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Thermodynamic analysis of a gas turbine cycle equipped with a non-ideal adiabatic model for a double acting Stirling engine
Energy Conversion and Management 147 (2017) 120–134
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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Thermodynamic analysis of a gas turbine cycle equipped with a non-ideal adiabatic model for a double acting Stirling engine Mahmood Korlu a, Jamasb Pirkandi a,⇑, Arman Maroufi b a b
Department of Aerospace Engineering, Malek Ashtar University of Technology, Tehran, Iran Faculty of Industrial and Mechanical Engineering, Islamic Azad University, Qazvin Branch, Qazvin, Iran
a r t i c l e
i n f o
Article history: Received 25 January 2017 Received in revised form 24 March 2017 Accepted 13 April 2017
Keywords: Stirling engine Gas turbine Hybrid cycle Non-ideal adiabatic
a b s t r a c t The aim of this study is to investigate the thermodynamic performance of a gas turbine cycle equipped with a double acting Stirling engine. A portion of gas turbine exhaust gases are allocated to providing the heat required for the Stirling engine. Employing this hybrid cycle improves gas turbine performance and power generation. The double acting Stirling engine is used in this study and the non-ideal adiabatic model is used to numerical solution. The regenerator’s net enthalpy loss, the regenerator’s wall heat leakage, the energy dissipation caused by pressure drops in heat exchangers and regenerator are the losses that were taken into account for the Stirling engine. The hybrid cycle, gas turbine governing equations and Stirling engine analyses are carried out using the Matlab software. The pressure ratio of the compressor, the inlet temperature of turbine, the porosity, length and diameter of the regenerator were chosen as essential parameters in this article. Also the hybrid cycle effects, efficiency and power outputs are investigated. The results show that the hybrid gas turbine and Stirling engine improves the efficiency from 23.6 to 38.8%. Ó 2017 Published by Elsevier Ltd.
1. Introduction Finding new energy sources seems to be inevitable because the ever increasing worldwide energy demand, the limited fossil fuel resources, and the environmental pollution. Gas turbine is one of the energy sources with many applications in various industry such as aerospace, auxiliary power units in airplanes, industrial projects, transport industry and cogeneration systems. Every gas turbine consists of a compressor, combustion chamber where air and fuel are mixed and burned and a turbine that converts the hot and compressed gases energy into mechanical energy. A portion of the mechanical energy produced by the turbine is consumed to rotate the compressor and the rest of it is transported to the generator to produce electricity. Gas turbine’s works based on the Brayton cycle. The cycle was first proposed by John Barber [1], an English inventor. The researches of the recent 7 decades have increased the gas turbine efficiency. Also researchers try to find new methods to improve the efficiency without an overhaul. One way to improve the gas turbine efficiency achieved by changing operational conditions (temperature and pressure) and various cooling methods, the other way is to combine the gas turbine with
⇑ Corresponding author. E-mail address: [email protected] (J. Pirkandi). http://dx.doi.org/10.1016/j.enconman.2017.04.049 0196-8904/Ó 2017 Published by Elsevier Ltd.
another thermal engine, which can significantly improve power and efficiency. Combing the gas turbine with the Rankine and the Stirling cycles is an effective way of increasing efficiency. The gas turbine cycle and the Stirling cycle combination results a wholly different configuration. In this type of hybrid cycle, exhaust gases enter Stirling engine heat exchanger and provide the heat required by the Stirling engine [2]. In 2005, Pollikas estimated, for the first time, that using a bottoming Stirling cycle fed by exhaust gases of an RB211 Rolls-Royce gas turbine (27.5 MW electrical power) can recover 9 MWs. Adding a Stirling engine to a gas turbine can make a 47.7% overall efficiency[3]. Many researchers interested in Stirling engines due to low pollutant emissions, low sound production and the ability to operate with various fuels [4–6]. In fact, a Stirling engine is an external combustion engine and can use any external heat source and convert it into mechanical energy. The studies carried out by researchers on Stirling engine design and applications have produced very promising results. Construction of a solar engine with a 10 kW axial power is one of the most significant advances in Stirling technology. Combined heat and power generation is one of the new ideas that have been developed by Stirling manufacturers and are used in power plants. Recently, there are new ideas for the Stirling engine applications such as satellite power supply and being proposed as an alternative to steam turbines in nuclear power plants. Stirling engines’ physical structure consists of five
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Nomenclature A Cp Cv d f g h k l LHV m M n_ NTU P Pr Q_ Q R Re rp St T TIT V _ W W
area [m2 ] specific heat at constant pressure [J/kg K] specific heat at constant volume [J/kg K] hydraulic diameter [m] friction factor coefficient mass flux [kg/m2 s] enthalpy [kJ/kmol] thermal conductivity [W/m K] length [m] lower heating value [kJ/mol] mass [kg] mass of working gas in the Stirling engine [kg] molar flow rate [mol/s] number of transfer units pressure [bar] non-dimensional Prandtl number heat transfer rate [kW] heat [kJ] universal gas constant [8.314 J/kmol K] non-dimensional Reynolds number compressor pressure ratio non-dimensional Stanton number temperature [K] turbine inlet temperature [K] volume [m3] power [kW] work [kJ]
Greek symbols density [kg/m3 ] efficiency h crank rotational angle [°] c ideal gas specific heat ratio e heat exchangers and regenerator effectiveness l viscosity [kg/m s]
q g
subsystems, and each the subsystem is considered a control volume. The engine consists of two spaces with varying volumes, named expansion space and compression space, and three heat exchangers with constant volume, named heater, cooler and regenerator. Many researchers study the Stirling engine ever since its invention by Robert Stirling. The first acceptable mathematical analysis of the Stirling cycle was developed by Schmidt, fifty years after its invention [7]. Schmidt’s analysis was based on isotherm expansion and compression spaces, in his approach the thermodynamic models were linearized and therefore, the initial power and engine efficiency calculations were fairly simple. Finkelstein [8] improved Schmidt’s thermodynamic analysis and proposed the first adiabatic analyses. Urieli and Berchowitz [9] used the thermodynamic model to calculate power output and efficiency of Stirling engines. Fermosa and Despesse [10] modelled the engine using an isotherm model to investigate dead spaces’ effects on the engine power output and efficiency. Popescu et al. [11] shows that the low performance is essentially caused by the non-ideal thermal regenerator. Kaushik, Cun-quan and Wu [12–14] proved that thermal regenerator effectiveness factor, thermal conductivity between engine and reservoir have the highest impacts on Stirling engines performance. Kongtragool and Wongwises [15] studied the effects of regenerator effectiveness and dead spaces on Stirling engine inlet heat and efficiency. Costa et al. [16] and Timoumi et al. [17–20]investigated Stirling engine losses and irreversibilities through adiabatic modeling. Thombare and Verma [4] have con-
Subscripts a actual process a.c air compressor c compression space cc combustion chamber ck interface between compression space and cooler clc compression clearance volume cle expansion clearance volume diss dissipation e expansion space elec electrical f.c fuel compressor gen generator gt gas turbine g gas h heater he interface between heater and expansion space k cooler kr interface between cooler and regenerator loss loss r regenerator rh interface between regenerator and heater rloss net enthalpy loss s isentropic process suth Sutherland st Stirling engine swc compression swept swe expansion swept w wall wg wetted area wh heater wall wk cooler wall wr regenerator wall wrloss losses from the regenerator walls
ducted a series of studies in which they gathered the available technologies and recent advances in Stirling engine analyzation and provided some suggestions as how to use them. Tavvakolpour et al. [21] investigated the gamma type Stirling engine, they used the Schmidt theory to solve the equations in isotherm form and used flat panel to absorb solar energy as the hot thermal source. Gostante and Invernizzi [22] modelled the Stirling engine and then investigated the effects of various gases on the engine output and efficiency. In the present study, the energy of a gas turbine exhaust gases are used as the hot source in a double acting Stirling engine. The thermodynamic equations of every single component of the gas turbine and double acting Stirling engine hybrid cycle are simulated by a code written in Matlab. The non-ideal adiabatic model is used to model the Stirling engine. Then, with parametric study for the mentioned hybrid system, the effect of gas turbine inlet temperature, compressor pressure ratio, regenerator porosity and regenerator length and diameter on gas turbine, Stirling engine and hybrid cycle performance are investiagated.
2. System configuration A schematic of the proposed hybrid system can be seen in Fig. 1. As can be seen, the hot exhaust gases enter a heat exchanger where it heats up Stirling engine working fluid. This method can prevent exhaust gases’ thermal losses since some of the heat will be used in the Stirling engine and electric power generation. In the proposed
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Fig. 1. The proposal for the gas turbines and Stirling engines hybrid system.
system both components generate electric power, which improves system efficiency. The proposed system consists of a gas turbine, air compressor, fuel compressor, combustion chamber, double acting Stirling engine, four heat exchangers and one thermal regenerator. The fuel used in the system with a composition of 97% methane, 1.5% carbon dioxide and 1.5% nitrogen. The air used in the study consisted of 21% oxygen and 79% nitrogen. In this system, the fuel enters the compressor where its pressure increases, then it preheated and is injected into the combustion chamber. The air follows a similar path, first, it’s compressed in the compressor followed by heating up in the heat exchanger and then it’s mixed with the fuel in combustion chamber. After chemical reactions in combustion chamber, the hot exhaust gases enter the turbine and generate work by expansion. After exiting the turbine, the hot gases continue on their path to the Stirling engine and provide the heat that the engine requires to operate. The Stirling engine generates some extra power by compressing and expanding the working fluid, thus increasing system efficiency. Finally, the hot gases that exit the Stirling engine heat exchangers pass through the fuel and air heat exchangers and exit the system. The fourth heat exchanger is used in the Stirling engine and is usually used for cooling purposes.
The processes are considered to be reversible. The engine used in the system, is a double acting Stirling engine. The volume changes in expansion and compression spaces of the Stirling engine are assumed to be sinusoidal. It is assumed that the Stirling engine’s works with a constant speed. The compression and expansion processes are adiabatic. The hydraulic and thermal losses related to the Stirling engine are included in the numerical solution. 4. The governing equations In this section, using available equations, the workings of every component is introduced and then the components are investigated and analyzed separately and under steady state conditions. After that, a computer code is written in Matlab software then the system’s thermodynamic performance is investigated by changing various parameters, e.g. gas turbine inlet temperature, compressor pressure ratio and the Stirling engine effective parameters such as regenerator porosity, regenerator length and diameter.
3. Assumptions
4.1. Modeling the gas turbine
The following assumptions are made in modeling and analyzing the hybrid system:
The thermodynamic performance of each of the gas turbine components introduced in the preceding section will be analyzed herein. The first laws of thermodynamics are thus employed under the assumption of steady flow for the entire cycle.
Steady state flow in all of the components. Changes in kinetic and potential energies are disregarded. All of the gases present in the system are assumed to behave as ideal gases. The flow inside the compressor and turbine is assumed to be adiabatic. The ambient conditions (temperature and pressure) are assumed to be equal to compressor inlet conditions. There is no leakage and the mass of working fluid remains constant.
4.1.1. Air compressor As shown in Fig. 1 the ambient air enters the compressor in a defined pressure and temperature and exits it after compression at a higher temperature and pressure. The exiting air’s adiabatic temperature can be calculated if the compressor pressure ratio and inlet and outlet temperatures are known. Assuming an adiabatic compression process and knowing compressor pressure ratio and isentropic efficiency, the specific heat of air and flow rate of
M. Korlu et al. / Energy Conversion and Management 147 (2017) 120–134
compressor air, one can calculate the compressor outlet gases’ temperature and the actual required work [23]:
T 2s ¼ T1
c1 c1 P2 c ¼ rp c P1
ð1Þ
ga:c ¼
2s h 1 T 2s T 1 wa:c;s h ¼ ¼ wa:c;a T2 T1 h2 h1
ð2Þ
2 h 1 Þ _ a:c ¼ n_ 1 ðh W
ð3Þ
4.1.2. Fuel compressor The fuel compressor calculations are similar to the air compressor. The compressor outlet temperature and actual work are calculated using the mentioned equations. 4.1.3. Heat exchangers Heat exchangers are used to maximize thermal energy retrieved from turbine exhaust gases and achieving a higher thermal efficiency. In this system, a heat exchanger is used at the combustion chamber inlet to increase the compressed air temperature. More hot air means better air and fuel mixing, more complete combustion and lower fuel requirements for reaching the desired turbine intake conditions. Therefore, more hot air, improves efficiency by decreasing the fuel consumption. This system employs three heat exchangers. The first heat exchanger is placed after the air compressor, in the compressed air path. The second exchanger is placed after the fuel compressor, in the combustion chamber fuel intake path. The third exchanger is positioned after the turbine and is used to provide heat required by the Stirling engine. The temperature obtained for point 12 is equal to the Stirling engine heater temperature. The heat exchangers effectiveness can be defined as:
e1 ¼
T3 T2 T9 T2
ð4Þ
e2 ¼
T6 T5 T 10 T 5
ð5Þ
e3 ¼
T9 T8 T 12 T 8
ð6Þ
Q_ loss;cc ¼ n_ 4 ð1 gcc Þ LHV
123
ð11Þ
4.1.5. Turbine The turbine is one of the most important components of the proposed system which operates by exploiting the energy of hot gases produced by combustion reactions while they are leaving the combustion chamber. A portion of the mechanical energy produced in the turbine is consumed to rotate the compressor, while the rest of it is used to generate electric energy. The turbine’s work and output temperature can be calculated by Eqs. (12)–(14) as long as we have the turbine inlet temperature, turbine pressure ratio and isentropic efficiency [23]:
7 h 8 Þ _ gt ¼ n_ 7 ðh W
ð12Þ
cgc1 P7 g T7 ¼ P8 T 8s
ð13Þ
ggt ¼
7 h 8 h wgt;s T7 T8 ¼ ¼ wgt;a h7 h8s T 7 T 8s
ð14Þ
4.2. Modeling the Stirling engine The Stirling engines use the Stirling cycle which is different from the internal combustion engines’ cycles. The working fluid inside the Stirling engine don’t leave the engine and unlike diesel and petrol engines. Stirling engines don’t have an exhaust poppet valve to eject the high pressure gases inside the combustion chamber. Therefore, Stirling engines are fairly quiet. Any external heat source, from petrol to solar energy and plants’ burning heat, can be used in the Stirling cycles since no combustion takes place in the engines’ cylinders. The basic principle in Stirling engine is that there is a constant amount of gas inside the engine that is closed off and cannot leave the engine, then, through a series of expansion and compression processes, the Stirling cycle changes the gas’ pressure and converts it into work.
depends on the combustion chamber efficiency (gcc ) and lower heating value (LHV) of reactions that occur in the combustion chamber. Writing the energy conservation equations and having the combustion chamber efficiency, Eqs. (10) and (11) can be used to calculate inlet fuel flow rate and combustion chamber heat loss.
4.2.1. Ideal adiabatic model In this section, the equations governing the double acting Stirling engine in the ideal adiabatic view are presented. The adiabatic model assumes that the heater and cooler have unlimited heat transfer and isotherm conditions are applicable as well. Therefore, the fluid inside heat exchangers is always at either the maximum or minimum temperature which are T max and T min respectively. The working fluid temperature in cylinders throughout the cycle can be less or more than T max in the expansion space or T min in the compression space. Fig. 2 shows temperature in various engine components and temperature slope in five engine components [9]. In order to solve the ideal adiabatic model, total mass of the system should be assumed constant and then, using the energy equation and ideal gas equation of state, the equations required to measure the engine heat transfer, exerted work and engine efficiency can be obtained. Per convention, the singular labels in Fig. 2 represent the five components of the Stirling engine and dual labels represent the component intersections. Considering the coordinate system that is defined for the model, 22 variables and 16 differential equations need to be solved to reach a solution for the engine cycle. The starting point for this analysis is the assumption of constant total gas mass in the Stirling engine. The energy equation and ideal gas equation of state are used, in every volume, to obtain the coordinate system for the differential and algebraic equations of that particular volume. Finally, these equations are connected to each other by employing the continuity equation [9].
6 þ n_ 1 h 3 n_ 7 h 7 Q_ n_ 4 h loss;cc ¼ 0
mc þ mk þ mr þ mh þ me ¼ M
Exchangers’ outlet temperature can be obtained by using the energy conservation equation:
3 h 2 Þ ¼ n_ 7 ðh 9 h 10 Þ n_ 1 ðh
ð7Þ
6 h 5 Þ ¼ n_ 7 ðh 10 h 11 Þ n_ 4 ðh
ð8Þ
h Þ ¼ n_ ðh h Þ n_ st ðh 12 13 7 8 9
ð9Þ
4.1.4. Combustion chamber Fuel and air enter the combustion chamber after leaving heat exchangers, where they mix up and react with each other. The reactions that occur in the combustion chambers are exothermal, therefor, temperature of gases that leave the combustion chamber are higher than those which enter it. In the following equation Q_ loss;cc is the heat loss of the combustion chamber and its value
ð10Þ
ð15Þ
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Fig. 2. Temperature distribution in different parts of the Stirling engine based on ideal adiabatic model [9].
The differential form of equation of state for the heat exchanger (heater, regenerator and cooler) can be written as Eq. (16) since temperature and volume are both constant:
Dmc þ Dme þ Dpðmk =P þ mr =P þ mh =PÞ ¼ 0
ð16Þ
The temperature of the engine regenerator, which is where hot and cold fluids exchange heat, can be calculated by Eq. (24):
Tr ¼
Th Tk ln TT h
ð24Þ
k
To obtain an explicit equation for Dp, D mc and D me must be eliminated from Eq. (16). To this end, the energy equation is written for the compression space according to Fig. 2:
Masses related to each of the five Stirling engine components can be calculated by using equation of state.
DQ c C p T ck Dmc ¼ DW c þ C v Dðmc T c Þ
m ¼ pV=RT
ð17Þ
Eq. (18) is obtained by considering the adiabatic assumption for the compression space and substituting the pDV c = DW c equation into Eq. (17):
Dmc ¼ ðpDV c þ V c Dp=cÞ=RT ck
ð18Þ
The expansion space calculations are similar to the compression space and can be calculated through Eq. (19):
Dme ¼ ðpDV e þ V e Dp=cÞ=RT he
ð19Þ
Substituting Eqs. (18) and (19) into Eq. (16) and using the ideal gas equation of state, we obtain Eq. (20):
Dp ¼ h
Vc T ck
c e cp DV þ DV T ck T he i þ c VT k þ VT rr þ VT h þ TV e k
h
ð20Þ
he
The expansion and compression spaces’ volumes can be obtained by knowing the crankshaft angle, h. The piston is at the top of the cylinder when the crankshaft angle is zero.
V e ¼ V cle þ 0:5:V swe ½1 þ cosðh þ aÞ
ð21Þ
The above equation, V cle is the expansion clearance volume, V swe is the expansion swept volume and a is the phase angle. Similar equations can be obtained for compression space.
V c ¼ V clc þ 0:5:V swc ½1 þ cosðhÞ
ð22Þ
It is noteworthy, V clc is the compression clearance volume, V swc is the compression swept volume and h is the crankshaft angle. The total engine volume is calculated by Eq. (23):
V ¼ Ve þ Vr þ Vc þ Vk þ Vh
ð23Þ
ð25Þ
As shown in Eqs. (26)–(29), the mass flow rate of each control volume obtained from law of conservation of mass:
gAck ¼ Dmc
ð26Þ
gAkr ¼ gAck Dmk
ð27Þ
gAhe ¼ gArh Dmh
ð28Þ
gArh ¼ gAhe Dmh
ð29Þ
In this method, T ck and T he are used to represent temperatures at intersections of the compression space and cooler and the expansion space and heater respectively. The values of these parameters depend on the direction of the mass flow and are defined as per Eqs. (30) and (31):
if gAck > 0 then T ck
T c otherwise T ck
Tk
ð30Þ
if gAhe > 0 then T he
T h otherwise T he
Te
ð31Þ
The energy differentials for 3 sections, heater, cooler and regenerator, are obtained by Eq. (17) and using the ideal gas equations of state, Eqs. (32)–(34):
DQ k ¼ V k DpC v =R C p ðT ck gAck T kr gAkr Þ
ð32Þ
DQ r ¼ V r DpC v =R C p ðT kr gAkr T rh gArh Þ
ð33Þ
DQ h ¼ V h DpC v =R C p ðT rh gArh T he gAhe Þ
ð34Þ
After obtaining the temperature, pressure and volume of various components, the work differential of each component can be calculated by Eqs. (35)–(37):
M. Korlu et al. / Energy Conversion and Management 147 (2017) 120–134
DW c ¼ pDV c
ð35Þ
DW e ¼ pDV e
ð36Þ
DW net ¼ DW c þ DW e
ð37Þ
In the ideal adiabatic model, the Stirling engine efficiency can be calculated by Eq. (38):
gst ¼
W net Qh
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length. The thermal losses from the regenerator walls can be obtained from Eq. (39) [9]:
Q wrloss ¼
kwr awr ðT wh T wk Þ lr
ð39Þ
It is noteworthy, kwr , awr , lr , T wh and T wk represent Regenerator wall conduction heat transfer coefficient, total regenerator tube wall area, effective regenerator length, heater and cooler wall temperatures.
ð38Þ
In order to numerically solve the equations obtained from the ideal adiabatic model, the equations of pressure, energy and mass change in expansion and compression spaces have to be solved simultaneously. Using the initial values method would be the best way of solving these equations numerically. In this method, the values of all variables are defined at the starting point and the equations are then solved using these values to reach the final values. The obtained functions include all of the variables and the functions related to engine volume change at various crankshaft angles. First, the y vector is defined as a function of the variables, for example yp represents the gas pressure in the system and ymc represents the mass of the gas in the compression space. With the initial values available, the y vector can be defined as y(t0 ) = y0 Using the y vector and the Dy = F(t, y) system of differential equations, we obtain the y(t) values which, satisfies both differential equations and initial values. In fact, in this numerical method the initial values are obtained for time t 0 , then new values are calculated after a small time step, which means the new values are calculated at t 1 = t0 + Dt. Therefore, there is a set of direct loops at various times which give the integer values of y(t). In the adiabatic method, the initial pressure and mass change in the compression space must be substituted into the equations along with other known values. The engine used in this study was an alpha-type double acting Stirling engine. From a power output viewpoint, alpha-type Stirling engines fall into two categories, the two piston alpha-type engines (simple) and four piston alpha-type engines (double acting engines). Compared to simple engines, the double acting engines reach higher power outputs. Using the geometric and performance specifications of the Ford-Philips 4-215 Stirling engine and the initial values, we can obtain the heater, cooler and regenerator heat transfer values and amount of engine’s work done and efficiency. The numerical code is capable of applying the desired geometric and physical changes and obtaining the changes in engine’s operating power and efficiency. 4.2.2. Non-ideal adiabatic model In this section, effects of non-ideal heat exchanger performance on the Stirling engine are investigated. In real Stirling engines some of heat is stored inside the regenerator when the gas is moving from heater to cooler and is not passed on to the gas when it is returning from the cooler. In other words, there is some convection heat transfer between the gas and exchanger walls, which occurs due to differences between the wall and average gas temperatures. The Urieli adiabatic model [9], developed in 1984, is used in this section for problem analysis. In the rest of this section we will investigate the effects of heat exchangers performances on the system as a whole. It should be noted that equations related to heat and hydraulic losses are included in the numerical code. 4.2.2.1. Regenerator wall heat leakage. Physically, the regenerator is placed between the heater and cooler. The temperature difference between the two heat exchangers causes an undesirable heat loss. The amount of this heat loss depends on thermal conductivity of regenerator, effective conduction area and regenerator effective
4.2.2.2. Regenerator net enthalpy loss. Regenerator net enthalpy loss occurs in the non-ideal regenerator and flow of the working gas through the regenerator. e, the regenerator effectiveness, is defined as the actual energy returned to the working fluid divided by the returned energy value in the ideal adiabatic model. Therefore, the energy stored by the regenerator when the gas is passing through on its way from the expansion space to compression space is not returned to the returning working gas completely. In ideal cases the effectiveness coefficient is equal to one (e = 1). Regenerator net enthalpy loss can be obtained from Eq. (40) [9]:
Q rloss ¼ ð1 eÞ ðQ rmax Q rmin Þ
ð40Þ
In Eq. (40), Q rmax and Q rmax symbols represent the maximum and minimum regenerator heat transfers. The regenerator effectiveness is obtained from Eq. (41):
e¼
NTU NTU þ 1
ð41Þ
The number of transfer units (NTU) in Eq. (46) is obtained from Eq. (42):
NTU ¼
St awgr 2ar
ð42Þ
where awgr and ar represent the regen internal wetted area and the regen internal free flow area [9]. The values of Stanton (St) and Prandtl (Pr) numbers are obtained by Eq. (43) from Kays and London’s [24] results:
St ¼
0:46 Re0:4 Pr
Pr ¼ 0:7
ð43Þ
The Reynolds number (Re) is obtained from Eq. (44) [9]:
Re ¼
jgj d
ð44Þ
l
l ¼ lo
1:5 to þ tsuth t to t þ tsuth
ð45Þ
In Eqs. (44) and (50), g, d, l, lo, to, tsuth and t represent the fluid mass flux in the regenerator, regenerator hydraulic diameter, dynamic viscosity at the gas temperature, dynamic viscosity at reference temperature, reference temperature, Sutherland constant and gas temperature respectively. 4.2.2.3. Energy dissipation by pressure drops in heat exchangers and regenerator. In the Stirling engine, the regenerator porosity causes some pressure drop in the working fluid. In turn, this pressure drop decreases the engine power output. The working fluid flow through the regenerator also generates heat and decreases the heat required by the engine. The pressure variations in the regenerator can be calculated by Eq. (46):
dp ¼
2l V g l f r 2
md
ð46Þ
where fr, l, V, g, l, m and d symbols represent the Reynolds friction factor, dynamic viscosity, void volume in the heat exchangers/
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M. Korlu et al. / Energy Conversion and Management 147 (2017) 120–134
regenerator, fluid mass flux in the heat exchangers/regenerator, effective length of the heat exchangers/regenerator, fluid mass in the heat exchangers/regenerator and hydraulic diameter of the heat exchangers/regenerator respectively and the value of l is calculated by Eq. (45). The value of fr for heat exchangers and regenerator can be obtained by Eq. (47):
f r ¼ 54 þ 1:43Re0:78
f r ¼ 0:0791Re0:75
ð47Þ
The generated heat and work loss in heat exchangers and regenerator can be obtained by substituting the pressure drop values into Eqs. (48) and (49):
Q diss ¼
dp g a
q
dW work ¼ dp dV e
7. Results and discussion
ð48Þ ð49Þ
Finally, Eqs. (50)–(52) are used to calculate the actual power, actual required heat and engine efficiency:
actW power ¼ W power dW work
ð50Þ
actQ hpower ¼ Q hpower þ Q rloss þ Q wrloss Q hdiss Q rdiss
ð51Þ
gst ¼
actW power actQ hpower
ð52Þ
The Q hpower and W power , in Eqs. (50)–(52), represent the required heat and generated power of the Stirling engine in the ideal adiabatic method and are calculated from Eqs. (35) and (37) [9]: 4.3. Equations governing the hybrid system In this section, the whole system is treated as a single control volume then the electric efficiency and net power output are calculated using Eqs. (53)–(55):
_ W
be seen Table 1, while Table 2 contains the gas turbine and the rest of the components’ information. Table 3 compares the results of Urieli’s code [9] with results of the present study which uses the Ford-Philips 4-215 engine. The difference between the present numerical code and Urieli results has arisen from the fact that the present code includes energy dissipation by pressure drops in heat exchangers and regenerator. The presented results show that the numerical code is capable of predicting the Stirling engine power and efficiency.
gele;net ¼ _ net n4 LHV
ð53Þ
_ net ¼ W _ ac;gt þ W _ st W _ a:c W _ f :c W
ð54Þ
_ ac;gt ¼ g _ W in;gen W gt
ð55Þ
_ st can be calculated using Eq. (50). Where, g The value of W in;gen , _ _ _ W a:c , W f :c and W gt represent the DC–AC inverter efficiency, air compressor power consumption, fuel compressor power consumption and gas turbine power generation respectively. 5. Solving method A computer program is written using the aforementioned equations and follows the flowchart shown in Fig. 3. As can be seen in the flowchart, the Stirling engine calculations are carried out by specifying an initial value for heater and cooler temperatures. The gas turbine, and related parameters, calculations are carried out after finding the heat transfer required by the Stirling engine. 6. Verification To verify the computer code, the Stirling engine and turbine gas are verified separately since there are not many studies investigating the system proposed in the present paper. As previously explained this study uses the Ford-Philips 4-215 Stirling engine. This engine was introduced by a Dutch scientist named Philips and was developed and manufactured later by Ford [25]. Nowadays, this engine is known as Philips. The engine parameters can
In the present study, the turbine inlet temperature, compressor pressure ratio, regenerator porosity, Stirling engine regenerator length and diameter are taken as the parameters affecting the hybrid system performance. The aim of the present study was to investigate the effect of aforementioned parameters on gas turbine, Stirling engine and hybrid cycle power and efficiency. The figures for pressure variation against Stirling engine total volume and temperature variation, pressure drop variation in heat exchangers and regenerator, accumulated heat transferred and work done by various Stirling engine components against crankshaft angle are presented for further investigation. It should be noted that calculations for the above figures assumed that gas turbine inlet air flow rate and inlet temperature are equal to 300 kilomole per hour and 1273 K respectively. By assuming above operational conditions for the gas turbine and using the information available in Table 2, the temperature of 1023 K (according to information contained in Table 1) is obtained for the Stirling engine heater. Fig. 4 shows pressure variations against Stirling engine total volume. It should be noted that the work done in each cycle is equal to the area under the P-V graph. The Stirling engine is based on the Carnot cycle, but it can be seen that the Stirling engine have significant differences with the Carnot cycle which is composed of two isothermal and two constant volume processes. These differences have arisen from the fact that, unlike the Carnot cycle, actual conditions are applied to the present numerical code. Fig. 5 illustrates the temperature variations of various Stirling sections against crankshaft angle. As can be seen, there is some convection heat transfer between the wall and working fluid since this model has applied actual conditions and takes the temperature difference of heat exchanger wall and working fluid into account. The convectional heat transfer induces some thermal loss. According to the figure, the heater temperature is lower than 1023 K (the ideal adiabatic model’s heater temperature) and the cooler temperature is higher than 337 K (the ideal adiabatic model’s cooler temperature), which causes the lower performance in the non-ideal adiabatic model. The theta-energy graph, which represents the accumulated heat transferred and work done in various sections in the Stirling engine against crankshaft angle, is shown in Fig. 6. According to the figure, the Stirling engine’s work starts with the expansion process (and positive slope) and continues through the compression process, it finishes the cycle in an expansion process, the Stirling engine’s work assumes positive values at the end of the cycle. Interestingly, the regenerator heavily affects the engine performance and has the highest heat transfer in the engine. It has to be mentioned that the expansion space and heater section’s works are not equal at the beginning of the cycle but converge at the end of the cycle, similar results can be seen for the compression space and cooler. This section will discuss the relation of crankshaft angle with heat exchanger and regenerator pressure drop variations. In the Stirling engine the flow begins from the expansion space and enters heater on its way to the regenerator. The flow of the fluid through the porous environment of the regenerator and its impacts
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Fig. 3. Flowchart for hybrid system.
Table 1 Ford-Philips 4-215 engine data [9]. Number of cooler tubes/cylinder Cooler tube internal diameter Cooler tube length Regenerator diameter Regenerator length Regenerator matrix wire diameter Regenerator matrix mesh size Regenerator matrix porosity Number of regenerator units per cylinder Number of heater tubes/cylinder Heater tube internal diameter Heater tube length Compression clearance volume Expansion clearance volume Compression swept volume Expansion swept volume Total volume of engine Bore Stroke Mean pressure Angular Velocity Operating gas Hot space temperature Cold space temperature
Table 2 Gas turbine performance parameters. 742 0.9 mm 87 mm 73 mm 34 mm 36 lm 200 0.62 2 22 4 mm 462 mm 214.2 cm3 214.2 cm3 870.6 cm3 870.6 cm3 670 cm3 73 mm 52 mm 15 MPa 3300 rpm Hydrogen 1023 K 337 K
with regenerator matrix induces pressure drop in the fluid in addition to transferring heat from the working fluid to regenerator matrix. The working fluid subsequently enters the cooler followed by the compression space and then starts returning back to the expansion space. On its way back, the working fluid passes through the regenerator, where its pressure decreases once again due to
Air flow rate Compressors inlet temperature Compressors inlet pressure Pressure losses in heat exchangers Heat exchangers effectiveness Fuel heat exchanger outlet pressure Turbine inlet temperature AC generator efficiency Pressure losses in the combustion chamber Compressors efficiency Turbine efficiency Combustion chamber efficiency
300 kmol/h 288 K 1 bar 4% 90% 1 bar 1273 K 95% 5% 81% 84% 95%
Table 3 Stirling engine performance parameters.
Power output (kW) Heat transferred to the heater (kW) Thermal efficiency (%)
Urieli and Berchowitz [9]
Present research
Percentage error
192.93 362.01
192.93 351.5
0 3
53.3
54.9
3
absorbing the heat that itself had previously stored in the regenerator matrix. The minus sign is used to convey ‘‘opposite of the working fluid’s flow direction” and is used here to complete the Stirling cycle. Fig. 7 shows significant pressure drops in the thermal regenerator compared to the heater and cooler.
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220
Pressure (bar)
200
180
160
140
120
100 2000
2200
2400
2600
2800
3000
3200
3400
Total volume (cc) Fig. 4. The variation of pressure with Stirling engine total volume.
1100 1000
Tc Te Tk Tr Th Twk Twh
Temperature (K)
900 800 700 600 500 400 300
0
50
100
150
200
250
300
350
400
Crank anle (derees) Fig. 5. The variation of temperatures of the in Stirling engine with crank angle.
While previous sections discussed the parameters affecting the Stirling engine analysis, this section moves on to parameters that affect the Stirling engine, gas turbine and hybrid cycles. The effects of the compressor pressure ratio and turbine gases’ inlet temperature on the Stirling engine, gas turbine and hybrid cycle power output are investigated first. As presented in Figs. 8 and 9, the increasing the compressor pressure ratio increases the gas turbine net power at first but then decreases it. In fact, this happens because increasing the compressor pressure ratio, increases the turbine outlet and inlet enthalpy difference at a higher rate than the compressor, this continues until a certain pressure, after which this process is reversed and the compressor’s enthalpy difference becomes dominant and decreases the gas turbine cycle net power output. On the other hand, the results show that turbine inlet temperature is directly proportional to gas turbine power generation, while it has a negligible relation with the compressor’s power consumption. Furthermore, increasing the compressor pressure ratio decreases the turbine outlet temperature, therefore, the Stirling engine power output decreases as well, since the heat transferred to the engine is a function of the turbine outlet temperature. From
the results, it can be concluded that the hybrid cycle net power output increases with an increase in pressure ratio, but decreases at high pressures ratio. It should be noted that the above analysis is carried out for Stirling engine heater temperatures between 823 and 1023 K. The graphs depicting the Stirling engine efficiency and gas turbine and hybrid cycles electric efficiencies against compressor pressure ratio, for various gas turbine inlet temperatures, are shown in Figs. 10 and 11. As can be seen increasing the compressor pressure ratio increases the gas turbine cycle efficiency at first but decreases it at higher values. This occurs due to direct proportionality of electric efficiency and gas turbine cycle net power output. According to both graphs, increasing compressor pressure ratio decreases the Stirling engine efficiency, which occurs due to a decrease in Stirling engine net power output. From these results, it can be concluded that by an increase in pressure ratio, increases the hybrid cycle electric efficiency at first but then decreases it after a certain ratio and mimics the hybrid cycle net power output results. It should be noted that the above analysis is carried out for Stirling engine heater temperatures between 823 and 1023 K.
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20
10
Energy [kj]
0
-10
Qk Qr
-20
Qh
-30
Wcompression
W Wexpansion -40
0
50
100
150
200
250
300
350
400
Crank angle (degrees) Fig. 6. The variation of accumulated heat transferred and work done in various sections in the Stirling engine with crank angle.
250
cooler pressure drop
Heat exchanger pressure drop [kPa]
200
heater pressure drop
150
regenerator pressure drop
100 50 0 -50 -100 -150 -200 -250
0
50
100
150
200
250
300
350
400
Crank angle (degrees) Fig. 7. The variation of pressure drop across heat exchangers and regenerator in the Stirling engine with crank angle.
This section investigates the effects of Stirling engine parameters, e.g. regenerator porosity, regenerator length and diameter, on performance of the Stirling engine and hybrid cycle. The investigations are carried out for turbine inlet flow rate of 300 kilomole per hour and turbine inlet and Stirling heater temperatures of 1273 and 1023 K respectively. The effect of regenerator length on regenerator loss is presented in Fig. 12. According to the figure the regenerator net enthalpy loss decreases when regenerator length is increased. To explain this it must be noted that increasing the regenerator length, increases the regenerator effectiveness, which in turn lowers regenerator net enthalpy loss. Regenerator wall heat leakage is also reduced, which is expected since it is inversely proportional to the regenerator length. Overall it seems that energy dissipation by pressure drops in regenerator increase alongside regenerator length, again this is as expected since pressure drop us directly proportional to regenerator length. The effect of regenerator length on Stirling engine and hybrid cycle power output and efficiency is presented in Fig. 13. The figure
shows that the Stirling engines power output decreases as regenerator length increases, this is caused by the increased pressure drop, which in turn increases work loss in the regenerator and heat exchangers. Increasing the regenerator length, increases the hybrid cycle power output due to decreased Stirling engine heater heat transfer and increased gas turbine power output. The increased hybrid cycle power output itself causes an improvement in hybrid cycle electric efficiency. It should be noted that the improved Stirling engine efficiency can be explained by the decrease in regenerator-related losses. The Effect of engine diameter on regenerator loss is shown in Fig. 14. According to the figure the regenerator net enthalpy loss is reduced with increasing regenerator diameter. To explain this it must be noted that increasing the regenerator diameter, increases the regenerator effectiveness, which in turn lowers regenerator net enthalpy loss. Regenerator wall heat leakage is also reduced since it is inversely proportional to regenerator diameter. Also, energy dissipation by pressure drops in regenerator decreases with an increase in regenerator diameter. This can be justified by
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500
430
Gas Turbin
Power output (kW)
Stirling engine 360
Stirling engine + Gas Turbin
290
220
150
80
1
2
3
4
5
6
7
Compressor pressure ratio Fig. 8. The variation of Stirling engine, gas turbine and hybrid cycle power output with compressor pressure ratio (TIT = 1273).
480
Power output (kW)
400
320
240
160
Gas Turbin Stirling engine Stirling engine + Gas Turbin
80
0
1
2
3
4
5
6
Compressor pressure ratio Fig. 9. The variation of Stirling engine, gas turbine and hybrid cycle power output with compressor pressure ratio (TIT = 1223).
58
50
Gas Turbine 40
54
Stirling engine + Gas Turbin
30
50
20
46
10
1
2
3
4
5
6
7
Efficiency (%)
Electrical efficiency (%)
Stirling engine
42
Compressor pressure ratio Fig. 10. The variation of Stirling engine, gas turbine and hybrid cycle efficiency with compressor pressure ratio (TIT = 1273).
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50
58
Gas Turbine Stirling engine Stirling engine + Gas Turbin
54
30 50 20 46
10
0
Efficiency (%)
Electrical efficiency (%)
40
1
2
3
4
5
6
42
Compressor pressure ratio Fig. 11. The variation of Stirling engine, gas turbine and hybrid cycle efficiency with compressor pressure ratio (TIT = 1223).
35 30
Regenerator net enthalpy loss Regenerator wall heat leakage
25
Losses[kW]
Regenerator Dissipation 20 15 10 5 0 10
20
30
40
50
60
70
Regenerator length (mm)
330
58
290
52
Actual ST power output Actual GT+ST power output
250
46
Actual ST efficiency
Efficiency (%)
Power output [kW]
Fig. 12. The variation of regenerator losses with regenerator length.
Actual GT+ST electrical efficiency 210
170 22
40
28
34
40
34 46
Regenerator length (mm) Fig. 13. The variation of Stirling engine and hybrid cycle efficiency and power output with regenerator length.
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20
Regenerator Dissipation Regenerator wall heat leakage
Losses[kW]
16
Regenerator net enthalpy loss
12
8
4
0 58
63
68
73
78
83
88
Regenerator diameter (mm)
350
58
300
52
Actual ST power output Actual GT+ST power output 250
46
Actual ST efficiency Actual GT+ST electrical efficiency
200
150 60
Efficiency (%)
Power output [kW]
Fig. 14. The variation of regenerator losses with regenerator diameter.
40
65
70
75
80
34 85
Regenerator diameter (mm) Fig. 15. The variation of Stirling engine and hybrid cycle efficiency and power output with regenerator diameter.
the decreased pressure drop which in turn is inversely proportional to regenerator diameter. Fig. 15 shows the variations in Stirling engine and hybrid cycle power outputs and efficiencies against regenerator diameter. As can be seen, the Stirling power output decreases with an increase in regenerator diameter, this occurs due to the increased, pressured drop induced, work loss in heat exchangers and regenerator. Increasing regenerator diameter, increases hybrid cycle power generation due to decreased heat transfer in Stirling heater and increased gas turbine power generation. The increased hybrid cycle power output itself causes an improvement in hybrid cycle electric efficiency. The reason for improved Stirling engine efficiency, after an increase in regenerator diameter, is a decrease in regeneratorrelated losses. The graph of regenerator porosity against regenerator losses is shown in Fig. 16. As can be seen in the figure, regenerator net enthalpy loss increases with regenerator porosity. This can be justified by considering that, according to the equation of regenerator net enthalpy loss, increasing regenerator porosity decreases regen-
erator efficiency. Regenerator wall heat leakage is independent of regenerator porosity, therefore is does not change. Energy dissipation by pressure drops in regenerator decrease when there is an increase in regenerator porosity. This is due to the increase in hydraulic diameter for regenerator which is inversely proportional to the regenerator pressure drop. The effect of regenerator porosity variations on the Stirling engine and hybrid cycle efficiencies and power outputs is illustrated in Fig. 17. As can be seen in the figure, Stirling engine and hybrid cycle power generations both increase and then decrease with regenerator porosity. This occurs since the work loss in regenerator and heat exchangers initially approaches the optimal value, but then increases after a certain value and therefore decreases both power and efficiency. The Stirling engine efficiency and overall system efficiency both follow the behavior of their respective power output. The results show that in lower pressure ratios most of the power is generated by the Stirling engine and the gas turbine cycle produces relatively smaller amounts. Yet, the share of gas in power
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200
Regenerator net enthalpy loss Regenerator wall heat leakage
Losses[kW]
150
Regenerator Dissipation
100
50
0 0.35
0.55
0.75
0.95
Porosity Fig. 16. The variation of regenerator losses with regenerator porosity.
340
56
50
260
220
Actual ST power output Actual GT+ST power output Actual ST efficiency Actual GT+ST Electrical efficiency
44
38
180
140 0.38
Efficiency (%)
Power output [kW]
300
0.46
0.54
0.62
0.7
0.78
32 0.86
Porosity Fig. 17. The variation of Stirling engine and hybrid cycle efficiency and power output with regenerator porosity.
Fig. 18. Comparison of Stirling engine and hybrid cycle efficiency and power output.
generation improves by increasing the pressure ratio. From the figures it can be concluded that the adding a Stirling engine to gas turbine cycle can improve a simple gas turbine cycle efficiency
and power generation. Fig. 18 shows a comparison between a simple gas turbine cycle and the proposed hybrid cycle. According to the figure, the generated power had increased from 263.4 kW for
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the simple gas turbine to 432.4 kW for the hybrid cycle, which is a significant improvement. Subsequently, the electric efficiency is also improved significantly, form 23.6% to 38.85%. It should be noted that the investigation was carried out for the following conditions, gas turbine inlet temperature of 300 kilomole per hour, turbine inlet temperature of 1273 K and Stirling engine heater temperature of 923 K. These conditions result in the highest efficiency possible for the hybrid cycle. Considering the aforementioned results it can be concluded that the Stirling engine is very suitable for extracting energy from gas turbine exhaust gases. 8. Conclusions In this study, we have tried to introduce a new subject for the use of gas turbine exhaust and combining this system with Stirling engine for producing more power. To accomplish this goal, a portion of gas turbine exhaust gases are allocated to providing the heat required for the Stirling engine. The double acting Stirling engine is used in this study and the non-ideal adiabatic model is used to numerical solution. The influence of several effective parameters (such as the pressure ratio of the compressor, the inlet temperature of turbine, the porosity, length and diameter of the regenerator) on hybrid system performance was studied. The following points are presented as conclusion from the presented study The area below the pressure-volume graph of the Stirling engine is significantly different from the Carnot cycle. The results show that most of the energy transfer in the Stirling engine occurs in the regenerator, which indicates that the regenerator has significant influence on Stirling engine efficiency. The highest pressure drop has occurred in the regenerator, followed by heater and cooler. The Stirling engine power generation and efficiency decrease, when the compressor pressure ratio increases. The hybrid cycle power generation and efficiency increases and then decreases with increasing pressure ratio. It is also noteworthy that an increase in the turbine inlet temperature increases both power generation and efficiency of the hybrid cycle. The results show that increasing the regenerator length from 24 to 44 mm, causes increases of 2 and 1% in the efficiency of the Stirling engine and hybrid cycle respectively. The results show that increasing the regenerator diameter from 63 to 84 mm, causes increases of 2.5 and 1% in the efficiency of the Stirling engine and hybrid cycle respectively. The results show that with increasing regenerator porosity from 0.42 to 0.82 can be achieved regenerator porosity optimum point, Stirling engine and hybrid system maximum efficiency about 55 and 36.7% respectively.
Combining the gas turbine with the Stirling engine increases the net power output from 263.4 kW for the gas turbine to 432.4 kW for the hybrid cycle and increases efficiency from 23.6% for the gas turbine to 38.85% for the hybrid cycle.
References [1] John Barber, Obtaining and Applying Motive Power, & c. A Method of Rising Inflammable Air for the Purposes of Procuring Motion, and Facilitating Metallurgical Operations (The Gas Turbine invention), UK patent no. 1833; 1971. [2] Milewski J, Badyda K, Miller A. Gas turbines in unconventional applications. In: Konstantin V, editor. Performance and robustness of gas turbines. Croatia: InTech; 2012. p. 121–64. [3] Poullikkas A. An overview of current and future sustainable gas turbine technologies. J Renew Sustain Energy Rev 2005;9:409–43. [4] Thombare DG, Verma SK. Technological development in the Stirling cycle engines. J Renew Sustain Energy Rev 2008;12:1–38. [5] Li T, Tang D, Li Z, Du J, Zhou T, Jia Y. Development and test of a Stirling engine driven by waste gases for the micro-CHP system. J Appl Therm Eng 2012;33– 34:119–23. [6] Kuhn V, Klemeš J, Bulatov I. MicroCHP: overview of selected technologies, products and field test results. J Appl Therm Eng 2008;28:2039–48. [7] Schmidt G. Classical analysis of operation of Stirling engine, a report published in German Engineering Union, vol. XV; 1871. p. 1–12 [original German]. [8] Finkellstein T. Computer analysis of Stirling engines. In: 10th IECEC 759140. p. 933–41. [9] Urieli I, Berchowitz DM. Stirling cycle engine analysis. Oxford: Oxford University Press; 1984. [10] Formosa F, Despesse G. Analytical model for Stirling cycle machine design. J Energy Convers Manage 2010;51:1855–63. [11] Popescu G, Radcenco V, Costea M, Feidt M. Thermodynamic optimization in the finished time of Stirling engine. Int J Therm Sci 1996;35:656–61. [12] Kaushik SC, Kumar S. Finite time thermodynamic analysis of endoreversible Stirling heat engine with regenerative losses. J Energy 2000;25:989–1003. [13] Cun-quan Z, Yi-nong W, Guo-lin J, Dong-yu L, Lie X. Dynamic simulation of one-stage Oxford split-Stirling cryocooler and comparison with experiment. J Cryog 2002;42:377–586. [14] Wu F, Chen L, Wu C, Sun F. Optimum performance of irreversible Stirling engine with imperfect regeneration. J Energy Convers Manage 1998;39:727–32. [15] Kongtragool B, Wongwises S. A review of solar powered Stirling engines and low temperature differential Stirling engines. J Renew Sustain Energy Rev 2003;7:131–54. [16] Costa M, Petrescu S, Harman C. The effect of irreversibilities on solar Stirling engine cycle performance. J Energy Convers Manage 1999;40:1723–31. [17] Timoumi Y, Nasrallah SB, Tlili I. Numerical simulation and losses analysis in a Stirling engine. Int J. Heat Technol 2006;24:97–105. [18] Timoumi Y, Tlili I, Nasrallah SB. Reduction of energy losses in a Stirling engine. Int J Heat Technol 2007;25:81–90. [19] Timoumi Y, Nasrallah SB, Tlili I. Thermodynamic analysis of the Stirling heat engine with regenerative losses and internal irreversibilities. Int J Engine Res 2007;9:45–56. [20] Timoumi Y, Tlili I, Nasrallah SB. Design and performance optimization of GPU3 Stirling engines. J Energy 2008;33:1100–14. [21] Tavakolpour A, Zomorodian A, Golneshan AA. Simulation, construction and testing of a two cylinder solar Stirling engine powered by a flat plate solar collector without regenerator. J Renew Energy 2008;33:77–87. [22] Gostante M, Invernizzi A. Stirling engines using working fluids with strong real gas effects. J Appl Therm Eng 2010;30:1703–10. [23] Haseli Y, Dincer I, Naterer GF. Thermodynamic modeling of a gas turbine cycle combined with a solid oxide fuel cell. Int J Hydrogen Energy 2008;33:5811–22. [24] Kays WM, London AL. Compact heat exchangers. 3rd ed. New York: McGraw Hill; 1984. [25] Van Giessel R, Reinink F. Design of the 4-215 D.A. Automotive Stirling engine. Conference of the SAE International Automotive Engineering Congress and Exposition, SAE Technical Paper 770082, Detroit, USA; 1977.
Contents lists available at ScienceDirect
Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Thermodynamic analysis of a gas turbine cycle equipped with a non-ideal adiabatic model for a double acting Stirling engine Mahmood Korlu a, Jamasb Pirkandi a,⇑, Arman Maroufi b a b
Department of Aerospace Engineering, Malek Ashtar University of Technology, Tehran, Iran Faculty of Industrial and Mechanical Engineering, Islamic Azad University, Qazvin Branch, Qazvin, Iran
a r t i c l e
i n f o
Article history: Received 25 January 2017 Received in revised form 24 March 2017 Accepted 13 April 2017
Keywords: Stirling engine Gas turbine Hybrid cycle Non-ideal adiabatic
a b s t r a c t The aim of this study is to investigate the thermodynamic performance of a gas turbine cycle equipped with a double acting Stirling engine. A portion of gas turbine exhaust gases are allocated to providing the heat required for the Stirling engine. Employing this hybrid cycle improves gas turbine performance and power generation. The double acting Stirling engine is used in this study and the non-ideal adiabatic model is used to numerical solution. The regenerator’s net enthalpy loss, the regenerator’s wall heat leakage, the energy dissipation caused by pressure drops in heat exchangers and regenerator are the losses that were taken into account for the Stirling engine. The hybrid cycle, gas turbine governing equations and Stirling engine analyses are carried out using the Matlab software. The pressure ratio of the compressor, the inlet temperature of turbine, the porosity, length and diameter of the regenerator were chosen as essential parameters in this article. Also the hybrid cycle effects, efficiency and power outputs are investigated. The results show that the hybrid gas turbine and Stirling engine improves the efficiency from 23.6 to 38.8%. Ó 2017 Published by Elsevier Ltd.
1. Introduction Finding new energy sources seems to be inevitable because the ever increasing worldwide energy demand, the limited fossil fuel resources, and the environmental pollution. Gas turbine is one of the energy sources with many applications in various industry such as aerospace, auxiliary power units in airplanes, industrial projects, transport industry and cogeneration systems. Every gas turbine consists of a compressor, combustion chamber where air and fuel are mixed and burned and a turbine that converts the hot and compressed gases energy into mechanical energy. A portion of the mechanical energy produced by the turbine is consumed to rotate the compressor and the rest of it is transported to the generator to produce electricity. Gas turbine’s works based on the Brayton cycle. The cycle was first proposed by John Barber [1], an English inventor. The researches of the recent 7 decades have increased the gas turbine efficiency. Also researchers try to find new methods to improve the efficiency without an overhaul. One way to improve the gas turbine efficiency achieved by changing operational conditions (temperature and pressure) and various cooling methods, the other way is to combine the gas turbine with
⇑ Corresponding author. E-mail address: [email protected] (J. Pirkandi). http://dx.doi.org/10.1016/j.enconman.2017.04.049 0196-8904/Ó 2017 Published by Elsevier Ltd.
another thermal engine, which can significantly improve power and efficiency. Combing the gas turbine with the Rankine and the Stirling cycles is an effective way of increasing efficiency. The gas turbine cycle and the Stirling cycle combination results a wholly different configuration. In this type of hybrid cycle, exhaust gases enter Stirling engine heat exchanger and provide the heat required by the Stirling engine [2]. In 2005, Pollikas estimated, for the first time, that using a bottoming Stirling cycle fed by exhaust gases of an RB211 Rolls-Royce gas turbine (27.5 MW electrical power) can recover 9 MWs. Adding a Stirling engine to a gas turbine can make a 47.7% overall efficiency[3]. Many researchers interested in Stirling engines due to low pollutant emissions, low sound production and the ability to operate with various fuels [4–6]. In fact, a Stirling engine is an external combustion engine and can use any external heat source and convert it into mechanical energy. The studies carried out by researchers on Stirling engine design and applications have produced very promising results. Construction of a solar engine with a 10 kW axial power is one of the most significant advances in Stirling technology. Combined heat and power generation is one of the new ideas that have been developed by Stirling manufacturers and are used in power plants. Recently, there are new ideas for the Stirling engine applications such as satellite power supply and being proposed as an alternative to steam turbines in nuclear power plants. Stirling engines’ physical structure consists of five
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Nomenclature A Cp Cv d f g h k l LHV m M n_ NTU P Pr Q_ Q R Re rp St T TIT V _ W W
area [m2 ] specific heat at constant pressure [J/kg K] specific heat at constant volume [J/kg K] hydraulic diameter [m] friction factor coefficient mass flux [kg/m2 s] enthalpy [kJ/kmol] thermal conductivity [W/m K] length [m] lower heating value [kJ/mol] mass [kg] mass of working gas in the Stirling engine [kg] molar flow rate [mol/s] number of transfer units pressure [bar] non-dimensional Prandtl number heat transfer rate [kW] heat [kJ] universal gas constant [8.314 J/kmol K] non-dimensional Reynolds number compressor pressure ratio non-dimensional Stanton number temperature [K] turbine inlet temperature [K] volume [m3] power [kW] work [kJ]
Greek symbols density [kg/m3 ] efficiency h crank rotational angle [°] c ideal gas specific heat ratio e heat exchangers and regenerator effectiveness l viscosity [kg/m s]
q g
subsystems, and each the subsystem is considered a control volume. The engine consists of two spaces with varying volumes, named expansion space and compression space, and three heat exchangers with constant volume, named heater, cooler and regenerator. Many researchers study the Stirling engine ever since its invention by Robert Stirling. The first acceptable mathematical analysis of the Stirling cycle was developed by Schmidt, fifty years after its invention [7]. Schmidt’s analysis was based on isotherm expansion and compression spaces, in his approach the thermodynamic models were linearized and therefore, the initial power and engine efficiency calculations were fairly simple. Finkelstein [8] improved Schmidt’s thermodynamic analysis and proposed the first adiabatic analyses. Urieli and Berchowitz [9] used the thermodynamic model to calculate power output and efficiency of Stirling engines. Fermosa and Despesse [10] modelled the engine using an isotherm model to investigate dead spaces’ effects on the engine power output and efficiency. Popescu et al. [11] shows that the low performance is essentially caused by the non-ideal thermal regenerator. Kaushik, Cun-quan and Wu [12–14] proved that thermal regenerator effectiveness factor, thermal conductivity between engine and reservoir have the highest impacts on Stirling engines performance. Kongtragool and Wongwises [15] studied the effects of regenerator effectiveness and dead spaces on Stirling engine inlet heat and efficiency. Costa et al. [16] and Timoumi et al. [17–20]investigated Stirling engine losses and irreversibilities through adiabatic modeling. Thombare and Verma [4] have con-
Subscripts a actual process a.c air compressor c compression space cc combustion chamber ck interface between compression space and cooler clc compression clearance volume cle expansion clearance volume diss dissipation e expansion space elec electrical f.c fuel compressor gen generator gt gas turbine g gas h heater he interface between heater and expansion space k cooler kr interface between cooler and regenerator loss loss r regenerator rh interface between regenerator and heater rloss net enthalpy loss s isentropic process suth Sutherland st Stirling engine swc compression swept swe expansion swept w wall wg wetted area wh heater wall wk cooler wall wr regenerator wall wrloss losses from the regenerator walls
ducted a series of studies in which they gathered the available technologies and recent advances in Stirling engine analyzation and provided some suggestions as how to use them. Tavvakolpour et al. [21] investigated the gamma type Stirling engine, they used the Schmidt theory to solve the equations in isotherm form and used flat panel to absorb solar energy as the hot thermal source. Gostante and Invernizzi [22] modelled the Stirling engine and then investigated the effects of various gases on the engine output and efficiency. In the present study, the energy of a gas turbine exhaust gases are used as the hot source in a double acting Stirling engine. The thermodynamic equations of every single component of the gas turbine and double acting Stirling engine hybrid cycle are simulated by a code written in Matlab. The non-ideal adiabatic model is used to model the Stirling engine. Then, with parametric study for the mentioned hybrid system, the effect of gas turbine inlet temperature, compressor pressure ratio, regenerator porosity and regenerator length and diameter on gas turbine, Stirling engine and hybrid cycle performance are investiagated.
2. System configuration A schematic of the proposed hybrid system can be seen in Fig. 1. As can be seen, the hot exhaust gases enter a heat exchanger where it heats up Stirling engine working fluid. This method can prevent exhaust gases’ thermal losses since some of the heat will be used in the Stirling engine and electric power generation. In the proposed
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Fig. 1. The proposal for the gas turbines and Stirling engines hybrid system.
system both components generate electric power, which improves system efficiency. The proposed system consists of a gas turbine, air compressor, fuel compressor, combustion chamber, double acting Stirling engine, four heat exchangers and one thermal regenerator. The fuel used in the system with a composition of 97% methane, 1.5% carbon dioxide and 1.5% nitrogen. The air used in the study consisted of 21% oxygen and 79% nitrogen. In this system, the fuel enters the compressor where its pressure increases, then it preheated and is injected into the combustion chamber. The air follows a similar path, first, it’s compressed in the compressor followed by heating up in the heat exchanger and then it’s mixed with the fuel in combustion chamber. After chemical reactions in combustion chamber, the hot exhaust gases enter the turbine and generate work by expansion. After exiting the turbine, the hot gases continue on their path to the Stirling engine and provide the heat that the engine requires to operate. The Stirling engine generates some extra power by compressing and expanding the working fluid, thus increasing system efficiency. Finally, the hot gases that exit the Stirling engine heat exchangers pass through the fuel and air heat exchangers and exit the system. The fourth heat exchanger is used in the Stirling engine and is usually used for cooling purposes.
The processes are considered to be reversible. The engine used in the system, is a double acting Stirling engine. The volume changes in expansion and compression spaces of the Stirling engine are assumed to be sinusoidal. It is assumed that the Stirling engine’s works with a constant speed. The compression and expansion processes are adiabatic. The hydraulic and thermal losses related to the Stirling engine are included in the numerical solution. 4. The governing equations In this section, using available equations, the workings of every component is introduced and then the components are investigated and analyzed separately and under steady state conditions. After that, a computer code is written in Matlab software then the system’s thermodynamic performance is investigated by changing various parameters, e.g. gas turbine inlet temperature, compressor pressure ratio and the Stirling engine effective parameters such as regenerator porosity, regenerator length and diameter.
3. Assumptions
4.1. Modeling the gas turbine
The following assumptions are made in modeling and analyzing the hybrid system:
The thermodynamic performance of each of the gas turbine components introduced in the preceding section will be analyzed herein. The first laws of thermodynamics are thus employed under the assumption of steady flow for the entire cycle.
Steady state flow in all of the components. Changes in kinetic and potential energies are disregarded. All of the gases present in the system are assumed to behave as ideal gases. The flow inside the compressor and turbine is assumed to be adiabatic. The ambient conditions (temperature and pressure) are assumed to be equal to compressor inlet conditions. There is no leakage and the mass of working fluid remains constant.
4.1.1. Air compressor As shown in Fig. 1 the ambient air enters the compressor in a defined pressure and temperature and exits it after compression at a higher temperature and pressure. The exiting air’s adiabatic temperature can be calculated if the compressor pressure ratio and inlet and outlet temperatures are known. Assuming an adiabatic compression process and knowing compressor pressure ratio and isentropic efficiency, the specific heat of air and flow rate of
M. Korlu et al. / Energy Conversion and Management 147 (2017) 120–134
compressor air, one can calculate the compressor outlet gases’ temperature and the actual required work [23]:
T 2s ¼ T1
c1 c1 P2 c ¼ rp c P1
ð1Þ
ga:c ¼
2s h 1 T 2s T 1 wa:c;s h ¼ ¼ wa:c;a T2 T1 h2 h1
ð2Þ
2 h 1 Þ _ a:c ¼ n_ 1 ðh W
ð3Þ
4.1.2. Fuel compressor The fuel compressor calculations are similar to the air compressor. The compressor outlet temperature and actual work are calculated using the mentioned equations. 4.1.3. Heat exchangers Heat exchangers are used to maximize thermal energy retrieved from turbine exhaust gases and achieving a higher thermal efficiency. In this system, a heat exchanger is used at the combustion chamber inlet to increase the compressed air temperature. More hot air means better air and fuel mixing, more complete combustion and lower fuel requirements for reaching the desired turbine intake conditions. Therefore, more hot air, improves efficiency by decreasing the fuel consumption. This system employs three heat exchangers. The first heat exchanger is placed after the air compressor, in the compressed air path. The second exchanger is placed after the fuel compressor, in the combustion chamber fuel intake path. The third exchanger is positioned after the turbine and is used to provide heat required by the Stirling engine. The temperature obtained for point 12 is equal to the Stirling engine heater temperature. The heat exchangers effectiveness can be defined as:
e1 ¼
T3 T2 T9 T2
ð4Þ
e2 ¼
T6 T5 T 10 T 5
ð5Þ
e3 ¼
T9 T8 T 12 T 8
ð6Þ
Q_ loss;cc ¼ n_ 4 ð1 gcc Þ LHV
123
ð11Þ
4.1.5. Turbine The turbine is one of the most important components of the proposed system which operates by exploiting the energy of hot gases produced by combustion reactions while they are leaving the combustion chamber. A portion of the mechanical energy produced in the turbine is consumed to rotate the compressor, while the rest of it is used to generate electric energy. The turbine’s work and output temperature can be calculated by Eqs. (12)–(14) as long as we have the turbine inlet temperature, turbine pressure ratio and isentropic efficiency [23]:
7 h 8 Þ _ gt ¼ n_ 7 ðh W
ð12Þ
cgc1 P7 g T7 ¼ P8 T 8s
ð13Þ
ggt ¼
7 h 8 h wgt;s T7 T8 ¼ ¼ wgt;a h7 h8s T 7 T 8s
ð14Þ
4.2. Modeling the Stirling engine The Stirling engines use the Stirling cycle which is different from the internal combustion engines’ cycles. The working fluid inside the Stirling engine don’t leave the engine and unlike diesel and petrol engines. Stirling engines don’t have an exhaust poppet valve to eject the high pressure gases inside the combustion chamber. Therefore, Stirling engines are fairly quiet. Any external heat source, from petrol to solar energy and plants’ burning heat, can be used in the Stirling cycles since no combustion takes place in the engines’ cylinders. The basic principle in Stirling engine is that there is a constant amount of gas inside the engine that is closed off and cannot leave the engine, then, through a series of expansion and compression processes, the Stirling cycle changes the gas’ pressure and converts it into work.
depends on the combustion chamber efficiency (gcc ) and lower heating value (LHV) of reactions that occur in the combustion chamber. Writing the energy conservation equations and having the combustion chamber efficiency, Eqs. (10) and (11) can be used to calculate inlet fuel flow rate and combustion chamber heat loss.
4.2.1. Ideal adiabatic model In this section, the equations governing the double acting Stirling engine in the ideal adiabatic view are presented. The adiabatic model assumes that the heater and cooler have unlimited heat transfer and isotherm conditions are applicable as well. Therefore, the fluid inside heat exchangers is always at either the maximum or minimum temperature which are T max and T min respectively. The working fluid temperature in cylinders throughout the cycle can be less or more than T max in the expansion space or T min in the compression space. Fig. 2 shows temperature in various engine components and temperature slope in five engine components [9]. In order to solve the ideal adiabatic model, total mass of the system should be assumed constant and then, using the energy equation and ideal gas equation of state, the equations required to measure the engine heat transfer, exerted work and engine efficiency can be obtained. Per convention, the singular labels in Fig. 2 represent the five components of the Stirling engine and dual labels represent the component intersections. Considering the coordinate system that is defined for the model, 22 variables and 16 differential equations need to be solved to reach a solution for the engine cycle. The starting point for this analysis is the assumption of constant total gas mass in the Stirling engine. The energy equation and ideal gas equation of state are used, in every volume, to obtain the coordinate system for the differential and algebraic equations of that particular volume. Finally, these equations are connected to each other by employing the continuity equation [9].
6 þ n_ 1 h 3 n_ 7 h 7 Q_ n_ 4 h loss;cc ¼ 0
mc þ mk þ mr þ mh þ me ¼ M
Exchangers’ outlet temperature can be obtained by using the energy conservation equation:
3 h 2 Þ ¼ n_ 7 ðh 9 h 10 Þ n_ 1 ðh
ð7Þ
6 h 5 Þ ¼ n_ 7 ðh 10 h 11 Þ n_ 4 ðh
ð8Þ
h Þ ¼ n_ ðh h Þ n_ st ðh 12 13 7 8 9
ð9Þ
4.1.4. Combustion chamber Fuel and air enter the combustion chamber after leaving heat exchangers, where they mix up and react with each other. The reactions that occur in the combustion chambers are exothermal, therefor, temperature of gases that leave the combustion chamber are higher than those which enter it. In the following equation Q_ loss;cc is the heat loss of the combustion chamber and its value
ð10Þ
ð15Þ
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Fig. 2. Temperature distribution in different parts of the Stirling engine based on ideal adiabatic model [9].
The differential form of equation of state for the heat exchanger (heater, regenerator and cooler) can be written as Eq. (16) since temperature and volume are both constant:
Dmc þ Dme þ Dpðmk =P þ mr =P þ mh =PÞ ¼ 0
ð16Þ
The temperature of the engine regenerator, which is where hot and cold fluids exchange heat, can be calculated by Eq. (24):
Tr ¼
Th Tk ln TT h
ð24Þ
k
To obtain an explicit equation for Dp, D mc and D me must be eliminated from Eq. (16). To this end, the energy equation is written for the compression space according to Fig. 2:
Masses related to each of the five Stirling engine components can be calculated by using equation of state.
DQ c C p T ck Dmc ¼ DW c þ C v Dðmc T c Þ
m ¼ pV=RT
ð17Þ
Eq. (18) is obtained by considering the adiabatic assumption for the compression space and substituting the pDV c = DW c equation into Eq. (17):
Dmc ¼ ðpDV c þ V c Dp=cÞ=RT ck
ð18Þ
The expansion space calculations are similar to the compression space and can be calculated through Eq. (19):
Dme ¼ ðpDV e þ V e Dp=cÞ=RT he
ð19Þ
Substituting Eqs. (18) and (19) into Eq. (16) and using the ideal gas equation of state, we obtain Eq. (20):
Dp ¼ h
Vc T ck
c e cp DV þ DV T ck T he i þ c VT k þ VT rr þ VT h þ TV e k
h
ð20Þ
he
The expansion and compression spaces’ volumes can be obtained by knowing the crankshaft angle, h. The piston is at the top of the cylinder when the crankshaft angle is zero.
V e ¼ V cle þ 0:5:V swe ½1 þ cosðh þ aÞ
ð21Þ
The above equation, V cle is the expansion clearance volume, V swe is the expansion swept volume and a is the phase angle. Similar equations can be obtained for compression space.
V c ¼ V clc þ 0:5:V swc ½1 þ cosðhÞ
ð22Þ
It is noteworthy, V clc is the compression clearance volume, V swc is the compression swept volume and h is the crankshaft angle. The total engine volume is calculated by Eq. (23):
V ¼ Ve þ Vr þ Vc þ Vk þ Vh
ð23Þ
ð25Þ
As shown in Eqs. (26)–(29), the mass flow rate of each control volume obtained from law of conservation of mass:
gAck ¼ Dmc
ð26Þ
gAkr ¼ gAck Dmk
ð27Þ
gAhe ¼ gArh Dmh
ð28Þ
gArh ¼ gAhe Dmh
ð29Þ
In this method, T ck and T he are used to represent temperatures at intersections of the compression space and cooler and the expansion space and heater respectively. The values of these parameters depend on the direction of the mass flow and are defined as per Eqs. (30) and (31):
if gAck > 0 then T ck
T c otherwise T ck
Tk
ð30Þ
if gAhe > 0 then T he
T h otherwise T he
Te
ð31Þ
The energy differentials for 3 sections, heater, cooler and regenerator, are obtained by Eq. (17) and using the ideal gas equations of state, Eqs. (32)–(34):
DQ k ¼ V k DpC v =R C p ðT ck gAck T kr gAkr Þ
ð32Þ
DQ r ¼ V r DpC v =R C p ðT kr gAkr T rh gArh Þ
ð33Þ
DQ h ¼ V h DpC v =R C p ðT rh gArh T he gAhe Þ
ð34Þ
After obtaining the temperature, pressure and volume of various components, the work differential of each component can be calculated by Eqs. (35)–(37):
M. Korlu et al. / Energy Conversion and Management 147 (2017) 120–134
DW c ¼ pDV c
ð35Þ
DW e ¼ pDV e
ð36Þ
DW net ¼ DW c þ DW e
ð37Þ
In the ideal adiabatic model, the Stirling engine efficiency can be calculated by Eq. (38):
gst ¼
W net Qh
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length. The thermal losses from the regenerator walls can be obtained from Eq. (39) [9]:
Q wrloss ¼
kwr awr ðT wh T wk Þ lr
ð39Þ
It is noteworthy, kwr , awr , lr , T wh and T wk represent Regenerator wall conduction heat transfer coefficient, total regenerator tube wall area, effective regenerator length, heater and cooler wall temperatures.
ð38Þ
In order to numerically solve the equations obtained from the ideal adiabatic model, the equations of pressure, energy and mass change in expansion and compression spaces have to be solved simultaneously. Using the initial values method would be the best way of solving these equations numerically. In this method, the values of all variables are defined at the starting point and the equations are then solved using these values to reach the final values. The obtained functions include all of the variables and the functions related to engine volume change at various crankshaft angles. First, the y vector is defined as a function of the variables, for example yp represents the gas pressure in the system and ymc represents the mass of the gas in the compression space. With the initial values available, the y vector can be defined as y(t0 ) = y0 Using the y vector and the Dy = F(t, y) system of differential equations, we obtain the y(t) values which, satisfies both differential equations and initial values. In fact, in this numerical method the initial values are obtained for time t 0 , then new values are calculated after a small time step, which means the new values are calculated at t 1 = t0 + Dt. Therefore, there is a set of direct loops at various times which give the integer values of y(t). In the adiabatic method, the initial pressure and mass change in the compression space must be substituted into the equations along with other known values. The engine used in this study was an alpha-type double acting Stirling engine. From a power output viewpoint, alpha-type Stirling engines fall into two categories, the two piston alpha-type engines (simple) and four piston alpha-type engines (double acting engines). Compared to simple engines, the double acting engines reach higher power outputs. Using the geometric and performance specifications of the Ford-Philips 4-215 Stirling engine and the initial values, we can obtain the heater, cooler and regenerator heat transfer values and amount of engine’s work done and efficiency. The numerical code is capable of applying the desired geometric and physical changes and obtaining the changes in engine’s operating power and efficiency. 4.2.2. Non-ideal adiabatic model In this section, effects of non-ideal heat exchanger performance on the Stirling engine are investigated. In real Stirling engines some of heat is stored inside the regenerator when the gas is moving from heater to cooler and is not passed on to the gas when it is returning from the cooler. In other words, there is some convection heat transfer between the gas and exchanger walls, which occurs due to differences between the wall and average gas temperatures. The Urieli adiabatic model [9], developed in 1984, is used in this section for problem analysis. In the rest of this section we will investigate the effects of heat exchangers performances on the system as a whole. It should be noted that equations related to heat and hydraulic losses are included in the numerical code. 4.2.2.1. Regenerator wall heat leakage. Physically, the regenerator is placed between the heater and cooler. The temperature difference between the two heat exchangers causes an undesirable heat loss. The amount of this heat loss depends on thermal conductivity of regenerator, effective conduction area and regenerator effective
4.2.2.2. Regenerator net enthalpy loss. Regenerator net enthalpy loss occurs in the non-ideal regenerator and flow of the working gas through the regenerator. e, the regenerator effectiveness, is defined as the actual energy returned to the working fluid divided by the returned energy value in the ideal adiabatic model. Therefore, the energy stored by the regenerator when the gas is passing through on its way from the expansion space to compression space is not returned to the returning working gas completely. In ideal cases the effectiveness coefficient is equal to one (e = 1). Regenerator net enthalpy loss can be obtained from Eq. (40) [9]:
Q rloss ¼ ð1 eÞ ðQ rmax Q rmin Þ
ð40Þ
In Eq. (40), Q rmax and Q rmax symbols represent the maximum and minimum regenerator heat transfers. The regenerator effectiveness is obtained from Eq. (41):
e¼
NTU NTU þ 1
ð41Þ
The number of transfer units (NTU) in Eq. (46) is obtained from Eq. (42):
NTU ¼
St awgr 2ar
ð42Þ
where awgr and ar represent the regen internal wetted area and the regen internal free flow area [9]. The values of Stanton (St) and Prandtl (Pr) numbers are obtained by Eq. (43) from Kays and London’s [24] results:
St ¼
0:46 Re0:4 Pr
Pr ¼ 0:7
ð43Þ
The Reynolds number (Re) is obtained from Eq. (44) [9]:
Re ¼
jgj d
ð44Þ
l
l ¼ lo
1:5 to þ tsuth t to t þ tsuth
ð45Þ
In Eqs. (44) and (50), g, d, l, lo, to, tsuth and t represent the fluid mass flux in the regenerator, regenerator hydraulic diameter, dynamic viscosity at the gas temperature, dynamic viscosity at reference temperature, reference temperature, Sutherland constant and gas temperature respectively. 4.2.2.3. Energy dissipation by pressure drops in heat exchangers and regenerator. In the Stirling engine, the regenerator porosity causes some pressure drop in the working fluid. In turn, this pressure drop decreases the engine power output. The working fluid flow through the regenerator also generates heat and decreases the heat required by the engine. The pressure variations in the regenerator can be calculated by Eq. (46):
dp ¼
2l V g l f r 2
md
ð46Þ
where fr, l, V, g, l, m and d symbols represent the Reynolds friction factor, dynamic viscosity, void volume in the heat exchangers/
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regenerator, fluid mass flux in the heat exchangers/regenerator, effective length of the heat exchangers/regenerator, fluid mass in the heat exchangers/regenerator and hydraulic diameter of the heat exchangers/regenerator respectively and the value of l is calculated by Eq. (45). The value of fr for heat exchangers and regenerator can be obtained by Eq. (47):
f r ¼ 54 þ 1:43Re0:78
f r ¼ 0:0791Re0:75
ð47Þ
The generated heat and work loss in heat exchangers and regenerator can be obtained by substituting the pressure drop values into Eqs. (48) and (49):
Q diss ¼
dp g a
q
dW work ¼ dp dV e
7. Results and discussion
ð48Þ ð49Þ
Finally, Eqs. (50)–(52) are used to calculate the actual power, actual required heat and engine efficiency:
actW power ¼ W power dW work
ð50Þ
actQ hpower ¼ Q hpower þ Q rloss þ Q wrloss Q hdiss Q rdiss
ð51Þ
gst ¼
actW power actQ hpower
ð52Þ
The Q hpower and W power , in Eqs. (50)–(52), represent the required heat and generated power of the Stirling engine in the ideal adiabatic method and are calculated from Eqs. (35) and (37) [9]: 4.3. Equations governing the hybrid system In this section, the whole system is treated as a single control volume then the electric efficiency and net power output are calculated using Eqs. (53)–(55):
_ W
be seen Table 1, while Table 2 contains the gas turbine and the rest of the components’ information. Table 3 compares the results of Urieli’s code [9] with results of the present study which uses the Ford-Philips 4-215 engine. The difference between the present numerical code and Urieli results has arisen from the fact that the present code includes energy dissipation by pressure drops in heat exchangers and regenerator. The presented results show that the numerical code is capable of predicting the Stirling engine power and efficiency.
gele;net ¼ _ net n4 LHV
ð53Þ
_ net ¼ W _ ac;gt þ W _ st W _ a:c W _ f :c W
ð54Þ
_ ac;gt ¼ g _ W in;gen W gt
ð55Þ
_ st can be calculated using Eq. (50). Where, g The value of W in;gen , _ _ _ W a:c , W f :c and W gt represent the DC–AC inverter efficiency, air compressor power consumption, fuel compressor power consumption and gas turbine power generation respectively. 5. Solving method A computer program is written using the aforementioned equations and follows the flowchart shown in Fig. 3. As can be seen in the flowchart, the Stirling engine calculations are carried out by specifying an initial value for heater and cooler temperatures. The gas turbine, and related parameters, calculations are carried out after finding the heat transfer required by the Stirling engine. 6. Verification To verify the computer code, the Stirling engine and turbine gas are verified separately since there are not many studies investigating the system proposed in the present paper. As previously explained this study uses the Ford-Philips 4-215 Stirling engine. This engine was introduced by a Dutch scientist named Philips and was developed and manufactured later by Ford [25]. Nowadays, this engine is known as Philips. The engine parameters can
In the present study, the turbine inlet temperature, compressor pressure ratio, regenerator porosity, Stirling engine regenerator length and diameter are taken as the parameters affecting the hybrid system performance. The aim of the present study was to investigate the effect of aforementioned parameters on gas turbine, Stirling engine and hybrid cycle power and efficiency. The figures for pressure variation against Stirling engine total volume and temperature variation, pressure drop variation in heat exchangers and regenerator, accumulated heat transferred and work done by various Stirling engine components against crankshaft angle are presented for further investigation. It should be noted that calculations for the above figures assumed that gas turbine inlet air flow rate and inlet temperature are equal to 300 kilomole per hour and 1273 K respectively. By assuming above operational conditions for the gas turbine and using the information available in Table 2, the temperature of 1023 K (according to information contained in Table 1) is obtained for the Stirling engine heater. Fig. 4 shows pressure variations against Stirling engine total volume. It should be noted that the work done in each cycle is equal to the area under the P-V graph. The Stirling engine is based on the Carnot cycle, but it can be seen that the Stirling engine have significant differences with the Carnot cycle which is composed of two isothermal and two constant volume processes. These differences have arisen from the fact that, unlike the Carnot cycle, actual conditions are applied to the present numerical code. Fig. 5 illustrates the temperature variations of various Stirling sections against crankshaft angle. As can be seen, there is some convection heat transfer between the wall and working fluid since this model has applied actual conditions and takes the temperature difference of heat exchanger wall and working fluid into account. The convectional heat transfer induces some thermal loss. According to the figure, the heater temperature is lower than 1023 K (the ideal adiabatic model’s heater temperature) and the cooler temperature is higher than 337 K (the ideal adiabatic model’s cooler temperature), which causes the lower performance in the non-ideal adiabatic model. The theta-energy graph, which represents the accumulated heat transferred and work done in various sections in the Stirling engine against crankshaft angle, is shown in Fig. 6. According to the figure, the Stirling engine’s work starts with the expansion process (and positive slope) and continues through the compression process, it finishes the cycle in an expansion process, the Stirling engine’s work assumes positive values at the end of the cycle. Interestingly, the regenerator heavily affects the engine performance and has the highest heat transfer in the engine. It has to be mentioned that the expansion space and heater section’s works are not equal at the beginning of the cycle but converge at the end of the cycle, similar results can be seen for the compression space and cooler. This section will discuss the relation of crankshaft angle with heat exchanger and regenerator pressure drop variations. In the Stirling engine the flow begins from the expansion space and enters heater on its way to the regenerator. The flow of the fluid through the porous environment of the regenerator and its impacts
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Fig. 3. Flowchart for hybrid system.
Table 1 Ford-Philips 4-215 engine data [9]. Number of cooler tubes/cylinder Cooler tube internal diameter Cooler tube length Regenerator diameter Regenerator length Regenerator matrix wire diameter Regenerator matrix mesh size Regenerator matrix porosity Number of regenerator units per cylinder Number of heater tubes/cylinder Heater tube internal diameter Heater tube length Compression clearance volume Expansion clearance volume Compression swept volume Expansion swept volume Total volume of engine Bore Stroke Mean pressure Angular Velocity Operating gas Hot space temperature Cold space temperature
Table 2 Gas turbine performance parameters. 742 0.9 mm 87 mm 73 mm 34 mm 36 lm 200 0.62 2 22 4 mm 462 mm 214.2 cm3 214.2 cm3 870.6 cm3 870.6 cm3 670 cm3 73 mm 52 mm 15 MPa 3300 rpm Hydrogen 1023 K 337 K
with regenerator matrix induces pressure drop in the fluid in addition to transferring heat from the working fluid to regenerator matrix. The working fluid subsequently enters the cooler followed by the compression space and then starts returning back to the expansion space. On its way back, the working fluid passes through the regenerator, where its pressure decreases once again due to
Air flow rate Compressors inlet temperature Compressors inlet pressure Pressure losses in heat exchangers Heat exchangers effectiveness Fuel heat exchanger outlet pressure Turbine inlet temperature AC generator efficiency Pressure losses in the combustion chamber Compressors efficiency Turbine efficiency Combustion chamber efficiency
300 kmol/h 288 K 1 bar 4% 90% 1 bar 1273 K 95% 5% 81% 84% 95%
Table 3 Stirling engine performance parameters.
Power output (kW) Heat transferred to the heater (kW) Thermal efficiency (%)
Urieli and Berchowitz [9]
Present research
Percentage error
192.93 362.01
192.93 351.5
0 3
53.3
54.9
3
absorbing the heat that itself had previously stored in the regenerator matrix. The minus sign is used to convey ‘‘opposite of the working fluid’s flow direction” and is used here to complete the Stirling cycle. Fig. 7 shows significant pressure drops in the thermal regenerator compared to the heater and cooler.
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220
Pressure (bar)
200
180
160
140
120
100 2000
2200
2400
2600
2800
3000
3200
3400
Total volume (cc) Fig. 4. The variation of pressure with Stirling engine total volume.
1100 1000
Tc Te Tk Tr Th Twk Twh
Temperature (K)
900 800 700 600 500 400 300
0
50
100
150
200
250
300
350
400
Crank anle (derees) Fig. 5. The variation of temperatures of the in Stirling engine with crank angle.
While previous sections discussed the parameters affecting the Stirling engine analysis, this section moves on to parameters that affect the Stirling engine, gas turbine and hybrid cycles. The effects of the compressor pressure ratio and turbine gases’ inlet temperature on the Stirling engine, gas turbine and hybrid cycle power output are investigated first. As presented in Figs. 8 and 9, the increasing the compressor pressure ratio increases the gas turbine net power at first but then decreases it. In fact, this happens because increasing the compressor pressure ratio, increases the turbine outlet and inlet enthalpy difference at a higher rate than the compressor, this continues until a certain pressure, after which this process is reversed and the compressor’s enthalpy difference becomes dominant and decreases the gas turbine cycle net power output. On the other hand, the results show that turbine inlet temperature is directly proportional to gas turbine power generation, while it has a negligible relation with the compressor’s power consumption. Furthermore, increasing the compressor pressure ratio decreases the turbine outlet temperature, therefore, the Stirling engine power output decreases as well, since the heat transferred to the engine is a function of the turbine outlet temperature. From
the results, it can be concluded that the hybrid cycle net power output increases with an increase in pressure ratio, but decreases at high pressures ratio. It should be noted that the above analysis is carried out for Stirling engine heater temperatures between 823 and 1023 K. The graphs depicting the Stirling engine efficiency and gas turbine and hybrid cycles electric efficiencies against compressor pressure ratio, for various gas turbine inlet temperatures, are shown in Figs. 10 and 11. As can be seen increasing the compressor pressure ratio increases the gas turbine cycle efficiency at first but decreases it at higher values. This occurs due to direct proportionality of electric efficiency and gas turbine cycle net power output. According to both graphs, increasing compressor pressure ratio decreases the Stirling engine efficiency, which occurs due to a decrease in Stirling engine net power output. From these results, it can be concluded that by an increase in pressure ratio, increases the hybrid cycle electric efficiency at first but then decreases it after a certain ratio and mimics the hybrid cycle net power output results. It should be noted that the above analysis is carried out for Stirling engine heater temperatures between 823 and 1023 K.
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20
10
Energy [kj]
0
-10
Qk Qr
-20
Qh
-30
Wcompression
W Wexpansion -40
0
50
100
150
200
250
300
350
400
Crank angle (degrees) Fig. 6. The variation of accumulated heat transferred and work done in various sections in the Stirling engine with crank angle.
250
cooler pressure drop
Heat exchanger pressure drop [kPa]
200
heater pressure drop
150
regenerator pressure drop
100 50 0 -50 -100 -150 -200 -250
0
50
100
150
200
250
300
350
400
Crank angle (degrees) Fig. 7. The variation of pressure drop across heat exchangers and regenerator in the Stirling engine with crank angle.
This section investigates the effects of Stirling engine parameters, e.g. regenerator porosity, regenerator length and diameter, on performance of the Stirling engine and hybrid cycle. The investigations are carried out for turbine inlet flow rate of 300 kilomole per hour and turbine inlet and Stirling heater temperatures of 1273 and 1023 K respectively. The effect of regenerator length on regenerator loss is presented in Fig. 12. According to the figure the regenerator net enthalpy loss decreases when regenerator length is increased. To explain this it must be noted that increasing the regenerator length, increases the regenerator effectiveness, which in turn lowers regenerator net enthalpy loss. Regenerator wall heat leakage is also reduced, which is expected since it is inversely proportional to the regenerator length. Overall it seems that energy dissipation by pressure drops in regenerator increase alongside regenerator length, again this is as expected since pressure drop us directly proportional to regenerator length. The effect of regenerator length on Stirling engine and hybrid cycle power output and efficiency is presented in Fig. 13. The figure
shows that the Stirling engines power output decreases as regenerator length increases, this is caused by the increased pressure drop, which in turn increases work loss in the regenerator and heat exchangers. Increasing the regenerator length, increases the hybrid cycle power output due to decreased Stirling engine heater heat transfer and increased gas turbine power output. The increased hybrid cycle power output itself causes an improvement in hybrid cycle electric efficiency. It should be noted that the improved Stirling engine efficiency can be explained by the decrease in regenerator-related losses. The Effect of engine diameter on regenerator loss is shown in Fig. 14. According to the figure the regenerator net enthalpy loss is reduced with increasing regenerator diameter. To explain this it must be noted that increasing the regenerator diameter, increases the regenerator effectiveness, which in turn lowers regenerator net enthalpy loss. Regenerator wall heat leakage is also reduced since it is inversely proportional to regenerator diameter. Also, energy dissipation by pressure drops in regenerator decreases with an increase in regenerator diameter. This can be justified by
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500
430
Gas Turbin
Power output (kW)
Stirling engine 360
Stirling engine + Gas Turbin
290
220
150
80
1
2
3
4
5
6
7
Compressor pressure ratio Fig. 8. The variation of Stirling engine, gas turbine and hybrid cycle power output with compressor pressure ratio (TIT = 1273).
480
Power output (kW)
400
320
240
160
Gas Turbin Stirling engine Stirling engine + Gas Turbin
80
0
1
2
3
4
5
6
Compressor pressure ratio Fig. 9. The variation of Stirling engine, gas turbine and hybrid cycle power output with compressor pressure ratio (TIT = 1223).
58
50
Gas Turbine 40
54
Stirling engine + Gas Turbin
30
50
20
46
10
1
2
3
4
5
6
7
Efficiency (%)
Electrical efficiency (%)
Stirling engine
42
Compressor pressure ratio Fig. 10. The variation of Stirling engine, gas turbine and hybrid cycle efficiency with compressor pressure ratio (TIT = 1273).
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50
58
Gas Turbine Stirling engine Stirling engine + Gas Turbin
54
30 50 20 46
10
0
Efficiency (%)
Electrical efficiency (%)
40
1
2
3
4
5
6
42
Compressor pressure ratio Fig. 11. The variation of Stirling engine, gas turbine and hybrid cycle efficiency with compressor pressure ratio (TIT = 1223).
35 30
Regenerator net enthalpy loss Regenerator wall heat leakage
25
Losses[kW]
Regenerator Dissipation 20 15 10 5 0 10
20
30
40
50
60
70
Regenerator length (mm)
330
58
290
52
Actual ST power output Actual GT+ST power output
250
46
Actual ST efficiency
Efficiency (%)
Power output [kW]
Fig. 12. The variation of regenerator losses with regenerator length.
Actual GT+ST electrical efficiency 210
170 22
40
28
34
40
34 46
Regenerator length (mm) Fig. 13. The variation of Stirling engine and hybrid cycle efficiency and power output with regenerator length.
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20
Regenerator Dissipation Regenerator wall heat leakage
Losses[kW]
16
Regenerator net enthalpy loss
12
8
4
0 58
63
68
73
78
83
88
Regenerator diameter (mm)
350
58
300
52
Actual ST power output Actual GT+ST power output 250
46
Actual ST efficiency Actual GT+ST electrical efficiency
200
150 60
Efficiency (%)
Power output [kW]
Fig. 14. The variation of regenerator losses with regenerator diameter.
40
65
70
75
80
34 85
Regenerator diameter (mm) Fig. 15. The variation of Stirling engine and hybrid cycle efficiency and power output with regenerator diameter.
the decreased pressure drop which in turn is inversely proportional to regenerator diameter. Fig. 15 shows the variations in Stirling engine and hybrid cycle power outputs and efficiencies against regenerator diameter. As can be seen, the Stirling power output decreases with an increase in regenerator diameter, this occurs due to the increased, pressured drop induced, work loss in heat exchangers and regenerator. Increasing regenerator diameter, increases hybrid cycle power generation due to decreased heat transfer in Stirling heater and increased gas turbine power generation. The increased hybrid cycle power output itself causes an improvement in hybrid cycle electric efficiency. The reason for improved Stirling engine efficiency, after an increase in regenerator diameter, is a decrease in regeneratorrelated losses. The graph of regenerator porosity against regenerator losses is shown in Fig. 16. As can be seen in the figure, regenerator net enthalpy loss increases with regenerator porosity. This can be justified by considering that, according to the equation of regenerator net enthalpy loss, increasing regenerator porosity decreases regen-
erator efficiency. Regenerator wall heat leakage is independent of regenerator porosity, therefore is does not change. Energy dissipation by pressure drops in regenerator decrease when there is an increase in regenerator porosity. This is due to the increase in hydraulic diameter for regenerator which is inversely proportional to the regenerator pressure drop. The effect of regenerator porosity variations on the Stirling engine and hybrid cycle efficiencies and power outputs is illustrated in Fig. 17. As can be seen in the figure, Stirling engine and hybrid cycle power generations both increase and then decrease with regenerator porosity. This occurs since the work loss in regenerator and heat exchangers initially approaches the optimal value, but then increases after a certain value and therefore decreases both power and efficiency. The Stirling engine efficiency and overall system efficiency both follow the behavior of their respective power output. The results show that in lower pressure ratios most of the power is generated by the Stirling engine and the gas turbine cycle produces relatively smaller amounts. Yet, the share of gas in power
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200
Regenerator net enthalpy loss Regenerator wall heat leakage
Losses[kW]
150
Regenerator Dissipation
100
50
0 0.35
0.55
0.75
0.95
Porosity Fig. 16. The variation of regenerator losses with regenerator porosity.
340
56
50
260
220
Actual ST power output Actual GT+ST power output Actual ST efficiency Actual GT+ST Electrical efficiency
44
38
180
140 0.38
Efficiency (%)
Power output [kW]
300
0.46
0.54
0.62
0.7
0.78
32 0.86
Porosity Fig. 17. The variation of Stirling engine and hybrid cycle efficiency and power output with regenerator porosity.
Fig. 18. Comparison of Stirling engine and hybrid cycle efficiency and power output.
generation improves by increasing the pressure ratio. From the figures it can be concluded that the adding a Stirling engine to gas turbine cycle can improve a simple gas turbine cycle efficiency
and power generation. Fig. 18 shows a comparison between a simple gas turbine cycle and the proposed hybrid cycle. According to the figure, the generated power had increased from 263.4 kW for
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the simple gas turbine to 432.4 kW for the hybrid cycle, which is a significant improvement. Subsequently, the electric efficiency is also improved significantly, form 23.6% to 38.85%. It should be noted that the investigation was carried out for the following conditions, gas turbine inlet temperature of 300 kilomole per hour, turbine inlet temperature of 1273 K and Stirling engine heater temperature of 923 K. These conditions result in the highest efficiency possible for the hybrid cycle. Considering the aforementioned results it can be concluded that the Stirling engine is very suitable for extracting energy from gas turbine exhaust gases. 8. Conclusions In this study, we have tried to introduce a new subject for the use of gas turbine exhaust and combining this system with Stirling engine for producing more power. To accomplish this goal, a portion of gas turbine exhaust gases are allocated to providing the heat required for the Stirling engine. The double acting Stirling engine is used in this study and the non-ideal adiabatic model is used to numerical solution. The influence of several effective parameters (such as the pressure ratio of the compressor, the inlet temperature of turbine, the porosity, length and diameter of the regenerator) on hybrid system performance was studied. The following points are presented as conclusion from the presented study The area below the pressure-volume graph of the Stirling engine is significantly different from the Carnot cycle. The results show that most of the energy transfer in the Stirling engine occurs in the regenerator, which indicates that the regenerator has significant influence on Stirling engine efficiency. The highest pressure drop has occurred in the regenerator, followed by heater and cooler. The Stirling engine power generation and efficiency decrease, when the compressor pressure ratio increases. The hybrid cycle power generation and efficiency increases and then decreases with increasing pressure ratio. It is also noteworthy that an increase in the turbine inlet temperature increases both power generation and efficiency of the hybrid cycle. The results show that increasing the regenerator length from 24 to 44 mm, causes increases of 2 and 1% in the efficiency of the Stirling engine and hybrid cycle respectively. The results show that increasing the regenerator diameter from 63 to 84 mm, causes increases of 2.5 and 1% in the efficiency of the Stirling engine and hybrid cycle respectively. The results show that with increasing regenerator porosity from 0.42 to 0.82 can be achieved regenerator porosity optimum point, Stirling engine and hybrid system maximum efficiency about 55 and 36.7% respectively.
Combining the gas turbine with the Stirling engine increases the net power output from 263.4 kW for the gas turbine to 432.4 kW for the hybrid cycle and increases efficiency from 23.6% for the gas turbine to 38.85% for the hybrid cycle.
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