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Dynamic modelling of the expansion cylinder of an open Joule cycle Ericsson engine: A bond graph approach
Energy 102 (2016) 31e43
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Dynamic modelling of the expansion cylinder of an open Joule cycle Ericsson engine: A bond graph approach M. Creyx a, *, E. Delacourt a, b, C. Morin a, b, B. Desmet b a b
LAMIH (UMR CNRS 8201 Laboratory), UVHC, Le Mont Houy, F-59313 Valenciennes Cedex 9, France ENSIAME (Engineering School), UVHC, Le Mont Houy, F-59313 Valenciennes Cedex 9, France
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 November 2014 Received in revised form 7 November 2015 Accepted 31 January 2016 Available online xxx
A dynamic model using the bond graph formalism of the expansion cylinder of an open Joule cycle Ericsson engine intended for a biomass-fuelled micro-CHP system is presented. Dynamic phenomena, such as the thermodynamic evolution of air, the instantaneous air mass flow rates linked to pressure drops crossing the valves, the heat transferred through the expansion cylinder wall and the mechanical friction losses, are included in the model. The influence on the Ericsson engine performances of the main operating conditions (intake air pressure and temperature, timing of intake and exhaust valve closing, rotational speed, mechanical friction losses and heat transfer at expansion cylinder wall) is studied. The operating conditions maximizing the performances of the Ericsson engine used in the a biomass-fuelled micro-CHP unit are an intake air pressure between 6 and 8 bar, a maximized intake air temperature, an adjustment of the intake and exhaust valve closing corresponding to an expansion cycle close to the theoretical Joule cycle, a rotational speed close to 800 rpm. The heat transfer at the expansion cylinder wall reduces the engine performances. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Ericsson engine Dynamic model Bond graphs Cogeneration Hot air engines
1. Introduction Nowadays, the Ericsson hot air engine is mainly developed for micro-combined heat and electrical power (micro-CHP) units [1e3]. The Ericsson engine is adapted to several heat sources: combustion from any fuel (for example natural gas [1], solid biomass [2]), solar energy [3] and geothermal energy. In the solid biomass-fuelled micro-CHP systems, the cogeneration engines currently developed require an external heat supply, in particular the turbines with Organic Rankine Cycle (ORC) [4], the Stirling [5,6] and the Ericsson [2] hot air engines. Such engines allow, by means of a heat exchanger, to separate the flue gas associated to combustion and containing soot particles from the working fluid of the cogeneration device that converts thermal energy into mechanical energy. The Ericsson engines have several advantages compared with the ORC turbines and the Stirling engines [2]. They can work with an open cycle (no cold source required). Contrary to the
de * Corresponding author. Laboratoire LAMIH (UMR CNRS 8201), Universite sis, Le Mont Houy, 59313 Valenciennes Cedex Valenciennes et du Hainaut-Cambre 09, France. Tel.: þ33 3 27 51 19 83; fax: þ33 3 27 51 19 61. E-mail addresses: [email protected] (M. Creyx), [email protected] (E. Delacourt), [email protected] (C. Morin), [email protected] (B. Desmet). http://dx.doi.org/10.1016/j.energy.2016.01.106 0360-5442/© 2016 Elsevier Ltd. All rights reserved.
Stirling engines, no compromise between a large area and a small volume of the heat exchanger is imposed [7]. For open cycle configurations, the pressure level of the working fluid (air) is lower than for the Stirling engines, limiting the tightness constraints and consequently the costs. For the power scale of micro-CHP (under 50 kW), the performances of the Ericsson and Stirling engines present similar levels [8], higher than in the case of ORC turbines [9]. To describe the thermodynamic processes of air supplying the Ericsson engine, the Joule cycle (adiabatic compression and expansion, isobaric intake and exhaust phases) is the most realistic [2]. Several authors [2,8,10e14] numerically studied Joule cycle Ericsson engines with closed [8] or open [2,10e14] cycles and with internal [10,11] or external combustion [2,8,12e14]. The configurations of Ericsson engine considered include a compression cylinder, a heat exchanger, an expansion cylinder and sometimes a preheater [8,10e12] or a combustion chamber [11]. The working conditions studied are various [2]. Table 1 details the contents of these models (steady-state or dynamic operating conditions, phenomena modelled, performances observed). Creyx et al. [2], Moss et al. [14] developed thermoet al. [10], Bell et al. [11] and Toure dynamic models in a steady-state, whereas Wojewoda et al. [8], Lontsi et al. [13] and Fula et al. [12] implemented dynamic models allowing the simulation of transient phases. These models consider
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Nomenclature Acyl Aveff C Cd Cf cv Dp F h H_ hwall IMP IMPe IMEP L lv Ma mcycle mcyl m_ in m_ ex m_ N p Pm Q_ H
Qwall Q_ wall r rcrank rv Se Sf t T u Up v V Veevc Vlivc
exchange surface area of the expansion cylinder wall, m2 effective air cross-section of the valve, m2 torque, N.m discharge coefficient friction torque, N.m air specific heat capacity at constant volume, J/kg.K piston diameter, m force, N specific enthalpy of air, J/kg total enthalpy flux of air, W heat transfer coefficient of air, W/m2.K indicated mean pressure, Pa indicated mean pressure of expansion cycle, Pa indicated mean effective pressure, Pa connecting rod length, m valve lift, m Mach number air intake mass per cycle, kg/cycle mass of air in the expansion chamber, kg intake air mass flow rate, kg/s exhaust air mass flow rate, kg/s air mass flow rate through a valve, kg/s rotational speed, rpm pressure, Pa mechanical power, W thermal power transmitted in the heat exchanger, W heat crossing the expansion cylinder wall during a cycle, J thermal power exchanged at expansion cylinder wall, W specific gas constant, J/kg.K crank length, m valve head radius, m effort source flow source time, s temperature, K specific internal energy of air, J/kg piston velocity, m/s air speed through the valve, m/s volume, m3 expansion chamber volume at time of exhaust valve closing, m3 expansion chamber volume at time of intake valve closing, m3
several approaches of the Ericsson engine: thermodynamic processes, or pressure drops across the valves, or mechanical friction losses, or heat transfers at each cylinder wall, or coupled processes (combination of the previous phenomena). The authors observe various performances: specific indicated work wi, indicated mean pressure IMP, pressureevolume diagram, mechanical power Pm, thermodynamic efficiency hth. All these elements are not considered simultaneously in a model and the disparity of the models limits the comparison of the performances between the different configurations of Ericsson engine studied. A dynamic model of the expansion cylinder of an open Joule cycle Ericsson engine is presented in this paper. The expansion
Weff,c Weff,e Wi wi Wic Wie _ W x_
ie
effective work of the compression cycle, J effective work of the expansion cycle, J indicated work of the Ericsson engine, J specific indicated work of the Ericsson engine, J/kg indicated work of the compression cycle, J indicated work of the expansion cycle, J indicated power of the expansion cycle, W piston velocity, m/s
Greeks symbols early exhaust valve closing coefficient late intake valve closing coefficient Laplace coefficient of air valve opening angle, rad global efficiency, % thermodynamic efficiency, % crankshaft angle, rad q_ rotational speed, rad/s 4 angle formed by the connecting rod and the piston symmetry axis, rad 4_ derivative of angle 4, rad/s u rotational speed, rad/s
aeevc alivc g dv hglobal hth q
Subscripts atm atmospheric c compression cycle conrod crank and connecting rod system of the expansion cylinder crankcase crankcase of the expansion cylinder cyl expansion chamber e expansion cycle ex exhaust h intake of the expansion cylinder in intake M maximum m minimum p, piston piston of the expansion cylinder throat minimum cross-section of a valve upstream upstream of a valve v valve wall internal wall of the expansion cylinder 0 initial step Abbreviations BDC bottom dead center TDC top dead center
cylinder, studied separately, corresponds to a part of a modular test bench of a biomass-fuelled micro-CHP system set up in the laboratory. The model includes the following dynamic phenomena: the thermodynamic processes of air, the mechanical losses (evaluation of instantaneous air mass flow rates linked to pressure drops across the valves, friction) and the heat transfer at the expansion cylinder wall. This model is implemented using the bond graph formalism with the software 20sim. This tool allows to simplify the representation of various coupled physical phenomena, including the variable causality, by using unified variables for all the physical domains (effort and flux) whose product equals the transmitted power [15]. It allows an evolving modelling that facilitates the
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Table 1 Contents of the models of Joule cycle Ericsson engine from the literature. Author
Creyx et al. [2] [14] Toure Wojewoda et al. [8] Bell et al. [11] Moss et al. [10] Lontsi et al. [13] Fula et al. [12]
Model
Steady-state Steady-state Dynamic Steady-state Steady-state Dynamic Dynamic
Performances observed
wi, hth, IMP Pm, hth Pm, hth, PV diagram wi, hth
hth PV diagram PV diagram
Phenomenon modelled Thermodynamic processes
Pressure drops
X X X X X X X
X X X X X
integration of new equations or blocks in an existing model, using a system approach. The bond graph representation also allows to model dynamic phenomena and transient operating phases. In the following, the configuration of the Ericsson engine studied is described. The dynamic model is detailed. Then, a verification of the working conditions (evolution of pressure and temperature of the expansion chamber during a cycle, intake mass per cycle) and the performances (indicated work, indicated power, indicated mean pressure of the expansion cycle, heat quantity required during a cycle) obtained with the simulation is presented. Finally, the results of the expansion cylinder dynamic model are described with the influence on the Ericsson engine performances of several parameters: air pressure and temperature at the inlet of the cylinder, times of intake and exhaust valve closing, rotational speed, mechanical friction losses and heat transfer at the expansion cylinder wall.
2. Configuration of the Ericsson engine The Ericsson engine studied, similar to the engine described in Ref. [2], is composed of a compression cylinder, a heat exchanger and an expansion cylinder (cf. Fig. 1) and works with an open Joule cycle. The compression and expansion cylinders are supposed
Frictional losses
Heat transfer at cylinder wall
X X X X
connected to the same crankshaft. The focus is on the expansion cylinder. Its geometrical dimensions and main operating parameters, corresponding to the test bench of the biomass-fuelled microCHP unit set up in the laboratory, are given in Table 2. The distribution is performed by a valve train with rocker arm. To model the physical phenomena occurring in the expansion cylinder (thermodynamic processes of air, instantaneous air mass flow rates linked to pressure drops across the valves, mechanical friction losses, heat transfers at the cylinder wall), a decomposition with a bond graph representation of every cylinder part is performed (cf. Fig. 2): intake and exhaust valves, expansion chamber, expansion cylinder wall, piston, crank and connecting rod system, internal volume of the crankcase, mechanical friction losses. To evaluate the performances of the complete Ericsson engine, the compression cylinder is theoretically modelled, with dynamic phenomena neglected. The air thermodynamic processes in the compression cylinder are considered only with a theoretical Joule cycle ensuring the same air intake mass per cycle than the one flowing through the expansion cylinder. Under these hypotheses, the performances of the Ericsson engine are slightly overestimated, since the dynamic losses of the compression cylinder are neglected. The heat exchanger is modelled by fixed input parameters (air temperature and pressure at the inlet of the expansion cylinder), the pressure drops being neglected. The dynamic model of the expansion cycle is described in this paper, allowing the determination of temporal evolution of the variables in the expansion cylinder and the evaluation of the Ericsson engine performances including the losses linked to dynamic phenomena in the expansion cylinder.
3. Theoretical model of the expansion cylinder The expansion cylinder is represented by the bond graph model of Fig. 2, where every block corresponds to a sub-component. The blocks are linked by arrows symbolizing the power exchanges and the associated effort and flow variables. The equations corresponding to every block and those required to evaluate the global
Table 2 Geometrical dimensions and main operating parameters of the expansion cylinder.
Fig. 1. Ericsson engine configuration [2].
Cycle Swept volume Unswept volume Stroke Bore Connecting rod length Crank length Intake air pressure of expansion cylinder Intake air temperature of expansion cylinder Air volume flow rate Rotational speed
2-stroke 160 cm3 20 cm3 45 mm 68 mm 76 mm 22.5 mm 4e8 bar 450e650 C 2e10 m3/h (N) 100e1400 rpm
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Fig. 2. Global bond graph scheme of the expansion cylinder.
performances of the Ericsson engine are detailed in the following. The resolution method of the set of equations is then described.
the valves, to the piston movement and to the heat exchanges at the cylinder wall. The mass balance is written as follows:
3.1. Equations implemented in the bond graph blocks
d mcyl ¼ m_ in m_ ex dt
3.1.1. Expansion chamber The expansion chamber block contains a mass balance and an energy balance of the air present in the chamber between two successive moments (cf. Fig. 3). The energy exchanges with the air inside the chamber are linked to the air intake and exhaust across
(1)
with t the time, mcyl the mass of air in the expansion chamber, m_ in and m_ ex the intake and exhaust air mass flow rates of the expansion chamber. The energy balance of the air inside the expansion chamber is the following:
Fig. 3. Details of the bond graph representation of the expansion chamber.
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dT dmcyl cyl ucyl þ mcyl cv Tcyl dt dt ! dVcyl v2in v2 _ þ m_ in hin þ ¼ Q wall pcyl m_ ex hex þ ex dt 2 2
(2)
with Tcyl, pcyl and Vcyl respectively the air temperature, pressure and volume in the chamber, ucyl the specific internal energy of air, cv the specific heat capacity of air at constant volume, Q_ wall the thermal power exchanged at the cylinder wall, hin and hex the specific enthalpies of intake and exhaust air, vin and vex the velocities of intake and exhaust air. The reference conditions are a temperature of 25 C and the atmospheric pressure 101,315 Pa. By convention, a power “entering” the expansion chamber is supposed positive. The potential energy due to gravity of the intake and exhaust air is neglected in the energy balance.
vthroat
35
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 u !g1 1 u g u g r Tupstream @ pthroat A 1 ¼ t2 g1 pupstream
(6)
with r the specific constant of air. In the case of subsonic conditions, the air temperature at the throat and the air mass flow rate are evaluated as follows:
Tthroat ¼ Tupstream
pthroat pupstream
!g1 g
(7)
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u !2 0 !g1 1 u g g pupstream u 2g pthroat p throat @1 A m_ ¼ Aveff pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t pupstream r Tupstream g 1 pupstream (8)
3.1.2. Intake and exhaust valves The blocks of the intake and exhaust valves are similar. They model the instantaneous air mass flow rate linked to pressure drops across the valves and link the temperature and pressure conditions at the upstream and at the minimum cross-section of each valve [16]. The instantaneous enthalpy fluxes crossing the valves are also evaluated. The effective air cross-section is evaluated as follows:
Aveff ¼ Cd p lv sin dv ð2 rv þ lv sin dv cos dv Þ
(4)
with Tupstream the upstream air temperature, Tthroat the air temperature at the minimum cross-section of the valve, g the Laplace coefficient of air, Ma the Mach number. The Laplace coefficient g is supposed constant in the model. The thermodynamic process of air between the upstream and the valve throat being adiabatic reversible, the condition corresponding to a sonic flow is written:
pthroat pupstream
2 gþ1
Tthroat ¼ Tupstream
2 gþ1
pupstream m_ ¼ Aveff pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r Tupstream
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gþ1 g1 2 g gþ1
(9)
(10)
(3)
with Cd the discharge coefficient [17], lv the valve lift, dv the valve seat angle, rv the valve head radius. In this equation, the function describing the valve lift lv depending on the crankshaft angle is sinusoidal. The velocity and the temperature of air at the minimum crosssection of the valve together with the air mass flow rate crossing the valve are evaluated for subsonic and sonic flow conditions. The model of a convergent nozzle is used to represent the flow across the valve. The velocities of air upstream and downstream of the valve are neglected. The flow is supposed quasi-steady and adiabatic reversible between the upstream and the minimum crosssection of the valve (equivalent to a nozzle throat). The air is considered as an ideal gas. With these conditions, the SainteVenant equation leads to:
Tupstream g1 Ma ¼1þ 2 Tthroat
In the case of sonic conditions, the flow is blocked at the minimum air cross-section of the valve. At this position, the air temperature and the air mass flow rate are written:
g g1
(5)
with pthroat and pupstream the air pressures respectively of the valve throat and of the upstream of the valve. In applying the SainteVenant theorem (conservation of total enthalpy), the speed of air at the throat is expressed, for subsonic and sonic conditions, as follows:
3.1.3. Heat transfer at the cylinder wall The heat flux at the cylinder wall is evaluated from the heat transfer coefficient of air at the wall of the expansion chamber hwall, by considering a constant wall temperature Twall:
Q_ wall ¼ hwall Acyl Tcyl Twall
(11)
with Q_ wall the heat flux crossing the expansion chamber wall, Acyl the surface of the expansion chamber in contact with air, Tcyl the air temperature in the expansion chamber. The heat transfer coefficient of air at the wall of the expansion chamber is estimated with the Hohenberg correlation [18], which gives intermediate values of this heat transfer coefficient compared with other correlations [12]:
0;06 0;8 0;4 hwall ¼ 0; 0001$130$Vcyl $pcyl $Tcyl $ U p þ 1; 4
(12)
with Vcyl, pcyl, Tcyl respectively the volume, pressure and temperature of air in the expansion chamber, U p the mean piston velocity. 3.1.4. Crank and connecting rod system The Crank and connecting rod system is represented in Fig. 4. The derivative of geometric relations in the triangle OBP leads to the following equations:
x_ ¼ rcrank q_ sin q L 4_ sin 4
(13)
L 4_ cos 4 rcrank q_ cos q ¼ 0
(14)
with x_ the piston velocity (also noted Up), rcrank and L the lengths of the crank and the connecting rod respectively, q the crankshaft R angle (qðtÞ ¼ u$dt), q_ the crankshaft angular rotational speed (also noted u), 4 and 4_ respectively the angle and the time derivative of the angle formed by the connecting rod and the piston
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Fig. 4. Piston Rod Crank scheme of the expansion cylinder.
symmetry axis. The inertial loads of the crank and connecting rod system and of the crankshaft are neglected. Eq. (13) and Eq. (14) are transformed in Bond Graph formulation by the block presented in Fig. 5, referenced as “Crank and connecting Rod system” in Fig. 2. Fconrod and Cconrod are respectively the force applied on the top face of the piston and the indicated torque on the crank shaft. If u is fixed, Up(t) will be defined. The model associated at the output of the Crank-Rod system has to solve the effort “Fconrod(t)” from the flux “Up(t)”. Then the value of Cconrod(t) will be solved with the conservation of the power between the input ant the output of the block.
3.1.5. Piston The piston block evaluates the force Fpiston applied on the piston from the pressure ppiston applied by the air inside the expansion _ chamber on the piston and the derivative of the volume VðtÞ:
p Fpiston ¼ D2p ppiston 4
(15)
with Dp the piston diameter. The variation of air volume with time is defined by:
p VðtÞ ¼ Vc þ $D2p $ðxTDC xðtÞÞ 4
(16)
With Vc the clearance volume and xTDC the position of the piston at the top dead center. _ The flux VðtÞ is then defined by the following expression:
p _ _ with x_ ¼ Up VðtÞ ¼ qv ¼ $D2p $xðtÞ 4
(17)
3.1.6. Internal volume of the crankcase The air at atmospheric pressure patm situated in the internal volume of the crankcase applies a force Fcrankcase under the piston head:
p Fcrankcase ¼ D2p patm 4
(18)
This force and the force applied by the connecting rod on the piston Fconrod are linked to the force applied by the air inside the expansion chamber on the piston Fpiston (neglecting the inertial load of the piston):
Fpiston ¼ Fcrankcase þ Fconrod
(19)
3.1.7. Mechanical friction losses The mean global friction torque of the expansion cylinder applied on the crankshaft Cf is estimated with an experimental correlation obtained from measurements on the engine test bench: the crankshaft of the expansion cylinder at ambient temperature is driven by an electric engine with several rotational speeds (100e1000 rpm) and the mean friction torque applied on the crankshaft is measured with a calibrated force sensor set on a lever arm adapted for torque between 0 and 100 N. From a linear interpolation on experimental data [17], the mean friction torque is described by the following correlation:
Cf ¼ 2; 253$104 $N þ 0; 5207 with determination coefficient R2 ¼ 0:88 (20) with N the rotational speed (in rpm). The algebraic mean global friction torque is added to the indicated torque to obtain the effective torque applied on the crankshaft (cf Fig. 2):
Ccrankshaft ¼ Cconrod þ Cf
(21)
3.2. Equations of global performances The indicated power received by the air in the expansion chamber is calculated from Eq. (2):
_ ¼ p dVcyl W ie cyl dt Fig. 5. Bond Graph block of the crank and connecting rod system.
(22)
The indicated work of the expansion cycle is evaluated from the time integral of the indicated power during a cycle:
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37
Fig. 7. Evolution of the IMP depending on the intake air temperature Th obtained with the thermodynamic model [2] and with the dynamic model for several intake air pressures of 4, 6 and 8 bar.
The indicated work and the specific indicated work of the Ericsson engine are written respectively:
Fig. 6. Evolution of the air state variables of the expansion chamber depending on the crankshaft angle for the bond graph and scilab dynamic models with N ¼ 600 rpm, ph ¼ 6 bar, Th ¼ 550 C, intake and exhaust valves opened for the range of crankshaft angle respectively from 10 to 80 and from 160 to 360 .
Z Wie ¼
_ dt W ie
(23)
1 cycle
The effective work is deduced from the indicated work and from the mean global friction torque:
Weff ;e ¼ Wie 2p$Cf
(24)
Wi ¼ jWie j Wic
(26)
. wi ¼ ðjWie j Wic Þ mcycle
(27)
with mcycle the mass of air entering the cylinder per cycle. The indicated mean pressure (IMP) of the Ericsson engine is evaluated with the following formula:
IMP ¼ ðjWie j Wic Þ=ðVMe Vme Þ
(28)
with VMe and Vme respectively the maximum and minimum volumes of the expansion chamber during a cycle. The indicated mean effective pressure (IMEP) is written:
. ðVMe Vme Þ IMEP ¼ Weff ;e Weff ;c
(29)
with u the crankshaft rotational speed (in rad/s). The indicated work Wic of the compression cycle of the Ericsson engine is evaluated considering a theoretical Joule cycle [2]. The mean global friction torque of the compression cylinder is considered similar to the one of the expansion cylinder (mechanical systems of compression and expansion cylinder with similar dimensions and working conditions), leading to the following effective work of the compression cycle:
Weff ;c ¼ Wic 2p$Cf
(25)
Table 3 Comparison of the performances evaluated with the bond graph and scilab models for N ¼ 600 rpm, ph ¼ 6 bar, Th ¼ 550 C, intake and exhaust valves opened for the range of crankshaft angle respectively from 10 to 80 and from 160 to 360 . Bond graph model
Scilab model
Difference (%)
mcycle (g/cycle) Wie (J) _ (W) W
0.2157 56.88 568.83
0.2160 57.02 570.20
0.15% 0.24% 0.24%
IMPe (bar) Qwall (J/cycle)
3.4807 17.61
3.4888 18.85
0.23% 6.56%
ie
Fig. 8. Evolution of the thermodynamic efficiency depending on the intake air temperature Th obtained with the thermodynamic model [2] and with the dynamic model for several intake air pressures of 4, 6 and 8 bar.
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Fig. 9. Evolution of the air intake mass per cycle depending on the intake air temperature Th obtained with the thermodynamic model [2] and with the dynamic model for several intake air pressures of 4, 6 and 8 bar.
The thermodynamic efficiency of the Ericsson engine is evaluated from the difference of indicated works of the compression and expansion cycles of the Ericsson engine and from the heat transmitted to the air by the heat exchanger between the exhaust of the compression cylinder and the intake of the expansion cylinder:
. hth ¼ ðjWie j Wic Þ Q_ H
(30)
with Q_ H the heat quantity transferred to air in the heat exchanger supplying the Ericsson engine. The global efficiency is obtained similarly, considering the mechanical friction losses:
. hglobal ¼ Weff ;e Weff ;c Q_ H
(31)
Fig. 10. Comparison of the pressureevolume diagrams obtained with the thermodynamic model [2] and with the dynamic model for N ¼ 600 rpm, ph ¼ 6 bar, Th ¼ 550 C, intake and exhaust valves opened for the range of crankshaft angle respectively from 10 to 80 and from 160 to 325 .
4. Verification of the dynamic model implemented The validation of the dynamic model of the expansion cylinder implemented is limited because of the lack of numerical or experimental data in the literature. The modelling of the expansion cylinder presented here, with a bond graph decomposition of the equations including the thermodynamic processes, the instantaneous air mass flow rate linked to pressure drops across the valves, the mechanical friction losses and the heat transfer at the expansion cylinder wall, is innovative. The models of the literature do not include all of these aspects simultaneously. To verify the results obtained with the bond graph model, a second dynamic model has been implemented as a script in the software scilab with a more common approach to solve the set of equations, considering the steady-state operating cycle of the Ericsson engine expansion cycle. The equations previously described are considered depending on the crankshaft angle q. The mass and energy balances of Eqs. (1) and (2) are expressed as follows in the second dynamic model:
3.3. Resolution method of the equations In the bond graph model of the expansion cylinder, the values of the following variables are imposed: rotational speed, air pressure and temperature at the inlet of the expansion cylinder, atmospheric pressure at the outlet. Initially, the piston is sup-
dT ¼ dq
Q_ wall u
"
dV pcyl dqcyl
þ
dm rTcyl dqcyl
þ
1 u
! v m_ in hin þ 2in hcyl 2
dmcyl 1 ¼ ðm_ in m_ ex Þ u dq
(32)
# v2ex _ mex hex þ 2 hcyl
mcyl cv Tcyl
posed at the top dead center (TDC), with an angular position of the crankshaft of 0 . The reference temperature is at 25 C for the calculations of specific internal energy and enthalpy. The set of equations implemented with the bond graph software is solved using an implicit method: the modified backward differentiation formula method (modified BDF method), using a calculation step of 1.108 s. The simulation is performed during two stabilized operating cycles in a steady-state (the transient operating phases are not studied here).
(33)
The resolution of the differential equations is performed using a predictor-corrector scheme with a second-order precision. The results of the bond graph and the dynamic models with a steady-state cycle are presented in Fig. 6 and Table 3 for particular operating conditions of the Ericsson engine: rotational speed of 600 rpm, pressure and temperature at the intake of the expansion cylinder of respectively 6 bar and 550 C, intake valve opened between q ¼ 10 and q ¼ 80 , exhaust valve opened between
M. Creyx et al. / Energy 102 (2016) 31e43
q ¼ 160 and q ¼ 360 . The friction torque and the heat transfer at the expansion cylinder wall are considered in these simulations. The evolutions of the state variables (volume, pressure, temperature of air in the expansion chamber) are close for both simulations (cf. Fig. 6). For the range of crankshaft angles corresponding to the exhaust phase, the maximum difference of air temperatures obtained between the bond graph model and the second model is of 6.2% (lower value for the bond graph model). This result is explained by the way the specific internal energy and enthalpy are implemented in both models (the integrals of heat capacity depending on temperature with a 3rd order polynomial fit are obtained with an exact calculation in the scilab model contrary to the bond graph model where these integrals are linearized). The performances of the expansion cylinder (indicated work, indicated power, IMPe of the expansion cycle, heat transmitted through the expansion cylinder wall) obtained with both models are presented in Table 3. The differences of the performances obtained, resulting of the differences of the air temperature in the expansion chamber, are very low. The most important difference of 6.6% is noticed for the heat crossing the expansion cylinder wall, which is linked to the temperature difference of air in the expansion chamber obtained with both models during the exhaust phase. The match of the results of both models implemented and solved in different ways allows to check the appropriateness of these models and is a first step to validate the bond graph model.
39
5. Results and discussion The results obtained with the bond graph dynamic model are presented below. The influence on the state variables (pressure, temperature, volume of air in the expansion chamber) and on the Ericsson engine performances of several parameters and phenomena (pressure and temperature at the inlet of the expansion cylinder, times of intake and exhaust valve closing, rotational speed, mechanical friction losses, heat transfer at the expansion cylinder wall) are considered. The mechanical friction losses and the heat transfer at the expansion cylinder wall are considered in the dynamic model for the results presented. The temperature of the expansion cylinder wall is supposed of 200 C (maximum temperature of the lubricating oil). 5.1. Influence of the air pressure and temperature at the inlet of the expansion cylinder The influence on the indicated mean pressure IMP, the thermodynamic efficiency hth and the air intake mass per cycle mcycle of the air pressure ph and the air temperature Th at the inlet of the expansion cylinder are shown in Fig. 7, Fig. 8 and Fig. 9 respectively. The results obtained with the thermodynamic model [2] and with the dynamic model, considering the data of Table 2, a rotational speed of 600 rpm, an expansion cycle with early exhaust valve
Fig. 11. Influence of the timings of valve closing on the air cross-sections of the intake valve (b) and of the exhaust valve (c) on the pressureevolume diagram of the expansion cycle (a), on the air pressure (d) and air temperature (e) in the expansion chamber for ph ¼ 6 bar, Th ¼ 550 C and N ¼ 600 rpm.
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closing allowing to reach the intake air pressure at TDC and a timing of intake valve closing maximizing the IMP (timing adjusted with the dynamic model and considered similar for the thermodynamic model) are presented. The indicated work of the expansion cycle varies little with the temperature of intake air, with a maximum variation of 1.7% on the intake temperature range from 450 to 650 C. The IMP and the thermodynamic efficiency rise with the intake temperature (cf. Figs. 7 and 8 respectively) because of the decreasing of the intake mass per cycle (cf. Fig. 9) which induces a reduction of the work required for the air compression cycle. The results of the dynamic and thermodynamic models [2] show that the maximum IMP is obtained for an intake pressure situated between 6 and 8 bar and that the intake mass per cycle decreases with the rise of the intake pressure.
For the same intake air pressure and temperature, the values of IMP, thermodynamic efficiency and intake mass per cycle obtained with the thermodynamic [2] and dynamic models vary significantly (maximum variation of 40% for the IMP and of 51% for the thermodynamic efficiency between the two models). These differences are linked to the evaluation of the instantaneous air mass flow rate linked to pressure drops across the valve considered in the dynamic model (sinusoidal valve lift), contrary to the thermodynamic model (no pressure drop which is equivalent to consider a rectangular valve lift with a maximum valve cross-section area). For the same operating conditions (in particular the timing of intake and exhaust valve closing), the intake phase is extended with the thermodynamic model [2] compared with the dynamic model (cf. Fig. 10), enhancing the air intake mass per cycle (cf. Fig. 9). The exhaust phase is also extended (cf. Fig. 10), limiting the air pressure in the
Fig. 12. Influence of the late intake valve closing coefficient alivc and the early exhaust valve closing coefficient aeevc on the IMP, the thermodynamic efficiency and the air intake mass per cycle for several intake air pressures of 4, 6 and 8 bar, with an intake air temperature of 550 C and a rotational speed of 600 rpm.
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41
transfer at expansion cylinder wall and mechanical friction) is situated between 0.6 and 1.2 bar and the thermodynamic efficiency between 11 and 33%. So, the thermodynamic performances of the Ericsson engine are maintained at high values when considering the dynamic losses, compared to the results of the thermodynamic study [2]. 5.2. Influence of the closing times of intake and exhaust valves
Fig. 13. Evolution of the IMP, the thermodynamic efficiency and the intake mass per cycle depending on the rotational speed with and without heat transfer across the expansion cylinder wall for air intake conditions of 6 bar and 550 C.
expansion chamber at TDC and allowing an air intake during the pressure balance after the opening of the intake valve [2]. The air compression and expansion phases in the expansion chamber which occur with different pressureevolume conditions for both models (cf. Fig. 10) explain the variation of the results obtained. For the intake air temperature and pressure ranges respectively from 450 to 650 C and from 4 to 8 bar, the IMP estimated with the dynamic model (including the evaluation of the instantaneous air mass flow rate linked to pressure drops across the valves, heat
Fig. 14. Evolution of the indicated mean pressure IMP and the indicated mean effective pressure IMEP depending on the rotational speed for air intake conditions of 6 bar and 550 C.
The influence of the times of intake and exhaust valve closing on the state variables of the air in the expansion chamber is presented in Fig. 11. At the opening time of the exhaust valve (cf. Fig. 11c), a quantity of air present in the expansion chamber is irreversibly expanded until reaching equal pressures in the chamber and in the discharge pipe (cf. Fig. 11d). For the case A, the exhaust phase is extended until reaching the TDC corresponding to the crankshaft angle q ¼ 0 (or 360 ) (cf. Fig. 11c). The early exhaust valve closing (cf. Fig. 11c) allows a compression of the air present in the clearance volume until reaching the intake air pressure for the case B and an intermediate pressure in the case C (cf. Fig. 11d). The early exhaust valve closing of the case B (cf. Fig. 11c) induces a reduction of the intake mass per cycle (cf. Fig. 11a) and an enhancement of the air temperature in the chamber at the end of the exhaust phase (cf. Fig. 11e). The opening of the intake valve (cf. Fig. 11b) induces a pressure balance corresponding to a compression of the air contained in the clearance volume until reaching the intake air pressure (cf. Fig. 11d) with an additional entering air mass. This compression phase leads to an increase in the air temperature of the chamber linked to the irreversibilities (cf. Fig. 11e). An extension of the intake phase is shown with case C, compared with cases A and B (cf. Fig. 11a and b), linked with a late intake valve closing which induces an enhanced intake air mass per cycle (cf. Fig. 11a). To represent the timings of intake and exhaust valve closing, two adimensional coefficients corresponding to volume ratio linked to the swept volume of the expansion cylinder are introduced [2]:
alivc ¼ ðVlivc Vme Þ=ðVMe Vme Þ
(34)
aeevc ¼ ðVeevc Vme Þ=ðVMe Vme Þ
(35)
with Vlivc and Veevc the volumes of the expansion chamber at the time respectively of intake and of exhaust valve closing. The minimum value of the late intake valve closing coefficient alivc corresponds to the case of an expansion allowing to reach the atmospheric pressure at bottom dead center (BDC). The maximum value of alivc is equal to 1. The early exhaust valve closing coefficient aeevc is situated between 0 and a value below 1 allowing to reach the intake air pressure at TDC after compression of the air in the clearance volume. The IMP, the thermodynamic efficiency and the air intake mass per cycle depending on alivc and aeevc coefficients are represented in Fig. 12 for several intake air pressures (4, 6 and 8 bar), with an intake air temperature of 550 C and a rotational speed of 600 rpm. The range of valid values of alivc and aeevc coefficients (positive indicated work of the Ericsson engine, cf. Eq. (26)) depends strongly on the intake air pressure. With a higher intake air pressure, the valid alivc coefficients are shifted to lower values and the range of valid aeevc coefficients is enhanced. The maximum IMP and thermodynamic efficiencies are reached for pressures situated between 6 and 8 bar, as shown previously, for small late intake valve closing and for small early exhaust valve closing, that confirms the results obtained in Ref. [2] with a thermodynamic model. The values of alivc and aeevc coefficients maximizing the IMP and the thermodynamic efficiency correspond to expansion cycles close to the theoretical
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M. Creyx et al. / Energy 102 (2016) 31e43
Joule cycle. The intake air mass per cycle depends little on the early exhaust valve closing coefficient aeevc and increases with the late intake valve closing coefficient alivc. So, the highest performances of the Ericsson engine are obtained for the cycles with the smallest air intake mass. In the following, the aeevc coefficient corresponds to a compression of the air in the clearance volume reaching the intake air pressure at TDC and the alivc coefficient is adjusted to maximize the IMP.
for an adiabatic expansion cylinder wall, which can be explained by the thermal power lost through the wall when a heat transfer occurs. The intake mass per cycle is higher when a heat transfer crosses the expansion cylinder wall, which is explained by the cooling of air in the expansion chamber through the wall (for similar pressureevolume conditions in the expansion chamber, a lower temperature induces a higher mass of air). 5.4. Influence of the mechanical friction losses
5.3. Influence of the rotational speed Fig. 13 presents the influence of the rotational speed on the IMP, on the thermodynamic efficiency hth and on the intake mass per cycle for the cases with and without heat transfer at expansion cylinder wall and considering the air intake conditions of 6 bar and 550 C. The intake air mass per cycle decreases with the increase of the rotational speed, which is caused by the increase of pressure drops through the valves with the rotational speed. The IMP decreases with the rotational speed, with a very slight slope when a heat transfer occurs through the expansion cylinder wall. The thermodynamic efficiency is reduced when the rotational speed rises for an adiabatic expansion cylinder wall. The reduction of IMP and thermodynamic efficiencies at high rotational speed is linked to higher dynamic losses (in particular mechanical friction losses). The thermodynamic efficiency remains stable when a heat transfer occurs through the cylinder wall, which might be explained by the important decrease of intake mass per cycle in this case, inducing a decrease of the heat supplied per cycle in the Ericsson engine, that might compensate the increase of mechanical friction losses with the rotational speed which reduces the effective work of the engine (cf. Eq. (30)). The IMP and the thermodynamic efficiency are higher
The indicated mean pressure IMP and the indicated mean effective pressure IMEP are presented in Fig. 14 for intake conditions of 6 bar and 550 C. The IMEP corresponds to the IMP reduced by the mechanical friction losses. These losses represent from 56 to 72% of the IMP for the range of rotational speed considered and rise with the rotational speed, which is linked to the correlation of the friction torque (cf. Eq. (20)). For the operating conditions chosen in Fig. 14, the rotational speed maximizing the IMP and the IMEP is close to 800 rpm, the IMP is close to its maximum values and the IMEP lies between 0.25 and 0.4 bar. A higher intake air temperature (cf. Fig. 7) or a reduction of the mechanical friction losses may improve this result. 5.5. Effect of heat transfer at the expansion cylinder wall The influence of a heat transfer at the expansion cylinder wall on the air state variables in the expansion chamber is presented in Fig. 15 for a wall temperature of 200 C, intake air conditions of 6 bar and 550 C and a rotational speed of 600 rpm. The modelling of the heat transfer through the expansion cylinder wall has a little impact on the air pressure in the expansion chamber, even if the
Fig. 15. Evolution of the air pressure (a), of the air temperature (b) in the expansion cylinder and of the shape of the expansion cycle (c) with and without heat transfer across the expansion cylinder wall for a wall temperature of 200 C, intake conditions of 6 bar and 550 C and a rotational speed of 600 rpm.
M. Creyx et al. / Energy 102 (2016) 31e43
expansion process becomes non isentropic. The effect on temperature is more important: the hypothesis of an adiabatic wall induces an overestimation of the air temperature in the chamber of about 155e225 C for the operating conditions of Fig. 15. The thermal heat losses across the wall are reduced when the rotational speed is enhanced, so that the differences of IMP and of intake air mass per cycle with and without heat transfer are reduced at high rotational speed (cf. Fig. 13). This phenomenon is linked to the duration decrease of heat transfer between the air of the expansion chamber and the wall during an expansion cycle when the rotational speed increases. 6. Conclusion A dynamic model of the expansion cylinder of an open Joule cycle Ericsson engine was encoded and simulated with a bond graph formalism. The working conditions and the design of the expansion cylinder have been chosen in agreement with results obtained with a steady-state thermodynamic study described by Creyx et al. [2]. The Ericsson engine will be coupled subsequently to an air-gas heat exchanger built in a domestic biomass boiler. Several dynamic phenomena are modelled: thermodynamic processes in the expansion chamber, instantaneous air mass flow rate across the valves, mechanical friction losses, heat transfers at the expansion cylinder wall. Their influence on the state variables evolution of the air inside the expansion chamber (pressure and temperature) and on the Ericsson engine performances (IMP and thermodynamic efficiency) is then studied. The effects of the air intake pressure and temperature conditions and those of the timing of intake and exhaust valve closing are studied with the dynamic model and compared to the results obtained with a steady-state thermodynamic model [2]. Steady-state and present dynamic study show that the intake air pressure maximizing the Ericsson engine IMP is situated between 6 and 8 bar and that the highest temperatures enhance the Ericsson engine performances. The IMP and the thermodynamic efficiency are little dependent on the rotational speed of the engine. The mechanical friction losses represent an important part of the IMP. The IMEP is situated between 0.25 and 0.4 bar for the operating conditions considered. Considering a hypothesis of adiabatic expansion cylinder wall, compared with the modelling of the heat transfer through the expansion chamber wall, the values of the state variables of the air in the expansion chamber vary little but a strong impact is observed on the Ericsson engine performances (IMP and thermodynamic efficiency). Acknowledgement This work is performed in the framework of the regional project Sylwatt, in partnership with the laboratories LAMIH (Valenciennes, France), PC2A (Lille, France), CCM (Dunkerque, France) and received
43
gion Nord-Pas-de-Calais, from the a financial support from the Re French National Association of Research and Technology ANRT (doctoral scholarship) and from the company Enerbiom. The authors are grateful to Marc Lippert and Jesse Schiffler from the laboratory LAMIH who set up the test bench for the friction torque measurement of the Ericsson engine expansion cylinder.
References [1] Bonnet S, Alaphilippe M, Stouffs P. Energy, exergy and cost analysis of a microcogeneration system based on an Ericsson engine. Int J Therm Sci 2005;44: 1161e8. http://dx.doi.org/10.1016/j.ijthermalsci.2005.09.005. [2] Creyx M, Delacourt E, Morin C, Desmet B, Peultier P. Energetic optimization of the performances of a hot air engine for micro-CHP systems working with a Joule or an Ericsson cycle. Energy 2013;49:229e39. http://dx.doi.org/10.1016/ j.energy.2012.10.061. [3] Alaphilippe M, Bonnet S, Stouffs P. Low power thermodynamic solar energy conversion: coupling of a parabolic trough concentrator and an Ericsson engine. Int J Thermodyn 2007;10:37e45. http://dx.doi.org/10.5541/ ijot.1034000186. [4] Bouvenot JB, Latour B, Siroux M, Flament B, Stabat P, Marchio D. Appl Therm Eng 2014;73:1039e52. http://dx.doi.org/10.1016/j.applthermaleng. 2014.08.073. [5] Boucher J, Lanzetta F, Nika P. Optimization of a dual free piston Stirling engine. Appl Therm Eng 2007;27:802e11. http://dx.doi.org/10.1016/ j.applthermaleng.2006.10.021. [6] Bouvenot JB, Andlauer B, Stabat P, Marchio D, Flament B, Latour B, et al. Gas Stirling engine mCHP boiler experimental data driven model for building energy simulation. Energy Build 2014;84:117e31. http://dx.doi.org/10.1016/ j.enbuild.2014.08.023. [7] Costea M, Feidt M. The effect of the overall heat transfer coefficient variation on the optimal distribution of the heat transfer surface conductance or area in a Stirling engine. Energy Convers Manag 1998;39:1753e61. http://dx.doi.org/ 10.1016/S0196-8904(98)00063-6. [8] Wojewoda J, Kazimierski Z. Numerical model and investigations of the externally heated valve Joule engine. Energy 2010;35:2099e108. http:// dx.doi.org/10.1016/j.energy.2010.01.028. [9] Thiers S, Aoun B, Peuportier B. Experimental characterization, modelling and simulation of a wood pellet micro-combined heat and power unit used as a heat source for a residential building. Energy Build 2010;42:896e903. http:// dx.doi.org/10.1016/j.enbuild.2009.12.011. [10] Moss RW, Roskilly AP, Nanda SK. Reciprocating Joule-cycle engine for domestic CHP systems. Appl Energy 2005;80:169e85. http://dx.doi.org/10.1016/ j.apenergy.2004.03.007. [11] Bell MA, Partridge T. Thermodynamic design of a reciprocating Joule cycle engine. Proceedings of the Institution of mechanical Engineers, Part H. J Power Energy 2003;217:239e46. http://dx.doi.org/10.1243/095765003322066475. [12] Fula A, Stouffs P, Sierra F. In-cylinder heat transfer in an Ericsson engine prototype. In: Proc. of the Int. Conf. on renewable energies and power quality ICREPQ’13, Bilbao, Spain; March 2013. p. 1e6. [13] Lontsi F, Hamandjoda O, Fozao K, Stouffs P. Dynamic simulation of a small modified Joule cycle reciprocating Ericsson engine for micro-cogeneration systems. Energy 2013;63:309e16. http://dx.doi.org/10.1016/j.energy. 2013.10.061. A, Stouffs P. Modeling of the Ericsson engine. Energy 2014;76:445e52. [14] Toure http://dx.doi.org/10.1016/j.energy.2014.08.030. [15] Ould Bouamama B. Bond graph approach as analysis tool in thermofluid model library conception. J Frankl Inst 2003;340:1e23. http://dx.doi.org/ 10.1016/S0016-0032(02)00051-0. [16] Moran MJ, Shapiro HN. Fundamentals of engineering thermodynamics. 5th ed. John Wiley and Sons; 2007. [17] Heywood JB. Internal combustion engine fundamentals. McGraw-Hill; 1988. [18] Hohenberg GF. Advanced approaches for heat transfer calculations. SAE Paper no 790825. 1979.
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Energy journal homepage: www.elsevier.com/locate/energy
Dynamic modelling of the expansion cylinder of an open Joule cycle Ericsson engine: A bond graph approach M. Creyx a, *, E. Delacourt a, b, C. Morin a, b, B. Desmet b a b
LAMIH (UMR CNRS 8201 Laboratory), UVHC, Le Mont Houy, F-59313 Valenciennes Cedex 9, France ENSIAME (Engineering School), UVHC, Le Mont Houy, F-59313 Valenciennes Cedex 9, France
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 November 2014 Received in revised form 7 November 2015 Accepted 31 January 2016 Available online xxx
A dynamic model using the bond graph formalism of the expansion cylinder of an open Joule cycle Ericsson engine intended for a biomass-fuelled micro-CHP system is presented. Dynamic phenomena, such as the thermodynamic evolution of air, the instantaneous air mass flow rates linked to pressure drops crossing the valves, the heat transferred through the expansion cylinder wall and the mechanical friction losses, are included in the model. The influence on the Ericsson engine performances of the main operating conditions (intake air pressure and temperature, timing of intake and exhaust valve closing, rotational speed, mechanical friction losses and heat transfer at expansion cylinder wall) is studied. The operating conditions maximizing the performances of the Ericsson engine used in the a biomass-fuelled micro-CHP unit are an intake air pressure between 6 and 8 bar, a maximized intake air temperature, an adjustment of the intake and exhaust valve closing corresponding to an expansion cycle close to the theoretical Joule cycle, a rotational speed close to 800 rpm. The heat transfer at the expansion cylinder wall reduces the engine performances. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Ericsson engine Dynamic model Bond graphs Cogeneration Hot air engines
1. Introduction Nowadays, the Ericsson hot air engine is mainly developed for micro-combined heat and electrical power (micro-CHP) units [1e3]. The Ericsson engine is adapted to several heat sources: combustion from any fuel (for example natural gas [1], solid biomass [2]), solar energy [3] and geothermal energy. In the solid biomass-fuelled micro-CHP systems, the cogeneration engines currently developed require an external heat supply, in particular the turbines with Organic Rankine Cycle (ORC) [4], the Stirling [5,6] and the Ericsson [2] hot air engines. Such engines allow, by means of a heat exchanger, to separate the flue gas associated to combustion and containing soot particles from the working fluid of the cogeneration device that converts thermal energy into mechanical energy. The Ericsson engines have several advantages compared with the ORC turbines and the Stirling engines [2]. They can work with an open cycle (no cold source required). Contrary to the
de * Corresponding author. Laboratoire LAMIH (UMR CNRS 8201), Universite sis, Le Mont Houy, 59313 Valenciennes Cedex Valenciennes et du Hainaut-Cambre 09, France. Tel.: þ33 3 27 51 19 83; fax: þ33 3 27 51 19 61. E-mail addresses: [email protected] (M. Creyx), [email protected] (E. Delacourt), [email protected] (C. Morin), [email protected] (B. Desmet). http://dx.doi.org/10.1016/j.energy.2016.01.106 0360-5442/© 2016 Elsevier Ltd. All rights reserved.
Stirling engines, no compromise between a large area and a small volume of the heat exchanger is imposed [7]. For open cycle configurations, the pressure level of the working fluid (air) is lower than for the Stirling engines, limiting the tightness constraints and consequently the costs. For the power scale of micro-CHP (under 50 kW), the performances of the Ericsson and Stirling engines present similar levels [8], higher than in the case of ORC turbines [9]. To describe the thermodynamic processes of air supplying the Ericsson engine, the Joule cycle (adiabatic compression and expansion, isobaric intake and exhaust phases) is the most realistic [2]. Several authors [2,8,10e14] numerically studied Joule cycle Ericsson engines with closed [8] or open [2,10e14] cycles and with internal [10,11] or external combustion [2,8,12e14]. The configurations of Ericsson engine considered include a compression cylinder, a heat exchanger, an expansion cylinder and sometimes a preheater [8,10e12] or a combustion chamber [11]. The working conditions studied are various [2]. Table 1 details the contents of these models (steady-state or dynamic operating conditions, phenomena modelled, performances observed). Creyx et al. [2], Moss et al. [14] developed thermoet al. [10], Bell et al. [11] and Toure dynamic models in a steady-state, whereas Wojewoda et al. [8], Lontsi et al. [13] and Fula et al. [12] implemented dynamic models allowing the simulation of transient phases. These models consider
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Nomenclature Acyl Aveff C Cd Cf cv Dp F h H_ hwall IMP IMPe IMEP L lv Ma mcycle mcyl m_ in m_ ex m_ N p Pm Q_ H
Qwall Q_ wall r rcrank rv Se Sf t T u Up v V Veevc Vlivc
exchange surface area of the expansion cylinder wall, m2 effective air cross-section of the valve, m2 torque, N.m discharge coefficient friction torque, N.m air specific heat capacity at constant volume, J/kg.K piston diameter, m force, N specific enthalpy of air, J/kg total enthalpy flux of air, W heat transfer coefficient of air, W/m2.K indicated mean pressure, Pa indicated mean pressure of expansion cycle, Pa indicated mean effective pressure, Pa connecting rod length, m valve lift, m Mach number air intake mass per cycle, kg/cycle mass of air in the expansion chamber, kg intake air mass flow rate, kg/s exhaust air mass flow rate, kg/s air mass flow rate through a valve, kg/s rotational speed, rpm pressure, Pa mechanical power, W thermal power transmitted in the heat exchanger, W heat crossing the expansion cylinder wall during a cycle, J thermal power exchanged at expansion cylinder wall, W specific gas constant, J/kg.K crank length, m valve head radius, m effort source flow source time, s temperature, K specific internal energy of air, J/kg piston velocity, m/s air speed through the valve, m/s volume, m3 expansion chamber volume at time of exhaust valve closing, m3 expansion chamber volume at time of intake valve closing, m3
several approaches of the Ericsson engine: thermodynamic processes, or pressure drops across the valves, or mechanical friction losses, or heat transfers at each cylinder wall, or coupled processes (combination of the previous phenomena). The authors observe various performances: specific indicated work wi, indicated mean pressure IMP, pressureevolume diagram, mechanical power Pm, thermodynamic efficiency hth. All these elements are not considered simultaneously in a model and the disparity of the models limits the comparison of the performances between the different configurations of Ericsson engine studied. A dynamic model of the expansion cylinder of an open Joule cycle Ericsson engine is presented in this paper. The expansion
Weff,c Weff,e Wi wi Wic Wie _ W x_
ie
effective work of the compression cycle, J effective work of the expansion cycle, J indicated work of the Ericsson engine, J specific indicated work of the Ericsson engine, J/kg indicated work of the compression cycle, J indicated work of the expansion cycle, J indicated power of the expansion cycle, W piston velocity, m/s
Greeks symbols early exhaust valve closing coefficient late intake valve closing coefficient Laplace coefficient of air valve opening angle, rad global efficiency, % thermodynamic efficiency, % crankshaft angle, rad q_ rotational speed, rad/s 4 angle formed by the connecting rod and the piston symmetry axis, rad 4_ derivative of angle 4, rad/s u rotational speed, rad/s
aeevc alivc g dv hglobal hth q
Subscripts atm atmospheric c compression cycle conrod crank and connecting rod system of the expansion cylinder crankcase crankcase of the expansion cylinder cyl expansion chamber e expansion cycle ex exhaust h intake of the expansion cylinder in intake M maximum m minimum p, piston piston of the expansion cylinder throat minimum cross-section of a valve upstream upstream of a valve v valve wall internal wall of the expansion cylinder 0 initial step Abbreviations BDC bottom dead center TDC top dead center
cylinder, studied separately, corresponds to a part of a modular test bench of a biomass-fuelled micro-CHP system set up in the laboratory. The model includes the following dynamic phenomena: the thermodynamic processes of air, the mechanical losses (evaluation of instantaneous air mass flow rates linked to pressure drops across the valves, friction) and the heat transfer at the expansion cylinder wall. This model is implemented using the bond graph formalism with the software 20sim. This tool allows to simplify the representation of various coupled physical phenomena, including the variable causality, by using unified variables for all the physical domains (effort and flux) whose product equals the transmitted power [15]. It allows an evolving modelling that facilitates the
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33
Table 1 Contents of the models of Joule cycle Ericsson engine from the literature. Author
Creyx et al. [2] [14] Toure Wojewoda et al. [8] Bell et al. [11] Moss et al. [10] Lontsi et al. [13] Fula et al. [12]
Model
Steady-state Steady-state Dynamic Steady-state Steady-state Dynamic Dynamic
Performances observed
wi, hth, IMP Pm, hth Pm, hth, PV diagram wi, hth
hth PV diagram PV diagram
Phenomenon modelled Thermodynamic processes
Pressure drops
X X X X X X X
X X X X X
integration of new equations or blocks in an existing model, using a system approach. The bond graph representation also allows to model dynamic phenomena and transient operating phases. In the following, the configuration of the Ericsson engine studied is described. The dynamic model is detailed. Then, a verification of the working conditions (evolution of pressure and temperature of the expansion chamber during a cycle, intake mass per cycle) and the performances (indicated work, indicated power, indicated mean pressure of the expansion cycle, heat quantity required during a cycle) obtained with the simulation is presented. Finally, the results of the expansion cylinder dynamic model are described with the influence on the Ericsson engine performances of several parameters: air pressure and temperature at the inlet of the cylinder, times of intake and exhaust valve closing, rotational speed, mechanical friction losses and heat transfer at the expansion cylinder wall.
2. Configuration of the Ericsson engine The Ericsson engine studied, similar to the engine described in Ref. [2], is composed of a compression cylinder, a heat exchanger and an expansion cylinder (cf. Fig. 1) and works with an open Joule cycle. The compression and expansion cylinders are supposed
Frictional losses
Heat transfer at cylinder wall
X X X X
connected to the same crankshaft. The focus is on the expansion cylinder. Its geometrical dimensions and main operating parameters, corresponding to the test bench of the biomass-fuelled microCHP unit set up in the laboratory, are given in Table 2. The distribution is performed by a valve train with rocker arm. To model the physical phenomena occurring in the expansion cylinder (thermodynamic processes of air, instantaneous air mass flow rates linked to pressure drops across the valves, mechanical friction losses, heat transfers at the cylinder wall), a decomposition with a bond graph representation of every cylinder part is performed (cf. Fig. 2): intake and exhaust valves, expansion chamber, expansion cylinder wall, piston, crank and connecting rod system, internal volume of the crankcase, mechanical friction losses. To evaluate the performances of the complete Ericsson engine, the compression cylinder is theoretically modelled, with dynamic phenomena neglected. The air thermodynamic processes in the compression cylinder are considered only with a theoretical Joule cycle ensuring the same air intake mass per cycle than the one flowing through the expansion cylinder. Under these hypotheses, the performances of the Ericsson engine are slightly overestimated, since the dynamic losses of the compression cylinder are neglected. The heat exchanger is modelled by fixed input parameters (air temperature and pressure at the inlet of the expansion cylinder), the pressure drops being neglected. The dynamic model of the expansion cycle is described in this paper, allowing the determination of temporal evolution of the variables in the expansion cylinder and the evaluation of the Ericsson engine performances including the losses linked to dynamic phenomena in the expansion cylinder.
3. Theoretical model of the expansion cylinder The expansion cylinder is represented by the bond graph model of Fig. 2, where every block corresponds to a sub-component. The blocks are linked by arrows symbolizing the power exchanges and the associated effort and flow variables. The equations corresponding to every block and those required to evaluate the global
Table 2 Geometrical dimensions and main operating parameters of the expansion cylinder.
Fig. 1. Ericsson engine configuration [2].
Cycle Swept volume Unswept volume Stroke Bore Connecting rod length Crank length Intake air pressure of expansion cylinder Intake air temperature of expansion cylinder Air volume flow rate Rotational speed
2-stroke 160 cm3 20 cm3 45 mm 68 mm 76 mm 22.5 mm 4e8 bar 450e650 C 2e10 m3/h (N) 100e1400 rpm
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Fig. 2. Global bond graph scheme of the expansion cylinder.
performances of the Ericsson engine are detailed in the following. The resolution method of the set of equations is then described.
the valves, to the piston movement and to the heat exchanges at the cylinder wall. The mass balance is written as follows:
3.1. Equations implemented in the bond graph blocks
d mcyl ¼ m_ in m_ ex dt
3.1.1. Expansion chamber The expansion chamber block contains a mass balance and an energy balance of the air present in the chamber between two successive moments (cf. Fig. 3). The energy exchanges with the air inside the chamber are linked to the air intake and exhaust across
(1)
with t the time, mcyl the mass of air in the expansion chamber, m_ in and m_ ex the intake and exhaust air mass flow rates of the expansion chamber. The energy balance of the air inside the expansion chamber is the following:
Fig. 3. Details of the bond graph representation of the expansion chamber.
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dT dmcyl cyl ucyl þ mcyl cv Tcyl dt dt ! dVcyl v2in v2 _ þ m_ in hin þ ¼ Q wall pcyl m_ ex hex þ ex dt 2 2
(2)
with Tcyl, pcyl and Vcyl respectively the air temperature, pressure and volume in the chamber, ucyl the specific internal energy of air, cv the specific heat capacity of air at constant volume, Q_ wall the thermal power exchanged at the cylinder wall, hin and hex the specific enthalpies of intake and exhaust air, vin and vex the velocities of intake and exhaust air. The reference conditions are a temperature of 25 C and the atmospheric pressure 101,315 Pa. By convention, a power “entering” the expansion chamber is supposed positive. The potential energy due to gravity of the intake and exhaust air is neglected in the energy balance.
vthroat
35
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 u !g1 1 u g u g r Tupstream @ pthroat A 1 ¼ t2 g1 pupstream
(6)
with r the specific constant of air. In the case of subsonic conditions, the air temperature at the throat and the air mass flow rate are evaluated as follows:
Tthroat ¼ Tupstream
pthroat pupstream
!g1 g
(7)
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u !2 0 !g1 1 u g g pupstream u 2g pthroat p throat @1 A m_ ¼ Aveff pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t pupstream r Tupstream g 1 pupstream (8)
3.1.2. Intake and exhaust valves The blocks of the intake and exhaust valves are similar. They model the instantaneous air mass flow rate linked to pressure drops across the valves and link the temperature and pressure conditions at the upstream and at the minimum cross-section of each valve [16]. The instantaneous enthalpy fluxes crossing the valves are also evaluated. The effective air cross-section is evaluated as follows:
Aveff ¼ Cd p lv sin dv ð2 rv þ lv sin dv cos dv Þ
(4)
with Tupstream the upstream air temperature, Tthroat the air temperature at the minimum cross-section of the valve, g the Laplace coefficient of air, Ma the Mach number. The Laplace coefficient g is supposed constant in the model. The thermodynamic process of air between the upstream and the valve throat being adiabatic reversible, the condition corresponding to a sonic flow is written:
pthroat pupstream
2 gþ1
Tthroat ¼ Tupstream
2 gþ1
pupstream m_ ¼ Aveff pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r Tupstream
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gþ1 g1 2 g gþ1
(9)
(10)
(3)
with Cd the discharge coefficient [17], lv the valve lift, dv the valve seat angle, rv the valve head radius. In this equation, the function describing the valve lift lv depending on the crankshaft angle is sinusoidal. The velocity and the temperature of air at the minimum crosssection of the valve together with the air mass flow rate crossing the valve are evaluated for subsonic and sonic flow conditions. The model of a convergent nozzle is used to represent the flow across the valve. The velocities of air upstream and downstream of the valve are neglected. The flow is supposed quasi-steady and adiabatic reversible between the upstream and the minimum crosssection of the valve (equivalent to a nozzle throat). The air is considered as an ideal gas. With these conditions, the SainteVenant equation leads to:
Tupstream g1 Ma ¼1þ 2 Tthroat
In the case of sonic conditions, the flow is blocked at the minimum air cross-section of the valve. At this position, the air temperature and the air mass flow rate are written:
g g1
(5)
with pthroat and pupstream the air pressures respectively of the valve throat and of the upstream of the valve. In applying the SainteVenant theorem (conservation of total enthalpy), the speed of air at the throat is expressed, for subsonic and sonic conditions, as follows:
3.1.3. Heat transfer at the cylinder wall The heat flux at the cylinder wall is evaluated from the heat transfer coefficient of air at the wall of the expansion chamber hwall, by considering a constant wall temperature Twall:
Q_ wall ¼ hwall Acyl Tcyl Twall
(11)
with Q_ wall the heat flux crossing the expansion chamber wall, Acyl the surface of the expansion chamber in contact with air, Tcyl the air temperature in the expansion chamber. The heat transfer coefficient of air at the wall of the expansion chamber is estimated with the Hohenberg correlation [18], which gives intermediate values of this heat transfer coefficient compared with other correlations [12]:
0;06 0;8 0;4 hwall ¼ 0; 0001$130$Vcyl $pcyl $Tcyl $ U p þ 1; 4
(12)
with Vcyl, pcyl, Tcyl respectively the volume, pressure and temperature of air in the expansion chamber, U p the mean piston velocity. 3.1.4. Crank and connecting rod system The Crank and connecting rod system is represented in Fig. 4. The derivative of geometric relations in the triangle OBP leads to the following equations:
x_ ¼ rcrank q_ sin q L 4_ sin 4
(13)
L 4_ cos 4 rcrank q_ cos q ¼ 0
(14)
with x_ the piston velocity (also noted Up), rcrank and L the lengths of the crank and the connecting rod respectively, q the crankshaft R angle (qðtÞ ¼ u$dt), q_ the crankshaft angular rotational speed (also noted u), 4 and 4_ respectively the angle and the time derivative of the angle formed by the connecting rod and the piston
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M. Creyx et al. / Energy 102 (2016) 31e43
Fig. 4. Piston Rod Crank scheme of the expansion cylinder.
symmetry axis. The inertial loads of the crank and connecting rod system and of the crankshaft are neglected. Eq. (13) and Eq. (14) are transformed in Bond Graph formulation by the block presented in Fig. 5, referenced as “Crank and connecting Rod system” in Fig. 2. Fconrod and Cconrod are respectively the force applied on the top face of the piston and the indicated torque on the crank shaft. If u is fixed, Up(t) will be defined. The model associated at the output of the Crank-Rod system has to solve the effort “Fconrod(t)” from the flux “Up(t)”. Then the value of Cconrod(t) will be solved with the conservation of the power between the input ant the output of the block.
3.1.5. Piston The piston block evaluates the force Fpiston applied on the piston from the pressure ppiston applied by the air inside the expansion _ chamber on the piston and the derivative of the volume VðtÞ:
p Fpiston ¼ D2p ppiston 4
(15)
with Dp the piston diameter. The variation of air volume with time is defined by:
p VðtÞ ¼ Vc þ $D2p $ðxTDC xðtÞÞ 4
(16)
With Vc the clearance volume and xTDC the position of the piston at the top dead center. _ The flux VðtÞ is then defined by the following expression:
p _ _ with x_ ¼ Up VðtÞ ¼ qv ¼ $D2p $xðtÞ 4
(17)
3.1.6. Internal volume of the crankcase The air at atmospheric pressure patm situated in the internal volume of the crankcase applies a force Fcrankcase under the piston head:
p Fcrankcase ¼ D2p patm 4
(18)
This force and the force applied by the connecting rod on the piston Fconrod are linked to the force applied by the air inside the expansion chamber on the piston Fpiston (neglecting the inertial load of the piston):
Fpiston ¼ Fcrankcase þ Fconrod
(19)
3.1.7. Mechanical friction losses The mean global friction torque of the expansion cylinder applied on the crankshaft Cf is estimated with an experimental correlation obtained from measurements on the engine test bench: the crankshaft of the expansion cylinder at ambient temperature is driven by an electric engine with several rotational speeds (100e1000 rpm) and the mean friction torque applied on the crankshaft is measured with a calibrated force sensor set on a lever arm adapted for torque between 0 and 100 N. From a linear interpolation on experimental data [17], the mean friction torque is described by the following correlation:
Cf ¼ 2; 253$104 $N þ 0; 5207 with determination coefficient R2 ¼ 0:88 (20) with N the rotational speed (in rpm). The algebraic mean global friction torque is added to the indicated torque to obtain the effective torque applied on the crankshaft (cf Fig. 2):
Ccrankshaft ¼ Cconrod þ Cf
(21)
3.2. Equations of global performances The indicated power received by the air in the expansion chamber is calculated from Eq. (2):
_ ¼ p dVcyl W ie cyl dt Fig. 5. Bond Graph block of the crank and connecting rod system.
(22)
The indicated work of the expansion cycle is evaluated from the time integral of the indicated power during a cycle:
M. Creyx et al. / Energy 102 (2016) 31e43
37
Fig. 7. Evolution of the IMP depending on the intake air temperature Th obtained with the thermodynamic model [2] and with the dynamic model for several intake air pressures of 4, 6 and 8 bar.
The indicated work and the specific indicated work of the Ericsson engine are written respectively:
Fig. 6. Evolution of the air state variables of the expansion chamber depending on the crankshaft angle for the bond graph and scilab dynamic models with N ¼ 600 rpm, ph ¼ 6 bar, Th ¼ 550 C, intake and exhaust valves opened for the range of crankshaft angle respectively from 10 to 80 and from 160 to 360 .
Z Wie ¼
_ dt W ie
(23)
1 cycle
The effective work is deduced from the indicated work and from the mean global friction torque:
Weff ;e ¼ Wie 2p$Cf
(24)
Wi ¼ jWie j Wic
(26)
. wi ¼ ðjWie j Wic Þ mcycle
(27)
with mcycle the mass of air entering the cylinder per cycle. The indicated mean pressure (IMP) of the Ericsson engine is evaluated with the following formula:
IMP ¼ ðjWie j Wic Þ=ðVMe Vme Þ
(28)
with VMe and Vme respectively the maximum and minimum volumes of the expansion chamber during a cycle. The indicated mean effective pressure (IMEP) is written:
. ðVMe Vme Þ IMEP ¼ Weff ;e Weff ;c
(29)
with u the crankshaft rotational speed (in rad/s). The indicated work Wic of the compression cycle of the Ericsson engine is evaluated considering a theoretical Joule cycle [2]. The mean global friction torque of the compression cylinder is considered similar to the one of the expansion cylinder (mechanical systems of compression and expansion cylinder with similar dimensions and working conditions), leading to the following effective work of the compression cycle:
Weff ;c ¼ Wic 2p$Cf
(25)
Table 3 Comparison of the performances evaluated with the bond graph and scilab models for N ¼ 600 rpm, ph ¼ 6 bar, Th ¼ 550 C, intake and exhaust valves opened for the range of crankshaft angle respectively from 10 to 80 and from 160 to 360 . Bond graph model
Scilab model
Difference (%)
mcycle (g/cycle) Wie (J) _ (W) W
0.2157 56.88 568.83
0.2160 57.02 570.20
0.15% 0.24% 0.24%
IMPe (bar) Qwall (J/cycle)
3.4807 17.61
3.4888 18.85
0.23% 6.56%
ie
Fig. 8. Evolution of the thermodynamic efficiency depending on the intake air temperature Th obtained with the thermodynamic model [2] and with the dynamic model for several intake air pressures of 4, 6 and 8 bar.
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M. Creyx et al. / Energy 102 (2016) 31e43
Fig. 9. Evolution of the air intake mass per cycle depending on the intake air temperature Th obtained with the thermodynamic model [2] and with the dynamic model for several intake air pressures of 4, 6 and 8 bar.
The thermodynamic efficiency of the Ericsson engine is evaluated from the difference of indicated works of the compression and expansion cycles of the Ericsson engine and from the heat transmitted to the air by the heat exchanger between the exhaust of the compression cylinder and the intake of the expansion cylinder:
. hth ¼ ðjWie j Wic Þ Q_ H
(30)
with Q_ H the heat quantity transferred to air in the heat exchanger supplying the Ericsson engine. The global efficiency is obtained similarly, considering the mechanical friction losses:
. hglobal ¼ Weff ;e Weff ;c Q_ H
(31)
Fig. 10. Comparison of the pressureevolume diagrams obtained with the thermodynamic model [2] and with the dynamic model for N ¼ 600 rpm, ph ¼ 6 bar, Th ¼ 550 C, intake and exhaust valves opened for the range of crankshaft angle respectively from 10 to 80 and from 160 to 325 .
4. Verification of the dynamic model implemented The validation of the dynamic model of the expansion cylinder implemented is limited because of the lack of numerical or experimental data in the literature. The modelling of the expansion cylinder presented here, with a bond graph decomposition of the equations including the thermodynamic processes, the instantaneous air mass flow rate linked to pressure drops across the valves, the mechanical friction losses and the heat transfer at the expansion cylinder wall, is innovative. The models of the literature do not include all of these aspects simultaneously. To verify the results obtained with the bond graph model, a second dynamic model has been implemented as a script in the software scilab with a more common approach to solve the set of equations, considering the steady-state operating cycle of the Ericsson engine expansion cycle. The equations previously described are considered depending on the crankshaft angle q. The mass and energy balances of Eqs. (1) and (2) are expressed as follows in the second dynamic model:
3.3. Resolution method of the equations In the bond graph model of the expansion cylinder, the values of the following variables are imposed: rotational speed, air pressure and temperature at the inlet of the expansion cylinder, atmospheric pressure at the outlet. Initially, the piston is sup-
dT ¼ dq
Q_ wall u
"
dV pcyl dqcyl
þ
dm rTcyl dqcyl
þ
1 u
! v m_ in hin þ 2in hcyl 2
dmcyl 1 ¼ ðm_ in m_ ex Þ u dq
(32)
# v2ex _ mex hex þ 2 hcyl
mcyl cv Tcyl
posed at the top dead center (TDC), with an angular position of the crankshaft of 0 . The reference temperature is at 25 C for the calculations of specific internal energy and enthalpy. The set of equations implemented with the bond graph software is solved using an implicit method: the modified backward differentiation formula method (modified BDF method), using a calculation step of 1.108 s. The simulation is performed during two stabilized operating cycles in a steady-state (the transient operating phases are not studied here).
(33)
The resolution of the differential equations is performed using a predictor-corrector scheme with a second-order precision. The results of the bond graph and the dynamic models with a steady-state cycle are presented in Fig. 6 and Table 3 for particular operating conditions of the Ericsson engine: rotational speed of 600 rpm, pressure and temperature at the intake of the expansion cylinder of respectively 6 bar and 550 C, intake valve opened between q ¼ 10 and q ¼ 80 , exhaust valve opened between
M. Creyx et al. / Energy 102 (2016) 31e43
q ¼ 160 and q ¼ 360 . The friction torque and the heat transfer at the expansion cylinder wall are considered in these simulations. The evolutions of the state variables (volume, pressure, temperature of air in the expansion chamber) are close for both simulations (cf. Fig. 6). For the range of crankshaft angles corresponding to the exhaust phase, the maximum difference of air temperatures obtained between the bond graph model and the second model is of 6.2% (lower value for the bond graph model). This result is explained by the way the specific internal energy and enthalpy are implemented in both models (the integrals of heat capacity depending on temperature with a 3rd order polynomial fit are obtained with an exact calculation in the scilab model contrary to the bond graph model where these integrals are linearized). The performances of the expansion cylinder (indicated work, indicated power, IMPe of the expansion cycle, heat transmitted through the expansion cylinder wall) obtained with both models are presented in Table 3. The differences of the performances obtained, resulting of the differences of the air temperature in the expansion chamber, are very low. The most important difference of 6.6% is noticed for the heat crossing the expansion cylinder wall, which is linked to the temperature difference of air in the expansion chamber obtained with both models during the exhaust phase. The match of the results of both models implemented and solved in different ways allows to check the appropriateness of these models and is a first step to validate the bond graph model.
39
5. Results and discussion The results obtained with the bond graph dynamic model are presented below. The influence on the state variables (pressure, temperature, volume of air in the expansion chamber) and on the Ericsson engine performances of several parameters and phenomena (pressure and temperature at the inlet of the expansion cylinder, times of intake and exhaust valve closing, rotational speed, mechanical friction losses, heat transfer at the expansion cylinder wall) are considered. The mechanical friction losses and the heat transfer at the expansion cylinder wall are considered in the dynamic model for the results presented. The temperature of the expansion cylinder wall is supposed of 200 C (maximum temperature of the lubricating oil). 5.1. Influence of the air pressure and temperature at the inlet of the expansion cylinder The influence on the indicated mean pressure IMP, the thermodynamic efficiency hth and the air intake mass per cycle mcycle of the air pressure ph and the air temperature Th at the inlet of the expansion cylinder are shown in Fig. 7, Fig. 8 and Fig. 9 respectively. The results obtained with the thermodynamic model [2] and with the dynamic model, considering the data of Table 2, a rotational speed of 600 rpm, an expansion cycle with early exhaust valve
Fig. 11. Influence of the timings of valve closing on the air cross-sections of the intake valve (b) and of the exhaust valve (c) on the pressureevolume diagram of the expansion cycle (a), on the air pressure (d) and air temperature (e) in the expansion chamber for ph ¼ 6 bar, Th ¼ 550 C and N ¼ 600 rpm.
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M. Creyx et al. / Energy 102 (2016) 31e43
closing allowing to reach the intake air pressure at TDC and a timing of intake valve closing maximizing the IMP (timing adjusted with the dynamic model and considered similar for the thermodynamic model) are presented. The indicated work of the expansion cycle varies little with the temperature of intake air, with a maximum variation of 1.7% on the intake temperature range from 450 to 650 C. The IMP and the thermodynamic efficiency rise with the intake temperature (cf. Figs. 7 and 8 respectively) because of the decreasing of the intake mass per cycle (cf. Fig. 9) which induces a reduction of the work required for the air compression cycle. The results of the dynamic and thermodynamic models [2] show that the maximum IMP is obtained for an intake pressure situated between 6 and 8 bar and that the intake mass per cycle decreases with the rise of the intake pressure.
For the same intake air pressure and temperature, the values of IMP, thermodynamic efficiency and intake mass per cycle obtained with the thermodynamic [2] and dynamic models vary significantly (maximum variation of 40% for the IMP and of 51% for the thermodynamic efficiency between the two models). These differences are linked to the evaluation of the instantaneous air mass flow rate linked to pressure drops across the valve considered in the dynamic model (sinusoidal valve lift), contrary to the thermodynamic model (no pressure drop which is equivalent to consider a rectangular valve lift with a maximum valve cross-section area). For the same operating conditions (in particular the timing of intake and exhaust valve closing), the intake phase is extended with the thermodynamic model [2] compared with the dynamic model (cf. Fig. 10), enhancing the air intake mass per cycle (cf. Fig. 9). The exhaust phase is also extended (cf. Fig. 10), limiting the air pressure in the
Fig. 12. Influence of the late intake valve closing coefficient alivc and the early exhaust valve closing coefficient aeevc on the IMP, the thermodynamic efficiency and the air intake mass per cycle for several intake air pressures of 4, 6 and 8 bar, with an intake air temperature of 550 C and a rotational speed of 600 rpm.
M. Creyx et al. / Energy 102 (2016) 31e43
41
transfer at expansion cylinder wall and mechanical friction) is situated between 0.6 and 1.2 bar and the thermodynamic efficiency between 11 and 33%. So, the thermodynamic performances of the Ericsson engine are maintained at high values when considering the dynamic losses, compared to the results of the thermodynamic study [2]. 5.2. Influence of the closing times of intake and exhaust valves
Fig. 13. Evolution of the IMP, the thermodynamic efficiency and the intake mass per cycle depending on the rotational speed with and without heat transfer across the expansion cylinder wall for air intake conditions of 6 bar and 550 C.
expansion chamber at TDC and allowing an air intake during the pressure balance after the opening of the intake valve [2]. The air compression and expansion phases in the expansion chamber which occur with different pressureevolume conditions for both models (cf. Fig. 10) explain the variation of the results obtained. For the intake air temperature and pressure ranges respectively from 450 to 650 C and from 4 to 8 bar, the IMP estimated with the dynamic model (including the evaluation of the instantaneous air mass flow rate linked to pressure drops across the valves, heat
Fig. 14. Evolution of the indicated mean pressure IMP and the indicated mean effective pressure IMEP depending on the rotational speed for air intake conditions of 6 bar and 550 C.
The influence of the times of intake and exhaust valve closing on the state variables of the air in the expansion chamber is presented in Fig. 11. At the opening time of the exhaust valve (cf. Fig. 11c), a quantity of air present in the expansion chamber is irreversibly expanded until reaching equal pressures in the chamber and in the discharge pipe (cf. Fig. 11d). For the case A, the exhaust phase is extended until reaching the TDC corresponding to the crankshaft angle q ¼ 0 (or 360 ) (cf. Fig. 11c). The early exhaust valve closing (cf. Fig. 11c) allows a compression of the air present in the clearance volume until reaching the intake air pressure for the case B and an intermediate pressure in the case C (cf. Fig. 11d). The early exhaust valve closing of the case B (cf. Fig. 11c) induces a reduction of the intake mass per cycle (cf. Fig. 11a) and an enhancement of the air temperature in the chamber at the end of the exhaust phase (cf. Fig. 11e). The opening of the intake valve (cf. Fig. 11b) induces a pressure balance corresponding to a compression of the air contained in the clearance volume until reaching the intake air pressure (cf. Fig. 11d) with an additional entering air mass. This compression phase leads to an increase in the air temperature of the chamber linked to the irreversibilities (cf. Fig. 11e). An extension of the intake phase is shown with case C, compared with cases A and B (cf. Fig. 11a and b), linked with a late intake valve closing which induces an enhanced intake air mass per cycle (cf. Fig. 11a). To represent the timings of intake and exhaust valve closing, two adimensional coefficients corresponding to volume ratio linked to the swept volume of the expansion cylinder are introduced [2]:
alivc ¼ ðVlivc Vme Þ=ðVMe Vme Þ
(34)
aeevc ¼ ðVeevc Vme Þ=ðVMe Vme Þ
(35)
with Vlivc and Veevc the volumes of the expansion chamber at the time respectively of intake and of exhaust valve closing. The minimum value of the late intake valve closing coefficient alivc corresponds to the case of an expansion allowing to reach the atmospheric pressure at bottom dead center (BDC). The maximum value of alivc is equal to 1. The early exhaust valve closing coefficient aeevc is situated between 0 and a value below 1 allowing to reach the intake air pressure at TDC after compression of the air in the clearance volume. The IMP, the thermodynamic efficiency and the air intake mass per cycle depending on alivc and aeevc coefficients are represented in Fig. 12 for several intake air pressures (4, 6 and 8 bar), with an intake air temperature of 550 C and a rotational speed of 600 rpm. The range of valid values of alivc and aeevc coefficients (positive indicated work of the Ericsson engine, cf. Eq. (26)) depends strongly on the intake air pressure. With a higher intake air pressure, the valid alivc coefficients are shifted to lower values and the range of valid aeevc coefficients is enhanced. The maximum IMP and thermodynamic efficiencies are reached for pressures situated between 6 and 8 bar, as shown previously, for small late intake valve closing and for small early exhaust valve closing, that confirms the results obtained in Ref. [2] with a thermodynamic model. The values of alivc and aeevc coefficients maximizing the IMP and the thermodynamic efficiency correspond to expansion cycles close to the theoretical
42
M. Creyx et al. / Energy 102 (2016) 31e43
Joule cycle. The intake air mass per cycle depends little on the early exhaust valve closing coefficient aeevc and increases with the late intake valve closing coefficient alivc. So, the highest performances of the Ericsson engine are obtained for the cycles with the smallest air intake mass. In the following, the aeevc coefficient corresponds to a compression of the air in the clearance volume reaching the intake air pressure at TDC and the alivc coefficient is adjusted to maximize the IMP.
for an adiabatic expansion cylinder wall, which can be explained by the thermal power lost through the wall when a heat transfer occurs. The intake mass per cycle is higher when a heat transfer crosses the expansion cylinder wall, which is explained by the cooling of air in the expansion chamber through the wall (for similar pressureevolume conditions in the expansion chamber, a lower temperature induces a higher mass of air). 5.4. Influence of the mechanical friction losses
5.3. Influence of the rotational speed Fig. 13 presents the influence of the rotational speed on the IMP, on the thermodynamic efficiency hth and on the intake mass per cycle for the cases with and without heat transfer at expansion cylinder wall and considering the air intake conditions of 6 bar and 550 C. The intake air mass per cycle decreases with the increase of the rotational speed, which is caused by the increase of pressure drops through the valves with the rotational speed. The IMP decreases with the rotational speed, with a very slight slope when a heat transfer occurs through the expansion cylinder wall. The thermodynamic efficiency is reduced when the rotational speed rises for an adiabatic expansion cylinder wall. The reduction of IMP and thermodynamic efficiencies at high rotational speed is linked to higher dynamic losses (in particular mechanical friction losses). The thermodynamic efficiency remains stable when a heat transfer occurs through the cylinder wall, which might be explained by the important decrease of intake mass per cycle in this case, inducing a decrease of the heat supplied per cycle in the Ericsson engine, that might compensate the increase of mechanical friction losses with the rotational speed which reduces the effective work of the engine (cf. Eq. (30)). The IMP and the thermodynamic efficiency are higher
The indicated mean pressure IMP and the indicated mean effective pressure IMEP are presented in Fig. 14 for intake conditions of 6 bar and 550 C. The IMEP corresponds to the IMP reduced by the mechanical friction losses. These losses represent from 56 to 72% of the IMP for the range of rotational speed considered and rise with the rotational speed, which is linked to the correlation of the friction torque (cf. Eq. (20)). For the operating conditions chosen in Fig. 14, the rotational speed maximizing the IMP and the IMEP is close to 800 rpm, the IMP is close to its maximum values and the IMEP lies between 0.25 and 0.4 bar. A higher intake air temperature (cf. Fig. 7) or a reduction of the mechanical friction losses may improve this result. 5.5. Effect of heat transfer at the expansion cylinder wall The influence of a heat transfer at the expansion cylinder wall on the air state variables in the expansion chamber is presented in Fig. 15 for a wall temperature of 200 C, intake air conditions of 6 bar and 550 C and a rotational speed of 600 rpm. The modelling of the heat transfer through the expansion cylinder wall has a little impact on the air pressure in the expansion chamber, even if the
Fig. 15. Evolution of the air pressure (a), of the air temperature (b) in the expansion cylinder and of the shape of the expansion cycle (c) with and without heat transfer across the expansion cylinder wall for a wall temperature of 200 C, intake conditions of 6 bar and 550 C and a rotational speed of 600 rpm.
M. Creyx et al. / Energy 102 (2016) 31e43
expansion process becomes non isentropic. The effect on temperature is more important: the hypothesis of an adiabatic wall induces an overestimation of the air temperature in the chamber of about 155e225 C for the operating conditions of Fig. 15. The thermal heat losses across the wall are reduced when the rotational speed is enhanced, so that the differences of IMP and of intake air mass per cycle with and without heat transfer are reduced at high rotational speed (cf. Fig. 13). This phenomenon is linked to the duration decrease of heat transfer between the air of the expansion chamber and the wall during an expansion cycle when the rotational speed increases. 6. Conclusion A dynamic model of the expansion cylinder of an open Joule cycle Ericsson engine was encoded and simulated with a bond graph formalism. The working conditions and the design of the expansion cylinder have been chosen in agreement with results obtained with a steady-state thermodynamic study described by Creyx et al. [2]. The Ericsson engine will be coupled subsequently to an air-gas heat exchanger built in a domestic biomass boiler. Several dynamic phenomena are modelled: thermodynamic processes in the expansion chamber, instantaneous air mass flow rate across the valves, mechanical friction losses, heat transfers at the expansion cylinder wall. Their influence on the state variables evolution of the air inside the expansion chamber (pressure and temperature) and on the Ericsson engine performances (IMP and thermodynamic efficiency) is then studied. The effects of the air intake pressure and temperature conditions and those of the timing of intake and exhaust valve closing are studied with the dynamic model and compared to the results obtained with a steady-state thermodynamic model [2]. Steady-state and present dynamic study show that the intake air pressure maximizing the Ericsson engine IMP is situated between 6 and 8 bar and that the highest temperatures enhance the Ericsson engine performances. The IMP and the thermodynamic efficiency are little dependent on the rotational speed of the engine. The mechanical friction losses represent an important part of the IMP. The IMEP is situated between 0.25 and 0.4 bar for the operating conditions considered. Considering a hypothesis of adiabatic expansion cylinder wall, compared with the modelling of the heat transfer through the expansion chamber wall, the values of the state variables of the air in the expansion chamber vary little but a strong impact is observed on the Ericsson engine performances (IMP and thermodynamic efficiency). Acknowledgement This work is performed in the framework of the regional project Sylwatt, in partnership with the laboratories LAMIH (Valenciennes, France), PC2A (Lille, France), CCM (Dunkerque, France) and received
43
gion Nord-Pas-de-Calais, from the a financial support from the Re French National Association of Research and Technology ANRT (doctoral scholarship) and from the company Enerbiom. The authors are grateful to Marc Lippert and Jesse Schiffler from the laboratory LAMIH who set up the test bench for the friction torque measurement of the Ericsson engine expansion cylinder.
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