Article Use of Stirling Engine for Waste Heat Recovery
Peter Durcansky * , Radovan Nosek and Jozef Jandacka Department of Power Engineering, Faculty of Mechanical Engineering, University of Zilina, 010 26 Zilina, Slovakia; [email protected] (R.N.); [email protected] (J.J.) * Correspondence: [email protected]
Received: 2 June 2020; Accepted: 4 August 2020; Published: 10 August 2020
Abstract: Even though this discovery dates back to 1816, the greatest advancement in technology and understanding of Stirling-cycle devices has occurred in the last 50 years. Although their mass production is currently limited to special-purpose machines, its prospective use is in combination with renewable sources and indicates a potential for commercial purposes. The lack of commercial success, despite obvious advantages, is probably due to a lack of appropriate modeling techniques and theoretical predictions of what these devices can achieve. Nowadays the Stirling engine has found its use mainly in solar power plants, where it represents the only piston engine converting solar energy into mechanical and then electricity with relatively high efficiency. The Stirling engine also appears to be suitable for recovering waste heat, especially in heavy industry. The numerical model was adapted for the existing Cleanergy Stirling engine, to evaluate the possibilities of this one engine for waste heat recovery. This paper also deals with application options and individual parameters that affect the efficiency of this Stirling engine for waste heat recovery. The analysis showed that this kind of engine is capable of recovering and utilizing heat above 300 ◦C, which determines its possible use with solar energy.
Keywords: waste heat; Stirling engine; heat recovery
1. Introduction A Stirling engine is one of the few representatives of hot-air engines. At the time of invention, its efficiency was comparable to the steam engine. This engine forms a closed thermodynamic system, using heat energy supplied from an external source and converts it to mechanical [1]. Its construction consists of two pistons and one or two cylinders, with a regenerator placed between. A regenerator can be used as a wire or ceramic grid, or any kind of porous material, which has a high heat capacity and low thermal conductivity. The regenerator serves for the temporary storage of thermal energy [2]. A great advantage of the Stirling engine, but also other hot air engines, is the variety of heat sources that can be used [3]. The simpler design compared to internal combustion engines with only a few moving parts is a great advantage. Single-phase working medium, without production of any carbon fouling, allows an extended service interval. Very low noise, smooth operation and low vibration level make these engines also suitable for domestic use. The engine does not necessary need oxygen for burning process, where the heat can be supplied from any heat source, allowing its use in space applications. One of the best described engines is the GM GPU3 Stirling engine, which served as the basis for many projects. This engine has 236 cm3, rhombic drive, and works with pressure from 1.2 to 6.9 MPa [4,5]. The impact of the working medium is very significant, but no less important is the regenerator located between the two cylinders, where heat is transferred from the hot to the cold cylinder, which should be supplied to the medium during further isochoric compression [6]. As working medium can be used air, but the most common is helium [7]. A comparison to other engines and some new models is shown in Table1. This table presents engines prepared for production, but also experimental engines. The most suitable experimental engines for heat recovery are those with lower heater temperatures [7].
Energies 2020, 13, 4133; doi:10.3390/en13164133 www.mdpi.com/journal/energies Energies 2020, 13, x FOR PEER REVIEW 2 of 15
also experimental engines. The most suitable experimental engines for heat recovery are those with lower heater temperatures [7]. Energies 2020, 13, 4133 2 of 14 Table 1. List of Stirling engines.
Mean Pressure Heater Temperature Efficiency Engine Manufacturer Table 1. List of Stirling engines. Power (kW) (MPa) (°C) (%) Mean Pressure Heater EngineUnited Manufacturer Stirling [8] 14.5 691Power (kW) E35ffi ciency (%) 30 (MPa) Temperature (◦C) Philips [7] 14.2 649 23 38 United Stirling [8] 14.5 691 35 30 GMPhilips GPU-3 [7] [8] 14.2Up to 6.9 649816 238.1 38 39 GMFarwell GPU-3 [8 ][8] Up to 6.90.1 816594 8.10.1 39 16.9 KarabulutFarwell [8 ]et al. [9] 0.10.4 594260 0.10.183 16.9 N/A Karabulut et al. [9] 0.4 260 0.183 N/A KwankaomengKwankaomeng [10] [10] 0.10.1 150150 0.10.1 5.6 5.6 Cooke-YarboroughCooke-Yarborough [11]N [11] /AN/A 292292 0.1070.107 13.7 13.7 SoloSolo Stirling Stirling [12]N [12] /AN/A Min. 540Min. 540 7.37.3 25.4 25.4 WhisperGen [wikipedia] N/A Min. 550 Up to 1 kW 11 WhisperGenViessmann Vitotwin [wikipedia] N/A Min. 550 Up to 1 kW 11 N/A Min. 500 Up to 1 kW 30 Viessmann[viessmann.de] Vitotwin [viessmann.de] N/A Min. 500 Up to 1 kW 30 CleanergyCleanergy [12] [12] 1.5–91.5–9 N/AN/A 2–92–9 25 25
OurOur laboratorylaboratory isis workingworking withwith thethe CleanergyCleanergy StirlingStirling engine,engine, soso toto analyzeanalyze andand evaluateevaluate thethe propertiesproperties of of thisthis StirlingStirling engineengine forfor applicationapplication inin thethe heatheat recoveryrecovery system,system, wewe havehave chosenchosen toto evolveevolve aa numericalnumerical model.model. The engine is αα-type,-type, withwith 160160 cmcm33, and the cross-section is shown in FigureFigure1 1.. TheThe aimaim ofof thisthis workwork isis toto give information fromfrom experimentsexperiments withwith thisthis commerciallycommercially availableavailable engineengine underunder variousvarious conditionsconditions andand itsits capabilitycapability forfor thethe heatheat recoveryrecovery system. system.
Figure 1. Cleanergy α—type Stirling engine, adaptation from [11]. Figure 1. Cleanergy α—type Stirling engine, adaptation from [11]. 2. Numerical Model 2. Numerical Model Stirling engine cycle is composed of isothermal compression, expansion, constant volume heating, and coolingStirling [12 engine]. One ofcycle the firstis composed analyses ofof the isothe Stirlingrmal engine compression, is the Schmidt expansion, analysis, constant made by volume Gustaf Schmidtheating, and in 1871 cooling [13]. [12]. He One made of somethe first assumptions, analyses of the for Stirling example, engine all parts is the are Schmidt moving analysis, sinusoidally, made constantby Gustaf temperatures Schmidt in in 1871 all parts [13]. of He the made engine, some and perfectassumptions, gas law for is appliedexample, to workingall parts fluids.are moving Input sinusoidally, constant temperatures in all parts of the engine, and perfect gas law is applied to parameters for application engines according to Schmidt analyses are the volumes Vswe, Vcle, Vr, Vswc, working fluids. Input parameters for application engines according to Schmidt analyses are the Vclc working temperatures Th, Tk, as shown in Figure2[12]. volumes V , V , V , V , V working temperatures T , T , as shown in Figure 2 [12].
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FigureFigure 2.2. SchmidtSchmidt analysis,analysis, adaptedadapted fromfrom [[12].12].
AccordingAccording toto thesethese assumptionsassumptions [[13],13], thethe volumesvolumes areare defineddefined as:as:
𝑉 Vswe 𝑉 =𝑉Ve = V+ + ∙[1−𝑐𝑜𝑠[1 cos((𝜑ϕ))]] (1)(1) cle 2 2 · − ForFor αα—type,—type, the the compression compression space space is is defined defined as: as: 𝑉 h i 𝑉 =𝑉 + V∙[1−𝑐𝑜𝑠 𝜑−𝛼swc ] (2) Vc = V + 1 cos ϕ α (2) clc 2 2 · − − f The main variables are pressure, volume, and temperature difference. When assessing efficiency The main variables are pressure, volume, and temperature difference. When assessing efficiency in terms of temperatures, it is necessary to take into account the real possibilities and the technology in terms of temperatures, it is necessary to take into account the real possibilities and the technology available [14]. It is also necessary to adapt the maximum cycle temperatures to the pressure ratios available [14]. It is also necessary to adapt the maximum cycle temperatures to the pressure ratios and permissible operating pressures [15,16]. A slightly different condition in terms of heat transfer is and permissible operating pressures [15,16]. A slightly different condition in terms of heat transfer is achievable with the highest heater and lowest cooler temperatures—the highest determined by the achievable with the highest heater and lowest cooler temperatures—the highest determined by the heat heat source temperature and the lowest determined by the cooling method [17]. More advanced source temperature and the lowest determined by the cooling method [17]. More advanced numerical numerical model was presented by Urieli [18], with adiabatic engine assumptions. Another analytical model was presented by Urieli [18], with adiabatic engine assumptions. Another analytical model model for estimating the performance of Stirling engines is known as finite speed thermodynamic for estimating the performance of Stirling engines is known as finite speed thermodynamic analysis, analysis, which is based on the basic laws of thermodynamics. In last years the finite time which is based on the basic laws of thermodynamics. In last years the finite time thermodynamic thermodynamic analysis was used for an endoreversible Stirling engine [17]. analysis was used for an endoreversible Stirling engine [17]. To use the Stirling engine for waste heat recovery, the maximal temperature is determined by To use the Stirling engine for waste heat recovery, the maximal temperature is determined by the the source, in the form of gas, or liquid [19]. So it would seem most appropriate to use, or rather source, in the form of gas, or liquid [19]. So it would seem most appropriate to use, or rather design a design a device with the highest possible mass flow to ensure sufficient power. Here, however, the device with the highest possible mass flow to ensure sufficient power. Here, however, the opposite opposite side of the problem must be realized. As the amount of working fluid flowing through the side of the problem must be realized. As the amount of working fluid flowing through the constant constant cross-section area or pipe increases, the speed increases, and the flow can become critical cross-section area or pipe increases, the speed increases, and the flow can become critical [20]. Thus, [20]. Thus, with increasing quantities, the entire volume must increase and the cost of the mechanical with increasing quantities, the entire volume must increase and the cost of the mechanical parts of parts of the engine is disproportionately increasing as it is a piston machine. Other main reasons the engine is disproportionately increasing as it is a piston machine. Other main reasons influencing influencing the resulting cycle efficiency are harmful space of cylinders, low regenerator efficiency, the resulting cycle efficiency are harmful space of cylinders, low regenerator efficiency, real pressure real pressure losses in the regenerator, in pipes and heat exchangers, as well as mechanical friction losses in the regenerator, in pipes and heat exchangers, as well as mechanical friction losses of moving losses of moving parts and also heat losses. The regenerator, as the main energy conservation part, parts and also heat losses. The regenerator, as the main energy conservation part, has a significant role has a significant role and the change in the inner heat transfer coefficient can influence total effectivity and the change in the inner heat transfer coefficient can influence total effectivity more than friction more than friction losses [21]. However, the regenerator quality can be also expressed in terms of losses [21]. However, the regenerator quality can be also expressed in terms of enthalpy difference [22]. enthalpy difference [22]. Our numerical model is based on [18] and adiabatic engine assumptions. Our numerical model is based on [18] and adiabatic engine assumptions. According to the model, According to the model, the engine is theoretically divided into five characteristic control volumes: the engine is theoretically divided into five characteristic control volumes: expansion chamber (e), expansion chamber (e), heater (h), cooler (k), regenerator (r), and compression chamber (k). The heater (h), cooler (k), regenerator (r), and compression chamber (k). The model is based on the basic model is based on the basic equations of mass and energy equilibrium, with a state equation applied
Energies 2020, 13, 4133 4 of 14 Energies 2020, 13, x FOR PEER REVIEW 4 of 15 toequations each control of mass volume and energy [18]. Figure equilibrium, 3 shows with the ageneral state equation control appliedvolume toand each the control relevant volume variables [18]. usedFigure in3 equations.shows the general control volume and the relevant variables used in equations.
FigureFigure 3. 3.Common Common control control volume volume [ 17[17].]. By applying the mass balance equation to the above control volume, the mass change according By applying the mass balance equation to the above control volume, the mass change according to the piston crank position (ϕ) can be expressed as: to the piston crank position (ϕ) can be expressed as: . . 𝑑𝑚dm 𝑚 −𝑚mi m=o = (3)(3) − 𝑑𝜑dϕ By including the equation of energy equilibrium and neglecting the kinetic components wewe getget 𝑑𝑄 𝑑𝑊 𝑑(𝑚𝑇) +𝑐dQ 𝑇 𝑚 −𝑐. 𝑇 𝑚 . = dW+𝑐 d(mT) (4) + c T m cpo To mo = + cv (4) 𝑑𝜑 dϕ pi i i − 𝑑𝜑dϕ 𝑑𝜑dϕ The evaluation of the pressure change in the engine operating space is calculated taking into The evaluation of the pressure change in the engine operating space is calculated taking into account the mass balance applied to the whole engine. account the mass balance applied to the whole engine. 𝑀=𝑚 +𝑚 +𝑚 +𝑚 +𝑚 (5) M = mc + mk + mr + mh + me (5) And after supplying mass balance into the form of pressure equation: And after supplying mass balance into the form𝑀𝑟 of pressure equation: 𝑝= 𝑉 𝑉 𝑉 𝑉 𝑉 (6) + + Mr+ + p =𝑇 𝑇 𝑇 𝑇 𝑇 (6) Vc + Vk + Vr + Vh + Ve The equation takes into account the Tpressurec Tk withTr theTh volumeTe contained through the working fluid Thein each equation control takes volume. into The account volumes the pressure associated with with the the volume heater, contained cooler, and through regenerator the working remain constantfluid in each during control the cycle. volume. OnThe the volumesother hand, associated the volume with in the the heater, compression cooler, and and regenerator expansion remainspaces changesconstant continuously during the cycle. and thus On thedetermines other hand, the pressure the volume fluctuation in the compression during the cycle. and expansion Volume changes spaces arechanges determined continuously by the and dynamics thus determines of the crank the pressure mechanism, fluctuation which during determines the cycle. the Volumeposition changes of the pistonsare determined at different by thecrank dynamics angle (φ of) thevalues. crank These mechanism, variations which are cons determinesidered sinusoidal the position and of thethus pistons allow theat di derivationfferent crank of different angle (ϕ expressi) values.ons These for all variations engine configurations. are considered sinusoidal and thus allow the derivationThe expected of different temperature expressions changes for all in engine the individu configurations.al volumes are shown in Figure 4. Cooler and heaterThe temperatures expected temperature are assumed changes to be inconstant the individual, with the volumes temperatures are shown of the in compression Figure4. Cooler and expansionand heater spaces temperatures varying areover assumed the cycle to depend be constant,ing on withadiabatic the temperaturesexpansion and of compression. the compression The compressionand expansion and spaces expansion varying work over is thecalculated cycle depending from the energy on adiabatic balance expansion equation andfor the compression. respective workingThe compression volumes, and which expansion are considered work is calculated adiabatic from[18]. the energy balance equation for the respective working volumes, which are considered adiabatic [18].
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Figure 4. Temperature change in control volumes, adapted from [17]. Figure 4. Temperature change in control volumes, adapted from [17]. Mass of working gas is defined by ideal isothermal Schmidt analysis. This analysis assumes isothermalMass of working working spaces, gas is i.e., defined the temperatures by ideal isothermal in the compression Schmidt analysis. and expansion This analysis space are assumes equal to isothermalthe temperatures working of spaces, the cooler i.e., and the heatertemperatures [22–25]. in These the compression assumptions and make expansion it possible space to obtainare equal the tofollowing the temperatures analytical of expression. the cooler and heater [22–25]. These assumptions make it possible to obtain the following analytical expression. Z ! 2π p V V V V V 𝑝 𝑉 c 𝑉 k𝑉 𝑉r 𝑉 h e 𝑀= M = + + + ++ ++ +𝑑𝜑 dϕ (7)(7) r Tk Tk Tr Th Th 0 𝑟 𝑇 𝑇 𝑇 𝑇 𝑇 TheThe analytical analytical solution solution according according to to [17] [17] for for the the alpaa alpaa configuration: configuration:
𝑝 (𝑠√1−𝑏√ ) 2 𝑀= pm s 1 b (8) M = 𝑟 − (8) r where parameters “s” and “b” are defined as: where parameters “s” and “b” are defined as: 𝑐 𝑏= (9) 𝑠 c b = (9) s 1 s𝑉 𝑉 𝑉 𝑉 !2 !2 (10) 𝑐= 1 V +2 V V𝑐𝑜𝑠 𝛼 + V 2 𝑇 swe 𝑇 swe𝑇 swc 𝑇 swc c = + 2 cos α f + (10) 2 Th Th Tk Tk 𝑇 Th 𝑉 𝑙𝑛V 𝑉V 𝑉V 𝑉 V r𝑇ln T 𝑉 V𝑉 V𝑉 V s𝑠== swc++ clc++ +k + +k ++ h + + swe + cle (11)(11) 2𝑇2T 𝑇T 𝑇 T 𝑇 −𝑇T T 𝑇 T2𝑇 2T𝑇 Te k k k h − k h h TheThe working working medium medium mass mass at at each each control control volume volume and and the the mass mass flow flow rate rate at at the the control control volume volume interfacesinterfaces (m (mckck,, mmkrkr,, mmrhrh,, mmhehe)) are are evaluated evaluated at at each each crank crank angle angle position. position. TheThe mass mass components components “ “j”j” representing representing the the set set jj == {{kk, ,rr, ,hh}} concerning concerning the the crank crank angle angle for for heat heat exchangers,exchangers, which which are are considered considered to to be be isot isothermal,hermal, are are calculated calculated from the equation: 𝑑𝑚 𝑚 𝜕𝑝 ! dm j mj ∂p = = (12)(12) 𝑑𝜑dϕ 𝑝 p𝜕𝜑 ∂ϕ Mass components for compression and expansion volume, which are considered not to be Mass components for compression and expansion volume, which are considered not to be isothermal, can be expressed as: isothermal, can be expressed as: 𝜕𝑝 ∂p V ∂V 𝑉 c ∂ϕ 𝜕𝑉p c + 𝜕𝜑 dm 𝑝 +∂ϕ κ (13) 𝑑𝑚 c =𝜕𝜑 𝜅 (13) d=ϕ rT 𝑑𝜑 𝑟𝑇 ck
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∂p Ve ∂ϕ p ∂Ve + dm ∂ϕ κ e = (14) dϕ rThe Mass flow at each control volume boundary:
. ∂m m = c (15) ck − ∂ϕ
. ∂m m = e (16) he − ∂ϕ
. . ∂m m = m k (17) kr ck − ∂ϕ
. . ∂m m = m + h (18) rh he ∂ϕ Furthermore, by substituting mass components into pressure change equation:
∂Vc ∂Ve ! κp ∂ϕ + ∂ϕ dp Tck The = V V (19) dϕ − Vc + κ k + Vr + h + Ve Tck Tk Tr Th The
The regenerator temperature change is linear and according to [22] can be defined as mean effective value: Th Tk Tr = − (20) ln Th Tk The temperature change for compression and expansion volume during the cycle can be expressed via a differential form of state equation for working gas:
∂p ∂Vc ∂mc dTc ∂ϕ ∂ϕ ∂ϕ = Tc + (21) dϕ p Vc − mc
∂p ∂Ve ∂me dTe ∂ϕ ∂ϕ ∂ϕ = Te + (22) dϕ p Ve − me
Boundary temperatures Tck and The are dependent on working medium flow direction. By applying the energy balance equation to each heat exchanger, taking into account that these are isothermal spaces, we obtain, according to [18] the following expressions: ∂p Vk cv dQk ∂ϕ = cp(T m T m ) (23) dϕ r − ck ck − kr kr ∂p Vr cv dQr ∂ϕ = cp(T m T m ) (24) dϕ r − kr kr − rh rh ∂p Vh cv dQh ∂ϕ = cp(T m T m ) (25) dϕ r − rh rh − he he Energies 2020, 13, 4133 7 of 14
Compression and expansion work can be expressed from the energy balance equation for every control volume, which is considered as adiabatic: ! dW ∂V c = p c (26) dϕ ∂ϕ ! dW ∂V e = p e (27) dϕ ∂ϕ Then the total cycle work, depending on the crank angle, is obtained as the sum of the individual components of the compression and expansion work.
dW dW dW = c + e (28) dϕ dϕ dϕ
The system is solved as a quasi-static flow, so the mass flows in all control volumes (heater, cooler, regenerator) is constant for each integration interval. For the calculation, it is necessary to know at least the initial values of working temperatures for compression and expansion cylinder, which are the result of adiabatic processes and enthalpy flow. The only clue to their correct selection is that their end-of-cycle values should be equal to their respective values at the start of the cycle. The pressure variation in the working cylinder, according to Ipci and Karabulut [21], can be described as a function of temperature. Thermal resistance was also expressed according to Cheng [23]. Due to the cyclical nature of the system, the problem is solved by entering the initial conditions and integrating them over several cycles, with the optimization method based on [26–28]. The example calculation is performed for the Cleanergy C9G Stirling engine, using Matlab, partly the Matlab PDE toolbox [29]. Some solutions of differential equations were used according to [30]. Due to the model simplicity, one computational cycle took only few minutes to solve. Boundary conditions were established from experiments and from [31], where the same engine was examined. In this study [31], author examined fuel impact on engine effectivity and proved possible fuel change from natural gas to renewable sources. In Figure5 are presented the temperatures changes in the compression and expansion volumes during one cycle. The dashed lines represent the values of temperatures Th and Tk, which represent a constant average value of the temperature of the heater and cooler heads, respectively. The Te and Tc represent temperatures in expansion and compression volume. In this case, the values Th = 313.15 K, Tk = 973.15 K, the mean value of the working pressure pm = 150 bar at the rotational frequency f = 25 Hz were chosen. These values simulate the real operation of the device at the maximum pressure of the working medium. The temperatures of the expansion volume change in a wider temperature range, where the maximum absolute value of the change is equal to dTe = 72.41 K, while at a compression volume dTc = 16.92 K. This phenomenon is caused by the conversion of the thermal energy into mechanical energy. Energy change during one cycle is shown in Figure6. EnergiesEnergies2020 2020, 13, 13, 4133, x FOR PEER REVIEW 8 of8 15 of 14 Energies 2020, 13, x FOR PEER REVIEW 8 of 15
Figure 5. The temperatures during compression and expansion. Figure 5. TheThe temperatures temperatures during co compressionmpression and expansion.
FigureFigure 6.6. Energy change change during during one one cycle. cycle.
3. Experimental Measurements 3. Experimental Measurements Figure 6. Energy change during one cycle. TheThe measurement measurement and and calculation calculation were were made made for thefor Stirlingthe Stirling engine, engine, which which is part is of part the Cleanergy of the 3.C9G CleanergyExperimental cogeneration C9G Measurements cogeneration unit. The measurementunit. The measurement was carried was out carried at di outfferent at different engine operatingengine operating pressure settingsEnergiespressure 2020 using settings, 13, naturalx FOR using PEER gas natural REVIEW as fuel. gas The as layoutfuel. The is layout shown is in shown Figure in7 .Figure 7. 9 of 15 The measurement and calculation were made for the Stirling engine, which is part of the Cleanergy C9G cogeneration unit. The measurement was carried out at different engine operating pressure settings using natural gas as fuel. The layout is shown in Figure 7.
Figure 7. The experimental setup. Figure 7. The experimental setup.
The measured parameters were the inlet and outlet temperature from the water heat exchanger, which is a part of the engine. To measure them, wells were created on the inlet and outlet pipes, in which the sensors were placed. Temperature measurements were performed with PT100 resistance thermometers, flow with magnetic flow meters, and pressure with pressure gauges. The internal engine parameters such as the temperature of the radiator, heater, combustion chamber, and helium working pressure are recorded by the engine control panel. These values are recorded on a computer using MODBUS technology. The cogeneration unit was connected to an exchanger control station, which cooled the excess thermal energy via an external cooler with a capacity of 50 kW. The working gas supply and the associated working gas pressure in the engine were provided by an electromagnetic valve located on the helium storage bottle. The electrical power produced was displayed on the control panel. Natural gas consumption was measured by a gas meter with remote reading. The engine has its own control system, which allows to control and monitor the engine in real time, without computer connection. After pressing the start button, the control system checks working medium pressure, fuel gas pressure and starts to heat up the burning chamber. After reaching approx. 400 °C, the engine starts and continues the heating process until it reaches 520 °C in the combustion chamber, when the engine starter disconnects and the engine starts to produce electricity. After few test experiments we have found out, that there is a huge difference between hot and cold start. For cold start, where the engine and cooling circuit have ambient temperature, the heating process lasts at least 15 min, electricity is produced after 7 min and after another 8 min the electricity production is stable. For a hot start, where cooling circuit is pre-heated to 80 °C and engine is not cooled down, with hot cylinder temperature of 200 °C, the starting process took only 7 min, where electricity production occurred after 1.5 min. Due to this we have always started the measurement with cold engine, with cooled down heating circuit. Every experiment consisted of 4 periods—starting process, engine pressure setup, measurement, cooling down. The main measurement was performed for 2 h, under stable conditions, where data were collected via ModBus: engine working pressure, engine heater temperature, engine cooler temperature, electric output at generator, external cooling system temperatures and mass flow. From these data, the average value was expressed and standard deviation estimation was determined, according to [31–33], as: 1 𝑥̅ = 𝑥 (29) 𝑛
∑ 𝑥 −𝑛𝑥̅ 𝑠 = (30) ̅ 𝑛(𝑛 − 1)
where n is the number of measured values, in our case it is 360, xi is the measured value and 𝑠 ̅ is standard deviation.
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The measured parameters were the inlet and outlet temperature from the water heat exchanger, which is a part of the engine. To measure them, wells were created on the inlet and outlet pipes, in which the sensors were placed. Temperature measurements were performed with PT100 resistance thermometers, flow with magnetic flow meters, and pressure with pressure gauges. The internal engine parameters such as the temperature of the radiator, heater, combustion chamber, and helium working pressure are recorded by the engine control panel. These values are recorded on a computer using MODBUS technology. The cogeneration unit was connected to an exchanger control station, which cooled the excess thermal energy via an external cooler with a capacity of 50 kW. The working gas supply and the associated working gas pressure in the engine were provided by an electromagnetic valve located on the helium storage bottle. The electrical power produced was displayed on the control panel. Natural gas consumption was measured by a gas meter with remote reading. The engine has its own control system, which allows to control and monitor the engine in real time, without computer connection. After pressing the start button, the control system checks working medium pressure, fuel gas pressure and starts to heat up the burning chamber. After reaching approx. 400 ◦C, the engine starts and continues the heating process until it reaches 520 ◦C in the combustion chamber, when the engine starter disconnects and the engine starts to produce electricity. After few test experiments we have found out, that there is a huge difference between hot and cold start. For cold start, where the engine and cooling circuit have ambient temperature, the heating process lasts at least 15 min, electricity is produced after 7 min and after another 8 min the electricity production is stable. For a hot start, where cooling circuit is pre-heated to 80 ◦C and engine is not cooled down, with hot cylinder temperature of 200 ◦C, the starting process took only 7 min, where electricity production occurred after 1.5 min. Due to this we have always started the measurement with cold engine, with cooled down heating circuit. Every experiment consisted of 4 periods—starting process, engine pressure setup, measurement, cooling down. The main measurement was performed for 2 h, under stable conditions, where data were collected via ModBus: engine working pressure, engine heater temperature, engine cooler temperature, electric output at generator, external cooling system temperatures and mass flow. From these data, the average value was expressed and standard deviation estimation was determined, according to [31–33], as: n 1 X x = x (29) n i i=1 s Pn 2 2 i=1 x nx s = i − (30) x n(n 1) − where n is the number of measured values, in our case it is 360, xi is the measured value and sx is standard deviation. For indirect measured values, standard deviation was determined according to [31–33] as:
Xn !2 2 ∂y sy = si (31) ∂xi i=1
The results of measurement and standard deviation of the arithmetic mean value estimation are presented in the Table2. These results served as sample for comparison with proposed numerical model.
Table 2. Measured electric output with standard deviation.
Electric Output—Average Standard Deviation of the Arithmetic Working Pressure (bar) Efficiency (%) Value (W) Mean Value Estimation (W) 48.2 2480 16.7 6.5 ± 52.1 2730 17.1 4.8 ± 56.3 2990 17.6 5.8 ± 59.9 3240 18.8 4.1 ± 65 3510 19.5 4.9 ± 70 3820 20.0 5.0 ± 80.9 4060 20.9 5.7 ± Energies 2020, 13, x FOR PEER REVIEW 10 of 15
For indirect measured values, standard deviation was determined according to [31–33] as: