Applied Thermal Engineering Volume 83 Issue 2015 [Doi 10.1016_2Fj.applthermaleng.2015.03.006] Araoz,...

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

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2015 Elsevier Ltd All rights reserved

Applied Thermal Engineering 83 (2015) 16e30



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)

JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 30 17



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]

JA Araoz et al Applied Thermal Engineering 83 (2015) 16 e 3018



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




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






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]




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)




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




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




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




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


de 0005 M Tubes external diameter

di 00031 M Tubes internal diameterkw 142 Wm K Material conductivity

len 0149 m Tubes length



Table A3

Inputs for the regenerator in ACM


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




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


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


Table A5

Working and cooling 1047298uid inputs in ACM


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


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




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 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




[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




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)

























































below















model





























min mout frac14 dm

d4(1)


dQ


dW

d4thorn cv

dethmTTHORN

d4(2)


PV frac14 mRT (3)










[23]

hm frac14 Ws

Wi(4)




hm frac14 E

1

E E

W

Wi(5)











W frac14


















hb frac14 Ws

Q hcfrac14

Wi

Q hc

Ws

Wifrac14 hthm (7)





























Table 1














Table 2





































dV e frac14 V swe


dV c

frac14 dV e

V swc

2 sineth4THORN (11)




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





























4 Model validation









Fig 8




















Table 3



1388 K

Table 4



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




































calculations






presents














































Table 5


















53 Work 1047298ow






































Fig 15 Heat 1047298


Table 6














Fig 14 Work 1047298








article

54 Heat 1047298ow




























6 Conclusions



































Acknowledgements







Table 7



5358 184535 1610 1810 290

Table A1









Table A2








Table A3












Appendix B



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 eThe


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





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


dQ rd4

frac14

V r

vPv4

Cv


dQ hd4

frac14V hvPv4

Cv


dWc

d4frac14 P

vV cv4

dWe

d4frac14 P

vV ev4



Q h frac14 1



Q k frac14 1





3frac14 NTU

1 thorn NTU




Table A4









freq 5 Hz Frequency








Table A5






Table A6








Q h frac14 11

hrhAohthorn R foh

ethTad TwohTHORN


T2ad thorn T2

woh



1

hokAokthorn

1

2mwaterCpwater

hok frac14 11

hwaterthorn R fok

Energy losses


DP frac14

f

dhy

1

2 rv

2

l



DP frac14 Cf nr

2u2


Re



Wploss frac14

Z 2p0

Xifrac143

ifrac141

DPi dV edq

$dq



JL dethTe TcTHORN



hm frac14 Ws

Wi


and forced work

hm frac14 E

1

E E

W

Wi

Forced work

W frac14



Brake ef 1047297

ciency

hb frac14 Ws

Q htfrac14

Wi

Q ht

Ws

Wifrac14 hthm

References








































































































below















model





























min mout frac14 dm

d4(1)


dQ


dW

d4thorn cv

dethmTTHORN

d4(2)


PV frac14 mRT (3)










[23]

hm frac14 Ws

Wi(4)




hm frac14 E

1

E E

W

Wi(5)











W frac14


















hb frac14 Ws

Q hcfrac14

Wi

Q hc

Ws

Wifrac14 hthm (7)





























Table 1














Table 2





































dV e frac14 V swe


dV c

frac14 dV e

V swc

2 sineth4THORN (11)




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





























4 Model validation









Fig 8




















Table 3



1388 K

Table 4



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




































calculations






presents














































Table 5


















53 Work 1047298ow






































Fig 15 Heat 1047298


Table 6














Fig 14 Work 1047298








article

54 Heat 1047298ow




























6 Conclusions



































Acknowledgements







Table 7



5358 184535 1610 1810 290

Table A1









Table A2








Table A3












Appendix B



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 eThe


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





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


dQ rd4

frac14

V r

vPv4

Cv


dQ hd4

frac14V hvPv4

Cv


dWc

d4frac14 P

vV cv4

dWe

d4frac14 P

vV ev4



Q h frac14 1



Q k frac14 1





3frac14 NTU

1 thorn NTU




Table A4









freq 5 Hz Frequency








Table A5






Table A6








Q h frac14 11

hrhAohthorn R foh

ethTad TwohTHORN


T2ad thorn T2

woh



1

hokAokthorn

1

2mwaterCpwater

hok frac14 11

hwaterthorn R fok

Energy losses


DP frac14

f

dhy

1

2 rv

2

l



DP frac14 Cf nr

2u2


Re



Wploss frac14

Z 2p0

Xifrac143

ifrac141

DPi dV edq

$dq



JL dethTe TcTHORN



hm frac14 Ws

Wi


and forced work

hm frac14 E

1

E E

W

Wi

Forced work

W frac14



Brake ef 1047297

ciency

hb frac14 Ws

Q htfrac14

Wi

Q ht

Ws

Wifrac14 hthm

References




















































min mout frac14 dm

d4(1)


dQ


dW

d4thorn cv

dethmTTHORN

d4(2)


PV frac14 mRT (3)










[23]

hm frac14 Ws

Wi(4)




hm frac14 E

1

E E

W

Wi(5)











W frac14


















hb frac14 Ws

Q hcfrac14

Wi

Q hc

Ws

Wifrac14 hthm (7)





























Table 1














Table 2





































dV e frac14 V swe


dV c

frac14 dV e

V swc

2 sineth4THORN (11)




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





























4 Model validation









Fig 8




















Table 3



1388 K

Table 4



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




































calculations






presents














































Table 5


















53 Work 1047298ow






































Fig 15 Heat 1047298


Table 6














Fig 14 Work 1047298








article

54 Heat 1047298ow




























6 Conclusions



































Acknowledgements







Table 7



5358 184535 1610 1810 290

Table A1









Table A2








Table A3












Appendix B



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 eThe


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





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


dQ rd4

frac14

V r

vPv4

Cv


dQ hd4

frac14V hvPv4

Cv


dWc

d4frac14 P

vV cv4

dWe

d4frac14 P

vV ev4



Q h frac14 1



Q k frac14 1





3frac14 NTU

1 thorn NTU




Table A4









freq 5 Hz Frequency








Table A5






Table A6








Q h frac14 11

hrhAohthorn R foh

ethTad TwohTHORN


T2ad thorn T2

woh



1

hokAokthorn

1

2mwaterCpwater

hok frac14 11

hwaterthorn R fok

Energy losses


DP frac14

f

dhy

1

2 rv

2

l



DP frac14 Cf nr

2u2


Re



Wploss frac14

Z 2p0

Xifrac143

ifrac141

DPi dV edq

$dq



JL dethTe TcTHORN



hm frac14 Ws

Wi


and forced work

hm frac14 E

1

E E

W

Wi

Forced work

W frac14



Brake ef 1047297

ciency

hb frac14 Ws

Q htfrac14

Wi

Q ht

Ws

Wifrac14 hthm

References











































































Table 1














Table 2





































dV e frac14 V swe


dV c

frac14 dV e

V swc

2 sineth4THORN (11)




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





























4 Model validation









Fig 8




















Table 3



1388 K

Table 4



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




































calculations






presents














































Table 5


















53 Work 1047298ow






































Fig 15 Heat 1047298


Table 6














Fig 14 Work 1047298








article

54 Heat 1047298ow




























6 Conclusions



































Acknowledgements







Table 7



5358 184535 1610 1810 290

Table A1









Table A2








Table A3












Appendix B



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 eThe


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





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


dQ rd4

frac14

V r

vPv4

Cv


dQ hd4

frac14V hvPv4

Cv


dWc

d4frac14 P

vV cv4

dWe

d4frac14 P

vV ev4



Q h frac14 1



Q k frac14 1





3frac14 NTU

1 thorn NTU




Table A4









freq 5 Hz Frequency








Table A5






Table A6








Q h frac14 11

hrhAohthorn R foh

ethTad TwohTHORN


T2ad thorn T2

woh



1

hokAokthorn

1

2mwaterCpwater

hok frac14 11

hwaterthorn R fok

Energy losses


DP frac14

f

dhy

1

2 rv

2

l



DP frac14 Cf nr

2u2


Re



Wploss frac14

Z 2p0

Xifrac143

ifrac141

DPi dV edq

$dq



JL dethTe TcTHORN



hm frac14 Ws

Wi


and forced work

hm frac14 E

1

E E

W

Wi

Forced work

W frac14



Brake ef 1047297

ciency

hb frac14 Ws

Q htfrac14

Wi

Q ht

Ws

Wifrac14 hthm

References






















































Table 2





































dV e frac14 V swe


dV c

frac14 dV e

V swc

2 sineth4THORN (11)




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





























4 Model validation









Fig 8




















Table 3



1388 K

Table 4



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




































calculations






presents














































Table 5


















53 Work 1047298ow






































Fig 15 Heat 1047298


Table 6














Fig 14 Work 1047298








article

54 Heat 1047298ow




























6 Conclusions



































Acknowledgements







Table 7



5358 184535 1610 1810 290

Table A1









Table A2








Table A3












Appendix B



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 eThe


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





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


dQ rd4

frac14

V r

vPv4

Cv


dQ hd4

frac14V hvPv4

Cv


dWc

d4frac14 P

vV cv4

dWe

d4frac14 P

vV ev4



Q h frac14 1



Q k frac14 1





3frac14 NTU

1 thorn NTU




Table A4









freq 5 Hz Frequency








Table A5






Table A6








Q h frac14 11

hrhAohthorn R foh

ethTad TwohTHORN


T2ad thorn T2

woh



1

hokAokthorn

1

2mwaterCpwater

hok frac14 11

hwaterthorn R fok

Energy losses


DP frac14

f

dhy

1

2 rv

2

l



DP frac14 Cf nr

2u2


Re



Wploss frac14

Z 2p0

Xifrac143

ifrac141

DPi dV edq

$dq



JL dethTe TcTHORN



hm frac14 Ws

Wi


and forced work

hm frac14 E

1

E E

W

Wi

Forced work

W frac14



Brake ef 1047297

ciency

hb frac14 Ws

Q htfrac14

Wi

Q ht

Ws

Wifrac14 hthm

References






































































dV e frac14 V swe


dV c

frac14 dV e

V swc

2 sineth4THORN (11)




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





























4 Model validation









Fig 8




















Table 3



1388 K

Table 4



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




































calculations






presents














































Table 5


















53 Work 1047298ow






































Fig 15 Heat 1047298


Table 6














Fig 14 Work 1047298








article

54 Heat 1047298ow




























6 Conclusions



































Acknowledgements







Table 7



5358 184535 1610 1810 290

Table A1









Table A2








Table A3












Appendix B



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 eThe


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





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


dQ rd4

frac14

V r

vPv4

Cv


dQ hd4

frac14V hvPv4

Cv


dWc

d4frac14 P

vV cv4

dWe

d4frac14 P

vV ev4



Q h frac14 1



Q k frac14 1





3frac14 NTU

1 thorn NTU




Table A4









freq 5 Hz Frequency








Table A5






Table A6








Q h frac14 11

hrhAohthorn R foh

ethTad TwohTHORN


T2ad thorn T2

woh



1

hokAokthorn

1

2mwaterCpwater

hok frac14 11

hwaterthorn R fok

Energy losses


DP frac14

f

dhy

1

2 rv

2

l



DP frac14 Cf nr

2u2


Re



Wploss frac14

Z 2p0

Xifrac143

ifrac141

DPi dV edq

$dq



JL dethTe TcTHORN



hm frac14 Ws

Wi


and forced work

hm frac14 E

1

E E

W

Wi

Forced work

W frac14



Brake ef 1047297

ciency

hb frac14 Ws

Q htfrac14

Wi

Q ht

Ws

Wifrac14 hthm

References











































































4 Model validation









Fig 8




















Table 3



1388 K

Table 4



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




































calculations






presents














































Table 5


















53 Work 1047298ow






































Fig 15 Heat 1047298


Table 6














Fig 14 Work 1047298








article

54 Heat 1047298ow




























6 Conclusions



































Acknowledgements







Table 7



5358 184535 1610 1810 290

Table A1









Table A2








Table A3












Appendix B



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 eThe


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





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


dQ rd4

frac14

V r

vPv4

Cv


dQ hd4

frac14V hvPv4

Cv


dWc

d4frac14 P

vV cv4

dWe

d4frac14 P

vV ev4



Q h frac14 1



Q k frac14 1





3frac14 NTU

1 thorn NTU




Table A4









freq 5 Hz Frequency








Table A5






Table A6








Q h frac14 11

hrhAohthorn R foh

ethTad TwohTHORN


T2ad thorn T2

woh



1

hokAokthorn

1

2mwaterCpwater

hok frac14 11

hwaterthorn R fok

Energy losses


DP frac14

f

dhy

1

2 rv

2

l



DP frac14 Cf nr

2u2


Re



Wploss frac14

Z 2p0

Xifrac143

ifrac141

DPi dV edq

$dq



JL dethTe TcTHORN



hm frac14 Ws

Wi


and forced work

hm frac14 E

1

E E

W

Wi

Forced work

W frac14



Brake ef 1047297

ciency

hb frac14 Ws

Q htfrac14

Wi

Q ht

Ws

Wifrac14 hthm

References


















































































calculations






presents














































Table 5


















53 Work 1047298ow






































Fig 15 Heat 1047298


Table 6














Fig 14 Work 1047298








article

54 Heat 1047298ow




























6 Conclusions



































Acknowledgements







Table 7



5358 184535 1610 1810 290

Table A1









Table A2








Table A3












Appendix B



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 eThe


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





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


dQ rd4

frac14

V r

vPv4

Cv


dQ hd4

frac14V hvPv4

Cv


dWc

d4frac14 P

vV cv4

dWe

d4frac14 P

vV ev4



Q h frac14 1



Q k frac14 1





3frac14 NTU

1 thorn NTU




Table A4









freq 5 Hz Frequency








Table A5






Table A6








Q h frac14 11

hrhAohthorn R foh

ethTad TwohTHORN


T2ad thorn T2

woh



1

hokAokthorn

1

2mwaterCpwater

hok frac14 11

hwaterthorn R fok

Energy losses


DP frac14

f

dhy

1

2 rv

2

l



DP frac14 Cf nr

2u2


Re



Wploss frac14

Z 2p0

Xifrac143

ifrac141

DPi dV edq

$dq



JL dethTe TcTHORN



hm frac14 Ws

Wi


and forced work

hm frac14 E

1

E E

W

Wi

Forced work

W frac14



Brake ef 1047297

ciency

hb frac14 Ws

Q htfrac14

Wi

Q ht

Ws

Wifrac14 hthm

References





















































53 Work 1047298ow






































Fig 15 Heat 1047298


Table 6














Fig 14 Work 1047298








article

54 Heat 1047298ow




























6 Conclusions



































Acknowledgements







Table 7



5358 184535 1610 1810 290

Table A1









Table A2








Table A3












Appendix B



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 eThe


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





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


dQ rd4

frac14

V r

vPv4

Cv


dQ hd4

frac14V hvPv4

Cv


dWc

d4frac14 P

vV cv4

dWe

d4frac14 P

vV ev4



Q h frac14 1



Q k frac14 1





3frac14 NTU

1 thorn NTU




Table A4









freq 5 Hz Frequency








Table A5






Table A6








Q h frac14 11

hrhAohthorn R foh

ethTad TwohTHORN


T2ad thorn T2

woh



1

hokAokthorn

1

2mwaterCpwater

hok frac14 11

hwaterthorn R fok

Energy losses


DP frac14

f

dhy

1

2 rv

2

l



DP frac14 Cf nr

2u2


Re



Wploss frac14

Z 2p0

Xifrac143

ifrac141

DPi dV edq

$dq



JL dethTe TcTHORN



hm frac14 Ws

Wi


and forced work

hm frac14 E

1

E E

W

Wi

Forced work

W frac14



Brake ef 1047297

ciency

hb frac14 Ws

Q htfrac14

Wi

Q ht

Ws

Wifrac14 hthm

References





















































article

54 Heat 1047298ow




























6 Conclusions



































Acknowledgements







Table 7



5358 184535 1610 1810 290

Table A1









Table A2








Table A3












Appendix B



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 eThe


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





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


dQ rd4

frac14

V r

vPv4

Cv


dQ hd4

frac14V hvPv4

Cv


dWc

d4frac14 P

vV cv4

dWe

d4frac14 P

vV ev4



Q h frac14 1



Q k frac14 1





3frac14 NTU

1 thorn NTU




Table A4









freq 5 Hz Frequency








Table A5






Table A6








Q h frac14 11

hrhAohthorn R foh

ethTad TwohTHORN


T2ad thorn T2

woh



1

hokAokthorn

1

2mwaterCpwater

hok frac14 11

hwaterthorn R fok

Energy losses


DP frac14

f

dhy

1

2 rv

2

l



DP frac14 Cf nr

2u2


Re



Wploss frac14

Z 2p0

Xifrac143

ifrac141

DPi dV edq

$dq



JL dethTe TcTHORN



hm frac14 Ws

Wi


and forced work

hm frac14 E

1

E E

W

Wi

Forced work

W frac14



Brake ef 1047297

ciency

hb frac14 Ws

Q htfrac14

Wi

Q ht

Ws

Wifrac14 hthm

References



















































Appendix B



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 eThe


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





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


dQ rd4

frac14

V r

vPv4

Cv


dQ hd4

frac14V hvPv4

Cv


dWc

d4frac14 P

vV cv4

dWe

d4frac14 P

vV ev4



Q h frac14 1



Q k frac14 1





3frac14 NTU

1 thorn NTU




Table A4









freq 5 Hz Frequency








Table A5






Table A6








Q h frac14 11

hrhAohthorn R foh

ethTad TwohTHORN


T2ad thorn T2

woh



1

hokAokthorn

1

2mwaterCpwater

hok frac14 11

hwaterthorn R fok

Energy losses


DP frac14

f

dhy

1

2 rv

2

l



DP frac14 Cf nr

2u2


Re



Wploss frac14

Z 2p0

Xifrac143

ifrac141

DPi dV edq

$dq



JL dethTe TcTHORN



hm frac14 Ws

Wi


and forced work

hm frac14 E

1

E E

W

Wi

Forced work

W frac14



Brake ef 1047297

ciency

hb frac14 Ws

Q htfrac14

Wi

Q ht

Ws

Wifrac14 hthm

References



















































Q h frac14 11

hrhAohthorn R foh

ethTad TwohTHORN


T2ad thorn T2

woh



1

hokAokthorn

1

2mwaterCpwater

hok frac14 11

hwaterthorn R fok

Energy losses


DP frac14

f

dhy

1

2 rv

2

l



DP frac14 Cf nr

2u2


Re



Wploss frac14

Z 2p0

Xifrac143

ifrac141

DPi dV edq

$dq



JL dethTe TcTHORN



hm frac14 Ws

Wi


and forced work

hm frac14 E

1

E E

W

Wi

Forced work

W frac14



Brake ef 1047297

ciency

hb frac14 Ws

Q htfrac14

Wi

Q ht

Ws

Wifrac14 hthm

References

















































Applied Thermal Engineering Volume 83 Issue 2015 [Doi 10.1016_2Fj.applthermaleng.2015.03.006] Araoz,...

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Transcript of Applied Thermal Engineering Volume 83 Issue 2015 [Doi 10.1016_2Fj.applthermaleng.2015.03.006] Araoz,...