EVO 2

Lv c;in ¼

ð5Þ

Lv c;ex

ð6Þ

One assumes that the compressor is operating adiabatically, so that Q_ c ¼ 0: The instantaneous mechanical power W_ c is computed as:

_ c ¼ Pc dV c W ds

ð7Þ

105

friction, the pressure decreases linearly along the flow. Assume that the convection heat transfer occurs under a constant heat flux boundary condition (qs = const.). Then, the temperature increases nearly linearly along the tube. As a result, Ta,m and Pa,m may be hypothetically associated with the middle section of the absorber pipe. The mass of air accumulated in absorber pipe, Ma(s), is com_ c;out and m _ a;out ¼ m _ e;in . The tem_ a;in ¼ m puted with Eq. (1), where m perature Ta,m(s) is the unknown of Eq. (2). In Eq. (2) the involved heat flux is modeled as:

Q_ u ¼ pDabs;in Lptc ha ðT abs;m  T a;m Þ

ð11Þ

where Dabs,in is the inner diameter of the absorber tube and ha represents the convection heat transfer coefficient, while Tabs,m denotes the absorber inner wall temperature of the pipe middle section. Note that, in the frame of the boundary condition qs = const., the difference DTm = Tabs,m  Ta,m designates the relevant temperature gap for computing the convection heat flux. Finally, the pressure Pa,m is calculated with Eq. (3), where the volume Vabs of the absorber tube is employed. Its value is defined by the inner pipe diameter Dabs,in and pipe length, Lptc. Once the useful heat flux Q_ u is known, one needs to compute

Here VC is the instantaneous working volume. By using the crankrod kinematics, its variation is expressed by [19]:

the enthalpy and the temperature of air entering the expander. Pulsating flow and time delay phenomena are neglected. As a result, the energy conservation yields the enthalpy h3 as follows:

Vc ¼

_ h3 ¼ h2 þ Q_ u =m


qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V Sc 2 1 þ kc  cos h  k2c  sin h þ V Cc 2

ð8Þ

where VSc is the compressor swept volume and VCc is the compressor clearance volume. 3.1.3. Expander When the expander is considered, the unknowns are the instantaneous mass Me(s), temperature Te(s) and pressure Pe(s) of the air enclosed in the cylinder. As shown in Fig. 1, the expansion piston moves in reversal phase compared to the movement of the compressor piston. The opening of inlet and exhaust expander valves takes place at the top-dead-center and bottom-dead-center, respectively, while their closing is advanced by IVC (inlet valve closing) angle and EVC (exhaust valve closing) angle, respectively. Thus, the expansion process takes place for p-IVC < h < p, while the compression process takes place for 2p-EVC < h < 2p. During these processes, _ e;out ¼ 0: If the crank angle h _ e;in ; m the valves are closed, so that m is in the range 0 6 h 6 p  IVC or p 6 h 6 2p  EVC, the admission or the exhaust processes take places, respectively. During expansion and compression, the mass flow rate through the valves is determined by Eq. (4), where the valve lifts are computed as [7,8]:






Lv e;max h 1  cos 2p 2 p  IVC
  Lv e;max hp ¼ 1  cos 2p p  EVC 2

Lv e;in ¼ Lv e;ex



ð9Þ ð10Þ

Due to the adiabatic operation of the expander cylinder, Q_ e ¼ 0: _ e and the instantaneous volume, V e , are The mechanical power, W computed with relations similar to Eqs. (7) and (8), respectively. Of course, when Eq. (8) is used for the instantaneous volume of the expander, the expander swept volume VSe, the clearance volume VCe, and the ratio of connecting rod length to crank radius, ke, are used instead of the corresponding quantities associated with the compressor. 3.1.4. Absorber tube volume When the absorber is considered, the unknowns are the instantaneous mass Ma(s), instantaneous spatially mean temperature Ta,m(s) and pressure Pa,m(s) of the air inside the absorber tube. Due to

ð12Þ

_ is the mean mass flow rate of air crossing the absorber where m tube. Further, the pressure loss of the air flowing through the absorber tube is computed by the relation:

DP a ¼ f

w2 Lptc qa;m a;m Dabs;in 2

ð13Þ

where f is the friction coefficient, qa,m and wa,m represent the mean air density and velocity, respectively. Finally, the pressure loss is assumed to be equally distributed between the absorber tube ends. The values of ha, and f can be obtained by using specific correlation relationships. However, the model is undetermined since the temperature Tabs,m is still unknown. To find Tabs,m, one has to develop a model for PTC, which is shown in Section 3.2. 3.2. Model of PTC 3.2.1. Hypotheses and equations The following assumptions are adopted: (1) The mass flow rate of air passing through the absorber pipe is constant during one engine cycle. Its value equals the time average of the mass flow rate during that cycle, and may change from cycle to cycle. (2) Heat transfer analysis relies on a bulk model in which the heat fluxes are evaluated at the average temperatures along the length of receiver tube. (3) All heat fluxes are uniform around receiver tubes circumference and length. That is, the heat transfer is developing under the boundary condition of a constant heat flux and the temperature distributions are axisymmetric. (4) The absorbed flux of solar energy is treated as incoming heat flux, even if it acts as a surface source of heat. (5) The conduction in all directions inside the walls of the receiver pipes is neglected. In the absence of radial conduction and internal heat generation, the wall temperature is the same in each pipe cross-section. Assumption #2 is quite correct in evaluating the convection heat transfer inside the absorber tube. However, it introduces addi-

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106

tional errors in estimating the free convection and radiation heat fluxes since the corresponding available correlation relationships are developed under uniform temperature condition. Hypothesis #5 is more restrictive than those used in [20,21]. This hypothesis allows to reduce the number of unknowns and, implicitly, the number of differential equations to be solved. As a consequence, it is expected that the heat gain is overestimated and the heat losses are underestimated. A cross section through the tubular receiver and the heat flux definitions are shown in Fig. 2. The incoming solar energy flux, Q_ S is reflected by the parabolic trough mirror towards the tubular receiver (TR). Due to the effect of the incidence angle and other optical losses, the heat flux received by the tubular receiver, Q_ TR , ðaÞ is smaller than Q_ S . A fraction of Q_ TR , denoted Q_ gls , is absorbed by ðaÞ the glass envelope and another fraction, named Q_ , is transmitted abs

through the glass envelope and is finally absorbed by the absorber ðaÞ tube. The largest part of Q_ , namely Q_ usf , is transferred to the air abs

flowing inside the absorber pipe and represents the useful heat supplied to the engine. The remaining fraction is transmitted back towards the glass envelope through convection and radiation, ðcÞ respectively, and the corresponding fluxes are denoted Q_ absgls

and

ðrÞ Q_ absgls .

In turn, the glass envelope transfers to the sky the radiðrÞ ation heat flux Q_ , and to the environment, the convection heat glssky

ðcÞ flux Q_ glsenv .

The energy balance equations for the absorber and glass surfaces are as follows:

dðT abs;m Þ ðaÞ ðrÞ ðcÞ ¼ Q_ abs  Q_ usf  Q_ absgls  Q_ absgls ds dðT gls;m Þ ðaÞ ðrÞ ðcÞ ðrÞ ðcÞ ¼ Q_ gls þ Q_ absgls þ Q_ absgsl  Q_ glssky  Q_ glsenv C gls ds

C abs

ð14Þ ð15Þ

where Cabs = Mabscp,abs and Cgls = Mglscp,gls represent the heat capacity of absorber pipe and glass envelope tube, respectively. The energy fluxes appearing in the right-hand side of Eqs. (14) and (15) are described in Sections 3.2.3–3.2.6. 3.2.2. Incident solar irradiance The total flux of solar energy incident on a solar collector consists of two components: (i) beam (or direct) solar energy flux, GB, which is received from the Sun without having been scattered by the atmosphere; (ii) diffuse solar energy flux, GD, which is received from the celestial vault. However, in case of solar collectors having higher concentration ratio, the diffuse part of solar

radiation is less important, so that only the beam solar energy flux GB is considered here. Many models have been proposed to estimate GB under clear sky conditions. A review of these models can be found in Ref. [22]. Here, the Hottel and Woertz clear sky model [23] is used. 3.2.3. Absorption of solar energy The solar energy flux incident on the collector aperture is:

Q_ S ¼ Hptc Lptc GB

ð16Þ

where Hptc and Lptc are the aperture width and length, respectively

(see Fig. 1). Due to optical losses, a lower energy flux, Q_ S , is incident on the receiver tube. The optical losses are taken into account by the effective optical efficiency of the glass envelope, genv, which is computed according to Refs. [20,21,24]. As a result, the energy flux collected by the tubular receiver is expressed by:

Q_ TR ¼ genv Q_ S

ð17Þ

The heat fluxes absorbed by the glass envelope and by the absorber tube, respectively, are given by the following relations: ðaÞ Q_ gls ¼ agls Q_ TR

ð18Þ

ðaÞ Q_ abs ¼ sgls aabs Q_ TR

ð19Þ

where agls and sgls represent the absorption and transmission coefficient of the glass envelope, respectively, while aabs stands for the absorption coefficient of the absorber tube. 3.2.4. Convection heat transfer between absorber tube and working fluid The heat flux transferred between absorber tube and working fluid is computed by using Eq. (11), for which ha has to be modeled. It is well known that the value of ha strongly influences the PTC heat losses. The higher the values of ha are, the lower are the PTC heat losses. Typically, the inner surface of the absorber tube is smooth, hence the values of ha are quite small. To improve the PTC efficiency, a heat transfer enhancement technique should be employed. Here, the roughness of heat transfer surface is increased by using transverse or helical repeated ribs of several configurations. The ribs are created by inserting a wire coil inside the absorber tube. The geometry of the proposed arrangement is shown in Fig. 3. For this ribbed (augmented) tube, the convection heat transfer coefficient and friction factor are computed by using the Ravigururajan and Bergles correlations [27]:

Fig. 2. Model for the PTC tubular receiver.

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107

According to Swinbank [30], the sky temperature is computed as Tsky = 0.0552Tenv1.5. Convection heat transfer takes place between the glass envelope and the environmental air. The convection heat flux is computed with the relation: ðcÞ Q_ glsenv ¼ pDgls;ex Lptc hglsenv ðT gls;m  T env Þ

Fig.3. Geometry of wire coil insertion inside the absorber tube. p-rib pitch, e-rib height, a-helix angle.

ðrbÞ

NuDabs;in ðsmÞ

NuDabs;in (

¼

8 < :

"



1 þ 2:64ReD0:036 abs;in




0:212 

e Dabs;in



f au e ¼ 1 þ 29:1ReaDReabs;in f sm Dabs;in

0:21 

p Dabs;in

ae 

90

15 )16 15 a aa 16 Sf 90

ap 

p

a 0:29

Dabs;in

Pr 0:024

#7 91=7 = ; ð20Þ ð21Þ

where superscripts (rb) and (sm) denote the ribbed and smooth surfaces, respectively. Here we are using smooth tube correlations proposed by Petukhov and Popov [25] for heat transfer, by Filonenko [26] for friction factor; relationships to compute the exponents aRe, ap and aa can be found in Ref. [19]. Eqs. (20) and (21) apply for ReD = 5000–250,000, Pr = 0.66–37.5 and e/D = 0.01–0.2, p/ D = 0.1–7, a/90 = 0.3–1. The following values of the tube parameters are used here: e/D = 0.047, p/D = 0.747, a/90 = 0.844. 3.2.5. Heat transfer between absorber pipe and glass envelope The heat transfer between absorber pipe and glass envelope occurs by radiation and convection. By assuming the diffuse-gray model, the net radiation heat flux exchanged between the absorber tube and glass envelope is [28]:



ðrÞ Q_ absgls

¼ pDabs;ex Lptc

r T 4abs;m  T 4gls;m 1

eabs

D

þ Dabs;ex gls;int



1egls





ð22Þ

egls 8

In the above relation, r ¼ 5:67  10 is Stefan-Boltzmann constant while Dabs,ex and Dgls,in represent the outer diameter of the absorber pipe and the inner diameter of the glass envelope, respectively. To reduce the radiation heat flux, the outer surface is covered by a selective coating with low emittance in the long wave length spectrum. The convection heat transfer between the receiver pipe and glass envelope depends on the gas pressure in the annulus bordered by the receiver pipe and the glass envelope. When the pressure exceeds 1 torr (133.3 Pa), the convection heat transfer is ruled out by the free convection mechanism. Below this threshold pressure, the mechanism of molecular convection occurs. The correlation of Ratzel et al. [29] shows that the annulus gas pressure is smaller when the convection heat transfer coefficient is lower. Here we assume very small annulus pressures (less than 0.0001 torr or 0.013 Pa), so that the heat flux transferred by convection is neglected. 3.2.6. Heat transfer from glass envelope to atmosphere Heat transfer from the glass envelope to the atmosphere occurs by radiation and convection. Note that both heat fluxes stand for tubular receiver losses. Radiation heat transfer happens between glass envelope and sky. The radiation heat flux is given by: ðrÞ Q_ glssky ¼ rpDgls;ex Lptc egls

h i T 4gls;m  T 4sky

ð23Þ

ð24Þ

Depending on wind condition, the convection is either forced (wind is present) or natural (no wind). In the case of no wind, the free convection mechanism is responsible for the heat exchange. As a result, the convection coefficient, hglsenv is computed by using Churchill and Chu correlation [31]:

( NuDgls;ex ¼

0:6 þ

)2

1=6

0:387RaDgls;ex ½1 þ ð0:559=PrÞ9=16 

8=27

ð25Þ

where the Rayleigh number is expressed by:

gbðT gls:m  T a ÞD3gls;ex

RaDgls;ex ¼

t2

Pr

ð26Þ

The physical properties of the air and the expansion coefficient, b, are evaluated at the mean temperature ðT gls;m þ T a Þ=2: The correlation holds true for 105 < RaDgls;ex < 1012 . When there is wind, the convection heat transfer from the glass envelope to the environment occurs by forced convection. Consequently, the Nusselt number may be estimated with the correlation relationships for external forced convection around an isothermal cylinder. Since the paper goal is to reveal the potential of PTC-JCREE system, this case is not considered here. 3.3. Mean cycle properties and performances The average per cycle of a property (say x) is determined by using the relation:

x ¼

1

sc

Z sc

xds

ð27Þ

0

Except is the mean specific enthalpy for which the mass average formula is employed:

¼ 1 h _ m

Z sc

_ s mhd

ð28Þ

0

The engine output power is computed as:

_ _ _ W eng ¼ W e  W c

ð29Þ

while the engine thermal efficiency is given by:

_ W

geng ¼ _ eng Q

ð30Þ

usf

The thermal efficiency of the PTC is calculated by taking the intercepted solar heat flux as a reference, so that:

gptc ¼

_ Q usf K h Q_ s

ð31Þ

The overall efficiency of the PTC-JCREE system is given by the ratio between the engine output power and the intercepted solar energy flux, resulting that:

gov erall ¼ geng gptc

ð32Þ

3.4. Physical and optical properties The temperature variation of air specific heat at constant pressure was modeled through a fourth order polynomial function [32].

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By using it, the link between temperature and specific enthalpy was also established. The adiabatic exponent k employed by Eq. (4) was computed at the high side temperature Thigh. The dependence of air viscosity and thermal conductivity on temperature was modeled with Sutherland laws [33]. To complete the model, some manufacturing materials for the PTC and their optical properties should be specified. The assumptions are as follows. The absorber pipe is made of stainless steel 304L (q = 8030 kg/m3, cp = 500 J/kg K) and its outer surface is covered by a Luz Cerment coating type. The glass envelope is built of Pyrex glass (q = 2230 kg/m3, cp = 750 J/kg K). The optical properties required by model include the glass envelope transmissivity, absorptivity and emissivity and the selective coating absorptivity and emissivity. The glass envelope absorptivity and emissivity are independent of temperature and selective coating type [20]. The values used in the model are agls ¼ 0:02, and egls ¼ 0:86. The other quantities depend on the type of selective coating. For Luz Cerment coating, the envelope transmissivity and the coating absorptivity does not depend on temperature and have the values sgls ¼ 0:935 and aabs ¼ 0:92, respectively. The coating emissivity is temperature dependent according to the relation eabs ¼ 0:000327  T  0:065971, which is valid between 373 K and 673 K [20].

file and the value of solar irradiance is updated. The third loop performs the previous calculation for a time interval of fifteen minutes. At the end of the loop various quantities are saved in another output EXCEL file for post-processing. Finally, the fourth loop performs the calculations for a time interval prescribed by the user, which is a multiple of fifteen minutes. The start-up module is intended to provide the correct initial value solution for the ODE system, to be used when the operation of the PTC-JCREE system begins. It is a separate code and consists of the first two inner loops previously presented. In this case, the guess values are set by assuming the air temperature and pressure inside the cylinders and absorber tube equal to the atmospheric temperature and pressure, respectively, and by estimating the initial absorber and glass temperatures as well as the cycle mean and heat transfer parameters. To get a quick engine pressurization, the expander inlet and exhaust valves were kept closed and opened respectively, until a pressure of about 300 kP was established in the absorber tube [8]. Depending on the accuracy of these estimations, some convergence problems may appear during firsts cycle computations. Once the stable operation regime was reached (meaning that the mass flow rate entering the compressor equals the mass flow rate exiting the expander), the resulting variable values are saved in files and used in the main computations.

4. Numerical procedure 5. System set-up procedure The model consists of eight ordinary differential equations (ODEs) [i.e., Eqs. (1) and (2) applied for the compressor, expander and absorber tube volume, respectively, and Eqs. (14) and (15) used for the PTC absorber pipe and glass envelope, respectively] with eight unknown, namely Mc, Tc, Ma, Ta,m, Me, Te, Tabs,m, Tgls,m. This ODE system and the associated initial values constitute an initial value problem (IVP) and should be integrated over the cycle time interval scycle = 1/N (or over a full rotation of the crank shaft). At every integration time step, the pressure values in different engine components were computed by using Eq. (3). As a result, a set of eight values, computed at each integration time step during the time interval [0, scycle], quantifies the operation of the PTCJCREE system. The incident solar irradiance was updated at time intervals of one minute, by using Hottel and Woertz clear-sky model. The ODE system was implemented in the MATLAB environment [34]. Several routines (m-functions), were written to code each part of the mathematical model described in previous section. Some of them employ various MATLAB built-in functions. Thus, the function ODE 23 was used for solving the ODE system with a relative error tolerance of 106; the function FZERO is employed to solve the equation h3 = h(T3) in order to find the temperature T3 of the air entering the expander; the function TRAPZ was used for evaluating the integral needed to find mean properties per cycle. Other routines deal with the computation mass flow rate passing the compressor and expander valves, the temperature variation of air physical properties and the specific enthalpy calculation for a given temperature, the time dependence of solar beam radiance, the computation of heat transfer coefficients and heat fluxes characterizing the PTC collector, etc. These routines numerically describe the operation of JCEE-PTC system and are used as basic sequence in both main and startup modules of the program. The main program module consists of four embedded loops. The inner loop solves the ODE system for a full engine cycle. The mean cycle properties and the heat transfer coefficients, are computed at the end of this loop. The second loop repeats the cycle computation during a minute. The final solution of the previous cycle as well as the mean cycle properties constitute the initial values for the next cycle computation. At the end of this second loop, the engine performances and mean cycle properties are saved in an output EXCEL

The set-up procedure starts by using guess values for some parameters of the system. To obtain reliable values for the geometrical parameters of the PTC module, the volume of the absorber tube was set at a value of 0.0125 m3. Since the turbulent air flow regime inside the absorber tube was targeted, an inner diameter of 0.051 m was chosen, resulting a PTC length of Lptc = 6.1 m. These sizes are similar to that of ISP PT-1 collector [10]. The PTC is kept at a fixed tilt angle of b = 21°. This value was chosen to get the maximum solar irradiation near the noon [35]. The compressor swept and clearance volumes were fixed at values VSc = 22.6  104 m3 and VCc = 0.11Vcs respectively, as used in Ref. [7,8]. These values were adopted in order to obtain an exhaust compressor air pressure of about 400 kP. To set up the other parameters of system, as well as the engine start up time, several numerical simulations were performed. During this procedure, some malfunction operation regimes were identified. The malfunction is characterized by undesired reversed gaps between the air pressure at the end of the compression or expansion process occurring in compressor or expander cylinder, on one hand, and the pressure in the absorber tube or the atmospheric pressure, on the other hand. For example, Fig. 4 reveals two specific malfunctions of the expander. At the end of the compression process, the air pressure inside the expander cylinder is greater by about 25 kPa than that in the absorber tube. As a result, during the first stage of the admission process, the air is flowing from the expander towards the absorber and the air mass inside the cylinder is decreasing instead of increasing. Besides, at the end of expansion, the pressure inside the expander is lower that the atmospheric pressure by about 8 kPa, so that, for a certain time interval during the exhaust process, the atmospheric air is entering the expander. Consequently, during this time interval, the air mass enclosed in the expander cylinder is increasing instead of decreasing. In the following, these malfunction phenomena will be named gas over-compression and gas over-expansion inside the expander cylinder, respectively. Note that gas under-compression and gas under-expansion malfunctions can occur inside the compressor cylinder, too. The over-compression and over-expansion malfunctions can be fixed either by diminishing the values of some geometrical param-

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109

Fig. 4. Malfunction operation regime of the expander. (a) Air pressures distributions inside the compressor (c), expander (e) and absorber tube (a, m); (b) air mass variation inside the compressor (c) and expander (e) cylinders. Circles show undesirable situations.

eters of the expander cylinder or by growing the air temperature inside the absorber tube. Three ways to increase the value of Ta,m are considered here: (i) by delaying the morning starting time of the engine; (ii) by increasing the aperture area of PTC and/or by adjusting the closing advance angles of the expander valves; (iii) by reducing the engine rotation speed. In the first two cases, the heat flux received by the absorber tube is increasing, while the engine rotation speed remains unchanged, as well as the average mass flow rate of air crossing the system. In the third case, the average mass flow rate is decreasing, for the same heat flux received by the absorber tube. The results obtained for this last case are illustrated in Fig. 5, which shows the air pressure variation inside various components of the system, for the same engine configuration and solar irradiance value used to draw Fig. 4, but for lower rotation speed. The increase of the air temperature inside the absorber tube leads to the increase of pressure and also to the decrease of air mass entering the expander cylinder during

the admission process. Therefore, even if the rotation speed is now smaller the engine power is higher. Note that during the simulations performed to set-up the design parameters of the system, most of the above actions were employed. Based on simulation results, the morning starting time of the engine was set to 8:46 AM and the rotation speed was set to Nr = 480 rpm. It has been found that, for the adopted values of VSc and VCc, Nr = 480 rpm is the lowest rotation speed for which the engine properly operates around the noon (i.e., without gas under-compression malfunction occurring in the compressor cylinder). The final design values of the PTC-JCREE parameters are shown in Table 1. Due to the increase that solar irradiance has in the morning, the air temperature and pressure inside the absorber tube are gradually increasing. As a result, at the beginning of the expander admission, the air pressure gap existing between the admission manifold and the expander cylinder is increasing. The same phenomenon,

Fig. 5. Proper operation regime of expander. (a) Air pressures distributions in compressor (c), expander (e) and absorber tube (a, m); (b) air mass variation inside the compressor (c) and expander (e) cylinders.

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110 Table 1 Design parameters of PTC-JCREE components. Component

Design parameters

Compressor

Swept volume, VSc = 22.6  104 m3 Clearance volume VCc = 0.11Vcs Ratio of connecting rod to crank radius kc = 3 Delay of inlet valve opening: IVO = 35° Delay of exhaust valve opening: EVO = 118°

Parabolic trough collector

Expander

Length of aperture, Lptc = 6.1 m Width of aperture, Hptc = 2.9 m Inner diameter of glass envelope, Dabs,in = 0.051 m Thickness of glass envelope, dgls = 1 mm Inner diameter of absorber tube Dgls,in = 0.075 m Thickness of absorber tube, dabs = 1 mm Volume of the absorber tube, Vabs = 0.0125 m3 Swept volume, VSe = 35.5  104 m3 Clearance volume, VCe = 0.09VSe Ratio of connecting rod to crank radius, ke = 3 Advance of inlet valve closing: IVC = 111° Advance of exhaust valve closing: EVC = 25°

but at smaller scale, is happening at the start of expander exhaust process. These behaviors are illustrated by Fig. 6, which shows the air pressure variation in various JCREE components at 10:31 AM, for Nr = 480 rot/min. To reduce these pressure differences, the air temperature inside the absorber tube has to be diminished. This can be achieved by increasing the mass flow rate of air entering the absorber tube, i.e. by increasing the engine rotation speed. Fig. 7 reveal the improvements in air pressure variation achieved for Nr = 600 rot/min. Numerical simulations proved that, for a fixed configuration of the components, the JCREE-PCT system can operate properly (i.e. without encountering malfunction operation regimes) only if the absorber mean air temperature ranges between specified lower and upper bounds. If the temperature is below the lower bound, a reversed air flow occurs during the expander admission and/or exhaust processes (over-compression and/or over-expansion malfunction). If the temperature is above the higher bound, a reversed air flow occurs during the air exhaust from the compressor (undercompression malfunction). Moreover, inside the allowed air temperature interval, the value of the JCREE output power depends on the engine rotation speed. In order to reveal the full potential

of the PTC-JCREE system to generate work a series of rotation speed values has to be considered. 6. Operation strategy for higher performance Now we focus on engine operation. First, the effect of the engine rotation speed on the output power is analyzed (Section 6.1). Second, a strategy to increase the total work supplied during the whole operation time interval is proposed (Section 6.2). Third, results are compared and discussed (Section 6.3). 6.1. Constant rotation speed Four operation regimes, characterized by constant engine rotation speed, have been considered, namely Nr = 480, 540, 600 and 660 rot/min. For each value of the engine rotation speed, the time variation of the ECREE-FTC system parameters and performances is investigated and the output work is computed. The engine operation starts in early morning. The precise starting time is the moment when the engine reaches the first proper operation regime (i.e. without malfunction). That moment is a function of the rotation speed. Once the stabilized operation regime was reached, the full numerical procedure is employed and the solar irradiation is changed from minute to minute according to Hottel and Woertz model. To avoid the gas over-expansion in the expander cylinder, in the afternoon the engine operates the same amount of time as in the morning. Thus, malfunction is avoided during the whole day and the engine works in a proper way. Note that the starting time is delayed when the rotation speed Nr is augmented. Therefore, the operation time interval of the system is diminishing during a day if the rotation speed is increasing. The longest operation time interval corresponds to the lowest Nr value. The main parameters influencing the performance of PTC-JCREE system are the solar irradiance GB, the engine rotation speed, Nr,  a;m . However, these _ T 3 , and P and the cycle mean parameters, m, parameters are not all independent. Indeed: (i) the mass flow rate entering the engine depends on the engine rotation speed; (ii) the heat exchange between the absorber tube and the flowing air takes place under constant flux boundary condition; (iii) the temperature T 3 is varying with respect to the solar irradiance GB and air

Fig. 6. Pressure profiles for Nr = 480 rot/min; time: 10:31 AM; Engine output power: 1762.13 W. (a) Variation of compressor (c), absorber tube (a, m) and expander (e) air pressures with respect of crank angle. (b) P-V diagram for compressor and expander cylinder.

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111

Fig. 7. Pressure profiles for Nr = 600 rot/min; time: 10:31 AM; Engine output power: 1817.44 W. (a) Variation of compressor (c), absorber tube (a, m) and expander (e) air pressures with respect of crank angle. (d) P-V diagram for compressor and expander cylinder.

Fig. 8. Time variation of mean parameters describing the JCREE cycle. (a) Mean temperature of air entering the expander. (b) Mean pressure of the air entering the absorber tube.

_ is mainly influenced by Nr mass flow rate. Thus, the variation of m only, while the mean air temperature at absorber outlet, T 3 , and  a;m , depend on both the mean air pressure inside the absorber, P Nr and GB (or, in other words, on the solar time). As a result, the independent parameters of the PTC-JCREE system are Nr and the solar time.  a;m as a function of the solar Fig. 8 shows the variation of T 3 and P time, for different values of the engine rotation speed. At Nr = const., the air mass flow rate entering the absorber is quite constant, so the profile of T 3 depends on the solar irradiance GB, whose value increases from sunrise to noon and symmetrically decrease from noon to sunset. For given solar time value (or, in other words, for given value of GB), the increase of Nr determines _ and, further, the reduction of T 3 (due to the the increase of m boundary condition qs = const.). As a result, T 3 increases by increasing the solar irradiance and decreases by increasing the engine rotation speed. Fig. 8a also reveals the range of T 3 values for which this specific configuration of PTC-JCREE system can properly oper a;m is related to Ma and T a;m . Since Ma is ate. Eq. (3) shows that P quite the same for all operation regimes (around 0.028 kg), the  a;m depends only on the time variation of T a;m , time variation of P  a;m increases which is obvious similar to that of T 3 . As a result, P by increasing solar irradiance, but decreases by increasing engine rotation speed.

Fig. 9a reveals the time variation of the engine power with respect to solar time and engine rotation speed. The PTC-JCREE system performs like an open Joule cycle. Then, the engine power can _ eng ¼ mw, _ be expressed as W where w is the engine specific work. Usually, the specific work depends on both compressor exhaust air pressure and expander inlet air temperature. As previously  a;m and T 3 are linked together so that only T 3 acts as an shown, P independent variable. At constant rotation speed (Nr = const.), the time variation of the engine power is mainly ruled by the dependency between w and T 3 . As a result, the output power increases toward the noon and decreases afterwards. However, due to the thermal inertia of the absorber tube and glass envelope, the highest value of the engine power occurs shortly after the solar noon. For given solar time value (or, in other words, for GB = const.), _ and further, the the increase of Nr determines the increase of m reduction of T 3 . Accordingly, the specific work decreases, but the engine power may increase due to the increase of the air mass flow rate. This is happening, for instance, when the rotation speed is changed from Nr = 480 to Nr = 540 rpm (see Fig. 9a). In the morning and afternoon the difference between the two power values is very small, but it grows up as the solar time approaches to the noon. Changing the engine rotation speed from Nr = 540 to Nr = 600 rot/ min brings, for a short time interval in the morning and afternoon, an opposite behavior of the power gap variation. During a time

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The time variation of the PTC thermal efficiency, gptc is shown in Fig. 10. The PTC thermal efficiency mainly depends on the glass envelope mean temperature, T gls;m , which determines the amount of PTC heat loss. Thus, higher values of T gls;m yield lower values of gPTC. The time variation of T gls;m strictly follow the time variation of T a;m , which is similar to that of T 3 (see Fig. 8a). As a result, at con-

stant rotation speed (Nr = const.), gPTC decreases until the noon, when it reaches a minimum value and increases afterwards. When Nr is increased, T gls;m decreases due to the mass flow rate augmentation, so that, at the same value of GB, the PTC efficiency grows up. The time distribution of the overall efficiency of the PTC-JCREE system, gov erall , is shown in Fig. 11. Due to the opposite variations of geng and gptc, some local extreme values exist. Nevertheless, the differences between the maximum and minimum values are small. These differences become smaller by increasing Nr and they vanish at the highest rotation speed value. Consequently, for Nr = 660 rpm, only a maximum point exists. Besides, for engine rotation speeds beyond 540 rpm the operation time interval is narrow and the values of the overall efficiency are almost constant in time. The engine output power at constant rotation speed was integrated over a time interval of fifteen minutes. Computations were performed for four different values of the rotation speed Nr, namely 480, 540, 600 and 660 rpm. Results are shown in Fig. 12. In a large interval of time, i.e. 10:46–13:15, the increase of Nr makes the output work to increase. However, the work difference (work gain) for a pair of two consecutive values of Nr slowly increases by increasing the irradiance and steadily decreases by increasing the rotation speed pair values. Thus, for the same pair of two consecutive values

Fig. 9. Engine performance as function of solar time and rotation speed. (a) Engine output power and (b) thermal efficiency.

interval of about 5 min, the power associated with the latter value of Nr (i.e. 600 rot/min) is lower than that developed by the former value (i.e. 540 rot/min). Outside this time interval, the power obtained at Nr = 600 rpm is greater than that associated with Nr = 540 rpm, but the power difference is smaller than in the previous case (i.e. when the effects of the rotation speeds Nr = 480 to Nr = 540 rpm have been compared). The change in the rotation speed from 600 rpm to 660 rpm narrows the time interval when the engine power rises and enlarges the interval of time when _ accordingly, the power decreases. Clearly, the increase of Nr, and m is able to compensate the reduction of T 3 only at lower values of Nr. When Nr increases, the time interval when the system develops higher power decreases and the power gain diminishes. Consequently, for the configuration of PTC-JCREE system considered here, a rotation speed greater than 660 rpm is not suitable, because, as revealed by Fig. 9a, it is not able to provides a power increase. The variation of thermal engine efficiency, geng , with respect to the solar time and engine rotation speed is presented in fig. 9b. It is well known that in case of the common internal combustion Joule cycle, geng depends only on the compressor exhaust pressure. As a result, the time variation of geng may be explained by using the  a;m (see Fig. 8b). However, the predicted values time variation of P

Fig. 10. Time variation of PTC thermal efficiency with respect to the solar time and engine rotation speed.

of the engine efficiency are significantly lower than those obtained from the classical relation, geng ¼ 1  ðP1 =Pa;m Þðk1Þ=k . This is explained by the fact that the present model takes into account the temperature variation of the specific heats, as well as some irreversibilities associated with the air flow through valves and absorber tube. The present approach is more realistic than the common academic approach.

Fig. 11. Time variation of the overall efficiency of the PTC-JCREE system with respect to the solar time and engine rotation speed.

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113

Fig. 12. Engine output work computed on time intervals of fifteen minutes as function of engine rotation speed and solar time.

Table 2 Output work and overall efficiency during the time interval of proper operation, for four values of the rotation speed. Nr [rot/min]

Operation time interval [h:min]

Output work EW [kJ]

Input heat EQ [kJ]

Overall efficiency

480 560 600 660

8:46–15:15 9:16–14:45 9:46–14:15 10:16–13:45

39269.5 35267.4 30128.1 24105.7

292211.8 257288.8 217554.7 173696.8

0.134 0.137 0.138 0.139

of Nr, lower gain of output work is obtained on the time interval 10:46–11:00 than on the time interval 11:01–11:45; and, on the same time interval of fifteen minutes, higher amount of work is gained when changing Nr from 480 to 540 rpm than when changing it from 540 to 600 rpm. For solar time earlier than 10:46, or later than 13:15, the increase of Nr does not necessary lead to an increase of the work production. For example, between 10:16 and 10:31 the rotation speed 660 rpm provides lower work than that delivered by the rotation speed 600 rpm. The values of the output work, computed for each considered rotation speed on its entire (or allowable) operation time interval, are presented in Table 2. There, the solar energy intercepted by the PTC mirror and the overall efficiency of the PTC-JCREE system are also shown. The highest output work is obtained for the lowest value of the rotation speed. This means that the increase of the output work, obtained by the increase of the engine rotation speed on the time intervals of fifteen minutes, does not exceed the loss of the work production caused by the reduction of the entire operation time interval. However, the overall efficiency is almost the same for all four rotation speeds considered here. 6.2. Time-variable rotation speed Results presented in Section 6.1 suggest that switching at appropriate time between different values of the rotation speed could increase the output work provided by the PTC-JCREE system during the maximum possible operation time interval (i.e. 8:46– 15:15). Such a switching procedure was implemented here. The switching times were estimated as follows. For given fifteen min-

utes interval, the output work has been computed for all the considered values of Nr, and the largest value of the output work has been selected. That largest work value was associated with the operating value of the rotation speed, Nr,opr. The procedure is repeated for the next fifteen minutes interval. The switching time comes out from the change of the value of Nr,opr for two adjacent intervals of fifteen minutes. The sequence of switching is presented in Table 3. From the point of view of the numerical solution, the rotation speed was updated at each switching time and the former numerical solution was used as the initial condition for the solution corresponding to the new Nr value. Figs. 13 and 14 present the time variation of the mean cycle temperature T3 and the performance indicators of the PTC-JCREE system, respectively. The sudden change of the engine rotation speed determines peculiar unsteadiness of these quantities, which are also influenced by the thermal inertia of the tubular receiver components. This unsteadiness is attenuated within a time interval of about ten minutes. Afterwards, the time variation becomes identical to that shown in Figs. 8–10 for specific values of Nr and time intervals. The value of Nr,opr is associated to the largest output work value for a given period of time, but it also corresponds to the highest power value supplied by the system during that time period. With few exceptions, this rotation speed is the greatest of all available. As a result, the engine power lies inside the entire power range (see fig. 14a), i.e. 1200–2000 W. However, the intervals of variation of T3, geng, gptc and gov erall (see Figs. 13 and 14b–d) is smaller than those shown in Figs. 8a and 9–11. By far, the most important adjustments occur in case of the efficiencies geng, gptc, goverall,

Table 3 Time intervals of constant rotation speed considered for increasing the engine supplied work. Nr,opr [rot/min]

480

540

600

660

600

540

480

Operation time interval

8:46–9:15

9:16–9:45

9:46–10:45

10:46–13:15

13:16–14:15

14:16–14:45

14:46–15:15

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Fig. 13. Time variation of cycle mean highest temperature for different switching time of engine rotation speed.

which have very narrow variation during the day. Thus, except for some peaks determined by the change of Nr, the engine thermal efficiency is varying between 0.25 and 0.262, while the PTC efficiency lies between 0.65 and 0.675. Since the overall efficiency is the product of the other two efficiencies, its variation interval is also small, between 0.134 and 0.14, the highest values occurring at noon. Practically, one may assume constant mean values over the day, for all these efficiencies. This is an important result. Fig. 15 shows the time variation of the work supplied on time intervals of fifteen minutes, obtained by using the switching procedure between different values of the rotation speed. Small differences appear when comparing these results with those shown in Fig. 12, for the same value of Nr. These differences occur only on the first time interval of fifteen minutes after Nr is switching. The differences are due to the thermal inertia of the receiver tube components, whose time scale is less than 15 min. By summing up the amounts of work obtained by using the switching procedure, one finds that the engine supplies during

the entire operation interval an amount of work of 40,380 kJ, with a mean overall efficiency of about 0.138. One may think that switching among four values of engine rotation speed is quite complicated. Fig. 12 suggests that the work gain obtained by changing Nr from 600 to 660 rot/min is less important. So, if one keeps Nr at a value of 600 rot/min (instead of 660 rot/min) during the time interval 10:46–13:15, one obtains during the total operation time an amount of work of 40,322 kJ, with an overall efficiency quite similar with that obtained in previous case. Obviously, the decrease of work production compared with 40,380 kJ does not justify the technical complication associated with the Nr switching. Further, if one restricts the engine operation interval from 9:16 to 14.46, only two values of the rotation speed are involved in the switching procedure (i.e. 540 and 600 rot/min). Then, the supplied work is 35,586 kJ, which is 88% of the work generated in the first case, while the mean overall efficiency is kept at the same value 0.138. Under these circumstances, switching among 480, 540 and 600 rpm is the best strategy. 6.3. Constant rotation speed vs time-variable rotation speed Now, compare the results obtained in Sections 6.1 and 6.2. When using the constant rotation speed strategy, one gets an output work of 39269.5 kJ at Nr = 480 rpm. In the case of switching the rotation speed between among 480, 540 and 600 rpm, the produced work is 40,322 kJ. Note that in both cases the engine operates on the whole allowable time interval (i.e. 8:46–15:15). Fig. 16 shows the time dependence of work production, computed from the start-up time to current time, for four values of Nr and for the switching strategy, respectively. In each case, at the end of its allowable operation time interval, one retrieves the results reported in Sections 6.1 and 6.2. For the strategy Nr = constant, the rotation speed of 480 rpm is the best choice, since it offers at any time the highest output work. Clearly, the work gain

Fig. 14. Time variation for several performance indicators of the PTC-JCREE system and the appropriate switching time of engine rotation speed. (a) Engine power; (b) engine thermal efficiency; (c) PTC thermal efficiency; (d) overall efficiency of the PTC-JCREE system.

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Fig. 15. Time variation of the highest work supplied on time intervals of fifteen minutes in case of using the switching procedure between different values of the rotation speed.

Fig. 16. Time dependence of work production for different constant values of the rotation speed and for the swiching strategy among 480 rpm, 540 rpm and 600 rpm.

due to an early start-up is responsible for this behavior, despite of the fact that the engine power increases by increasing Nr. The work gain obtained by using the Nr swiching strategy, compared to that obtained in case of constant value Nr = 480 rpm, increases along the day; however, even at the end of the operation interval, this work gain is not too large (i.e. 1052.5 kJ only). On the other hand, the work loss due to mechanical friction is not taken into account in this paper. Since this work loss increases with increasing Nr, the output work difference between these two cases is clearly smaller in practice. As a result, one may consider that a constant value of Nr = 480 rpm is the best choice if the operation time can lie on the entire allowable time interval (i.e. 8:46–15:15). Now, assume that clear sky conditions are not always met along the entire allowable time interval of operation, but only on a specific (shorter) time interval within it. Then, Fig. 16 can be used to estimate the work produced on this specific interval of time, by computing the difference between the values of the output work at its endpoints. In this case, the largest amount of work may be obtained for another value of Nr than that of 480 rpm. Two examples are given now. In the first example, the engine starts-up later than 8:46 but it shuts down at the end of the allowable operation time interval. As shown in Table 2 (see column 2), the time when

the engine shut down depends on Nr, so it is 15:15 for 480 rpm, 14:45 for 540 rpm, etc. The largest amount of work is obtained, again, at Nr = 480 rpm, because among all values of rotation speed considered here, this value assures the longest operating time interval. The second example refers to an operation interval symmetric around the noon, but shorter than the entire allowable time interval. In this case, higher rotation speeds yield larger values of work. Thus, for the time interval lying between 9:30–14:30, the amount of work 32612.4 kJ is obtained at Nr = 540 rpm, while between 10:00–14:00, the amount of work 27169.3 kJ corresponds to Nr = 600 rpm. 7. Conclusions Ericsson engines, in contrast to Stirling engines, are not very often treated in literature. A solar-driven Joule cycle reciprocating Ericsson engine (JCREE) designed for small scale applications has been studied in this paper. The JCREE is coupled with a parabolic trough collector (PTC), oriented along E-W axis. The two main contributions of this work are as follows. First, several malfunction phenomena may occur during JCREE operation. Examples are the gas under-compression occurring in com-

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pressor cylinder and the over-compression and/or over-expansion of the gas happening in the expander cylinder. These phenomena have not been specifically treated in previous studies. Here, a time dependent model has been developed and used to carefully design the PTC-JCREE system, allowing to avoid improper JCREE operation. Second, the work supplied during a day by the JCREE depends on the value of the engine rotation speed. Detailed dynamic simulations emphasized an operation strategy able to increase the output work during a day. The strategy is based on a switching procedure between different values of the rotation speed. Specific results are reported for the design and operation of a PTC-JCREE system placed at Bucharest latitude (44°2500 ). A clear sky summer (June 21th) has been considered. The time dependence of the beam solar irradiance is considered and the PTC tilt angle is kept constant during the day at a value which ensures a maximum solar irradiance interception at noon. At the reference (constant) engine rotation speed of 480 rpm, the specific PTC-JCREE system treated here generates in a day the work 39,270 kJ at overall efficiency 0.135. When the rotation speed increases, the operation time interval diminishes and, as a consequence, the output work decreases. A better operation strategy consists of switching among three values of the engine rotation speed, namely 480, 540 and 600 rpm. In this case, the output work per day is 40,322 kJ and the overall efficiency is 0.137. Taking into account the small increase of performance brought by this improved strategy in comparison with the reference strategy and the technical complications associated with the change of the rotation speed during the operation time, one may conclude that the constant rotation speed of may represent a suitable choice. Note that in both cases the overall efficiency is around 0.135 and is quite constant along the whole operation time interval, which ranges from 8:46 to 15:15.

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