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New insight into regenerated air heat pump cycle
Energy 91 (2015) 226e234
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Energy journal homepage: www.elsevier.com/locate/energy
New insight into regenerated air heat pump cycle Chun-Lu Zhang*, Han Yuan, Xiang Cao Institute of Refrigeration and Cyrogenics, School of Mechanical Engineering, Tongji University, Shanghai 201804, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 February 2015 Received in revised form 20 July 2015 Accepted 2 August 2015 Available online 8 September 2015
Regenerated air (reverse Brayton) cycle has unique potentials in heat pump applications compared to conventional vapor-compression cycles. To better understand the regenerated air heat pump cycle characteristics, a thermodynamic model with new equivalent parameters was developed in this paper. Equivalent temperature ratio and equivalent isentropic efficiency of expander were introduced to represent the effect of regenerator, which made the regenerated air cycle in the same mathematical expressions as the basic air cycle and created an easy way to prove some important features that regenerated air cycle inherits from the basic one. Moreover, we proved in theory that the regenerator does not always improve the air cycle efficiency. Larger temperature ratio and lower effectiveness of regenerator could make the regenerated air cycle even worse than the basic air cycle. Lastly, we found that only under certain conditions the cycle could get remarkable benefits from a well-sized regenerator. These results would enable further study of the regenerated air cycle from a different perspective. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Heat pump Air cycle Regenerated air cycle Regenerator Model Optimization
1. Introduction Given the increasing environmental concerns, air is absolutely natural, safe and free working fluid for refrigeration or heat pump applications. Therefore, the reverse Brayton cycle using air as the refrigerant has been investigated for a long time. The air cycle machine was primarily served as the aircraft air conditioner [1,2] and was widely applied in cryogenic engineering [3e5]. In addition, the applicability of air cycle in transportation air conditioning [6e9], refrigerated storage [10e12], integrated air conditioning and desiccant system [13], domestic heating [14], heat pump clothes dryer [15], and heat pump water heater [16] was investigated as well. Nevertheless, the fairly low energy efficiency still constrains the air cycle from extensive applications in normal temperature range. Limited by the commercial applications, more air cycle investigations were conducted with thermodynamic modeling not experiments. There were three main thermodynamic modeling approaches in the literature. One is the classical thermodynamics [17e21], the second is the finite-time thermodynamics [22e35], and the last is the recently developed entransy theory [36,37]. Although the assumptions or constraints taken in those investigations were more or less different, all studies reached
* Corresponding author. Tel.: þ86 136 71825 133. E-mail address: [email protected] (C.-L. Zhang). http://dx.doi.org/10.1016/j.energy.2015.08.052 0360-5442/© 2015 Elsevier Ltd. All rights reserved.
consensus that air cycles should be thermodynamically optimized to get higher energy efficiency. In the previous work [18], we found and proved that at optimal pressure ratio the basic air cycle can make the heating capacity in line with the heating load. It's an important feature for heat pump off-design operations, which vapor-compression heat pumps cannot do. Later on, we investigated the regenerated air cycle and did numerical simulation on the cycle performance at optimal pressure ratio [19]. Effected by the regenerator, the regenerated air cycle seemed more energy efficient than the basic one and could keep the optimal pressure ratio much more stable. Meanwhile, at the optimal pressure ratio the heating capacity is in line with the heating load. Therefore, the regenerated air cycle has higher potentials in heat pump applications. However, the existence of regenerator also introduced more complexity to the thermodynamic cycle model so that most conclusions were drawn by simulations not a solid theoretical proof. In this paper, we develop a new theoretical approach to prove and reveal some important features of the regenerated air cycle. Two new parameters, so-called the equivalent temperature ratio and the equivalent expander efficiency, are introduced to represent the effect of regenerator so that the regenerated cycle can be in the same mathematical expressions as the basic one. Moreover, we can therefore have an in-depth look into what exactly the regenerator contributes to the cycle performance.
C.-L. Zhang et al. / Energy 91 (2015) 226e234
Nomenclature cp COP h hfg qL qL,v qH qH,v QH p pr R T V_ w0 wc
specific heat at constant pressure, J kg1 K1 coefficient of performance enthalpy, J kg1 latent heat of vaporization, J kg1 heat absorption per unit mass flow from lowtemperature environment, J kg1 volumetric heat absorption, heat absorption per unit volume flow at compressor inlet, J m3 heating capacity per unit mass flow, J kg1 volumetric heating capacity, heating capacity per unit volume flow at compressor inlet, J m3 heating capacity, W pressure, Pa pressure ratio gas constant (¼ 8.314), J K1 mol1 temperature, C, K volume flow rate, m3 s1 net power consumption per unit mass flow, J kg1 compressor power consumption per unit mass flow, J kg1
2. Thermodynamic model of regenerated air heat pump cycle A typical regenerated air cycle heat pump is as shown in Fig. 1. Compared to the basic air cycle, a regenerator is added to transfer heat from hot air after the hot side heat exchanger to the compressor suction line. To reduce the heat transfer loss and eliminate the frosting/defrosting issue in heat pump applications, a semi-open air cycle without the cold side heat exchanger is preferred and investigated in this work. Fig. 2 illustrates the thermodynamic processes which air undergoes in the regenerated air heat pump cycle. Curves 2e3 and 5e6 represent compression and expansion processes, respectively. Curve 3e4 demonstrates the isobaric heat rejection process in the hot side heat exchanger. Process 6-1 is open for the semi-open cycle. Curves 4e5 and 1e2 are the processes in hot and cold sides of the regenerator, respectively.
we X Y
227
expander power recovery per unit mass flow, J kg1 intermediate variable defined in equation (58) intermediate variable defined in equation (59)
Greek symbols εr effectiveness of regenerator hc isentropic efficiency of compressor he isentropic efficiency of expander h0e equivalent isentropic efficiency of expander defined in equation (22) p derived pressure ratio defined in equation (17) q temperature ratio defined in equation (18) Q equivalent temperature ratio defined in equation (19) Subscripts in inlet H high-pressure side; hot side; heating L low-pressure side; cold side opt optimal out outlet v volumetric
The assumptions and irreversibilities taken into account for model development of the regenerated cycle are as follows. 1) The air is ideal gas. 2) All processes in heat exchangers are isobaric. 3) Constant isentropic efficiencies are specified for compressor and expander. 4) The effectiveness of regenerator is given. 5) Temperature difference between the hot side inlet of regenerator and high-temperature heat sink (so-called exit temperature difference) is given. 6) Temperature difference between the cold side inlet of regenerator and low-temperature heat source is zero for the semi-open cycle. Accordingly, we can develop the thermodynamic model hereinbelow.
qH 3
Hot side heat exchanger
Compressor
4 5
2
wc
Regenerator
we 1
Expander
6
Cold side heat exchanger
qL Fig. 1. Schematic of regenerated air cycle heat pump.
Fig. 2. T-s diagram of regenerated air cycle.
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C.-L. Zhang et al. / Energy 91 (2015) 226e234
Heat absorption per unit mass flow
qL ¼ h1 h6 ¼ cp ðT1 T6 Þ
(1)
Heat absorption per unit volume flow or volumetric heat absorption (i.e. volumetric cooling capacity for cooling cycle)
qL;v ¼
h1 h6 cp ðT1 T6 Þ ¼ v2 v2
(2)
Heating capacity per unit mass flow
qH ¼ h3 h4 ¼ cp ðT3 T4 Þ
(3)
Volumetric heating capacity
Accordingly, the compressor suction temperature T2 and the expander entering temperature T5 of the regenerated cycle can be found.
T2 ¼ T1 þ εr ðT4 T1 Þ
(15)
T5 ¼ T4 εr ðT4 T1 Þ
(16)
In comparison with the basic air cycle, the compressor suction temperature is raised by the regenerator, meanwhile the expander entering temperature falls. To get equivalent expressions as the basic cycle, we define the following dimensionless parameters. The derived pressure ratio is the same as that of the basic cycle. k1
qH;v
h h4 cp ¼ 3 ¼ ðT3 T4 Þ v2 v2
(4)
Compressor power consumption per unit mass flow
wc ¼ h3 h2 ¼ cp ðT3 T2 Þ ¼
h3s h2 cp ðT3s T2 Þ ¼ hc hc
(5)
Expander power recovery per unit mass flow
we ¼ h5 h6 ¼ cp ðT5 T6 Þ ¼ he ðh5 h6s Þ ¼ cp ðT5 T6s Þhe (6) Accordingly, net power consumption per unit mass flow of the cycle
cp ðT3s T2 Þ w0 ¼ wc we ¼ cp ðT5 T6s Þhe hc
(7)
qH T3 T4 ¼ wc we T3s T2 ðT5 T6s Þhe
(8)
hc
According to the ideal gas law,
pv ¼ RT
(9)
From equation (5), we have
T3 ¼ T2 þ
T3s T2 hc
(10)
From equation (6), we have
T6 ¼ T5 ðT5 T6s Þhe
(11)
The isentropic compression process yields
T3s ¼ T2
pH pL
k1 k
k1
¼ T2 pr k
(12)
(17)
The temperature ratio, which is similar to that of the basic cycle, characterizes the ratio of heat sink temperature to the actual heat source temperature.
q¼
T4 T1
(18)
But effected by the regenerator, the actual compressor suction temperature rises so that the compressor seems to pump heat from a higher temperature heat source. Therefore, the equivalent temperature ratio would make more sense here.
Q¼
T4 T4 q ¼ ¼ T2 T1 þ εr ðT4 T1 Þ 1 þ εr ðq 1Þ
(19)
Since the effectiveness of regenerator 0 εr < 1, we have
1
Heating COP (coefficient of performance)
COPH ¼
p ¼ pr k
(20)
When there is no regenerator, εr ¼ 0. Accordingly, the equivalent temperature ratio Q reduces to the temperature ratio q and the regenerated cycle reduces to the basic one. Oppositely, when εr approaches unity, Q gets close to unity as well. Higher εr results in lower temperature ratio, which leads to higher cycle efficiency. On the other side, air temperature leaving the hot side heat exchanger is lowered by the regenerator before entering the expander. Afterwards, air temperature continues to fall inside the expander. Compared to the basic cycle, this side of the regenerator can be regarded as part of the expander. In terms of equations (6), (13) and (16)e(18), we obtain
1 h we ¼ cp T1 ½q εr ðq 1Þ 1 p e 1 q εr ðq 1Þ he ¼ cp T1 q 1 p q
(21)
If we define the equivalent isentropic efficiency of expander as below, the expander power recovery of regenerated cycle would be in the same expression as that of basic cycle [18].
The isentropic expansion process yields
T6s ¼ T5
pH pL
1k k
1k k
¼ T5 pr
h0e ¼ (13)
The main difference between the regenerated and basic cycles is caused by the regenerator. In the regenerator, the energy balance yields
T2 T1 ¼ T4 T5 ¼ εr ðT4 T1 Þ where εr is the effectiveness of regenerator.
(14)
q εr ðq 1Þ he q
(22)
Since 0 εr < 1, we have
he < h0e he q
(23)
When there is no regenerator, the regenerated cycle reduces to the basic one and the equivalent expander efficiency reduces to the original efficiency. Oppositely, higher εr results in lower expander efficiency, which leads to lower cycle efficiency. Therefore, the
C.-L. Zhang et al. / Energy 91 (2015) 226e234
impact of regenerator on the cycle performance has two sides and needs further investigation. With the newly introduced equivalent temperature ratio and equivalent expander efficiency, we can treat the regenerated air cycle as the basic one and rewrite the thermodynamic model of regenerated air cycle in the same expressions of basic cycle [18]. Heat absorption per unit mass flow
1 qL ¼ cp T2 1 Q þ Qh0e 1 p
Qh0e þ
popt ¼
cp pL 1 1 Q þ Qh0e 1 p R
(24)
(25)
COPH;opt ¼
Volumetric heating capacity
qH;v
c p pL ¼ R
p1 1Qþ hc
(27)
cp T2 ðp 1Þ hc
ðp 1Þ p Qh0e hc phc
qH phc ð1 QÞ þ pðp 1Þ ¼ w0 ðp 1Þ p Qh0e hc
cp pL R
1Qþ
popt 1 hc
(38) (39)
popt 1 popt Qh0e hc popt hc
(30)
4. Features of regenerated air cycle at optimal heating COP
(31)
(32)
and
Qh0e p> > Q > max Qh0e hc ; 1 1 Q þ Qh0e
popt 1 qH ¼ cP T2 1 Q þ hc
(37)
(29)
Similar to the basic cycle [18], the heat absorption and net power consumption must be positive, which yields
1 Q þ Qh0e > 0
cp pL 1 1 Q þ Qh0e 1 popt R
Based on numerical verification, the above equations are consistent with previous results [19]. The main advantage of current results is that uniform expressions were obtained for both regenerated cycle and basic cycle. Consequently, the further investigation on the regenerated cycle can leverage what we have learned from the basic cycle.
Heating COP
COPH ¼
(36)
w0 ¼ cP T2
Net power consumption per unit mass flow of the cycle
w0 ¼ wc we ¼ cp T2
1 popt
(28)
Expander power recovery per unit mass flow
1 0 h we ¼ cp T2 Q 1 p e
qL;v ¼
qH;v ¼
Compressor power consumption per unit mass flow
wc ¼
(35)
qL ¼ cp T2 1 Q þ Qh0e 1
(26)
(34)
Moreover, the heat pump cycle performance indices at optimal pressure ratio can be expressed in the same way.
popt hc ð1 QÞ þ popt popt 1 popt 1 popt Qhc h0e 2 1 Q þ Qh0e hc þ1 ¼ nqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffio2 Qh0e 1 hc þ Qhc 1 h0e þ Q 1
Heating capacity per unit mass flow
p1 qH ¼ cp T2 1 Q þ hc
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Qh0e ðQ 1Þ 1 hc þ Qhc 1 h0e 1 Q þ Qh0e
Volumetric heat absorption
qL;v ¼
229
(33)
The above inequalities will be explicitly or implicitly invoked in further analysis hereinafter.
As we revealed in previous work of the basic air cycle [18], air cycle has some unique features in heat pump applications. Particularly, air cycle could be more competitive in the applications of large temperature difference. Therefore, we continue to figure out how the regenerated air cycle performance varies with the temperature ratio. In addition, how the regenerated air cycle performs with the regenerator will be further explored. 4.1. Performance variation with actual temperature ratio First of all, we would find out how the equivalent temperature ratio and the equivalent expander efficiency vary with the actual temperature ratio q. Equations (19) and (22) yields
vQ ¼ vq
v
q 1þεr ðq1Þ
Similar to the basic cycle, heating COP of the regenerated cycle will reaches its maximum at optimal pressure ratio [19]. Again, with the equivalent temperature ratio and equivalent expander efficiency, we can treat the regenerated air cycle as the basic one. Similar to the basic cycle's derivations [18], the optimal pressure ratio can be found, which is in the same expression as the basic one as well.
vh0e ¼ vq
¼
vq
3. Optimization of regenerated air heat pump cycle
(40)
v
qεr ðq1Þ q
vq
1 εr ½1 þ εr ðq 1Þ2
>0
(41)
he ¼
εr q2
he < 0
(42)
When the actual temperature ratio increases, the equivalent temperature ratio increases as expected, and the equivalent expander efficiency decreases because larger temperature ratio results in more temperature drop in the hot side of regenerator. Relations (41) and (42) can be further used to analyze the cycle performance. As for the optimal pressure ratio,
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C.-L. Zhang et al. / Energy 91 (2015) 226e234
vpopt vpopt vQ vpopt vh0e þ ¼ vq vQ vq vh0 e vq
(43)
vCOPH;opt vCOPH;opt vQ vCOPH;opt vh0e þ ¼ <0 vq vq vQ vh0 e vq
(54)
vpopt >0 vQ
(44)
Therefore the optimal heating COP decreases when the actual temperature ratio increases. It behaves the same as the basic cycle.
vpopt <0 vh0e
(45)
4.2. Performance variation with effectiveness of regenerator
According to the basic air cycle analysis [18], we knew
Relations (41) through (45) yield
vpopt >0 vq
(46)
Therefore the optimal pressure ratio increases with the actual temperature ratio. It's the same as the basic cycle. As for the volumetric heating capacity,
vqH;v vqH;v vQ vqH;v vh0e þ ¼ vq vQ vq vh0e vq
(47)
In terms of the basic air cycle analysis [18], we had
vqH;v >0 vQ
(48)
Meanwhile,
vqH;v cp p1 vpopt ¼ <0 vh0e Rhc vh0e
(49)
Relations (41), (42), (47)e(49) yield
vqH;v >0 vq
(50)
Therefore the volumetric heating capacity at optimal pressure ratio increases with the actual temperature ratio, which means the regenerated cycle heat pump can provide more heating capacity when the ambient temperature goes down or the heat rejection temperature goes up. It is also the important feature we captured in the basic cycle [18]. As for the optimal heating COP,
vCOPH;opt vCOPH;opt vQ vCOPH;opt vh0e þ ¼ vq vq vQ vh0 e vq
(51)
(52)
Furthermore,
vCOPH;opt vh0e " # popt hc ð1 QÞ þ popt popt 1 vpopt v ¼ vpopt vh0e popt 1 popt Qhc h0e " # v popt hc ð1 QÞ þ popt popt 1 þ 0 vhe popt 1 popt Qhc h0e " # vpopt v popt hc ð1 QÞ þ popt popt 1 þ 0 ¼0 vh0e vhe popt 1 popt Qhc h0e popt hc ð1 QÞ þ popt popt 1 ¼ 2 Qhc popt 1 popt Qhc h0e Qhc ¼ COPH;opt >0 p Qhc h0e Relations (41), (42), (51)e(53) yield
vQ qðq 1Þ QðQ 1Þ ¼ ¼ <0 vεr 1 εr ½1 þ εr ðq 1Þ2
(55)
vh0e q1 h0 he ¼ q e < 0 ¼ q vεr q1 εr
(56)
Both the equivalent temperature ratio and the equivalent expander efficiency goes down as the effectiveness of regenerator goes up. Moreover, the lower the equivalent temperature ratio is, the lower the optimal pressure ratio will be. Oppositely, the lower the equivalent expander efficiency is, the higher the optimal pressure ratio will be. Therefore the regenerator has a dual character, which leads to a further proof below.
vpopt vpopt vQ vpopt vh0e ¼ þ vεr vQ vεr vh0e vεr
(57)
Let
X ¼ 1 Q þ Qh0e
(58)
Y ¼ Qh0e
(59)
The optimal pressure ratio or equation (34) turns into
popt ¼
Yþ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi YðY XÞð1 hc XÞ X
(60)
Since
The basic air cycle analysis [18] yields
vCOPH;opt <0 vQ
First of all, we would figure out how the equivalent temperature ratio and the equivalent expander efficiency vary with the effectiveness of regenerator or the size of regenerator. According to equations (19) and (22), we have
" pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# vpopt v Y þ YðY XÞð1 hc XÞ ¼ vX vX X " # Y Y X þ Yð1 hc XÞ ¼ 2 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < 0; 2 YðY XÞð1 hc XÞ X " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# vpopt v Y þ YðY XÞð1 hc XÞ ¼ vY vY X " # 1 ð2Y XÞð1 hc XÞ 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 0; ¼ X 2 YðY XÞð1 hc XÞ
(53) we have
vpopt Y vpopt hc YðY XÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ¼ <0 X vY vX 2X YðY XÞð1 hc XÞ
(61)
Furthermore,
vpopt 0 vpopt vpopt vX vpopt vY vpopt 0 þ ¼ ¼ he 1 þ h vQ vX vQ vY vQ vX vY e
(62)
C.-L. Zhang et al. / Energy 91 (2015) 226e234
vpopt vpopt vX vpopt vY vpopt vpopt Qþ Q ¼ þ ¼ vh0e vX vh0e vY vh0e vX vY
(63)
Rearranging equations (62) and (63) yields
vpopt h0e vpopt vpopt ¼ vX Q vh0e vQ
(64)
vpopt 1 h0e vpopt vpopt ¼ þ vY Q vh0e vQ
(65)
231
Input: Efficiencies of compressor, expander and regenerator; heat source temperatures, exit temperature difference of hot side heat exchanger
Calculate optimal pressure ratio using Eqs.(34), (19) and (22)
Substituting equations (64) and (65) into equation (61) yields
vpopt Y vpopt h0e vpopt 1 þ ¼ 0 X vY vX Q 1 Q þ Qhe vh0 e vpopt Q1 < 0; þ 0 1 Q þ Qhe vQ
Calculate cycle performance at optimal pressure ratio using Eqs.(35) ~ (40)
which can be reduced to
QðQ 1Þ vpopt vpopt þ <0 h0e vQ vh0e
(66)
In terms of relations (55) and (56), multiplying inequality (66) yields
0 vQ vhe vεr vεr
to
"
# h0e vQ QðQ 1Þ vpopt QðQ 1Þ vh0e vpopt þ <0 q vε 1 εr vεr vQ vh0e h0 e εr r q1
⇔
1 εr vpopt vQ vpopt vh0e þ >0 q vQ vεr vh0 e vεr εr q1 (67) When εr / 0 and q / þ∞,
1 εr vpopt vQ vpopt vh0e vpopt vQ vpopt vh0e vpopt þ / þ ¼ q vQ vεr vh0 e vεr vQ vεr vh0e vεr vεr q1 εr
Fig. 3. Flowchart of air cycle performance calculation.
Numerical simulations can provide visible details for further discussions. Two main aspects are considered in simulation. One is verification of important features we found hereinbefore, another is the applicability of regenerator since we found the regenerator has a dual character. Note that in following simulations except the last case, the isentropic efficiencies of compressor and expander are set to 0.85 and 0.9, respectively in terms of engineering practice [16], which could impact the numbers not the trends. Fig. 5 illustrates how the equivalent expander efficiency changes with the effectiveness of regenerator and the actual temperature h0 ratio. In accordance with equation (22), he ¼ 1 εr 1 1q , which e
(68) Relations (67) and (68) indicate that under extreme conditions vp we could have vεopt > 0. In other words, a regenerator may not alr ways reduce the optimal pressure ratio. Numerical simulations in next section will verify this point. As for the optimal heating COP,
vCOPH;opt vCOPH;opt vQ vCOPH;opt vh0e ¼ þ vεr vεr vQ vh0e vεr
Output: Cycle performance
indicates the equivalent expander efficiency is a linear function of the regenerator effectiveness. Higher effectiveness of regenerator would result in lower equivalent expander efficiency. Meanwhile it is also a linear function of the reciprocal of actual temperature ratio. When the effectiveness of regenerator increases from 0 to 1, higher temperature ratio would lead to more loss of equivalent expander efficiency.
(69)
According to the basic air cycle analysis [18] and relations (55) (56), we knew that Therefore, the sign of
vCOPH;opt vQ vCOPH;opt vεr
< 0,
vCOPH;opt vQ vh0 e vεr < 0,
> 0 and
vh0e vεr
< 0.
can be conditional. For brevity, nu-
merical verification instead of lengthy proof will be provided in next section. In summary, air cycle performance is not always improved by a regenerator even though it looked like this way in previous researches. 5. Model validation and simulation To figure out the regenerated air heat pump cycle performance, a basic flowchart of model calculation is given in Fig. 3. Experimental data from the literature [16] are used to validate the thermodynamic model. As shown in Fig. 4, the model predictions are in good agreement with experimental data.
Fig. 4. Thermodynamic model validation.
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C.-L. Zhang et al. / Energy 91 (2015) 226e234
Fig. 6 demonstrates how the equivalent temperature ratio varies with the effectiveness of regenerator and the actual temperature ratio. According to equation (19),
1 1 1 1 ¼ εr 1 þ ¼ ð1 εr Þ þ εr Q q q q
Fig. 5. Change of equivalent expander efficiency.
Fig. 6. Change of equivalent temperature ratio.
(70)
which indicates the reciprocal of equivalent temperature ratio is a linear function of the effectiveness of regenerator as well as the reciprocal of actual temperature ratio. Furthermore, the effectiveness of regenerator and the reciprocal of actual temperature ratio are interchangeable in equation (70), which means the two variables play an equal role in influencing the equivalent temperature ratio. Particularly, when the effectiveness of regenerator is high enough, the actual temperature ratio will have marginal impact on the equivalent temperature ratio. Likewise, when the actual temperature ratio is low enough, the effectiveness of regenerator will have marginal impact on the equivalent temperature ratio. Fig. 7 shows how the pressure ratio varies with the effectiveness of regenerator and the actual temperature ratio. As plotted in Fig. 7(a), when the actual temperature ratio is not very high (typically it is not greater than 1.5 in usual heat pump applications), the pressure ratio can be reduced by a larger regenerator. However, as we predicted in theory and as shown in Fig. 7(b), under certain extreme conditions, a larger regenerator might lead to even higher pressure ratio. Generally speaking, researchers invested the regenerated cycle for cycle efficiency improvement. However, in terms of equation (69), the result might not be as good as expected. Thus two main concerns are raised. The first concern is whether a regenerator always improves the cycle efficiency. Fig. 8 gives the answer. As shown in Fig. 8(a), when the actual temperature ratio is not greater than 1.5, a regenerator can improve the heating COP of cycle. The larger regenerator we use, the higher COP we can get. As shown in Fig. 8(b), however, when the temperature ratio is 1.7 or higher, a regenerator cannot improve the heating COP until it's large enough. The second concern is how to better size the regenerator for cycle efficiency improvement. In Fig. 8(a), on each curve we can locate the points of 10% COP increase (compared to the COP of basic cycle) and 20% COP increase, respectively. To achieve 10% COP increase, the effectiveness of regenerator should be greater than 0.55. To achieve 20% COP increase, the effectiveness of regenerator should be in the range of 0.8e0.9. The higher the temperature ratio
Fig. 7. Change of optimal pressure ratio.
C.-L. Zhang et al. / Energy 91 (2015) 226e234
233
Fig. 8. Change of optimal heating COP.
is, the higher the effectiveness of regenerator should be. Therefore, the regenerated cycle is more suitable for low temperature ratio applications. Since the above simulations and discussions were based on the given efficiencies of compressor and expander, at last we study how the regenerator works with compressor and expander of different efficiencies. In this case, we set the effectiveness of regenerator to 0.85. Simulation results of heating COP increase from regenerator (compared to the COP of basic cycle) are shown in Fig. 9. Two main conclusions can be drawn from Fig. 9. One is that there is a combination of compressor and expander efficiencies making the most energy efficient use of regenerator. Another is that the heating COP increase from regenerator is more sensitive to the compressor efficiency. Note that although the cycle performance of such heat pump system significantly depends on the isentropic efficiencies of compressor and expander, the objective of the work is to give a new theoretical proof of some important features of regenerated air heat pump cycle so that the readers could have better understanding of the cycle from a new perspective. The calculations of cycle performance here were only used to illustrate and verify the conclusions made from the theoretical proof and analysis. The cycle performance itself is not the main concern of this work. Detailed
discussions on the cycle performance can be found in our previous work [19]. 6. Conclusions A thermodynamic model of regenerated air heat pump cycle was developed for theoretical and numerical analysis of the cycle characteristics. Two new parameters, equivalent temperature ratio and equivalent isentropic efficiency of expander, were introduced to make the regenerated cycle consistent with the basic cycle in mathematical expressions. Based on this new approach, the following main conclusions were drawn. In the regenerated air heat pump cycle, the maximum heating COP can be achieved at optimal pressure ratio. The formulae of optimal pressure ratio and COP were in the same expression as the basic cycle when we applied the equivalent temperature ratio and equivalent isentropic efficiency of expander. It's proved that in theory the regenerated air heat pump cycle can also make the heating capacity in line with the heating load at optimal pressure ratio. Meanwhile, the heating COP of regenerated cycle could be more than 20% higher than that of basic cycle. These indicate the regenerated air cycle has more potentials in heat pump applications. It's proved that a regenerator does not always improve the cycle performance. Larger temperature ratio and lower effectiveness of regenerator could make the regenerated cycle even worse than the basic cycle. The regenerator should be carefully assessed under actual operating conditions and well sized to gain remarkable heating COP increase. This work would enable further study of the regenerated air cycle from a different perspective. References
Fig. 9. Optimal heating COP increase vs. compressor/expander efficiencies.
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[21] Park SK, Ahn JH, Kim TS. Off-design operating characteristics of an open-cycle air refrigeration system. Int J Refrig 2012;35(8):2311e20. [22] Zhang W, Chen L, Sun F. Thermodynamic optimisation for open regenerated inverse Brayton cycle (refrigeration/heat pump cycle) part 2-performance optimisation. J Energy Inst 2012;85(2):96e102. [23] Chen L, Ni N, Sun F, Wu C. Performance of real regenerated air heat pumps. Int J Power & Energy Syst 1999;19(3):231e8. [24] Bi Y, Chen L, Sun F. Heating load, heating-load density and COP optimizations of an endoreversible air heat-pump. Appl Energy 2008;85(7):607e17. [25] Bi YH, Chen LG, Sun FR. Heating load, heating load density and COP optimisations for an endoreversible variable temperature heat reservoir air heat pump. J Energy Inst 2009;82(1):43e7. [26] Bi YH, Xie GZ, Chen LE, Sun FR. Heating load density optimization of an irreversible simple Brayton cycle heat pump coupled to counter-flow heat exchangers. Appl Math Model 2012;36(5):1854e63. [27] Bi YH, Chen LG, Sun FR. Comparative performance analysis for endoreversible simple air heat pump cycles considering ecological, exergetic efficiency and heating load objectives. Int J Exergy 2009;6(4):550e66. [28] Bi Y, Chen L, Sun F. Ecological, exergetic efficiency and heating load optimizations for endoreversible variable-temperature heat reservoir simple air heat pump cycles. Int J Low-Carbon Technol 2010;5(1):7e17. [29] Bi Y, Chen L, Sun F. Exergetic efficiency optimization for an irreversible heat pump working on reversed Brayton cycle. Pramana J Phys 2010;74(3): 351e63. [30] Bi Y, Chen L, Sun F. Ecological, exergetic efficiency and heating load optimizations for irreversible variable-temperature heat reservoir simple air heat pump cycles. Indian J Pure Appl Phys 2009;47(12):852e62. [31] Bi Y, Chen L, Sun F. Exergy-based ecological optimization for an endoreversible variable-temperature heat reservoir air heat pump cycle. Riv Mex Fis 2009;55(2):112e9. [32] Bi Y, Chen L, Sun F, Wu C. Exergy-based ecological optimisation for an endoreversible air heat pump cycle. Int J Ambient Energy 2009;30(1):45e52. [33] Bi Y, Chen L, Sun F. Optimum allocation of heat exchanger inventory of irreversible air heat pump cycles. Int J Sustain Energy 2010;29(3):133e41. [34] Chen L, Ni N, Wu C, Sun F. Heating load vs. COP characteristics for irreversible air-heat pump cycles. Int J Power Energy Syst 2001;21(2):105e11. [35] Chen L, Ni N, Wu C, Sun F. Performance analysis of a closed regenerated Brayton heat pump with internal irreversibilities. Int J Energy Res 1999;23(12):1039e50. [36] Chen Q, Xu Y-C, Hao J-H. An optimization method for gas refrigeration cycle based on the combination of both thermodynamics and entransy theory. Appl Energy 2014;113:982e9. [37] Hao J-H, Chen Q, Xu Y-C. A global optimization method for regenerative air refrigeration systems. Appl Therm Eng 2014;65(1):255e61.
Contents lists available at ScienceDirect
Energy journal homepage: www.elsevier.com/locate/energy
New insight into regenerated air heat pump cycle Chun-Lu Zhang*, Han Yuan, Xiang Cao Institute of Refrigeration and Cyrogenics, School of Mechanical Engineering, Tongji University, Shanghai 201804, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 February 2015 Received in revised form 20 July 2015 Accepted 2 August 2015 Available online 8 September 2015
Regenerated air (reverse Brayton) cycle has unique potentials in heat pump applications compared to conventional vapor-compression cycles. To better understand the regenerated air heat pump cycle characteristics, a thermodynamic model with new equivalent parameters was developed in this paper. Equivalent temperature ratio and equivalent isentropic efficiency of expander were introduced to represent the effect of regenerator, which made the regenerated air cycle in the same mathematical expressions as the basic air cycle and created an easy way to prove some important features that regenerated air cycle inherits from the basic one. Moreover, we proved in theory that the regenerator does not always improve the air cycle efficiency. Larger temperature ratio and lower effectiveness of regenerator could make the regenerated air cycle even worse than the basic air cycle. Lastly, we found that only under certain conditions the cycle could get remarkable benefits from a well-sized regenerator. These results would enable further study of the regenerated air cycle from a different perspective. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Heat pump Air cycle Regenerated air cycle Regenerator Model Optimization
1. Introduction Given the increasing environmental concerns, air is absolutely natural, safe and free working fluid for refrigeration or heat pump applications. Therefore, the reverse Brayton cycle using air as the refrigerant has been investigated for a long time. The air cycle machine was primarily served as the aircraft air conditioner [1,2] and was widely applied in cryogenic engineering [3e5]. In addition, the applicability of air cycle in transportation air conditioning [6e9], refrigerated storage [10e12], integrated air conditioning and desiccant system [13], domestic heating [14], heat pump clothes dryer [15], and heat pump water heater [16] was investigated as well. Nevertheless, the fairly low energy efficiency still constrains the air cycle from extensive applications in normal temperature range. Limited by the commercial applications, more air cycle investigations were conducted with thermodynamic modeling not experiments. There were three main thermodynamic modeling approaches in the literature. One is the classical thermodynamics [17e21], the second is the finite-time thermodynamics [22e35], and the last is the recently developed entransy theory [36,37]. Although the assumptions or constraints taken in those investigations were more or less different, all studies reached
* Corresponding author. Tel.: þ86 136 71825 133. E-mail address: [email protected] (C.-L. Zhang). http://dx.doi.org/10.1016/j.energy.2015.08.052 0360-5442/© 2015 Elsevier Ltd. All rights reserved.
consensus that air cycles should be thermodynamically optimized to get higher energy efficiency. In the previous work [18], we found and proved that at optimal pressure ratio the basic air cycle can make the heating capacity in line with the heating load. It's an important feature for heat pump off-design operations, which vapor-compression heat pumps cannot do. Later on, we investigated the regenerated air cycle and did numerical simulation on the cycle performance at optimal pressure ratio [19]. Effected by the regenerator, the regenerated air cycle seemed more energy efficient than the basic one and could keep the optimal pressure ratio much more stable. Meanwhile, at the optimal pressure ratio the heating capacity is in line with the heating load. Therefore, the regenerated air cycle has higher potentials in heat pump applications. However, the existence of regenerator also introduced more complexity to the thermodynamic cycle model so that most conclusions were drawn by simulations not a solid theoretical proof. In this paper, we develop a new theoretical approach to prove and reveal some important features of the regenerated air cycle. Two new parameters, so-called the equivalent temperature ratio and the equivalent expander efficiency, are introduced to represent the effect of regenerator so that the regenerated cycle can be in the same mathematical expressions as the basic one. Moreover, we can therefore have an in-depth look into what exactly the regenerator contributes to the cycle performance.
C.-L. Zhang et al. / Energy 91 (2015) 226e234
Nomenclature cp COP h hfg qL qL,v qH qH,v QH p pr R T V_ w0 wc
specific heat at constant pressure, J kg1 K1 coefficient of performance enthalpy, J kg1 latent heat of vaporization, J kg1 heat absorption per unit mass flow from lowtemperature environment, J kg1 volumetric heat absorption, heat absorption per unit volume flow at compressor inlet, J m3 heating capacity per unit mass flow, J kg1 volumetric heating capacity, heating capacity per unit volume flow at compressor inlet, J m3 heating capacity, W pressure, Pa pressure ratio gas constant (¼ 8.314), J K1 mol1 temperature, C, K volume flow rate, m3 s1 net power consumption per unit mass flow, J kg1 compressor power consumption per unit mass flow, J kg1
2. Thermodynamic model of regenerated air heat pump cycle A typical regenerated air cycle heat pump is as shown in Fig. 1. Compared to the basic air cycle, a regenerator is added to transfer heat from hot air after the hot side heat exchanger to the compressor suction line. To reduce the heat transfer loss and eliminate the frosting/defrosting issue in heat pump applications, a semi-open air cycle without the cold side heat exchanger is preferred and investigated in this work. Fig. 2 illustrates the thermodynamic processes which air undergoes in the regenerated air heat pump cycle. Curves 2e3 and 5e6 represent compression and expansion processes, respectively. Curve 3e4 demonstrates the isobaric heat rejection process in the hot side heat exchanger. Process 6-1 is open for the semi-open cycle. Curves 4e5 and 1e2 are the processes in hot and cold sides of the regenerator, respectively.
we X Y
227
expander power recovery per unit mass flow, J kg1 intermediate variable defined in equation (58) intermediate variable defined in equation (59)
Greek symbols εr effectiveness of regenerator hc isentropic efficiency of compressor he isentropic efficiency of expander h0e equivalent isentropic efficiency of expander defined in equation (22) p derived pressure ratio defined in equation (17) q temperature ratio defined in equation (18) Q equivalent temperature ratio defined in equation (19) Subscripts in inlet H high-pressure side; hot side; heating L low-pressure side; cold side opt optimal out outlet v volumetric
The assumptions and irreversibilities taken into account for model development of the regenerated cycle are as follows. 1) The air is ideal gas. 2) All processes in heat exchangers are isobaric. 3) Constant isentropic efficiencies are specified for compressor and expander. 4) The effectiveness of regenerator is given. 5) Temperature difference between the hot side inlet of regenerator and high-temperature heat sink (so-called exit temperature difference) is given. 6) Temperature difference between the cold side inlet of regenerator and low-temperature heat source is zero for the semi-open cycle. Accordingly, we can develop the thermodynamic model hereinbelow.
qH 3
Hot side heat exchanger
Compressor
4 5
2
wc
Regenerator
we 1
Expander
6
Cold side heat exchanger
qL Fig. 1. Schematic of regenerated air cycle heat pump.
Fig. 2. T-s diagram of regenerated air cycle.
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Heat absorption per unit mass flow
qL ¼ h1 h6 ¼ cp ðT1 T6 Þ
(1)
Heat absorption per unit volume flow or volumetric heat absorption (i.e. volumetric cooling capacity for cooling cycle)
qL;v ¼
h1 h6 cp ðT1 T6 Þ ¼ v2 v2
(2)
Heating capacity per unit mass flow
qH ¼ h3 h4 ¼ cp ðT3 T4 Þ
(3)
Volumetric heating capacity
Accordingly, the compressor suction temperature T2 and the expander entering temperature T5 of the regenerated cycle can be found.
T2 ¼ T1 þ εr ðT4 T1 Þ
(15)
T5 ¼ T4 εr ðT4 T1 Þ
(16)
In comparison with the basic air cycle, the compressor suction temperature is raised by the regenerator, meanwhile the expander entering temperature falls. To get equivalent expressions as the basic cycle, we define the following dimensionless parameters. The derived pressure ratio is the same as that of the basic cycle. k1
qH;v
h h4 cp ¼ 3 ¼ ðT3 T4 Þ v2 v2
(4)
Compressor power consumption per unit mass flow
wc ¼ h3 h2 ¼ cp ðT3 T2 Þ ¼
h3s h2 cp ðT3s T2 Þ ¼ hc hc
(5)
Expander power recovery per unit mass flow
we ¼ h5 h6 ¼ cp ðT5 T6 Þ ¼ he ðh5 h6s Þ ¼ cp ðT5 T6s Þhe (6) Accordingly, net power consumption per unit mass flow of the cycle
cp ðT3s T2 Þ w0 ¼ wc we ¼ cp ðT5 T6s Þhe hc
(7)
qH T3 T4 ¼ wc we T3s T2 ðT5 T6s Þhe
(8)
hc
According to the ideal gas law,
pv ¼ RT
(9)
From equation (5), we have
T3 ¼ T2 þ
T3s T2 hc
(10)
From equation (6), we have
T6 ¼ T5 ðT5 T6s Þhe
(11)
The isentropic compression process yields
T3s ¼ T2
pH pL
k1 k
k1
¼ T2 pr k
(12)
(17)
The temperature ratio, which is similar to that of the basic cycle, characterizes the ratio of heat sink temperature to the actual heat source temperature.
q¼
T4 T1
(18)
But effected by the regenerator, the actual compressor suction temperature rises so that the compressor seems to pump heat from a higher temperature heat source. Therefore, the equivalent temperature ratio would make more sense here.
Q¼
T4 T4 q ¼ ¼ T2 T1 þ εr ðT4 T1 Þ 1 þ εr ðq 1Þ
(19)
Since the effectiveness of regenerator 0 εr < 1, we have
1
Heating COP (coefficient of performance)
COPH ¼
p ¼ pr k
(20)
When there is no regenerator, εr ¼ 0. Accordingly, the equivalent temperature ratio Q reduces to the temperature ratio q and the regenerated cycle reduces to the basic one. Oppositely, when εr approaches unity, Q gets close to unity as well. Higher εr results in lower temperature ratio, which leads to higher cycle efficiency. On the other side, air temperature leaving the hot side heat exchanger is lowered by the regenerator before entering the expander. Afterwards, air temperature continues to fall inside the expander. Compared to the basic cycle, this side of the regenerator can be regarded as part of the expander. In terms of equations (6), (13) and (16)e(18), we obtain
1 h we ¼ cp T1 ½q εr ðq 1Þ 1 p e 1 q εr ðq 1Þ he ¼ cp T1 q 1 p q
(21)
If we define the equivalent isentropic efficiency of expander as below, the expander power recovery of regenerated cycle would be in the same expression as that of basic cycle [18].
The isentropic expansion process yields
T6s ¼ T5
pH pL
1k k
1k k
¼ T5 pr
h0e ¼ (13)
The main difference between the regenerated and basic cycles is caused by the regenerator. In the regenerator, the energy balance yields
T2 T1 ¼ T4 T5 ¼ εr ðT4 T1 Þ where εr is the effectiveness of regenerator.
(14)
q εr ðq 1Þ he q
(22)
Since 0 εr < 1, we have
he < h0e he q
(23)
When there is no regenerator, the regenerated cycle reduces to the basic one and the equivalent expander efficiency reduces to the original efficiency. Oppositely, higher εr results in lower expander efficiency, which leads to lower cycle efficiency. Therefore, the
C.-L. Zhang et al. / Energy 91 (2015) 226e234
impact of regenerator on the cycle performance has two sides and needs further investigation. With the newly introduced equivalent temperature ratio and equivalent expander efficiency, we can treat the regenerated air cycle as the basic one and rewrite the thermodynamic model of regenerated air cycle in the same expressions of basic cycle [18]. Heat absorption per unit mass flow
1 qL ¼ cp T2 1 Q þ Qh0e 1 p
Qh0e þ
popt ¼
cp pL 1 1 Q þ Qh0e 1 p R
(24)
(25)
COPH;opt ¼
Volumetric heating capacity
qH;v
c p pL ¼ R
p1 1Qþ hc
(27)
cp T2 ðp 1Þ hc
ðp 1Þ p Qh0e hc phc
qH phc ð1 QÞ þ pðp 1Þ ¼ w0 ðp 1Þ p Qh0e hc
cp pL R
1Qþ
popt 1 hc
(38) (39)
popt 1 popt Qh0e hc popt hc
(30)
4. Features of regenerated air cycle at optimal heating COP
(31)
(32)
and
Qh0e p> > Q > max Qh0e hc ; 1 1 Q þ Qh0e
popt 1 qH ¼ cP T2 1 Q þ hc
(37)
(29)
Similar to the basic cycle [18], the heat absorption and net power consumption must be positive, which yields
1 Q þ Qh0e > 0
cp pL 1 1 Q þ Qh0e 1 popt R
Based on numerical verification, the above equations are consistent with previous results [19]. The main advantage of current results is that uniform expressions were obtained for both regenerated cycle and basic cycle. Consequently, the further investigation on the regenerated cycle can leverage what we have learned from the basic cycle.
Heating COP
COPH ¼
(36)
w0 ¼ cP T2
Net power consumption per unit mass flow of the cycle
w0 ¼ wc we ¼ cp T2
1 popt
(28)
Expander power recovery per unit mass flow
1 0 h we ¼ cp T2 Q 1 p e
qL;v ¼
qH;v ¼
Compressor power consumption per unit mass flow
wc ¼
(35)
qL ¼ cp T2 1 Q þ Qh0e 1
(26)
(34)
Moreover, the heat pump cycle performance indices at optimal pressure ratio can be expressed in the same way.
popt hc ð1 QÞ þ popt popt 1 popt 1 popt Qhc h0e 2 1 Q þ Qh0e hc þ1 ¼ nqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffio2 Qh0e 1 hc þ Qhc 1 h0e þ Q 1
Heating capacity per unit mass flow
p1 qH ¼ cp T2 1 Q þ hc
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Qh0e ðQ 1Þ 1 hc þ Qhc 1 h0e 1 Q þ Qh0e
Volumetric heat absorption
qL;v ¼
229
(33)
The above inequalities will be explicitly or implicitly invoked in further analysis hereinafter.
As we revealed in previous work of the basic air cycle [18], air cycle has some unique features in heat pump applications. Particularly, air cycle could be more competitive in the applications of large temperature difference. Therefore, we continue to figure out how the regenerated air cycle performance varies with the temperature ratio. In addition, how the regenerated air cycle performs with the regenerator will be further explored. 4.1. Performance variation with actual temperature ratio First of all, we would find out how the equivalent temperature ratio and the equivalent expander efficiency vary with the actual temperature ratio q. Equations (19) and (22) yields
vQ ¼ vq
v
q 1þεr ðq1Þ
Similar to the basic cycle, heating COP of the regenerated cycle will reaches its maximum at optimal pressure ratio [19]. Again, with the equivalent temperature ratio and equivalent expander efficiency, we can treat the regenerated air cycle as the basic one. Similar to the basic cycle's derivations [18], the optimal pressure ratio can be found, which is in the same expression as the basic one as well.
vh0e ¼ vq
¼
vq
3. Optimization of regenerated air heat pump cycle
(40)
v
qεr ðq1Þ q
vq
1 εr ½1 þ εr ðq 1Þ2
>0
(41)
he ¼
εr q2
he < 0
(42)
When the actual temperature ratio increases, the equivalent temperature ratio increases as expected, and the equivalent expander efficiency decreases because larger temperature ratio results in more temperature drop in the hot side of regenerator. Relations (41) and (42) can be further used to analyze the cycle performance. As for the optimal pressure ratio,
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C.-L. Zhang et al. / Energy 91 (2015) 226e234
vpopt vpopt vQ vpopt vh0e þ ¼ vq vQ vq vh0 e vq
(43)
vCOPH;opt vCOPH;opt vQ vCOPH;opt vh0e þ ¼ <0 vq vq vQ vh0 e vq
(54)
vpopt >0 vQ
(44)
Therefore the optimal heating COP decreases when the actual temperature ratio increases. It behaves the same as the basic cycle.
vpopt <0 vh0e
(45)
4.2. Performance variation with effectiveness of regenerator
According to the basic air cycle analysis [18], we knew
Relations (41) through (45) yield
vpopt >0 vq
(46)
Therefore the optimal pressure ratio increases with the actual temperature ratio. It's the same as the basic cycle. As for the volumetric heating capacity,
vqH;v vqH;v vQ vqH;v vh0e þ ¼ vq vQ vq vh0e vq
(47)
In terms of the basic air cycle analysis [18], we had
vqH;v >0 vQ
(48)
Meanwhile,
vqH;v cp p1 vpopt ¼ <0 vh0e Rhc vh0e
(49)
Relations (41), (42), (47)e(49) yield
vqH;v >0 vq
(50)
Therefore the volumetric heating capacity at optimal pressure ratio increases with the actual temperature ratio, which means the regenerated cycle heat pump can provide more heating capacity when the ambient temperature goes down or the heat rejection temperature goes up. It is also the important feature we captured in the basic cycle [18]. As for the optimal heating COP,
vCOPH;opt vCOPH;opt vQ vCOPH;opt vh0e þ ¼ vq vq vQ vh0 e vq
(51)
(52)
Furthermore,
vCOPH;opt vh0e " # popt hc ð1 QÞ þ popt popt 1 vpopt v ¼ vpopt vh0e popt 1 popt Qhc h0e " # v popt hc ð1 QÞ þ popt popt 1 þ 0 vhe popt 1 popt Qhc h0e " # vpopt v popt hc ð1 QÞ þ popt popt 1 þ 0 ¼0 vh0e vhe popt 1 popt Qhc h0e popt hc ð1 QÞ þ popt popt 1 ¼ 2 Qhc popt 1 popt Qhc h0e Qhc ¼ COPH;opt >0 p Qhc h0e Relations (41), (42), (51)e(53) yield
vQ qðq 1Þ QðQ 1Þ ¼ ¼ <0 vεr 1 εr ½1 þ εr ðq 1Þ2
(55)
vh0e q1 h0 he ¼ q e < 0 ¼ q vεr q1 εr
(56)
Both the equivalent temperature ratio and the equivalent expander efficiency goes down as the effectiveness of regenerator goes up. Moreover, the lower the equivalent temperature ratio is, the lower the optimal pressure ratio will be. Oppositely, the lower the equivalent expander efficiency is, the higher the optimal pressure ratio will be. Therefore the regenerator has a dual character, which leads to a further proof below.
vpopt vpopt vQ vpopt vh0e ¼ þ vεr vQ vεr vh0e vεr
(57)
Let
X ¼ 1 Q þ Qh0e
(58)
Y ¼ Qh0e
(59)
The optimal pressure ratio or equation (34) turns into
popt ¼
Yþ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi YðY XÞð1 hc XÞ X
(60)
Since
The basic air cycle analysis [18] yields
vCOPH;opt <0 vQ
First of all, we would figure out how the equivalent temperature ratio and the equivalent expander efficiency vary with the effectiveness of regenerator or the size of regenerator. According to equations (19) and (22), we have
" pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# vpopt v Y þ YðY XÞð1 hc XÞ ¼ vX vX X " # Y Y X þ Yð1 hc XÞ ¼ 2 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < 0; 2 YðY XÞð1 hc XÞ X " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# vpopt v Y þ YðY XÞð1 hc XÞ ¼ vY vY X " # 1 ð2Y XÞð1 hc XÞ 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 0; ¼ X 2 YðY XÞð1 hc XÞ
(53) we have
vpopt Y vpopt hc YðY XÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ¼ <0 X vY vX 2X YðY XÞð1 hc XÞ
(61)
Furthermore,
vpopt 0 vpopt vpopt vX vpopt vY vpopt 0 þ ¼ ¼ he 1 þ h vQ vX vQ vY vQ vX vY e
(62)
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vpopt vpopt vX vpopt vY vpopt vpopt Qþ Q ¼ þ ¼ vh0e vX vh0e vY vh0e vX vY
(63)
Rearranging equations (62) and (63) yields
vpopt h0e vpopt vpopt ¼ vX Q vh0e vQ
(64)
vpopt 1 h0e vpopt vpopt ¼ þ vY Q vh0e vQ
(65)
231
Input: Efficiencies of compressor, expander and regenerator; heat source temperatures, exit temperature difference of hot side heat exchanger
Calculate optimal pressure ratio using Eqs.(34), (19) and (22)
Substituting equations (64) and (65) into equation (61) yields
vpopt Y vpopt h0e vpopt 1 þ ¼ 0 X vY vX Q 1 Q þ Qhe vh0 e vpopt Q1 < 0; þ 0 1 Q þ Qhe vQ
Calculate cycle performance at optimal pressure ratio using Eqs.(35) ~ (40)
which can be reduced to
QðQ 1Þ vpopt vpopt þ <0 h0e vQ vh0e
(66)
In terms of relations (55) and (56), multiplying inequality (66) yields
0 vQ vhe vεr vεr
to
"
# h0e vQ QðQ 1Þ vpopt QðQ 1Þ vh0e vpopt þ <0 q vε 1 εr vεr vQ vh0e h0 e εr r q1
⇔
1 εr vpopt vQ vpopt vh0e þ >0 q vQ vεr vh0 e vεr εr q1 (67) When εr / 0 and q / þ∞,
1 εr vpopt vQ vpopt vh0e vpopt vQ vpopt vh0e vpopt þ / þ ¼ q vQ vεr vh0 e vεr vQ vεr vh0e vεr vεr q1 εr
Fig. 3. Flowchart of air cycle performance calculation.
Numerical simulations can provide visible details for further discussions. Two main aspects are considered in simulation. One is verification of important features we found hereinbefore, another is the applicability of regenerator since we found the regenerator has a dual character. Note that in following simulations except the last case, the isentropic efficiencies of compressor and expander are set to 0.85 and 0.9, respectively in terms of engineering practice [16], which could impact the numbers not the trends. Fig. 5 illustrates how the equivalent expander efficiency changes with the effectiveness of regenerator and the actual temperature h0 ratio. In accordance with equation (22), he ¼ 1 εr 1 1q , which e
(68) Relations (67) and (68) indicate that under extreme conditions vp we could have vεopt > 0. In other words, a regenerator may not alr ways reduce the optimal pressure ratio. Numerical simulations in next section will verify this point. As for the optimal heating COP,
vCOPH;opt vCOPH;opt vQ vCOPH;opt vh0e ¼ þ vεr vεr vQ vh0e vεr
Output: Cycle performance
indicates the equivalent expander efficiency is a linear function of the regenerator effectiveness. Higher effectiveness of regenerator would result in lower equivalent expander efficiency. Meanwhile it is also a linear function of the reciprocal of actual temperature ratio. When the effectiveness of regenerator increases from 0 to 1, higher temperature ratio would lead to more loss of equivalent expander efficiency.
(69)
According to the basic air cycle analysis [18] and relations (55) (56), we knew that Therefore, the sign of
vCOPH;opt vQ vCOPH;opt vεr
< 0,
vCOPH;opt vQ vh0 e vεr < 0,
> 0 and
vh0e vεr
< 0.
can be conditional. For brevity, nu-
merical verification instead of lengthy proof will be provided in next section. In summary, air cycle performance is not always improved by a regenerator even though it looked like this way in previous researches. 5. Model validation and simulation To figure out the regenerated air heat pump cycle performance, a basic flowchart of model calculation is given in Fig. 3. Experimental data from the literature [16] are used to validate the thermodynamic model. As shown in Fig. 4, the model predictions are in good agreement with experimental data.
Fig. 4. Thermodynamic model validation.
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Fig. 6 demonstrates how the equivalent temperature ratio varies with the effectiveness of regenerator and the actual temperature ratio. According to equation (19),
1 1 1 1 ¼ εr 1 þ ¼ ð1 εr Þ þ εr Q q q q
Fig. 5. Change of equivalent expander efficiency.
Fig. 6. Change of equivalent temperature ratio.
(70)
which indicates the reciprocal of equivalent temperature ratio is a linear function of the effectiveness of regenerator as well as the reciprocal of actual temperature ratio. Furthermore, the effectiveness of regenerator and the reciprocal of actual temperature ratio are interchangeable in equation (70), which means the two variables play an equal role in influencing the equivalent temperature ratio. Particularly, when the effectiveness of regenerator is high enough, the actual temperature ratio will have marginal impact on the equivalent temperature ratio. Likewise, when the actual temperature ratio is low enough, the effectiveness of regenerator will have marginal impact on the equivalent temperature ratio. Fig. 7 shows how the pressure ratio varies with the effectiveness of regenerator and the actual temperature ratio. As plotted in Fig. 7(a), when the actual temperature ratio is not very high (typically it is not greater than 1.5 in usual heat pump applications), the pressure ratio can be reduced by a larger regenerator. However, as we predicted in theory and as shown in Fig. 7(b), under certain extreme conditions, a larger regenerator might lead to even higher pressure ratio. Generally speaking, researchers invested the regenerated cycle for cycle efficiency improvement. However, in terms of equation (69), the result might not be as good as expected. Thus two main concerns are raised. The first concern is whether a regenerator always improves the cycle efficiency. Fig. 8 gives the answer. As shown in Fig. 8(a), when the actual temperature ratio is not greater than 1.5, a regenerator can improve the heating COP of cycle. The larger regenerator we use, the higher COP we can get. As shown in Fig. 8(b), however, when the temperature ratio is 1.7 or higher, a regenerator cannot improve the heating COP until it's large enough. The second concern is how to better size the regenerator for cycle efficiency improvement. In Fig. 8(a), on each curve we can locate the points of 10% COP increase (compared to the COP of basic cycle) and 20% COP increase, respectively. To achieve 10% COP increase, the effectiveness of regenerator should be greater than 0.55. To achieve 20% COP increase, the effectiveness of regenerator should be in the range of 0.8e0.9. The higher the temperature ratio
Fig. 7. Change of optimal pressure ratio.
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233
Fig. 8. Change of optimal heating COP.
is, the higher the effectiveness of regenerator should be. Therefore, the regenerated cycle is more suitable for low temperature ratio applications. Since the above simulations and discussions were based on the given efficiencies of compressor and expander, at last we study how the regenerator works with compressor and expander of different efficiencies. In this case, we set the effectiveness of regenerator to 0.85. Simulation results of heating COP increase from regenerator (compared to the COP of basic cycle) are shown in Fig. 9. Two main conclusions can be drawn from Fig. 9. One is that there is a combination of compressor and expander efficiencies making the most energy efficient use of regenerator. Another is that the heating COP increase from regenerator is more sensitive to the compressor efficiency. Note that although the cycle performance of such heat pump system significantly depends on the isentropic efficiencies of compressor and expander, the objective of the work is to give a new theoretical proof of some important features of regenerated air heat pump cycle so that the readers could have better understanding of the cycle from a new perspective. The calculations of cycle performance here were only used to illustrate and verify the conclusions made from the theoretical proof and analysis. The cycle performance itself is not the main concern of this work. Detailed
discussions on the cycle performance can be found in our previous work [19]. 6. Conclusions A thermodynamic model of regenerated air heat pump cycle was developed for theoretical and numerical analysis of the cycle characteristics. Two new parameters, equivalent temperature ratio and equivalent isentropic efficiency of expander, were introduced to make the regenerated cycle consistent with the basic cycle in mathematical expressions. Based on this new approach, the following main conclusions were drawn. In the regenerated air heat pump cycle, the maximum heating COP can be achieved at optimal pressure ratio. The formulae of optimal pressure ratio and COP were in the same expression as the basic cycle when we applied the equivalent temperature ratio and equivalent isentropic efficiency of expander. It's proved that in theory the regenerated air heat pump cycle can also make the heating capacity in line with the heating load at optimal pressure ratio. Meanwhile, the heating COP of regenerated cycle could be more than 20% higher than that of basic cycle. These indicate the regenerated air cycle has more potentials in heat pump applications. It's proved that a regenerator does not always improve the cycle performance. Larger temperature ratio and lower effectiveness of regenerator could make the regenerated cycle even worse than the basic cycle. The regenerator should be carefully assessed under actual operating conditions and well sized to gain remarkable heating COP increase. This work would enable further study of the regenerated air cycle from a different perspective. References
Fig. 9. Optimal heating COP increase vs. compressor/expander efficiencies.
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