CV rh

PROCEEDINGS OF THE INSTITUTE OF VEHICLES 2(106)/2016
Adrian Chmielewski1, Robert Gumiński2, Jędrzej Mączak3
ANALYSIS OF ISOTHERMAL THERMODYNAMIC PROCESSES
IN THE STIRLING ENGINE
Nomenclature:
cp -specific heat at constant pressure [J/kgK]
cv -specific heat at constant volume [J/kgK]
κ - isentropic exponent (κ=cp/cv)
lc - length of compression space [m]
lco - length of the cooler [m]
lr - length of the regenerator [m]
lh - length of the heater [m]
lh - length of the expansion space [m]
dmcor - the flow of the working gas mass at the cor control
border [kg/s]
dmhe - the flow of the working gas mass at the he control
border [kg/s]
dmr - the flow of the working gas mass at the regenerator
[kg/s]
dmrc - the flow of the working gas mass at the rc control
border [kg/s]
dmrh - the flow of the working gas mass at the rh control
border [kg/s]
dQh - rate of heat transfer to heater [J/s]
dQc - rate of heat transfer to cooler [J/s]
dQr - rate of heat transfer to regenerator [J/s]
p - working gas pressure [Pa]
dp - the pressure derivative [Pa/s]
mtot - total mass of the working gas mass in the all spaces [kg]
mc - working gas mass in the compression space [kg]
mr - working gas mass in the regenerator [kg]
mh - working gas mass in the heater [kg]
mco - working gas mass in the cooler [kg]
me - working gas mass in the expansion space [kg]
R - individual gas constant [J/kgK]
Vc - working gas volume in the compression space [m3]
Ve - working gas volume in the expansion space [m3]
Vco - working gas in the cooler [m3]
Vr - working gas volume in the regenerator [m3]
Vh - working gas volume in the heater [m3]
Te.- expansion space temperature [K]
Tco.- compression space temperature [K]
Trc - temperature on the regenerator on the side of the cooler
[K]
Trh - temperature on the regenerator on the side of the heater
[K]
τ - temperature indicator (Tc/Th) [-]
Tr - temperature of the regenerator element
Tr=(Th-Tc)/(lnTh/Tc) [K]
cco - the control border between the compression space and
the cooler [–]
cor - the control border between the cooler and the
regenerator [–]
rh - the control border between the regenerator and the
heater [–]
he - the control border between the heater and the expansion
space [–]
1. Introduction
This work focuses on the isothermal thermodynamic analysis of the Stirling engine.
The Stirling engine is an external combustion heat machine using heat supplied from the
outside. It operates in the closed thermodynamic cycle with the intermittent isothermal
and isochoric compression and expansion of the working gas in the working chamber
with the defined temperature difference between the lower and upper heat sources [1, 2].
The work prediction [3] of such an engine, using the thermodynamic analysis, is
particularly vital. The forerunners of thermodynamic analyses and mathematical
descriptions of the Stirling engines were Walter [4], Urieli [5], Berchowitz [6],
Shoureshi [7], Organ [8], Martini [9], Żmudzki [10]. In theoretical thermodynamic
analyses, where there is the ideal heat regeneration, thermal efficiency of the ideal closed
Stirling cycle is equal to the Carnot cycle efficiency [4-11]. Increasing the cycle’s
efficiency is possible through increasing the temperature difference between the lower
and the upper heat sources, which can be achieved by, among others: decreasing the
temperature of the lower heat source or increasing the temperature of the upper heat
1
Research Assistant Adrian Chmielewski, Institute of Vehicles, Warsaw University of Technology,
Assistant Professor Robert Gumiński, Institute of Vehicles, Warsaw University of Technology,
3
Associate Professor Jędrzej Mączak, Institute of Vehicles, Warsaw University of Technology.
2
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source at the constant temperature of the lower heat source. Increasing the temperature
of the upper heat source is limited by the thermal fatigue enduring properties of the hightemperature heat exchangers.
Moving on to the techniques of conducting analyses for Stirling engines, it should be
emphasised that they are divided, according to Martini’s nomenclature, into analyses
from the zero- to the fourth-order [1-4].
In these mathematical models, the whole engine is divided into several or over a
dozen of sections (control volume units), therefore, they are practically zero- or onedimensional models [4-17]. The analysis of thermodynamic processes itself can be
conducted at different levels of difficulty and with different degrees of simplification,
starting from the simplest: isothermal analyses [4-12] (constant temperature values in
control spaces), adiabatic analyses [4-9] (no heat exchange in the compression and
expansion spaces), quasi-adiabatic, to more advanced polytropic considerations [17].
In this work, in Chapter 2, the authors focused on theoretical foundations and the
description of the ideal isothermal thermodynamic processes taking place in the Stirling
engine.
2. Isothermal analysis of thermodynamic processes
This chapter discusses isothermal analysis of the thermodynamic processes taking
place in the Stirling engine. The analysis was carried out with the assumption that the
temperatures in the cooler space (Vco) and in the compression space (Vc) are equal to
each other and constant Tc = Tco = const. Similar assumption for the isothermal
transformations was adopted for both the temperature in the heater space (Vh), and in the
expansion space (Ve), where, respectively Th= Te = const. (Fig. 1). Figure 1 shows the
division of the working space into sections, which correspond to the volume unitsof the
heat exchangers (Vco - volume of the cooler, Vr – volume of the regenerator, and Vhvolume of the heater). The total mass of the working gas in the working space is
constant, mt=const. and it is a sum of the mass of the working gas in individual sections,
which can be written as follows:
mt  mc  mco  mr  mh  me
(1)
Using the Clapeyron equation pV = mRT allows for formulating the relation for the
pressure change in the working space for isothermal analyses:
p
Vc
Tc

Vco
Tco
mt R
mt R

Vr Vh Ve
1
Vc  Vco   Vr  1 Vh  Ve 



Tc
Tr Th
Tr Th Te
(2)
Figure 1 presents the temperature change in the selected sections of the working
space. In the case of the regenerator, whose activity is accumulating energy (collecting
heat from the working gas while it is transferred from the heater, and supplying the heat
accumulated in the regenerator matrix during transferring the working gas from the
cooler).
22
Fig. 1. Diagram of the working space divided into sections
The effect of the transferring process of the working gas is temperature change on
the regenerator’s length lr from the temperature Tc (from the cold side) to the
temperature Th from the hot side, which can be written as:
Tr ( x)  x(Th  Tc ) / l r  Tc
(3)
Analysing Figure 1, a relation describing the working gas mass in the regenerator can
be written:
mr 
Ar l r
R
lr
 p (T
0
h
T
Vr
1
dx  p
ln h
 Tc ) x  Tc l r
R(Th  Tc ) Tc
(4)
Substituting for equation (3) the relation mr = pVr / TrR, we obtain a relation
describing the mean temperature at the regenerator:
Tr  (Th  Tc ) / ln(Th / Tc )
(5)
Figure 2 shows the space of energy exchange in one of the working sections
(Fig. 1.), including: the compression space or the expansion space.
23
minp,Tinp
mout,Tout
m, p, T, V
W
Q
T=Tinp=Tout
Fig. 2. Space of energy exchange for isothermal processes
An equation of state for a system considered in such a way (Fig. 2) can be put as
follows:


dQ  c p Tinpdminp  c pToutdmout  dW  cv d (mT )
(6)
Given that T = Tinp = Tout, the following can be written:


dQ  c pT dminp  dmout  pdV  cv d (mT )

(7)
dm
Analysis of the heat exchange at the heat exchangers (heater, cooler), (Fig. 3),
shows that pdV = 0, thus the relation (6) takes the form:
dQ  c pTdm  cv d (mT )  dQ  dmT (cv  c p )



(8)
R
Cooler
cco
dmcco
Regenerator
cor
Heater
rh
dmcor mr, p, Tr, Vr dmrh
dmr
dQco
dQr
he
dmhe
dQh
Fig. 3. Mass flow and heat exchange at the heat exchangers
With the knowledge of dm = Vdp / RT, the following can be written for the cooler:
dQco  Vco dp
And for the heater:
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(9)
dQh  Vh dp
(10)
The last element of the considered heat exchangers is the regenerator. Figure 4
presents the mass flow of the working gas, as well as the heat exchange taking place at
the regenerator.
Regenerator
cor
rh
dminp=dmcormr, p, Tr, Vr dmout=dmrh
dmr
dQr
Fig. 4. Mass flow and heat exchange at the regenerator
In the case of the regenerator, it is known as a fact that:
dmr  dminp  dmout  dmcor  dmrh
(11)
Respectively, dmcor amounts to:
dmc or  dmc  dmco  
1
 pdVc  dpVc  Vco 
RTc
(12)
Respectively, dmrh amounts to:
dmrh  dme  dmh 
1
 pdVe  dpVe  Vh  (13)
RTh
The heat exchange taking place at the regenerator can be written as:
dQr  cv dmrTr  c p (Tc dmcor  Th dmrh )
(14)
After implementation of the relations (11-13) to (14) and simple transformations, we
obtain:
1
dQr  cvVr dp  c p dpVc  Vco V hVe   pdVe  dVc 
(15)
R


The last unknown is a pressure derivative, which can be determined by
differentiating the relation (1):
0  dmc  dme  dmh  dmco  dmr  dmc  dme 
Respectively, dmc amounts to:
25
dp Vh Vr Vc 
 
 
R  Th Tr Tc 
(16)
dmc 
pdVc Vc dp

RTc
RTc
(17)
dme 
pdVe Ve dp

RTe
RTe
(18)
Whereas dme amounts to:
Inserting relations (17) and (18) into relation (16), and taking into account that
  Tc / Th , we obtain the relation for dp:
dp  
pdVc  dVe 
pdVc  dVe 
(19)


1
 1

1

Vc  Vco 
ln Vr   Vh  Ve 
Vc  Vco   
ln Vr  Vh  Ve 
1    
1     

3. Summary
In this paper, a complex derivation of the relations which will serve the purpose of
mathematical modelling of the isothermal thermodynamic processes taking place in the
Stirling engine has been presented. The results of the analyses will be extended in the
future in the advanced, combined quasi-adiabatic model using the dynamic model of the
piston-crankshaft assembly [18-20], and the results of experimental and operational
research [21-26] on the Alpha-type Stirling engine. The authors of this work plan to use
an ANFIS (Adaptive Neuro-Fuzzy Inference System) neuro-fuzzy controller [27-34] to
control such a system.
References:
[1]
Cinar C., Yucesu S., Topgul T., Okur M.: Beta-type Stirling engine operating at
atmospheric pressure, Applied Energy Vol. 81, pp. 351–357, 2005.
[2]
Shazly J.H., Hafez A.Z., El Shenawy E.T., Eteiba M.B.: Simulation, design and
thermal analysis of a solar Stirling engine using MATLAB, Energy Conversion
and Management, Vol. 79, pp. 626–639, 2014.
[3]
Chmielewski A., Gontarz S., Gumiński R., Mączak J., Szulim P. Analiza wpływu
parametrów eksploatacyjnych na drgania układu mikrokogeneracyjnego. Przegląd
Elektrotechniczny No. 1, pp.45-53 Vol. 2016.
[4]
Walter G. Stirling Engines. Oxford University Press, New York, 1980.
[5]
Urieli I., Berchowitz D.M. Stirling cycle engine analysis. Adam Hilger Ltd.
Bristol 1984.
[6]
Berchowitz D. M. Stirling cycle engine design and optimisation., Doctor of
Philosophy Thesis, Ohio, August 1986.
[7]
Shoureshi R. Analysis and design of Stirling Engines for Waste-Heat Recovery.
Massachusetts Institute of Technology, June 1981.
[8]
Organ A.J. The Regenerator and the Stirling Engine. Mechanical Engineering
Publications Limited, 1997.
26
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
Martini W.R., Stirling Engine Design Manual. National Aeronautics and Space
Administration (NASA); CR-168088,1983.
Żmudzki S. Silniki Stirlinga (Stirling Engines). WNT Warsaw, 1993.
Chmielewski A., Gumiński R., Radkowski S., Szulim P., Experimental research
and application possibilities of microcogeneration system with Stirling engine,
Journal of Power Technologies, 95 (Polish Energy Mix) (2015), 14–22.
Chmielewski A., Gumiński R., Dynamic model of a free-piston Stirling engine
with four degrees of freedom combined with the thermodynamic submodel. 21st
International Conference on Methods and Models in Automation and Robotics
MMAR 2016 Międzyzdroje IEEE [In Print].
Chen W. L., Wong K. L., Chang Y. F., A computational fluid dynamics study on
the heat transfer characteristics of the working cycle of a low-temperaturedifferential c-type Stirling engine, International Journal of Heat and Mass
Transfer Vol. 75, 145–155, 2014.
Cheng C.H., Yang H. S., Analytical model for predicting the effect of operating
speed on shaft power output of Stirling engines. Energy Vol. 36, pp. 5899-5908,
2011.
Thombarea D.G., Verma S.K.: Technological development in the Stirling cycle
engines. Renewable and Sustainable Energy Reviews Vol. 12, pp. 1–38, 2008.
Cheng, C., H., Yang, H., S., Keong, L. Theoretical and experimental study of a
300W beta–type Stirling engine. Energy,Vol. 59, pp. 590–599, 2013.
Babaelahi M., Sayyaadi H. A new thermal model based on polytropic numerical
simulation of Stirling engines. Applied Energy 2015; 141: 143–159.
Chmielewski A., Gumiński R., Radkowski S. Chosen properties of a dynamic
model of crankshaft assembly with three degrees of freedom. 20th International
Conference On Methods and Models in Automation and Robotics (MMAR),
IEEE, pp. 1038-1043, 2015. ISBN: 978-1-4799-8700.
Chmielewski A., Bogucki Ł., Gumiński R., Mączak J. Dynamic model of a
crankshaft assembly with two degrees of freedom. Journal of KONES, No. 3, Vol.
22, pp.275-2822, 2015.
Chmielewski A., Gumiński R., Mączak J., Selected properties of the adiabatic
model of the Stirling engine combined with the model of the piston-crankshaft
system. 21st International Conference on Methods and Models in Automation and
Robotics MMAR 2016 Międzyzdroje IEEE [In Print].
Chmielewski A., Gumiński R., Mączak J., Szulim P., Model-based research on
the micro cogeneration system with Stirling engine. Journal of Power
Technologies 2016 [In Print].
Chmielewski A., Gontarz S., Gumiński R., Mączak J., Szulim P. Research study
of the micro cogeneration system with automatic loading unit. Proceedings of
AUTOMATION-2016, March 2-4, 2016, Warsaw, Poland, Springer International
Publishing Switzerland 2016, Challenges in Automation, Robotics and
Measurement Techniques, Advances in Intelligent Systems and Computing, DOI
10.1007/978-3-319-29357-8_34, Vol. 440, pp. 375-386, 2016.
Chmielewski A., Gontarz S., Gumiński R., Mączak J., Szulim P. Research on a
micro cogeneration system with an automatic load-applying entity. Proceedings
of AUTOMATION-2016, March 2-4, 2016, Warsaw, Poland, Springer
International Publishing Switzerland 2016, Challenges in Automation, Robotics
27
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
and Measurement Techniques, Advances in Intelligent Systems and Computing,
DOI 10.1007/978-3-319-29357-8_35, Vol. 440, pp. 387-395, 2016.
Chmielewski A., Gumiński R., Lubikowski K., Mączak J., Szulim P. Badania
układu mikrokogeneracyjnego z silnikiem Stirlinga. Część I. Rynek Energii Nr 4
(119), pp. 42-48, 2015.
Chmielewski A., Gumiński R., Mączak J., Szulim P. Badania układu
mikrokogeneracyjnego z silnikiem Stirlinga. Część II. Rynek Energii Nr 5(120),
pp.53-60, 2015.
Chmielewski A., Gumiński R., Mączak J., Radkowski S., Szulim P. Aspects of
balanced development of RES and distributed micro cogeneration use in Poland:
case study of a µCHP with Stirling engine. Elsevier, Renewable and Sustainable
Energy Reviews Vol. 60, pp. 930-952, 2016.
Chmielewski A., Paweł Maciąg, Gumiński R., Mączak J. The use of Fuzzy Logic
in the control of an inverted pendulum. Springer Proceedings in Mathematics and
Statistics (Dynamical Systems - Modelling), 2016 [In Print].
Braz-César M., Barros R. Neuro-fuzzy control of structures with MR dampers.
Dynamical Sytems Control and Stability ISBN 978-83-7283-708-0, pp. 95-106,
2015.
Reddy M. J. B., Mohanta D.K. Performance Evaluation of an Adaptive-NetworkBased Fuzzy Inference System Approach for Location of Faults on Transmission
Lines Using Monte Carlo Simulation. IEEE Transactions On Fuzzy Systems,
VOL. 16, NO. 4, pp. 909-919, August 2008.
Szabłowski Ł., Milewski J., Badyda K. Utilisation of a set of distributed
generation sources controlled by artificial neural network to meet electricity
demand of public utility buildings. Proceedings of ECOS 2015 - The 28th
International Conference on Efficiency, Cost, Optimization, Simulation and
Environmental Impact of Energy Systems 2015.
Szabłowski, Ł., Milewski, J., Kuta J. Internal Combustion Engine controlled by
Artificial Neural Network. SL: Structural Longevity, Vol.7, No.3, pp.187-202,
2012.
Milewski J., Szabłowski Ł., Wołowicz M., Miller A. Procedury uczenia
sztucznych sieci neuronowych modelu węglanowego ogniwa paliwowego. Rynek
Energii, No. 2, Vol. 117, pp. 99-106, 2015.
Szabłowski Ł., Milewski J., Kuta J., Optimal control strategy of µ-turbine as a
DG unit obtained by an utilization of Artificial Neural Network. Proceedings of
3rd International Conference Contemporary Problems of Thermal Engineering
CPOTE2012, pp. 259-260, 2012.
Milewski J., Zahadat P., Modeling electrical behavior of solid oxide electrolyzer
cells by using artificial neural network. International Journal of Hydrogen Energy
40 (23), pp. 7246-7251, 2015.
Abstract
This work presents a thermodynamic analysis for thermodynamic processes taking
place in the Stirling engine working space. The working space was divided into
operational sections, which corresponded to the analysed control volume units,
including: the compression space, cooler, regenerator, heater, and the expansion space.
On the basis of the conducted thermodynamic analysis, useful relations were derived,
which will be used in the future to build the advanced, combined model in which energy
28
and heat losses are taken into consideration, as well as the Stirling engine dynamics
during the work cycle. Among the most important thermodynamic processes presented
in this work, are: heat exchange at the heat exchangers (the cooler, regenerator, heater),
and the isothermal heat exchange in the compression and expansion spaces.
Keywords: thermodynamic analysis, isothermal processes, Stirling engine.
ANALIZA IZOTERMICZNYCH PROCESÓW TERMODYNAMICZNYCH
ZACHODZĄCYCH W SILNIKU STIRLINGA
Streszczenie
W niniejszej pracy przedstawiono analizę termodynamiczną dla procesów
termodynamicznych zachodzących w przestrzeni roboczej silnika Stirlinga. Przestrzeń
robocza podzielona została na sekcje robocze, które odpowiadały analizowanym
objętościom kontrolnym, m.in: przestrzeni sprężania, chłodnicy, regeneratora,
nagrzewnicy oraz przestrzeni rozprężania. Na podstawie przeprowadzonej analizy
termodynamicznej wyprowadzono użyteczne zależności, które zostaną w przyszłości
wykorzystane do budowy zaawansowanego kombinowanego modelu uwzględniającego
straty energii, ciepła oraz dynamikę silnika Stirlinga podczas realizacji cyklu roboczego.
Do najważniejszych procesów termodynamicznych przedstawionych w niniejszej pracy
zaliczyć należy: wymianę ciepła na wymiennikach ciepła (chłodnicy, regeneratorze,
nagrzewnicy) oraz izotermiczną wymianę ciepła w przestrzeniach sprężania oraz
rozprężania.
Słowa kluczowe: analiza termodynamiczna, procesy izotermiczne, silnik Stirlinga.
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