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7212019 Applied Thermal Engineering Volume 83 Issue 2015 [Doi 101016_2Fjapplthermaleng201503006] Araoz Joseph hellip
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Research paper
Development and validation of a thermodynamic model for the
performance analysis of a gamma Stirling engine prototype
Joseph A Araoz a b Evelyn Cardozo a b Marianne Salomon a Lucio Alejo bTorsten H Fransson a
a Department of Energy Technology School of Industrial Technology and Management (ITM) Royal Institute of Technology (KTH) 100 44 Stockholm Swedenb Facultad de Ciencias y Tecnologiacutea (FCyT) Universidad Mayor de San Simon (UMSS) Cochabamba Bolivia
h i g h l i g h t s
A numerical model for a Stirling engine was developed
A mechanical ef 1047297ciency analysis was included in the model
The model was validated with experimental data of a novel prototype
The model results permit a deeper insight into the engine operation
a r t i c l e i n f o
Article history
Received 20 August 2014
Accepted 4 March 2015
Available online 14 March 2015
Keywords
Stirling engineSimulation and modelling
Thermodynamic analysis
Energy technology
a b s t r a c t
This work presents the development and validation of a numerical model that represents the perfor-
mance of a gamma Stirling engine prototype The model follows a modular approach considering ideal
adiabatic working spaces limited internal and external heat transfer through the heat exchangers and
mechanical and thermal losses during the cycle In addition it includes the calculation of the mechanical
ef 1047297ciency taking into account the crank mechanism effectiveness and the forced work during the cycle
Consequently the model aims to predict the work that can be effectively taken from the shaft The modelwas compared with experimental data obtained in an experimental rig built for the engine prototype
The results showed an acceptable degree of accuracy when comparing with the experimental data with
errors ranging from plusmn1 to plusmn8 for the temperature in the heater side less than plusmn1 error for the cooler
temperatures and plusmn1 to plusmn8 for the brake power calculations Therefore the model was probed
adequate for study of the prototype performance In addition the results of the simulation re1047298ected the
limited performance obtained during the prototype experiments and a 1047297rst analysis of the results
attributed this to the forced work during the cycle The implemented model is the basis for a subsequent
parametric analysis that will complement the results presented
copy 2015 Elsevier Ltd All rights reserved
1 Introduction
Actual energy demand and environmental problems require
intensive research for the development of ef 1047297cient and sustainable
energy solutions In this scenario the Stirling engine technology
appears as a renewed solution [1] with the potential to meet the
requirements at small-scale [2] thanks to its known theoreticalcapabilities However actual designs are far from meeting the ef 1047297-
ciency requirements needed to be commercially viable as shown by
Thomas [3] Dong [4] and Gonzales-Pino [5] This heightened the
need for engineering tools like numerical simulation that could
assess design improvements together with test measurements in
order to optimize the engine performance before implementing
them in the engine
Different prototypes have been developed guided by simulation
analysis The simulation studies varied in complexity from simu-
lation based on 1047297rst order [6] second order analysis as reported by
Cheng [7] Mehdizadeh [8] Parlak [9] Strauss [10] and Tlili [11] and
Corresponding author Department of Energy Technology School of Industrial
Technology and Management (ITM) Royal Institute of Technology (KTH) 100 44
Stockholm Sweden Tel thorn46 704014380 fax thorn46 (0)8 790 7477
E-mail address araozkthse (JA Araoz)
Contents lists available at ScienceDirect
Applied Thermal Engineering
j o u r n a l h o m e p a g e w w w e l s e v i e r c o m l o c a t e a p t h e r m e n g
httpdxdoiorg101016japplthermaleng201503006
1359-4311copy
2015 Elsevier Ltd All rights reserved
Applied Thermal Engineering 83 (2015) 16e30
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computer 1047298uid dynamics (CFD) analysis that include the works of
Mahkamov [12] Ibrahim [13] and Wilson [14] Among these
methods1047297rstorder methods aresimple andlimited to estimate the
power output and engine ef 1047297ciency under ideal assumptions On
the other extreme CFD analyses are very complex and require
intensive computing resources [6] Therefore second order ana-
lyses have been preferred for 1047297rst design and optimization studies
of the engine considering a compromise between prediction
accuracy and computational requirements These second order
methods include the mass and energy balances through the
different spaces of the engine and also evaluate the friction and
thermal losses using a decoupled approach
Different studies have guided the development of Stirling en-
gines prototypes These include novel con1047297gurations in the
regenerator [15] the heat exchangers [16] the crank mechanism
[17] and optimization studies [10] However there is still a need to
Nomenclature
A area (m2)
Ao external wet area of the tube (m2)
Cf non-dimensional friction coef 1047297cient
Cfd form drag coef 1047297cient
Csf skin friction coef 1047297cient
Cp constant pressure speci1047297c heat (Jkg K)
Cpwater constant pressure speci1047297c heat for inlet water (Jkg K)
Cv constant volume speci1047297c heat (Jkg K)
d diameter (m)
dhy hydraulic diameter (m)
E crank mechanism effectiveness
Err error tolerance
Error1 absolute error calculated for Tc and Te
Error2 absolute error calculated for Tk and Th
Error3 absolute error calculated for Twk and Twh
f friction factor coef 1047297cient
freq engine frequency (Hz)
FR view factor
h convective heat transfer coef 1047297cient (Wm2 K)
hr radiation heat transfer coef 1047297cient (Wm2 K)hwater water 1047297lm heat transfer coef 1047297cient (Wm2 K)
k thermal conductivity (Wm K)
K piston to displacer swept volume ratio length (m)
m mass (kg)
n number of 1047298ow resistance layers
mwater mass 1047298ow of the inlet water (kgs)
M total mass of the working gas (kg)
NTU number of transfer units
P pressure level (Pa)
Pch engine charging pressure (bar)
Pbr engine brake power (W)
Q heat transfer rate (W)
Q hc heater heat transfer rate by cycle (Jcycle)
Q kc cooler heat transfer rate by cycle (Jcycle)Q rc regenerator heat transfer rate by cycle (Jcycle)
Q ht total heating requirement for the engine (W)
Q kt total cooling requirement for the engine (W)
Q lossr heat loss due to imperfect regenerator (W)
Q lk heat loss due to internal conduction (W)
Q lsh heat loss due to shuttle conduction (W)
R gas constant (Jkg K)
R ci conductive thermal resistance for tubes wall(KW)
R 1047297 fouling thermal resistance inside the tubes (KW)
R fo fouling thermal resistance outside the tubes (KW)
R hi convective thermal resistance inside the tubes (KW)
t time (s)
T temperature (K)
Tad adiabatic 1047298ame temperature of the fuel (K)TfM measured 1047298ame temperature (K)
Tratio cold to heat temperature ratio
Twi temperature at the internal wall of the tubes (K)
Two temperature at the outer wall of the tubes (K)
Twater_in inlet temperature of the water (K)
v mean velocity (ms)
V volume (m3)
V de total dead volume (m3)
V swe expansion space swept volume (m3)
V swc compression space swept volume(m3)
W work 1047298ow per cycle (Jcycle)
Wi engine indicated work (Jcycle)
Ws engine shaft work (Jcycle)
Wploss energy loss due to pressure drop (Jcycle)
W engine forced work (Jcycle)
X dead volume ratio
Acronyms
ACM Aspen Custom Modeller
CHP Combined Heat and Power
SE Stirling Engine
Subscripts
b buffer space
c compression space
d displacere expansion space
f 1047297nal value
h heater space
hous regenerator housing space
i inside section in
in let 1047298ow
k cooler space
M measured values
o outside section
out outlet 1047298ow
r regenerator space
w wall
whe heater wall
wk cooler wall0 initial value
Superscripts
thorn positive variation
negative variation
Greek symbols
a phase shift angle (rad)
as surface absorptivity
g adiabatic constant
hb brake ef 1047297ciency
hb mechanical ef 1047297ciency
hb thermal ef 1047297ciency
s Stefane
Boltzmann constant (Wm2
K4
) 3 regenerator effectiveness
r 1047298uid density (kgm3)
4 Crank rotational angle (rad)
m viscosity (kgm s)
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develop improved engines that should present higher ef 1047297ciency
levels fuel 1047298exibility and should also be easy to integrate within
combined heat and power systems (CHP) It is especially important
the mentioned integration capability because of the great potential
that combined heat and power systems presents as decentralized
solutions based on renewable energy [18] Some works that
explored this integrations include Paringlsson and Carlsen [19] Nish-
iyama [20] and Sato [21]
In this sense the objective of this paper is the development of a
thermodynamic-numerical model of a Stirling engine that should
represent the performance of a new 1 kW gamma engine prototype
built by GENOA Stirling Company in Italy This model aims to assess
through numerical simulation analysis the performance improve-ment of the GENOA engine prototype and it is centred on a second
order thermodynamic analysis implemented in Aspen Custom
Modellerreg The numerical model is based on Urielli approach [22]
it considers ideal adiabatic working spaces limited internal and
external heat transfer through the heat exchangers and mechan-
ical and thermal losses during the cycle In addition it includes the
numerical evaluation of the mechanical ef 1047297ciency taking into ac-
count the crank mechanism effectiveness and the forced work
during the cycle according to Senft methodology [23] Therefore
the model combines Urielli and Senft approaches into a restruc-
tured numerical analysis that computes the work that can be
effectively taken from the shaft The model was validated with data
obtained from an experimental rig built for the engine The details
about the methods used for the measurements are reported inCardozo et al [24]
2 Mathematical model
A mathematical model for the simulation of Stirling engine
systems was developed in a previous work [25] This consisted on
four main modules named ideal adiabatic internal heat transfer
external heat transfer and energy losses This paper improves the
model by adding the evaluation of the mechanical ef 1047297ciency of the
system thus the improved model contains 5 modules The 1047297rst
module corresponds to an ideal Stirling engine adiabatic model
which assumes ideal adiabatic compression and expansion spaces
to estimate the main engine variables The derivation of the
equations that govern this system are explained in Urielli [22] Theoutputs of this module are coupled to the internal heat transfer
module which through appropriate correlations evaluate the heat
transfer the temperature and the thermodynamic properties of the
working 1047298uid inside the heat exchangers The variation of the
thermodynamic properties with the temperature is considered at
every time step of the system The next module external heat
transfer module couples the heat transfer between the external
walls at the hot and cold side of the engine This is done through
energy balances and heat transfer correlations described in detail
in Araoz et al [25] The following module energy losses module
evaluates the losses due to pressure drop axial conduction shuttle
heat transfer and imperfect regeneration once the cyclic steady
state conditions were reached Finally the mechanical ef 1047297ciency
module permits to estimate the effect of forced work during the
cycle and the effect that the design for the crank mechanism have
on the performance of the engine
The main variables that connect the modules are described
below
- External heat transfer module This module considers the
adiabatic 1047298ame temperature and the inlet temperature of the
cooling 1047298uid on the hot and cold side respectively Therefore the
heat source (Q h) and the heat sink (Q k) are used to estimate the
wall temperatures (Twoh Twok) This approach is proposed to
couple the Stirling engine within the external heat and cooling
sources respectively
- Internal heat transfer module The internal working gas tem-
peratures (Th Tk) in the heater and cooler respectively are
calculated using heat transfer correlations for steady state in-
ternal forced convective 1047298ow [26] On the other hand the
regenerator analysis proposes the use of cyclic 1047298ow heat transfer
correlations which are more suitable for the 1047298ow conditions onthis space [27] Therefore with these correlations the effect of
limited heat transfer inside the engine is introduced in the
model
- Ideal adiabatic module The main operative variables such as
net shaft work (Ws) heat and cooling demands (Q h Q k) are
calculated considering the internal working 1047298uid temperature
distribution and the engine geometric characteristics following
Uriellis [22] approach
- Energy losses module The losses inside the engine are esti-
mated to correct the ideal adiabatic outputs This module con-
siders the losses due to pressure drop axial conduction shuttle
heat transfer and imperfect regeneration
- Mechanical ef 1047297ciency module The losses due to forced
compression and expansion are evaluated considering the
buffer pressure (Pb) the shape of the cycle and the crank
mechanism effectiveness (E)
The relationships between the modules are shown in Fig 1 The
loops represent the iterative calculationsto achieve the steady state
cyclic conditions The detailed report of the 1047297rst four modules can
be found in Araoz et al [25] and the detailed description of the new
mechanical ef 1047297ciency module is presented in the next section
21 Governing equations
The equations included in the model are based in the mass
energy balances and the equation of state for the working gas
These balances were applied to the control volumes shown in Fig 2
Fig 1 Block diagram for the Stirling model
Fig 2 Control volumes for Stirling engine based on Urielli [22]
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The mass balance is expressed as
min mout frac14 dm
d4(1)
The energy balance neglecting the energy kinetic terms
dQ
d4thorn cpinTinmin cpoutToutmout frac14
dW
d4thorn cv
dethmTTHORN
d4(2)
The equation of state for the gas in the control volume
PV frac14 mRT (3)
The balances were applied to each control volume to obtain a set
of algebraic differential equations This set was complemented with
correlations for the heat transfer in the heat exchangers and the
losses of the engine The details of the model development are
presented in Araoz [25] However a summary of the equations is
presented in Appendix B
22 Mechanical ef 1047297ciency and shaft work
The mechanical ef 1047297ciency of an engine measures the amount of the work produced by the thermodynamic cycle (indicated work
Wi) that can be effectively taken from the shaft shaft work (Ws)
[23]
hm frac14 Ws
Wi(4)
The mechanical ef 1047297ciency is evaluated with the fundamental
ef 1047297ciency theorem considering a constant mechanism effective-
ness (E) as developed by Senft [23]
hm frac14 E
1
E E
W
Wi(5)
where W represents the forced work This is the work that the
crank mechanism must deliver to the piston to make it move in
opposition to the pressure difference across it [23] For example
during the expansion process when the pressure of the gas inside
the working space is lower than the opposite buffer pressure then
the expansion process is forced In a similar way during the
compression process when the pressure inside the working space
is higher than the opposite buffer pressure then the compression is
forced Therefore this forced work depends mainly on the cycleshape and the buffer pressure level (Pb) and its calculated with the
following expression [23]
W frac14
I ethP PbTHORNthorndV thorn
I ethP PbTHORNdV thorn (6)
The superscripts difference the two types of forced work the
1047297rst one during the compression (dV ) when the buffer pressure is
below the working space pressure (P Pb)thorn and the second during
the expansion (dV thorn) when the buffer pressure is above the working
space pressure (P Pb)
The modi1047297ed model includes a numerical integration of Eq (6)
and the evaluation of both the mechanical ef 1047297ciency from Eq (5)
and the shaft work from Eq (4)
23 Brake thermal ef 1047297ciency
The overall ef 1047297ciency or brake thermal ef 1047297ciency is de1047297ned as
the ratio of the shaft work Ws and the net heat input of the engine
Q hc This can be calculated by the product of the thermal ef 1047297ciency
and the mechanical ef 1047297ciency as shown in Eq (7) The additional
module includes the estimation of the mechanical ef 1047297ciency and
the brake ef 1047297ciency
hb frac14 Ws
Q hcfrac14
Wi
Q hc
Ws
Wifrac14 hthm (7)
Fig 3 Genoa Stirling scheme
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3 Simulation of the Genoa engine
31 System description
The Genoa Stirling is a two cylinder gamma type engine built as
a prototype for research studies by GENOA Stirling SRL company
from Italy [28] According to its speci1047297cations it is capable to pro-
duce up to 1 kW electrical output with air as working 1047298uid at
600 rpm rotational speed and with the heater temperature around
750 C [28] The main components of the engine such as the
crankcase the crank mechanism with the balancing 1047298ywheel the
heat exchangers and the generator of the engine areshownin Fig 3Additional pictures for the heater cooler and regenerator heat
exchangers are shown in Fig 4
The gamma Stirling engine consists of two identical piston-
displacer cylinders connected to a common shaft under similar
operational conditions Therefore it is assumed that both cylinders
present similar thermodynamic cycles and consequently the dou-
ble cylinder thermodynamic analysis is simpli1047297ed to one cylinder
analysis The validity of the similarity on both cylinders is a com-
mon approach on Stirling simulation studies [1129e32] In addi-
tion the model assumes adiabatic expansion and compression
spaces and that the steady state cyclic conditions are reached
The Stirling engine was used in an experimental rig built at the
Energy department Royal Institute of Technology (KTH) Stock-
holm Sweden This rig consisted on the engine coupled to a pellet
Fig 4 Heat exchangers of the engine prototype
Table 1
Main parameters for the engine simulation
Parameter Value De1047297nition Description
freq 5 Hz Frequency of the engine
X 13353 V deV swe Dead volume ratio
K 03684 V swcV swe Piston to displacer swept volume ratio
Tratio 023 TadTwater_in Cold to heat temperature ratio
Pch 125 bar e Engine charging pressure
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Fig 6 Layout Stirling engine model in ACM
Table 2
Description of the blocks for the ACM model
Block name Description
Comp-Exp The block contains the data that describes the volume variation inside the engine The swept dead volumes crank mechanism and the
characteristics of the pistons
Cooler The block contains the geometrical data for the cooler heat exchanger
Heater T he b lock c ont ai ns t he geometr ic al data for t he heat exch an ger
Regenerator The block contains the geometrical data for the regenerator and the details of the matrix porosity and material
Ext-heat The characteristics of the external heat source are contained in this block
Mech_Ef 1047297ciency The block contains the parameters for the calculation of the engine mechanical ef 1047297ciency
CoolingFluid The characteristics of the external cooling 1047298uid are contained in the block
WorkingGAS The block contains the parameters for the calculation of the properties for the working gas inside the engine
Stirling This is the main block and contains the main equations that describe the thermodynamic analysis of the engine
Fig 7 a) Schematic set-up of the Stirling engine integrated with a combustion chamber and a boiler [27] b) Temperature measurement points for the working gas in the Stirling
engine T2 hot side T10 cold side T11 T12 hot and cold side of the regenerator [24]
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burner in order to produce heat and power simultaneously as
shown in Fig 7a This con1047297guration had technical limitations that
are still being studied in order to improve both power and thermal
outputs But despite of these limitations experimental results were
obtained and these were compared with the model
32 Inputs for the model
The main inputs for the engine simulation are shown in Table 1
Supplementary inputs that include the design and operational
characteristics of the engine are presented in Appendix A
The model also needs to consider the relation of the crank
mechanism and the variation of the volumes inside the working
spaces Therefore considering that the engine has gamma type
con1047297guration the following relations for the expansion and
compression spaces were included [23]
V e frac14 V cle thornV swe
2 eth1 thorn coseth4 thorn aTHORNTHORN (8)
V c frac14 V clc thorn ethV swe V eTHORN thornV swc
2 eth1 thorn coseth4THORNTHORN (9)
Furthermore the following volume derivatives were evaluated
dV e frac14 V swe
2 sineth4 thorn aTHORN (10)
dV c
frac14 dV e
V swc
2 sineth4THORN (11)
Fig 8 Measurement points for the CHP-Stirling experimental rig [24]
Table 3Comparison of the measured and predicted temperatures along the engine
Time (s) TfM (K) ThM (K) Th (K) Error TkM (K) Tk (K) Error TrM (K) Tr (K) Error
3780e3900 13878 8164 8184 025 3224 3211 041 6018 5316 116
3900e4020 13829 8196 8075 147 3218 3214 012 6006 5277 1215
4020e4140 13931 8232 8142 109 3216 3215 004 6012 5302 118
4140e4200 13778 8308 7981 394 3216 3216 001 6036 5243 1314
4200e4380 13835 8374 8063 371 3224 3214 031 6075 5272 1321
4380e4560 13777 8518 7957 659 3218 3217 003 6142 5234 1478
4560e4680 13857 8536 8071 545 3217 3215 007 6154 5276 1426
4680e4800 13844 8464 8021 523 3216 3217 001 6135 5258 143
4800e4980 13669 8433 7708 859 3221 3223 005 6129 5144 1607
Fig 9 Temperature variation along the heat exchangers and regenerator temperature assumed by the model (T r)
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33 Numerical solution
The system consists of a set of algebraic differential equations
which are shown in Appendix B These consider as boundary con-
ditions that the temperatures of the working gas at the end of the
cycle must be equal to the temperatures at the beginning of the
cycle once cyclic steady state conditions are reached Therefore an
iterative shooting method [33] using a fourth order Runge Kutta
scheme for the time discretization was implemented for the nu-
merical solution The iteration process was done until cyclic steadystate conditions which is numerically reached when the difference
between the assumed initial values and the values calculated at the
end of the cycle are lower than a de1047297ned error After the cyclic
steady state solution was reached the energy losses and the forced
work were evaluated The forced work was calculated using the
classical Simpson 38 numerical integration rule [34] The scheme
in Fig 5 summarizes the iterative steps for the solution
The numerical solution was implemented in Aspen Custom
Modellerreg (ACM) [35] which is a product from Aspen Plusreg that
permits the elaboration of customized models [36] This software
has its own modelling language and can also be coupled with Cthornthorn
procedures The layout of the model in ACM is shown in Fig 6 The
blocks were programmed with the equations shown in the
Appendix B and then the solution of the system was obtained withthe algorithm previously described
The descriptions of the blocks are shown in Table 2 Additional
details of the block inputs are given in Appendix A
4 Model validation
The geometrical and operational characteristics for the Genoa
engine are described in Table 1 and Appendix A The engine was
mounted in the experimental rig shown in Fig 7a In addition the
temperatures of the working gas were measured at the different
points of the engine shown in Fig 7b
The experimental rig used wood pellets as fuel Additional
temperatures measured for the validation were The temperature
close to the 1047298ame (T1) the water inlet temperature (T8) the wateroutlet temperature (T9) Other measurements are also as shown in
Fig 8
The temperature T1 was measured using a type K empty 15 mm
Inconel 600 thermocouple The additional temperatures shown in
Fig 8 were measured using type K empty 10 mm thermocouples
Considering the type of thermocouples the expanded uncertainty
was plusmn32 C with a coverage factor of 2 The speed of the engine
crankshaft was monitored by a pulse sensor and a frequency to
analog converter (OMROM E2A and Red Lion IFMA) with an un-
certaintyplusmn 02 The pressure inside the engine was measured with
a pressure transducer (RS type 46) with analog signal and an un-
certainty of plusmn01 bar All the measurements were recorded from the
beginning to the end of the test using a data logger Additional
details of the measurements are reported in Cardozo et al [24]The engine was run during long periods and the data was
measured constantly However for the validation purposes only the
periods were stability is reached were considered In this case the
steady state condition was dif 1047297cult to reach due to the constant
variation of the 1047298ame temperature [24] Therefore average values
for the measurements within certain stability periods were taken
These are compared with the values calculated by the model at the
different values measured for the 1047298ame temperature shown in
Table 3
Fig 10 Temperature variation along the engine 1047298
ame temperature Tad frac14
1388 K
Table 4
Measured and predicted brake power
Time (s) TfM (K) Measured frequency (Hz) Measured pressure (bar) Brake power (W) experimental Brake power (W) calculated Error
3780e3900 13878 517 1250 5472 5359 206
3900e4020 13829 526 1250 5539 5208 597
4020e4140 13930 527 1250 5561 5349 381
4140e4200 13778 533 1250 4635 5003 794
4200e4380 13835 528 1250 5359 5197 302
4380e
4560 13777 536 1250 5091 5033 1144560e4680 13857 529 1250 5096 5163 131
4680e4800 13843 534 1250 559 5153 782
4800e4980 13669 556 1254 4713 4613 212
Fig 11 Volumes variation during the engine cycle
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From Table 3 the model presents good accuracy for the pre-
diction of the cooler temperatures (Tk) with the maximum error of
the order of plusmn041 In addition the calculations for the heater
temperatures (Th) present reasonable accuracy at initial times but
then the error increases This growth may be explained with the
thermal inertia that constantly increments the measured temper-
ature even on periods where the1047298ame temperature decreases This
thermal inertia is neglected by the model since it assumes steady
state heat transfer conditions On the other hand the prediction of
the mean temperature in the regenerator space (Tr) presents
higher differences This is analysed with the Fig 9 below which
shows the variations of the temperatures inside the heat ex-
changers assumed by the model
From Fig 9 it can be seen that the model assumes that thetemperatures at the interfaces heater-regenerator and cooler-
regenerator were equal to the temperatures at the cooler (T k) and
heater (Th) spaces respectively Therefore the average temperature
at the regenerator (Tr) was calculated with these values This
assumption neglects the axial temperature variation along the
heater and cooler which is re1047298ected on the measurements taken at
the exact interfaces positions T11 and T12This explains the differ-
ence between the average regenerator temperature calculated with
the measured temperatures (TrM) and the calculated with the es-
timations of the model Tr as it is shown in Table 3 However
considering that the model was capable to calculate within a good
degree of accuracy the power output measured during the exper-
imental runs it can be inferred that the error for the regenerator
temperature estimation have little in1047298
uence on the brake powercalculation This is shown in Table 4 where the values for the
measured and calculated brake power are compared at different
operating conditions The percentage error ranges from plusmn131 to
plusmn794 which is an acceptable approximation for 1047297rst design
calculations
5 Results and discussion
This section presents additionally results for the simulation of
the engine under the experimental conditions described before
This aims to completely describe the thermodynamic performance
of the engine and thus identify the main limitations that the engine
presents
51 Temperature variation
Fig10 shows the temperature variation in the differentspaces of
the engine cylinder once the cyclic steady state conditions are
reached This 1047297gure displays the sinusoidal variation of the tem-
peratures inside the compression (Tc) and expansion (Te) spaces It
can also be seen that the expansion space presents periods with
elevated temperatures which results into a high thermal stress for
the material and therefore further engine deterioration In addition
the 1047297gure also shows that the mean temperatures for the working
1047298uid inside the heater (Th) and cooler (Tk) are close to the heat
exchangers walls temperature (Twk Twhe) This indicates a good
heat transfer rate on both heat exchangers and consequently a
good thermal performance based on the model assumptions
However it is important to notice that this performance will
decrease with the time due to the fouling on the heat exchangers
which is not accounted for in the engine model
52 Mass distribution and volumes variation
The mass distribution and volumes variation for the engine
during a complete cycle are shown in Figs 11 and 12 respectivelyThese variations permit to analyse the engine dynamics during the
compression and expansion processes
Fig 11 permits to identify the following processes the
compression characterized by the decrease in the total volume
from the time around t frac14 001 to t frac14 004 the heating process
when the total volume variation is not pronounced and the tem-
peratures increase around t frac14 004 to t frac14 006 the expansion
process when the total volume increases around t frac14 006 to
t frac14 009 and the cooling process when the volume stays almost
constant and the temperatures decrease at the times around
t frac14 009 to t frac14 010 and t frac14 0 to t frac14 001
The compressionperiod starts with the increment of the mass in
the compression space and a decrease of the mass in the expansion
space as shown in Fig 12 The decreasing mass in the expansionspace indicates a good dynamic for the compression process
because it is desirable to keep low the hotter portion of the mass
during this period However the mass on the compression space is
too high which is not desirable since this will be re1047298ected in a large
negative compression work In addition the expansion process also
presents a reduced performance due to the low values for the mass
in the expansion space during the expansion process This repre-
sents an expansion with low hotter mass and thus a low working
output to the shaft Furthermore the low mass in the expansion
space during the heating period might be the main cause for the
high temperatures reached Therefore the volumes and mass 1047298ow
dynamics of the reference case should be improved to reach higher
work outputs and avoid the overheating of the expansion chamber
Fig 12 also shows that the mass in the heater and cooler arereally small compared with the mass in the regenerator during the
Table 5
Engine work 1047298ow per cycle
Model output per cycle Aspen Custom Modeller (ACM)
Expansion Work (We Jcycle) 5262
Compression Work (Wc Jcycle) 2339
Pre ssure drop lost heater (J cycle) 021
Pre ssure drop lost cooler (J cycle) 007
Pressure drop lost regenerator (Jcycle) 028
Total lost due to pressure drop (Jcycle) 056
Net indicated work (Wi Jcycle) 2867
Forced work (W Jcycle) 2349
Brake Work Output (Wbr Jcycle) 518
Fig 12 Mass variation inside the engine spaces during a complete cycle
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complete cycle This re1047298ects the high importance of the regenerator
ef 1047297ciency for the engine performance
53 Work 1047298ow
Table 5 shows the simulation results for the compression and
expansion work during a single cycle This table also presents the
different work losses estimated for the system
The temperatures measured and the temperatures calculated
show a good thermal performance of the engine But the measured
brake power was very low Different problems on the engine
design and operational conditions may explain these very lowresults However additional experimental instrumentation is
needed for a detailed design study For this reason the present
analysis considers a theoretical approach that may be later com-
plemented with experimental studies This theoretical approach
considers Eq (7) From this equation and considering that the
thermal performance was found acceptable the main losses should
correspond to a low mechanical ef 1047297ciency of the prototype This
mechanical ef 1047297ciency is reduced by the presence of forced work
during the cycle and mechanical friction on the crank mechanism
Fig 13 presents the evaluation of the forced work in a pressure
volume diagram for the gas cycle inside the gamma prototype
From this it can be seen that the forced work (W) is mainly due to
the forced expansion process This means that at the experimentalconditions large part of the cyclic work may have been used to
complete the forced expansion process and thus the real engine
output is smaller than expected
The results discussed above are complemented with the vari-
ation of the compression (Wc) expansion (We) and net indicated
work (Wi) during the cycle shown in Fig 14
Fig 14 shows that during the 1047297rst part of the cycle from t frac14 0 to
t frac14 0045 the compression and expansion spaces present
exchanged roles This means that an increment of the volume is
presented in the compression space and a decrement of the volume
is present in the expansion one This reduced the engine perfor-
mance but it cannot be avoided since the gas needs to pass from
one space to another Regarding the second part of the cycle from
t frac14 0045 to t frac14 0095 the expansion and compression are shapedas expected and thus indicate a better dynamic during this period
However considering that large part of the expansion process is
forced the net brake work is low as reported in Table 5
From the previous analysis it can be concluded that a detailed
evaluation of the volumes dynamics the cranks mechanism effec-
tiveness and the forced work during the cycle must be considered
Fig 15 Heat 1047298
ow variation during the engine cycle
Table 6
Heat 1047298ow and heat loses during the cycle
Heat 1047298ow (Jcycle)
Heat exchanger space
Heater 1047298ow (Q hcJcycle) 5282
Cooler 1047298ow (Q kc Jcycle) 2356
Regenerator 1047298ow (Q rc Jcycle) 005
Heat lossesInternal conduction losses (Q lkc Jcycle) 2698
Shuttle conduction losses (Q lshc Jcycle) 8004
Regenerator losses during heating (Q lossrc Jcycle) 1862
Regenerator losses during cooling (Q lossrc Jcycle) 1862
Total heat requirements
Heating requirements (Q htc Jcycle) 17847
Cooling requirements (Q ktc Jcycle) 4218
Fig 14 Work 1047298
ow during the engine cycle
Fig 13 Pressureevolume diagram and forced work during the cycle
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in order to re-design the engine for a better performance This will
be covered on a detailed parametric study to be reported on a next
article
54 Heat 1047298ow
Table 6 presents the results for the heat 1047298ow and corresponding
heat losses through the heat exchangers calculated at the end of a
single cycle [25] As it can be seen the total heat requirements are
almost three times the requirements calculated without consid-
ering the losses It can also be seen that the shuttle conduction
losses represent the main heat loss during the cycle These corre-
spond to the losses due to the oscillation of the hot displacer across
the temperature gradient in the working spaces of the engine
The cyclic variation for the heat 1047298ow is additionally shown in
Fig 15The heat requirements for the heater and cooler present
slight variations during the entire cycle On the other hand the
regenerator presents high variations managing large quantities of
heat This con1047297rms the large importance of this heat exchanger on
the engine performance
55 Brake power and brake ef 1047297ciency
The engine brake power is de1047297ned as the net brake work per
cycle (Ws) times the engine frequency (freq)
Pbr frac14 Ws freq (12)
The net brake work and the total heat requirement presented
on Tables 5 and 6 respectively are doubled considering the double
cylinder engine These values are reported on Table 7 which alsoreports the thermal and brake ef 1047297ciencies for the engine
The results re1047298ect the low performance of the engine under the
experimental conditions This was mainly attributed to the forced
work and the mechanical ef 1047297ciency as it was analysed in the pre-
vious section In addition complementary works will broad this
analysis with the aim of propose improvements on the engine
design and operational parameters
6 Conclusions
In the present work a thermodynamic model for a Stirling en-
gine was improved by including the numerical evaluation of the
forced work and the mechanical ef 1047297ciency then validated against
experimental data and 1047297nally implemented for the simulation of an engine prototype The numerical model developed considered
the analytic approach proposed by Senft [23] but extended its
application for the case of the more realistic adiabatic working
spaces assumptions Consequently the effective work taken from
the shaft is better estimated and thus used for a more complete
analysis of the thermal and mechanical performance of an engine
For this article the analysis considered a novel gamma engine
prototype under the experimental conditions of a micro scale
combined heat and power system fuelled by wood pellets
The simulation results were compared with the experimental
data measured during long time runs of the system The model
performance was very good for the prediction of the temperatures
in the different spaces of the engine In addition the estimations for
the net brake power also presented results similar to the measured
values However additional experimental work should be per-
formed to obtain data to validate the calculation of the different
losses through the engine
According to the results obtained the thermal performance of
the engine was found acceptable and thus the low power output
measured is preliminary attributed to a reduced mechanical ef 1047297-
ciency of the system The possible reasons for this low performance
were further analysed with the different results for the tempera-
tures variation mass and volume variation pressure drops and the
pressure volume diagrams obtained with the model According to
these analyses the dynamics of the volumes variation and the
crank mechanism may also be improved in order to obtain higher
network during the cycle In addition it was found that the engine
performance is very sensitive to the effect of the buffer pressure
These results will be extended with a sensitivity analysis for the
system on a complementary work that aims to identify better the
effect of the different parameters on the engine performance
Acknowledgements
This work was possible thanks to the 1047297nancial support of the
Swedish International Development Cooperation Agency the di-
vision of Heat and Power Technology Department of Energy
Technology at Royal Institute of Technology (KTH) in Sweden and
Universidad Mayor de San Simon (UMSS) in Bolivia
Appendix A Detailed Stirling engine parameters
Table 7
Power output and ef 1047297ciency of the engine
Brake power (W) Heat requirement (W) Thermal ef 1047297ciency () Mechanical ef 1047297ciency () Brake ef 1047297ciency ()
5358 184535 1610 1810 290
Table A1
Inputs for the cooler in ACM
Variable Value Units Description
do 0005 m Tubes external diameter
di 0003 m Tubes internal diameter
kw 14200 Wm K Material conductivity
L 0032 m Tubes length
num 162 e Number of tubes
sl 0005 m Space between tubes
Table A2
Inputs for the heater in ACM
Variable Value Units Description
de 0005 M Tubes external diameter
di 00031 M Tubes internal diameterkw 142 Wm K Material conductivity
len 0149 m Tubes length
num 360 e Number of tubes
sl 0005 m Space between tubes
Table A3
Inputs for the regenerator in ACM
Variable Value Units Description
Din 0078 m Regenerator housing internal diameter
dout 0 107 m Regenera tor h ousin g extern al diameter
dwir e 2 1e-0 04 m Wi re di ameter of t he ma tri x
kwr 270 Wm K Thermal conductivity of the matrix material
Lr 007 m Length of the regenerator housing
Porosity 087 Matrix porosity
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Appendix B
Main equations for the Stirling engine model
Stirling engine module
Mean pressure
P frac14 MR
V cTc
thorn V kTk
thorn V rTr
thorn V hTh
thorn V eTe
Pressure variation
dP
d4frac14
gP
0BB
vV cv4
Tck
thorn
vV ev4
The
1CCA
V cTckthorn g
V kTk
thorn V rTrthorn V hTh
thorn V eThe
Mass of the working gas in the different spaces
mc frac14 p
V c
RTc
mk frac14 p
V k
RTk
mr frac14 p
V r
RTr
mh frac14 p
V h
RTh
me frac14 p
V e
RTe
Mass accumulation
dmk
d4frac14
mk
P
vP
v4
dmh
d4frac14
mh
P
vP
v4
dmr
d4frac14
mr
P
vP
v4
dmc
d4frac14
P
vV cv4
thorn
V c
vPv4
g
RTck
dme
d4frac14
P
vV ev4
thorn
V e
vPv4
g
RThe
Mass 1047298ow
mck frac14 dmc mhe frac14 dme mkr frac14 mck dmkmrh frac14 mhe thorn dmh
Conditional temperatures
If mck gt 0 then Tck frac14 Tc else Tck frac14 Tk
If mhe gt 0 then The frac14 Th else The frac14 Te
Temperatures
dTc
d4frac14 Tc
0BBvPv4
P
thorn
vV cv4
V c
vmc
v4
mc
1CCA
dTe
d4frac14 Te
0BBvPv4
P thorn
vV ev4
V e
vmev4
me
1CCAEnergy
dQ kd4
frac14
V k
vPv4
Cv
R CpethTckmck TkrmkrTHORN
dQ rd4
frac14
V r
vPv4
Cv
R C pethTkrmkr TrhmrhTHORN
dQ hd4
frac14V hvPv4
Cv
R CpethTrhmrh ThemheTHORN
dWc
d4frac14 P
vV cv4
dWe
d4frac14 P
vV ev4
Internal heat transfer module
Heat transfer from the heater wall to the working gas
Q h frac14 1
R cih thorn R hih thorn R fihethTwoh ThTHORN
Heat transfer from the cooler wall to the working gas
Q k frac14 1
R cik thorn R hik thorn R fikethTwik TkTHORN
Heat loss during the regenerator process
Q lossr frac14 eth1 3THORN Q r
Regenerator effectiveness
3frac14 NTU
1 thorn NTU
External heat transfer module
Heat transfer from the 1047298
ame to the external wall of the heater
Table A4
Inputs for the expansion-compression spaces and crank mechanism
Variable Value Units Description
vclc 44e-006 m3 Compression space clearance volume
vcle 26e-005 m3 Expansion space clearance volume
vswc 926e-005 m3 Compression space swept volume
vswe 25134e-004 m3 Expansion space swept volume
dispd 0062 m Displacer diameter
displ 007 m Displacer lengthef fmek 08 e Mechanism effectiveness
freq 5 Hz Frequency
jgap 0006 M Gap between cylinder displacer and wall
kpist 1627 Wm K Piston conductivity
pbuff 12ethorn006 Pa Buffer pressure
phase 900 deg Phase angle advance
pmean 1 25ethorn00 6 Pa Mean o pera tin g pressure
strk 0035 m Displacer stroke
dispd 0062 m Displacer diameter
Table A5
Working and cooling 1047298uid inputs in ACM
Variable Value Units Description
Working Fluid Air e Working 1047298uid inside the engine
Cooling FLUID Water e Cooling 1047298uid through the engine cooler
Tcooling 2 88 K Inl et tempera ture of th e co ol in g 1047298uid
Table A6
Fouling factors and external combustion inputs in ACM
Variable Value Units Description
T ad 1 38 7 K Fla me temper at ur e i n t he c omb ustio n ch amb er
absorp 070 e Absorptivity of the heater material
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Q h frac14 11
hrhAohthorn R foh
ethTad TwohTHORN
hrh frac14 assAohFR ethTad thorn TwohTHORN
T2ad thorn T2
woh
Estimation of the outlet temperature of the cooling 1047298uid
Twok frac14 Twater in thorn Q k
1
hokAokthorn
1
2mwaterCpwater
hok frac14 11
hwaterthorn R fok
Energy losses
Pressure drop in the heat exchangers
DP frac14
f
dhy
1
2 rv
2
l
Pressure drop in the regenerator based on the correlations of
Thomas and Pittman [37]
DP frac14 Cf nr
2u2
Cf frac14 Cfd thornCsf
Re
Csf frac14 68556 Cfd frac14 05274 wire screensCsf frac14 70035 Cfd frac14 09307 metal felts
Total pumping losses
Wploss frac14
Z 2p0
Xifrac143
ifrac141
DPi dV edq
$dq
Energy losses due to shuttle conduction
Q lsh frac14 04Z2KpistDd
JL dethTe TcTHORN
Mechanical ef 1047297ciency module
Mechanical ef 1047297ciency
hm frac14 Ws
Wi
Mechanical ef 1047297ciency considering the mechanism effectiveness
and forced work
hm frac14 E
1
E E
W
Wi
Forced work
W frac14
I ethP PbTHORNthorndV thorn
I ethP PbTHORNdV thorn
Brake ef 1047297
ciency
hb frac14 Ws
Q htfrac14
Wi
Q ht
Ws
Wifrac14 hthm
References
[1] DG Thombare SK Verma Technological development in the Stirling cycle
engines Renew Sustain Energy Rev 12 (2008) 1e
38 httpdxdoiorg101016jrser200607001
[2] AD Peacock M Newborough Impact of micro-CHP systems on domesticsector CO2 emissions Appl Therm Eng 25 (2005) 2653e2676
[3] B Thomas Benchmark testing of Micro-CHP units Appl Therm Eng 28(2008) 2049e2054 httpdxdoiorg101016japplthermaleng200803010
[4] L Dong H Liu S Riffat Development of small-scale and micro-scale biomass-fuelled CHP systems e a literature review Appl Therm Eng 29 (2009)2119e2126 httpdxdoiorg101016japplthermaleng200812004
[5] I Gonzalez-Pino A Campos-Celador E Perez-Iribarren J Teres-Zubiaga JM Sala Parametric study of the operational and economic feasibility of Stirling micro-cogeneration devices in Spain Appl Therm Eng (2013) httpdxdoiorg101016japplthermaleng201312020
[6] R Dyson S Wilson R Tew Review of computational Stirling analysismethods in 2nd International Energy Conversion Engineering ConferenceAmerican Institute of Aeronautics and Astronautics 2004 httpdxdoiorg10251462004-5582
[7] C-H Cheng Y-J Yu Numerical model for predicting thermodynamic cycleand thermal ef 1047297ciency of a beta-type Stirling engine with rhombic-drivemechanism Renew Energy 35 (2010) 2590e2601 httpdxdoiorg101016jrenene201004002
[8] N Seraj Mehdizadeh P Stouffs Simulation of a Martini displacer free pistonStirling engine for electric power generation Int J Thermodyn 3 (1) (2000)27e34 httpdxdoiorg105541ijot30
[9] N Parlak A Wagner M Elsner HS Soyhan Thermodynamic analysis of agamma type Stirling engine in non-ideal adiabatic conditions Renew Energy34 (2009) 266e273 httpdxdoiorg101016jrenene200802030
[10] JM Strauss RT Dobson Evaluation of a second order simulation for Sterlingengine design and optimisation J Energy South Afr 21 (2010) 17e29
[11] I Tlili Y Timoumi SB Nasrallah Thermodynamic analysis of the Stirling heatengine with regenerative losses and internal irreversibilities Int J Engine Res9 (2008) 45e56 httpdxdoiorg10124314680874JER01707
[12] K Mahkamov D Djumanov Three-dimensional CFD modeling of a Stirlingengine in Proceedings of the 11th international Stirling Engine Conferencevol 19 2003
[13] M Ibrahim R Tew Z Zhang D Gedeon T Simon CFD Modeling of Free-piston Stirling Engines Cleveland Ohio 2001
[14] S Wilson R Dyson Multi-D CFD modeling of a free-piston Stirling convertorat NASA GRC in Proc 2nd International Energy Conversion EngineeringConference vol 5673 2004
[15] E Eid Performance of a beta-con1047297guration heat engine having a regenerativedisplacer Renew Energy 34 (2009) 2404e2413 httpdxdoiorg101016
jrenene200903016[16] AA El-Ehwany GM Hennes EI Eid EA El-Kenany Development of the
performance of an alpha-type heat engine by using elbow-bend transposed-1047298uids heat exchanger as a heater and a cooler Energy Convers Manag 52(2011) 1010e1019 httpdxdoiorg101016jenconman201008029
[17] C-H Cheng Y-J Yu Dynamic simulation of a beta-type Stirling engine withcam-drive mechanism via the combination of the thermodynamic and dy-namic models Renew Energy 36 (2011) 714e725
[18] E Entchev J Gusdorf M Swinton M Bell F Szadkowski W Kalb1047298eisch et alMicro-generation technology assessment for housing technology EnergyBuild 36 (2004) 925e931 httpdxdoiorg101016jenbuild200403004
[19] M Paringlsson H Carlsen Development of a wood powder fuelled 35 kW StirlingCHP unit in Proceedings of the 11th ISEC (International Stirling EngineConference) 2003 pp 221e230
[20] a Nishiyama H Shimojima a Ishikawa Y Itaya S Kambara H Moritomi etal Fuel and emissions properties of Stirling engine operated with woodpowder Fuel 86 (2007) 2333e2342 httpdxdoiorg101016
jfuel200701040[21] K Sato N Ohiwa A Ishikawa H Shimojima A Nishiyama Y Moriya
Development of small-scale CHP Plant with a wood powder-fueled Stirlingengine J Power Energy Syst 2 (2008) 1221e1231 httpdxdoiorg101299
jpes21221[22] I Urieli DM Berchowitz Stirling Cycle Engine Analysis Hilger Ltd Bristol
1984[23] JR Senft Mechanical Ef 1047297ciency of Heat Engines Cambridge University Press
2007[24] E Cardozo C Erlich A Malmquist L Alejo Integration of a wood pellet
burner and a Stirling engine to produce residential heat and power ApplTherm Eng (2014) httpdxdoiorg101016japplthermaleng201408024
[25] JA Araoz M Salomon L Alejo TH Fransson Non-ideal Stirling enginethermodynamic model suitable for the integration into overall energy sys-tems Appl Therm Eng 73 (2014) 203e219 httpdxdoiorg101016
japplthermaleng201407050
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[26] FP Incropera AS Lavine DP DeWitt Fundamentals of Heat and MassTransfer John Wiley amp Sons 2011
[27] B Thomas D Pittman Update on the evaluation of different correlations forthe 1047298ow friction factor and heat transfer of Stirling engine regenerators inCollection of Technical Papers 35th Intersociety Energy Conversion Engi-neering Conference and Exhibit (IECEC) (Cat No00CH37022) American InstAeronaut amp Astronautics 2000 pp 76e84 httpdxdoiorg101109IECEC2000870632
[28] Genoastirling Srl 2014 wwwgenoastirlingcom[29] RC Tew K Jefferies D Miao USD of ED of TE Conservation LR Center
A Stirling Engine Computer Model for Performance Calculations (GoogleEBook) Department of Energy Of 1047297ce of Conservation and Solar ApplicationsDivision of Transportation Energy Conservation 1978
[30] I Urieli CJ Rallis DM Berchowitz Computer simulation of Stirling cyclemachines in 12th Intersociety Energy Conversion Engineering Conferencevol 1 American Nuclear Society Washington DC 1977 pp 1512e1521
[31] K Mahkamov Design improvements to a biomass Stirling engine usingmathematical analysis and 3D CFD modeling J Energy Resour Technol 128(2006) 203 httpdxdoiorg10111512213273
[32] H Snyman Second Order Analyses Methods for Stirling Engine Design Uni-versity of Stellenbosch Stellenbosch 2007
[33] G Sewell Numerical Solution of Ordinary and Partial Differential Equationssecond ed John Wiley amp Sons 2005
[34] DD Do Richard G Rice Applied Mathematics and Modeling for ChemicalEngineers [Hardcover] second ed Wiley-AIChE 2012
[35] Aspentech Aspen Custom Modelerreg AspenTech 2015
[36] Aspentech Chemical Process Optimization Software d
Chemical ProcessDesign Aspen Plus 2015
[37] B Thomas D Pittman AIAA-2000-2812 Update on the Evaluation of DifferentCorrelations for the Flow Friction Factor and Heat Transfer of Stirling EngineRegenerators 2000 pp 76e84
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computer 1047298uid dynamics (CFD) analysis that include the works of
Mahkamov [12] Ibrahim [13] and Wilson [14] Among these
methods1047297rstorder methods aresimple andlimited to estimate the
power output and engine ef 1047297ciency under ideal assumptions On
the other extreme CFD analyses are very complex and require
intensive computing resources [6] Therefore second order ana-
lyses have been preferred for 1047297rst design and optimization studies
of the engine considering a compromise between prediction
accuracy and computational requirements These second order
methods include the mass and energy balances through the
different spaces of the engine and also evaluate the friction and
thermal losses using a decoupled approach
Different studies have guided the development of Stirling en-
gines prototypes These include novel con1047297gurations in the
regenerator [15] the heat exchangers [16] the crank mechanism
[17] and optimization studies [10] However there is still a need to
Nomenclature
A area (m2)
Ao external wet area of the tube (m2)
Cf non-dimensional friction coef 1047297cient
Cfd form drag coef 1047297cient
Csf skin friction coef 1047297cient
Cp constant pressure speci1047297c heat (Jkg K)
Cpwater constant pressure speci1047297c heat for inlet water (Jkg K)
Cv constant volume speci1047297c heat (Jkg K)
d diameter (m)
dhy hydraulic diameter (m)
E crank mechanism effectiveness
Err error tolerance
Error1 absolute error calculated for Tc and Te
Error2 absolute error calculated for Tk and Th
Error3 absolute error calculated for Twk and Twh
f friction factor coef 1047297cient
freq engine frequency (Hz)
FR view factor
h convective heat transfer coef 1047297cient (Wm2 K)
hr radiation heat transfer coef 1047297cient (Wm2 K)hwater water 1047297lm heat transfer coef 1047297cient (Wm2 K)
k thermal conductivity (Wm K)
K piston to displacer swept volume ratio length (m)
m mass (kg)
n number of 1047298ow resistance layers
mwater mass 1047298ow of the inlet water (kgs)
M total mass of the working gas (kg)
NTU number of transfer units
P pressure level (Pa)
Pch engine charging pressure (bar)
Pbr engine brake power (W)
Q heat transfer rate (W)
Q hc heater heat transfer rate by cycle (Jcycle)
Q kc cooler heat transfer rate by cycle (Jcycle)Q rc regenerator heat transfer rate by cycle (Jcycle)
Q ht total heating requirement for the engine (W)
Q kt total cooling requirement for the engine (W)
Q lossr heat loss due to imperfect regenerator (W)
Q lk heat loss due to internal conduction (W)
Q lsh heat loss due to shuttle conduction (W)
R gas constant (Jkg K)
R ci conductive thermal resistance for tubes wall(KW)
R 1047297 fouling thermal resistance inside the tubes (KW)
R fo fouling thermal resistance outside the tubes (KW)
R hi convective thermal resistance inside the tubes (KW)
t time (s)
T temperature (K)
Tad adiabatic 1047298ame temperature of the fuel (K)TfM measured 1047298ame temperature (K)
Tratio cold to heat temperature ratio
Twi temperature at the internal wall of the tubes (K)
Two temperature at the outer wall of the tubes (K)
Twater_in inlet temperature of the water (K)
v mean velocity (ms)
V volume (m3)
V de total dead volume (m3)
V swe expansion space swept volume (m3)
V swc compression space swept volume(m3)
W work 1047298ow per cycle (Jcycle)
Wi engine indicated work (Jcycle)
Ws engine shaft work (Jcycle)
Wploss energy loss due to pressure drop (Jcycle)
W engine forced work (Jcycle)
X dead volume ratio
Acronyms
ACM Aspen Custom Modeller
CHP Combined Heat and Power
SE Stirling Engine
Subscripts
b buffer space
c compression space
d displacere expansion space
f 1047297nal value
h heater space
hous regenerator housing space
i inside section in
in let 1047298ow
k cooler space
M measured values
o outside section
out outlet 1047298ow
r regenerator space
w wall
whe heater wall
wk cooler wall0 initial value
Superscripts
thorn positive variation
negative variation
Greek symbols
a phase shift angle (rad)
as surface absorptivity
g adiabatic constant
hb brake ef 1047297ciency
hb mechanical ef 1047297ciency
hb thermal ef 1047297ciency
s Stefane
Boltzmann constant (Wm2
K4
) 3 regenerator effectiveness
r 1047298uid density (kgm3)
4 Crank rotational angle (rad)
m viscosity (kgm s)
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develop improved engines that should present higher ef 1047297ciency
levels fuel 1047298exibility and should also be easy to integrate within
combined heat and power systems (CHP) It is especially important
the mentioned integration capability because of the great potential
that combined heat and power systems presents as decentralized
solutions based on renewable energy [18] Some works that
explored this integrations include Paringlsson and Carlsen [19] Nish-
iyama [20] and Sato [21]
In this sense the objective of this paper is the development of a
thermodynamic-numerical model of a Stirling engine that should
represent the performance of a new 1 kW gamma engine prototype
built by GENOA Stirling Company in Italy This model aims to assess
through numerical simulation analysis the performance improve-ment of the GENOA engine prototype and it is centred on a second
order thermodynamic analysis implemented in Aspen Custom
Modellerreg The numerical model is based on Urielli approach [22]
it considers ideal adiabatic working spaces limited internal and
external heat transfer through the heat exchangers and mechan-
ical and thermal losses during the cycle In addition it includes the
numerical evaluation of the mechanical ef 1047297ciency taking into ac-
count the crank mechanism effectiveness and the forced work
during the cycle according to Senft methodology [23] Therefore
the model combines Urielli and Senft approaches into a restruc-
tured numerical analysis that computes the work that can be
effectively taken from the shaft The model was validated with data
obtained from an experimental rig built for the engine The details
about the methods used for the measurements are reported inCardozo et al [24]
2 Mathematical model
A mathematical model for the simulation of Stirling engine
systems was developed in a previous work [25] This consisted on
four main modules named ideal adiabatic internal heat transfer
external heat transfer and energy losses This paper improves the
model by adding the evaluation of the mechanical ef 1047297ciency of the
system thus the improved model contains 5 modules The 1047297rst
module corresponds to an ideal Stirling engine adiabatic model
which assumes ideal adiabatic compression and expansion spaces
to estimate the main engine variables The derivation of the
equations that govern this system are explained in Urielli [22] Theoutputs of this module are coupled to the internal heat transfer
module which through appropriate correlations evaluate the heat
transfer the temperature and the thermodynamic properties of the
working 1047298uid inside the heat exchangers The variation of the
thermodynamic properties with the temperature is considered at
every time step of the system The next module external heat
transfer module couples the heat transfer between the external
walls at the hot and cold side of the engine This is done through
energy balances and heat transfer correlations described in detail
in Araoz et al [25] The following module energy losses module
evaluates the losses due to pressure drop axial conduction shuttle
heat transfer and imperfect regeneration once the cyclic steady
state conditions were reached Finally the mechanical ef 1047297ciency
module permits to estimate the effect of forced work during the
cycle and the effect that the design for the crank mechanism have
on the performance of the engine
The main variables that connect the modules are described
below
- External heat transfer module This module considers the
adiabatic 1047298ame temperature and the inlet temperature of the
cooling 1047298uid on the hot and cold side respectively Therefore the
heat source (Q h) and the heat sink (Q k) are used to estimate the
wall temperatures (Twoh Twok) This approach is proposed to
couple the Stirling engine within the external heat and cooling
sources respectively
- Internal heat transfer module The internal working gas tem-
peratures (Th Tk) in the heater and cooler respectively are
calculated using heat transfer correlations for steady state in-
ternal forced convective 1047298ow [26] On the other hand the
regenerator analysis proposes the use of cyclic 1047298ow heat transfer
correlations which are more suitable for the 1047298ow conditions onthis space [27] Therefore with these correlations the effect of
limited heat transfer inside the engine is introduced in the
model
- Ideal adiabatic module The main operative variables such as
net shaft work (Ws) heat and cooling demands (Q h Q k) are
calculated considering the internal working 1047298uid temperature
distribution and the engine geometric characteristics following
Uriellis [22] approach
- Energy losses module The losses inside the engine are esti-
mated to correct the ideal adiabatic outputs This module con-
siders the losses due to pressure drop axial conduction shuttle
heat transfer and imperfect regeneration
- Mechanical ef 1047297ciency module The losses due to forced
compression and expansion are evaluated considering the
buffer pressure (Pb) the shape of the cycle and the crank
mechanism effectiveness (E)
The relationships between the modules are shown in Fig 1 The
loops represent the iterative calculationsto achieve the steady state
cyclic conditions The detailed report of the 1047297rst four modules can
be found in Araoz et al [25] and the detailed description of the new
mechanical ef 1047297ciency module is presented in the next section
21 Governing equations
The equations included in the model are based in the mass
energy balances and the equation of state for the working gas
These balances were applied to the control volumes shown in Fig 2
Fig 1 Block diagram for the Stirling model
Fig 2 Control volumes for Stirling engine based on Urielli [22]
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The mass balance is expressed as
min mout frac14 dm
d4(1)
The energy balance neglecting the energy kinetic terms
dQ
d4thorn cpinTinmin cpoutToutmout frac14
dW
d4thorn cv
dethmTTHORN
d4(2)
The equation of state for the gas in the control volume
PV frac14 mRT (3)
The balances were applied to each control volume to obtain a set
of algebraic differential equations This set was complemented with
correlations for the heat transfer in the heat exchangers and the
losses of the engine The details of the model development are
presented in Araoz [25] However a summary of the equations is
presented in Appendix B
22 Mechanical ef 1047297ciency and shaft work
The mechanical ef 1047297ciency of an engine measures the amount of the work produced by the thermodynamic cycle (indicated work
Wi) that can be effectively taken from the shaft shaft work (Ws)
[23]
hm frac14 Ws
Wi(4)
The mechanical ef 1047297ciency is evaluated with the fundamental
ef 1047297ciency theorem considering a constant mechanism effective-
ness (E) as developed by Senft [23]
hm frac14 E
1
E E
W
Wi(5)
where W represents the forced work This is the work that the
crank mechanism must deliver to the piston to make it move in
opposition to the pressure difference across it [23] For example
during the expansion process when the pressure of the gas inside
the working space is lower than the opposite buffer pressure then
the expansion process is forced In a similar way during the
compression process when the pressure inside the working space
is higher than the opposite buffer pressure then the compression is
forced Therefore this forced work depends mainly on the cycleshape and the buffer pressure level (Pb) and its calculated with the
following expression [23]
W frac14
I ethP PbTHORNthorndV thorn
I ethP PbTHORNdV thorn (6)
The superscripts difference the two types of forced work the
1047297rst one during the compression (dV ) when the buffer pressure is
below the working space pressure (P Pb)thorn and the second during
the expansion (dV thorn) when the buffer pressure is above the working
space pressure (P Pb)
The modi1047297ed model includes a numerical integration of Eq (6)
and the evaluation of both the mechanical ef 1047297ciency from Eq (5)
and the shaft work from Eq (4)
23 Brake thermal ef 1047297ciency
The overall ef 1047297ciency or brake thermal ef 1047297ciency is de1047297ned as
the ratio of the shaft work Ws and the net heat input of the engine
Q hc This can be calculated by the product of the thermal ef 1047297ciency
and the mechanical ef 1047297ciency as shown in Eq (7) The additional
module includes the estimation of the mechanical ef 1047297ciency and
the brake ef 1047297ciency
hb frac14 Ws
Q hcfrac14
Wi
Q hc
Ws
Wifrac14 hthm (7)
Fig 3 Genoa Stirling scheme
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3 Simulation of the Genoa engine
31 System description
The Genoa Stirling is a two cylinder gamma type engine built as
a prototype for research studies by GENOA Stirling SRL company
from Italy [28] According to its speci1047297cations it is capable to pro-
duce up to 1 kW electrical output with air as working 1047298uid at
600 rpm rotational speed and with the heater temperature around
750 C [28] The main components of the engine such as the
crankcase the crank mechanism with the balancing 1047298ywheel the
heat exchangers and the generator of the engine areshownin Fig 3Additional pictures for the heater cooler and regenerator heat
exchangers are shown in Fig 4
The gamma Stirling engine consists of two identical piston-
displacer cylinders connected to a common shaft under similar
operational conditions Therefore it is assumed that both cylinders
present similar thermodynamic cycles and consequently the dou-
ble cylinder thermodynamic analysis is simpli1047297ed to one cylinder
analysis The validity of the similarity on both cylinders is a com-
mon approach on Stirling simulation studies [1129e32] In addi-
tion the model assumes adiabatic expansion and compression
spaces and that the steady state cyclic conditions are reached
The Stirling engine was used in an experimental rig built at the
Energy department Royal Institute of Technology (KTH) Stock-
holm Sweden This rig consisted on the engine coupled to a pellet
Fig 4 Heat exchangers of the engine prototype
Table 1
Main parameters for the engine simulation
Parameter Value De1047297nition Description
freq 5 Hz Frequency of the engine
X 13353 V deV swe Dead volume ratio
K 03684 V swcV swe Piston to displacer swept volume ratio
Tratio 023 TadTwater_in Cold to heat temperature ratio
Pch 125 bar e Engine charging pressure
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Fig 6 Layout Stirling engine model in ACM
Table 2
Description of the blocks for the ACM model
Block name Description
Comp-Exp The block contains the data that describes the volume variation inside the engine The swept dead volumes crank mechanism and the
characteristics of the pistons
Cooler The block contains the geometrical data for the cooler heat exchanger
Heater T he b lock c ont ai ns t he geometr ic al data for t he heat exch an ger
Regenerator The block contains the geometrical data for the regenerator and the details of the matrix porosity and material
Ext-heat The characteristics of the external heat source are contained in this block
Mech_Ef 1047297ciency The block contains the parameters for the calculation of the engine mechanical ef 1047297ciency
CoolingFluid The characteristics of the external cooling 1047298uid are contained in the block
WorkingGAS The block contains the parameters for the calculation of the properties for the working gas inside the engine
Stirling This is the main block and contains the main equations that describe the thermodynamic analysis of the engine
Fig 7 a) Schematic set-up of the Stirling engine integrated with a combustion chamber and a boiler [27] b) Temperature measurement points for the working gas in the Stirling
engine T2 hot side T10 cold side T11 T12 hot and cold side of the regenerator [24]
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burner in order to produce heat and power simultaneously as
shown in Fig 7a This con1047297guration had technical limitations that
are still being studied in order to improve both power and thermal
outputs But despite of these limitations experimental results were
obtained and these were compared with the model
32 Inputs for the model
The main inputs for the engine simulation are shown in Table 1
Supplementary inputs that include the design and operational
characteristics of the engine are presented in Appendix A
The model also needs to consider the relation of the crank
mechanism and the variation of the volumes inside the working
spaces Therefore considering that the engine has gamma type
con1047297guration the following relations for the expansion and
compression spaces were included [23]
V e frac14 V cle thornV swe
2 eth1 thorn coseth4 thorn aTHORNTHORN (8)
V c frac14 V clc thorn ethV swe V eTHORN thornV swc
2 eth1 thorn coseth4THORNTHORN (9)
Furthermore the following volume derivatives were evaluated
dV e frac14 V swe
2 sineth4 thorn aTHORN (10)
dV c
frac14 dV e
V swc
2 sineth4THORN (11)
Fig 8 Measurement points for the CHP-Stirling experimental rig [24]
Table 3Comparison of the measured and predicted temperatures along the engine
Time (s) TfM (K) ThM (K) Th (K) Error TkM (K) Tk (K) Error TrM (K) Tr (K) Error
3780e3900 13878 8164 8184 025 3224 3211 041 6018 5316 116
3900e4020 13829 8196 8075 147 3218 3214 012 6006 5277 1215
4020e4140 13931 8232 8142 109 3216 3215 004 6012 5302 118
4140e4200 13778 8308 7981 394 3216 3216 001 6036 5243 1314
4200e4380 13835 8374 8063 371 3224 3214 031 6075 5272 1321
4380e4560 13777 8518 7957 659 3218 3217 003 6142 5234 1478
4560e4680 13857 8536 8071 545 3217 3215 007 6154 5276 1426
4680e4800 13844 8464 8021 523 3216 3217 001 6135 5258 143
4800e4980 13669 8433 7708 859 3221 3223 005 6129 5144 1607
Fig 9 Temperature variation along the heat exchangers and regenerator temperature assumed by the model (T r)
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33 Numerical solution
The system consists of a set of algebraic differential equations
which are shown in Appendix B These consider as boundary con-
ditions that the temperatures of the working gas at the end of the
cycle must be equal to the temperatures at the beginning of the
cycle once cyclic steady state conditions are reached Therefore an
iterative shooting method [33] using a fourth order Runge Kutta
scheme for the time discretization was implemented for the nu-
merical solution The iteration process was done until cyclic steadystate conditions which is numerically reached when the difference
between the assumed initial values and the values calculated at the
end of the cycle are lower than a de1047297ned error After the cyclic
steady state solution was reached the energy losses and the forced
work were evaluated The forced work was calculated using the
classical Simpson 38 numerical integration rule [34] The scheme
in Fig 5 summarizes the iterative steps for the solution
The numerical solution was implemented in Aspen Custom
Modellerreg (ACM) [35] which is a product from Aspen Plusreg that
permits the elaboration of customized models [36] This software
has its own modelling language and can also be coupled with Cthornthorn
procedures The layout of the model in ACM is shown in Fig 6 The
blocks were programmed with the equations shown in the
Appendix B and then the solution of the system was obtained withthe algorithm previously described
The descriptions of the blocks are shown in Table 2 Additional
details of the block inputs are given in Appendix A
4 Model validation
The geometrical and operational characteristics for the Genoa
engine are described in Table 1 and Appendix A The engine was
mounted in the experimental rig shown in Fig 7a In addition the
temperatures of the working gas were measured at the different
points of the engine shown in Fig 7b
The experimental rig used wood pellets as fuel Additional
temperatures measured for the validation were The temperature
close to the 1047298ame (T1) the water inlet temperature (T8) the wateroutlet temperature (T9) Other measurements are also as shown in
Fig 8
The temperature T1 was measured using a type K empty 15 mm
Inconel 600 thermocouple The additional temperatures shown in
Fig 8 were measured using type K empty 10 mm thermocouples
Considering the type of thermocouples the expanded uncertainty
was plusmn32 C with a coverage factor of 2 The speed of the engine
crankshaft was monitored by a pulse sensor and a frequency to
analog converter (OMROM E2A and Red Lion IFMA) with an un-
certaintyplusmn 02 The pressure inside the engine was measured with
a pressure transducer (RS type 46) with analog signal and an un-
certainty of plusmn01 bar All the measurements were recorded from the
beginning to the end of the test using a data logger Additional
details of the measurements are reported in Cardozo et al [24]The engine was run during long periods and the data was
measured constantly However for the validation purposes only the
periods were stability is reached were considered In this case the
steady state condition was dif 1047297cult to reach due to the constant
variation of the 1047298ame temperature [24] Therefore average values
for the measurements within certain stability periods were taken
These are compared with the values calculated by the model at the
different values measured for the 1047298ame temperature shown in
Table 3
Fig 10 Temperature variation along the engine 1047298
ame temperature Tad frac14
1388 K
Table 4
Measured and predicted brake power
Time (s) TfM (K) Measured frequency (Hz) Measured pressure (bar) Brake power (W) experimental Brake power (W) calculated Error
3780e3900 13878 517 1250 5472 5359 206
3900e4020 13829 526 1250 5539 5208 597
4020e4140 13930 527 1250 5561 5349 381
4140e4200 13778 533 1250 4635 5003 794
4200e4380 13835 528 1250 5359 5197 302
4380e
4560 13777 536 1250 5091 5033 1144560e4680 13857 529 1250 5096 5163 131
4680e4800 13843 534 1250 559 5153 782
4800e4980 13669 556 1254 4713 4613 212
Fig 11 Volumes variation during the engine cycle
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From Table 3 the model presents good accuracy for the pre-
diction of the cooler temperatures (Tk) with the maximum error of
the order of plusmn041 In addition the calculations for the heater
temperatures (Th) present reasonable accuracy at initial times but
then the error increases This growth may be explained with the
thermal inertia that constantly increments the measured temper-
ature even on periods where the1047298ame temperature decreases This
thermal inertia is neglected by the model since it assumes steady
state heat transfer conditions On the other hand the prediction of
the mean temperature in the regenerator space (Tr) presents
higher differences This is analysed with the Fig 9 below which
shows the variations of the temperatures inside the heat ex-
changers assumed by the model
From Fig 9 it can be seen that the model assumes that thetemperatures at the interfaces heater-regenerator and cooler-
regenerator were equal to the temperatures at the cooler (T k) and
heater (Th) spaces respectively Therefore the average temperature
at the regenerator (Tr) was calculated with these values This
assumption neglects the axial temperature variation along the
heater and cooler which is re1047298ected on the measurements taken at
the exact interfaces positions T11 and T12This explains the differ-
ence between the average regenerator temperature calculated with
the measured temperatures (TrM) and the calculated with the es-
timations of the model Tr as it is shown in Table 3 However
considering that the model was capable to calculate within a good
degree of accuracy the power output measured during the exper-
imental runs it can be inferred that the error for the regenerator
temperature estimation have little in1047298
uence on the brake powercalculation This is shown in Table 4 where the values for the
measured and calculated brake power are compared at different
operating conditions The percentage error ranges from plusmn131 to
plusmn794 which is an acceptable approximation for 1047297rst design
calculations
5 Results and discussion
This section presents additionally results for the simulation of
the engine under the experimental conditions described before
This aims to completely describe the thermodynamic performance
of the engine and thus identify the main limitations that the engine
presents
51 Temperature variation
Fig10 shows the temperature variation in the differentspaces of
the engine cylinder once the cyclic steady state conditions are
reached This 1047297gure displays the sinusoidal variation of the tem-
peratures inside the compression (Tc) and expansion (Te) spaces It
can also be seen that the expansion space presents periods with
elevated temperatures which results into a high thermal stress for
the material and therefore further engine deterioration In addition
the 1047297gure also shows that the mean temperatures for the working
1047298uid inside the heater (Th) and cooler (Tk) are close to the heat
exchangers walls temperature (Twk Twhe) This indicates a good
heat transfer rate on both heat exchangers and consequently a
good thermal performance based on the model assumptions
However it is important to notice that this performance will
decrease with the time due to the fouling on the heat exchangers
which is not accounted for in the engine model
52 Mass distribution and volumes variation
The mass distribution and volumes variation for the engine
during a complete cycle are shown in Figs 11 and 12 respectivelyThese variations permit to analyse the engine dynamics during the
compression and expansion processes
Fig 11 permits to identify the following processes the
compression characterized by the decrease in the total volume
from the time around t frac14 001 to t frac14 004 the heating process
when the total volume variation is not pronounced and the tem-
peratures increase around t frac14 004 to t frac14 006 the expansion
process when the total volume increases around t frac14 006 to
t frac14 009 and the cooling process when the volume stays almost
constant and the temperatures decrease at the times around
t frac14 009 to t frac14 010 and t frac14 0 to t frac14 001
The compressionperiod starts with the increment of the mass in
the compression space and a decrease of the mass in the expansion
space as shown in Fig 12 The decreasing mass in the expansionspace indicates a good dynamic for the compression process
because it is desirable to keep low the hotter portion of the mass
during this period However the mass on the compression space is
too high which is not desirable since this will be re1047298ected in a large
negative compression work In addition the expansion process also
presents a reduced performance due to the low values for the mass
in the expansion space during the expansion process This repre-
sents an expansion with low hotter mass and thus a low working
output to the shaft Furthermore the low mass in the expansion
space during the heating period might be the main cause for the
high temperatures reached Therefore the volumes and mass 1047298ow
dynamics of the reference case should be improved to reach higher
work outputs and avoid the overheating of the expansion chamber
Fig 12 also shows that the mass in the heater and cooler arereally small compared with the mass in the regenerator during the
Table 5
Engine work 1047298ow per cycle
Model output per cycle Aspen Custom Modeller (ACM)
Expansion Work (We Jcycle) 5262
Compression Work (Wc Jcycle) 2339
Pre ssure drop lost heater (J cycle) 021
Pre ssure drop lost cooler (J cycle) 007
Pressure drop lost regenerator (Jcycle) 028
Total lost due to pressure drop (Jcycle) 056
Net indicated work (Wi Jcycle) 2867
Forced work (W Jcycle) 2349
Brake Work Output (Wbr Jcycle) 518
Fig 12 Mass variation inside the engine spaces during a complete cycle
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complete cycle This re1047298ects the high importance of the regenerator
ef 1047297ciency for the engine performance
53 Work 1047298ow
Table 5 shows the simulation results for the compression and
expansion work during a single cycle This table also presents the
different work losses estimated for the system
The temperatures measured and the temperatures calculated
show a good thermal performance of the engine But the measured
brake power was very low Different problems on the engine
design and operational conditions may explain these very lowresults However additional experimental instrumentation is
needed for a detailed design study For this reason the present
analysis considers a theoretical approach that may be later com-
plemented with experimental studies This theoretical approach
considers Eq (7) From this equation and considering that the
thermal performance was found acceptable the main losses should
correspond to a low mechanical ef 1047297ciency of the prototype This
mechanical ef 1047297ciency is reduced by the presence of forced work
during the cycle and mechanical friction on the crank mechanism
Fig 13 presents the evaluation of the forced work in a pressure
volume diagram for the gas cycle inside the gamma prototype
From this it can be seen that the forced work (W) is mainly due to
the forced expansion process This means that at the experimentalconditions large part of the cyclic work may have been used to
complete the forced expansion process and thus the real engine
output is smaller than expected
The results discussed above are complemented with the vari-
ation of the compression (Wc) expansion (We) and net indicated
work (Wi) during the cycle shown in Fig 14
Fig 14 shows that during the 1047297rst part of the cycle from t frac14 0 to
t frac14 0045 the compression and expansion spaces present
exchanged roles This means that an increment of the volume is
presented in the compression space and a decrement of the volume
is present in the expansion one This reduced the engine perfor-
mance but it cannot be avoided since the gas needs to pass from
one space to another Regarding the second part of the cycle from
t frac14 0045 to t frac14 0095 the expansion and compression are shapedas expected and thus indicate a better dynamic during this period
However considering that large part of the expansion process is
forced the net brake work is low as reported in Table 5
From the previous analysis it can be concluded that a detailed
evaluation of the volumes dynamics the cranks mechanism effec-
tiveness and the forced work during the cycle must be considered
Fig 15 Heat 1047298
ow variation during the engine cycle
Table 6
Heat 1047298ow and heat loses during the cycle
Heat 1047298ow (Jcycle)
Heat exchanger space
Heater 1047298ow (Q hcJcycle) 5282
Cooler 1047298ow (Q kc Jcycle) 2356
Regenerator 1047298ow (Q rc Jcycle) 005
Heat lossesInternal conduction losses (Q lkc Jcycle) 2698
Shuttle conduction losses (Q lshc Jcycle) 8004
Regenerator losses during heating (Q lossrc Jcycle) 1862
Regenerator losses during cooling (Q lossrc Jcycle) 1862
Total heat requirements
Heating requirements (Q htc Jcycle) 17847
Cooling requirements (Q ktc Jcycle) 4218
Fig 14 Work 1047298
ow during the engine cycle
Fig 13 Pressureevolume diagram and forced work during the cycle
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in order to re-design the engine for a better performance This will
be covered on a detailed parametric study to be reported on a next
article
54 Heat 1047298ow
Table 6 presents the results for the heat 1047298ow and corresponding
heat losses through the heat exchangers calculated at the end of a
single cycle [25] As it can be seen the total heat requirements are
almost three times the requirements calculated without consid-
ering the losses It can also be seen that the shuttle conduction
losses represent the main heat loss during the cycle These corre-
spond to the losses due to the oscillation of the hot displacer across
the temperature gradient in the working spaces of the engine
The cyclic variation for the heat 1047298ow is additionally shown in
Fig 15The heat requirements for the heater and cooler present
slight variations during the entire cycle On the other hand the
regenerator presents high variations managing large quantities of
heat This con1047297rms the large importance of this heat exchanger on
the engine performance
55 Brake power and brake ef 1047297ciency
The engine brake power is de1047297ned as the net brake work per
cycle (Ws) times the engine frequency (freq)
Pbr frac14 Ws freq (12)
The net brake work and the total heat requirement presented
on Tables 5 and 6 respectively are doubled considering the double
cylinder engine These values are reported on Table 7 which alsoreports the thermal and brake ef 1047297ciencies for the engine
The results re1047298ect the low performance of the engine under the
experimental conditions This was mainly attributed to the forced
work and the mechanical ef 1047297ciency as it was analysed in the pre-
vious section In addition complementary works will broad this
analysis with the aim of propose improvements on the engine
design and operational parameters
6 Conclusions
In the present work a thermodynamic model for a Stirling en-
gine was improved by including the numerical evaluation of the
forced work and the mechanical ef 1047297ciency then validated against
experimental data and 1047297nally implemented for the simulation of an engine prototype The numerical model developed considered
the analytic approach proposed by Senft [23] but extended its
application for the case of the more realistic adiabatic working
spaces assumptions Consequently the effective work taken from
the shaft is better estimated and thus used for a more complete
analysis of the thermal and mechanical performance of an engine
For this article the analysis considered a novel gamma engine
prototype under the experimental conditions of a micro scale
combined heat and power system fuelled by wood pellets
The simulation results were compared with the experimental
data measured during long time runs of the system The model
performance was very good for the prediction of the temperatures
in the different spaces of the engine In addition the estimations for
the net brake power also presented results similar to the measured
values However additional experimental work should be per-
formed to obtain data to validate the calculation of the different
losses through the engine
According to the results obtained the thermal performance of
the engine was found acceptable and thus the low power output
measured is preliminary attributed to a reduced mechanical ef 1047297-
ciency of the system The possible reasons for this low performance
were further analysed with the different results for the tempera-
tures variation mass and volume variation pressure drops and the
pressure volume diagrams obtained with the model According to
these analyses the dynamics of the volumes variation and the
crank mechanism may also be improved in order to obtain higher
network during the cycle In addition it was found that the engine
performance is very sensitive to the effect of the buffer pressure
These results will be extended with a sensitivity analysis for the
system on a complementary work that aims to identify better the
effect of the different parameters on the engine performance
Acknowledgements
This work was possible thanks to the 1047297nancial support of the
Swedish International Development Cooperation Agency the di-
vision of Heat and Power Technology Department of Energy
Technology at Royal Institute of Technology (KTH) in Sweden and
Universidad Mayor de San Simon (UMSS) in Bolivia
Appendix A Detailed Stirling engine parameters
Table 7
Power output and ef 1047297ciency of the engine
Brake power (W) Heat requirement (W) Thermal ef 1047297ciency () Mechanical ef 1047297ciency () Brake ef 1047297ciency ()
5358 184535 1610 1810 290
Table A1
Inputs for the cooler in ACM
Variable Value Units Description
do 0005 m Tubes external diameter
di 0003 m Tubes internal diameter
kw 14200 Wm K Material conductivity
L 0032 m Tubes length
num 162 e Number of tubes
sl 0005 m Space between tubes
Table A2
Inputs for the heater in ACM
Variable Value Units Description
de 0005 M Tubes external diameter
di 00031 M Tubes internal diameterkw 142 Wm K Material conductivity
len 0149 m Tubes length
num 360 e Number of tubes
sl 0005 m Space between tubes
Table A3
Inputs for the regenerator in ACM
Variable Value Units Description
Din 0078 m Regenerator housing internal diameter
dout 0 107 m Regenera tor h ousin g extern al diameter
dwir e 2 1e-0 04 m Wi re di ameter of t he ma tri x
kwr 270 Wm K Thermal conductivity of the matrix material
Lr 007 m Length of the regenerator housing
Porosity 087 Matrix porosity
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Appendix B
Main equations for the Stirling engine model
Stirling engine module
Mean pressure
P frac14 MR
V cTc
thorn V kTk
thorn V rTr
thorn V hTh
thorn V eTe
Pressure variation
dP
d4frac14
gP
0BB
vV cv4
Tck
thorn
vV ev4
The
1CCA
V cTckthorn g
V kTk
thorn V rTrthorn V hTh
thorn V eThe
Mass of the working gas in the different spaces
mc frac14 p
V c
RTc
mk frac14 p
V k
RTk
mr frac14 p
V r
RTr
mh frac14 p
V h
RTh
me frac14 p
V e
RTe
Mass accumulation
dmk
d4frac14
mk
P
vP
v4
dmh
d4frac14
mh
P
vP
v4
dmr
d4frac14
mr
P
vP
v4
dmc
d4frac14
P
vV cv4
thorn
V c
vPv4
g
RTck
dme
d4frac14
P
vV ev4
thorn
V e
vPv4
g
RThe
Mass 1047298ow
mck frac14 dmc mhe frac14 dme mkr frac14 mck dmkmrh frac14 mhe thorn dmh
Conditional temperatures
If mck gt 0 then Tck frac14 Tc else Tck frac14 Tk
If mhe gt 0 then The frac14 Th else The frac14 Te
Temperatures
dTc
d4frac14 Tc
0BBvPv4
P
thorn
vV cv4
V c
vmc
v4
mc
1CCA
dTe
d4frac14 Te
0BBvPv4
P thorn
vV ev4
V e
vmev4
me
1CCAEnergy
dQ kd4
frac14
V k
vPv4
Cv
R CpethTckmck TkrmkrTHORN
dQ rd4
frac14
V r
vPv4
Cv
R C pethTkrmkr TrhmrhTHORN
dQ hd4
frac14V hvPv4
Cv
R CpethTrhmrh ThemheTHORN
dWc
d4frac14 P
vV cv4
dWe
d4frac14 P
vV ev4
Internal heat transfer module
Heat transfer from the heater wall to the working gas
Q h frac14 1
R cih thorn R hih thorn R fihethTwoh ThTHORN
Heat transfer from the cooler wall to the working gas
Q k frac14 1
R cik thorn R hik thorn R fikethTwik TkTHORN
Heat loss during the regenerator process
Q lossr frac14 eth1 3THORN Q r
Regenerator effectiveness
3frac14 NTU
1 thorn NTU
External heat transfer module
Heat transfer from the 1047298
ame to the external wall of the heater
Table A4
Inputs for the expansion-compression spaces and crank mechanism
Variable Value Units Description
vclc 44e-006 m3 Compression space clearance volume
vcle 26e-005 m3 Expansion space clearance volume
vswc 926e-005 m3 Compression space swept volume
vswe 25134e-004 m3 Expansion space swept volume
dispd 0062 m Displacer diameter
displ 007 m Displacer lengthef fmek 08 e Mechanism effectiveness
freq 5 Hz Frequency
jgap 0006 M Gap between cylinder displacer and wall
kpist 1627 Wm K Piston conductivity
pbuff 12ethorn006 Pa Buffer pressure
phase 900 deg Phase angle advance
pmean 1 25ethorn00 6 Pa Mean o pera tin g pressure
strk 0035 m Displacer stroke
dispd 0062 m Displacer diameter
Table A5
Working and cooling 1047298uid inputs in ACM
Variable Value Units Description
Working Fluid Air e Working 1047298uid inside the engine
Cooling FLUID Water e Cooling 1047298uid through the engine cooler
Tcooling 2 88 K Inl et tempera ture of th e co ol in g 1047298uid
Table A6
Fouling factors and external combustion inputs in ACM
Variable Value Units Description
T ad 1 38 7 K Fla me temper at ur e i n t he c omb ustio n ch amb er
absorp 070 e Absorptivity of the heater material
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Q h frac14 11
hrhAohthorn R foh
ethTad TwohTHORN
hrh frac14 assAohFR ethTad thorn TwohTHORN
T2ad thorn T2
woh
Estimation of the outlet temperature of the cooling 1047298uid
Twok frac14 Twater in thorn Q k
1
hokAokthorn
1
2mwaterCpwater
hok frac14 11
hwaterthorn R fok
Energy losses
Pressure drop in the heat exchangers
DP frac14
f
dhy
1
2 rv
2
l
Pressure drop in the regenerator based on the correlations of
Thomas and Pittman [37]
DP frac14 Cf nr
2u2
Cf frac14 Cfd thornCsf
Re
Csf frac14 68556 Cfd frac14 05274 wire screensCsf frac14 70035 Cfd frac14 09307 metal felts
Total pumping losses
Wploss frac14
Z 2p0
Xifrac143
ifrac141
DPi dV edq
$dq
Energy losses due to shuttle conduction
Q lsh frac14 04Z2KpistDd
JL dethTe TcTHORN
Mechanical ef 1047297ciency module
Mechanical ef 1047297ciency
hm frac14 Ws
Wi
Mechanical ef 1047297ciency considering the mechanism effectiveness
and forced work
hm frac14 E
1
E E
W
Wi
Forced work
W frac14
I ethP PbTHORNthorndV thorn
I ethP PbTHORNdV thorn
Brake ef 1047297
ciency
hb frac14 Ws
Q htfrac14
Wi
Q ht
Ws
Wifrac14 hthm
References
[1] DG Thombare SK Verma Technological development in the Stirling cycle
engines Renew Sustain Energy Rev 12 (2008) 1e
38 httpdxdoiorg101016jrser200607001
[2] AD Peacock M Newborough Impact of micro-CHP systems on domesticsector CO2 emissions Appl Therm Eng 25 (2005) 2653e2676
[3] B Thomas Benchmark testing of Micro-CHP units Appl Therm Eng 28(2008) 2049e2054 httpdxdoiorg101016japplthermaleng200803010
[4] L Dong H Liu S Riffat Development of small-scale and micro-scale biomass-fuelled CHP systems e a literature review Appl Therm Eng 29 (2009)2119e2126 httpdxdoiorg101016japplthermaleng200812004
[5] I Gonzalez-Pino A Campos-Celador E Perez-Iribarren J Teres-Zubiaga JM Sala Parametric study of the operational and economic feasibility of Stirling micro-cogeneration devices in Spain Appl Therm Eng (2013) httpdxdoiorg101016japplthermaleng201312020
[6] R Dyson S Wilson R Tew Review of computational Stirling analysismethods in 2nd International Energy Conversion Engineering ConferenceAmerican Institute of Aeronautics and Astronautics 2004 httpdxdoiorg10251462004-5582
[7] C-H Cheng Y-J Yu Numerical model for predicting thermodynamic cycleand thermal ef 1047297ciency of a beta-type Stirling engine with rhombic-drivemechanism Renew Energy 35 (2010) 2590e2601 httpdxdoiorg101016jrenene201004002
[8] N Seraj Mehdizadeh P Stouffs Simulation of a Martini displacer free pistonStirling engine for electric power generation Int J Thermodyn 3 (1) (2000)27e34 httpdxdoiorg105541ijot30
[9] N Parlak A Wagner M Elsner HS Soyhan Thermodynamic analysis of agamma type Stirling engine in non-ideal adiabatic conditions Renew Energy34 (2009) 266e273 httpdxdoiorg101016jrenene200802030
[10] JM Strauss RT Dobson Evaluation of a second order simulation for Sterlingengine design and optimisation J Energy South Afr 21 (2010) 17e29
[11] I Tlili Y Timoumi SB Nasrallah Thermodynamic analysis of the Stirling heatengine with regenerative losses and internal irreversibilities Int J Engine Res9 (2008) 45e56 httpdxdoiorg10124314680874JER01707
[12] K Mahkamov D Djumanov Three-dimensional CFD modeling of a Stirlingengine in Proceedings of the 11th international Stirling Engine Conferencevol 19 2003
[13] M Ibrahim R Tew Z Zhang D Gedeon T Simon CFD Modeling of Free-piston Stirling Engines Cleveland Ohio 2001
[14] S Wilson R Dyson Multi-D CFD modeling of a free-piston Stirling convertorat NASA GRC in Proc 2nd International Energy Conversion EngineeringConference vol 5673 2004
[15] E Eid Performance of a beta-con1047297guration heat engine having a regenerativedisplacer Renew Energy 34 (2009) 2404e2413 httpdxdoiorg101016
jrenene200903016[16] AA El-Ehwany GM Hennes EI Eid EA El-Kenany Development of the
performance of an alpha-type heat engine by using elbow-bend transposed-1047298uids heat exchanger as a heater and a cooler Energy Convers Manag 52(2011) 1010e1019 httpdxdoiorg101016jenconman201008029
[17] C-H Cheng Y-J Yu Dynamic simulation of a beta-type Stirling engine withcam-drive mechanism via the combination of the thermodynamic and dy-namic models Renew Energy 36 (2011) 714e725
[18] E Entchev J Gusdorf M Swinton M Bell F Szadkowski W Kalb1047298eisch et alMicro-generation technology assessment for housing technology EnergyBuild 36 (2004) 925e931 httpdxdoiorg101016jenbuild200403004
[19] M Paringlsson H Carlsen Development of a wood powder fuelled 35 kW StirlingCHP unit in Proceedings of the 11th ISEC (International Stirling EngineConference) 2003 pp 221e230
[20] a Nishiyama H Shimojima a Ishikawa Y Itaya S Kambara H Moritomi etal Fuel and emissions properties of Stirling engine operated with woodpowder Fuel 86 (2007) 2333e2342 httpdxdoiorg101016
jfuel200701040[21] K Sato N Ohiwa A Ishikawa H Shimojima A Nishiyama Y Moriya
Development of small-scale CHP Plant with a wood powder-fueled Stirlingengine J Power Energy Syst 2 (2008) 1221e1231 httpdxdoiorg101299
jpes21221[22] I Urieli DM Berchowitz Stirling Cycle Engine Analysis Hilger Ltd Bristol
1984[23] JR Senft Mechanical Ef 1047297ciency of Heat Engines Cambridge University Press
2007[24] E Cardozo C Erlich A Malmquist L Alejo Integration of a wood pellet
burner and a Stirling engine to produce residential heat and power ApplTherm Eng (2014) httpdxdoiorg101016japplthermaleng201408024
[25] JA Araoz M Salomon L Alejo TH Fransson Non-ideal Stirling enginethermodynamic model suitable for the integration into overall energy sys-tems Appl Therm Eng 73 (2014) 203e219 httpdxdoiorg101016
japplthermaleng201407050
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 30 29
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httpslidepdfcomreaderfullapplied-thermal-engineering-volume-83-issue-2015-doi-1010162fjapplthermaleng201503006 1515
[26] FP Incropera AS Lavine DP DeWitt Fundamentals of Heat and MassTransfer John Wiley amp Sons 2011
[27] B Thomas D Pittman Update on the evaluation of different correlations forthe 1047298ow friction factor and heat transfer of Stirling engine regenerators inCollection of Technical Papers 35th Intersociety Energy Conversion Engi-neering Conference and Exhibit (IECEC) (Cat No00CH37022) American InstAeronaut amp Astronautics 2000 pp 76e84 httpdxdoiorg101109IECEC2000870632
[28] Genoastirling Srl 2014 wwwgenoastirlingcom[29] RC Tew K Jefferies D Miao USD of ED of TE Conservation LR Center
A Stirling Engine Computer Model for Performance Calculations (GoogleEBook) Department of Energy Of 1047297ce of Conservation and Solar ApplicationsDivision of Transportation Energy Conservation 1978
[30] I Urieli CJ Rallis DM Berchowitz Computer simulation of Stirling cyclemachines in 12th Intersociety Energy Conversion Engineering Conferencevol 1 American Nuclear Society Washington DC 1977 pp 1512e1521
[31] K Mahkamov Design improvements to a biomass Stirling engine usingmathematical analysis and 3D CFD modeling J Energy Resour Technol 128(2006) 203 httpdxdoiorg10111512213273
[32] H Snyman Second Order Analyses Methods for Stirling Engine Design Uni-versity of Stellenbosch Stellenbosch 2007
[33] G Sewell Numerical Solution of Ordinary and Partial Differential Equationssecond ed John Wiley amp Sons 2005
[34] DD Do Richard G Rice Applied Mathematics and Modeling for ChemicalEngineers [Hardcover] second ed Wiley-AIChE 2012
[35] Aspentech Aspen Custom Modelerreg AspenTech 2015
[36] Aspentech Chemical Process Optimization Software d
Chemical ProcessDesign Aspen Plus 2015
[37] B Thomas D Pittman AIAA-2000-2812 Update on the Evaluation of DifferentCorrelations for the Flow Friction Factor and Heat Transfer of Stirling EngineRegenerators 2000 pp 76e84
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 3030
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develop improved engines that should present higher ef 1047297ciency
levels fuel 1047298exibility and should also be easy to integrate within
combined heat and power systems (CHP) It is especially important
the mentioned integration capability because of the great potential
that combined heat and power systems presents as decentralized
solutions based on renewable energy [18] Some works that
explored this integrations include Paringlsson and Carlsen [19] Nish-
iyama [20] and Sato [21]
In this sense the objective of this paper is the development of a
thermodynamic-numerical model of a Stirling engine that should
represent the performance of a new 1 kW gamma engine prototype
built by GENOA Stirling Company in Italy This model aims to assess
through numerical simulation analysis the performance improve-ment of the GENOA engine prototype and it is centred on a second
order thermodynamic analysis implemented in Aspen Custom
Modellerreg The numerical model is based on Urielli approach [22]
it considers ideal adiabatic working spaces limited internal and
external heat transfer through the heat exchangers and mechan-
ical and thermal losses during the cycle In addition it includes the
numerical evaluation of the mechanical ef 1047297ciency taking into ac-
count the crank mechanism effectiveness and the forced work
during the cycle according to Senft methodology [23] Therefore
the model combines Urielli and Senft approaches into a restruc-
tured numerical analysis that computes the work that can be
effectively taken from the shaft The model was validated with data
obtained from an experimental rig built for the engine The details
about the methods used for the measurements are reported inCardozo et al [24]
2 Mathematical model
A mathematical model for the simulation of Stirling engine
systems was developed in a previous work [25] This consisted on
four main modules named ideal adiabatic internal heat transfer
external heat transfer and energy losses This paper improves the
model by adding the evaluation of the mechanical ef 1047297ciency of the
system thus the improved model contains 5 modules The 1047297rst
module corresponds to an ideal Stirling engine adiabatic model
which assumes ideal adiabatic compression and expansion spaces
to estimate the main engine variables The derivation of the
equations that govern this system are explained in Urielli [22] Theoutputs of this module are coupled to the internal heat transfer
module which through appropriate correlations evaluate the heat
transfer the temperature and the thermodynamic properties of the
working 1047298uid inside the heat exchangers The variation of the
thermodynamic properties with the temperature is considered at
every time step of the system The next module external heat
transfer module couples the heat transfer between the external
walls at the hot and cold side of the engine This is done through
energy balances and heat transfer correlations described in detail
in Araoz et al [25] The following module energy losses module
evaluates the losses due to pressure drop axial conduction shuttle
heat transfer and imperfect regeneration once the cyclic steady
state conditions were reached Finally the mechanical ef 1047297ciency
module permits to estimate the effect of forced work during the
cycle and the effect that the design for the crank mechanism have
on the performance of the engine
The main variables that connect the modules are described
below
- External heat transfer module This module considers the
adiabatic 1047298ame temperature and the inlet temperature of the
cooling 1047298uid on the hot and cold side respectively Therefore the
heat source (Q h) and the heat sink (Q k) are used to estimate the
wall temperatures (Twoh Twok) This approach is proposed to
couple the Stirling engine within the external heat and cooling
sources respectively
- Internal heat transfer module The internal working gas tem-
peratures (Th Tk) in the heater and cooler respectively are
calculated using heat transfer correlations for steady state in-
ternal forced convective 1047298ow [26] On the other hand the
regenerator analysis proposes the use of cyclic 1047298ow heat transfer
correlations which are more suitable for the 1047298ow conditions onthis space [27] Therefore with these correlations the effect of
limited heat transfer inside the engine is introduced in the
model
- Ideal adiabatic module The main operative variables such as
net shaft work (Ws) heat and cooling demands (Q h Q k) are
calculated considering the internal working 1047298uid temperature
distribution and the engine geometric characteristics following
Uriellis [22] approach
- Energy losses module The losses inside the engine are esti-
mated to correct the ideal adiabatic outputs This module con-
siders the losses due to pressure drop axial conduction shuttle
heat transfer and imperfect regeneration
- Mechanical ef 1047297ciency module The losses due to forced
compression and expansion are evaluated considering the
buffer pressure (Pb) the shape of the cycle and the crank
mechanism effectiveness (E)
The relationships between the modules are shown in Fig 1 The
loops represent the iterative calculationsto achieve the steady state
cyclic conditions The detailed report of the 1047297rst four modules can
be found in Araoz et al [25] and the detailed description of the new
mechanical ef 1047297ciency module is presented in the next section
21 Governing equations
The equations included in the model are based in the mass
energy balances and the equation of state for the working gas
These balances were applied to the control volumes shown in Fig 2
Fig 1 Block diagram for the Stirling model
Fig 2 Control volumes for Stirling engine based on Urielli [22]
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The mass balance is expressed as
min mout frac14 dm
d4(1)
The energy balance neglecting the energy kinetic terms
dQ
d4thorn cpinTinmin cpoutToutmout frac14
dW
d4thorn cv
dethmTTHORN
d4(2)
The equation of state for the gas in the control volume
PV frac14 mRT (3)
The balances were applied to each control volume to obtain a set
of algebraic differential equations This set was complemented with
correlations for the heat transfer in the heat exchangers and the
losses of the engine The details of the model development are
presented in Araoz [25] However a summary of the equations is
presented in Appendix B
22 Mechanical ef 1047297ciency and shaft work
The mechanical ef 1047297ciency of an engine measures the amount of the work produced by the thermodynamic cycle (indicated work
Wi) that can be effectively taken from the shaft shaft work (Ws)
[23]
hm frac14 Ws
Wi(4)
The mechanical ef 1047297ciency is evaluated with the fundamental
ef 1047297ciency theorem considering a constant mechanism effective-
ness (E) as developed by Senft [23]
hm frac14 E
1
E E
W
Wi(5)
where W represents the forced work This is the work that the
crank mechanism must deliver to the piston to make it move in
opposition to the pressure difference across it [23] For example
during the expansion process when the pressure of the gas inside
the working space is lower than the opposite buffer pressure then
the expansion process is forced In a similar way during the
compression process when the pressure inside the working space
is higher than the opposite buffer pressure then the compression is
forced Therefore this forced work depends mainly on the cycleshape and the buffer pressure level (Pb) and its calculated with the
following expression [23]
W frac14
I ethP PbTHORNthorndV thorn
I ethP PbTHORNdV thorn (6)
The superscripts difference the two types of forced work the
1047297rst one during the compression (dV ) when the buffer pressure is
below the working space pressure (P Pb)thorn and the second during
the expansion (dV thorn) when the buffer pressure is above the working
space pressure (P Pb)
The modi1047297ed model includes a numerical integration of Eq (6)
and the evaluation of both the mechanical ef 1047297ciency from Eq (5)
and the shaft work from Eq (4)
23 Brake thermal ef 1047297ciency
The overall ef 1047297ciency or brake thermal ef 1047297ciency is de1047297ned as
the ratio of the shaft work Ws and the net heat input of the engine
Q hc This can be calculated by the product of the thermal ef 1047297ciency
and the mechanical ef 1047297ciency as shown in Eq (7) The additional
module includes the estimation of the mechanical ef 1047297ciency and
the brake ef 1047297ciency
hb frac14 Ws
Q hcfrac14
Wi
Q hc
Ws
Wifrac14 hthm (7)
Fig 3 Genoa Stirling scheme
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3 Simulation of the Genoa engine
31 System description
The Genoa Stirling is a two cylinder gamma type engine built as
a prototype for research studies by GENOA Stirling SRL company
from Italy [28] According to its speci1047297cations it is capable to pro-
duce up to 1 kW electrical output with air as working 1047298uid at
600 rpm rotational speed and with the heater temperature around
750 C [28] The main components of the engine such as the
crankcase the crank mechanism with the balancing 1047298ywheel the
heat exchangers and the generator of the engine areshownin Fig 3Additional pictures for the heater cooler and regenerator heat
exchangers are shown in Fig 4
The gamma Stirling engine consists of two identical piston-
displacer cylinders connected to a common shaft under similar
operational conditions Therefore it is assumed that both cylinders
present similar thermodynamic cycles and consequently the dou-
ble cylinder thermodynamic analysis is simpli1047297ed to one cylinder
analysis The validity of the similarity on both cylinders is a com-
mon approach on Stirling simulation studies [1129e32] In addi-
tion the model assumes adiabatic expansion and compression
spaces and that the steady state cyclic conditions are reached
The Stirling engine was used in an experimental rig built at the
Energy department Royal Institute of Technology (KTH) Stock-
holm Sweden This rig consisted on the engine coupled to a pellet
Fig 4 Heat exchangers of the engine prototype
Table 1
Main parameters for the engine simulation
Parameter Value De1047297nition Description
freq 5 Hz Frequency of the engine
X 13353 V deV swe Dead volume ratio
K 03684 V swcV swe Piston to displacer swept volume ratio
Tratio 023 TadTwater_in Cold to heat temperature ratio
Pch 125 bar e Engine charging pressure
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Fig 6 Layout Stirling engine model in ACM
Table 2
Description of the blocks for the ACM model
Block name Description
Comp-Exp The block contains the data that describes the volume variation inside the engine The swept dead volumes crank mechanism and the
characteristics of the pistons
Cooler The block contains the geometrical data for the cooler heat exchanger
Heater T he b lock c ont ai ns t he geometr ic al data for t he heat exch an ger
Regenerator The block contains the geometrical data for the regenerator and the details of the matrix porosity and material
Ext-heat The characteristics of the external heat source are contained in this block
Mech_Ef 1047297ciency The block contains the parameters for the calculation of the engine mechanical ef 1047297ciency
CoolingFluid The characteristics of the external cooling 1047298uid are contained in the block
WorkingGAS The block contains the parameters for the calculation of the properties for the working gas inside the engine
Stirling This is the main block and contains the main equations that describe the thermodynamic analysis of the engine
Fig 7 a) Schematic set-up of the Stirling engine integrated with a combustion chamber and a boiler [27] b) Temperature measurement points for the working gas in the Stirling
engine T2 hot side T10 cold side T11 T12 hot and cold side of the regenerator [24]
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burner in order to produce heat and power simultaneously as
shown in Fig 7a This con1047297guration had technical limitations that
are still being studied in order to improve both power and thermal
outputs But despite of these limitations experimental results were
obtained and these were compared with the model
32 Inputs for the model
The main inputs for the engine simulation are shown in Table 1
Supplementary inputs that include the design and operational
characteristics of the engine are presented in Appendix A
The model also needs to consider the relation of the crank
mechanism and the variation of the volumes inside the working
spaces Therefore considering that the engine has gamma type
con1047297guration the following relations for the expansion and
compression spaces were included [23]
V e frac14 V cle thornV swe
2 eth1 thorn coseth4 thorn aTHORNTHORN (8)
V c frac14 V clc thorn ethV swe V eTHORN thornV swc
2 eth1 thorn coseth4THORNTHORN (9)
Furthermore the following volume derivatives were evaluated
dV e frac14 V swe
2 sineth4 thorn aTHORN (10)
dV c
frac14 dV e
V swc
2 sineth4THORN (11)
Fig 8 Measurement points for the CHP-Stirling experimental rig [24]
Table 3Comparison of the measured and predicted temperatures along the engine
Time (s) TfM (K) ThM (K) Th (K) Error TkM (K) Tk (K) Error TrM (K) Tr (K) Error
3780e3900 13878 8164 8184 025 3224 3211 041 6018 5316 116
3900e4020 13829 8196 8075 147 3218 3214 012 6006 5277 1215
4020e4140 13931 8232 8142 109 3216 3215 004 6012 5302 118
4140e4200 13778 8308 7981 394 3216 3216 001 6036 5243 1314
4200e4380 13835 8374 8063 371 3224 3214 031 6075 5272 1321
4380e4560 13777 8518 7957 659 3218 3217 003 6142 5234 1478
4560e4680 13857 8536 8071 545 3217 3215 007 6154 5276 1426
4680e4800 13844 8464 8021 523 3216 3217 001 6135 5258 143
4800e4980 13669 8433 7708 859 3221 3223 005 6129 5144 1607
Fig 9 Temperature variation along the heat exchangers and regenerator temperature assumed by the model (T r)
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33 Numerical solution
The system consists of a set of algebraic differential equations
which are shown in Appendix B These consider as boundary con-
ditions that the temperatures of the working gas at the end of the
cycle must be equal to the temperatures at the beginning of the
cycle once cyclic steady state conditions are reached Therefore an
iterative shooting method [33] using a fourth order Runge Kutta
scheme for the time discretization was implemented for the nu-
merical solution The iteration process was done until cyclic steadystate conditions which is numerically reached when the difference
between the assumed initial values and the values calculated at the
end of the cycle are lower than a de1047297ned error After the cyclic
steady state solution was reached the energy losses and the forced
work were evaluated The forced work was calculated using the
classical Simpson 38 numerical integration rule [34] The scheme
in Fig 5 summarizes the iterative steps for the solution
The numerical solution was implemented in Aspen Custom
Modellerreg (ACM) [35] which is a product from Aspen Plusreg that
permits the elaboration of customized models [36] This software
has its own modelling language and can also be coupled with Cthornthorn
procedures The layout of the model in ACM is shown in Fig 6 The
blocks were programmed with the equations shown in the
Appendix B and then the solution of the system was obtained withthe algorithm previously described
The descriptions of the blocks are shown in Table 2 Additional
details of the block inputs are given in Appendix A
4 Model validation
The geometrical and operational characteristics for the Genoa
engine are described in Table 1 and Appendix A The engine was
mounted in the experimental rig shown in Fig 7a In addition the
temperatures of the working gas were measured at the different
points of the engine shown in Fig 7b
The experimental rig used wood pellets as fuel Additional
temperatures measured for the validation were The temperature
close to the 1047298ame (T1) the water inlet temperature (T8) the wateroutlet temperature (T9) Other measurements are also as shown in
Fig 8
The temperature T1 was measured using a type K empty 15 mm
Inconel 600 thermocouple The additional temperatures shown in
Fig 8 were measured using type K empty 10 mm thermocouples
Considering the type of thermocouples the expanded uncertainty
was plusmn32 C with a coverage factor of 2 The speed of the engine
crankshaft was monitored by a pulse sensor and a frequency to
analog converter (OMROM E2A and Red Lion IFMA) with an un-
certaintyplusmn 02 The pressure inside the engine was measured with
a pressure transducer (RS type 46) with analog signal and an un-
certainty of plusmn01 bar All the measurements were recorded from the
beginning to the end of the test using a data logger Additional
details of the measurements are reported in Cardozo et al [24]The engine was run during long periods and the data was
measured constantly However for the validation purposes only the
periods were stability is reached were considered In this case the
steady state condition was dif 1047297cult to reach due to the constant
variation of the 1047298ame temperature [24] Therefore average values
for the measurements within certain stability periods were taken
These are compared with the values calculated by the model at the
different values measured for the 1047298ame temperature shown in
Table 3
Fig 10 Temperature variation along the engine 1047298
ame temperature Tad frac14
1388 K
Table 4
Measured and predicted brake power
Time (s) TfM (K) Measured frequency (Hz) Measured pressure (bar) Brake power (W) experimental Brake power (W) calculated Error
3780e3900 13878 517 1250 5472 5359 206
3900e4020 13829 526 1250 5539 5208 597
4020e4140 13930 527 1250 5561 5349 381
4140e4200 13778 533 1250 4635 5003 794
4200e4380 13835 528 1250 5359 5197 302
4380e
4560 13777 536 1250 5091 5033 1144560e4680 13857 529 1250 5096 5163 131
4680e4800 13843 534 1250 559 5153 782
4800e4980 13669 556 1254 4713 4613 212
Fig 11 Volumes variation during the engine cycle
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From Table 3 the model presents good accuracy for the pre-
diction of the cooler temperatures (Tk) with the maximum error of
the order of plusmn041 In addition the calculations for the heater
temperatures (Th) present reasonable accuracy at initial times but
then the error increases This growth may be explained with the
thermal inertia that constantly increments the measured temper-
ature even on periods where the1047298ame temperature decreases This
thermal inertia is neglected by the model since it assumes steady
state heat transfer conditions On the other hand the prediction of
the mean temperature in the regenerator space (Tr) presents
higher differences This is analysed with the Fig 9 below which
shows the variations of the temperatures inside the heat ex-
changers assumed by the model
From Fig 9 it can be seen that the model assumes that thetemperatures at the interfaces heater-regenerator and cooler-
regenerator were equal to the temperatures at the cooler (T k) and
heater (Th) spaces respectively Therefore the average temperature
at the regenerator (Tr) was calculated with these values This
assumption neglects the axial temperature variation along the
heater and cooler which is re1047298ected on the measurements taken at
the exact interfaces positions T11 and T12This explains the differ-
ence between the average regenerator temperature calculated with
the measured temperatures (TrM) and the calculated with the es-
timations of the model Tr as it is shown in Table 3 However
considering that the model was capable to calculate within a good
degree of accuracy the power output measured during the exper-
imental runs it can be inferred that the error for the regenerator
temperature estimation have little in1047298
uence on the brake powercalculation This is shown in Table 4 where the values for the
measured and calculated brake power are compared at different
operating conditions The percentage error ranges from plusmn131 to
plusmn794 which is an acceptable approximation for 1047297rst design
calculations
5 Results and discussion
This section presents additionally results for the simulation of
the engine under the experimental conditions described before
This aims to completely describe the thermodynamic performance
of the engine and thus identify the main limitations that the engine
presents
51 Temperature variation
Fig10 shows the temperature variation in the differentspaces of
the engine cylinder once the cyclic steady state conditions are
reached This 1047297gure displays the sinusoidal variation of the tem-
peratures inside the compression (Tc) and expansion (Te) spaces It
can also be seen that the expansion space presents periods with
elevated temperatures which results into a high thermal stress for
the material and therefore further engine deterioration In addition
the 1047297gure also shows that the mean temperatures for the working
1047298uid inside the heater (Th) and cooler (Tk) are close to the heat
exchangers walls temperature (Twk Twhe) This indicates a good
heat transfer rate on both heat exchangers and consequently a
good thermal performance based on the model assumptions
However it is important to notice that this performance will
decrease with the time due to the fouling on the heat exchangers
which is not accounted for in the engine model
52 Mass distribution and volumes variation
The mass distribution and volumes variation for the engine
during a complete cycle are shown in Figs 11 and 12 respectivelyThese variations permit to analyse the engine dynamics during the
compression and expansion processes
Fig 11 permits to identify the following processes the
compression characterized by the decrease in the total volume
from the time around t frac14 001 to t frac14 004 the heating process
when the total volume variation is not pronounced and the tem-
peratures increase around t frac14 004 to t frac14 006 the expansion
process when the total volume increases around t frac14 006 to
t frac14 009 and the cooling process when the volume stays almost
constant and the temperatures decrease at the times around
t frac14 009 to t frac14 010 and t frac14 0 to t frac14 001
The compressionperiod starts with the increment of the mass in
the compression space and a decrease of the mass in the expansion
space as shown in Fig 12 The decreasing mass in the expansionspace indicates a good dynamic for the compression process
because it is desirable to keep low the hotter portion of the mass
during this period However the mass on the compression space is
too high which is not desirable since this will be re1047298ected in a large
negative compression work In addition the expansion process also
presents a reduced performance due to the low values for the mass
in the expansion space during the expansion process This repre-
sents an expansion with low hotter mass and thus a low working
output to the shaft Furthermore the low mass in the expansion
space during the heating period might be the main cause for the
high temperatures reached Therefore the volumes and mass 1047298ow
dynamics of the reference case should be improved to reach higher
work outputs and avoid the overheating of the expansion chamber
Fig 12 also shows that the mass in the heater and cooler arereally small compared with the mass in the regenerator during the
Table 5
Engine work 1047298ow per cycle
Model output per cycle Aspen Custom Modeller (ACM)
Expansion Work (We Jcycle) 5262
Compression Work (Wc Jcycle) 2339
Pre ssure drop lost heater (J cycle) 021
Pre ssure drop lost cooler (J cycle) 007
Pressure drop lost regenerator (Jcycle) 028
Total lost due to pressure drop (Jcycle) 056
Net indicated work (Wi Jcycle) 2867
Forced work (W Jcycle) 2349
Brake Work Output (Wbr Jcycle) 518
Fig 12 Mass variation inside the engine spaces during a complete cycle
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complete cycle This re1047298ects the high importance of the regenerator
ef 1047297ciency for the engine performance
53 Work 1047298ow
Table 5 shows the simulation results for the compression and
expansion work during a single cycle This table also presents the
different work losses estimated for the system
The temperatures measured and the temperatures calculated
show a good thermal performance of the engine But the measured
brake power was very low Different problems on the engine
design and operational conditions may explain these very lowresults However additional experimental instrumentation is
needed for a detailed design study For this reason the present
analysis considers a theoretical approach that may be later com-
plemented with experimental studies This theoretical approach
considers Eq (7) From this equation and considering that the
thermal performance was found acceptable the main losses should
correspond to a low mechanical ef 1047297ciency of the prototype This
mechanical ef 1047297ciency is reduced by the presence of forced work
during the cycle and mechanical friction on the crank mechanism
Fig 13 presents the evaluation of the forced work in a pressure
volume diagram for the gas cycle inside the gamma prototype
From this it can be seen that the forced work (W) is mainly due to
the forced expansion process This means that at the experimentalconditions large part of the cyclic work may have been used to
complete the forced expansion process and thus the real engine
output is smaller than expected
The results discussed above are complemented with the vari-
ation of the compression (Wc) expansion (We) and net indicated
work (Wi) during the cycle shown in Fig 14
Fig 14 shows that during the 1047297rst part of the cycle from t frac14 0 to
t frac14 0045 the compression and expansion spaces present
exchanged roles This means that an increment of the volume is
presented in the compression space and a decrement of the volume
is present in the expansion one This reduced the engine perfor-
mance but it cannot be avoided since the gas needs to pass from
one space to another Regarding the second part of the cycle from
t frac14 0045 to t frac14 0095 the expansion and compression are shapedas expected and thus indicate a better dynamic during this period
However considering that large part of the expansion process is
forced the net brake work is low as reported in Table 5
From the previous analysis it can be concluded that a detailed
evaluation of the volumes dynamics the cranks mechanism effec-
tiveness and the forced work during the cycle must be considered
Fig 15 Heat 1047298
ow variation during the engine cycle
Table 6
Heat 1047298ow and heat loses during the cycle
Heat 1047298ow (Jcycle)
Heat exchanger space
Heater 1047298ow (Q hcJcycle) 5282
Cooler 1047298ow (Q kc Jcycle) 2356
Regenerator 1047298ow (Q rc Jcycle) 005
Heat lossesInternal conduction losses (Q lkc Jcycle) 2698
Shuttle conduction losses (Q lshc Jcycle) 8004
Regenerator losses during heating (Q lossrc Jcycle) 1862
Regenerator losses during cooling (Q lossrc Jcycle) 1862
Total heat requirements
Heating requirements (Q htc Jcycle) 17847
Cooling requirements (Q ktc Jcycle) 4218
Fig 14 Work 1047298
ow during the engine cycle
Fig 13 Pressureevolume diagram and forced work during the cycle
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in order to re-design the engine for a better performance This will
be covered on a detailed parametric study to be reported on a next
article
54 Heat 1047298ow
Table 6 presents the results for the heat 1047298ow and corresponding
heat losses through the heat exchangers calculated at the end of a
single cycle [25] As it can be seen the total heat requirements are
almost three times the requirements calculated without consid-
ering the losses It can also be seen that the shuttle conduction
losses represent the main heat loss during the cycle These corre-
spond to the losses due to the oscillation of the hot displacer across
the temperature gradient in the working spaces of the engine
The cyclic variation for the heat 1047298ow is additionally shown in
Fig 15The heat requirements for the heater and cooler present
slight variations during the entire cycle On the other hand the
regenerator presents high variations managing large quantities of
heat This con1047297rms the large importance of this heat exchanger on
the engine performance
55 Brake power and brake ef 1047297ciency
The engine brake power is de1047297ned as the net brake work per
cycle (Ws) times the engine frequency (freq)
Pbr frac14 Ws freq (12)
The net brake work and the total heat requirement presented
on Tables 5 and 6 respectively are doubled considering the double
cylinder engine These values are reported on Table 7 which alsoreports the thermal and brake ef 1047297ciencies for the engine
The results re1047298ect the low performance of the engine under the
experimental conditions This was mainly attributed to the forced
work and the mechanical ef 1047297ciency as it was analysed in the pre-
vious section In addition complementary works will broad this
analysis with the aim of propose improvements on the engine
design and operational parameters
6 Conclusions
In the present work a thermodynamic model for a Stirling en-
gine was improved by including the numerical evaluation of the
forced work and the mechanical ef 1047297ciency then validated against
experimental data and 1047297nally implemented for the simulation of an engine prototype The numerical model developed considered
the analytic approach proposed by Senft [23] but extended its
application for the case of the more realistic adiabatic working
spaces assumptions Consequently the effective work taken from
the shaft is better estimated and thus used for a more complete
analysis of the thermal and mechanical performance of an engine
For this article the analysis considered a novel gamma engine
prototype under the experimental conditions of a micro scale
combined heat and power system fuelled by wood pellets
The simulation results were compared with the experimental
data measured during long time runs of the system The model
performance was very good for the prediction of the temperatures
in the different spaces of the engine In addition the estimations for
the net brake power also presented results similar to the measured
values However additional experimental work should be per-
formed to obtain data to validate the calculation of the different
losses through the engine
According to the results obtained the thermal performance of
the engine was found acceptable and thus the low power output
measured is preliminary attributed to a reduced mechanical ef 1047297-
ciency of the system The possible reasons for this low performance
were further analysed with the different results for the tempera-
tures variation mass and volume variation pressure drops and the
pressure volume diagrams obtained with the model According to
these analyses the dynamics of the volumes variation and the
crank mechanism may also be improved in order to obtain higher
network during the cycle In addition it was found that the engine
performance is very sensitive to the effect of the buffer pressure
These results will be extended with a sensitivity analysis for the
system on a complementary work that aims to identify better the
effect of the different parameters on the engine performance
Acknowledgements
This work was possible thanks to the 1047297nancial support of the
Swedish International Development Cooperation Agency the di-
vision of Heat and Power Technology Department of Energy
Technology at Royal Institute of Technology (KTH) in Sweden and
Universidad Mayor de San Simon (UMSS) in Bolivia
Appendix A Detailed Stirling engine parameters
Table 7
Power output and ef 1047297ciency of the engine
Brake power (W) Heat requirement (W) Thermal ef 1047297ciency () Mechanical ef 1047297ciency () Brake ef 1047297ciency ()
5358 184535 1610 1810 290
Table A1
Inputs for the cooler in ACM
Variable Value Units Description
do 0005 m Tubes external diameter
di 0003 m Tubes internal diameter
kw 14200 Wm K Material conductivity
L 0032 m Tubes length
num 162 e Number of tubes
sl 0005 m Space between tubes
Table A2
Inputs for the heater in ACM
Variable Value Units Description
de 0005 M Tubes external diameter
di 00031 M Tubes internal diameterkw 142 Wm K Material conductivity
len 0149 m Tubes length
num 360 e Number of tubes
sl 0005 m Space between tubes
Table A3
Inputs for the regenerator in ACM
Variable Value Units Description
Din 0078 m Regenerator housing internal diameter
dout 0 107 m Regenera tor h ousin g extern al diameter
dwir e 2 1e-0 04 m Wi re di ameter of t he ma tri x
kwr 270 Wm K Thermal conductivity of the matrix material
Lr 007 m Length of the regenerator housing
Porosity 087 Matrix porosity
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Appendix B
Main equations for the Stirling engine model
Stirling engine module
Mean pressure
P frac14 MR
V cTc
thorn V kTk
thorn V rTr
thorn V hTh
thorn V eTe
Pressure variation
dP
d4frac14
gP
0BB
vV cv4
Tck
thorn
vV ev4
The
1CCA
V cTckthorn g
V kTk
thorn V rTrthorn V hTh
thorn V eThe
Mass of the working gas in the different spaces
mc frac14 p
V c
RTc
mk frac14 p
V k
RTk
mr frac14 p
V r
RTr
mh frac14 p
V h
RTh
me frac14 p
V e
RTe
Mass accumulation
dmk
d4frac14
mk
P
vP
v4
dmh
d4frac14
mh
P
vP
v4
dmr
d4frac14
mr
P
vP
v4
dmc
d4frac14
P
vV cv4
thorn
V c
vPv4
g
RTck
dme
d4frac14
P
vV ev4
thorn
V e
vPv4
g
RThe
Mass 1047298ow
mck frac14 dmc mhe frac14 dme mkr frac14 mck dmkmrh frac14 mhe thorn dmh
Conditional temperatures
If mck gt 0 then Tck frac14 Tc else Tck frac14 Tk
If mhe gt 0 then The frac14 Th else The frac14 Te
Temperatures
dTc
d4frac14 Tc
0BBvPv4
P
thorn
vV cv4
V c
vmc
v4
mc
1CCA
dTe
d4frac14 Te
0BBvPv4
P thorn
vV ev4
V e
vmev4
me
1CCAEnergy
dQ kd4
frac14
V k
vPv4
Cv
R CpethTckmck TkrmkrTHORN
dQ rd4
frac14
V r
vPv4
Cv
R C pethTkrmkr TrhmrhTHORN
dQ hd4
frac14V hvPv4
Cv
R CpethTrhmrh ThemheTHORN
dWc
d4frac14 P
vV cv4
dWe
d4frac14 P
vV ev4
Internal heat transfer module
Heat transfer from the heater wall to the working gas
Q h frac14 1
R cih thorn R hih thorn R fihethTwoh ThTHORN
Heat transfer from the cooler wall to the working gas
Q k frac14 1
R cik thorn R hik thorn R fikethTwik TkTHORN
Heat loss during the regenerator process
Q lossr frac14 eth1 3THORN Q r
Regenerator effectiveness
3frac14 NTU
1 thorn NTU
External heat transfer module
Heat transfer from the 1047298
ame to the external wall of the heater
Table A4
Inputs for the expansion-compression spaces and crank mechanism
Variable Value Units Description
vclc 44e-006 m3 Compression space clearance volume
vcle 26e-005 m3 Expansion space clearance volume
vswc 926e-005 m3 Compression space swept volume
vswe 25134e-004 m3 Expansion space swept volume
dispd 0062 m Displacer diameter
displ 007 m Displacer lengthef fmek 08 e Mechanism effectiveness
freq 5 Hz Frequency
jgap 0006 M Gap between cylinder displacer and wall
kpist 1627 Wm K Piston conductivity
pbuff 12ethorn006 Pa Buffer pressure
phase 900 deg Phase angle advance
pmean 1 25ethorn00 6 Pa Mean o pera tin g pressure
strk 0035 m Displacer stroke
dispd 0062 m Displacer diameter
Table A5
Working and cooling 1047298uid inputs in ACM
Variable Value Units Description
Working Fluid Air e Working 1047298uid inside the engine
Cooling FLUID Water e Cooling 1047298uid through the engine cooler
Tcooling 2 88 K Inl et tempera ture of th e co ol in g 1047298uid
Table A6
Fouling factors and external combustion inputs in ACM
Variable Value Units Description
T ad 1 38 7 K Fla me temper at ur e i n t he c omb ustio n ch amb er
absorp 070 e Absorptivity of the heater material
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Q h frac14 11
hrhAohthorn R foh
ethTad TwohTHORN
hrh frac14 assAohFR ethTad thorn TwohTHORN
T2ad thorn T2
woh
Estimation of the outlet temperature of the cooling 1047298uid
Twok frac14 Twater in thorn Q k
1
hokAokthorn
1
2mwaterCpwater
hok frac14 11
hwaterthorn R fok
Energy losses
Pressure drop in the heat exchangers
DP frac14
f
dhy
1
2 rv
2
l
Pressure drop in the regenerator based on the correlations of
Thomas and Pittman [37]
DP frac14 Cf nr
2u2
Cf frac14 Cfd thornCsf
Re
Csf frac14 68556 Cfd frac14 05274 wire screensCsf frac14 70035 Cfd frac14 09307 metal felts
Total pumping losses
Wploss frac14
Z 2p0
Xifrac143
ifrac141
DPi dV edq
$dq
Energy losses due to shuttle conduction
Q lsh frac14 04Z2KpistDd
JL dethTe TcTHORN
Mechanical ef 1047297ciency module
Mechanical ef 1047297ciency
hm frac14 Ws
Wi
Mechanical ef 1047297ciency considering the mechanism effectiveness
and forced work
hm frac14 E
1
E E
W
Wi
Forced work
W frac14
I ethP PbTHORNthorndV thorn
I ethP PbTHORNdV thorn
Brake ef 1047297
ciency
hb frac14 Ws
Q htfrac14
Wi
Q ht
Ws
Wifrac14 hthm
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[15] E Eid Performance of a beta-con1047297guration heat engine having a regenerativedisplacer Renew Energy 34 (2009) 2404e2413 httpdxdoiorg101016
jrenene200903016[16] AA El-Ehwany GM Hennes EI Eid EA El-Kenany Development of the
performance of an alpha-type heat engine by using elbow-bend transposed-1047298uids heat exchanger as a heater and a cooler Energy Convers Manag 52(2011) 1010e1019 httpdxdoiorg101016jenconman201008029
[17] C-H Cheng Y-J Yu Dynamic simulation of a beta-type Stirling engine withcam-drive mechanism via the combination of the thermodynamic and dy-namic models Renew Energy 36 (2011) 714e725
[18] E Entchev J Gusdorf M Swinton M Bell F Szadkowski W Kalb1047298eisch et alMicro-generation technology assessment for housing technology EnergyBuild 36 (2004) 925e931 httpdxdoiorg101016jenbuild200403004
[19] M Paringlsson H Carlsen Development of a wood powder fuelled 35 kW StirlingCHP unit in Proceedings of the 11th ISEC (International Stirling EngineConference) 2003 pp 221e230
[20] a Nishiyama H Shimojima a Ishikawa Y Itaya S Kambara H Moritomi etal Fuel and emissions properties of Stirling engine operated with woodpowder Fuel 86 (2007) 2333e2342 httpdxdoiorg101016
jfuel200701040[21] K Sato N Ohiwa A Ishikawa H Shimojima A Nishiyama Y Moriya
Development of small-scale CHP Plant with a wood powder-fueled Stirlingengine J Power Energy Syst 2 (2008) 1221e1231 httpdxdoiorg101299
jpes21221[22] I Urieli DM Berchowitz Stirling Cycle Engine Analysis Hilger Ltd Bristol
1984[23] JR Senft Mechanical Ef 1047297ciency of Heat Engines Cambridge University Press
2007[24] E Cardozo C Erlich A Malmquist L Alejo Integration of a wood pellet
burner and a Stirling engine to produce residential heat and power ApplTherm Eng (2014) httpdxdoiorg101016japplthermaleng201408024
[25] JA Araoz M Salomon L Alejo TH Fransson Non-ideal Stirling enginethermodynamic model suitable for the integration into overall energy sys-tems Appl Therm Eng 73 (2014) 203e219 httpdxdoiorg101016
japplthermaleng201407050
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[26] FP Incropera AS Lavine DP DeWitt Fundamentals of Heat and MassTransfer John Wiley amp Sons 2011
[27] B Thomas D Pittman Update on the evaluation of different correlations forthe 1047298ow friction factor and heat transfer of Stirling engine regenerators inCollection of Technical Papers 35th Intersociety Energy Conversion Engi-neering Conference and Exhibit (IECEC) (Cat No00CH37022) American InstAeronaut amp Astronautics 2000 pp 76e84 httpdxdoiorg101109IECEC2000870632
[28] Genoastirling Srl 2014 wwwgenoastirlingcom[29] RC Tew K Jefferies D Miao USD of ED of TE Conservation LR Center
A Stirling Engine Computer Model for Performance Calculations (GoogleEBook) Department of Energy Of 1047297ce of Conservation and Solar ApplicationsDivision of Transportation Energy Conservation 1978
[30] I Urieli CJ Rallis DM Berchowitz Computer simulation of Stirling cyclemachines in 12th Intersociety Energy Conversion Engineering Conferencevol 1 American Nuclear Society Washington DC 1977 pp 1512e1521
[31] K Mahkamov Design improvements to a biomass Stirling engine usingmathematical analysis and 3D CFD modeling J Energy Resour Technol 128(2006) 203 httpdxdoiorg10111512213273
[32] H Snyman Second Order Analyses Methods for Stirling Engine Design Uni-versity of Stellenbosch Stellenbosch 2007
[33] G Sewell Numerical Solution of Ordinary and Partial Differential Equationssecond ed John Wiley amp Sons 2005
[34] DD Do Richard G Rice Applied Mathematics and Modeling for ChemicalEngineers [Hardcover] second ed Wiley-AIChE 2012
[35] Aspentech Aspen Custom Modelerreg AspenTech 2015
[36] Aspentech Chemical Process Optimization Software d
Chemical ProcessDesign Aspen Plus 2015
[37] B Thomas D Pittman AIAA-2000-2812 Update on the Evaluation of DifferentCorrelations for the Flow Friction Factor and Heat Transfer of Stirling EngineRegenerators 2000 pp 76e84
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The mass balance is expressed as
min mout frac14 dm
d4(1)
The energy balance neglecting the energy kinetic terms
dQ
d4thorn cpinTinmin cpoutToutmout frac14
dW
d4thorn cv
dethmTTHORN
d4(2)
The equation of state for the gas in the control volume
PV frac14 mRT (3)
The balances were applied to each control volume to obtain a set
of algebraic differential equations This set was complemented with
correlations for the heat transfer in the heat exchangers and the
losses of the engine The details of the model development are
presented in Araoz [25] However a summary of the equations is
presented in Appendix B
22 Mechanical ef 1047297ciency and shaft work
The mechanical ef 1047297ciency of an engine measures the amount of the work produced by the thermodynamic cycle (indicated work
Wi) that can be effectively taken from the shaft shaft work (Ws)
[23]
hm frac14 Ws
Wi(4)
The mechanical ef 1047297ciency is evaluated with the fundamental
ef 1047297ciency theorem considering a constant mechanism effective-
ness (E) as developed by Senft [23]
hm frac14 E
1
E E
W
Wi(5)
where W represents the forced work This is the work that the
crank mechanism must deliver to the piston to make it move in
opposition to the pressure difference across it [23] For example
during the expansion process when the pressure of the gas inside
the working space is lower than the opposite buffer pressure then
the expansion process is forced In a similar way during the
compression process when the pressure inside the working space
is higher than the opposite buffer pressure then the compression is
forced Therefore this forced work depends mainly on the cycleshape and the buffer pressure level (Pb) and its calculated with the
following expression [23]
W frac14
I ethP PbTHORNthorndV thorn
I ethP PbTHORNdV thorn (6)
The superscripts difference the two types of forced work the
1047297rst one during the compression (dV ) when the buffer pressure is
below the working space pressure (P Pb)thorn and the second during
the expansion (dV thorn) when the buffer pressure is above the working
space pressure (P Pb)
The modi1047297ed model includes a numerical integration of Eq (6)
and the evaluation of both the mechanical ef 1047297ciency from Eq (5)
and the shaft work from Eq (4)
23 Brake thermal ef 1047297ciency
The overall ef 1047297ciency or brake thermal ef 1047297ciency is de1047297ned as
the ratio of the shaft work Ws and the net heat input of the engine
Q hc This can be calculated by the product of the thermal ef 1047297ciency
and the mechanical ef 1047297ciency as shown in Eq (7) The additional
module includes the estimation of the mechanical ef 1047297ciency and
the brake ef 1047297ciency
hb frac14 Ws
Q hcfrac14
Wi
Q hc
Ws
Wifrac14 hthm (7)
Fig 3 Genoa Stirling scheme
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3 Simulation of the Genoa engine
31 System description
The Genoa Stirling is a two cylinder gamma type engine built as
a prototype for research studies by GENOA Stirling SRL company
from Italy [28] According to its speci1047297cations it is capable to pro-
duce up to 1 kW electrical output with air as working 1047298uid at
600 rpm rotational speed and with the heater temperature around
750 C [28] The main components of the engine such as the
crankcase the crank mechanism with the balancing 1047298ywheel the
heat exchangers and the generator of the engine areshownin Fig 3Additional pictures for the heater cooler and regenerator heat
exchangers are shown in Fig 4
The gamma Stirling engine consists of two identical piston-
displacer cylinders connected to a common shaft under similar
operational conditions Therefore it is assumed that both cylinders
present similar thermodynamic cycles and consequently the dou-
ble cylinder thermodynamic analysis is simpli1047297ed to one cylinder
analysis The validity of the similarity on both cylinders is a com-
mon approach on Stirling simulation studies [1129e32] In addi-
tion the model assumes adiabatic expansion and compression
spaces and that the steady state cyclic conditions are reached
The Stirling engine was used in an experimental rig built at the
Energy department Royal Institute of Technology (KTH) Stock-
holm Sweden This rig consisted on the engine coupled to a pellet
Fig 4 Heat exchangers of the engine prototype
Table 1
Main parameters for the engine simulation
Parameter Value De1047297nition Description
freq 5 Hz Frequency of the engine
X 13353 V deV swe Dead volume ratio
K 03684 V swcV swe Piston to displacer swept volume ratio
Tratio 023 TadTwater_in Cold to heat temperature ratio
Pch 125 bar e Engine charging pressure
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Fig 6 Layout Stirling engine model in ACM
Table 2
Description of the blocks for the ACM model
Block name Description
Comp-Exp The block contains the data that describes the volume variation inside the engine The swept dead volumes crank mechanism and the
characteristics of the pistons
Cooler The block contains the geometrical data for the cooler heat exchanger
Heater T he b lock c ont ai ns t he geometr ic al data for t he heat exch an ger
Regenerator The block contains the geometrical data for the regenerator and the details of the matrix porosity and material
Ext-heat The characteristics of the external heat source are contained in this block
Mech_Ef 1047297ciency The block contains the parameters for the calculation of the engine mechanical ef 1047297ciency
CoolingFluid The characteristics of the external cooling 1047298uid are contained in the block
WorkingGAS The block contains the parameters for the calculation of the properties for the working gas inside the engine
Stirling This is the main block and contains the main equations that describe the thermodynamic analysis of the engine
Fig 7 a) Schematic set-up of the Stirling engine integrated with a combustion chamber and a boiler [27] b) Temperature measurement points for the working gas in the Stirling
engine T2 hot side T10 cold side T11 T12 hot and cold side of the regenerator [24]
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burner in order to produce heat and power simultaneously as
shown in Fig 7a This con1047297guration had technical limitations that
are still being studied in order to improve both power and thermal
outputs But despite of these limitations experimental results were
obtained and these were compared with the model
32 Inputs for the model
The main inputs for the engine simulation are shown in Table 1
Supplementary inputs that include the design and operational
characteristics of the engine are presented in Appendix A
The model also needs to consider the relation of the crank
mechanism and the variation of the volumes inside the working
spaces Therefore considering that the engine has gamma type
con1047297guration the following relations for the expansion and
compression spaces were included [23]
V e frac14 V cle thornV swe
2 eth1 thorn coseth4 thorn aTHORNTHORN (8)
V c frac14 V clc thorn ethV swe V eTHORN thornV swc
2 eth1 thorn coseth4THORNTHORN (9)
Furthermore the following volume derivatives were evaluated
dV e frac14 V swe
2 sineth4 thorn aTHORN (10)
dV c
frac14 dV e
V swc
2 sineth4THORN (11)
Fig 8 Measurement points for the CHP-Stirling experimental rig [24]
Table 3Comparison of the measured and predicted temperatures along the engine
Time (s) TfM (K) ThM (K) Th (K) Error TkM (K) Tk (K) Error TrM (K) Tr (K) Error
3780e3900 13878 8164 8184 025 3224 3211 041 6018 5316 116
3900e4020 13829 8196 8075 147 3218 3214 012 6006 5277 1215
4020e4140 13931 8232 8142 109 3216 3215 004 6012 5302 118
4140e4200 13778 8308 7981 394 3216 3216 001 6036 5243 1314
4200e4380 13835 8374 8063 371 3224 3214 031 6075 5272 1321
4380e4560 13777 8518 7957 659 3218 3217 003 6142 5234 1478
4560e4680 13857 8536 8071 545 3217 3215 007 6154 5276 1426
4680e4800 13844 8464 8021 523 3216 3217 001 6135 5258 143
4800e4980 13669 8433 7708 859 3221 3223 005 6129 5144 1607
Fig 9 Temperature variation along the heat exchangers and regenerator temperature assumed by the model (T r)
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33 Numerical solution
The system consists of a set of algebraic differential equations
which are shown in Appendix B These consider as boundary con-
ditions that the temperatures of the working gas at the end of the
cycle must be equal to the temperatures at the beginning of the
cycle once cyclic steady state conditions are reached Therefore an
iterative shooting method [33] using a fourth order Runge Kutta
scheme for the time discretization was implemented for the nu-
merical solution The iteration process was done until cyclic steadystate conditions which is numerically reached when the difference
between the assumed initial values and the values calculated at the
end of the cycle are lower than a de1047297ned error After the cyclic
steady state solution was reached the energy losses and the forced
work were evaluated The forced work was calculated using the
classical Simpson 38 numerical integration rule [34] The scheme
in Fig 5 summarizes the iterative steps for the solution
The numerical solution was implemented in Aspen Custom
Modellerreg (ACM) [35] which is a product from Aspen Plusreg that
permits the elaboration of customized models [36] This software
has its own modelling language and can also be coupled with Cthornthorn
procedures The layout of the model in ACM is shown in Fig 6 The
blocks were programmed with the equations shown in the
Appendix B and then the solution of the system was obtained withthe algorithm previously described
The descriptions of the blocks are shown in Table 2 Additional
details of the block inputs are given in Appendix A
4 Model validation
The geometrical and operational characteristics for the Genoa
engine are described in Table 1 and Appendix A The engine was
mounted in the experimental rig shown in Fig 7a In addition the
temperatures of the working gas were measured at the different
points of the engine shown in Fig 7b
The experimental rig used wood pellets as fuel Additional
temperatures measured for the validation were The temperature
close to the 1047298ame (T1) the water inlet temperature (T8) the wateroutlet temperature (T9) Other measurements are also as shown in
Fig 8
The temperature T1 was measured using a type K empty 15 mm
Inconel 600 thermocouple The additional temperatures shown in
Fig 8 were measured using type K empty 10 mm thermocouples
Considering the type of thermocouples the expanded uncertainty
was plusmn32 C with a coverage factor of 2 The speed of the engine
crankshaft was monitored by a pulse sensor and a frequency to
analog converter (OMROM E2A and Red Lion IFMA) with an un-
certaintyplusmn 02 The pressure inside the engine was measured with
a pressure transducer (RS type 46) with analog signal and an un-
certainty of plusmn01 bar All the measurements were recorded from the
beginning to the end of the test using a data logger Additional
details of the measurements are reported in Cardozo et al [24]The engine was run during long periods and the data was
measured constantly However for the validation purposes only the
periods were stability is reached were considered In this case the
steady state condition was dif 1047297cult to reach due to the constant
variation of the 1047298ame temperature [24] Therefore average values
for the measurements within certain stability periods were taken
These are compared with the values calculated by the model at the
different values measured for the 1047298ame temperature shown in
Table 3
Fig 10 Temperature variation along the engine 1047298
ame temperature Tad frac14
1388 K
Table 4
Measured and predicted brake power
Time (s) TfM (K) Measured frequency (Hz) Measured pressure (bar) Brake power (W) experimental Brake power (W) calculated Error
3780e3900 13878 517 1250 5472 5359 206
3900e4020 13829 526 1250 5539 5208 597
4020e4140 13930 527 1250 5561 5349 381
4140e4200 13778 533 1250 4635 5003 794
4200e4380 13835 528 1250 5359 5197 302
4380e
4560 13777 536 1250 5091 5033 1144560e4680 13857 529 1250 5096 5163 131
4680e4800 13843 534 1250 559 5153 782
4800e4980 13669 556 1254 4713 4613 212
Fig 11 Volumes variation during the engine cycle
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From Table 3 the model presents good accuracy for the pre-
diction of the cooler temperatures (Tk) with the maximum error of
the order of plusmn041 In addition the calculations for the heater
temperatures (Th) present reasonable accuracy at initial times but
then the error increases This growth may be explained with the
thermal inertia that constantly increments the measured temper-
ature even on periods where the1047298ame temperature decreases This
thermal inertia is neglected by the model since it assumes steady
state heat transfer conditions On the other hand the prediction of
the mean temperature in the regenerator space (Tr) presents
higher differences This is analysed with the Fig 9 below which
shows the variations of the temperatures inside the heat ex-
changers assumed by the model
From Fig 9 it can be seen that the model assumes that thetemperatures at the interfaces heater-regenerator and cooler-
regenerator were equal to the temperatures at the cooler (T k) and
heater (Th) spaces respectively Therefore the average temperature
at the regenerator (Tr) was calculated with these values This
assumption neglects the axial temperature variation along the
heater and cooler which is re1047298ected on the measurements taken at
the exact interfaces positions T11 and T12This explains the differ-
ence between the average regenerator temperature calculated with
the measured temperatures (TrM) and the calculated with the es-
timations of the model Tr as it is shown in Table 3 However
considering that the model was capable to calculate within a good
degree of accuracy the power output measured during the exper-
imental runs it can be inferred that the error for the regenerator
temperature estimation have little in1047298
uence on the brake powercalculation This is shown in Table 4 where the values for the
measured and calculated brake power are compared at different
operating conditions The percentage error ranges from plusmn131 to
plusmn794 which is an acceptable approximation for 1047297rst design
calculations
5 Results and discussion
This section presents additionally results for the simulation of
the engine under the experimental conditions described before
This aims to completely describe the thermodynamic performance
of the engine and thus identify the main limitations that the engine
presents
51 Temperature variation
Fig10 shows the temperature variation in the differentspaces of
the engine cylinder once the cyclic steady state conditions are
reached This 1047297gure displays the sinusoidal variation of the tem-
peratures inside the compression (Tc) and expansion (Te) spaces It
can also be seen that the expansion space presents periods with
elevated temperatures which results into a high thermal stress for
the material and therefore further engine deterioration In addition
the 1047297gure also shows that the mean temperatures for the working
1047298uid inside the heater (Th) and cooler (Tk) are close to the heat
exchangers walls temperature (Twk Twhe) This indicates a good
heat transfer rate on both heat exchangers and consequently a
good thermal performance based on the model assumptions
However it is important to notice that this performance will
decrease with the time due to the fouling on the heat exchangers
which is not accounted for in the engine model
52 Mass distribution and volumes variation
The mass distribution and volumes variation for the engine
during a complete cycle are shown in Figs 11 and 12 respectivelyThese variations permit to analyse the engine dynamics during the
compression and expansion processes
Fig 11 permits to identify the following processes the
compression characterized by the decrease in the total volume
from the time around t frac14 001 to t frac14 004 the heating process
when the total volume variation is not pronounced and the tem-
peratures increase around t frac14 004 to t frac14 006 the expansion
process when the total volume increases around t frac14 006 to
t frac14 009 and the cooling process when the volume stays almost
constant and the temperatures decrease at the times around
t frac14 009 to t frac14 010 and t frac14 0 to t frac14 001
The compressionperiod starts with the increment of the mass in
the compression space and a decrease of the mass in the expansion
space as shown in Fig 12 The decreasing mass in the expansionspace indicates a good dynamic for the compression process
because it is desirable to keep low the hotter portion of the mass
during this period However the mass on the compression space is
too high which is not desirable since this will be re1047298ected in a large
negative compression work In addition the expansion process also
presents a reduced performance due to the low values for the mass
in the expansion space during the expansion process This repre-
sents an expansion with low hotter mass and thus a low working
output to the shaft Furthermore the low mass in the expansion
space during the heating period might be the main cause for the
high temperatures reached Therefore the volumes and mass 1047298ow
dynamics of the reference case should be improved to reach higher
work outputs and avoid the overheating of the expansion chamber
Fig 12 also shows that the mass in the heater and cooler arereally small compared with the mass in the regenerator during the
Table 5
Engine work 1047298ow per cycle
Model output per cycle Aspen Custom Modeller (ACM)
Expansion Work (We Jcycle) 5262
Compression Work (Wc Jcycle) 2339
Pre ssure drop lost heater (J cycle) 021
Pre ssure drop lost cooler (J cycle) 007
Pressure drop lost regenerator (Jcycle) 028
Total lost due to pressure drop (Jcycle) 056
Net indicated work (Wi Jcycle) 2867
Forced work (W Jcycle) 2349
Brake Work Output (Wbr Jcycle) 518
Fig 12 Mass variation inside the engine spaces during a complete cycle
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complete cycle This re1047298ects the high importance of the regenerator
ef 1047297ciency for the engine performance
53 Work 1047298ow
Table 5 shows the simulation results for the compression and
expansion work during a single cycle This table also presents the
different work losses estimated for the system
The temperatures measured and the temperatures calculated
show a good thermal performance of the engine But the measured
brake power was very low Different problems on the engine
design and operational conditions may explain these very lowresults However additional experimental instrumentation is
needed for a detailed design study For this reason the present
analysis considers a theoretical approach that may be later com-
plemented with experimental studies This theoretical approach
considers Eq (7) From this equation and considering that the
thermal performance was found acceptable the main losses should
correspond to a low mechanical ef 1047297ciency of the prototype This
mechanical ef 1047297ciency is reduced by the presence of forced work
during the cycle and mechanical friction on the crank mechanism
Fig 13 presents the evaluation of the forced work in a pressure
volume diagram for the gas cycle inside the gamma prototype
From this it can be seen that the forced work (W) is mainly due to
the forced expansion process This means that at the experimentalconditions large part of the cyclic work may have been used to
complete the forced expansion process and thus the real engine
output is smaller than expected
The results discussed above are complemented with the vari-
ation of the compression (Wc) expansion (We) and net indicated
work (Wi) during the cycle shown in Fig 14
Fig 14 shows that during the 1047297rst part of the cycle from t frac14 0 to
t frac14 0045 the compression and expansion spaces present
exchanged roles This means that an increment of the volume is
presented in the compression space and a decrement of the volume
is present in the expansion one This reduced the engine perfor-
mance but it cannot be avoided since the gas needs to pass from
one space to another Regarding the second part of the cycle from
t frac14 0045 to t frac14 0095 the expansion and compression are shapedas expected and thus indicate a better dynamic during this period
However considering that large part of the expansion process is
forced the net brake work is low as reported in Table 5
From the previous analysis it can be concluded that a detailed
evaluation of the volumes dynamics the cranks mechanism effec-
tiveness and the forced work during the cycle must be considered
Fig 15 Heat 1047298
ow variation during the engine cycle
Table 6
Heat 1047298ow and heat loses during the cycle
Heat 1047298ow (Jcycle)
Heat exchanger space
Heater 1047298ow (Q hcJcycle) 5282
Cooler 1047298ow (Q kc Jcycle) 2356
Regenerator 1047298ow (Q rc Jcycle) 005
Heat lossesInternal conduction losses (Q lkc Jcycle) 2698
Shuttle conduction losses (Q lshc Jcycle) 8004
Regenerator losses during heating (Q lossrc Jcycle) 1862
Regenerator losses during cooling (Q lossrc Jcycle) 1862
Total heat requirements
Heating requirements (Q htc Jcycle) 17847
Cooling requirements (Q ktc Jcycle) 4218
Fig 14 Work 1047298
ow during the engine cycle
Fig 13 Pressureevolume diagram and forced work during the cycle
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in order to re-design the engine for a better performance This will
be covered on a detailed parametric study to be reported on a next
article
54 Heat 1047298ow
Table 6 presents the results for the heat 1047298ow and corresponding
heat losses through the heat exchangers calculated at the end of a
single cycle [25] As it can be seen the total heat requirements are
almost three times the requirements calculated without consid-
ering the losses It can also be seen that the shuttle conduction
losses represent the main heat loss during the cycle These corre-
spond to the losses due to the oscillation of the hot displacer across
the temperature gradient in the working spaces of the engine
The cyclic variation for the heat 1047298ow is additionally shown in
Fig 15The heat requirements for the heater and cooler present
slight variations during the entire cycle On the other hand the
regenerator presents high variations managing large quantities of
heat This con1047297rms the large importance of this heat exchanger on
the engine performance
55 Brake power and brake ef 1047297ciency
The engine brake power is de1047297ned as the net brake work per
cycle (Ws) times the engine frequency (freq)
Pbr frac14 Ws freq (12)
The net brake work and the total heat requirement presented
on Tables 5 and 6 respectively are doubled considering the double
cylinder engine These values are reported on Table 7 which alsoreports the thermal and brake ef 1047297ciencies for the engine
The results re1047298ect the low performance of the engine under the
experimental conditions This was mainly attributed to the forced
work and the mechanical ef 1047297ciency as it was analysed in the pre-
vious section In addition complementary works will broad this
analysis with the aim of propose improvements on the engine
design and operational parameters
6 Conclusions
In the present work a thermodynamic model for a Stirling en-
gine was improved by including the numerical evaluation of the
forced work and the mechanical ef 1047297ciency then validated against
experimental data and 1047297nally implemented for the simulation of an engine prototype The numerical model developed considered
the analytic approach proposed by Senft [23] but extended its
application for the case of the more realistic adiabatic working
spaces assumptions Consequently the effective work taken from
the shaft is better estimated and thus used for a more complete
analysis of the thermal and mechanical performance of an engine
For this article the analysis considered a novel gamma engine
prototype under the experimental conditions of a micro scale
combined heat and power system fuelled by wood pellets
The simulation results were compared with the experimental
data measured during long time runs of the system The model
performance was very good for the prediction of the temperatures
in the different spaces of the engine In addition the estimations for
the net brake power also presented results similar to the measured
values However additional experimental work should be per-
formed to obtain data to validate the calculation of the different
losses through the engine
According to the results obtained the thermal performance of
the engine was found acceptable and thus the low power output
measured is preliminary attributed to a reduced mechanical ef 1047297-
ciency of the system The possible reasons for this low performance
were further analysed with the different results for the tempera-
tures variation mass and volume variation pressure drops and the
pressure volume diagrams obtained with the model According to
these analyses the dynamics of the volumes variation and the
crank mechanism may also be improved in order to obtain higher
network during the cycle In addition it was found that the engine
performance is very sensitive to the effect of the buffer pressure
These results will be extended with a sensitivity analysis for the
system on a complementary work that aims to identify better the
effect of the different parameters on the engine performance
Acknowledgements
This work was possible thanks to the 1047297nancial support of the
Swedish International Development Cooperation Agency the di-
vision of Heat and Power Technology Department of Energy
Technology at Royal Institute of Technology (KTH) in Sweden and
Universidad Mayor de San Simon (UMSS) in Bolivia
Appendix A Detailed Stirling engine parameters
Table 7
Power output and ef 1047297ciency of the engine
Brake power (W) Heat requirement (W) Thermal ef 1047297ciency () Mechanical ef 1047297ciency () Brake ef 1047297ciency ()
5358 184535 1610 1810 290
Table A1
Inputs for the cooler in ACM
Variable Value Units Description
do 0005 m Tubes external diameter
di 0003 m Tubes internal diameter
kw 14200 Wm K Material conductivity
L 0032 m Tubes length
num 162 e Number of tubes
sl 0005 m Space between tubes
Table A2
Inputs for the heater in ACM
Variable Value Units Description
de 0005 M Tubes external diameter
di 00031 M Tubes internal diameterkw 142 Wm K Material conductivity
len 0149 m Tubes length
num 360 e Number of tubes
sl 0005 m Space between tubes
Table A3
Inputs for the regenerator in ACM
Variable Value Units Description
Din 0078 m Regenerator housing internal diameter
dout 0 107 m Regenera tor h ousin g extern al diameter
dwir e 2 1e-0 04 m Wi re di ameter of t he ma tri x
kwr 270 Wm K Thermal conductivity of the matrix material
Lr 007 m Length of the regenerator housing
Porosity 087 Matrix porosity
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Appendix B
Main equations for the Stirling engine model
Stirling engine module
Mean pressure
P frac14 MR
V cTc
thorn V kTk
thorn V rTr
thorn V hTh
thorn V eTe
Pressure variation
dP
d4frac14
gP
0BB
vV cv4
Tck
thorn
vV ev4
The
1CCA
V cTckthorn g
V kTk
thorn V rTrthorn V hTh
thorn V eThe
Mass of the working gas in the different spaces
mc frac14 p
V c
RTc
mk frac14 p
V k
RTk
mr frac14 p
V r
RTr
mh frac14 p
V h
RTh
me frac14 p
V e
RTe
Mass accumulation
dmk
d4frac14
mk
P
vP
v4
dmh
d4frac14
mh
P
vP
v4
dmr
d4frac14
mr
P
vP
v4
dmc
d4frac14
P
vV cv4
thorn
V c
vPv4
g
RTck
dme
d4frac14
P
vV ev4
thorn
V e
vPv4
g
RThe
Mass 1047298ow
mck frac14 dmc mhe frac14 dme mkr frac14 mck dmkmrh frac14 mhe thorn dmh
Conditional temperatures
If mck gt 0 then Tck frac14 Tc else Tck frac14 Tk
If mhe gt 0 then The frac14 Th else The frac14 Te
Temperatures
dTc
d4frac14 Tc
0BBvPv4
P
thorn
vV cv4
V c
vmc
v4
mc
1CCA
dTe
d4frac14 Te
0BBvPv4
P thorn
vV ev4
V e
vmev4
me
1CCAEnergy
dQ kd4
frac14
V k
vPv4
Cv
R CpethTckmck TkrmkrTHORN
dQ rd4
frac14
V r
vPv4
Cv
R C pethTkrmkr TrhmrhTHORN
dQ hd4
frac14V hvPv4
Cv
R CpethTrhmrh ThemheTHORN
dWc
d4frac14 P
vV cv4
dWe
d4frac14 P
vV ev4
Internal heat transfer module
Heat transfer from the heater wall to the working gas
Q h frac14 1
R cih thorn R hih thorn R fihethTwoh ThTHORN
Heat transfer from the cooler wall to the working gas
Q k frac14 1
R cik thorn R hik thorn R fikethTwik TkTHORN
Heat loss during the regenerator process
Q lossr frac14 eth1 3THORN Q r
Regenerator effectiveness
3frac14 NTU
1 thorn NTU
External heat transfer module
Heat transfer from the 1047298
ame to the external wall of the heater
Table A4
Inputs for the expansion-compression spaces and crank mechanism
Variable Value Units Description
vclc 44e-006 m3 Compression space clearance volume
vcle 26e-005 m3 Expansion space clearance volume
vswc 926e-005 m3 Compression space swept volume
vswe 25134e-004 m3 Expansion space swept volume
dispd 0062 m Displacer diameter
displ 007 m Displacer lengthef fmek 08 e Mechanism effectiveness
freq 5 Hz Frequency
jgap 0006 M Gap between cylinder displacer and wall
kpist 1627 Wm K Piston conductivity
pbuff 12ethorn006 Pa Buffer pressure
phase 900 deg Phase angle advance
pmean 1 25ethorn00 6 Pa Mean o pera tin g pressure
strk 0035 m Displacer stroke
dispd 0062 m Displacer diameter
Table A5
Working and cooling 1047298uid inputs in ACM
Variable Value Units Description
Working Fluid Air e Working 1047298uid inside the engine
Cooling FLUID Water e Cooling 1047298uid through the engine cooler
Tcooling 2 88 K Inl et tempera ture of th e co ol in g 1047298uid
Table A6
Fouling factors and external combustion inputs in ACM
Variable Value Units Description
T ad 1 38 7 K Fla me temper at ur e i n t he c omb ustio n ch amb er
absorp 070 e Absorptivity of the heater material
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Q h frac14 11
hrhAohthorn R foh
ethTad TwohTHORN
hrh frac14 assAohFR ethTad thorn TwohTHORN
T2ad thorn T2
woh
Estimation of the outlet temperature of the cooling 1047298uid
Twok frac14 Twater in thorn Q k
1
hokAokthorn
1
2mwaterCpwater
hok frac14 11
hwaterthorn R fok
Energy losses
Pressure drop in the heat exchangers
DP frac14
f
dhy
1
2 rv
2
l
Pressure drop in the regenerator based on the correlations of
Thomas and Pittman [37]
DP frac14 Cf nr
2u2
Cf frac14 Cfd thornCsf
Re
Csf frac14 68556 Cfd frac14 05274 wire screensCsf frac14 70035 Cfd frac14 09307 metal felts
Total pumping losses
Wploss frac14
Z 2p0
Xifrac143
ifrac141
DPi dV edq
$dq
Energy losses due to shuttle conduction
Q lsh frac14 04Z2KpistDd
JL dethTe TcTHORN
Mechanical ef 1047297ciency module
Mechanical ef 1047297ciency
hm frac14 Ws
Wi
Mechanical ef 1047297ciency considering the mechanism effectiveness
and forced work
hm frac14 E
1
E E
W
Wi
Forced work
W frac14
I ethP PbTHORNthorndV thorn
I ethP PbTHORNdV thorn
Brake ef 1047297
ciency
hb frac14 Ws
Q htfrac14
Wi
Q ht
Ws
Wifrac14 hthm
References
[1] DG Thombare SK Verma Technological development in the Stirling cycle
engines Renew Sustain Energy Rev 12 (2008) 1e
38 httpdxdoiorg101016jrser200607001
[2] AD Peacock M Newborough Impact of micro-CHP systems on domesticsector CO2 emissions Appl Therm Eng 25 (2005) 2653e2676
[3] B Thomas Benchmark testing of Micro-CHP units Appl Therm Eng 28(2008) 2049e2054 httpdxdoiorg101016japplthermaleng200803010
[4] L Dong H Liu S Riffat Development of small-scale and micro-scale biomass-fuelled CHP systems e a literature review Appl Therm Eng 29 (2009)2119e2126 httpdxdoiorg101016japplthermaleng200812004
[5] I Gonzalez-Pino A Campos-Celador E Perez-Iribarren J Teres-Zubiaga JM Sala Parametric study of the operational and economic feasibility of Stirling micro-cogeneration devices in Spain Appl Therm Eng (2013) httpdxdoiorg101016japplthermaleng201312020
[6] R Dyson S Wilson R Tew Review of computational Stirling analysismethods in 2nd International Energy Conversion Engineering ConferenceAmerican Institute of Aeronautics and Astronautics 2004 httpdxdoiorg10251462004-5582
[7] C-H Cheng Y-J Yu Numerical model for predicting thermodynamic cycleand thermal ef 1047297ciency of a beta-type Stirling engine with rhombic-drivemechanism Renew Energy 35 (2010) 2590e2601 httpdxdoiorg101016jrenene201004002
[8] N Seraj Mehdizadeh P Stouffs Simulation of a Martini displacer free pistonStirling engine for electric power generation Int J Thermodyn 3 (1) (2000)27e34 httpdxdoiorg105541ijot30
[9] N Parlak A Wagner M Elsner HS Soyhan Thermodynamic analysis of agamma type Stirling engine in non-ideal adiabatic conditions Renew Energy34 (2009) 266e273 httpdxdoiorg101016jrenene200802030
[10] JM Strauss RT Dobson Evaluation of a second order simulation for Sterlingengine design and optimisation J Energy South Afr 21 (2010) 17e29
[11] I Tlili Y Timoumi SB Nasrallah Thermodynamic analysis of the Stirling heatengine with regenerative losses and internal irreversibilities Int J Engine Res9 (2008) 45e56 httpdxdoiorg10124314680874JER01707
[12] K Mahkamov D Djumanov Three-dimensional CFD modeling of a Stirlingengine in Proceedings of the 11th international Stirling Engine Conferencevol 19 2003
[13] M Ibrahim R Tew Z Zhang D Gedeon T Simon CFD Modeling of Free-piston Stirling Engines Cleveland Ohio 2001
[14] S Wilson R Dyson Multi-D CFD modeling of a free-piston Stirling convertorat NASA GRC in Proc 2nd International Energy Conversion EngineeringConference vol 5673 2004
[15] E Eid Performance of a beta-con1047297guration heat engine having a regenerativedisplacer Renew Energy 34 (2009) 2404e2413 httpdxdoiorg101016
jrenene200903016[16] AA El-Ehwany GM Hennes EI Eid EA El-Kenany Development of the
performance of an alpha-type heat engine by using elbow-bend transposed-1047298uids heat exchanger as a heater and a cooler Energy Convers Manag 52(2011) 1010e1019 httpdxdoiorg101016jenconman201008029
[17] C-H Cheng Y-J Yu Dynamic simulation of a beta-type Stirling engine withcam-drive mechanism via the combination of the thermodynamic and dy-namic models Renew Energy 36 (2011) 714e725
[18] E Entchev J Gusdorf M Swinton M Bell F Szadkowski W Kalb1047298eisch et alMicro-generation technology assessment for housing technology EnergyBuild 36 (2004) 925e931 httpdxdoiorg101016jenbuild200403004
[19] M Paringlsson H Carlsen Development of a wood powder fuelled 35 kW StirlingCHP unit in Proceedings of the 11th ISEC (International Stirling EngineConference) 2003 pp 221e230
[20] a Nishiyama H Shimojima a Ishikawa Y Itaya S Kambara H Moritomi etal Fuel and emissions properties of Stirling engine operated with woodpowder Fuel 86 (2007) 2333e2342 httpdxdoiorg101016
jfuel200701040[21] K Sato N Ohiwa A Ishikawa H Shimojima A Nishiyama Y Moriya
Development of small-scale CHP Plant with a wood powder-fueled Stirlingengine J Power Energy Syst 2 (2008) 1221e1231 httpdxdoiorg101299
jpes21221[22] I Urieli DM Berchowitz Stirling Cycle Engine Analysis Hilger Ltd Bristol
1984[23] JR Senft Mechanical Ef 1047297ciency of Heat Engines Cambridge University Press
2007[24] E Cardozo C Erlich A Malmquist L Alejo Integration of a wood pellet
burner and a Stirling engine to produce residential heat and power ApplTherm Eng (2014) httpdxdoiorg101016japplthermaleng201408024
[25] JA Araoz M Salomon L Alejo TH Fransson Non-ideal Stirling enginethermodynamic model suitable for the integration into overall energy sys-tems Appl Therm Eng 73 (2014) 203e219 httpdxdoiorg101016
japplthermaleng201407050
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 30 29
7212019 Applied Thermal Engineering Volume 83 Issue 2015 [Doi 101016_2Fjapplthermaleng201503006] Araoz Joseph hellip
httpslidepdfcomreaderfullapplied-thermal-engineering-volume-83-issue-2015-doi-1010162fjapplthermaleng201503006 1515
[26] FP Incropera AS Lavine DP DeWitt Fundamentals of Heat and MassTransfer John Wiley amp Sons 2011
[27] B Thomas D Pittman Update on the evaluation of different correlations forthe 1047298ow friction factor and heat transfer of Stirling engine regenerators inCollection of Technical Papers 35th Intersociety Energy Conversion Engi-neering Conference and Exhibit (IECEC) (Cat No00CH37022) American InstAeronaut amp Astronautics 2000 pp 76e84 httpdxdoiorg101109IECEC2000870632
[28] Genoastirling Srl 2014 wwwgenoastirlingcom[29] RC Tew K Jefferies D Miao USD of ED of TE Conservation LR Center
A Stirling Engine Computer Model for Performance Calculations (GoogleEBook) Department of Energy Of 1047297ce of Conservation and Solar ApplicationsDivision of Transportation Energy Conservation 1978
[30] I Urieli CJ Rallis DM Berchowitz Computer simulation of Stirling cyclemachines in 12th Intersociety Energy Conversion Engineering Conferencevol 1 American Nuclear Society Washington DC 1977 pp 1512e1521
[31] K Mahkamov Design improvements to a biomass Stirling engine usingmathematical analysis and 3D CFD modeling J Energy Resour Technol 128(2006) 203 httpdxdoiorg10111512213273
[32] H Snyman Second Order Analyses Methods for Stirling Engine Design Uni-versity of Stellenbosch Stellenbosch 2007
[33] G Sewell Numerical Solution of Ordinary and Partial Differential Equationssecond ed John Wiley amp Sons 2005
[34] DD Do Richard G Rice Applied Mathematics and Modeling for ChemicalEngineers [Hardcover] second ed Wiley-AIChE 2012
[35] Aspentech Aspen Custom Modelerreg AspenTech 2015
[36] Aspentech Chemical Process Optimization Software d
Chemical ProcessDesign Aspen Plus 2015
[37] B Thomas D Pittman AIAA-2000-2812 Update on the Evaluation of DifferentCorrelations for the Flow Friction Factor and Heat Transfer of Stirling EngineRegenerators 2000 pp 76e84
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 3030
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3 Simulation of the Genoa engine
31 System description
The Genoa Stirling is a two cylinder gamma type engine built as
a prototype for research studies by GENOA Stirling SRL company
from Italy [28] According to its speci1047297cations it is capable to pro-
duce up to 1 kW electrical output with air as working 1047298uid at
600 rpm rotational speed and with the heater temperature around
750 C [28] The main components of the engine such as the
crankcase the crank mechanism with the balancing 1047298ywheel the
heat exchangers and the generator of the engine areshownin Fig 3Additional pictures for the heater cooler and regenerator heat
exchangers are shown in Fig 4
The gamma Stirling engine consists of two identical piston-
displacer cylinders connected to a common shaft under similar
operational conditions Therefore it is assumed that both cylinders
present similar thermodynamic cycles and consequently the dou-
ble cylinder thermodynamic analysis is simpli1047297ed to one cylinder
analysis The validity of the similarity on both cylinders is a com-
mon approach on Stirling simulation studies [1129e32] In addi-
tion the model assumes adiabatic expansion and compression
spaces and that the steady state cyclic conditions are reached
The Stirling engine was used in an experimental rig built at the
Energy department Royal Institute of Technology (KTH) Stock-
holm Sweden This rig consisted on the engine coupled to a pellet
Fig 4 Heat exchangers of the engine prototype
Table 1
Main parameters for the engine simulation
Parameter Value De1047297nition Description
freq 5 Hz Frequency of the engine
X 13353 V deV swe Dead volume ratio
K 03684 V swcV swe Piston to displacer swept volume ratio
Tratio 023 TadTwater_in Cold to heat temperature ratio
Pch 125 bar e Engine charging pressure
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Fig 6 Layout Stirling engine model in ACM
Table 2
Description of the blocks for the ACM model
Block name Description
Comp-Exp The block contains the data that describes the volume variation inside the engine The swept dead volumes crank mechanism and the
characteristics of the pistons
Cooler The block contains the geometrical data for the cooler heat exchanger
Heater T he b lock c ont ai ns t he geometr ic al data for t he heat exch an ger
Regenerator The block contains the geometrical data for the regenerator and the details of the matrix porosity and material
Ext-heat The characteristics of the external heat source are contained in this block
Mech_Ef 1047297ciency The block contains the parameters for the calculation of the engine mechanical ef 1047297ciency
CoolingFluid The characteristics of the external cooling 1047298uid are contained in the block
WorkingGAS The block contains the parameters for the calculation of the properties for the working gas inside the engine
Stirling This is the main block and contains the main equations that describe the thermodynamic analysis of the engine
Fig 7 a) Schematic set-up of the Stirling engine integrated with a combustion chamber and a boiler [27] b) Temperature measurement points for the working gas in the Stirling
engine T2 hot side T10 cold side T11 T12 hot and cold side of the regenerator [24]
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burner in order to produce heat and power simultaneously as
shown in Fig 7a This con1047297guration had technical limitations that
are still being studied in order to improve both power and thermal
outputs But despite of these limitations experimental results were
obtained and these were compared with the model
32 Inputs for the model
The main inputs for the engine simulation are shown in Table 1
Supplementary inputs that include the design and operational
characteristics of the engine are presented in Appendix A
The model also needs to consider the relation of the crank
mechanism and the variation of the volumes inside the working
spaces Therefore considering that the engine has gamma type
con1047297guration the following relations for the expansion and
compression spaces were included [23]
V e frac14 V cle thornV swe
2 eth1 thorn coseth4 thorn aTHORNTHORN (8)
V c frac14 V clc thorn ethV swe V eTHORN thornV swc
2 eth1 thorn coseth4THORNTHORN (9)
Furthermore the following volume derivatives were evaluated
dV e frac14 V swe
2 sineth4 thorn aTHORN (10)
dV c
frac14 dV e
V swc
2 sineth4THORN (11)
Fig 8 Measurement points for the CHP-Stirling experimental rig [24]
Table 3Comparison of the measured and predicted temperatures along the engine
Time (s) TfM (K) ThM (K) Th (K) Error TkM (K) Tk (K) Error TrM (K) Tr (K) Error
3780e3900 13878 8164 8184 025 3224 3211 041 6018 5316 116
3900e4020 13829 8196 8075 147 3218 3214 012 6006 5277 1215
4020e4140 13931 8232 8142 109 3216 3215 004 6012 5302 118
4140e4200 13778 8308 7981 394 3216 3216 001 6036 5243 1314
4200e4380 13835 8374 8063 371 3224 3214 031 6075 5272 1321
4380e4560 13777 8518 7957 659 3218 3217 003 6142 5234 1478
4560e4680 13857 8536 8071 545 3217 3215 007 6154 5276 1426
4680e4800 13844 8464 8021 523 3216 3217 001 6135 5258 143
4800e4980 13669 8433 7708 859 3221 3223 005 6129 5144 1607
Fig 9 Temperature variation along the heat exchangers and regenerator temperature assumed by the model (T r)
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33 Numerical solution
The system consists of a set of algebraic differential equations
which are shown in Appendix B These consider as boundary con-
ditions that the temperatures of the working gas at the end of the
cycle must be equal to the temperatures at the beginning of the
cycle once cyclic steady state conditions are reached Therefore an
iterative shooting method [33] using a fourth order Runge Kutta
scheme for the time discretization was implemented for the nu-
merical solution The iteration process was done until cyclic steadystate conditions which is numerically reached when the difference
between the assumed initial values and the values calculated at the
end of the cycle are lower than a de1047297ned error After the cyclic
steady state solution was reached the energy losses and the forced
work were evaluated The forced work was calculated using the
classical Simpson 38 numerical integration rule [34] The scheme
in Fig 5 summarizes the iterative steps for the solution
The numerical solution was implemented in Aspen Custom
Modellerreg (ACM) [35] which is a product from Aspen Plusreg that
permits the elaboration of customized models [36] This software
has its own modelling language and can also be coupled with Cthornthorn
procedures The layout of the model in ACM is shown in Fig 6 The
blocks were programmed with the equations shown in the
Appendix B and then the solution of the system was obtained withthe algorithm previously described
The descriptions of the blocks are shown in Table 2 Additional
details of the block inputs are given in Appendix A
4 Model validation
The geometrical and operational characteristics for the Genoa
engine are described in Table 1 and Appendix A The engine was
mounted in the experimental rig shown in Fig 7a In addition the
temperatures of the working gas were measured at the different
points of the engine shown in Fig 7b
The experimental rig used wood pellets as fuel Additional
temperatures measured for the validation were The temperature
close to the 1047298ame (T1) the water inlet temperature (T8) the wateroutlet temperature (T9) Other measurements are also as shown in
Fig 8
The temperature T1 was measured using a type K empty 15 mm
Inconel 600 thermocouple The additional temperatures shown in
Fig 8 were measured using type K empty 10 mm thermocouples
Considering the type of thermocouples the expanded uncertainty
was plusmn32 C with a coverage factor of 2 The speed of the engine
crankshaft was monitored by a pulse sensor and a frequency to
analog converter (OMROM E2A and Red Lion IFMA) with an un-
certaintyplusmn 02 The pressure inside the engine was measured with
a pressure transducer (RS type 46) with analog signal and an un-
certainty of plusmn01 bar All the measurements were recorded from the
beginning to the end of the test using a data logger Additional
details of the measurements are reported in Cardozo et al [24]The engine was run during long periods and the data was
measured constantly However for the validation purposes only the
periods were stability is reached were considered In this case the
steady state condition was dif 1047297cult to reach due to the constant
variation of the 1047298ame temperature [24] Therefore average values
for the measurements within certain stability periods were taken
These are compared with the values calculated by the model at the
different values measured for the 1047298ame temperature shown in
Table 3
Fig 10 Temperature variation along the engine 1047298
ame temperature Tad frac14
1388 K
Table 4
Measured and predicted brake power
Time (s) TfM (K) Measured frequency (Hz) Measured pressure (bar) Brake power (W) experimental Brake power (W) calculated Error
3780e3900 13878 517 1250 5472 5359 206
3900e4020 13829 526 1250 5539 5208 597
4020e4140 13930 527 1250 5561 5349 381
4140e4200 13778 533 1250 4635 5003 794
4200e4380 13835 528 1250 5359 5197 302
4380e
4560 13777 536 1250 5091 5033 1144560e4680 13857 529 1250 5096 5163 131
4680e4800 13843 534 1250 559 5153 782
4800e4980 13669 556 1254 4713 4613 212
Fig 11 Volumes variation during the engine cycle
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From Table 3 the model presents good accuracy for the pre-
diction of the cooler temperatures (Tk) with the maximum error of
the order of plusmn041 In addition the calculations for the heater
temperatures (Th) present reasonable accuracy at initial times but
then the error increases This growth may be explained with the
thermal inertia that constantly increments the measured temper-
ature even on periods where the1047298ame temperature decreases This
thermal inertia is neglected by the model since it assumes steady
state heat transfer conditions On the other hand the prediction of
the mean temperature in the regenerator space (Tr) presents
higher differences This is analysed with the Fig 9 below which
shows the variations of the temperatures inside the heat ex-
changers assumed by the model
From Fig 9 it can be seen that the model assumes that thetemperatures at the interfaces heater-regenerator and cooler-
regenerator were equal to the temperatures at the cooler (T k) and
heater (Th) spaces respectively Therefore the average temperature
at the regenerator (Tr) was calculated with these values This
assumption neglects the axial temperature variation along the
heater and cooler which is re1047298ected on the measurements taken at
the exact interfaces positions T11 and T12This explains the differ-
ence between the average regenerator temperature calculated with
the measured temperatures (TrM) and the calculated with the es-
timations of the model Tr as it is shown in Table 3 However
considering that the model was capable to calculate within a good
degree of accuracy the power output measured during the exper-
imental runs it can be inferred that the error for the regenerator
temperature estimation have little in1047298
uence on the brake powercalculation This is shown in Table 4 where the values for the
measured and calculated brake power are compared at different
operating conditions The percentage error ranges from plusmn131 to
plusmn794 which is an acceptable approximation for 1047297rst design
calculations
5 Results and discussion
This section presents additionally results for the simulation of
the engine under the experimental conditions described before
This aims to completely describe the thermodynamic performance
of the engine and thus identify the main limitations that the engine
presents
51 Temperature variation
Fig10 shows the temperature variation in the differentspaces of
the engine cylinder once the cyclic steady state conditions are
reached This 1047297gure displays the sinusoidal variation of the tem-
peratures inside the compression (Tc) and expansion (Te) spaces It
can also be seen that the expansion space presents periods with
elevated temperatures which results into a high thermal stress for
the material and therefore further engine deterioration In addition
the 1047297gure also shows that the mean temperatures for the working
1047298uid inside the heater (Th) and cooler (Tk) are close to the heat
exchangers walls temperature (Twk Twhe) This indicates a good
heat transfer rate on both heat exchangers and consequently a
good thermal performance based on the model assumptions
However it is important to notice that this performance will
decrease with the time due to the fouling on the heat exchangers
which is not accounted for in the engine model
52 Mass distribution and volumes variation
The mass distribution and volumes variation for the engine
during a complete cycle are shown in Figs 11 and 12 respectivelyThese variations permit to analyse the engine dynamics during the
compression and expansion processes
Fig 11 permits to identify the following processes the
compression characterized by the decrease in the total volume
from the time around t frac14 001 to t frac14 004 the heating process
when the total volume variation is not pronounced and the tem-
peratures increase around t frac14 004 to t frac14 006 the expansion
process when the total volume increases around t frac14 006 to
t frac14 009 and the cooling process when the volume stays almost
constant and the temperatures decrease at the times around
t frac14 009 to t frac14 010 and t frac14 0 to t frac14 001
The compressionperiod starts with the increment of the mass in
the compression space and a decrease of the mass in the expansion
space as shown in Fig 12 The decreasing mass in the expansionspace indicates a good dynamic for the compression process
because it is desirable to keep low the hotter portion of the mass
during this period However the mass on the compression space is
too high which is not desirable since this will be re1047298ected in a large
negative compression work In addition the expansion process also
presents a reduced performance due to the low values for the mass
in the expansion space during the expansion process This repre-
sents an expansion with low hotter mass and thus a low working
output to the shaft Furthermore the low mass in the expansion
space during the heating period might be the main cause for the
high temperatures reached Therefore the volumes and mass 1047298ow
dynamics of the reference case should be improved to reach higher
work outputs and avoid the overheating of the expansion chamber
Fig 12 also shows that the mass in the heater and cooler arereally small compared with the mass in the regenerator during the
Table 5
Engine work 1047298ow per cycle
Model output per cycle Aspen Custom Modeller (ACM)
Expansion Work (We Jcycle) 5262
Compression Work (Wc Jcycle) 2339
Pre ssure drop lost heater (J cycle) 021
Pre ssure drop lost cooler (J cycle) 007
Pressure drop lost regenerator (Jcycle) 028
Total lost due to pressure drop (Jcycle) 056
Net indicated work (Wi Jcycle) 2867
Forced work (W Jcycle) 2349
Brake Work Output (Wbr Jcycle) 518
Fig 12 Mass variation inside the engine spaces during a complete cycle
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complete cycle This re1047298ects the high importance of the regenerator
ef 1047297ciency for the engine performance
53 Work 1047298ow
Table 5 shows the simulation results for the compression and
expansion work during a single cycle This table also presents the
different work losses estimated for the system
The temperatures measured and the temperatures calculated
show a good thermal performance of the engine But the measured
brake power was very low Different problems on the engine
design and operational conditions may explain these very lowresults However additional experimental instrumentation is
needed for a detailed design study For this reason the present
analysis considers a theoretical approach that may be later com-
plemented with experimental studies This theoretical approach
considers Eq (7) From this equation and considering that the
thermal performance was found acceptable the main losses should
correspond to a low mechanical ef 1047297ciency of the prototype This
mechanical ef 1047297ciency is reduced by the presence of forced work
during the cycle and mechanical friction on the crank mechanism
Fig 13 presents the evaluation of the forced work in a pressure
volume diagram for the gas cycle inside the gamma prototype
From this it can be seen that the forced work (W) is mainly due to
the forced expansion process This means that at the experimentalconditions large part of the cyclic work may have been used to
complete the forced expansion process and thus the real engine
output is smaller than expected
The results discussed above are complemented with the vari-
ation of the compression (Wc) expansion (We) and net indicated
work (Wi) during the cycle shown in Fig 14
Fig 14 shows that during the 1047297rst part of the cycle from t frac14 0 to
t frac14 0045 the compression and expansion spaces present
exchanged roles This means that an increment of the volume is
presented in the compression space and a decrement of the volume
is present in the expansion one This reduced the engine perfor-
mance but it cannot be avoided since the gas needs to pass from
one space to another Regarding the second part of the cycle from
t frac14 0045 to t frac14 0095 the expansion and compression are shapedas expected and thus indicate a better dynamic during this period
However considering that large part of the expansion process is
forced the net brake work is low as reported in Table 5
From the previous analysis it can be concluded that a detailed
evaluation of the volumes dynamics the cranks mechanism effec-
tiveness and the forced work during the cycle must be considered
Fig 15 Heat 1047298
ow variation during the engine cycle
Table 6
Heat 1047298ow and heat loses during the cycle
Heat 1047298ow (Jcycle)
Heat exchanger space
Heater 1047298ow (Q hcJcycle) 5282
Cooler 1047298ow (Q kc Jcycle) 2356
Regenerator 1047298ow (Q rc Jcycle) 005
Heat lossesInternal conduction losses (Q lkc Jcycle) 2698
Shuttle conduction losses (Q lshc Jcycle) 8004
Regenerator losses during heating (Q lossrc Jcycle) 1862
Regenerator losses during cooling (Q lossrc Jcycle) 1862
Total heat requirements
Heating requirements (Q htc Jcycle) 17847
Cooling requirements (Q ktc Jcycle) 4218
Fig 14 Work 1047298
ow during the engine cycle
Fig 13 Pressureevolume diagram and forced work during the cycle
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in order to re-design the engine for a better performance This will
be covered on a detailed parametric study to be reported on a next
article
54 Heat 1047298ow
Table 6 presents the results for the heat 1047298ow and corresponding
heat losses through the heat exchangers calculated at the end of a
single cycle [25] As it can be seen the total heat requirements are
almost three times the requirements calculated without consid-
ering the losses It can also be seen that the shuttle conduction
losses represent the main heat loss during the cycle These corre-
spond to the losses due to the oscillation of the hot displacer across
the temperature gradient in the working spaces of the engine
The cyclic variation for the heat 1047298ow is additionally shown in
Fig 15The heat requirements for the heater and cooler present
slight variations during the entire cycle On the other hand the
regenerator presents high variations managing large quantities of
heat This con1047297rms the large importance of this heat exchanger on
the engine performance
55 Brake power and brake ef 1047297ciency
The engine brake power is de1047297ned as the net brake work per
cycle (Ws) times the engine frequency (freq)
Pbr frac14 Ws freq (12)
The net brake work and the total heat requirement presented
on Tables 5 and 6 respectively are doubled considering the double
cylinder engine These values are reported on Table 7 which alsoreports the thermal and brake ef 1047297ciencies for the engine
The results re1047298ect the low performance of the engine under the
experimental conditions This was mainly attributed to the forced
work and the mechanical ef 1047297ciency as it was analysed in the pre-
vious section In addition complementary works will broad this
analysis with the aim of propose improvements on the engine
design and operational parameters
6 Conclusions
In the present work a thermodynamic model for a Stirling en-
gine was improved by including the numerical evaluation of the
forced work and the mechanical ef 1047297ciency then validated against
experimental data and 1047297nally implemented for the simulation of an engine prototype The numerical model developed considered
the analytic approach proposed by Senft [23] but extended its
application for the case of the more realistic adiabatic working
spaces assumptions Consequently the effective work taken from
the shaft is better estimated and thus used for a more complete
analysis of the thermal and mechanical performance of an engine
For this article the analysis considered a novel gamma engine
prototype under the experimental conditions of a micro scale
combined heat and power system fuelled by wood pellets
The simulation results were compared with the experimental
data measured during long time runs of the system The model
performance was very good for the prediction of the temperatures
in the different spaces of the engine In addition the estimations for
the net brake power also presented results similar to the measured
values However additional experimental work should be per-
formed to obtain data to validate the calculation of the different
losses through the engine
According to the results obtained the thermal performance of
the engine was found acceptable and thus the low power output
measured is preliminary attributed to a reduced mechanical ef 1047297-
ciency of the system The possible reasons for this low performance
were further analysed with the different results for the tempera-
tures variation mass and volume variation pressure drops and the
pressure volume diagrams obtained with the model According to
these analyses the dynamics of the volumes variation and the
crank mechanism may also be improved in order to obtain higher
network during the cycle In addition it was found that the engine
performance is very sensitive to the effect of the buffer pressure
These results will be extended with a sensitivity analysis for the
system on a complementary work that aims to identify better the
effect of the different parameters on the engine performance
Acknowledgements
This work was possible thanks to the 1047297nancial support of the
Swedish International Development Cooperation Agency the di-
vision of Heat and Power Technology Department of Energy
Technology at Royal Institute of Technology (KTH) in Sweden and
Universidad Mayor de San Simon (UMSS) in Bolivia
Appendix A Detailed Stirling engine parameters
Table 7
Power output and ef 1047297ciency of the engine
Brake power (W) Heat requirement (W) Thermal ef 1047297ciency () Mechanical ef 1047297ciency () Brake ef 1047297ciency ()
5358 184535 1610 1810 290
Table A1
Inputs for the cooler in ACM
Variable Value Units Description
do 0005 m Tubes external diameter
di 0003 m Tubes internal diameter
kw 14200 Wm K Material conductivity
L 0032 m Tubes length
num 162 e Number of tubes
sl 0005 m Space between tubes
Table A2
Inputs for the heater in ACM
Variable Value Units Description
de 0005 M Tubes external diameter
di 00031 M Tubes internal diameterkw 142 Wm K Material conductivity
len 0149 m Tubes length
num 360 e Number of tubes
sl 0005 m Space between tubes
Table A3
Inputs for the regenerator in ACM
Variable Value Units Description
Din 0078 m Regenerator housing internal diameter
dout 0 107 m Regenera tor h ousin g extern al diameter
dwir e 2 1e-0 04 m Wi re di ameter of t he ma tri x
kwr 270 Wm K Thermal conductivity of the matrix material
Lr 007 m Length of the regenerator housing
Porosity 087 Matrix porosity
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Appendix B
Main equations for the Stirling engine model
Stirling engine module
Mean pressure
P frac14 MR
V cTc
thorn V kTk
thorn V rTr
thorn V hTh
thorn V eTe
Pressure variation
dP
d4frac14
gP
0BB
vV cv4
Tck
thorn
vV ev4
The
1CCA
V cTckthorn g
V kTk
thorn V rTrthorn V hTh
thorn V eThe
Mass of the working gas in the different spaces
mc frac14 p
V c
RTc
mk frac14 p
V k
RTk
mr frac14 p
V r
RTr
mh frac14 p
V h
RTh
me frac14 p
V e
RTe
Mass accumulation
dmk
d4frac14
mk
P
vP
v4
dmh
d4frac14
mh
P
vP
v4
dmr
d4frac14
mr
P
vP
v4
dmc
d4frac14
P
vV cv4
thorn
V c
vPv4
g
RTck
dme
d4frac14
P
vV ev4
thorn
V e
vPv4
g
RThe
Mass 1047298ow
mck frac14 dmc mhe frac14 dme mkr frac14 mck dmkmrh frac14 mhe thorn dmh
Conditional temperatures
If mck gt 0 then Tck frac14 Tc else Tck frac14 Tk
If mhe gt 0 then The frac14 Th else The frac14 Te
Temperatures
dTc
d4frac14 Tc
0BBvPv4
P
thorn
vV cv4
V c
vmc
v4
mc
1CCA
dTe
d4frac14 Te
0BBvPv4
P thorn
vV ev4
V e
vmev4
me
1CCAEnergy
dQ kd4
frac14
V k
vPv4
Cv
R CpethTckmck TkrmkrTHORN
dQ rd4
frac14
V r
vPv4
Cv
R C pethTkrmkr TrhmrhTHORN
dQ hd4
frac14V hvPv4
Cv
R CpethTrhmrh ThemheTHORN
dWc
d4frac14 P
vV cv4
dWe
d4frac14 P
vV ev4
Internal heat transfer module
Heat transfer from the heater wall to the working gas
Q h frac14 1
R cih thorn R hih thorn R fihethTwoh ThTHORN
Heat transfer from the cooler wall to the working gas
Q k frac14 1
R cik thorn R hik thorn R fikethTwik TkTHORN
Heat loss during the regenerator process
Q lossr frac14 eth1 3THORN Q r
Regenerator effectiveness
3frac14 NTU
1 thorn NTU
External heat transfer module
Heat transfer from the 1047298
ame to the external wall of the heater
Table A4
Inputs for the expansion-compression spaces and crank mechanism
Variable Value Units Description
vclc 44e-006 m3 Compression space clearance volume
vcle 26e-005 m3 Expansion space clearance volume
vswc 926e-005 m3 Compression space swept volume
vswe 25134e-004 m3 Expansion space swept volume
dispd 0062 m Displacer diameter
displ 007 m Displacer lengthef fmek 08 e Mechanism effectiveness
freq 5 Hz Frequency
jgap 0006 M Gap between cylinder displacer and wall
kpist 1627 Wm K Piston conductivity
pbuff 12ethorn006 Pa Buffer pressure
phase 900 deg Phase angle advance
pmean 1 25ethorn00 6 Pa Mean o pera tin g pressure
strk 0035 m Displacer stroke
dispd 0062 m Displacer diameter
Table A5
Working and cooling 1047298uid inputs in ACM
Variable Value Units Description
Working Fluid Air e Working 1047298uid inside the engine
Cooling FLUID Water e Cooling 1047298uid through the engine cooler
Tcooling 2 88 K Inl et tempera ture of th e co ol in g 1047298uid
Table A6
Fouling factors and external combustion inputs in ACM
Variable Value Units Description
T ad 1 38 7 K Fla me temper at ur e i n t he c omb ustio n ch amb er
absorp 070 e Absorptivity of the heater material
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Q h frac14 11
hrhAohthorn R foh
ethTad TwohTHORN
hrh frac14 assAohFR ethTad thorn TwohTHORN
T2ad thorn T2
woh
Estimation of the outlet temperature of the cooling 1047298uid
Twok frac14 Twater in thorn Q k
1
hokAokthorn
1
2mwaterCpwater
hok frac14 11
hwaterthorn R fok
Energy losses
Pressure drop in the heat exchangers
DP frac14
f
dhy
1
2 rv
2
l
Pressure drop in the regenerator based on the correlations of
Thomas and Pittman [37]
DP frac14 Cf nr
2u2
Cf frac14 Cfd thornCsf
Re
Csf frac14 68556 Cfd frac14 05274 wire screensCsf frac14 70035 Cfd frac14 09307 metal felts
Total pumping losses
Wploss frac14
Z 2p0
Xifrac143
ifrac141
DPi dV edq
$dq
Energy losses due to shuttle conduction
Q lsh frac14 04Z2KpistDd
JL dethTe TcTHORN
Mechanical ef 1047297ciency module
Mechanical ef 1047297ciency
hm frac14 Ws
Wi
Mechanical ef 1047297ciency considering the mechanism effectiveness
and forced work
hm frac14 E
1
E E
W
Wi
Forced work
W frac14
I ethP PbTHORNthorndV thorn
I ethP PbTHORNdV thorn
Brake ef 1047297
ciency
hb frac14 Ws
Q htfrac14
Wi
Q ht
Ws
Wifrac14 hthm
References
[1] DG Thombare SK Verma Technological development in the Stirling cycle
engines Renew Sustain Energy Rev 12 (2008) 1e
38 httpdxdoiorg101016jrser200607001
[2] AD Peacock M Newborough Impact of micro-CHP systems on domesticsector CO2 emissions Appl Therm Eng 25 (2005) 2653e2676
[3] B Thomas Benchmark testing of Micro-CHP units Appl Therm Eng 28(2008) 2049e2054 httpdxdoiorg101016japplthermaleng200803010
[4] L Dong H Liu S Riffat Development of small-scale and micro-scale biomass-fuelled CHP systems e a literature review Appl Therm Eng 29 (2009)2119e2126 httpdxdoiorg101016japplthermaleng200812004
[5] I Gonzalez-Pino A Campos-Celador E Perez-Iribarren J Teres-Zubiaga JM Sala Parametric study of the operational and economic feasibility of Stirling micro-cogeneration devices in Spain Appl Therm Eng (2013) httpdxdoiorg101016japplthermaleng201312020
[6] R Dyson S Wilson R Tew Review of computational Stirling analysismethods in 2nd International Energy Conversion Engineering ConferenceAmerican Institute of Aeronautics and Astronautics 2004 httpdxdoiorg10251462004-5582
[7] C-H Cheng Y-J Yu Numerical model for predicting thermodynamic cycleand thermal ef 1047297ciency of a beta-type Stirling engine with rhombic-drivemechanism Renew Energy 35 (2010) 2590e2601 httpdxdoiorg101016jrenene201004002
[8] N Seraj Mehdizadeh P Stouffs Simulation of a Martini displacer free pistonStirling engine for electric power generation Int J Thermodyn 3 (1) (2000)27e34 httpdxdoiorg105541ijot30
[9] N Parlak A Wagner M Elsner HS Soyhan Thermodynamic analysis of agamma type Stirling engine in non-ideal adiabatic conditions Renew Energy34 (2009) 266e273 httpdxdoiorg101016jrenene200802030
[10] JM Strauss RT Dobson Evaluation of a second order simulation for Sterlingengine design and optimisation J Energy South Afr 21 (2010) 17e29
[11] I Tlili Y Timoumi SB Nasrallah Thermodynamic analysis of the Stirling heatengine with regenerative losses and internal irreversibilities Int J Engine Res9 (2008) 45e56 httpdxdoiorg10124314680874JER01707
[12] K Mahkamov D Djumanov Three-dimensional CFD modeling of a Stirlingengine in Proceedings of the 11th international Stirling Engine Conferencevol 19 2003
[13] M Ibrahim R Tew Z Zhang D Gedeon T Simon CFD Modeling of Free-piston Stirling Engines Cleveland Ohio 2001
[14] S Wilson R Dyson Multi-D CFD modeling of a free-piston Stirling convertorat NASA GRC in Proc 2nd International Energy Conversion EngineeringConference vol 5673 2004
[15] E Eid Performance of a beta-con1047297guration heat engine having a regenerativedisplacer Renew Energy 34 (2009) 2404e2413 httpdxdoiorg101016
jrenene200903016[16] AA El-Ehwany GM Hennes EI Eid EA El-Kenany Development of the
performance of an alpha-type heat engine by using elbow-bend transposed-1047298uids heat exchanger as a heater and a cooler Energy Convers Manag 52(2011) 1010e1019 httpdxdoiorg101016jenconman201008029
[17] C-H Cheng Y-J Yu Dynamic simulation of a beta-type Stirling engine withcam-drive mechanism via the combination of the thermodynamic and dy-namic models Renew Energy 36 (2011) 714e725
[18] E Entchev J Gusdorf M Swinton M Bell F Szadkowski W Kalb1047298eisch et alMicro-generation technology assessment for housing technology EnergyBuild 36 (2004) 925e931 httpdxdoiorg101016jenbuild200403004
[19] M Paringlsson H Carlsen Development of a wood powder fuelled 35 kW StirlingCHP unit in Proceedings of the 11th ISEC (International Stirling EngineConference) 2003 pp 221e230
[20] a Nishiyama H Shimojima a Ishikawa Y Itaya S Kambara H Moritomi etal Fuel and emissions properties of Stirling engine operated with woodpowder Fuel 86 (2007) 2333e2342 httpdxdoiorg101016
jfuel200701040[21] K Sato N Ohiwa A Ishikawa H Shimojima A Nishiyama Y Moriya
Development of small-scale CHP Plant with a wood powder-fueled Stirlingengine J Power Energy Syst 2 (2008) 1221e1231 httpdxdoiorg101299
jpes21221[22] I Urieli DM Berchowitz Stirling Cycle Engine Analysis Hilger Ltd Bristol
1984[23] JR Senft Mechanical Ef 1047297ciency of Heat Engines Cambridge University Press
2007[24] E Cardozo C Erlich A Malmquist L Alejo Integration of a wood pellet
burner and a Stirling engine to produce residential heat and power ApplTherm Eng (2014) httpdxdoiorg101016japplthermaleng201408024
[25] JA Araoz M Salomon L Alejo TH Fransson Non-ideal Stirling enginethermodynamic model suitable for the integration into overall energy sys-tems Appl Therm Eng 73 (2014) 203e219 httpdxdoiorg101016
japplthermaleng201407050
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 30 29
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httpslidepdfcomreaderfullapplied-thermal-engineering-volume-83-issue-2015-doi-1010162fjapplthermaleng201503006 1515
[26] FP Incropera AS Lavine DP DeWitt Fundamentals of Heat and MassTransfer John Wiley amp Sons 2011
[27] B Thomas D Pittman Update on the evaluation of different correlations forthe 1047298ow friction factor and heat transfer of Stirling engine regenerators inCollection of Technical Papers 35th Intersociety Energy Conversion Engi-neering Conference and Exhibit (IECEC) (Cat No00CH37022) American InstAeronaut amp Astronautics 2000 pp 76e84 httpdxdoiorg101109IECEC2000870632
[28] Genoastirling Srl 2014 wwwgenoastirlingcom[29] RC Tew K Jefferies D Miao USD of ED of TE Conservation LR Center
A Stirling Engine Computer Model for Performance Calculations (GoogleEBook) Department of Energy Of 1047297ce of Conservation and Solar ApplicationsDivision of Transportation Energy Conservation 1978
[30] I Urieli CJ Rallis DM Berchowitz Computer simulation of Stirling cyclemachines in 12th Intersociety Energy Conversion Engineering Conferencevol 1 American Nuclear Society Washington DC 1977 pp 1512e1521
[31] K Mahkamov Design improvements to a biomass Stirling engine usingmathematical analysis and 3D CFD modeling J Energy Resour Technol 128(2006) 203 httpdxdoiorg10111512213273
[32] H Snyman Second Order Analyses Methods for Stirling Engine Design Uni-versity of Stellenbosch Stellenbosch 2007
[33] G Sewell Numerical Solution of Ordinary and Partial Differential Equationssecond ed John Wiley amp Sons 2005
[34] DD Do Richard G Rice Applied Mathematics and Modeling for ChemicalEngineers [Hardcover] second ed Wiley-AIChE 2012
[35] Aspentech Aspen Custom Modelerreg AspenTech 2015
[36] Aspentech Chemical Process Optimization Software d
Chemical ProcessDesign Aspen Plus 2015
[37] B Thomas D Pittman AIAA-2000-2812 Update on the Evaluation of DifferentCorrelations for the Flow Friction Factor and Heat Transfer of Stirling EngineRegenerators 2000 pp 76e84
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Fig 6 Layout Stirling engine model in ACM
Table 2
Description of the blocks for the ACM model
Block name Description
Comp-Exp The block contains the data that describes the volume variation inside the engine The swept dead volumes crank mechanism and the
characteristics of the pistons
Cooler The block contains the geometrical data for the cooler heat exchanger
Heater T he b lock c ont ai ns t he geometr ic al data for t he heat exch an ger
Regenerator The block contains the geometrical data for the regenerator and the details of the matrix porosity and material
Ext-heat The characteristics of the external heat source are contained in this block
Mech_Ef 1047297ciency The block contains the parameters for the calculation of the engine mechanical ef 1047297ciency
CoolingFluid The characteristics of the external cooling 1047298uid are contained in the block
WorkingGAS The block contains the parameters for the calculation of the properties for the working gas inside the engine
Stirling This is the main block and contains the main equations that describe the thermodynamic analysis of the engine
Fig 7 a) Schematic set-up of the Stirling engine integrated with a combustion chamber and a boiler [27] b) Temperature measurement points for the working gas in the Stirling
engine T2 hot side T10 cold side T11 T12 hot and cold side of the regenerator [24]
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burner in order to produce heat and power simultaneously as
shown in Fig 7a This con1047297guration had technical limitations that
are still being studied in order to improve both power and thermal
outputs But despite of these limitations experimental results were
obtained and these were compared with the model
32 Inputs for the model
The main inputs for the engine simulation are shown in Table 1
Supplementary inputs that include the design and operational
characteristics of the engine are presented in Appendix A
The model also needs to consider the relation of the crank
mechanism and the variation of the volumes inside the working
spaces Therefore considering that the engine has gamma type
con1047297guration the following relations for the expansion and
compression spaces were included [23]
V e frac14 V cle thornV swe
2 eth1 thorn coseth4 thorn aTHORNTHORN (8)
V c frac14 V clc thorn ethV swe V eTHORN thornV swc
2 eth1 thorn coseth4THORNTHORN (9)
Furthermore the following volume derivatives were evaluated
dV e frac14 V swe
2 sineth4 thorn aTHORN (10)
dV c
frac14 dV e
V swc
2 sineth4THORN (11)
Fig 8 Measurement points for the CHP-Stirling experimental rig [24]
Table 3Comparison of the measured and predicted temperatures along the engine
Time (s) TfM (K) ThM (K) Th (K) Error TkM (K) Tk (K) Error TrM (K) Tr (K) Error
3780e3900 13878 8164 8184 025 3224 3211 041 6018 5316 116
3900e4020 13829 8196 8075 147 3218 3214 012 6006 5277 1215
4020e4140 13931 8232 8142 109 3216 3215 004 6012 5302 118
4140e4200 13778 8308 7981 394 3216 3216 001 6036 5243 1314
4200e4380 13835 8374 8063 371 3224 3214 031 6075 5272 1321
4380e4560 13777 8518 7957 659 3218 3217 003 6142 5234 1478
4560e4680 13857 8536 8071 545 3217 3215 007 6154 5276 1426
4680e4800 13844 8464 8021 523 3216 3217 001 6135 5258 143
4800e4980 13669 8433 7708 859 3221 3223 005 6129 5144 1607
Fig 9 Temperature variation along the heat exchangers and regenerator temperature assumed by the model (T r)
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33 Numerical solution
The system consists of a set of algebraic differential equations
which are shown in Appendix B These consider as boundary con-
ditions that the temperatures of the working gas at the end of the
cycle must be equal to the temperatures at the beginning of the
cycle once cyclic steady state conditions are reached Therefore an
iterative shooting method [33] using a fourth order Runge Kutta
scheme for the time discretization was implemented for the nu-
merical solution The iteration process was done until cyclic steadystate conditions which is numerically reached when the difference
between the assumed initial values and the values calculated at the
end of the cycle are lower than a de1047297ned error After the cyclic
steady state solution was reached the energy losses and the forced
work were evaluated The forced work was calculated using the
classical Simpson 38 numerical integration rule [34] The scheme
in Fig 5 summarizes the iterative steps for the solution
The numerical solution was implemented in Aspen Custom
Modellerreg (ACM) [35] which is a product from Aspen Plusreg that
permits the elaboration of customized models [36] This software
has its own modelling language and can also be coupled with Cthornthorn
procedures The layout of the model in ACM is shown in Fig 6 The
blocks were programmed with the equations shown in the
Appendix B and then the solution of the system was obtained withthe algorithm previously described
The descriptions of the blocks are shown in Table 2 Additional
details of the block inputs are given in Appendix A
4 Model validation
The geometrical and operational characteristics for the Genoa
engine are described in Table 1 and Appendix A The engine was
mounted in the experimental rig shown in Fig 7a In addition the
temperatures of the working gas were measured at the different
points of the engine shown in Fig 7b
The experimental rig used wood pellets as fuel Additional
temperatures measured for the validation were The temperature
close to the 1047298ame (T1) the water inlet temperature (T8) the wateroutlet temperature (T9) Other measurements are also as shown in
Fig 8
The temperature T1 was measured using a type K empty 15 mm
Inconel 600 thermocouple The additional temperatures shown in
Fig 8 were measured using type K empty 10 mm thermocouples
Considering the type of thermocouples the expanded uncertainty
was plusmn32 C with a coverage factor of 2 The speed of the engine
crankshaft was monitored by a pulse sensor and a frequency to
analog converter (OMROM E2A and Red Lion IFMA) with an un-
certaintyplusmn 02 The pressure inside the engine was measured with
a pressure transducer (RS type 46) with analog signal and an un-
certainty of plusmn01 bar All the measurements were recorded from the
beginning to the end of the test using a data logger Additional
details of the measurements are reported in Cardozo et al [24]The engine was run during long periods and the data was
measured constantly However for the validation purposes only the
periods were stability is reached were considered In this case the
steady state condition was dif 1047297cult to reach due to the constant
variation of the 1047298ame temperature [24] Therefore average values
for the measurements within certain stability periods were taken
These are compared with the values calculated by the model at the
different values measured for the 1047298ame temperature shown in
Table 3
Fig 10 Temperature variation along the engine 1047298
ame temperature Tad frac14
1388 K
Table 4
Measured and predicted brake power
Time (s) TfM (K) Measured frequency (Hz) Measured pressure (bar) Brake power (W) experimental Brake power (W) calculated Error
3780e3900 13878 517 1250 5472 5359 206
3900e4020 13829 526 1250 5539 5208 597
4020e4140 13930 527 1250 5561 5349 381
4140e4200 13778 533 1250 4635 5003 794
4200e4380 13835 528 1250 5359 5197 302
4380e
4560 13777 536 1250 5091 5033 1144560e4680 13857 529 1250 5096 5163 131
4680e4800 13843 534 1250 559 5153 782
4800e4980 13669 556 1254 4713 4613 212
Fig 11 Volumes variation during the engine cycle
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From Table 3 the model presents good accuracy for the pre-
diction of the cooler temperatures (Tk) with the maximum error of
the order of plusmn041 In addition the calculations for the heater
temperatures (Th) present reasonable accuracy at initial times but
then the error increases This growth may be explained with the
thermal inertia that constantly increments the measured temper-
ature even on periods where the1047298ame temperature decreases This
thermal inertia is neglected by the model since it assumes steady
state heat transfer conditions On the other hand the prediction of
the mean temperature in the regenerator space (Tr) presents
higher differences This is analysed with the Fig 9 below which
shows the variations of the temperatures inside the heat ex-
changers assumed by the model
From Fig 9 it can be seen that the model assumes that thetemperatures at the interfaces heater-regenerator and cooler-
regenerator were equal to the temperatures at the cooler (T k) and
heater (Th) spaces respectively Therefore the average temperature
at the regenerator (Tr) was calculated with these values This
assumption neglects the axial temperature variation along the
heater and cooler which is re1047298ected on the measurements taken at
the exact interfaces positions T11 and T12This explains the differ-
ence between the average regenerator temperature calculated with
the measured temperatures (TrM) and the calculated with the es-
timations of the model Tr as it is shown in Table 3 However
considering that the model was capable to calculate within a good
degree of accuracy the power output measured during the exper-
imental runs it can be inferred that the error for the regenerator
temperature estimation have little in1047298
uence on the brake powercalculation This is shown in Table 4 where the values for the
measured and calculated brake power are compared at different
operating conditions The percentage error ranges from plusmn131 to
plusmn794 which is an acceptable approximation for 1047297rst design
calculations
5 Results and discussion
This section presents additionally results for the simulation of
the engine under the experimental conditions described before
This aims to completely describe the thermodynamic performance
of the engine and thus identify the main limitations that the engine
presents
51 Temperature variation
Fig10 shows the temperature variation in the differentspaces of
the engine cylinder once the cyclic steady state conditions are
reached This 1047297gure displays the sinusoidal variation of the tem-
peratures inside the compression (Tc) and expansion (Te) spaces It
can also be seen that the expansion space presents periods with
elevated temperatures which results into a high thermal stress for
the material and therefore further engine deterioration In addition
the 1047297gure also shows that the mean temperatures for the working
1047298uid inside the heater (Th) and cooler (Tk) are close to the heat
exchangers walls temperature (Twk Twhe) This indicates a good
heat transfer rate on both heat exchangers and consequently a
good thermal performance based on the model assumptions
However it is important to notice that this performance will
decrease with the time due to the fouling on the heat exchangers
which is not accounted for in the engine model
52 Mass distribution and volumes variation
The mass distribution and volumes variation for the engine
during a complete cycle are shown in Figs 11 and 12 respectivelyThese variations permit to analyse the engine dynamics during the
compression and expansion processes
Fig 11 permits to identify the following processes the
compression characterized by the decrease in the total volume
from the time around t frac14 001 to t frac14 004 the heating process
when the total volume variation is not pronounced and the tem-
peratures increase around t frac14 004 to t frac14 006 the expansion
process when the total volume increases around t frac14 006 to
t frac14 009 and the cooling process when the volume stays almost
constant and the temperatures decrease at the times around
t frac14 009 to t frac14 010 and t frac14 0 to t frac14 001
The compressionperiod starts with the increment of the mass in
the compression space and a decrease of the mass in the expansion
space as shown in Fig 12 The decreasing mass in the expansionspace indicates a good dynamic for the compression process
because it is desirable to keep low the hotter portion of the mass
during this period However the mass on the compression space is
too high which is not desirable since this will be re1047298ected in a large
negative compression work In addition the expansion process also
presents a reduced performance due to the low values for the mass
in the expansion space during the expansion process This repre-
sents an expansion with low hotter mass and thus a low working
output to the shaft Furthermore the low mass in the expansion
space during the heating period might be the main cause for the
high temperatures reached Therefore the volumes and mass 1047298ow
dynamics of the reference case should be improved to reach higher
work outputs and avoid the overheating of the expansion chamber
Fig 12 also shows that the mass in the heater and cooler arereally small compared with the mass in the regenerator during the
Table 5
Engine work 1047298ow per cycle
Model output per cycle Aspen Custom Modeller (ACM)
Expansion Work (We Jcycle) 5262
Compression Work (Wc Jcycle) 2339
Pre ssure drop lost heater (J cycle) 021
Pre ssure drop lost cooler (J cycle) 007
Pressure drop lost regenerator (Jcycle) 028
Total lost due to pressure drop (Jcycle) 056
Net indicated work (Wi Jcycle) 2867
Forced work (W Jcycle) 2349
Brake Work Output (Wbr Jcycle) 518
Fig 12 Mass variation inside the engine spaces during a complete cycle
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complete cycle This re1047298ects the high importance of the regenerator
ef 1047297ciency for the engine performance
53 Work 1047298ow
Table 5 shows the simulation results for the compression and
expansion work during a single cycle This table also presents the
different work losses estimated for the system
The temperatures measured and the temperatures calculated
show a good thermal performance of the engine But the measured
brake power was very low Different problems on the engine
design and operational conditions may explain these very lowresults However additional experimental instrumentation is
needed for a detailed design study For this reason the present
analysis considers a theoretical approach that may be later com-
plemented with experimental studies This theoretical approach
considers Eq (7) From this equation and considering that the
thermal performance was found acceptable the main losses should
correspond to a low mechanical ef 1047297ciency of the prototype This
mechanical ef 1047297ciency is reduced by the presence of forced work
during the cycle and mechanical friction on the crank mechanism
Fig 13 presents the evaluation of the forced work in a pressure
volume diagram for the gas cycle inside the gamma prototype
From this it can be seen that the forced work (W) is mainly due to
the forced expansion process This means that at the experimentalconditions large part of the cyclic work may have been used to
complete the forced expansion process and thus the real engine
output is smaller than expected
The results discussed above are complemented with the vari-
ation of the compression (Wc) expansion (We) and net indicated
work (Wi) during the cycle shown in Fig 14
Fig 14 shows that during the 1047297rst part of the cycle from t frac14 0 to
t frac14 0045 the compression and expansion spaces present
exchanged roles This means that an increment of the volume is
presented in the compression space and a decrement of the volume
is present in the expansion one This reduced the engine perfor-
mance but it cannot be avoided since the gas needs to pass from
one space to another Regarding the second part of the cycle from
t frac14 0045 to t frac14 0095 the expansion and compression are shapedas expected and thus indicate a better dynamic during this period
However considering that large part of the expansion process is
forced the net brake work is low as reported in Table 5
From the previous analysis it can be concluded that a detailed
evaluation of the volumes dynamics the cranks mechanism effec-
tiveness and the forced work during the cycle must be considered
Fig 15 Heat 1047298
ow variation during the engine cycle
Table 6
Heat 1047298ow and heat loses during the cycle
Heat 1047298ow (Jcycle)
Heat exchanger space
Heater 1047298ow (Q hcJcycle) 5282
Cooler 1047298ow (Q kc Jcycle) 2356
Regenerator 1047298ow (Q rc Jcycle) 005
Heat lossesInternal conduction losses (Q lkc Jcycle) 2698
Shuttle conduction losses (Q lshc Jcycle) 8004
Regenerator losses during heating (Q lossrc Jcycle) 1862
Regenerator losses during cooling (Q lossrc Jcycle) 1862
Total heat requirements
Heating requirements (Q htc Jcycle) 17847
Cooling requirements (Q ktc Jcycle) 4218
Fig 14 Work 1047298
ow during the engine cycle
Fig 13 Pressureevolume diagram and forced work during the cycle
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in order to re-design the engine for a better performance This will
be covered on a detailed parametric study to be reported on a next
article
54 Heat 1047298ow
Table 6 presents the results for the heat 1047298ow and corresponding
heat losses through the heat exchangers calculated at the end of a
single cycle [25] As it can be seen the total heat requirements are
almost three times the requirements calculated without consid-
ering the losses It can also be seen that the shuttle conduction
losses represent the main heat loss during the cycle These corre-
spond to the losses due to the oscillation of the hot displacer across
the temperature gradient in the working spaces of the engine
The cyclic variation for the heat 1047298ow is additionally shown in
Fig 15The heat requirements for the heater and cooler present
slight variations during the entire cycle On the other hand the
regenerator presents high variations managing large quantities of
heat This con1047297rms the large importance of this heat exchanger on
the engine performance
55 Brake power and brake ef 1047297ciency
The engine brake power is de1047297ned as the net brake work per
cycle (Ws) times the engine frequency (freq)
Pbr frac14 Ws freq (12)
The net brake work and the total heat requirement presented
on Tables 5 and 6 respectively are doubled considering the double
cylinder engine These values are reported on Table 7 which alsoreports the thermal and brake ef 1047297ciencies for the engine
The results re1047298ect the low performance of the engine under the
experimental conditions This was mainly attributed to the forced
work and the mechanical ef 1047297ciency as it was analysed in the pre-
vious section In addition complementary works will broad this
analysis with the aim of propose improvements on the engine
design and operational parameters
6 Conclusions
In the present work a thermodynamic model for a Stirling en-
gine was improved by including the numerical evaluation of the
forced work and the mechanical ef 1047297ciency then validated against
experimental data and 1047297nally implemented for the simulation of an engine prototype The numerical model developed considered
the analytic approach proposed by Senft [23] but extended its
application for the case of the more realistic adiabatic working
spaces assumptions Consequently the effective work taken from
the shaft is better estimated and thus used for a more complete
analysis of the thermal and mechanical performance of an engine
For this article the analysis considered a novel gamma engine
prototype under the experimental conditions of a micro scale
combined heat and power system fuelled by wood pellets
The simulation results were compared with the experimental
data measured during long time runs of the system The model
performance was very good for the prediction of the temperatures
in the different spaces of the engine In addition the estimations for
the net brake power also presented results similar to the measured
values However additional experimental work should be per-
formed to obtain data to validate the calculation of the different
losses through the engine
According to the results obtained the thermal performance of
the engine was found acceptable and thus the low power output
measured is preliminary attributed to a reduced mechanical ef 1047297-
ciency of the system The possible reasons for this low performance
were further analysed with the different results for the tempera-
tures variation mass and volume variation pressure drops and the
pressure volume diagrams obtained with the model According to
these analyses the dynamics of the volumes variation and the
crank mechanism may also be improved in order to obtain higher
network during the cycle In addition it was found that the engine
performance is very sensitive to the effect of the buffer pressure
These results will be extended with a sensitivity analysis for the
system on a complementary work that aims to identify better the
effect of the different parameters on the engine performance
Acknowledgements
This work was possible thanks to the 1047297nancial support of the
Swedish International Development Cooperation Agency the di-
vision of Heat and Power Technology Department of Energy
Technology at Royal Institute of Technology (KTH) in Sweden and
Universidad Mayor de San Simon (UMSS) in Bolivia
Appendix A Detailed Stirling engine parameters
Table 7
Power output and ef 1047297ciency of the engine
Brake power (W) Heat requirement (W) Thermal ef 1047297ciency () Mechanical ef 1047297ciency () Brake ef 1047297ciency ()
5358 184535 1610 1810 290
Table A1
Inputs for the cooler in ACM
Variable Value Units Description
do 0005 m Tubes external diameter
di 0003 m Tubes internal diameter
kw 14200 Wm K Material conductivity
L 0032 m Tubes length
num 162 e Number of tubes
sl 0005 m Space between tubes
Table A2
Inputs for the heater in ACM
Variable Value Units Description
de 0005 M Tubes external diameter
di 00031 M Tubes internal diameterkw 142 Wm K Material conductivity
len 0149 m Tubes length
num 360 e Number of tubes
sl 0005 m Space between tubes
Table A3
Inputs for the regenerator in ACM
Variable Value Units Description
Din 0078 m Regenerator housing internal diameter
dout 0 107 m Regenera tor h ousin g extern al diameter
dwir e 2 1e-0 04 m Wi re di ameter of t he ma tri x
kwr 270 Wm K Thermal conductivity of the matrix material
Lr 007 m Length of the regenerator housing
Porosity 087 Matrix porosity
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Appendix B
Main equations for the Stirling engine model
Stirling engine module
Mean pressure
P frac14 MR
V cTc
thorn V kTk
thorn V rTr
thorn V hTh
thorn V eTe
Pressure variation
dP
d4frac14
gP
0BB
vV cv4
Tck
thorn
vV ev4
The
1CCA
V cTckthorn g
V kTk
thorn V rTrthorn V hTh
thorn V eThe
Mass of the working gas in the different spaces
mc frac14 p
V c
RTc
mk frac14 p
V k
RTk
mr frac14 p
V r
RTr
mh frac14 p
V h
RTh
me frac14 p
V e
RTe
Mass accumulation
dmk
d4frac14
mk
P
vP
v4
dmh
d4frac14
mh
P
vP
v4
dmr
d4frac14
mr
P
vP
v4
dmc
d4frac14
P
vV cv4
thorn
V c
vPv4
g
RTck
dme
d4frac14
P
vV ev4
thorn
V e
vPv4
g
RThe
Mass 1047298ow
mck frac14 dmc mhe frac14 dme mkr frac14 mck dmkmrh frac14 mhe thorn dmh
Conditional temperatures
If mck gt 0 then Tck frac14 Tc else Tck frac14 Tk
If mhe gt 0 then The frac14 Th else The frac14 Te
Temperatures
dTc
d4frac14 Tc
0BBvPv4
P
thorn
vV cv4
V c
vmc
v4
mc
1CCA
dTe
d4frac14 Te
0BBvPv4
P thorn
vV ev4
V e
vmev4
me
1CCAEnergy
dQ kd4
frac14
V k
vPv4
Cv
R CpethTckmck TkrmkrTHORN
dQ rd4
frac14
V r
vPv4
Cv
R C pethTkrmkr TrhmrhTHORN
dQ hd4
frac14V hvPv4
Cv
R CpethTrhmrh ThemheTHORN
dWc
d4frac14 P
vV cv4
dWe
d4frac14 P
vV ev4
Internal heat transfer module
Heat transfer from the heater wall to the working gas
Q h frac14 1
R cih thorn R hih thorn R fihethTwoh ThTHORN
Heat transfer from the cooler wall to the working gas
Q k frac14 1
R cik thorn R hik thorn R fikethTwik TkTHORN
Heat loss during the regenerator process
Q lossr frac14 eth1 3THORN Q r
Regenerator effectiveness
3frac14 NTU
1 thorn NTU
External heat transfer module
Heat transfer from the 1047298
ame to the external wall of the heater
Table A4
Inputs for the expansion-compression spaces and crank mechanism
Variable Value Units Description
vclc 44e-006 m3 Compression space clearance volume
vcle 26e-005 m3 Expansion space clearance volume
vswc 926e-005 m3 Compression space swept volume
vswe 25134e-004 m3 Expansion space swept volume
dispd 0062 m Displacer diameter
displ 007 m Displacer lengthef fmek 08 e Mechanism effectiveness
freq 5 Hz Frequency
jgap 0006 M Gap between cylinder displacer and wall
kpist 1627 Wm K Piston conductivity
pbuff 12ethorn006 Pa Buffer pressure
phase 900 deg Phase angle advance
pmean 1 25ethorn00 6 Pa Mean o pera tin g pressure
strk 0035 m Displacer stroke
dispd 0062 m Displacer diameter
Table A5
Working and cooling 1047298uid inputs in ACM
Variable Value Units Description
Working Fluid Air e Working 1047298uid inside the engine
Cooling FLUID Water e Cooling 1047298uid through the engine cooler
Tcooling 2 88 K Inl et tempera ture of th e co ol in g 1047298uid
Table A6
Fouling factors and external combustion inputs in ACM
Variable Value Units Description
T ad 1 38 7 K Fla me temper at ur e i n t he c omb ustio n ch amb er
absorp 070 e Absorptivity of the heater material
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Q h frac14 11
hrhAohthorn R foh
ethTad TwohTHORN
hrh frac14 assAohFR ethTad thorn TwohTHORN
T2ad thorn T2
woh
Estimation of the outlet temperature of the cooling 1047298uid
Twok frac14 Twater in thorn Q k
1
hokAokthorn
1
2mwaterCpwater
hok frac14 11
hwaterthorn R fok
Energy losses
Pressure drop in the heat exchangers
DP frac14
f
dhy
1
2 rv
2
l
Pressure drop in the regenerator based on the correlations of
Thomas and Pittman [37]
DP frac14 Cf nr
2u2
Cf frac14 Cfd thornCsf
Re
Csf frac14 68556 Cfd frac14 05274 wire screensCsf frac14 70035 Cfd frac14 09307 metal felts
Total pumping losses
Wploss frac14
Z 2p0
Xifrac143
ifrac141
DPi dV edq
$dq
Energy losses due to shuttle conduction
Q lsh frac14 04Z2KpistDd
JL dethTe TcTHORN
Mechanical ef 1047297ciency module
Mechanical ef 1047297ciency
hm frac14 Ws
Wi
Mechanical ef 1047297ciency considering the mechanism effectiveness
and forced work
hm frac14 E
1
E E
W
Wi
Forced work
W frac14
I ethP PbTHORNthorndV thorn
I ethP PbTHORNdV thorn
Brake ef 1047297
ciency
hb frac14 Ws
Q htfrac14
Wi
Q ht
Ws
Wifrac14 hthm
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A Stirling Engine Computer Model for Performance Calculations (GoogleEBook) Department of Energy Of 1047297ce of Conservation and Solar ApplicationsDivision of Transportation Energy Conservation 1978
[30] I Urieli CJ Rallis DM Berchowitz Computer simulation of Stirling cyclemachines in 12th Intersociety Energy Conversion Engineering Conferencevol 1 American Nuclear Society Washington DC 1977 pp 1512e1521
[31] K Mahkamov Design improvements to a biomass Stirling engine usingmathematical analysis and 3D CFD modeling J Energy Resour Technol 128(2006) 203 httpdxdoiorg10111512213273
[32] H Snyman Second Order Analyses Methods for Stirling Engine Design Uni-versity of Stellenbosch Stellenbosch 2007
[33] G Sewell Numerical Solution of Ordinary and Partial Differential Equationssecond ed John Wiley amp Sons 2005
[34] DD Do Richard G Rice Applied Mathematics and Modeling for ChemicalEngineers [Hardcover] second ed Wiley-AIChE 2012
[35] Aspentech Aspen Custom Modelerreg AspenTech 2015
[36] Aspentech Chemical Process Optimization Software d
Chemical ProcessDesign Aspen Plus 2015
[37] B Thomas D Pittman AIAA-2000-2812 Update on the Evaluation of DifferentCorrelations for the Flow Friction Factor and Heat Transfer of Stirling EngineRegenerators 2000 pp 76e84
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Fig 6 Layout Stirling engine model in ACM
Table 2
Description of the blocks for the ACM model
Block name Description
Comp-Exp The block contains the data that describes the volume variation inside the engine The swept dead volumes crank mechanism and the
characteristics of the pistons
Cooler The block contains the geometrical data for the cooler heat exchanger
Heater T he b lock c ont ai ns t he geometr ic al data for t he heat exch an ger
Regenerator The block contains the geometrical data for the regenerator and the details of the matrix porosity and material
Ext-heat The characteristics of the external heat source are contained in this block
Mech_Ef 1047297ciency The block contains the parameters for the calculation of the engine mechanical ef 1047297ciency
CoolingFluid The characteristics of the external cooling 1047298uid are contained in the block
WorkingGAS The block contains the parameters for the calculation of the properties for the working gas inside the engine
Stirling This is the main block and contains the main equations that describe the thermodynamic analysis of the engine
Fig 7 a) Schematic set-up of the Stirling engine integrated with a combustion chamber and a boiler [27] b) Temperature measurement points for the working gas in the Stirling
engine T2 hot side T10 cold side T11 T12 hot and cold side of the regenerator [24]
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burner in order to produce heat and power simultaneously as
shown in Fig 7a This con1047297guration had technical limitations that
are still being studied in order to improve both power and thermal
outputs But despite of these limitations experimental results were
obtained and these were compared with the model
32 Inputs for the model
The main inputs for the engine simulation are shown in Table 1
Supplementary inputs that include the design and operational
characteristics of the engine are presented in Appendix A
The model also needs to consider the relation of the crank
mechanism and the variation of the volumes inside the working
spaces Therefore considering that the engine has gamma type
con1047297guration the following relations for the expansion and
compression spaces were included [23]
V e frac14 V cle thornV swe
2 eth1 thorn coseth4 thorn aTHORNTHORN (8)
V c frac14 V clc thorn ethV swe V eTHORN thornV swc
2 eth1 thorn coseth4THORNTHORN (9)
Furthermore the following volume derivatives were evaluated
dV e frac14 V swe
2 sineth4 thorn aTHORN (10)
dV c
frac14 dV e
V swc
2 sineth4THORN (11)
Fig 8 Measurement points for the CHP-Stirling experimental rig [24]
Table 3Comparison of the measured and predicted temperatures along the engine
Time (s) TfM (K) ThM (K) Th (K) Error TkM (K) Tk (K) Error TrM (K) Tr (K) Error
3780e3900 13878 8164 8184 025 3224 3211 041 6018 5316 116
3900e4020 13829 8196 8075 147 3218 3214 012 6006 5277 1215
4020e4140 13931 8232 8142 109 3216 3215 004 6012 5302 118
4140e4200 13778 8308 7981 394 3216 3216 001 6036 5243 1314
4200e4380 13835 8374 8063 371 3224 3214 031 6075 5272 1321
4380e4560 13777 8518 7957 659 3218 3217 003 6142 5234 1478
4560e4680 13857 8536 8071 545 3217 3215 007 6154 5276 1426
4680e4800 13844 8464 8021 523 3216 3217 001 6135 5258 143
4800e4980 13669 8433 7708 859 3221 3223 005 6129 5144 1607
Fig 9 Temperature variation along the heat exchangers and regenerator temperature assumed by the model (T r)
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33 Numerical solution
The system consists of a set of algebraic differential equations
which are shown in Appendix B These consider as boundary con-
ditions that the temperatures of the working gas at the end of the
cycle must be equal to the temperatures at the beginning of the
cycle once cyclic steady state conditions are reached Therefore an
iterative shooting method [33] using a fourth order Runge Kutta
scheme for the time discretization was implemented for the nu-
merical solution The iteration process was done until cyclic steadystate conditions which is numerically reached when the difference
between the assumed initial values and the values calculated at the
end of the cycle are lower than a de1047297ned error After the cyclic
steady state solution was reached the energy losses and the forced
work were evaluated The forced work was calculated using the
classical Simpson 38 numerical integration rule [34] The scheme
in Fig 5 summarizes the iterative steps for the solution
The numerical solution was implemented in Aspen Custom
Modellerreg (ACM) [35] which is a product from Aspen Plusreg that
permits the elaboration of customized models [36] This software
has its own modelling language and can also be coupled with Cthornthorn
procedures The layout of the model in ACM is shown in Fig 6 The
blocks were programmed with the equations shown in the
Appendix B and then the solution of the system was obtained withthe algorithm previously described
The descriptions of the blocks are shown in Table 2 Additional
details of the block inputs are given in Appendix A
4 Model validation
The geometrical and operational characteristics for the Genoa
engine are described in Table 1 and Appendix A The engine was
mounted in the experimental rig shown in Fig 7a In addition the
temperatures of the working gas were measured at the different
points of the engine shown in Fig 7b
The experimental rig used wood pellets as fuel Additional
temperatures measured for the validation were The temperature
close to the 1047298ame (T1) the water inlet temperature (T8) the wateroutlet temperature (T9) Other measurements are also as shown in
Fig 8
The temperature T1 was measured using a type K empty 15 mm
Inconel 600 thermocouple The additional temperatures shown in
Fig 8 were measured using type K empty 10 mm thermocouples
Considering the type of thermocouples the expanded uncertainty
was plusmn32 C with a coverage factor of 2 The speed of the engine
crankshaft was monitored by a pulse sensor and a frequency to
analog converter (OMROM E2A and Red Lion IFMA) with an un-
certaintyplusmn 02 The pressure inside the engine was measured with
a pressure transducer (RS type 46) with analog signal and an un-
certainty of plusmn01 bar All the measurements were recorded from the
beginning to the end of the test using a data logger Additional
details of the measurements are reported in Cardozo et al [24]The engine was run during long periods and the data was
measured constantly However for the validation purposes only the
periods were stability is reached were considered In this case the
steady state condition was dif 1047297cult to reach due to the constant
variation of the 1047298ame temperature [24] Therefore average values
for the measurements within certain stability periods were taken
These are compared with the values calculated by the model at the
different values measured for the 1047298ame temperature shown in
Table 3
Fig 10 Temperature variation along the engine 1047298
ame temperature Tad frac14
1388 K
Table 4
Measured and predicted brake power
Time (s) TfM (K) Measured frequency (Hz) Measured pressure (bar) Brake power (W) experimental Brake power (W) calculated Error
3780e3900 13878 517 1250 5472 5359 206
3900e4020 13829 526 1250 5539 5208 597
4020e4140 13930 527 1250 5561 5349 381
4140e4200 13778 533 1250 4635 5003 794
4200e4380 13835 528 1250 5359 5197 302
4380e
4560 13777 536 1250 5091 5033 1144560e4680 13857 529 1250 5096 5163 131
4680e4800 13843 534 1250 559 5153 782
4800e4980 13669 556 1254 4713 4613 212
Fig 11 Volumes variation during the engine cycle
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From Table 3 the model presents good accuracy for the pre-
diction of the cooler temperatures (Tk) with the maximum error of
the order of plusmn041 In addition the calculations for the heater
temperatures (Th) present reasonable accuracy at initial times but
then the error increases This growth may be explained with the
thermal inertia that constantly increments the measured temper-
ature even on periods where the1047298ame temperature decreases This
thermal inertia is neglected by the model since it assumes steady
state heat transfer conditions On the other hand the prediction of
the mean temperature in the regenerator space (Tr) presents
higher differences This is analysed with the Fig 9 below which
shows the variations of the temperatures inside the heat ex-
changers assumed by the model
From Fig 9 it can be seen that the model assumes that thetemperatures at the interfaces heater-regenerator and cooler-
regenerator were equal to the temperatures at the cooler (T k) and
heater (Th) spaces respectively Therefore the average temperature
at the regenerator (Tr) was calculated with these values This
assumption neglects the axial temperature variation along the
heater and cooler which is re1047298ected on the measurements taken at
the exact interfaces positions T11 and T12This explains the differ-
ence between the average regenerator temperature calculated with
the measured temperatures (TrM) and the calculated with the es-
timations of the model Tr as it is shown in Table 3 However
considering that the model was capable to calculate within a good
degree of accuracy the power output measured during the exper-
imental runs it can be inferred that the error for the regenerator
temperature estimation have little in1047298
uence on the brake powercalculation This is shown in Table 4 where the values for the
measured and calculated brake power are compared at different
operating conditions The percentage error ranges from plusmn131 to
plusmn794 which is an acceptable approximation for 1047297rst design
calculations
5 Results and discussion
This section presents additionally results for the simulation of
the engine under the experimental conditions described before
This aims to completely describe the thermodynamic performance
of the engine and thus identify the main limitations that the engine
presents
51 Temperature variation
Fig10 shows the temperature variation in the differentspaces of
the engine cylinder once the cyclic steady state conditions are
reached This 1047297gure displays the sinusoidal variation of the tem-
peratures inside the compression (Tc) and expansion (Te) spaces It
can also be seen that the expansion space presents periods with
elevated temperatures which results into a high thermal stress for
the material and therefore further engine deterioration In addition
the 1047297gure also shows that the mean temperatures for the working
1047298uid inside the heater (Th) and cooler (Tk) are close to the heat
exchangers walls temperature (Twk Twhe) This indicates a good
heat transfer rate on both heat exchangers and consequently a
good thermal performance based on the model assumptions
However it is important to notice that this performance will
decrease with the time due to the fouling on the heat exchangers
which is not accounted for in the engine model
52 Mass distribution and volumes variation
The mass distribution and volumes variation for the engine
during a complete cycle are shown in Figs 11 and 12 respectivelyThese variations permit to analyse the engine dynamics during the
compression and expansion processes
Fig 11 permits to identify the following processes the
compression characterized by the decrease in the total volume
from the time around t frac14 001 to t frac14 004 the heating process
when the total volume variation is not pronounced and the tem-
peratures increase around t frac14 004 to t frac14 006 the expansion
process when the total volume increases around t frac14 006 to
t frac14 009 and the cooling process when the volume stays almost
constant and the temperatures decrease at the times around
t frac14 009 to t frac14 010 and t frac14 0 to t frac14 001
The compressionperiod starts with the increment of the mass in
the compression space and a decrease of the mass in the expansion
space as shown in Fig 12 The decreasing mass in the expansionspace indicates a good dynamic for the compression process
because it is desirable to keep low the hotter portion of the mass
during this period However the mass on the compression space is
too high which is not desirable since this will be re1047298ected in a large
negative compression work In addition the expansion process also
presents a reduced performance due to the low values for the mass
in the expansion space during the expansion process This repre-
sents an expansion with low hotter mass and thus a low working
output to the shaft Furthermore the low mass in the expansion
space during the heating period might be the main cause for the
high temperatures reached Therefore the volumes and mass 1047298ow
dynamics of the reference case should be improved to reach higher
work outputs and avoid the overheating of the expansion chamber
Fig 12 also shows that the mass in the heater and cooler arereally small compared with the mass in the regenerator during the
Table 5
Engine work 1047298ow per cycle
Model output per cycle Aspen Custom Modeller (ACM)
Expansion Work (We Jcycle) 5262
Compression Work (Wc Jcycle) 2339
Pre ssure drop lost heater (J cycle) 021
Pre ssure drop lost cooler (J cycle) 007
Pressure drop lost regenerator (Jcycle) 028
Total lost due to pressure drop (Jcycle) 056
Net indicated work (Wi Jcycle) 2867
Forced work (W Jcycle) 2349
Brake Work Output (Wbr Jcycle) 518
Fig 12 Mass variation inside the engine spaces during a complete cycle
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complete cycle This re1047298ects the high importance of the regenerator
ef 1047297ciency for the engine performance
53 Work 1047298ow
Table 5 shows the simulation results for the compression and
expansion work during a single cycle This table also presents the
different work losses estimated for the system
The temperatures measured and the temperatures calculated
show a good thermal performance of the engine But the measured
brake power was very low Different problems on the engine
design and operational conditions may explain these very lowresults However additional experimental instrumentation is
needed for a detailed design study For this reason the present
analysis considers a theoretical approach that may be later com-
plemented with experimental studies This theoretical approach
considers Eq (7) From this equation and considering that the
thermal performance was found acceptable the main losses should
correspond to a low mechanical ef 1047297ciency of the prototype This
mechanical ef 1047297ciency is reduced by the presence of forced work
during the cycle and mechanical friction on the crank mechanism
Fig 13 presents the evaluation of the forced work in a pressure
volume diagram for the gas cycle inside the gamma prototype
From this it can be seen that the forced work (W) is mainly due to
the forced expansion process This means that at the experimentalconditions large part of the cyclic work may have been used to
complete the forced expansion process and thus the real engine
output is smaller than expected
The results discussed above are complemented with the vari-
ation of the compression (Wc) expansion (We) and net indicated
work (Wi) during the cycle shown in Fig 14
Fig 14 shows that during the 1047297rst part of the cycle from t frac14 0 to
t frac14 0045 the compression and expansion spaces present
exchanged roles This means that an increment of the volume is
presented in the compression space and a decrement of the volume
is present in the expansion one This reduced the engine perfor-
mance but it cannot be avoided since the gas needs to pass from
one space to another Regarding the second part of the cycle from
t frac14 0045 to t frac14 0095 the expansion and compression are shapedas expected and thus indicate a better dynamic during this period
However considering that large part of the expansion process is
forced the net brake work is low as reported in Table 5
From the previous analysis it can be concluded that a detailed
evaluation of the volumes dynamics the cranks mechanism effec-
tiveness and the forced work during the cycle must be considered
Fig 15 Heat 1047298
ow variation during the engine cycle
Table 6
Heat 1047298ow and heat loses during the cycle
Heat 1047298ow (Jcycle)
Heat exchanger space
Heater 1047298ow (Q hcJcycle) 5282
Cooler 1047298ow (Q kc Jcycle) 2356
Regenerator 1047298ow (Q rc Jcycle) 005
Heat lossesInternal conduction losses (Q lkc Jcycle) 2698
Shuttle conduction losses (Q lshc Jcycle) 8004
Regenerator losses during heating (Q lossrc Jcycle) 1862
Regenerator losses during cooling (Q lossrc Jcycle) 1862
Total heat requirements
Heating requirements (Q htc Jcycle) 17847
Cooling requirements (Q ktc Jcycle) 4218
Fig 14 Work 1047298
ow during the engine cycle
Fig 13 Pressureevolume diagram and forced work during the cycle
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in order to re-design the engine for a better performance This will
be covered on a detailed parametric study to be reported on a next
article
54 Heat 1047298ow
Table 6 presents the results for the heat 1047298ow and corresponding
heat losses through the heat exchangers calculated at the end of a
single cycle [25] As it can be seen the total heat requirements are
almost three times the requirements calculated without consid-
ering the losses It can also be seen that the shuttle conduction
losses represent the main heat loss during the cycle These corre-
spond to the losses due to the oscillation of the hot displacer across
the temperature gradient in the working spaces of the engine
The cyclic variation for the heat 1047298ow is additionally shown in
Fig 15The heat requirements for the heater and cooler present
slight variations during the entire cycle On the other hand the
regenerator presents high variations managing large quantities of
heat This con1047297rms the large importance of this heat exchanger on
the engine performance
55 Brake power and brake ef 1047297ciency
The engine brake power is de1047297ned as the net brake work per
cycle (Ws) times the engine frequency (freq)
Pbr frac14 Ws freq (12)
The net brake work and the total heat requirement presented
on Tables 5 and 6 respectively are doubled considering the double
cylinder engine These values are reported on Table 7 which alsoreports the thermal and brake ef 1047297ciencies for the engine
The results re1047298ect the low performance of the engine under the
experimental conditions This was mainly attributed to the forced
work and the mechanical ef 1047297ciency as it was analysed in the pre-
vious section In addition complementary works will broad this
analysis with the aim of propose improvements on the engine
design and operational parameters
6 Conclusions
In the present work a thermodynamic model for a Stirling en-
gine was improved by including the numerical evaluation of the
forced work and the mechanical ef 1047297ciency then validated against
experimental data and 1047297nally implemented for the simulation of an engine prototype The numerical model developed considered
the analytic approach proposed by Senft [23] but extended its
application for the case of the more realistic adiabatic working
spaces assumptions Consequently the effective work taken from
the shaft is better estimated and thus used for a more complete
analysis of the thermal and mechanical performance of an engine
For this article the analysis considered a novel gamma engine
prototype under the experimental conditions of a micro scale
combined heat and power system fuelled by wood pellets
The simulation results were compared with the experimental
data measured during long time runs of the system The model
performance was very good for the prediction of the temperatures
in the different spaces of the engine In addition the estimations for
the net brake power also presented results similar to the measured
values However additional experimental work should be per-
formed to obtain data to validate the calculation of the different
losses through the engine
According to the results obtained the thermal performance of
the engine was found acceptable and thus the low power output
measured is preliminary attributed to a reduced mechanical ef 1047297-
ciency of the system The possible reasons for this low performance
were further analysed with the different results for the tempera-
tures variation mass and volume variation pressure drops and the
pressure volume diagrams obtained with the model According to
these analyses the dynamics of the volumes variation and the
crank mechanism may also be improved in order to obtain higher
network during the cycle In addition it was found that the engine
performance is very sensitive to the effect of the buffer pressure
These results will be extended with a sensitivity analysis for the
system on a complementary work that aims to identify better the
effect of the different parameters on the engine performance
Acknowledgements
This work was possible thanks to the 1047297nancial support of the
Swedish International Development Cooperation Agency the di-
vision of Heat and Power Technology Department of Energy
Technology at Royal Institute of Technology (KTH) in Sweden and
Universidad Mayor de San Simon (UMSS) in Bolivia
Appendix A Detailed Stirling engine parameters
Table 7
Power output and ef 1047297ciency of the engine
Brake power (W) Heat requirement (W) Thermal ef 1047297ciency () Mechanical ef 1047297ciency () Brake ef 1047297ciency ()
5358 184535 1610 1810 290
Table A1
Inputs for the cooler in ACM
Variable Value Units Description
do 0005 m Tubes external diameter
di 0003 m Tubes internal diameter
kw 14200 Wm K Material conductivity
L 0032 m Tubes length
num 162 e Number of tubes
sl 0005 m Space between tubes
Table A2
Inputs for the heater in ACM
Variable Value Units Description
de 0005 M Tubes external diameter
di 00031 M Tubes internal diameterkw 142 Wm K Material conductivity
len 0149 m Tubes length
num 360 e Number of tubes
sl 0005 m Space between tubes
Table A3
Inputs for the regenerator in ACM
Variable Value Units Description
Din 0078 m Regenerator housing internal diameter
dout 0 107 m Regenera tor h ousin g extern al diameter
dwir e 2 1e-0 04 m Wi re di ameter of t he ma tri x
kwr 270 Wm K Thermal conductivity of the matrix material
Lr 007 m Length of the regenerator housing
Porosity 087 Matrix porosity
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Appendix B
Main equations for the Stirling engine model
Stirling engine module
Mean pressure
P frac14 MR
V cTc
thorn V kTk
thorn V rTr
thorn V hTh
thorn V eTe
Pressure variation
dP
d4frac14
gP
0BB
vV cv4
Tck
thorn
vV ev4
The
1CCA
V cTckthorn g
V kTk
thorn V rTrthorn V hTh
thorn V eThe
Mass of the working gas in the different spaces
mc frac14 p
V c
RTc
mk frac14 p
V k
RTk
mr frac14 p
V r
RTr
mh frac14 p
V h
RTh
me frac14 p
V e
RTe
Mass accumulation
dmk
d4frac14
mk
P
vP
v4
dmh
d4frac14
mh
P
vP
v4
dmr
d4frac14
mr
P
vP
v4
dmc
d4frac14
P
vV cv4
thorn
V c
vPv4
g
RTck
dme
d4frac14
P
vV ev4
thorn
V e
vPv4
g
RThe
Mass 1047298ow
mck frac14 dmc mhe frac14 dme mkr frac14 mck dmkmrh frac14 mhe thorn dmh
Conditional temperatures
If mck gt 0 then Tck frac14 Tc else Tck frac14 Tk
If mhe gt 0 then The frac14 Th else The frac14 Te
Temperatures
dTc
d4frac14 Tc
0BBvPv4
P
thorn
vV cv4
V c
vmc
v4
mc
1CCA
dTe
d4frac14 Te
0BBvPv4
P thorn
vV ev4
V e
vmev4
me
1CCAEnergy
dQ kd4
frac14
V k
vPv4
Cv
R CpethTckmck TkrmkrTHORN
dQ rd4
frac14
V r
vPv4
Cv
R C pethTkrmkr TrhmrhTHORN
dQ hd4
frac14V hvPv4
Cv
R CpethTrhmrh ThemheTHORN
dWc
d4frac14 P
vV cv4
dWe
d4frac14 P
vV ev4
Internal heat transfer module
Heat transfer from the heater wall to the working gas
Q h frac14 1
R cih thorn R hih thorn R fihethTwoh ThTHORN
Heat transfer from the cooler wall to the working gas
Q k frac14 1
R cik thorn R hik thorn R fikethTwik TkTHORN
Heat loss during the regenerator process
Q lossr frac14 eth1 3THORN Q r
Regenerator effectiveness
3frac14 NTU
1 thorn NTU
External heat transfer module
Heat transfer from the 1047298
ame to the external wall of the heater
Table A4
Inputs for the expansion-compression spaces and crank mechanism
Variable Value Units Description
vclc 44e-006 m3 Compression space clearance volume
vcle 26e-005 m3 Expansion space clearance volume
vswc 926e-005 m3 Compression space swept volume
vswe 25134e-004 m3 Expansion space swept volume
dispd 0062 m Displacer diameter
displ 007 m Displacer lengthef fmek 08 e Mechanism effectiveness
freq 5 Hz Frequency
jgap 0006 M Gap between cylinder displacer and wall
kpist 1627 Wm K Piston conductivity
pbuff 12ethorn006 Pa Buffer pressure
phase 900 deg Phase angle advance
pmean 1 25ethorn00 6 Pa Mean o pera tin g pressure
strk 0035 m Displacer stroke
dispd 0062 m Displacer diameter
Table A5
Working and cooling 1047298uid inputs in ACM
Variable Value Units Description
Working Fluid Air e Working 1047298uid inside the engine
Cooling FLUID Water e Cooling 1047298uid through the engine cooler
Tcooling 2 88 K Inl et tempera ture of th e co ol in g 1047298uid
Table A6
Fouling factors and external combustion inputs in ACM
Variable Value Units Description
T ad 1 38 7 K Fla me temper at ur e i n t he c omb ustio n ch amb er
absorp 070 e Absorptivity of the heater material
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Q h frac14 11
hrhAohthorn R foh
ethTad TwohTHORN
hrh frac14 assAohFR ethTad thorn TwohTHORN
T2ad thorn T2
woh
Estimation of the outlet temperature of the cooling 1047298uid
Twok frac14 Twater in thorn Q k
1
hokAokthorn
1
2mwaterCpwater
hok frac14 11
hwaterthorn R fok
Energy losses
Pressure drop in the heat exchangers
DP frac14
f
dhy
1
2 rv
2
l
Pressure drop in the regenerator based on the correlations of
Thomas and Pittman [37]
DP frac14 Cf nr
2u2
Cf frac14 Cfd thornCsf
Re
Csf frac14 68556 Cfd frac14 05274 wire screensCsf frac14 70035 Cfd frac14 09307 metal felts
Total pumping losses
Wploss frac14
Z 2p0
Xifrac143
ifrac141
DPi dV edq
$dq
Energy losses due to shuttle conduction
Q lsh frac14 04Z2KpistDd
JL dethTe TcTHORN
Mechanical ef 1047297ciency module
Mechanical ef 1047297ciency
hm frac14 Ws
Wi
Mechanical ef 1047297ciency considering the mechanism effectiveness
and forced work
hm frac14 E
1
E E
W
Wi
Forced work
W frac14
I ethP PbTHORNthorndV thorn
I ethP PbTHORNdV thorn
Brake ef 1047297
ciency
hb frac14 Ws
Q htfrac14
Wi
Q ht
Ws
Wifrac14 hthm
References
[1] DG Thombare SK Verma Technological development in the Stirling cycle
engines Renew Sustain Energy Rev 12 (2008) 1e
38 httpdxdoiorg101016jrser200607001
[2] AD Peacock M Newborough Impact of micro-CHP systems on domesticsector CO2 emissions Appl Therm Eng 25 (2005) 2653e2676
[3] B Thomas Benchmark testing of Micro-CHP units Appl Therm Eng 28(2008) 2049e2054 httpdxdoiorg101016japplthermaleng200803010
[4] L Dong H Liu S Riffat Development of small-scale and micro-scale biomass-fuelled CHP systems e a literature review Appl Therm Eng 29 (2009)2119e2126 httpdxdoiorg101016japplthermaleng200812004
[5] I Gonzalez-Pino A Campos-Celador E Perez-Iribarren J Teres-Zubiaga JM Sala Parametric study of the operational and economic feasibility of Stirling micro-cogeneration devices in Spain Appl Therm Eng (2013) httpdxdoiorg101016japplthermaleng201312020
[6] R Dyson S Wilson R Tew Review of computational Stirling analysismethods in 2nd International Energy Conversion Engineering ConferenceAmerican Institute of Aeronautics and Astronautics 2004 httpdxdoiorg10251462004-5582
[7] C-H Cheng Y-J Yu Numerical model for predicting thermodynamic cycleand thermal ef 1047297ciency of a beta-type Stirling engine with rhombic-drivemechanism Renew Energy 35 (2010) 2590e2601 httpdxdoiorg101016jrenene201004002
[8] N Seraj Mehdizadeh P Stouffs Simulation of a Martini displacer free pistonStirling engine for electric power generation Int J Thermodyn 3 (1) (2000)27e34 httpdxdoiorg105541ijot30
[9] N Parlak A Wagner M Elsner HS Soyhan Thermodynamic analysis of agamma type Stirling engine in non-ideal adiabatic conditions Renew Energy34 (2009) 266e273 httpdxdoiorg101016jrenene200802030
[10] JM Strauss RT Dobson Evaluation of a second order simulation for Sterlingengine design and optimisation J Energy South Afr 21 (2010) 17e29
[11] I Tlili Y Timoumi SB Nasrallah Thermodynamic analysis of the Stirling heatengine with regenerative losses and internal irreversibilities Int J Engine Res9 (2008) 45e56 httpdxdoiorg10124314680874JER01707
[12] K Mahkamov D Djumanov Three-dimensional CFD modeling of a Stirlingengine in Proceedings of the 11th international Stirling Engine Conferencevol 19 2003
[13] M Ibrahim R Tew Z Zhang D Gedeon T Simon CFD Modeling of Free-piston Stirling Engines Cleveland Ohio 2001
[14] S Wilson R Dyson Multi-D CFD modeling of a free-piston Stirling convertorat NASA GRC in Proc 2nd International Energy Conversion EngineeringConference vol 5673 2004
[15] E Eid Performance of a beta-con1047297guration heat engine having a regenerativedisplacer Renew Energy 34 (2009) 2404e2413 httpdxdoiorg101016
jrenene200903016[16] AA El-Ehwany GM Hennes EI Eid EA El-Kenany Development of the
performance of an alpha-type heat engine by using elbow-bend transposed-1047298uids heat exchanger as a heater and a cooler Energy Convers Manag 52(2011) 1010e1019 httpdxdoiorg101016jenconman201008029
[17] C-H Cheng Y-J Yu Dynamic simulation of a beta-type Stirling engine withcam-drive mechanism via the combination of the thermodynamic and dy-namic models Renew Energy 36 (2011) 714e725
[18] E Entchev J Gusdorf M Swinton M Bell F Szadkowski W Kalb1047298eisch et alMicro-generation technology assessment for housing technology EnergyBuild 36 (2004) 925e931 httpdxdoiorg101016jenbuild200403004
[19] M Paringlsson H Carlsen Development of a wood powder fuelled 35 kW StirlingCHP unit in Proceedings of the 11th ISEC (International Stirling EngineConference) 2003 pp 221e230
[20] a Nishiyama H Shimojima a Ishikawa Y Itaya S Kambara H Moritomi etal Fuel and emissions properties of Stirling engine operated with woodpowder Fuel 86 (2007) 2333e2342 httpdxdoiorg101016
jfuel200701040[21] K Sato N Ohiwa A Ishikawa H Shimojima A Nishiyama Y Moriya
Development of small-scale CHP Plant with a wood powder-fueled Stirlingengine J Power Energy Syst 2 (2008) 1221e1231 httpdxdoiorg101299
jpes21221[22] I Urieli DM Berchowitz Stirling Cycle Engine Analysis Hilger Ltd Bristol
1984[23] JR Senft Mechanical Ef 1047297ciency of Heat Engines Cambridge University Press
2007[24] E Cardozo C Erlich A Malmquist L Alejo Integration of a wood pellet
burner and a Stirling engine to produce residential heat and power ApplTherm Eng (2014) httpdxdoiorg101016japplthermaleng201408024
[25] JA Araoz M Salomon L Alejo TH Fransson Non-ideal Stirling enginethermodynamic model suitable for the integration into overall energy sys-tems Appl Therm Eng 73 (2014) 203e219 httpdxdoiorg101016
japplthermaleng201407050
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 30 29
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httpslidepdfcomreaderfullapplied-thermal-engineering-volume-83-issue-2015-doi-1010162fjapplthermaleng201503006 1515
[26] FP Incropera AS Lavine DP DeWitt Fundamentals of Heat and MassTransfer John Wiley amp Sons 2011
[27] B Thomas D Pittman Update on the evaluation of different correlations forthe 1047298ow friction factor and heat transfer of Stirling engine regenerators inCollection of Technical Papers 35th Intersociety Energy Conversion Engi-neering Conference and Exhibit (IECEC) (Cat No00CH37022) American InstAeronaut amp Astronautics 2000 pp 76e84 httpdxdoiorg101109IECEC2000870632
[28] Genoastirling Srl 2014 wwwgenoastirlingcom[29] RC Tew K Jefferies D Miao USD of ED of TE Conservation LR Center
A Stirling Engine Computer Model for Performance Calculations (GoogleEBook) Department of Energy Of 1047297ce of Conservation and Solar ApplicationsDivision of Transportation Energy Conservation 1978
[30] I Urieli CJ Rallis DM Berchowitz Computer simulation of Stirling cyclemachines in 12th Intersociety Energy Conversion Engineering Conferencevol 1 American Nuclear Society Washington DC 1977 pp 1512e1521
[31] K Mahkamov Design improvements to a biomass Stirling engine usingmathematical analysis and 3D CFD modeling J Energy Resour Technol 128(2006) 203 httpdxdoiorg10111512213273
[32] H Snyman Second Order Analyses Methods for Stirling Engine Design Uni-versity of Stellenbosch Stellenbosch 2007
[33] G Sewell Numerical Solution of Ordinary and Partial Differential Equationssecond ed John Wiley amp Sons 2005
[34] DD Do Richard G Rice Applied Mathematics and Modeling for ChemicalEngineers [Hardcover] second ed Wiley-AIChE 2012
[35] Aspentech Aspen Custom Modelerreg AspenTech 2015
[36] Aspentech Chemical Process Optimization Software d
Chemical ProcessDesign Aspen Plus 2015
[37] B Thomas D Pittman AIAA-2000-2812 Update on the Evaluation of DifferentCorrelations for the Flow Friction Factor and Heat Transfer of Stirling EngineRegenerators 2000 pp 76e84
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 3030
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burner in order to produce heat and power simultaneously as
shown in Fig 7a This con1047297guration had technical limitations that
are still being studied in order to improve both power and thermal
outputs But despite of these limitations experimental results were
obtained and these were compared with the model
32 Inputs for the model
The main inputs for the engine simulation are shown in Table 1
Supplementary inputs that include the design and operational
characteristics of the engine are presented in Appendix A
The model also needs to consider the relation of the crank
mechanism and the variation of the volumes inside the working
spaces Therefore considering that the engine has gamma type
con1047297guration the following relations for the expansion and
compression spaces were included [23]
V e frac14 V cle thornV swe
2 eth1 thorn coseth4 thorn aTHORNTHORN (8)
V c frac14 V clc thorn ethV swe V eTHORN thornV swc
2 eth1 thorn coseth4THORNTHORN (9)
Furthermore the following volume derivatives were evaluated
dV e frac14 V swe
2 sineth4 thorn aTHORN (10)
dV c
frac14 dV e
V swc
2 sineth4THORN (11)
Fig 8 Measurement points for the CHP-Stirling experimental rig [24]
Table 3Comparison of the measured and predicted temperatures along the engine
Time (s) TfM (K) ThM (K) Th (K) Error TkM (K) Tk (K) Error TrM (K) Tr (K) Error
3780e3900 13878 8164 8184 025 3224 3211 041 6018 5316 116
3900e4020 13829 8196 8075 147 3218 3214 012 6006 5277 1215
4020e4140 13931 8232 8142 109 3216 3215 004 6012 5302 118
4140e4200 13778 8308 7981 394 3216 3216 001 6036 5243 1314
4200e4380 13835 8374 8063 371 3224 3214 031 6075 5272 1321
4380e4560 13777 8518 7957 659 3218 3217 003 6142 5234 1478
4560e4680 13857 8536 8071 545 3217 3215 007 6154 5276 1426
4680e4800 13844 8464 8021 523 3216 3217 001 6135 5258 143
4800e4980 13669 8433 7708 859 3221 3223 005 6129 5144 1607
Fig 9 Temperature variation along the heat exchangers and regenerator temperature assumed by the model (T r)
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 30 23
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33 Numerical solution
The system consists of a set of algebraic differential equations
which are shown in Appendix B These consider as boundary con-
ditions that the temperatures of the working gas at the end of the
cycle must be equal to the temperatures at the beginning of the
cycle once cyclic steady state conditions are reached Therefore an
iterative shooting method [33] using a fourth order Runge Kutta
scheme for the time discretization was implemented for the nu-
merical solution The iteration process was done until cyclic steadystate conditions which is numerically reached when the difference
between the assumed initial values and the values calculated at the
end of the cycle are lower than a de1047297ned error After the cyclic
steady state solution was reached the energy losses and the forced
work were evaluated The forced work was calculated using the
classical Simpson 38 numerical integration rule [34] The scheme
in Fig 5 summarizes the iterative steps for the solution
The numerical solution was implemented in Aspen Custom
Modellerreg (ACM) [35] which is a product from Aspen Plusreg that
permits the elaboration of customized models [36] This software
has its own modelling language and can also be coupled with Cthornthorn
procedures The layout of the model in ACM is shown in Fig 6 The
blocks were programmed with the equations shown in the
Appendix B and then the solution of the system was obtained withthe algorithm previously described
The descriptions of the blocks are shown in Table 2 Additional
details of the block inputs are given in Appendix A
4 Model validation
The geometrical and operational characteristics for the Genoa
engine are described in Table 1 and Appendix A The engine was
mounted in the experimental rig shown in Fig 7a In addition the
temperatures of the working gas were measured at the different
points of the engine shown in Fig 7b
The experimental rig used wood pellets as fuel Additional
temperatures measured for the validation were The temperature
close to the 1047298ame (T1) the water inlet temperature (T8) the wateroutlet temperature (T9) Other measurements are also as shown in
Fig 8
The temperature T1 was measured using a type K empty 15 mm
Inconel 600 thermocouple The additional temperatures shown in
Fig 8 were measured using type K empty 10 mm thermocouples
Considering the type of thermocouples the expanded uncertainty
was plusmn32 C with a coverage factor of 2 The speed of the engine
crankshaft was monitored by a pulse sensor and a frequency to
analog converter (OMROM E2A and Red Lion IFMA) with an un-
certaintyplusmn 02 The pressure inside the engine was measured with
a pressure transducer (RS type 46) with analog signal and an un-
certainty of plusmn01 bar All the measurements were recorded from the
beginning to the end of the test using a data logger Additional
details of the measurements are reported in Cardozo et al [24]The engine was run during long periods and the data was
measured constantly However for the validation purposes only the
periods were stability is reached were considered In this case the
steady state condition was dif 1047297cult to reach due to the constant
variation of the 1047298ame temperature [24] Therefore average values
for the measurements within certain stability periods were taken
These are compared with the values calculated by the model at the
different values measured for the 1047298ame temperature shown in
Table 3
Fig 10 Temperature variation along the engine 1047298
ame temperature Tad frac14
1388 K
Table 4
Measured and predicted brake power
Time (s) TfM (K) Measured frequency (Hz) Measured pressure (bar) Brake power (W) experimental Brake power (W) calculated Error
3780e3900 13878 517 1250 5472 5359 206
3900e4020 13829 526 1250 5539 5208 597
4020e4140 13930 527 1250 5561 5349 381
4140e4200 13778 533 1250 4635 5003 794
4200e4380 13835 528 1250 5359 5197 302
4380e
4560 13777 536 1250 5091 5033 1144560e4680 13857 529 1250 5096 5163 131
4680e4800 13843 534 1250 559 5153 782
4800e4980 13669 556 1254 4713 4613 212
Fig 11 Volumes variation during the engine cycle
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From Table 3 the model presents good accuracy for the pre-
diction of the cooler temperatures (Tk) with the maximum error of
the order of plusmn041 In addition the calculations for the heater
temperatures (Th) present reasonable accuracy at initial times but
then the error increases This growth may be explained with the
thermal inertia that constantly increments the measured temper-
ature even on periods where the1047298ame temperature decreases This
thermal inertia is neglected by the model since it assumes steady
state heat transfer conditions On the other hand the prediction of
the mean temperature in the regenerator space (Tr) presents
higher differences This is analysed with the Fig 9 below which
shows the variations of the temperatures inside the heat ex-
changers assumed by the model
From Fig 9 it can be seen that the model assumes that thetemperatures at the interfaces heater-regenerator and cooler-
regenerator were equal to the temperatures at the cooler (T k) and
heater (Th) spaces respectively Therefore the average temperature
at the regenerator (Tr) was calculated with these values This
assumption neglects the axial temperature variation along the
heater and cooler which is re1047298ected on the measurements taken at
the exact interfaces positions T11 and T12This explains the differ-
ence between the average regenerator temperature calculated with
the measured temperatures (TrM) and the calculated with the es-
timations of the model Tr as it is shown in Table 3 However
considering that the model was capable to calculate within a good
degree of accuracy the power output measured during the exper-
imental runs it can be inferred that the error for the regenerator
temperature estimation have little in1047298
uence on the brake powercalculation This is shown in Table 4 where the values for the
measured and calculated brake power are compared at different
operating conditions The percentage error ranges from plusmn131 to
plusmn794 which is an acceptable approximation for 1047297rst design
calculations
5 Results and discussion
This section presents additionally results for the simulation of
the engine under the experimental conditions described before
This aims to completely describe the thermodynamic performance
of the engine and thus identify the main limitations that the engine
presents
51 Temperature variation
Fig10 shows the temperature variation in the differentspaces of
the engine cylinder once the cyclic steady state conditions are
reached This 1047297gure displays the sinusoidal variation of the tem-
peratures inside the compression (Tc) and expansion (Te) spaces It
can also be seen that the expansion space presents periods with
elevated temperatures which results into a high thermal stress for
the material and therefore further engine deterioration In addition
the 1047297gure also shows that the mean temperatures for the working
1047298uid inside the heater (Th) and cooler (Tk) are close to the heat
exchangers walls temperature (Twk Twhe) This indicates a good
heat transfer rate on both heat exchangers and consequently a
good thermal performance based on the model assumptions
However it is important to notice that this performance will
decrease with the time due to the fouling on the heat exchangers
which is not accounted for in the engine model
52 Mass distribution and volumes variation
The mass distribution and volumes variation for the engine
during a complete cycle are shown in Figs 11 and 12 respectivelyThese variations permit to analyse the engine dynamics during the
compression and expansion processes
Fig 11 permits to identify the following processes the
compression characterized by the decrease in the total volume
from the time around t frac14 001 to t frac14 004 the heating process
when the total volume variation is not pronounced and the tem-
peratures increase around t frac14 004 to t frac14 006 the expansion
process when the total volume increases around t frac14 006 to
t frac14 009 and the cooling process when the volume stays almost
constant and the temperatures decrease at the times around
t frac14 009 to t frac14 010 and t frac14 0 to t frac14 001
The compressionperiod starts with the increment of the mass in
the compression space and a decrease of the mass in the expansion
space as shown in Fig 12 The decreasing mass in the expansionspace indicates a good dynamic for the compression process
because it is desirable to keep low the hotter portion of the mass
during this period However the mass on the compression space is
too high which is not desirable since this will be re1047298ected in a large
negative compression work In addition the expansion process also
presents a reduced performance due to the low values for the mass
in the expansion space during the expansion process This repre-
sents an expansion with low hotter mass and thus a low working
output to the shaft Furthermore the low mass in the expansion
space during the heating period might be the main cause for the
high temperatures reached Therefore the volumes and mass 1047298ow
dynamics of the reference case should be improved to reach higher
work outputs and avoid the overheating of the expansion chamber
Fig 12 also shows that the mass in the heater and cooler arereally small compared with the mass in the regenerator during the
Table 5
Engine work 1047298ow per cycle
Model output per cycle Aspen Custom Modeller (ACM)
Expansion Work (We Jcycle) 5262
Compression Work (Wc Jcycle) 2339
Pre ssure drop lost heater (J cycle) 021
Pre ssure drop lost cooler (J cycle) 007
Pressure drop lost regenerator (Jcycle) 028
Total lost due to pressure drop (Jcycle) 056
Net indicated work (Wi Jcycle) 2867
Forced work (W Jcycle) 2349
Brake Work Output (Wbr Jcycle) 518
Fig 12 Mass variation inside the engine spaces during a complete cycle
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complete cycle This re1047298ects the high importance of the regenerator
ef 1047297ciency for the engine performance
53 Work 1047298ow
Table 5 shows the simulation results for the compression and
expansion work during a single cycle This table also presents the
different work losses estimated for the system
The temperatures measured and the temperatures calculated
show a good thermal performance of the engine But the measured
brake power was very low Different problems on the engine
design and operational conditions may explain these very lowresults However additional experimental instrumentation is
needed for a detailed design study For this reason the present
analysis considers a theoretical approach that may be later com-
plemented with experimental studies This theoretical approach
considers Eq (7) From this equation and considering that the
thermal performance was found acceptable the main losses should
correspond to a low mechanical ef 1047297ciency of the prototype This
mechanical ef 1047297ciency is reduced by the presence of forced work
during the cycle and mechanical friction on the crank mechanism
Fig 13 presents the evaluation of the forced work in a pressure
volume diagram for the gas cycle inside the gamma prototype
From this it can be seen that the forced work (W) is mainly due to
the forced expansion process This means that at the experimentalconditions large part of the cyclic work may have been used to
complete the forced expansion process and thus the real engine
output is smaller than expected
The results discussed above are complemented with the vari-
ation of the compression (Wc) expansion (We) and net indicated
work (Wi) during the cycle shown in Fig 14
Fig 14 shows that during the 1047297rst part of the cycle from t frac14 0 to
t frac14 0045 the compression and expansion spaces present
exchanged roles This means that an increment of the volume is
presented in the compression space and a decrement of the volume
is present in the expansion one This reduced the engine perfor-
mance but it cannot be avoided since the gas needs to pass from
one space to another Regarding the second part of the cycle from
t frac14 0045 to t frac14 0095 the expansion and compression are shapedas expected and thus indicate a better dynamic during this period
However considering that large part of the expansion process is
forced the net brake work is low as reported in Table 5
From the previous analysis it can be concluded that a detailed
evaluation of the volumes dynamics the cranks mechanism effec-
tiveness and the forced work during the cycle must be considered
Fig 15 Heat 1047298
ow variation during the engine cycle
Table 6
Heat 1047298ow and heat loses during the cycle
Heat 1047298ow (Jcycle)
Heat exchanger space
Heater 1047298ow (Q hcJcycle) 5282
Cooler 1047298ow (Q kc Jcycle) 2356
Regenerator 1047298ow (Q rc Jcycle) 005
Heat lossesInternal conduction losses (Q lkc Jcycle) 2698
Shuttle conduction losses (Q lshc Jcycle) 8004
Regenerator losses during heating (Q lossrc Jcycle) 1862
Regenerator losses during cooling (Q lossrc Jcycle) 1862
Total heat requirements
Heating requirements (Q htc Jcycle) 17847
Cooling requirements (Q ktc Jcycle) 4218
Fig 14 Work 1047298
ow during the engine cycle
Fig 13 Pressureevolume diagram and forced work during the cycle
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in order to re-design the engine for a better performance This will
be covered on a detailed parametric study to be reported on a next
article
54 Heat 1047298ow
Table 6 presents the results for the heat 1047298ow and corresponding
heat losses through the heat exchangers calculated at the end of a
single cycle [25] As it can be seen the total heat requirements are
almost three times the requirements calculated without consid-
ering the losses It can also be seen that the shuttle conduction
losses represent the main heat loss during the cycle These corre-
spond to the losses due to the oscillation of the hot displacer across
the temperature gradient in the working spaces of the engine
The cyclic variation for the heat 1047298ow is additionally shown in
Fig 15The heat requirements for the heater and cooler present
slight variations during the entire cycle On the other hand the
regenerator presents high variations managing large quantities of
heat This con1047297rms the large importance of this heat exchanger on
the engine performance
55 Brake power and brake ef 1047297ciency
The engine brake power is de1047297ned as the net brake work per
cycle (Ws) times the engine frequency (freq)
Pbr frac14 Ws freq (12)
The net brake work and the total heat requirement presented
on Tables 5 and 6 respectively are doubled considering the double
cylinder engine These values are reported on Table 7 which alsoreports the thermal and brake ef 1047297ciencies for the engine
The results re1047298ect the low performance of the engine under the
experimental conditions This was mainly attributed to the forced
work and the mechanical ef 1047297ciency as it was analysed in the pre-
vious section In addition complementary works will broad this
analysis with the aim of propose improvements on the engine
design and operational parameters
6 Conclusions
In the present work a thermodynamic model for a Stirling en-
gine was improved by including the numerical evaluation of the
forced work and the mechanical ef 1047297ciency then validated against
experimental data and 1047297nally implemented for the simulation of an engine prototype The numerical model developed considered
the analytic approach proposed by Senft [23] but extended its
application for the case of the more realistic adiabatic working
spaces assumptions Consequently the effective work taken from
the shaft is better estimated and thus used for a more complete
analysis of the thermal and mechanical performance of an engine
For this article the analysis considered a novel gamma engine
prototype under the experimental conditions of a micro scale
combined heat and power system fuelled by wood pellets
The simulation results were compared with the experimental
data measured during long time runs of the system The model
performance was very good for the prediction of the temperatures
in the different spaces of the engine In addition the estimations for
the net brake power also presented results similar to the measured
values However additional experimental work should be per-
formed to obtain data to validate the calculation of the different
losses through the engine
According to the results obtained the thermal performance of
the engine was found acceptable and thus the low power output
measured is preliminary attributed to a reduced mechanical ef 1047297-
ciency of the system The possible reasons for this low performance
were further analysed with the different results for the tempera-
tures variation mass and volume variation pressure drops and the
pressure volume diagrams obtained with the model According to
these analyses the dynamics of the volumes variation and the
crank mechanism may also be improved in order to obtain higher
network during the cycle In addition it was found that the engine
performance is very sensitive to the effect of the buffer pressure
These results will be extended with a sensitivity analysis for the
system on a complementary work that aims to identify better the
effect of the different parameters on the engine performance
Acknowledgements
This work was possible thanks to the 1047297nancial support of the
Swedish International Development Cooperation Agency the di-
vision of Heat and Power Technology Department of Energy
Technology at Royal Institute of Technology (KTH) in Sweden and
Universidad Mayor de San Simon (UMSS) in Bolivia
Appendix A Detailed Stirling engine parameters
Table 7
Power output and ef 1047297ciency of the engine
Brake power (W) Heat requirement (W) Thermal ef 1047297ciency () Mechanical ef 1047297ciency () Brake ef 1047297ciency ()
5358 184535 1610 1810 290
Table A1
Inputs for the cooler in ACM
Variable Value Units Description
do 0005 m Tubes external diameter
di 0003 m Tubes internal diameter
kw 14200 Wm K Material conductivity
L 0032 m Tubes length
num 162 e Number of tubes
sl 0005 m Space between tubes
Table A2
Inputs for the heater in ACM
Variable Value Units Description
de 0005 M Tubes external diameter
di 00031 M Tubes internal diameterkw 142 Wm K Material conductivity
len 0149 m Tubes length
num 360 e Number of tubes
sl 0005 m Space between tubes
Table A3
Inputs for the regenerator in ACM
Variable Value Units Description
Din 0078 m Regenerator housing internal diameter
dout 0 107 m Regenera tor h ousin g extern al diameter
dwir e 2 1e-0 04 m Wi re di ameter of t he ma tri x
kwr 270 Wm K Thermal conductivity of the matrix material
Lr 007 m Length of the regenerator housing
Porosity 087 Matrix porosity
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Appendix B
Main equations for the Stirling engine model
Stirling engine module
Mean pressure
P frac14 MR
V cTc
thorn V kTk
thorn V rTr
thorn V hTh
thorn V eTe
Pressure variation
dP
d4frac14
gP
0BB
vV cv4
Tck
thorn
vV ev4
The
1CCA
V cTckthorn g
V kTk
thorn V rTrthorn V hTh
thorn V eThe
Mass of the working gas in the different spaces
mc frac14 p
V c
RTc
mk frac14 p
V k
RTk
mr frac14 p
V r
RTr
mh frac14 p
V h
RTh
me frac14 p
V e
RTe
Mass accumulation
dmk
d4frac14
mk
P
vP
v4
dmh
d4frac14
mh
P
vP
v4
dmr
d4frac14
mr
P
vP
v4
dmc
d4frac14
P
vV cv4
thorn
V c
vPv4
g
RTck
dme
d4frac14
P
vV ev4
thorn
V e
vPv4
g
RThe
Mass 1047298ow
mck frac14 dmc mhe frac14 dme mkr frac14 mck dmkmrh frac14 mhe thorn dmh
Conditional temperatures
If mck gt 0 then Tck frac14 Tc else Tck frac14 Tk
If mhe gt 0 then The frac14 Th else The frac14 Te
Temperatures
dTc
d4frac14 Tc
0BBvPv4
P
thorn
vV cv4
V c
vmc
v4
mc
1CCA
dTe
d4frac14 Te
0BBvPv4
P thorn
vV ev4
V e
vmev4
me
1CCAEnergy
dQ kd4
frac14
V k
vPv4
Cv
R CpethTckmck TkrmkrTHORN
dQ rd4
frac14
V r
vPv4
Cv
R C pethTkrmkr TrhmrhTHORN
dQ hd4
frac14V hvPv4
Cv
R CpethTrhmrh ThemheTHORN
dWc
d4frac14 P
vV cv4
dWe
d4frac14 P
vV ev4
Internal heat transfer module
Heat transfer from the heater wall to the working gas
Q h frac14 1
R cih thorn R hih thorn R fihethTwoh ThTHORN
Heat transfer from the cooler wall to the working gas
Q k frac14 1
R cik thorn R hik thorn R fikethTwik TkTHORN
Heat loss during the regenerator process
Q lossr frac14 eth1 3THORN Q r
Regenerator effectiveness
3frac14 NTU
1 thorn NTU
External heat transfer module
Heat transfer from the 1047298
ame to the external wall of the heater
Table A4
Inputs for the expansion-compression spaces and crank mechanism
Variable Value Units Description
vclc 44e-006 m3 Compression space clearance volume
vcle 26e-005 m3 Expansion space clearance volume
vswc 926e-005 m3 Compression space swept volume
vswe 25134e-004 m3 Expansion space swept volume
dispd 0062 m Displacer diameter
displ 007 m Displacer lengthef fmek 08 e Mechanism effectiveness
freq 5 Hz Frequency
jgap 0006 M Gap between cylinder displacer and wall
kpist 1627 Wm K Piston conductivity
pbuff 12ethorn006 Pa Buffer pressure
phase 900 deg Phase angle advance
pmean 1 25ethorn00 6 Pa Mean o pera tin g pressure
strk 0035 m Displacer stroke
dispd 0062 m Displacer diameter
Table A5
Working and cooling 1047298uid inputs in ACM
Variable Value Units Description
Working Fluid Air e Working 1047298uid inside the engine
Cooling FLUID Water e Cooling 1047298uid through the engine cooler
Tcooling 2 88 K Inl et tempera ture of th e co ol in g 1047298uid
Table A6
Fouling factors and external combustion inputs in ACM
Variable Value Units Description
T ad 1 38 7 K Fla me temper at ur e i n t he c omb ustio n ch amb er
absorp 070 e Absorptivity of the heater material
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Q h frac14 11
hrhAohthorn R foh
ethTad TwohTHORN
hrh frac14 assAohFR ethTad thorn TwohTHORN
T2ad thorn T2
woh
Estimation of the outlet temperature of the cooling 1047298uid
Twok frac14 Twater in thorn Q k
1
hokAokthorn
1
2mwaterCpwater
hok frac14 11
hwaterthorn R fok
Energy losses
Pressure drop in the heat exchangers
DP frac14
f
dhy
1
2 rv
2
l
Pressure drop in the regenerator based on the correlations of
Thomas and Pittman [37]
DP frac14 Cf nr
2u2
Cf frac14 Cfd thornCsf
Re
Csf frac14 68556 Cfd frac14 05274 wire screensCsf frac14 70035 Cfd frac14 09307 metal felts
Total pumping losses
Wploss frac14
Z 2p0
Xifrac143
ifrac141
DPi dV edq
$dq
Energy losses due to shuttle conduction
Q lsh frac14 04Z2KpistDd
JL dethTe TcTHORN
Mechanical ef 1047297ciency module
Mechanical ef 1047297ciency
hm frac14 Ws
Wi
Mechanical ef 1047297ciency considering the mechanism effectiveness
and forced work
hm frac14 E
1
E E
W
Wi
Forced work
W frac14
I ethP PbTHORNthorndV thorn
I ethP PbTHORNdV thorn
Brake ef 1047297
ciency
hb frac14 Ws
Q htfrac14
Wi
Q ht
Ws
Wifrac14 hthm
References
[1] DG Thombare SK Verma Technological development in the Stirling cycle
engines Renew Sustain Energy Rev 12 (2008) 1e
38 httpdxdoiorg101016jrser200607001
[2] AD Peacock M Newborough Impact of micro-CHP systems on domesticsector CO2 emissions Appl Therm Eng 25 (2005) 2653e2676
[3] B Thomas Benchmark testing of Micro-CHP units Appl Therm Eng 28(2008) 2049e2054 httpdxdoiorg101016japplthermaleng200803010
[4] L Dong H Liu S Riffat Development of small-scale and micro-scale biomass-fuelled CHP systems e a literature review Appl Therm Eng 29 (2009)2119e2126 httpdxdoiorg101016japplthermaleng200812004
[5] I Gonzalez-Pino A Campos-Celador E Perez-Iribarren J Teres-Zubiaga JM Sala Parametric study of the operational and economic feasibility of Stirling micro-cogeneration devices in Spain Appl Therm Eng (2013) httpdxdoiorg101016japplthermaleng201312020
[6] R Dyson S Wilson R Tew Review of computational Stirling analysismethods in 2nd International Energy Conversion Engineering ConferenceAmerican Institute of Aeronautics and Astronautics 2004 httpdxdoiorg10251462004-5582
[7] C-H Cheng Y-J Yu Numerical model for predicting thermodynamic cycleand thermal ef 1047297ciency of a beta-type Stirling engine with rhombic-drivemechanism Renew Energy 35 (2010) 2590e2601 httpdxdoiorg101016jrenene201004002
[8] N Seraj Mehdizadeh P Stouffs Simulation of a Martini displacer free pistonStirling engine for electric power generation Int J Thermodyn 3 (1) (2000)27e34 httpdxdoiorg105541ijot30
[9] N Parlak A Wagner M Elsner HS Soyhan Thermodynamic analysis of agamma type Stirling engine in non-ideal adiabatic conditions Renew Energy34 (2009) 266e273 httpdxdoiorg101016jrenene200802030
[10] JM Strauss RT Dobson Evaluation of a second order simulation for Sterlingengine design and optimisation J Energy South Afr 21 (2010) 17e29
[11] I Tlili Y Timoumi SB Nasrallah Thermodynamic analysis of the Stirling heatengine with regenerative losses and internal irreversibilities Int J Engine Res9 (2008) 45e56 httpdxdoiorg10124314680874JER01707
[12] K Mahkamov D Djumanov Three-dimensional CFD modeling of a Stirlingengine in Proceedings of the 11th international Stirling Engine Conferencevol 19 2003
[13] M Ibrahim R Tew Z Zhang D Gedeon T Simon CFD Modeling of Free-piston Stirling Engines Cleveland Ohio 2001
[14] S Wilson R Dyson Multi-D CFD modeling of a free-piston Stirling convertorat NASA GRC in Proc 2nd International Energy Conversion EngineeringConference vol 5673 2004
[15] E Eid Performance of a beta-con1047297guration heat engine having a regenerativedisplacer Renew Energy 34 (2009) 2404e2413 httpdxdoiorg101016
jrenene200903016[16] AA El-Ehwany GM Hennes EI Eid EA El-Kenany Development of the
performance of an alpha-type heat engine by using elbow-bend transposed-1047298uids heat exchanger as a heater and a cooler Energy Convers Manag 52(2011) 1010e1019 httpdxdoiorg101016jenconman201008029
[17] C-H Cheng Y-J Yu Dynamic simulation of a beta-type Stirling engine withcam-drive mechanism via the combination of the thermodynamic and dy-namic models Renew Energy 36 (2011) 714e725
[18] E Entchev J Gusdorf M Swinton M Bell F Szadkowski W Kalb1047298eisch et alMicro-generation technology assessment for housing technology EnergyBuild 36 (2004) 925e931 httpdxdoiorg101016jenbuild200403004
[19] M Paringlsson H Carlsen Development of a wood powder fuelled 35 kW StirlingCHP unit in Proceedings of the 11th ISEC (International Stirling EngineConference) 2003 pp 221e230
[20] a Nishiyama H Shimojima a Ishikawa Y Itaya S Kambara H Moritomi etal Fuel and emissions properties of Stirling engine operated with woodpowder Fuel 86 (2007) 2333e2342 httpdxdoiorg101016
jfuel200701040[21] K Sato N Ohiwa A Ishikawa H Shimojima A Nishiyama Y Moriya
Development of small-scale CHP Plant with a wood powder-fueled Stirlingengine J Power Energy Syst 2 (2008) 1221e1231 httpdxdoiorg101299
jpes21221[22] I Urieli DM Berchowitz Stirling Cycle Engine Analysis Hilger Ltd Bristol
1984[23] JR Senft Mechanical Ef 1047297ciency of Heat Engines Cambridge University Press
2007[24] E Cardozo C Erlich A Malmquist L Alejo Integration of a wood pellet
burner and a Stirling engine to produce residential heat and power ApplTherm Eng (2014) httpdxdoiorg101016japplthermaleng201408024
[25] JA Araoz M Salomon L Alejo TH Fransson Non-ideal Stirling enginethermodynamic model suitable for the integration into overall energy sys-tems Appl Therm Eng 73 (2014) 203e219 httpdxdoiorg101016
japplthermaleng201407050
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 30 29
7212019 Applied Thermal Engineering Volume 83 Issue 2015 [Doi 101016_2Fjapplthermaleng201503006] Araoz Joseph hellip
httpslidepdfcomreaderfullapplied-thermal-engineering-volume-83-issue-2015-doi-1010162fjapplthermaleng201503006 1515
[26] FP Incropera AS Lavine DP DeWitt Fundamentals of Heat and MassTransfer John Wiley amp Sons 2011
[27] B Thomas D Pittman Update on the evaluation of different correlations forthe 1047298ow friction factor and heat transfer of Stirling engine regenerators inCollection of Technical Papers 35th Intersociety Energy Conversion Engi-neering Conference and Exhibit (IECEC) (Cat No00CH37022) American InstAeronaut amp Astronautics 2000 pp 76e84 httpdxdoiorg101109IECEC2000870632
[28] Genoastirling Srl 2014 wwwgenoastirlingcom[29] RC Tew K Jefferies D Miao USD of ED of TE Conservation LR Center
A Stirling Engine Computer Model for Performance Calculations (GoogleEBook) Department of Energy Of 1047297ce of Conservation and Solar ApplicationsDivision of Transportation Energy Conservation 1978
[30] I Urieli CJ Rallis DM Berchowitz Computer simulation of Stirling cyclemachines in 12th Intersociety Energy Conversion Engineering Conferencevol 1 American Nuclear Society Washington DC 1977 pp 1512e1521
[31] K Mahkamov Design improvements to a biomass Stirling engine usingmathematical analysis and 3D CFD modeling J Energy Resour Technol 128(2006) 203 httpdxdoiorg10111512213273
[32] H Snyman Second Order Analyses Methods for Stirling Engine Design Uni-versity of Stellenbosch Stellenbosch 2007
[33] G Sewell Numerical Solution of Ordinary and Partial Differential Equationssecond ed John Wiley amp Sons 2005
[34] DD Do Richard G Rice Applied Mathematics and Modeling for ChemicalEngineers [Hardcover] second ed Wiley-AIChE 2012
[35] Aspentech Aspen Custom Modelerreg AspenTech 2015
[36] Aspentech Chemical Process Optimization Software d
Chemical ProcessDesign Aspen Plus 2015
[37] B Thomas D Pittman AIAA-2000-2812 Update on the Evaluation of DifferentCorrelations for the Flow Friction Factor and Heat Transfer of Stirling EngineRegenerators 2000 pp 76e84
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 3030
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33 Numerical solution
The system consists of a set of algebraic differential equations
which are shown in Appendix B These consider as boundary con-
ditions that the temperatures of the working gas at the end of the
cycle must be equal to the temperatures at the beginning of the
cycle once cyclic steady state conditions are reached Therefore an
iterative shooting method [33] using a fourth order Runge Kutta
scheme for the time discretization was implemented for the nu-
merical solution The iteration process was done until cyclic steadystate conditions which is numerically reached when the difference
between the assumed initial values and the values calculated at the
end of the cycle are lower than a de1047297ned error After the cyclic
steady state solution was reached the energy losses and the forced
work were evaluated The forced work was calculated using the
classical Simpson 38 numerical integration rule [34] The scheme
in Fig 5 summarizes the iterative steps for the solution
The numerical solution was implemented in Aspen Custom
Modellerreg (ACM) [35] which is a product from Aspen Plusreg that
permits the elaboration of customized models [36] This software
has its own modelling language and can also be coupled with Cthornthorn
procedures The layout of the model in ACM is shown in Fig 6 The
blocks were programmed with the equations shown in the
Appendix B and then the solution of the system was obtained withthe algorithm previously described
The descriptions of the blocks are shown in Table 2 Additional
details of the block inputs are given in Appendix A
4 Model validation
The geometrical and operational characteristics for the Genoa
engine are described in Table 1 and Appendix A The engine was
mounted in the experimental rig shown in Fig 7a In addition the
temperatures of the working gas were measured at the different
points of the engine shown in Fig 7b
The experimental rig used wood pellets as fuel Additional
temperatures measured for the validation were The temperature
close to the 1047298ame (T1) the water inlet temperature (T8) the wateroutlet temperature (T9) Other measurements are also as shown in
Fig 8
The temperature T1 was measured using a type K empty 15 mm
Inconel 600 thermocouple The additional temperatures shown in
Fig 8 were measured using type K empty 10 mm thermocouples
Considering the type of thermocouples the expanded uncertainty
was plusmn32 C with a coverage factor of 2 The speed of the engine
crankshaft was monitored by a pulse sensor and a frequency to
analog converter (OMROM E2A and Red Lion IFMA) with an un-
certaintyplusmn 02 The pressure inside the engine was measured with
a pressure transducer (RS type 46) with analog signal and an un-
certainty of plusmn01 bar All the measurements were recorded from the
beginning to the end of the test using a data logger Additional
details of the measurements are reported in Cardozo et al [24]The engine was run during long periods and the data was
measured constantly However for the validation purposes only the
periods were stability is reached were considered In this case the
steady state condition was dif 1047297cult to reach due to the constant
variation of the 1047298ame temperature [24] Therefore average values
for the measurements within certain stability periods were taken
These are compared with the values calculated by the model at the
different values measured for the 1047298ame temperature shown in
Table 3
Fig 10 Temperature variation along the engine 1047298
ame temperature Tad frac14
1388 K
Table 4
Measured and predicted brake power
Time (s) TfM (K) Measured frequency (Hz) Measured pressure (bar) Brake power (W) experimental Brake power (W) calculated Error
3780e3900 13878 517 1250 5472 5359 206
3900e4020 13829 526 1250 5539 5208 597
4020e4140 13930 527 1250 5561 5349 381
4140e4200 13778 533 1250 4635 5003 794
4200e4380 13835 528 1250 5359 5197 302
4380e
4560 13777 536 1250 5091 5033 1144560e4680 13857 529 1250 5096 5163 131
4680e4800 13843 534 1250 559 5153 782
4800e4980 13669 556 1254 4713 4613 212
Fig 11 Volumes variation during the engine cycle
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From Table 3 the model presents good accuracy for the pre-
diction of the cooler temperatures (Tk) with the maximum error of
the order of plusmn041 In addition the calculations for the heater
temperatures (Th) present reasonable accuracy at initial times but
then the error increases This growth may be explained with the
thermal inertia that constantly increments the measured temper-
ature even on periods where the1047298ame temperature decreases This
thermal inertia is neglected by the model since it assumes steady
state heat transfer conditions On the other hand the prediction of
the mean temperature in the regenerator space (Tr) presents
higher differences This is analysed with the Fig 9 below which
shows the variations of the temperatures inside the heat ex-
changers assumed by the model
From Fig 9 it can be seen that the model assumes that thetemperatures at the interfaces heater-regenerator and cooler-
regenerator were equal to the temperatures at the cooler (T k) and
heater (Th) spaces respectively Therefore the average temperature
at the regenerator (Tr) was calculated with these values This
assumption neglects the axial temperature variation along the
heater and cooler which is re1047298ected on the measurements taken at
the exact interfaces positions T11 and T12This explains the differ-
ence between the average regenerator temperature calculated with
the measured temperatures (TrM) and the calculated with the es-
timations of the model Tr as it is shown in Table 3 However
considering that the model was capable to calculate within a good
degree of accuracy the power output measured during the exper-
imental runs it can be inferred that the error for the regenerator
temperature estimation have little in1047298
uence on the brake powercalculation This is shown in Table 4 where the values for the
measured and calculated brake power are compared at different
operating conditions The percentage error ranges from plusmn131 to
plusmn794 which is an acceptable approximation for 1047297rst design
calculations
5 Results and discussion
This section presents additionally results for the simulation of
the engine under the experimental conditions described before
This aims to completely describe the thermodynamic performance
of the engine and thus identify the main limitations that the engine
presents
51 Temperature variation
Fig10 shows the temperature variation in the differentspaces of
the engine cylinder once the cyclic steady state conditions are
reached This 1047297gure displays the sinusoidal variation of the tem-
peratures inside the compression (Tc) and expansion (Te) spaces It
can also be seen that the expansion space presents periods with
elevated temperatures which results into a high thermal stress for
the material and therefore further engine deterioration In addition
the 1047297gure also shows that the mean temperatures for the working
1047298uid inside the heater (Th) and cooler (Tk) are close to the heat
exchangers walls temperature (Twk Twhe) This indicates a good
heat transfer rate on both heat exchangers and consequently a
good thermal performance based on the model assumptions
However it is important to notice that this performance will
decrease with the time due to the fouling on the heat exchangers
which is not accounted for in the engine model
52 Mass distribution and volumes variation
The mass distribution and volumes variation for the engine
during a complete cycle are shown in Figs 11 and 12 respectivelyThese variations permit to analyse the engine dynamics during the
compression and expansion processes
Fig 11 permits to identify the following processes the
compression characterized by the decrease in the total volume
from the time around t frac14 001 to t frac14 004 the heating process
when the total volume variation is not pronounced and the tem-
peratures increase around t frac14 004 to t frac14 006 the expansion
process when the total volume increases around t frac14 006 to
t frac14 009 and the cooling process when the volume stays almost
constant and the temperatures decrease at the times around
t frac14 009 to t frac14 010 and t frac14 0 to t frac14 001
The compressionperiod starts with the increment of the mass in
the compression space and a decrease of the mass in the expansion
space as shown in Fig 12 The decreasing mass in the expansionspace indicates a good dynamic for the compression process
because it is desirable to keep low the hotter portion of the mass
during this period However the mass on the compression space is
too high which is not desirable since this will be re1047298ected in a large
negative compression work In addition the expansion process also
presents a reduced performance due to the low values for the mass
in the expansion space during the expansion process This repre-
sents an expansion with low hotter mass and thus a low working
output to the shaft Furthermore the low mass in the expansion
space during the heating period might be the main cause for the
high temperatures reached Therefore the volumes and mass 1047298ow
dynamics of the reference case should be improved to reach higher
work outputs and avoid the overheating of the expansion chamber
Fig 12 also shows that the mass in the heater and cooler arereally small compared with the mass in the regenerator during the
Table 5
Engine work 1047298ow per cycle
Model output per cycle Aspen Custom Modeller (ACM)
Expansion Work (We Jcycle) 5262
Compression Work (Wc Jcycle) 2339
Pre ssure drop lost heater (J cycle) 021
Pre ssure drop lost cooler (J cycle) 007
Pressure drop lost regenerator (Jcycle) 028
Total lost due to pressure drop (Jcycle) 056
Net indicated work (Wi Jcycle) 2867
Forced work (W Jcycle) 2349
Brake Work Output (Wbr Jcycle) 518
Fig 12 Mass variation inside the engine spaces during a complete cycle
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complete cycle This re1047298ects the high importance of the regenerator
ef 1047297ciency for the engine performance
53 Work 1047298ow
Table 5 shows the simulation results for the compression and
expansion work during a single cycle This table also presents the
different work losses estimated for the system
The temperatures measured and the temperatures calculated
show a good thermal performance of the engine But the measured
brake power was very low Different problems on the engine
design and operational conditions may explain these very lowresults However additional experimental instrumentation is
needed for a detailed design study For this reason the present
analysis considers a theoretical approach that may be later com-
plemented with experimental studies This theoretical approach
considers Eq (7) From this equation and considering that the
thermal performance was found acceptable the main losses should
correspond to a low mechanical ef 1047297ciency of the prototype This
mechanical ef 1047297ciency is reduced by the presence of forced work
during the cycle and mechanical friction on the crank mechanism
Fig 13 presents the evaluation of the forced work in a pressure
volume diagram for the gas cycle inside the gamma prototype
From this it can be seen that the forced work (W) is mainly due to
the forced expansion process This means that at the experimentalconditions large part of the cyclic work may have been used to
complete the forced expansion process and thus the real engine
output is smaller than expected
The results discussed above are complemented with the vari-
ation of the compression (Wc) expansion (We) and net indicated
work (Wi) during the cycle shown in Fig 14
Fig 14 shows that during the 1047297rst part of the cycle from t frac14 0 to
t frac14 0045 the compression and expansion spaces present
exchanged roles This means that an increment of the volume is
presented in the compression space and a decrement of the volume
is present in the expansion one This reduced the engine perfor-
mance but it cannot be avoided since the gas needs to pass from
one space to another Regarding the second part of the cycle from
t frac14 0045 to t frac14 0095 the expansion and compression are shapedas expected and thus indicate a better dynamic during this period
However considering that large part of the expansion process is
forced the net brake work is low as reported in Table 5
From the previous analysis it can be concluded that a detailed
evaluation of the volumes dynamics the cranks mechanism effec-
tiveness and the forced work during the cycle must be considered
Fig 15 Heat 1047298
ow variation during the engine cycle
Table 6
Heat 1047298ow and heat loses during the cycle
Heat 1047298ow (Jcycle)
Heat exchanger space
Heater 1047298ow (Q hcJcycle) 5282
Cooler 1047298ow (Q kc Jcycle) 2356
Regenerator 1047298ow (Q rc Jcycle) 005
Heat lossesInternal conduction losses (Q lkc Jcycle) 2698
Shuttle conduction losses (Q lshc Jcycle) 8004
Regenerator losses during heating (Q lossrc Jcycle) 1862
Regenerator losses during cooling (Q lossrc Jcycle) 1862
Total heat requirements
Heating requirements (Q htc Jcycle) 17847
Cooling requirements (Q ktc Jcycle) 4218
Fig 14 Work 1047298
ow during the engine cycle
Fig 13 Pressureevolume diagram and forced work during the cycle
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in order to re-design the engine for a better performance This will
be covered on a detailed parametric study to be reported on a next
article
54 Heat 1047298ow
Table 6 presents the results for the heat 1047298ow and corresponding
heat losses through the heat exchangers calculated at the end of a
single cycle [25] As it can be seen the total heat requirements are
almost three times the requirements calculated without consid-
ering the losses It can also be seen that the shuttle conduction
losses represent the main heat loss during the cycle These corre-
spond to the losses due to the oscillation of the hot displacer across
the temperature gradient in the working spaces of the engine
The cyclic variation for the heat 1047298ow is additionally shown in
Fig 15The heat requirements for the heater and cooler present
slight variations during the entire cycle On the other hand the
regenerator presents high variations managing large quantities of
heat This con1047297rms the large importance of this heat exchanger on
the engine performance
55 Brake power and brake ef 1047297ciency
The engine brake power is de1047297ned as the net brake work per
cycle (Ws) times the engine frequency (freq)
Pbr frac14 Ws freq (12)
The net brake work and the total heat requirement presented
on Tables 5 and 6 respectively are doubled considering the double
cylinder engine These values are reported on Table 7 which alsoreports the thermal and brake ef 1047297ciencies for the engine
The results re1047298ect the low performance of the engine under the
experimental conditions This was mainly attributed to the forced
work and the mechanical ef 1047297ciency as it was analysed in the pre-
vious section In addition complementary works will broad this
analysis with the aim of propose improvements on the engine
design and operational parameters
6 Conclusions
In the present work a thermodynamic model for a Stirling en-
gine was improved by including the numerical evaluation of the
forced work and the mechanical ef 1047297ciency then validated against
experimental data and 1047297nally implemented for the simulation of an engine prototype The numerical model developed considered
the analytic approach proposed by Senft [23] but extended its
application for the case of the more realistic adiabatic working
spaces assumptions Consequently the effective work taken from
the shaft is better estimated and thus used for a more complete
analysis of the thermal and mechanical performance of an engine
For this article the analysis considered a novel gamma engine
prototype under the experimental conditions of a micro scale
combined heat and power system fuelled by wood pellets
The simulation results were compared with the experimental
data measured during long time runs of the system The model
performance was very good for the prediction of the temperatures
in the different spaces of the engine In addition the estimations for
the net brake power also presented results similar to the measured
values However additional experimental work should be per-
formed to obtain data to validate the calculation of the different
losses through the engine
According to the results obtained the thermal performance of
the engine was found acceptable and thus the low power output
measured is preliminary attributed to a reduced mechanical ef 1047297-
ciency of the system The possible reasons for this low performance
were further analysed with the different results for the tempera-
tures variation mass and volume variation pressure drops and the
pressure volume diagrams obtained with the model According to
these analyses the dynamics of the volumes variation and the
crank mechanism may also be improved in order to obtain higher
network during the cycle In addition it was found that the engine
performance is very sensitive to the effect of the buffer pressure
These results will be extended with a sensitivity analysis for the
system on a complementary work that aims to identify better the
effect of the different parameters on the engine performance
Acknowledgements
This work was possible thanks to the 1047297nancial support of the
Swedish International Development Cooperation Agency the di-
vision of Heat and Power Technology Department of Energy
Technology at Royal Institute of Technology (KTH) in Sweden and
Universidad Mayor de San Simon (UMSS) in Bolivia
Appendix A Detailed Stirling engine parameters
Table 7
Power output and ef 1047297ciency of the engine
Brake power (W) Heat requirement (W) Thermal ef 1047297ciency () Mechanical ef 1047297ciency () Brake ef 1047297ciency ()
5358 184535 1610 1810 290
Table A1
Inputs for the cooler in ACM
Variable Value Units Description
do 0005 m Tubes external diameter
di 0003 m Tubes internal diameter
kw 14200 Wm K Material conductivity
L 0032 m Tubes length
num 162 e Number of tubes
sl 0005 m Space between tubes
Table A2
Inputs for the heater in ACM
Variable Value Units Description
de 0005 M Tubes external diameter
di 00031 M Tubes internal diameterkw 142 Wm K Material conductivity
len 0149 m Tubes length
num 360 e Number of tubes
sl 0005 m Space between tubes
Table A3
Inputs for the regenerator in ACM
Variable Value Units Description
Din 0078 m Regenerator housing internal diameter
dout 0 107 m Regenera tor h ousin g extern al diameter
dwir e 2 1e-0 04 m Wi re di ameter of t he ma tri x
kwr 270 Wm K Thermal conductivity of the matrix material
Lr 007 m Length of the regenerator housing
Porosity 087 Matrix porosity
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Appendix B
Main equations for the Stirling engine model
Stirling engine module
Mean pressure
P frac14 MR
V cTc
thorn V kTk
thorn V rTr
thorn V hTh
thorn V eTe
Pressure variation
dP
d4frac14
gP
0BB
vV cv4
Tck
thorn
vV ev4
The
1CCA
V cTckthorn g
V kTk
thorn V rTrthorn V hTh
thorn V eThe
Mass of the working gas in the different spaces
mc frac14 p
V c
RTc
mk frac14 p
V k
RTk
mr frac14 p
V r
RTr
mh frac14 p
V h
RTh
me frac14 p
V e
RTe
Mass accumulation
dmk
d4frac14
mk
P
vP
v4
dmh
d4frac14
mh
P
vP
v4
dmr
d4frac14
mr
P
vP
v4
dmc
d4frac14
P
vV cv4
thorn
V c
vPv4
g
RTck
dme
d4frac14
P
vV ev4
thorn
V e
vPv4
g
RThe
Mass 1047298ow
mck frac14 dmc mhe frac14 dme mkr frac14 mck dmkmrh frac14 mhe thorn dmh
Conditional temperatures
If mck gt 0 then Tck frac14 Tc else Tck frac14 Tk
If mhe gt 0 then The frac14 Th else The frac14 Te
Temperatures
dTc
d4frac14 Tc
0BBvPv4
P
thorn
vV cv4
V c
vmc
v4
mc
1CCA
dTe
d4frac14 Te
0BBvPv4
P thorn
vV ev4
V e
vmev4
me
1CCAEnergy
dQ kd4
frac14
V k
vPv4
Cv
R CpethTckmck TkrmkrTHORN
dQ rd4
frac14
V r
vPv4
Cv
R C pethTkrmkr TrhmrhTHORN
dQ hd4
frac14V hvPv4
Cv
R CpethTrhmrh ThemheTHORN
dWc
d4frac14 P
vV cv4
dWe
d4frac14 P
vV ev4
Internal heat transfer module
Heat transfer from the heater wall to the working gas
Q h frac14 1
R cih thorn R hih thorn R fihethTwoh ThTHORN
Heat transfer from the cooler wall to the working gas
Q k frac14 1
R cik thorn R hik thorn R fikethTwik TkTHORN
Heat loss during the regenerator process
Q lossr frac14 eth1 3THORN Q r
Regenerator effectiveness
3frac14 NTU
1 thorn NTU
External heat transfer module
Heat transfer from the 1047298
ame to the external wall of the heater
Table A4
Inputs for the expansion-compression spaces and crank mechanism
Variable Value Units Description
vclc 44e-006 m3 Compression space clearance volume
vcle 26e-005 m3 Expansion space clearance volume
vswc 926e-005 m3 Compression space swept volume
vswe 25134e-004 m3 Expansion space swept volume
dispd 0062 m Displacer diameter
displ 007 m Displacer lengthef fmek 08 e Mechanism effectiveness
freq 5 Hz Frequency
jgap 0006 M Gap between cylinder displacer and wall
kpist 1627 Wm K Piston conductivity
pbuff 12ethorn006 Pa Buffer pressure
phase 900 deg Phase angle advance
pmean 1 25ethorn00 6 Pa Mean o pera tin g pressure
strk 0035 m Displacer stroke
dispd 0062 m Displacer diameter
Table A5
Working and cooling 1047298uid inputs in ACM
Variable Value Units Description
Working Fluid Air e Working 1047298uid inside the engine
Cooling FLUID Water e Cooling 1047298uid through the engine cooler
Tcooling 2 88 K Inl et tempera ture of th e co ol in g 1047298uid
Table A6
Fouling factors and external combustion inputs in ACM
Variable Value Units Description
T ad 1 38 7 K Fla me temper at ur e i n t he c omb ustio n ch amb er
absorp 070 e Absorptivity of the heater material
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Q h frac14 11
hrhAohthorn R foh
ethTad TwohTHORN
hrh frac14 assAohFR ethTad thorn TwohTHORN
T2ad thorn T2
woh
Estimation of the outlet temperature of the cooling 1047298uid
Twok frac14 Twater in thorn Q k
1
hokAokthorn
1
2mwaterCpwater
hok frac14 11
hwaterthorn R fok
Energy losses
Pressure drop in the heat exchangers
DP frac14
f
dhy
1
2 rv
2
l
Pressure drop in the regenerator based on the correlations of
Thomas and Pittman [37]
DP frac14 Cf nr
2u2
Cf frac14 Cfd thornCsf
Re
Csf frac14 68556 Cfd frac14 05274 wire screensCsf frac14 70035 Cfd frac14 09307 metal felts
Total pumping losses
Wploss frac14
Z 2p0
Xifrac143
ifrac141
DPi dV edq
$dq
Energy losses due to shuttle conduction
Q lsh frac14 04Z2KpistDd
JL dethTe TcTHORN
Mechanical ef 1047297ciency module
Mechanical ef 1047297ciency
hm frac14 Ws
Wi
Mechanical ef 1047297ciency considering the mechanism effectiveness
and forced work
hm frac14 E
1
E E
W
Wi
Forced work
W frac14
I ethP PbTHORNthorndV thorn
I ethP PbTHORNdV thorn
Brake ef 1047297
ciency
hb frac14 Ws
Q htfrac14
Wi
Q ht
Ws
Wifrac14 hthm
References
[1] DG Thombare SK Verma Technological development in the Stirling cycle
engines Renew Sustain Energy Rev 12 (2008) 1e
38 httpdxdoiorg101016jrser200607001
[2] AD Peacock M Newborough Impact of micro-CHP systems on domesticsector CO2 emissions Appl Therm Eng 25 (2005) 2653e2676
[3] B Thomas Benchmark testing of Micro-CHP units Appl Therm Eng 28(2008) 2049e2054 httpdxdoiorg101016japplthermaleng200803010
[4] L Dong H Liu S Riffat Development of small-scale and micro-scale biomass-fuelled CHP systems e a literature review Appl Therm Eng 29 (2009)2119e2126 httpdxdoiorg101016japplthermaleng200812004
[5] I Gonzalez-Pino A Campos-Celador E Perez-Iribarren J Teres-Zubiaga JM Sala Parametric study of the operational and economic feasibility of Stirling micro-cogeneration devices in Spain Appl Therm Eng (2013) httpdxdoiorg101016japplthermaleng201312020
[6] R Dyson S Wilson R Tew Review of computational Stirling analysismethods in 2nd International Energy Conversion Engineering ConferenceAmerican Institute of Aeronautics and Astronautics 2004 httpdxdoiorg10251462004-5582
[7] C-H Cheng Y-J Yu Numerical model for predicting thermodynamic cycleand thermal ef 1047297ciency of a beta-type Stirling engine with rhombic-drivemechanism Renew Energy 35 (2010) 2590e2601 httpdxdoiorg101016jrenene201004002
[8] N Seraj Mehdizadeh P Stouffs Simulation of a Martini displacer free pistonStirling engine for electric power generation Int J Thermodyn 3 (1) (2000)27e34 httpdxdoiorg105541ijot30
[9] N Parlak A Wagner M Elsner HS Soyhan Thermodynamic analysis of agamma type Stirling engine in non-ideal adiabatic conditions Renew Energy34 (2009) 266e273 httpdxdoiorg101016jrenene200802030
[10] JM Strauss RT Dobson Evaluation of a second order simulation for Sterlingengine design and optimisation J Energy South Afr 21 (2010) 17e29
[11] I Tlili Y Timoumi SB Nasrallah Thermodynamic analysis of the Stirling heatengine with regenerative losses and internal irreversibilities Int J Engine Res9 (2008) 45e56 httpdxdoiorg10124314680874JER01707
[12] K Mahkamov D Djumanov Three-dimensional CFD modeling of a Stirlingengine in Proceedings of the 11th international Stirling Engine Conferencevol 19 2003
[13] M Ibrahim R Tew Z Zhang D Gedeon T Simon CFD Modeling of Free-piston Stirling Engines Cleveland Ohio 2001
[14] S Wilson R Dyson Multi-D CFD modeling of a free-piston Stirling convertorat NASA GRC in Proc 2nd International Energy Conversion EngineeringConference vol 5673 2004
[15] E Eid Performance of a beta-con1047297guration heat engine having a regenerativedisplacer Renew Energy 34 (2009) 2404e2413 httpdxdoiorg101016
jrenene200903016[16] AA El-Ehwany GM Hennes EI Eid EA El-Kenany Development of the
performance of an alpha-type heat engine by using elbow-bend transposed-1047298uids heat exchanger as a heater and a cooler Energy Convers Manag 52(2011) 1010e1019 httpdxdoiorg101016jenconman201008029
[17] C-H Cheng Y-J Yu Dynamic simulation of a beta-type Stirling engine withcam-drive mechanism via the combination of the thermodynamic and dy-namic models Renew Energy 36 (2011) 714e725
[18] E Entchev J Gusdorf M Swinton M Bell F Szadkowski W Kalb1047298eisch et alMicro-generation technology assessment for housing technology EnergyBuild 36 (2004) 925e931 httpdxdoiorg101016jenbuild200403004
[19] M Paringlsson H Carlsen Development of a wood powder fuelled 35 kW StirlingCHP unit in Proceedings of the 11th ISEC (International Stirling EngineConference) 2003 pp 221e230
[20] a Nishiyama H Shimojima a Ishikawa Y Itaya S Kambara H Moritomi etal Fuel and emissions properties of Stirling engine operated with woodpowder Fuel 86 (2007) 2333e2342 httpdxdoiorg101016
jfuel200701040[21] K Sato N Ohiwa A Ishikawa H Shimojima A Nishiyama Y Moriya
Development of small-scale CHP Plant with a wood powder-fueled Stirlingengine J Power Energy Syst 2 (2008) 1221e1231 httpdxdoiorg101299
jpes21221[22] I Urieli DM Berchowitz Stirling Cycle Engine Analysis Hilger Ltd Bristol
1984[23] JR Senft Mechanical Ef 1047297ciency of Heat Engines Cambridge University Press
2007[24] E Cardozo C Erlich A Malmquist L Alejo Integration of a wood pellet
burner and a Stirling engine to produce residential heat and power ApplTherm Eng (2014) httpdxdoiorg101016japplthermaleng201408024
[25] JA Araoz M Salomon L Alejo TH Fransson Non-ideal Stirling enginethermodynamic model suitable for the integration into overall energy sys-tems Appl Therm Eng 73 (2014) 203e219 httpdxdoiorg101016
japplthermaleng201407050
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 30 29
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[26] FP Incropera AS Lavine DP DeWitt Fundamentals of Heat and MassTransfer John Wiley amp Sons 2011
[27] B Thomas D Pittman Update on the evaluation of different correlations forthe 1047298ow friction factor and heat transfer of Stirling engine regenerators inCollection of Technical Papers 35th Intersociety Energy Conversion Engi-neering Conference and Exhibit (IECEC) (Cat No00CH37022) American InstAeronaut amp Astronautics 2000 pp 76e84 httpdxdoiorg101109IECEC2000870632
[28] Genoastirling Srl 2014 wwwgenoastirlingcom[29] RC Tew K Jefferies D Miao USD of ED of TE Conservation LR Center
A Stirling Engine Computer Model for Performance Calculations (GoogleEBook) Department of Energy Of 1047297ce of Conservation and Solar ApplicationsDivision of Transportation Energy Conservation 1978
[30] I Urieli CJ Rallis DM Berchowitz Computer simulation of Stirling cyclemachines in 12th Intersociety Energy Conversion Engineering Conferencevol 1 American Nuclear Society Washington DC 1977 pp 1512e1521
[31] K Mahkamov Design improvements to a biomass Stirling engine usingmathematical analysis and 3D CFD modeling J Energy Resour Technol 128(2006) 203 httpdxdoiorg10111512213273
[32] H Snyman Second Order Analyses Methods for Stirling Engine Design Uni-versity of Stellenbosch Stellenbosch 2007
[33] G Sewell Numerical Solution of Ordinary and Partial Differential Equationssecond ed John Wiley amp Sons 2005
[34] DD Do Richard G Rice Applied Mathematics and Modeling for ChemicalEngineers [Hardcover] second ed Wiley-AIChE 2012
[35] Aspentech Aspen Custom Modelerreg AspenTech 2015
[36] Aspentech Chemical Process Optimization Software d
Chemical ProcessDesign Aspen Plus 2015
[37] B Thomas D Pittman AIAA-2000-2812 Update on the Evaluation of DifferentCorrelations for the Flow Friction Factor and Heat Transfer of Stirling EngineRegenerators 2000 pp 76e84
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From Table 3 the model presents good accuracy for the pre-
diction of the cooler temperatures (Tk) with the maximum error of
the order of plusmn041 In addition the calculations for the heater
temperatures (Th) present reasonable accuracy at initial times but
then the error increases This growth may be explained with the
thermal inertia that constantly increments the measured temper-
ature even on periods where the1047298ame temperature decreases This
thermal inertia is neglected by the model since it assumes steady
state heat transfer conditions On the other hand the prediction of
the mean temperature in the regenerator space (Tr) presents
higher differences This is analysed with the Fig 9 below which
shows the variations of the temperatures inside the heat ex-
changers assumed by the model
From Fig 9 it can be seen that the model assumes that thetemperatures at the interfaces heater-regenerator and cooler-
regenerator were equal to the temperatures at the cooler (T k) and
heater (Th) spaces respectively Therefore the average temperature
at the regenerator (Tr) was calculated with these values This
assumption neglects the axial temperature variation along the
heater and cooler which is re1047298ected on the measurements taken at
the exact interfaces positions T11 and T12This explains the differ-
ence between the average regenerator temperature calculated with
the measured temperatures (TrM) and the calculated with the es-
timations of the model Tr as it is shown in Table 3 However
considering that the model was capable to calculate within a good
degree of accuracy the power output measured during the exper-
imental runs it can be inferred that the error for the regenerator
temperature estimation have little in1047298
uence on the brake powercalculation This is shown in Table 4 where the values for the
measured and calculated brake power are compared at different
operating conditions The percentage error ranges from plusmn131 to
plusmn794 which is an acceptable approximation for 1047297rst design
calculations
5 Results and discussion
This section presents additionally results for the simulation of
the engine under the experimental conditions described before
This aims to completely describe the thermodynamic performance
of the engine and thus identify the main limitations that the engine
presents
51 Temperature variation
Fig10 shows the temperature variation in the differentspaces of
the engine cylinder once the cyclic steady state conditions are
reached This 1047297gure displays the sinusoidal variation of the tem-
peratures inside the compression (Tc) and expansion (Te) spaces It
can also be seen that the expansion space presents periods with
elevated temperatures which results into a high thermal stress for
the material and therefore further engine deterioration In addition
the 1047297gure also shows that the mean temperatures for the working
1047298uid inside the heater (Th) and cooler (Tk) are close to the heat
exchangers walls temperature (Twk Twhe) This indicates a good
heat transfer rate on both heat exchangers and consequently a
good thermal performance based on the model assumptions
However it is important to notice that this performance will
decrease with the time due to the fouling on the heat exchangers
which is not accounted for in the engine model
52 Mass distribution and volumes variation
The mass distribution and volumes variation for the engine
during a complete cycle are shown in Figs 11 and 12 respectivelyThese variations permit to analyse the engine dynamics during the
compression and expansion processes
Fig 11 permits to identify the following processes the
compression characterized by the decrease in the total volume
from the time around t frac14 001 to t frac14 004 the heating process
when the total volume variation is not pronounced and the tem-
peratures increase around t frac14 004 to t frac14 006 the expansion
process when the total volume increases around t frac14 006 to
t frac14 009 and the cooling process when the volume stays almost
constant and the temperatures decrease at the times around
t frac14 009 to t frac14 010 and t frac14 0 to t frac14 001
The compressionperiod starts with the increment of the mass in
the compression space and a decrease of the mass in the expansion
space as shown in Fig 12 The decreasing mass in the expansionspace indicates a good dynamic for the compression process
because it is desirable to keep low the hotter portion of the mass
during this period However the mass on the compression space is
too high which is not desirable since this will be re1047298ected in a large
negative compression work In addition the expansion process also
presents a reduced performance due to the low values for the mass
in the expansion space during the expansion process This repre-
sents an expansion with low hotter mass and thus a low working
output to the shaft Furthermore the low mass in the expansion
space during the heating period might be the main cause for the
high temperatures reached Therefore the volumes and mass 1047298ow
dynamics of the reference case should be improved to reach higher
work outputs and avoid the overheating of the expansion chamber
Fig 12 also shows that the mass in the heater and cooler arereally small compared with the mass in the regenerator during the
Table 5
Engine work 1047298ow per cycle
Model output per cycle Aspen Custom Modeller (ACM)
Expansion Work (We Jcycle) 5262
Compression Work (Wc Jcycle) 2339
Pre ssure drop lost heater (J cycle) 021
Pre ssure drop lost cooler (J cycle) 007
Pressure drop lost regenerator (Jcycle) 028
Total lost due to pressure drop (Jcycle) 056
Net indicated work (Wi Jcycle) 2867
Forced work (W Jcycle) 2349
Brake Work Output (Wbr Jcycle) 518
Fig 12 Mass variation inside the engine spaces during a complete cycle
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complete cycle This re1047298ects the high importance of the regenerator
ef 1047297ciency for the engine performance
53 Work 1047298ow
Table 5 shows the simulation results for the compression and
expansion work during a single cycle This table also presents the
different work losses estimated for the system
The temperatures measured and the temperatures calculated
show a good thermal performance of the engine But the measured
brake power was very low Different problems on the engine
design and operational conditions may explain these very lowresults However additional experimental instrumentation is
needed for a detailed design study For this reason the present
analysis considers a theoretical approach that may be later com-
plemented with experimental studies This theoretical approach
considers Eq (7) From this equation and considering that the
thermal performance was found acceptable the main losses should
correspond to a low mechanical ef 1047297ciency of the prototype This
mechanical ef 1047297ciency is reduced by the presence of forced work
during the cycle and mechanical friction on the crank mechanism
Fig 13 presents the evaluation of the forced work in a pressure
volume diagram for the gas cycle inside the gamma prototype
From this it can be seen that the forced work (W) is mainly due to
the forced expansion process This means that at the experimentalconditions large part of the cyclic work may have been used to
complete the forced expansion process and thus the real engine
output is smaller than expected
The results discussed above are complemented with the vari-
ation of the compression (Wc) expansion (We) and net indicated
work (Wi) during the cycle shown in Fig 14
Fig 14 shows that during the 1047297rst part of the cycle from t frac14 0 to
t frac14 0045 the compression and expansion spaces present
exchanged roles This means that an increment of the volume is
presented in the compression space and a decrement of the volume
is present in the expansion one This reduced the engine perfor-
mance but it cannot be avoided since the gas needs to pass from
one space to another Regarding the second part of the cycle from
t frac14 0045 to t frac14 0095 the expansion and compression are shapedas expected and thus indicate a better dynamic during this period
However considering that large part of the expansion process is
forced the net brake work is low as reported in Table 5
From the previous analysis it can be concluded that a detailed
evaluation of the volumes dynamics the cranks mechanism effec-
tiveness and the forced work during the cycle must be considered
Fig 15 Heat 1047298
ow variation during the engine cycle
Table 6
Heat 1047298ow and heat loses during the cycle
Heat 1047298ow (Jcycle)
Heat exchanger space
Heater 1047298ow (Q hcJcycle) 5282
Cooler 1047298ow (Q kc Jcycle) 2356
Regenerator 1047298ow (Q rc Jcycle) 005
Heat lossesInternal conduction losses (Q lkc Jcycle) 2698
Shuttle conduction losses (Q lshc Jcycle) 8004
Regenerator losses during heating (Q lossrc Jcycle) 1862
Regenerator losses during cooling (Q lossrc Jcycle) 1862
Total heat requirements
Heating requirements (Q htc Jcycle) 17847
Cooling requirements (Q ktc Jcycle) 4218
Fig 14 Work 1047298
ow during the engine cycle
Fig 13 Pressureevolume diagram and forced work during the cycle
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in order to re-design the engine for a better performance This will
be covered on a detailed parametric study to be reported on a next
article
54 Heat 1047298ow
Table 6 presents the results for the heat 1047298ow and corresponding
heat losses through the heat exchangers calculated at the end of a
single cycle [25] As it can be seen the total heat requirements are
almost three times the requirements calculated without consid-
ering the losses It can also be seen that the shuttle conduction
losses represent the main heat loss during the cycle These corre-
spond to the losses due to the oscillation of the hot displacer across
the temperature gradient in the working spaces of the engine
The cyclic variation for the heat 1047298ow is additionally shown in
Fig 15The heat requirements for the heater and cooler present
slight variations during the entire cycle On the other hand the
regenerator presents high variations managing large quantities of
heat This con1047297rms the large importance of this heat exchanger on
the engine performance
55 Brake power and brake ef 1047297ciency
The engine brake power is de1047297ned as the net brake work per
cycle (Ws) times the engine frequency (freq)
Pbr frac14 Ws freq (12)
The net brake work and the total heat requirement presented
on Tables 5 and 6 respectively are doubled considering the double
cylinder engine These values are reported on Table 7 which alsoreports the thermal and brake ef 1047297ciencies for the engine
The results re1047298ect the low performance of the engine under the
experimental conditions This was mainly attributed to the forced
work and the mechanical ef 1047297ciency as it was analysed in the pre-
vious section In addition complementary works will broad this
analysis with the aim of propose improvements on the engine
design and operational parameters
6 Conclusions
In the present work a thermodynamic model for a Stirling en-
gine was improved by including the numerical evaluation of the
forced work and the mechanical ef 1047297ciency then validated against
experimental data and 1047297nally implemented for the simulation of an engine prototype The numerical model developed considered
the analytic approach proposed by Senft [23] but extended its
application for the case of the more realistic adiabatic working
spaces assumptions Consequently the effective work taken from
the shaft is better estimated and thus used for a more complete
analysis of the thermal and mechanical performance of an engine
For this article the analysis considered a novel gamma engine
prototype under the experimental conditions of a micro scale
combined heat and power system fuelled by wood pellets
The simulation results were compared with the experimental
data measured during long time runs of the system The model
performance was very good for the prediction of the temperatures
in the different spaces of the engine In addition the estimations for
the net brake power also presented results similar to the measured
values However additional experimental work should be per-
formed to obtain data to validate the calculation of the different
losses through the engine
According to the results obtained the thermal performance of
the engine was found acceptable and thus the low power output
measured is preliminary attributed to a reduced mechanical ef 1047297-
ciency of the system The possible reasons for this low performance
were further analysed with the different results for the tempera-
tures variation mass and volume variation pressure drops and the
pressure volume diagrams obtained with the model According to
these analyses the dynamics of the volumes variation and the
crank mechanism may also be improved in order to obtain higher
network during the cycle In addition it was found that the engine
performance is very sensitive to the effect of the buffer pressure
These results will be extended with a sensitivity analysis for the
system on a complementary work that aims to identify better the
effect of the different parameters on the engine performance
Acknowledgements
This work was possible thanks to the 1047297nancial support of the
Swedish International Development Cooperation Agency the di-
vision of Heat and Power Technology Department of Energy
Technology at Royal Institute of Technology (KTH) in Sweden and
Universidad Mayor de San Simon (UMSS) in Bolivia
Appendix A Detailed Stirling engine parameters
Table 7
Power output and ef 1047297ciency of the engine
Brake power (W) Heat requirement (W) Thermal ef 1047297ciency () Mechanical ef 1047297ciency () Brake ef 1047297ciency ()
5358 184535 1610 1810 290
Table A1
Inputs for the cooler in ACM
Variable Value Units Description
do 0005 m Tubes external diameter
di 0003 m Tubes internal diameter
kw 14200 Wm K Material conductivity
L 0032 m Tubes length
num 162 e Number of tubes
sl 0005 m Space between tubes
Table A2
Inputs for the heater in ACM
Variable Value Units Description
de 0005 M Tubes external diameter
di 00031 M Tubes internal diameterkw 142 Wm K Material conductivity
len 0149 m Tubes length
num 360 e Number of tubes
sl 0005 m Space between tubes
Table A3
Inputs for the regenerator in ACM
Variable Value Units Description
Din 0078 m Regenerator housing internal diameter
dout 0 107 m Regenera tor h ousin g extern al diameter
dwir e 2 1e-0 04 m Wi re di ameter of t he ma tri x
kwr 270 Wm K Thermal conductivity of the matrix material
Lr 007 m Length of the regenerator housing
Porosity 087 Matrix porosity
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Appendix B
Main equations for the Stirling engine model
Stirling engine module
Mean pressure
P frac14 MR
V cTc
thorn V kTk
thorn V rTr
thorn V hTh
thorn V eTe
Pressure variation
dP
d4frac14
gP
0BB
vV cv4
Tck
thorn
vV ev4
The
1CCA
V cTckthorn g
V kTk
thorn V rTrthorn V hTh
thorn V eThe
Mass of the working gas in the different spaces
mc frac14 p
V c
RTc
mk frac14 p
V k
RTk
mr frac14 p
V r
RTr
mh frac14 p
V h
RTh
me frac14 p
V e
RTe
Mass accumulation
dmk
d4frac14
mk
P
vP
v4
dmh
d4frac14
mh
P
vP
v4
dmr
d4frac14
mr
P
vP
v4
dmc
d4frac14
P
vV cv4
thorn
V c
vPv4
g
RTck
dme
d4frac14
P
vV ev4
thorn
V e
vPv4
g
RThe
Mass 1047298ow
mck frac14 dmc mhe frac14 dme mkr frac14 mck dmkmrh frac14 mhe thorn dmh
Conditional temperatures
If mck gt 0 then Tck frac14 Tc else Tck frac14 Tk
If mhe gt 0 then The frac14 Th else The frac14 Te
Temperatures
dTc
d4frac14 Tc
0BBvPv4
P
thorn
vV cv4
V c
vmc
v4
mc
1CCA
dTe
d4frac14 Te
0BBvPv4
P thorn
vV ev4
V e
vmev4
me
1CCAEnergy
dQ kd4
frac14
V k
vPv4
Cv
R CpethTckmck TkrmkrTHORN
dQ rd4
frac14
V r
vPv4
Cv
R C pethTkrmkr TrhmrhTHORN
dQ hd4
frac14V hvPv4
Cv
R CpethTrhmrh ThemheTHORN
dWc
d4frac14 P
vV cv4
dWe
d4frac14 P
vV ev4
Internal heat transfer module
Heat transfer from the heater wall to the working gas
Q h frac14 1
R cih thorn R hih thorn R fihethTwoh ThTHORN
Heat transfer from the cooler wall to the working gas
Q k frac14 1
R cik thorn R hik thorn R fikethTwik TkTHORN
Heat loss during the regenerator process
Q lossr frac14 eth1 3THORN Q r
Regenerator effectiveness
3frac14 NTU
1 thorn NTU
External heat transfer module
Heat transfer from the 1047298
ame to the external wall of the heater
Table A4
Inputs for the expansion-compression spaces and crank mechanism
Variable Value Units Description
vclc 44e-006 m3 Compression space clearance volume
vcle 26e-005 m3 Expansion space clearance volume
vswc 926e-005 m3 Compression space swept volume
vswe 25134e-004 m3 Expansion space swept volume
dispd 0062 m Displacer diameter
displ 007 m Displacer lengthef fmek 08 e Mechanism effectiveness
freq 5 Hz Frequency
jgap 0006 M Gap between cylinder displacer and wall
kpist 1627 Wm K Piston conductivity
pbuff 12ethorn006 Pa Buffer pressure
phase 900 deg Phase angle advance
pmean 1 25ethorn00 6 Pa Mean o pera tin g pressure
strk 0035 m Displacer stroke
dispd 0062 m Displacer diameter
Table A5
Working and cooling 1047298uid inputs in ACM
Variable Value Units Description
Working Fluid Air e Working 1047298uid inside the engine
Cooling FLUID Water e Cooling 1047298uid through the engine cooler
Tcooling 2 88 K Inl et tempera ture of th e co ol in g 1047298uid
Table A6
Fouling factors and external combustion inputs in ACM
Variable Value Units Description
T ad 1 38 7 K Fla me temper at ur e i n t he c omb ustio n ch amb er
absorp 070 e Absorptivity of the heater material
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 3028
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Q h frac14 11
hrhAohthorn R foh
ethTad TwohTHORN
hrh frac14 assAohFR ethTad thorn TwohTHORN
T2ad thorn T2
woh
Estimation of the outlet temperature of the cooling 1047298uid
Twok frac14 Twater in thorn Q k
1
hokAokthorn
1
2mwaterCpwater
hok frac14 11
hwaterthorn R fok
Energy losses
Pressure drop in the heat exchangers
DP frac14
f
dhy
1
2 rv
2
l
Pressure drop in the regenerator based on the correlations of
Thomas and Pittman [37]
DP frac14 Cf nr
2u2
Cf frac14 Cfd thornCsf
Re
Csf frac14 68556 Cfd frac14 05274 wire screensCsf frac14 70035 Cfd frac14 09307 metal felts
Total pumping losses
Wploss frac14
Z 2p0
Xifrac143
ifrac141
DPi dV edq
$dq
Energy losses due to shuttle conduction
Q lsh frac14 04Z2KpistDd
JL dethTe TcTHORN
Mechanical ef 1047297ciency module
Mechanical ef 1047297ciency
hm frac14 Ws
Wi
Mechanical ef 1047297ciency considering the mechanism effectiveness
and forced work
hm frac14 E
1
E E
W
Wi
Forced work
W frac14
I ethP PbTHORNthorndV thorn
I ethP PbTHORNdV thorn
Brake ef 1047297
ciency
hb frac14 Ws
Q htfrac14
Wi
Q ht
Ws
Wifrac14 hthm
References
[1] DG Thombare SK Verma Technological development in the Stirling cycle
engines Renew Sustain Energy Rev 12 (2008) 1e
38 httpdxdoiorg101016jrser200607001
[2] AD Peacock M Newborough Impact of micro-CHP systems on domesticsector CO2 emissions Appl Therm Eng 25 (2005) 2653e2676
[3] B Thomas Benchmark testing of Micro-CHP units Appl Therm Eng 28(2008) 2049e2054 httpdxdoiorg101016japplthermaleng200803010
[4] L Dong H Liu S Riffat Development of small-scale and micro-scale biomass-fuelled CHP systems e a literature review Appl Therm Eng 29 (2009)2119e2126 httpdxdoiorg101016japplthermaleng200812004
[5] I Gonzalez-Pino A Campos-Celador E Perez-Iribarren J Teres-Zubiaga JM Sala Parametric study of the operational and economic feasibility of Stirling micro-cogeneration devices in Spain Appl Therm Eng (2013) httpdxdoiorg101016japplthermaleng201312020
[6] R Dyson S Wilson R Tew Review of computational Stirling analysismethods in 2nd International Energy Conversion Engineering ConferenceAmerican Institute of Aeronautics and Astronautics 2004 httpdxdoiorg10251462004-5582
[7] C-H Cheng Y-J Yu Numerical model for predicting thermodynamic cycleand thermal ef 1047297ciency of a beta-type Stirling engine with rhombic-drivemechanism Renew Energy 35 (2010) 2590e2601 httpdxdoiorg101016jrenene201004002
[8] N Seraj Mehdizadeh P Stouffs Simulation of a Martini displacer free pistonStirling engine for electric power generation Int J Thermodyn 3 (1) (2000)27e34 httpdxdoiorg105541ijot30
[9] N Parlak A Wagner M Elsner HS Soyhan Thermodynamic analysis of agamma type Stirling engine in non-ideal adiabatic conditions Renew Energy34 (2009) 266e273 httpdxdoiorg101016jrenene200802030
[10] JM Strauss RT Dobson Evaluation of a second order simulation for Sterlingengine design and optimisation J Energy South Afr 21 (2010) 17e29
[11] I Tlili Y Timoumi SB Nasrallah Thermodynamic analysis of the Stirling heatengine with regenerative losses and internal irreversibilities Int J Engine Res9 (2008) 45e56 httpdxdoiorg10124314680874JER01707
[12] K Mahkamov D Djumanov Three-dimensional CFD modeling of a Stirlingengine in Proceedings of the 11th international Stirling Engine Conferencevol 19 2003
[13] M Ibrahim R Tew Z Zhang D Gedeon T Simon CFD Modeling of Free-piston Stirling Engines Cleveland Ohio 2001
[14] S Wilson R Dyson Multi-D CFD modeling of a free-piston Stirling convertorat NASA GRC in Proc 2nd International Energy Conversion EngineeringConference vol 5673 2004
[15] E Eid Performance of a beta-con1047297guration heat engine having a regenerativedisplacer Renew Energy 34 (2009) 2404e2413 httpdxdoiorg101016
jrenene200903016[16] AA El-Ehwany GM Hennes EI Eid EA El-Kenany Development of the
performance of an alpha-type heat engine by using elbow-bend transposed-1047298uids heat exchanger as a heater and a cooler Energy Convers Manag 52(2011) 1010e1019 httpdxdoiorg101016jenconman201008029
[17] C-H Cheng Y-J Yu Dynamic simulation of a beta-type Stirling engine withcam-drive mechanism via the combination of the thermodynamic and dy-namic models Renew Energy 36 (2011) 714e725
[18] E Entchev J Gusdorf M Swinton M Bell F Szadkowski W Kalb1047298eisch et alMicro-generation technology assessment for housing technology EnergyBuild 36 (2004) 925e931 httpdxdoiorg101016jenbuild200403004
[19] M Paringlsson H Carlsen Development of a wood powder fuelled 35 kW StirlingCHP unit in Proceedings of the 11th ISEC (International Stirling EngineConference) 2003 pp 221e230
[20] a Nishiyama H Shimojima a Ishikawa Y Itaya S Kambara H Moritomi etal Fuel and emissions properties of Stirling engine operated with woodpowder Fuel 86 (2007) 2333e2342 httpdxdoiorg101016
jfuel200701040[21] K Sato N Ohiwa A Ishikawa H Shimojima A Nishiyama Y Moriya
Development of small-scale CHP Plant with a wood powder-fueled Stirlingengine J Power Energy Syst 2 (2008) 1221e1231 httpdxdoiorg101299
jpes21221[22] I Urieli DM Berchowitz Stirling Cycle Engine Analysis Hilger Ltd Bristol
1984[23] JR Senft Mechanical Ef 1047297ciency of Heat Engines Cambridge University Press
2007[24] E Cardozo C Erlich A Malmquist L Alejo Integration of a wood pellet
burner and a Stirling engine to produce residential heat and power ApplTherm Eng (2014) httpdxdoiorg101016japplthermaleng201408024
[25] JA Araoz M Salomon L Alejo TH Fransson Non-ideal Stirling enginethermodynamic model suitable for the integration into overall energy sys-tems Appl Therm Eng 73 (2014) 203e219 httpdxdoiorg101016
japplthermaleng201407050
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 30 29
7212019 Applied Thermal Engineering Volume 83 Issue 2015 [Doi 101016_2Fjapplthermaleng201503006] Araoz Joseph hellip
httpslidepdfcomreaderfullapplied-thermal-engineering-volume-83-issue-2015-doi-1010162fjapplthermaleng201503006 1515
[26] FP Incropera AS Lavine DP DeWitt Fundamentals of Heat and MassTransfer John Wiley amp Sons 2011
[27] B Thomas D Pittman Update on the evaluation of different correlations forthe 1047298ow friction factor and heat transfer of Stirling engine regenerators inCollection of Technical Papers 35th Intersociety Energy Conversion Engi-neering Conference and Exhibit (IECEC) (Cat No00CH37022) American InstAeronaut amp Astronautics 2000 pp 76e84 httpdxdoiorg101109IECEC2000870632
[28] Genoastirling Srl 2014 wwwgenoastirlingcom[29] RC Tew K Jefferies D Miao USD of ED of TE Conservation LR Center
A Stirling Engine Computer Model for Performance Calculations (GoogleEBook) Department of Energy Of 1047297ce of Conservation and Solar ApplicationsDivision of Transportation Energy Conservation 1978
[30] I Urieli CJ Rallis DM Berchowitz Computer simulation of Stirling cyclemachines in 12th Intersociety Energy Conversion Engineering Conferencevol 1 American Nuclear Society Washington DC 1977 pp 1512e1521
[31] K Mahkamov Design improvements to a biomass Stirling engine usingmathematical analysis and 3D CFD modeling J Energy Resour Technol 128(2006) 203 httpdxdoiorg10111512213273
[32] H Snyman Second Order Analyses Methods for Stirling Engine Design Uni-versity of Stellenbosch Stellenbosch 2007
[33] G Sewell Numerical Solution of Ordinary and Partial Differential Equationssecond ed John Wiley amp Sons 2005
[34] DD Do Richard G Rice Applied Mathematics and Modeling for ChemicalEngineers [Hardcover] second ed Wiley-AIChE 2012
[35] Aspentech Aspen Custom Modelerreg AspenTech 2015
[36] Aspentech Chemical Process Optimization Software d
Chemical ProcessDesign Aspen Plus 2015
[37] B Thomas D Pittman AIAA-2000-2812 Update on the Evaluation of DifferentCorrelations for the Flow Friction Factor and Heat Transfer of Stirling EngineRegenerators 2000 pp 76e84
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 3030
7212019 Applied Thermal Engineering Volume 83 Issue 2015 [Doi 101016_2Fjapplthermaleng201503006] Araoz Joseph hellip
httpslidepdfcomreaderfullapplied-thermal-engineering-volume-83-issue-2015-doi-1010162fjapplthermaleng201503006 1115
complete cycle This re1047298ects the high importance of the regenerator
ef 1047297ciency for the engine performance
53 Work 1047298ow
Table 5 shows the simulation results for the compression and
expansion work during a single cycle This table also presents the
different work losses estimated for the system
The temperatures measured and the temperatures calculated
show a good thermal performance of the engine But the measured
brake power was very low Different problems on the engine
design and operational conditions may explain these very lowresults However additional experimental instrumentation is
needed for a detailed design study For this reason the present
analysis considers a theoretical approach that may be later com-
plemented with experimental studies This theoretical approach
considers Eq (7) From this equation and considering that the
thermal performance was found acceptable the main losses should
correspond to a low mechanical ef 1047297ciency of the prototype This
mechanical ef 1047297ciency is reduced by the presence of forced work
during the cycle and mechanical friction on the crank mechanism
Fig 13 presents the evaluation of the forced work in a pressure
volume diagram for the gas cycle inside the gamma prototype
From this it can be seen that the forced work (W) is mainly due to
the forced expansion process This means that at the experimentalconditions large part of the cyclic work may have been used to
complete the forced expansion process and thus the real engine
output is smaller than expected
The results discussed above are complemented with the vari-
ation of the compression (Wc) expansion (We) and net indicated
work (Wi) during the cycle shown in Fig 14
Fig 14 shows that during the 1047297rst part of the cycle from t frac14 0 to
t frac14 0045 the compression and expansion spaces present
exchanged roles This means that an increment of the volume is
presented in the compression space and a decrement of the volume
is present in the expansion one This reduced the engine perfor-
mance but it cannot be avoided since the gas needs to pass from
one space to another Regarding the second part of the cycle from
t frac14 0045 to t frac14 0095 the expansion and compression are shapedas expected and thus indicate a better dynamic during this period
However considering that large part of the expansion process is
forced the net brake work is low as reported in Table 5
From the previous analysis it can be concluded that a detailed
evaluation of the volumes dynamics the cranks mechanism effec-
tiveness and the forced work during the cycle must be considered
Fig 15 Heat 1047298
ow variation during the engine cycle
Table 6
Heat 1047298ow and heat loses during the cycle
Heat 1047298ow (Jcycle)
Heat exchanger space
Heater 1047298ow (Q hcJcycle) 5282
Cooler 1047298ow (Q kc Jcycle) 2356
Regenerator 1047298ow (Q rc Jcycle) 005
Heat lossesInternal conduction losses (Q lkc Jcycle) 2698
Shuttle conduction losses (Q lshc Jcycle) 8004
Regenerator losses during heating (Q lossrc Jcycle) 1862
Regenerator losses during cooling (Q lossrc Jcycle) 1862
Total heat requirements
Heating requirements (Q htc Jcycle) 17847
Cooling requirements (Q ktc Jcycle) 4218
Fig 14 Work 1047298
ow during the engine cycle
Fig 13 Pressureevolume diagram and forced work during the cycle
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 3026
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in order to re-design the engine for a better performance This will
be covered on a detailed parametric study to be reported on a next
article
54 Heat 1047298ow
Table 6 presents the results for the heat 1047298ow and corresponding
heat losses through the heat exchangers calculated at the end of a
single cycle [25] As it can be seen the total heat requirements are
almost three times the requirements calculated without consid-
ering the losses It can also be seen that the shuttle conduction
losses represent the main heat loss during the cycle These corre-
spond to the losses due to the oscillation of the hot displacer across
the temperature gradient in the working spaces of the engine
The cyclic variation for the heat 1047298ow is additionally shown in
Fig 15The heat requirements for the heater and cooler present
slight variations during the entire cycle On the other hand the
regenerator presents high variations managing large quantities of
heat This con1047297rms the large importance of this heat exchanger on
the engine performance
55 Brake power and brake ef 1047297ciency
The engine brake power is de1047297ned as the net brake work per
cycle (Ws) times the engine frequency (freq)
Pbr frac14 Ws freq (12)
The net brake work and the total heat requirement presented
on Tables 5 and 6 respectively are doubled considering the double
cylinder engine These values are reported on Table 7 which alsoreports the thermal and brake ef 1047297ciencies for the engine
The results re1047298ect the low performance of the engine under the
experimental conditions This was mainly attributed to the forced
work and the mechanical ef 1047297ciency as it was analysed in the pre-
vious section In addition complementary works will broad this
analysis with the aim of propose improvements on the engine
design and operational parameters
6 Conclusions
In the present work a thermodynamic model for a Stirling en-
gine was improved by including the numerical evaluation of the
forced work and the mechanical ef 1047297ciency then validated against
experimental data and 1047297nally implemented for the simulation of an engine prototype The numerical model developed considered
the analytic approach proposed by Senft [23] but extended its
application for the case of the more realistic adiabatic working
spaces assumptions Consequently the effective work taken from
the shaft is better estimated and thus used for a more complete
analysis of the thermal and mechanical performance of an engine
For this article the analysis considered a novel gamma engine
prototype under the experimental conditions of a micro scale
combined heat and power system fuelled by wood pellets
The simulation results were compared with the experimental
data measured during long time runs of the system The model
performance was very good for the prediction of the temperatures
in the different spaces of the engine In addition the estimations for
the net brake power also presented results similar to the measured
values However additional experimental work should be per-
formed to obtain data to validate the calculation of the different
losses through the engine
According to the results obtained the thermal performance of
the engine was found acceptable and thus the low power output
measured is preliminary attributed to a reduced mechanical ef 1047297-
ciency of the system The possible reasons for this low performance
were further analysed with the different results for the tempera-
tures variation mass and volume variation pressure drops and the
pressure volume diagrams obtained with the model According to
these analyses the dynamics of the volumes variation and the
crank mechanism may also be improved in order to obtain higher
network during the cycle In addition it was found that the engine
performance is very sensitive to the effect of the buffer pressure
These results will be extended with a sensitivity analysis for the
system on a complementary work that aims to identify better the
effect of the different parameters on the engine performance
Acknowledgements
This work was possible thanks to the 1047297nancial support of the
Swedish International Development Cooperation Agency the di-
vision of Heat and Power Technology Department of Energy
Technology at Royal Institute of Technology (KTH) in Sweden and
Universidad Mayor de San Simon (UMSS) in Bolivia
Appendix A Detailed Stirling engine parameters
Table 7
Power output and ef 1047297ciency of the engine
Brake power (W) Heat requirement (W) Thermal ef 1047297ciency () Mechanical ef 1047297ciency () Brake ef 1047297ciency ()
5358 184535 1610 1810 290
Table A1
Inputs for the cooler in ACM
Variable Value Units Description
do 0005 m Tubes external diameter
di 0003 m Tubes internal diameter
kw 14200 Wm K Material conductivity
L 0032 m Tubes length
num 162 e Number of tubes
sl 0005 m Space between tubes
Table A2
Inputs for the heater in ACM
Variable Value Units Description
de 0005 M Tubes external diameter
di 00031 M Tubes internal diameterkw 142 Wm K Material conductivity
len 0149 m Tubes length
num 360 e Number of tubes
sl 0005 m Space between tubes
Table A3
Inputs for the regenerator in ACM
Variable Value Units Description
Din 0078 m Regenerator housing internal diameter
dout 0 107 m Regenera tor h ousin g extern al diameter
dwir e 2 1e-0 04 m Wi re di ameter of t he ma tri x
kwr 270 Wm K Thermal conductivity of the matrix material
Lr 007 m Length of the regenerator housing
Porosity 087 Matrix porosity
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 30 27
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Appendix B
Main equations for the Stirling engine model
Stirling engine module
Mean pressure
P frac14 MR
V cTc
thorn V kTk
thorn V rTr
thorn V hTh
thorn V eTe
Pressure variation
dP
d4frac14
gP
0BB
vV cv4
Tck
thorn
vV ev4
The
1CCA
V cTckthorn g
V kTk
thorn V rTrthorn V hTh
thorn V eThe
Mass of the working gas in the different spaces
mc frac14 p
V c
RTc
mk frac14 p
V k
RTk
mr frac14 p
V r
RTr
mh frac14 p
V h
RTh
me frac14 p
V e
RTe
Mass accumulation
dmk
d4frac14
mk
P
vP
v4
dmh
d4frac14
mh
P
vP
v4
dmr
d4frac14
mr
P
vP
v4
dmc
d4frac14
P
vV cv4
thorn
V c
vPv4
g
RTck
dme
d4frac14
P
vV ev4
thorn
V e
vPv4
g
RThe
Mass 1047298ow
mck frac14 dmc mhe frac14 dme mkr frac14 mck dmkmrh frac14 mhe thorn dmh
Conditional temperatures
If mck gt 0 then Tck frac14 Tc else Tck frac14 Tk
If mhe gt 0 then The frac14 Th else The frac14 Te
Temperatures
dTc
d4frac14 Tc
0BBvPv4
P
thorn
vV cv4
V c
vmc
v4
mc
1CCA
dTe
d4frac14 Te
0BBvPv4
P thorn
vV ev4
V e
vmev4
me
1CCAEnergy
dQ kd4
frac14
V k
vPv4
Cv
R CpethTckmck TkrmkrTHORN
dQ rd4
frac14
V r
vPv4
Cv
R C pethTkrmkr TrhmrhTHORN
dQ hd4
frac14V hvPv4
Cv
R CpethTrhmrh ThemheTHORN
dWc
d4frac14 P
vV cv4
dWe
d4frac14 P
vV ev4
Internal heat transfer module
Heat transfer from the heater wall to the working gas
Q h frac14 1
R cih thorn R hih thorn R fihethTwoh ThTHORN
Heat transfer from the cooler wall to the working gas
Q k frac14 1
R cik thorn R hik thorn R fikethTwik TkTHORN
Heat loss during the regenerator process
Q lossr frac14 eth1 3THORN Q r
Regenerator effectiveness
3frac14 NTU
1 thorn NTU
External heat transfer module
Heat transfer from the 1047298
ame to the external wall of the heater
Table A4
Inputs for the expansion-compression spaces and crank mechanism
Variable Value Units Description
vclc 44e-006 m3 Compression space clearance volume
vcle 26e-005 m3 Expansion space clearance volume
vswc 926e-005 m3 Compression space swept volume
vswe 25134e-004 m3 Expansion space swept volume
dispd 0062 m Displacer diameter
displ 007 m Displacer lengthef fmek 08 e Mechanism effectiveness
freq 5 Hz Frequency
jgap 0006 M Gap between cylinder displacer and wall
kpist 1627 Wm K Piston conductivity
pbuff 12ethorn006 Pa Buffer pressure
phase 900 deg Phase angle advance
pmean 1 25ethorn00 6 Pa Mean o pera tin g pressure
strk 0035 m Displacer stroke
dispd 0062 m Displacer diameter
Table A5
Working and cooling 1047298uid inputs in ACM
Variable Value Units Description
Working Fluid Air e Working 1047298uid inside the engine
Cooling FLUID Water e Cooling 1047298uid through the engine cooler
Tcooling 2 88 K Inl et tempera ture of th e co ol in g 1047298uid
Table A6
Fouling factors and external combustion inputs in ACM
Variable Value Units Description
T ad 1 38 7 K Fla me temper at ur e i n t he c omb ustio n ch amb er
absorp 070 e Absorptivity of the heater material
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 3028
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Q h frac14 11
hrhAohthorn R foh
ethTad TwohTHORN
hrh frac14 assAohFR ethTad thorn TwohTHORN
T2ad thorn T2
woh
Estimation of the outlet temperature of the cooling 1047298uid
Twok frac14 Twater in thorn Q k
1
hokAokthorn
1
2mwaterCpwater
hok frac14 11
hwaterthorn R fok
Energy losses
Pressure drop in the heat exchangers
DP frac14
f
dhy
1
2 rv
2
l
Pressure drop in the regenerator based on the correlations of
Thomas and Pittman [37]
DP frac14 Cf nr
2u2
Cf frac14 Cfd thornCsf
Re
Csf frac14 68556 Cfd frac14 05274 wire screensCsf frac14 70035 Cfd frac14 09307 metal felts
Total pumping losses
Wploss frac14
Z 2p0
Xifrac143
ifrac141
DPi dV edq
$dq
Energy losses due to shuttle conduction
Q lsh frac14 04Z2KpistDd
JL dethTe TcTHORN
Mechanical ef 1047297ciency module
Mechanical ef 1047297ciency
hm frac14 Ws
Wi
Mechanical ef 1047297ciency considering the mechanism effectiveness
and forced work
hm frac14 E
1
E E
W
Wi
Forced work
W frac14
I ethP PbTHORNthorndV thorn
I ethP PbTHORNdV thorn
Brake ef 1047297
ciency
hb frac14 Ws
Q htfrac14
Wi
Q ht
Ws
Wifrac14 hthm
References
[1] DG Thombare SK Verma Technological development in the Stirling cycle
engines Renew Sustain Energy Rev 12 (2008) 1e
38 httpdxdoiorg101016jrser200607001
[2] AD Peacock M Newborough Impact of micro-CHP systems on domesticsector CO2 emissions Appl Therm Eng 25 (2005) 2653e2676
[3] B Thomas Benchmark testing of Micro-CHP units Appl Therm Eng 28(2008) 2049e2054 httpdxdoiorg101016japplthermaleng200803010
[4] L Dong H Liu S Riffat Development of small-scale and micro-scale biomass-fuelled CHP systems e a literature review Appl Therm Eng 29 (2009)2119e2126 httpdxdoiorg101016japplthermaleng200812004
[5] I Gonzalez-Pino A Campos-Celador E Perez-Iribarren J Teres-Zubiaga JM Sala Parametric study of the operational and economic feasibility of Stirling micro-cogeneration devices in Spain Appl Therm Eng (2013) httpdxdoiorg101016japplthermaleng201312020
[6] R Dyson S Wilson R Tew Review of computational Stirling analysismethods in 2nd International Energy Conversion Engineering ConferenceAmerican Institute of Aeronautics and Astronautics 2004 httpdxdoiorg10251462004-5582
[7] C-H Cheng Y-J Yu Numerical model for predicting thermodynamic cycleand thermal ef 1047297ciency of a beta-type Stirling engine with rhombic-drivemechanism Renew Energy 35 (2010) 2590e2601 httpdxdoiorg101016jrenene201004002
[8] N Seraj Mehdizadeh P Stouffs Simulation of a Martini displacer free pistonStirling engine for electric power generation Int J Thermodyn 3 (1) (2000)27e34 httpdxdoiorg105541ijot30
[9] N Parlak A Wagner M Elsner HS Soyhan Thermodynamic analysis of agamma type Stirling engine in non-ideal adiabatic conditions Renew Energy34 (2009) 266e273 httpdxdoiorg101016jrenene200802030
[10] JM Strauss RT Dobson Evaluation of a second order simulation for Sterlingengine design and optimisation J Energy South Afr 21 (2010) 17e29
[11] I Tlili Y Timoumi SB Nasrallah Thermodynamic analysis of the Stirling heatengine with regenerative losses and internal irreversibilities Int J Engine Res9 (2008) 45e56 httpdxdoiorg10124314680874JER01707
[12] K Mahkamov D Djumanov Three-dimensional CFD modeling of a Stirlingengine in Proceedings of the 11th international Stirling Engine Conferencevol 19 2003
[13] M Ibrahim R Tew Z Zhang D Gedeon T Simon CFD Modeling of Free-piston Stirling Engines Cleveland Ohio 2001
[14] S Wilson R Dyson Multi-D CFD modeling of a free-piston Stirling convertorat NASA GRC in Proc 2nd International Energy Conversion EngineeringConference vol 5673 2004
[15] E Eid Performance of a beta-con1047297guration heat engine having a regenerativedisplacer Renew Energy 34 (2009) 2404e2413 httpdxdoiorg101016
jrenene200903016[16] AA El-Ehwany GM Hennes EI Eid EA El-Kenany Development of the
performance of an alpha-type heat engine by using elbow-bend transposed-1047298uids heat exchanger as a heater and a cooler Energy Convers Manag 52(2011) 1010e1019 httpdxdoiorg101016jenconman201008029
[17] C-H Cheng Y-J Yu Dynamic simulation of a beta-type Stirling engine withcam-drive mechanism via the combination of the thermodynamic and dy-namic models Renew Energy 36 (2011) 714e725
[18] E Entchev J Gusdorf M Swinton M Bell F Szadkowski W Kalb1047298eisch et alMicro-generation technology assessment for housing technology EnergyBuild 36 (2004) 925e931 httpdxdoiorg101016jenbuild200403004
[19] M Paringlsson H Carlsen Development of a wood powder fuelled 35 kW StirlingCHP unit in Proceedings of the 11th ISEC (International Stirling EngineConference) 2003 pp 221e230
[20] a Nishiyama H Shimojima a Ishikawa Y Itaya S Kambara H Moritomi etal Fuel and emissions properties of Stirling engine operated with woodpowder Fuel 86 (2007) 2333e2342 httpdxdoiorg101016
jfuel200701040[21] K Sato N Ohiwa A Ishikawa H Shimojima A Nishiyama Y Moriya
Development of small-scale CHP Plant with a wood powder-fueled Stirlingengine J Power Energy Syst 2 (2008) 1221e1231 httpdxdoiorg101299
jpes21221[22] I Urieli DM Berchowitz Stirling Cycle Engine Analysis Hilger Ltd Bristol
1984[23] JR Senft Mechanical Ef 1047297ciency of Heat Engines Cambridge University Press
2007[24] E Cardozo C Erlich A Malmquist L Alejo Integration of a wood pellet
burner and a Stirling engine to produce residential heat and power ApplTherm Eng (2014) httpdxdoiorg101016japplthermaleng201408024
[25] JA Araoz M Salomon L Alejo TH Fransson Non-ideal Stirling enginethermodynamic model suitable for the integration into overall energy sys-tems Appl Therm Eng 73 (2014) 203e219 httpdxdoiorg101016
japplthermaleng201407050
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 30 29
7212019 Applied Thermal Engineering Volume 83 Issue 2015 [Doi 101016_2Fjapplthermaleng201503006] Araoz Joseph hellip
httpslidepdfcomreaderfullapplied-thermal-engineering-volume-83-issue-2015-doi-1010162fjapplthermaleng201503006 1515
[26] FP Incropera AS Lavine DP DeWitt Fundamentals of Heat and MassTransfer John Wiley amp Sons 2011
[27] B Thomas D Pittman Update on the evaluation of different correlations forthe 1047298ow friction factor and heat transfer of Stirling engine regenerators inCollection of Technical Papers 35th Intersociety Energy Conversion Engi-neering Conference and Exhibit (IECEC) (Cat No00CH37022) American InstAeronaut amp Astronautics 2000 pp 76e84 httpdxdoiorg101109IECEC2000870632
[28] Genoastirling Srl 2014 wwwgenoastirlingcom[29] RC Tew K Jefferies D Miao USD of ED of TE Conservation LR Center
A Stirling Engine Computer Model for Performance Calculations (GoogleEBook) Department of Energy Of 1047297ce of Conservation and Solar ApplicationsDivision of Transportation Energy Conservation 1978
[30] I Urieli CJ Rallis DM Berchowitz Computer simulation of Stirling cyclemachines in 12th Intersociety Energy Conversion Engineering Conferencevol 1 American Nuclear Society Washington DC 1977 pp 1512e1521
[31] K Mahkamov Design improvements to a biomass Stirling engine usingmathematical analysis and 3D CFD modeling J Energy Resour Technol 128(2006) 203 httpdxdoiorg10111512213273
[32] H Snyman Second Order Analyses Methods for Stirling Engine Design Uni-versity of Stellenbosch Stellenbosch 2007
[33] G Sewell Numerical Solution of Ordinary and Partial Differential Equationssecond ed John Wiley amp Sons 2005
[34] DD Do Richard G Rice Applied Mathematics and Modeling for ChemicalEngineers [Hardcover] second ed Wiley-AIChE 2012
[35] Aspentech Aspen Custom Modelerreg AspenTech 2015
[36] Aspentech Chemical Process Optimization Software d
Chemical ProcessDesign Aspen Plus 2015
[37] B Thomas D Pittman AIAA-2000-2812 Update on the Evaluation of DifferentCorrelations for the Flow Friction Factor and Heat Transfer of Stirling EngineRegenerators 2000 pp 76e84
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 3030
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in order to re-design the engine for a better performance This will
be covered on a detailed parametric study to be reported on a next
article
54 Heat 1047298ow
Table 6 presents the results for the heat 1047298ow and corresponding
heat losses through the heat exchangers calculated at the end of a
single cycle [25] As it can be seen the total heat requirements are
almost three times the requirements calculated without consid-
ering the losses It can also be seen that the shuttle conduction
losses represent the main heat loss during the cycle These corre-
spond to the losses due to the oscillation of the hot displacer across
the temperature gradient in the working spaces of the engine
The cyclic variation for the heat 1047298ow is additionally shown in
Fig 15The heat requirements for the heater and cooler present
slight variations during the entire cycle On the other hand the
regenerator presents high variations managing large quantities of
heat This con1047297rms the large importance of this heat exchanger on
the engine performance
55 Brake power and brake ef 1047297ciency
The engine brake power is de1047297ned as the net brake work per
cycle (Ws) times the engine frequency (freq)
Pbr frac14 Ws freq (12)
The net brake work and the total heat requirement presented
on Tables 5 and 6 respectively are doubled considering the double
cylinder engine These values are reported on Table 7 which alsoreports the thermal and brake ef 1047297ciencies for the engine
The results re1047298ect the low performance of the engine under the
experimental conditions This was mainly attributed to the forced
work and the mechanical ef 1047297ciency as it was analysed in the pre-
vious section In addition complementary works will broad this
analysis with the aim of propose improvements on the engine
design and operational parameters
6 Conclusions
In the present work a thermodynamic model for a Stirling en-
gine was improved by including the numerical evaluation of the
forced work and the mechanical ef 1047297ciency then validated against
experimental data and 1047297nally implemented for the simulation of an engine prototype The numerical model developed considered
the analytic approach proposed by Senft [23] but extended its
application for the case of the more realistic adiabatic working
spaces assumptions Consequently the effective work taken from
the shaft is better estimated and thus used for a more complete
analysis of the thermal and mechanical performance of an engine
For this article the analysis considered a novel gamma engine
prototype under the experimental conditions of a micro scale
combined heat and power system fuelled by wood pellets
The simulation results were compared with the experimental
data measured during long time runs of the system The model
performance was very good for the prediction of the temperatures
in the different spaces of the engine In addition the estimations for
the net brake power also presented results similar to the measured
values However additional experimental work should be per-
formed to obtain data to validate the calculation of the different
losses through the engine
According to the results obtained the thermal performance of
the engine was found acceptable and thus the low power output
measured is preliminary attributed to a reduced mechanical ef 1047297-
ciency of the system The possible reasons for this low performance
were further analysed with the different results for the tempera-
tures variation mass and volume variation pressure drops and the
pressure volume diagrams obtained with the model According to
these analyses the dynamics of the volumes variation and the
crank mechanism may also be improved in order to obtain higher
network during the cycle In addition it was found that the engine
performance is very sensitive to the effect of the buffer pressure
These results will be extended with a sensitivity analysis for the
system on a complementary work that aims to identify better the
effect of the different parameters on the engine performance
Acknowledgements
This work was possible thanks to the 1047297nancial support of the
Swedish International Development Cooperation Agency the di-
vision of Heat and Power Technology Department of Energy
Technology at Royal Institute of Technology (KTH) in Sweden and
Universidad Mayor de San Simon (UMSS) in Bolivia
Appendix A Detailed Stirling engine parameters
Table 7
Power output and ef 1047297ciency of the engine
Brake power (W) Heat requirement (W) Thermal ef 1047297ciency () Mechanical ef 1047297ciency () Brake ef 1047297ciency ()
5358 184535 1610 1810 290
Table A1
Inputs for the cooler in ACM
Variable Value Units Description
do 0005 m Tubes external diameter
di 0003 m Tubes internal diameter
kw 14200 Wm K Material conductivity
L 0032 m Tubes length
num 162 e Number of tubes
sl 0005 m Space between tubes
Table A2
Inputs for the heater in ACM
Variable Value Units Description
de 0005 M Tubes external diameter
di 00031 M Tubes internal diameterkw 142 Wm K Material conductivity
len 0149 m Tubes length
num 360 e Number of tubes
sl 0005 m Space between tubes
Table A3
Inputs for the regenerator in ACM
Variable Value Units Description
Din 0078 m Regenerator housing internal diameter
dout 0 107 m Regenera tor h ousin g extern al diameter
dwir e 2 1e-0 04 m Wi re di ameter of t he ma tri x
kwr 270 Wm K Thermal conductivity of the matrix material
Lr 007 m Length of the regenerator housing
Porosity 087 Matrix porosity
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 30 27
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Appendix B
Main equations for the Stirling engine model
Stirling engine module
Mean pressure
P frac14 MR
V cTc
thorn V kTk
thorn V rTr
thorn V hTh
thorn V eTe
Pressure variation
dP
d4frac14
gP
0BB
vV cv4
Tck
thorn
vV ev4
The
1CCA
V cTckthorn g
V kTk
thorn V rTrthorn V hTh
thorn V eThe
Mass of the working gas in the different spaces
mc frac14 p
V c
RTc
mk frac14 p
V k
RTk
mr frac14 p
V r
RTr
mh frac14 p
V h
RTh
me frac14 p
V e
RTe
Mass accumulation
dmk
d4frac14
mk
P
vP
v4
dmh
d4frac14
mh
P
vP
v4
dmr
d4frac14
mr
P
vP
v4
dmc
d4frac14
P
vV cv4
thorn
V c
vPv4
g
RTck
dme
d4frac14
P
vV ev4
thorn
V e
vPv4
g
RThe
Mass 1047298ow
mck frac14 dmc mhe frac14 dme mkr frac14 mck dmkmrh frac14 mhe thorn dmh
Conditional temperatures
If mck gt 0 then Tck frac14 Tc else Tck frac14 Tk
If mhe gt 0 then The frac14 Th else The frac14 Te
Temperatures
dTc
d4frac14 Tc
0BBvPv4
P
thorn
vV cv4
V c
vmc
v4
mc
1CCA
dTe
d4frac14 Te
0BBvPv4
P thorn
vV ev4
V e
vmev4
me
1CCAEnergy
dQ kd4
frac14
V k
vPv4
Cv
R CpethTckmck TkrmkrTHORN
dQ rd4
frac14
V r
vPv4
Cv
R C pethTkrmkr TrhmrhTHORN
dQ hd4
frac14V hvPv4
Cv
R CpethTrhmrh ThemheTHORN
dWc
d4frac14 P
vV cv4
dWe
d4frac14 P
vV ev4
Internal heat transfer module
Heat transfer from the heater wall to the working gas
Q h frac14 1
R cih thorn R hih thorn R fihethTwoh ThTHORN
Heat transfer from the cooler wall to the working gas
Q k frac14 1
R cik thorn R hik thorn R fikethTwik TkTHORN
Heat loss during the regenerator process
Q lossr frac14 eth1 3THORN Q r
Regenerator effectiveness
3frac14 NTU
1 thorn NTU
External heat transfer module
Heat transfer from the 1047298
ame to the external wall of the heater
Table A4
Inputs for the expansion-compression spaces and crank mechanism
Variable Value Units Description
vclc 44e-006 m3 Compression space clearance volume
vcle 26e-005 m3 Expansion space clearance volume
vswc 926e-005 m3 Compression space swept volume
vswe 25134e-004 m3 Expansion space swept volume
dispd 0062 m Displacer diameter
displ 007 m Displacer lengthef fmek 08 e Mechanism effectiveness
freq 5 Hz Frequency
jgap 0006 M Gap between cylinder displacer and wall
kpist 1627 Wm K Piston conductivity
pbuff 12ethorn006 Pa Buffer pressure
phase 900 deg Phase angle advance
pmean 1 25ethorn00 6 Pa Mean o pera tin g pressure
strk 0035 m Displacer stroke
dispd 0062 m Displacer diameter
Table A5
Working and cooling 1047298uid inputs in ACM
Variable Value Units Description
Working Fluid Air e Working 1047298uid inside the engine
Cooling FLUID Water e Cooling 1047298uid through the engine cooler
Tcooling 2 88 K Inl et tempera ture of th e co ol in g 1047298uid
Table A6
Fouling factors and external combustion inputs in ACM
Variable Value Units Description
T ad 1 38 7 K Fla me temper at ur e i n t he c omb ustio n ch amb er
absorp 070 e Absorptivity of the heater material
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 3028
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Q h frac14 11
hrhAohthorn R foh
ethTad TwohTHORN
hrh frac14 assAohFR ethTad thorn TwohTHORN
T2ad thorn T2
woh
Estimation of the outlet temperature of the cooling 1047298uid
Twok frac14 Twater in thorn Q k
1
hokAokthorn
1
2mwaterCpwater
hok frac14 11
hwaterthorn R fok
Energy losses
Pressure drop in the heat exchangers
DP frac14
f
dhy
1
2 rv
2
l
Pressure drop in the regenerator based on the correlations of
Thomas and Pittman [37]
DP frac14 Cf nr
2u2
Cf frac14 Cfd thornCsf
Re
Csf frac14 68556 Cfd frac14 05274 wire screensCsf frac14 70035 Cfd frac14 09307 metal felts
Total pumping losses
Wploss frac14
Z 2p0
Xifrac143
ifrac141
DPi dV edq
$dq
Energy losses due to shuttle conduction
Q lsh frac14 04Z2KpistDd
JL dethTe TcTHORN
Mechanical ef 1047297ciency module
Mechanical ef 1047297ciency
hm frac14 Ws
Wi
Mechanical ef 1047297ciency considering the mechanism effectiveness
and forced work
hm frac14 E
1
E E
W
Wi
Forced work
W frac14
I ethP PbTHORNthorndV thorn
I ethP PbTHORNdV thorn
Brake ef 1047297
ciency
hb frac14 Ws
Q htfrac14
Wi
Q ht
Ws
Wifrac14 hthm
References
[1] DG Thombare SK Verma Technological development in the Stirling cycle
engines Renew Sustain Energy Rev 12 (2008) 1e
38 httpdxdoiorg101016jrser200607001
[2] AD Peacock M Newborough Impact of micro-CHP systems on domesticsector CO2 emissions Appl Therm Eng 25 (2005) 2653e2676
[3] B Thomas Benchmark testing of Micro-CHP units Appl Therm Eng 28(2008) 2049e2054 httpdxdoiorg101016japplthermaleng200803010
[4] L Dong H Liu S Riffat Development of small-scale and micro-scale biomass-fuelled CHP systems e a literature review Appl Therm Eng 29 (2009)2119e2126 httpdxdoiorg101016japplthermaleng200812004
[5] I Gonzalez-Pino A Campos-Celador E Perez-Iribarren J Teres-Zubiaga JM Sala Parametric study of the operational and economic feasibility of Stirling micro-cogeneration devices in Spain Appl Therm Eng (2013) httpdxdoiorg101016japplthermaleng201312020
[6] R Dyson S Wilson R Tew Review of computational Stirling analysismethods in 2nd International Energy Conversion Engineering ConferenceAmerican Institute of Aeronautics and Astronautics 2004 httpdxdoiorg10251462004-5582
[7] C-H Cheng Y-J Yu Numerical model for predicting thermodynamic cycleand thermal ef 1047297ciency of a beta-type Stirling engine with rhombic-drivemechanism Renew Energy 35 (2010) 2590e2601 httpdxdoiorg101016jrenene201004002
[8] N Seraj Mehdizadeh P Stouffs Simulation of a Martini displacer free pistonStirling engine for electric power generation Int J Thermodyn 3 (1) (2000)27e34 httpdxdoiorg105541ijot30
[9] N Parlak A Wagner M Elsner HS Soyhan Thermodynamic analysis of agamma type Stirling engine in non-ideal adiabatic conditions Renew Energy34 (2009) 266e273 httpdxdoiorg101016jrenene200802030
[10] JM Strauss RT Dobson Evaluation of a second order simulation for Sterlingengine design and optimisation J Energy South Afr 21 (2010) 17e29
[11] I Tlili Y Timoumi SB Nasrallah Thermodynamic analysis of the Stirling heatengine with regenerative losses and internal irreversibilities Int J Engine Res9 (2008) 45e56 httpdxdoiorg10124314680874JER01707
[12] K Mahkamov D Djumanov Three-dimensional CFD modeling of a Stirlingengine in Proceedings of the 11th international Stirling Engine Conferencevol 19 2003
[13] M Ibrahim R Tew Z Zhang D Gedeon T Simon CFD Modeling of Free-piston Stirling Engines Cleveland Ohio 2001
[14] S Wilson R Dyson Multi-D CFD modeling of a free-piston Stirling convertorat NASA GRC in Proc 2nd International Energy Conversion EngineeringConference vol 5673 2004
[15] E Eid Performance of a beta-con1047297guration heat engine having a regenerativedisplacer Renew Energy 34 (2009) 2404e2413 httpdxdoiorg101016
jrenene200903016[16] AA El-Ehwany GM Hennes EI Eid EA El-Kenany Development of the
performance of an alpha-type heat engine by using elbow-bend transposed-1047298uids heat exchanger as a heater and a cooler Energy Convers Manag 52(2011) 1010e1019 httpdxdoiorg101016jenconman201008029
[17] C-H Cheng Y-J Yu Dynamic simulation of a beta-type Stirling engine withcam-drive mechanism via the combination of the thermodynamic and dy-namic models Renew Energy 36 (2011) 714e725
[18] E Entchev J Gusdorf M Swinton M Bell F Szadkowski W Kalb1047298eisch et alMicro-generation technology assessment for housing technology EnergyBuild 36 (2004) 925e931 httpdxdoiorg101016jenbuild200403004
[19] M Paringlsson H Carlsen Development of a wood powder fuelled 35 kW StirlingCHP unit in Proceedings of the 11th ISEC (International Stirling EngineConference) 2003 pp 221e230
[20] a Nishiyama H Shimojima a Ishikawa Y Itaya S Kambara H Moritomi etal Fuel and emissions properties of Stirling engine operated with woodpowder Fuel 86 (2007) 2333e2342 httpdxdoiorg101016
jfuel200701040[21] K Sato N Ohiwa A Ishikawa H Shimojima A Nishiyama Y Moriya
Development of small-scale CHP Plant with a wood powder-fueled Stirlingengine J Power Energy Syst 2 (2008) 1221e1231 httpdxdoiorg101299
jpes21221[22] I Urieli DM Berchowitz Stirling Cycle Engine Analysis Hilger Ltd Bristol
1984[23] JR Senft Mechanical Ef 1047297ciency of Heat Engines Cambridge University Press
2007[24] E Cardozo C Erlich A Malmquist L Alejo Integration of a wood pellet
burner and a Stirling engine to produce residential heat and power ApplTherm Eng (2014) httpdxdoiorg101016japplthermaleng201408024
[25] JA Araoz M Salomon L Alejo TH Fransson Non-ideal Stirling enginethermodynamic model suitable for the integration into overall energy sys-tems Appl Therm Eng 73 (2014) 203e219 httpdxdoiorg101016
japplthermaleng201407050
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 30 29
7212019 Applied Thermal Engineering Volume 83 Issue 2015 [Doi 101016_2Fjapplthermaleng201503006] Araoz Joseph hellip
httpslidepdfcomreaderfullapplied-thermal-engineering-volume-83-issue-2015-doi-1010162fjapplthermaleng201503006 1515
[26] FP Incropera AS Lavine DP DeWitt Fundamentals of Heat and MassTransfer John Wiley amp Sons 2011
[27] B Thomas D Pittman Update on the evaluation of different correlations forthe 1047298ow friction factor and heat transfer of Stirling engine regenerators inCollection of Technical Papers 35th Intersociety Energy Conversion Engi-neering Conference and Exhibit (IECEC) (Cat No00CH37022) American InstAeronaut amp Astronautics 2000 pp 76e84 httpdxdoiorg101109IECEC2000870632
[28] Genoastirling Srl 2014 wwwgenoastirlingcom[29] RC Tew K Jefferies D Miao USD of ED of TE Conservation LR Center
A Stirling Engine Computer Model for Performance Calculations (GoogleEBook) Department of Energy Of 1047297ce of Conservation and Solar ApplicationsDivision of Transportation Energy Conservation 1978
[30] I Urieli CJ Rallis DM Berchowitz Computer simulation of Stirling cyclemachines in 12th Intersociety Energy Conversion Engineering Conferencevol 1 American Nuclear Society Washington DC 1977 pp 1512e1521
[31] K Mahkamov Design improvements to a biomass Stirling engine usingmathematical analysis and 3D CFD modeling J Energy Resour Technol 128(2006) 203 httpdxdoiorg10111512213273
[32] H Snyman Second Order Analyses Methods for Stirling Engine Design Uni-versity of Stellenbosch Stellenbosch 2007
[33] G Sewell Numerical Solution of Ordinary and Partial Differential Equationssecond ed John Wiley amp Sons 2005
[34] DD Do Richard G Rice Applied Mathematics and Modeling for ChemicalEngineers [Hardcover] second ed Wiley-AIChE 2012
[35] Aspentech Aspen Custom Modelerreg AspenTech 2015
[36] Aspentech Chemical Process Optimization Software d
Chemical ProcessDesign Aspen Plus 2015
[37] B Thomas D Pittman AIAA-2000-2812 Update on the Evaluation of DifferentCorrelations for the Flow Friction Factor and Heat Transfer of Stirling EngineRegenerators 2000 pp 76e84
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 3030
7212019 Applied Thermal Engineering Volume 83 Issue 2015 [Doi 101016_2Fjapplthermaleng201503006] Araoz Joseph hellip
httpslidepdfcomreaderfullapplied-thermal-engineering-volume-83-issue-2015-doi-1010162fjapplthermaleng201503006 1315
Appendix B
Main equations for the Stirling engine model
Stirling engine module
Mean pressure
P frac14 MR
V cTc
thorn V kTk
thorn V rTr
thorn V hTh
thorn V eTe
Pressure variation
dP
d4frac14
gP
0BB
vV cv4
Tck
thorn
vV ev4
The
1CCA
V cTckthorn g
V kTk
thorn V rTrthorn V hTh
thorn V eThe
Mass of the working gas in the different spaces
mc frac14 p
V c
RTc
mk frac14 p
V k
RTk
mr frac14 p
V r
RTr
mh frac14 p
V h
RTh
me frac14 p
V e
RTe
Mass accumulation
dmk
d4frac14
mk
P
vP
v4
dmh
d4frac14
mh
P
vP
v4
dmr
d4frac14
mr
P
vP
v4
dmc
d4frac14
P
vV cv4
thorn
V c
vPv4
g
RTck
dme
d4frac14
P
vV ev4
thorn
V e
vPv4
g
RThe
Mass 1047298ow
mck frac14 dmc mhe frac14 dme mkr frac14 mck dmkmrh frac14 mhe thorn dmh
Conditional temperatures
If mck gt 0 then Tck frac14 Tc else Tck frac14 Tk
If mhe gt 0 then The frac14 Th else The frac14 Te
Temperatures
dTc
d4frac14 Tc
0BBvPv4
P
thorn
vV cv4
V c
vmc
v4
mc
1CCA
dTe
d4frac14 Te
0BBvPv4
P thorn
vV ev4
V e
vmev4
me
1CCAEnergy
dQ kd4
frac14
V k
vPv4
Cv
R CpethTckmck TkrmkrTHORN
dQ rd4
frac14
V r
vPv4
Cv
R C pethTkrmkr TrhmrhTHORN
dQ hd4
frac14V hvPv4
Cv
R CpethTrhmrh ThemheTHORN
dWc
d4frac14 P
vV cv4
dWe
d4frac14 P
vV ev4
Internal heat transfer module
Heat transfer from the heater wall to the working gas
Q h frac14 1
R cih thorn R hih thorn R fihethTwoh ThTHORN
Heat transfer from the cooler wall to the working gas
Q k frac14 1
R cik thorn R hik thorn R fikethTwik TkTHORN
Heat loss during the regenerator process
Q lossr frac14 eth1 3THORN Q r
Regenerator effectiveness
3frac14 NTU
1 thorn NTU
External heat transfer module
Heat transfer from the 1047298
ame to the external wall of the heater
Table A4
Inputs for the expansion-compression spaces and crank mechanism
Variable Value Units Description
vclc 44e-006 m3 Compression space clearance volume
vcle 26e-005 m3 Expansion space clearance volume
vswc 926e-005 m3 Compression space swept volume
vswe 25134e-004 m3 Expansion space swept volume
dispd 0062 m Displacer diameter
displ 007 m Displacer lengthef fmek 08 e Mechanism effectiveness
freq 5 Hz Frequency
jgap 0006 M Gap between cylinder displacer and wall
kpist 1627 Wm K Piston conductivity
pbuff 12ethorn006 Pa Buffer pressure
phase 900 deg Phase angle advance
pmean 1 25ethorn00 6 Pa Mean o pera tin g pressure
strk 0035 m Displacer stroke
dispd 0062 m Displacer diameter
Table A5
Working and cooling 1047298uid inputs in ACM
Variable Value Units Description
Working Fluid Air e Working 1047298uid inside the engine
Cooling FLUID Water e Cooling 1047298uid through the engine cooler
Tcooling 2 88 K Inl et tempera ture of th e co ol in g 1047298uid
Table A6
Fouling factors and external combustion inputs in ACM
Variable Value Units Description
T ad 1 38 7 K Fla me temper at ur e i n t he c omb ustio n ch amb er
absorp 070 e Absorptivity of the heater material
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 3028
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Q h frac14 11
hrhAohthorn R foh
ethTad TwohTHORN
hrh frac14 assAohFR ethTad thorn TwohTHORN
T2ad thorn T2
woh
Estimation of the outlet temperature of the cooling 1047298uid
Twok frac14 Twater in thorn Q k
1
hokAokthorn
1
2mwaterCpwater
hok frac14 11
hwaterthorn R fok
Energy losses
Pressure drop in the heat exchangers
DP frac14
f
dhy
1
2 rv
2
l
Pressure drop in the regenerator based on the correlations of
Thomas and Pittman [37]
DP frac14 Cf nr
2u2
Cf frac14 Cfd thornCsf
Re
Csf frac14 68556 Cfd frac14 05274 wire screensCsf frac14 70035 Cfd frac14 09307 metal felts
Total pumping losses
Wploss frac14
Z 2p0
Xifrac143
ifrac141
DPi dV edq
$dq
Energy losses due to shuttle conduction
Q lsh frac14 04Z2KpistDd
JL dethTe TcTHORN
Mechanical ef 1047297ciency module
Mechanical ef 1047297ciency
hm frac14 Ws
Wi
Mechanical ef 1047297ciency considering the mechanism effectiveness
and forced work
hm frac14 E
1
E E
W
Wi
Forced work
W frac14
I ethP PbTHORNthorndV thorn
I ethP PbTHORNdV thorn
Brake ef 1047297
ciency
hb frac14 Ws
Q htfrac14
Wi
Q ht
Ws
Wifrac14 hthm
References
[1] DG Thombare SK Verma Technological development in the Stirling cycle
engines Renew Sustain Energy Rev 12 (2008) 1e
38 httpdxdoiorg101016jrser200607001
[2] AD Peacock M Newborough Impact of micro-CHP systems on domesticsector CO2 emissions Appl Therm Eng 25 (2005) 2653e2676
[3] B Thomas Benchmark testing of Micro-CHP units Appl Therm Eng 28(2008) 2049e2054 httpdxdoiorg101016japplthermaleng200803010
[4] L Dong H Liu S Riffat Development of small-scale and micro-scale biomass-fuelled CHP systems e a literature review Appl Therm Eng 29 (2009)2119e2126 httpdxdoiorg101016japplthermaleng200812004
[5] I Gonzalez-Pino A Campos-Celador E Perez-Iribarren J Teres-Zubiaga JM Sala Parametric study of the operational and economic feasibility of Stirling micro-cogeneration devices in Spain Appl Therm Eng (2013) httpdxdoiorg101016japplthermaleng201312020
[6] R Dyson S Wilson R Tew Review of computational Stirling analysismethods in 2nd International Energy Conversion Engineering ConferenceAmerican Institute of Aeronautics and Astronautics 2004 httpdxdoiorg10251462004-5582
[7] C-H Cheng Y-J Yu Numerical model for predicting thermodynamic cycleand thermal ef 1047297ciency of a beta-type Stirling engine with rhombic-drivemechanism Renew Energy 35 (2010) 2590e2601 httpdxdoiorg101016jrenene201004002
[8] N Seraj Mehdizadeh P Stouffs Simulation of a Martini displacer free pistonStirling engine for electric power generation Int J Thermodyn 3 (1) (2000)27e34 httpdxdoiorg105541ijot30
[9] N Parlak A Wagner M Elsner HS Soyhan Thermodynamic analysis of agamma type Stirling engine in non-ideal adiabatic conditions Renew Energy34 (2009) 266e273 httpdxdoiorg101016jrenene200802030
[10] JM Strauss RT Dobson Evaluation of a second order simulation for Sterlingengine design and optimisation J Energy South Afr 21 (2010) 17e29
[11] I Tlili Y Timoumi SB Nasrallah Thermodynamic analysis of the Stirling heatengine with regenerative losses and internal irreversibilities Int J Engine Res9 (2008) 45e56 httpdxdoiorg10124314680874JER01707
[12] K Mahkamov D Djumanov Three-dimensional CFD modeling of a Stirlingengine in Proceedings of the 11th international Stirling Engine Conferencevol 19 2003
[13] M Ibrahim R Tew Z Zhang D Gedeon T Simon CFD Modeling of Free-piston Stirling Engines Cleveland Ohio 2001
[14] S Wilson R Dyson Multi-D CFD modeling of a free-piston Stirling convertorat NASA GRC in Proc 2nd International Energy Conversion EngineeringConference vol 5673 2004
[15] E Eid Performance of a beta-con1047297guration heat engine having a regenerativedisplacer Renew Energy 34 (2009) 2404e2413 httpdxdoiorg101016
jrenene200903016[16] AA El-Ehwany GM Hennes EI Eid EA El-Kenany Development of the
performance of an alpha-type heat engine by using elbow-bend transposed-1047298uids heat exchanger as a heater and a cooler Energy Convers Manag 52(2011) 1010e1019 httpdxdoiorg101016jenconman201008029
[17] C-H Cheng Y-J Yu Dynamic simulation of a beta-type Stirling engine withcam-drive mechanism via the combination of the thermodynamic and dy-namic models Renew Energy 36 (2011) 714e725
[18] E Entchev J Gusdorf M Swinton M Bell F Szadkowski W Kalb1047298eisch et alMicro-generation technology assessment for housing technology EnergyBuild 36 (2004) 925e931 httpdxdoiorg101016jenbuild200403004
[19] M Paringlsson H Carlsen Development of a wood powder fuelled 35 kW StirlingCHP unit in Proceedings of the 11th ISEC (International Stirling EngineConference) 2003 pp 221e230
[20] a Nishiyama H Shimojima a Ishikawa Y Itaya S Kambara H Moritomi etal Fuel and emissions properties of Stirling engine operated with woodpowder Fuel 86 (2007) 2333e2342 httpdxdoiorg101016
jfuel200701040[21] K Sato N Ohiwa A Ishikawa H Shimojima A Nishiyama Y Moriya
Development of small-scale CHP Plant with a wood powder-fueled Stirlingengine J Power Energy Syst 2 (2008) 1221e1231 httpdxdoiorg101299
jpes21221[22] I Urieli DM Berchowitz Stirling Cycle Engine Analysis Hilger Ltd Bristol
1984[23] JR Senft Mechanical Ef 1047297ciency of Heat Engines Cambridge University Press
2007[24] E Cardozo C Erlich A Malmquist L Alejo Integration of a wood pellet
burner and a Stirling engine to produce residential heat and power ApplTherm Eng (2014) httpdxdoiorg101016japplthermaleng201408024
[25] JA Araoz M Salomon L Alejo TH Fransson Non-ideal Stirling enginethermodynamic model suitable for the integration into overall energy sys-tems Appl Therm Eng 73 (2014) 203e219 httpdxdoiorg101016
japplthermaleng201407050
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 30 29
7212019 Applied Thermal Engineering Volume 83 Issue 2015 [Doi 101016_2Fjapplthermaleng201503006] Araoz Joseph hellip
httpslidepdfcomreaderfullapplied-thermal-engineering-volume-83-issue-2015-doi-1010162fjapplthermaleng201503006 1515
[26] FP Incropera AS Lavine DP DeWitt Fundamentals of Heat and MassTransfer John Wiley amp Sons 2011
[27] B Thomas D Pittman Update on the evaluation of different correlations forthe 1047298ow friction factor and heat transfer of Stirling engine regenerators inCollection of Technical Papers 35th Intersociety Energy Conversion Engi-neering Conference and Exhibit (IECEC) (Cat No00CH37022) American InstAeronaut amp Astronautics 2000 pp 76e84 httpdxdoiorg101109IECEC2000870632
[28] Genoastirling Srl 2014 wwwgenoastirlingcom[29] RC Tew K Jefferies D Miao USD of ED of TE Conservation LR Center
A Stirling Engine Computer Model for Performance Calculations (GoogleEBook) Department of Energy Of 1047297ce of Conservation and Solar ApplicationsDivision of Transportation Energy Conservation 1978
[30] I Urieli CJ Rallis DM Berchowitz Computer simulation of Stirling cyclemachines in 12th Intersociety Energy Conversion Engineering Conferencevol 1 American Nuclear Society Washington DC 1977 pp 1512e1521
[31] K Mahkamov Design improvements to a biomass Stirling engine usingmathematical analysis and 3D CFD modeling J Energy Resour Technol 128(2006) 203 httpdxdoiorg10111512213273
[32] H Snyman Second Order Analyses Methods for Stirling Engine Design Uni-versity of Stellenbosch Stellenbosch 2007
[33] G Sewell Numerical Solution of Ordinary and Partial Differential Equationssecond ed John Wiley amp Sons 2005
[34] DD Do Richard G Rice Applied Mathematics and Modeling for ChemicalEngineers [Hardcover] second ed Wiley-AIChE 2012
[35] Aspentech Aspen Custom Modelerreg AspenTech 2015
[36] Aspentech Chemical Process Optimization Software d
Chemical ProcessDesign Aspen Plus 2015
[37] B Thomas D Pittman AIAA-2000-2812 Update on the Evaluation of DifferentCorrelations for the Flow Friction Factor and Heat Transfer of Stirling EngineRegenerators 2000 pp 76e84
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 3030
7212019 Applied Thermal Engineering Volume 83 Issue 2015 [Doi 101016_2Fjapplthermaleng201503006] Araoz Joseph hellip
httpslidepdfcomreaderfullapplied-thermal-engineering-volume-83-issue-2015-doi-1010162fjapplthermaleng201503006 1415
Q h frac14 11
hrhAohthorn R foh
ethTad TwohTHORN
hrh frac14 assAohFR ethTad thorn TwohTHORN
T2ad thorn T2
woh
Estimation of the outlet temperature of the cooling 1047298uid
Twok frac14 Twater in thorn Q k
1
hokAokthorn
1
2mwaterCpwater
hok frac14 11
hwaterthorn R fok
Energy losses
Pressure drop in the heat exchangers
DP frac14
f
dhy
1
2 rv
2
l
Pressure drop in the regenerator based on the correlations of
Thomas and Pittman [37]
DP frac14 Cf nr
2u2
Cf frac14 Cfd thornCsf
Re
Csf frac14 68556 Cfd frac14 05274 wire screensCsf frac14 70035 Cfd frac14 09307 metal felts
Total pumping losses
Wploss frac14
Z 2p0
Xifrac143
ifrac141
DPi dV edq
$dq
Energy losses due to shuttle conduction
Q lsh frac14 04Z2KpistDd
JL dethTe TcTHORN
Mechanical ef 1047297ciency module
Mechanical ef 1047297ciency
hm frac14 Ws
Wi
Mechanical ef 1047297ciency considering the mechanism effectiveness
and forced work
hm frac14 E
1
E E
W
Wi
Forced work
W frac14
I ethP PbTHORNthorndV thorn
I ethP PbTHORNdV thorn
Brake ef 1047297
ciency
hb frac14 Ws
Q htfrac14
Wi
Q ht
Ws
Wifrac14 hthm
References
[1] DG Thombare SK Verma Technological development in the Stirling cycle
engines Renew Sustain Energy Rev 12 (2008) 1e
38 httpdxdoiorg101016jrser200607001
[2] AD Peacock M Newborough Impact of micro-CHP systems on domesticsector CO2 emissions Appl Therm Eng 25 (2005) 2653e2676
[3] B Thomas Benchmark testing of Micro-CHP units Appl Therm Eng 28(2008) 2049e2054 httpdxdoiorg101016japplthermaleng200803010
[4] L Dong H Liu S Riffat Development of small-scale and micro-scale biomass-fuelled CHP systems e a literature review Appl Therm Eng 29 (2009)2119e2126 httpdxdoiorg101016japplthermaleng200812004
[5] I Gonzalez-Pino A Campos-Celador E Perez-Iribarren J Teres-Zubiaga JM Sala Parametric study of the operational and economic feasibility of Stirling micro-cogeneration devices in Spain Appl Therm Eng (2013) httpdxdoiorg101016japplthermaleng201312020
[6] R Dyson S Wilson R Tew Review of computational Stirling analysismethods in 2nd International Energy Conversion Engineering ConferenceAmerican Institute of Aeronautics and Astronautics 2004 httpdxdoiorg10251462004-5582
[7] C-H Cheng Y-J Yu Numerical model for predicting thermodynamic cycleand thermal ef 1047297ciency of a beta-type Stirling engine with rhombic-drivemechanism Renew Energy 35 (2010) 2590e2601 httpdxdoiorg101016jrenene201004002
[8] N Seraj Mehdizadeh P Stouffs Simulation of a Martini displacer free pistonStirling engine for electric power generation Int J Thermodyn 3 (1) (2000)27e34 httpdxdoiorg105541ijot30
[9] N Parlak A Wagner M Elsner HS Soyhan Thermodynamic analysis of agamma type Stirling engine in non-ideal adiabatic conditions Renew Energy34 (2009) 266e273 httpdxdoiorg101016jrenene200802030
[10] JM Strauss RT Dobson Evaluation of a second order simulation for Sterlingengine design and optimisation J Energy South Afr 21 (2010) 17e29
[11] I Tlili Y Timoumi SB Nasrallah Thermodynamic analysis of the Stirling heatengine with regenerative losses and internal irreversibilities Int J Engine Res9 (2008) 45e56 httpdxdoiorg10124314680874JER01707
[12] K Mahkamov D Djumanov Three-dimensional CFD modeling of a Stirlingengine in Proceedings of the 11th international Stirling Engine Conferencevol 19 2003
[13] M Ibrahim R Tew Z Zhang D Gedeon T Simon CFD Modeling of Free-piston Stirling Engines Cleveland Ohio 2001
[14] S Wilson R Dyson Multi-D CFD modeling of a free-piston Stirling convertorat NASA GRC in Proc 2nd International Energy Conversion EngineeringConference vol 5673 2004
[15] E Eid Performance of a beta-con1047297guration heat engine having a regenerativedisplacer Renew Energy 34 (2009) 2404e2413 httpdxdoiorg101016
jrenene200903016[16] AA El-Ehwany GM Hennes EI Eid EA El-Kenany Development of the
performance of an alpha-type heat engine by using elbow-bend transposed-1047298uids heat exchanger as a heater and a cooler Energy Convers Manag 52(2011) 1010e1019 httpdxdoiorg101016jenconman201008029
[17] C-H Cheng Y-J Yu Dynamic simulation of a beta-type Stirling engine withcam-drive mechanism via the combination of the thermodynamic and dy-namic models Renew Energy 36 (2011) 714e725
[18] E Entchev J Gusdorf M Swinton M Bell F Szadkowski W Kalb1047298eisch et alMicro-generation technology assessment for housing technology EnergyBuild 36 (2004) 925e931 httpdxdoiorg101016jenbuild200403004
[19] M Paringlsson H Carlsen Development of a wood powder fuelled 35 kW StirlingCHP unit in Proceedings of the 11th ISEC (International Stirling EngineConference) 2003 pp 221e230
[20] a Nishiyama H Shimojima a Ishikawa Y Itaya S Kambara H Moritomi etal Fuel and emissions properties of Stirling engine operated with woodpowder Fuel 86 (2007) 2333e2342 httpdxdoiorg101016
jfuel200701040[21] K Sato N Ohiwa A Ishikawa H Shimojima A Nishiyama Y Moriya
Development of small-scale CHP Plant with a wood powder-fueled Stirlingengine J Power Energy Syst 2 (2008) 1221e1231 httpdxdoiorg101299
jpes21221[22] I Urieli DM Berchowitz Stirling Cycle Engine Analysis Hilger Ltd Bristol
1984[23] JR Senft Mechanical Ef 1047297ciency of Heat Engines Cambridge University Press
2007[24] E Cardozo C Erlich A Malmquist L Alejo Integration of a wood pellet
burner and a Stirling engine to produce residential heat and power ApplTherm Eng (2014) httpdxdoiorg101016japplthermaleng201408024
[25] JA Araoz M Salomon L Alejo TH Fransson Non-ideal Stirling enginethermodynamic model suitable for the integration into overall energy sys-tems Appl Therm Eng 73 (2014) 203e219 httpdxdoiorg101016
japplthermaleng201407050
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 30 29
7212019 Applied Thermal Engineering Volume 83 Issue 2015 [Doi 101016_2Fjapplthermaleng201503006] Araoz Joseph hellip
httpslidepdfcomreaderfullapplied-thermal-engineering-volume-83-issue-2015-doi-1010162fjapplthermaleng201503006 1515
[26] FP Incropera AS Lavine DP DeWitt Fundamentals of Heat and MassTransfer John Wiley amp Sons 2011
[27] B Thomas D Pittman Update on the evaluation of different correlations forthe 1047298ow friction factor and heat transfer of Stirling engine regenerators inCollection of Technical Papers 35th Intersociety Energy Conversion Engi-neering Conference and Exhibit (IECEC) (Cat No00CH37022) American InstAeronaut amp Astronautics 2000 pp 76e84 httpdxdoiorg101109IECEC2000870632
[28] Genoastirling Srl 2014 wwwgenoastirlingcom[29] RC Tew K Jefferies D Miao USD of ED of TE Conservation LR Center
A Stirling Engine Computer Model for Performance Calculations (GoogleEBook) Department of Energy Of 1047297ce of Conservation and Solar ApplicationsDivision of Transportation Energy Conservation 1978
[30] I Urieli CJ Rallis DM Berchowitz Computer simulation of Stirling cyclemachines in 12th Intersociety Energy Conversion Engineering Conferencevol 1 American Nuclear Society Washington DC 1977 pp 1512e1521
[31] K Mahkamov Design improvements to a biomass Stirling engine usingmathematical analysis and 3D CFD modeling J Energy Resour Technol 128(2006) 203 httpdxdoiorg10111512213273
[32] H Snyman Second Order Analyses Methods for Stirling Engine Design Uni-versity of Stellenbosch Stellenbosch 2007
[33] G Sewell Numerical Solution of Ordinary and Partial Differential Equationssecond ed John Wiley amp Sons 2005
[34] DD Do Richard G Rice Applied Mathematics and Modeling for ChemicalEngineers [Hardcover] second ed Wiley-AIChE 2012
[35] Aspentech Aspen Custom Modelerreg AspenTech 2015
[36] Aspentech Chemical Process Optimization Software d
Chemical ProcessDesign Aspen Plus 2015
[37] B Thomas D Pittman AIAA-2000-2812 Update on the Evaluation of DifferentCorrelations for the Flow Friction Factor and Heat Transfer of Stirling EngineRegenerators 2000 pp 76e84
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 3030
7212019 Applied Thermal Engineering Volume 83 Issue 2015 [Doi 101016_2Fjapplthermaleng201503006] Araoz Joseph hellip
httpslidepdfcomreaderfullapplied-thermal-engineering-volume-83-issue-2015-doi-1010162fjapplthermaleng201503006 1515
[26] FP Incropera AS Lavine DP DeWitt Fundamentals of Heat and MassTransfer John Wiley amp Sons 2011
[27] B Thomas D Pittman Update on the evaluation of different correlations forthe 1047298ow friction factor and heat transfer of Stirling engine regenerators inCollection of Technical Papers 35th Intersociety Energy Conversion Engi-neering Conference and Exhibit (IECEC) (Cat No00CH37022) American InstAeronaut amp Astronautics 2000 pp 76e84 httpdxdoiorg101109IECEC2000870632
[28] Genoastirling Srl 2014 wwwgenoastirlingcom[29] RC Tew K Jefferies D Miao USD of ED of TE Conservation LR Center
A Stirling Engine Computer Model for Performance Calculations (GoogleEBook) Department of Energy Of 1047297ce of Conservation and Solar ApplicationsDivision of Transportation Energy Conservation 1978
[30] I Urieli CJ Rallis DM Berchowitz Computer simulation of Stirling cyclemachines in 12th Intersociety Energy Conversion Engineering Conferencevol 1 American Nuclear Society Washington DC 1977 pp 1512e1521
[31] K Mahkamov Design improvements to a biomass Stirling engine usingmathematical analysis and 3D CFD modeling J Energy Resour Technol 128(2006) 203 httpdxdoiorg10111512213273
[32] H Snyman Second Order Analyses Methods for Stirling Engine Design Uni-versity of Stellenbosch Stellenbosch 2007
[33] G Sewell Numerical Solution of Ordinary and Partial Differential Equationssecond ed John Wiley amp Sons 2005
[34] DD Do Richard G Rice Applied Mathematics and Modeling for ChemicalEngineers [Hardcover] second ed Wiley-AIChE 2012
[35] Aspentech Aspen Custom Modelerreg AspenTech 2015
[36] Aspentech Chemical Process Optimization Software d
Chemical ProcessDesign Aspen Plus 2015
[37] B Thomas D Pittman AIAA-2000-2812 Update on the Evaluation of DifferentCorrelations for the Flow Friction Factor and Heat Transfer of Stirling EngineRegenerators 2000 pp 76e84
JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 3030
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