p ¼ mR Ve
TU
þ
Vh
TU
þ
VR
TR
þ
Vk
TL
þ
Vc
TL
ð3Þ
In which R is the considered gas constant per unit of mass and TR the
mean temperature of the regenerator.
The Schmidt analysis results [17] are recalled hereafter:
p ¼ mR
s
1
1 þ b cosð/ðtÞ hÞ ð4Þ
with:
b ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2sj cos a þ s2 þ j2 p
2m þ j þ s
tan h ¼ jsin a
jcos a þ s
1
s ¼ 2 TU
Vd
s
1
2m þ j þ s
m ¼ VCC
Vd
þ
VR
Vd
TL
TR
þ
VHC
Vd
s
ð5Þ
In which s = TL/TU.
Consequently, the main characteristic parameters of the engine
are given by the following analytic expressions:
Mean effective pressure
pmean ¼ mR
s
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 b2
q ð6Þ
Heat added (Qe) and heat rejected (Qc) for each cycle:
Qe ¼
Z
cycle
p dV HC ¼ pmean
pVd
b 1 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 b2
q
0
B@
1
CA
sin h ð7Þ
Qc ¼
Z
cycle
p dVCC ¼ pmean
jVd
b 1 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 b2
q
0
B@
1
CA
sinða hÞ ð8Þ
Qe stands for the heat added to the ‘‘hot part” of the engine and Qc
for the ‘‘cold part”.
Efficiency and power
The ideal thermodynamic efficiency is exactly the Carnot
efficiency:
gi ¼ 1 TL=TU ð9Þ
Useful mechanical work can be determined from Eqs. (7) and (8)
and varies notably as a function of the mean effective pressure:
Wi ¼ pmeanVd
pjðs 1Þ sin a
b ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
j2 þ s2 þ 2js cos a p 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 b2
q
ð10Þ
Thus, the indicated power can be expressed as the product of the
mechanical work by the operating frequency f:
Pi ¼ fWi ð11Þ
2.2. Thermal approach
Thermal power
The cycle average power turns out to be given in an alternative
way. According to the power balance of the machine as described
as in Fig. 1a, the cycle average power can expressed as:
Pth ¼ aTHð1 þ dC n ndsÞ ð12Þ
In which n = TU/TH is the temperature ratio between the heat source
and the highest temperature of the engine, d = b/a and C = TC/TH.
Thermal efficiency
The losses associated to the regenerator have the highest influence
on the performances. We anticipate that the shuttle losses as
well as the gas spring hysteresis have little influence on the performances.
By doing this, these losses are neglected compared to the
former ones.
The thermal efficiency of the engine is defined by the ratio of
the available power by the added thermal power:
gth ¼ Pth
Q_ h þ Q_ T þ Q_ R
ð13Þ
In which Q_ T is the conduction loss, and Q_ R is the thermal power related
to the regenerator inefficiency.
We define the various thermal impedances ratios:
qcond ¼ ccond=a ð14Þ
qR ¼ m_ RCv=a ð15Þ
where m_ R is the fluid mass rate inside the regenerator.
Hence:
Q_ T ¼ aqcondTUð1 sÞ ð16Þ
Q_ R ¼ ð1 eÞaqRTUð1 sÞ ð17Þ
The regenerator effectiveness e depends on the temperatures of the
engine. We define e ¼ T0
UTL
TUTL with T0
U < TU the highest actual outlet
temperature of the fluid from the regenerator. It must be noted that
as far as a symmetrical regenerator behaviour is assumed a single
effectiveness is defined and e can be conversely defined as
e ¼ T0
LTU
TLTU with T0
L > TL the actual lowest temperature of the fluid
from the regenerator.
Finally the thermal efficiency is expressed as a function of the
non-dimensional parameters:
gth ¼ 1 þ dc n nsd
1 n ðqcond þ ð1 eÞqRÞnð1 sÞ ð18Þ
Thermodynamic conditions
The second law of thermodynamics requires that the thermal
efficiency does not exceed the Carnot efficiency that is:
gth 6 1 s ð19Þ
For a machine which operates as an engine the available power Pth
is always positive. Then we set:
Pth P 0 ð20Þ
Therefore, using the two previous conditions (19) and (20), the permissible
values of n are within an hatched domain shown on Fig. 2.
In order to obtain the maximum performances an optimal case
is chosen. Thus, the inequality (19) can be switched to equality.
Consequently, the optimal ratio noptim between the heat source
temperature and the temperature of the expansion chamber of
the engine can be obtained. Using Eq. (18):
noptim ¼ dC þ s
ðd þ 1Þs ðqcond þ ð1 eÞqRÞð1 sÞ
2 ð21Þ
2.2.1. Expression of the thermal coefficients
According to the Newton’s law the coefficients a and b are assumed
to be representative of the heat exchange for the hot and
the cold end respectively. Thermal effectiveness of the heat
exchangers are dependant on the machine parameters and more
specifically the operating frequency f. Therefore, it is important
F. Formosa, G. Despesse / Energy Co
p ได้นาย¼TUþVhTUþVRTRþVkTLþVcTLð3Þในที่ R คือ ค่าคงก๊าซพิจารณาต่อหน่วยของมวล และ TRหมายความว่า อุณหภูมิของการกำเนิดใหม่ผลการวิเคราะห์ชมิดท์ [17] ถูกยกเลิกโดย:พี นาย¼s11 þ b hÞ cosð/ðtÞ ð4Þด้วย:บี¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2sj cos þ s2 þ j2 pโรงเจ 2m þþ sjsin h ¼ตันjcos þ s1s ¼ 2 ทูVds1โรงเจ 2m þþ sm ¼ VCCVdþVRVdTLTRþVHCVdsð5Þในที่ s = TL/TU.ดังนั้น ลักษณะพารามิเตอร์หลักของเครื่องยนต์ได้ โดยนิพจน์คู่ต่อไปนี้: หมายถึง ความดันที่มีประสิทธิภาพpmean ¼นายs1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 b2q ð6Þ ความร้อนและเพิ่มความร้อน (Qe) ปฏิเสธ (Qc) สำหรับแต่ละรอบ:Qe ¼ZวงจรdV p HC ¼ pmeanpVdb 1 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 b2q0บีแอท1CAh ð7Þ การทำบาปQc ¼Zวงจรpmean dVCC ¼ pjVdb 1 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 b2q0บีแอท1CAsinða hÞ ð8ÞQe หมายถึงความร้อนที่เพิ่ม ''ส่วนร้อน"ของเครื่องยนต์และ Qc''ส่วนเย็น" ประสิทธิภาพและพลังงานประสิทธิภาพทางอุณหพลศาสตร์เหมาะเป็นกับ Carnotประสิทธิภาพ:gi ¼ 1 TL = ð9Þ ทูสามารถระบุเครื่องจักรกลงานประโยชน์จาก Eqs (7) และ (8)และไปจนยวดเป็นฟังก์ชันของความดันที่มีประสิทธิภาพหมายถึง:Wi ¼ pmeanVdpjðs 1Þ sin ab ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij2 þ s2 þ 2js cos a p 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 b2 qð10ÞThus, the indicated power can be expressed as the product of themechanical work by the operating frequency f:Pi ¼ fWi ð11Þ2.2. Thermal approach Thermal powerThe cycle average power turns out to be given in an alternativeway. According to the power balance of the machine as describedas in Fig. 1a, the cycle average power can expressed as:Pth ¼ aTHð1 þ dC n ndsÞ ð12ÞIn which n = TU/TH is the temperature ratio between the heat sourceand the highest temperature of the engine, d = b/a and C = TC/TH. Thermal efficiencyThe losses associated to the regenerator have the highest influenceon the performances. We anticipate that the shuttle losses aswell as the gas spring hysteresis have little influence on the performances.By doing this, these losses are neglected compared to theformer ones.The thermal efficiency of the engine is defined by the ratio ofthe available power by the added thermal power:gth ¼ PthQ_ h þ Q_ T þ Q_ Rð13ÞIn which Q_ T is the conduction loss, and Q_ R is the thermal power relatedto the regenerator inefficiency.We define the various thermal impedances ratios:qcond ¼ ccond=a ð14ÞqR ¼ m_ RCv=a ð15Þwhere m_ R is the fluid mass rate inside the regenerator.Hence:Q_ T ¼ aqcondTUð1 sÞ ð16ÞQ_ R ¼ ð1 eÞaqRTUð1 sÞ ð17ÞThe regenerator effectiveness e depends on the temperatures of theengine. We define e ¼ T0UTLTUTL with T0U < TU the highest actual outlettemperature of the fluid from the regenerator. It must be noted thatas far as a symmetrical regenerator behaviour is assumed a singleeffectiveness is defined and e can be conversely defined ase ¼ T0LTUTLTU with T0L > TL the actual lowest temperature of the fluidfrom the regenerator.Finally the thermal efficiency is expressed as a function of thenon-dimensional parameters:gth ¼ 1 þ dc n nsd1 n ðqcond þ ð1 eÞqRÞnð1 sÞ ð18Þ Thermodynamic conditionsThe second law of thermodynamics requires that the thermalefficiency does not exceed the Carnot efficiency that is:gth 6 1 s ð19ÞFor a machine which operates as an engine the available power Pthis always positive. Then we set:Pth P 0 ð20ÞTherefore, using the two previous conditions (19) and (20), the permissiblevalues of n are within an hatched domain shown on Fig. 2.In order to obtain the maximum performances an optimal caseis chosen. Thus, the inequality (19) can be switched to equality.Consequently, the optimal ratio noptim between the heat sourcetemperature and the temperature of the expansion chamber ofthe engine can be obtained. Using Eq. (18):noptim ¼ dC þ sðd þ 1Þs ðqcond þ ð1 eÞqRÞð1 sÞ2 ð21Þ2.2.1. Expression of the thermal coefficients
According to the Newton’s law the coefficients a and b are assumed
to be representative of the heat exchange for the hot and
the cold end respectively. Thermal effectiveness of the heat
exchangers are dependant on the machine parameters and more
specifically the operating frequency f. Therefore, it is important
F. Formosa, G. Despesse / Energy Co
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