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Thermodynamic modeling and performance analysis of the variable-temperature heat reservoir absorption heat pump cycle
Physica A xx (xxxx) xxx–xxx
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Physica A journal homepage: www.elsevier.com/locate/physa
Thermodynamic modeling and performance analysis of the variable-temperature heat reservoir absorption heat pump cycle Q1
Xiaoyong Qin, Lingen Chen, Yanlin Ge ∗ , Fengrui Sun Institute of Thermal Science and Power Engineering, Naval University of Engineering, Wuhan 430033, PR China Military Key Laboratory for Naval Ship Power Engineering, Naval University of Engineering, Wuhan 430033, PR China College of Power Engineering, Naval University of Engineering, Wuhan 430033, PR China
highlights • • • •
An irreversible absorption heat pump cycle model is established. It is a variable-temperature heat reservoir four-temperature-level cycle. General relationship between heating load and COP is derived. Optimal performance characteristics between heating load and COP are obtained.
article
info
Article history: Received 31 January 2015 Received in revised form 27 April 2015 Available online xxxx Keywords: Finite-time-thermodynamics Absorption heat pump Variable-temperature heat reservoir Four-temperature-level
abstract For practical absorption heat pump (AHP) plants, not all external heat reservoir heat capacities are infinite. External heat reservoir heat capacity should be an effect factor in modeling and performance analysis of AHP cycles. A variable-temperature heat reservoir AHP cycle is modeled, in which internal working substance is working in four temperature levels and all irreversibility factors are considered. The irreversibility includes heat transfer irreversibility, internal dissipation irreversibility and heat leakage irreversibility. The general equations among coefficient of performance (COP), heating load and some key characteristic parameters are obtained. The general and optimal characteristics are obtained by using numerical calculations. Besides, the influences of heat capacities of heat reservoirs, internal dissipation irreversibility, and heat leakage irreversibility on cycle performance are analyzed. The conclusions can offer some guidelines for design and operation of AHP plants. © 2015 Published by Elsevier B.V.
1. Introduction Many low grade heats, for example geothermal energy, solar energy, discharged heat from various enterprises, etc., exist in our surroundings. AHP (the type I absorption heat pump) can utilize these heats, and at the same time, AHP can use environment friendly working substance. Thus, AHPs have active function for decreasing the environment pollution introduced by cycle working substance. Recent 20 years, many scholars have developed many researches about AHP for industrial uses [1–3].
∗ Corresponding author at: College of Power Engineering, Naval University of Engineering, Wuhan 430033, PR China. Tel.: +86 27 83615046; fax: +86 27 83638709. E-mail addresses: [email protected], [email protected] (Y. Ge). http://dx.doi.org/10.1016/j.physa.2015.05.081 0378-4371/© 2015 Published by Elsevier B.V.
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Nomenclature A total heat transfer surface area of all heat exchangers, m2 Ai (i = a, c , e, g ) heat transfer surface area of heat exchanger, m2 a the distribution ratio of heat rejection rate Ci (i = a, c , e, g ) heat capacity of heat reservoir, kW K−1 Ei (i = a, c , e, g ) effectiveness of heat exchanger I internal irreversibility factor Kil (i = a, c , e, g ) heat leakage coefficient of heat reservoir, kW K−1 Qi (i = a, c ) heat addition rate of heat reservoir, kW Qi (i = e, g ) heat rejection rate of heat reservoir, kW Qil (i = a, c , e, g ) heat leakage rate between heat reservoir and surrounding, kW Qi′ (i = a, c , e, g ) heat exchange rate between heat reservoir and internal working substance, kW Tiin (i = a, c , e, g ) inlet temperature of external heat reservoir, K Tiout (i = a, c , e, g ) outlet temperature of external heat reservoir, K Ti′ (i = a, c , e, g ) internal working substance temperature in heat exchanger, K Ui (i = a, c , e, g ) heat transfer coefficient, kW m−2 K−1 UA total heat exchanger inventory of all heat exchangers, kW K−1 Greek symbols
Π ψ
heating load, kW COP
Subscripts a c e g max s
ψ
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
absorber condenser evaporator generator maximum surrounding at maximum COP
The performance of AHP can be analyzed by classical thermodynamic and finite time thermodynamics [4–23]. Some new results have been obtained by using finite time thermodynamics analysis, which are not obtained or inconsistent with the results by using classical thermodynamic analysis. In accordance with endoreversible three-heat-reservoir (THR) model, Chen and Andresen [24] discussed performances of AHP cycles on Newtonian heat transfer law (Q ∝ (1T )). In accordance with irreversible THR model, Goktun [25], Lin and Yan [26], Wu et al. [27], and Ngouateu and Wouagfack [28] discussed characteristics of AHP cycles on Newtonian heat transfer law (Q ∝ (1T )). In accordance with endoreversible THR model, Su and Yan [29] discussed characteristics of AHP cycles on heat transfer law of Q ∝ (1T −1 ). The THR cycle model assumes that the external heat reservoir temperature and internal working substance temperature are the same in the condenser and absorber. But, internal working substance temperature cannot be the same or the external heat reservoir temperature cannot be the same in the condenser and absorber, in fact. Therefore, a model assumed that the absorber working substance temperature can be different to the condenser working substance is closer to an actual AHP cycle, which is called four-temperature-level (FTL) AHP cycle model [30–37]. In accordance with a FTL endoreversible cycle model, Qin et al. [30] and Ngouateu and Tchinda [31] analyzed characteristics of AHP cycles based on Newtonian heat transfer law. In accordance with a FTL irreversible cycle model, Chen [32], Huang et al. [33], Chen et al. [34], and Zhao et al. [35] analyzed performances of AHP cycles based on Newtonian heat transfer law. In accordance with four-temperature-level cycle models, Qin et al. [36,37] analyzed characteristics of AHP cycles based on heat transfer law of Q ∝ ∆(T n ). For many thermal energy systems, the heat reservoir heat capacities cannot all be infinite and the heat reservoir temperatures cannot all be constants (variable-temperature heat reservoirs). But, almost all of these studies [24–37] on AHP mentioned above were assumed that all heat reservoir heat capacities are infinite (constant-temperature heat reservoirs). Not all heat reservoir heat capacities of practical AHP plants are finite, too. The effects of heat capacities of heat reservoirs should be taken into account in finite time thermodynamic modeling and performance analyses for AHP cycles. Based on these achievements mentioned above [24–37], a variable-temperature heat reservoir irreversible AHP cycle model coupled to FTL will be established in this paper. The irreversibility induced by finite-rate heat transfer lied in the external heat reservoir and the internal working substance, the irreversibility induced by heat leakage losses between surroundings and the external heat reservoirs, and the irreversibility induced by internal working substance dissipation will be considered. The general equations including coefficient of performance (COP), heating load and some key irreversibility parameters of
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Fig. 1. A variable-temperature heat reservoir irreversible AHP model.
AHP cycles will be deduced based on this model. Using numerical calculations, the general and optimal performances among the COP, the heating load and key irreversibility parameters will also be gained. Furthermore, the influences of heat reservoir heat capacity, heat leakage irreversibility and internal irreversibility on cycle performance will be analyzed. 2. Physical model As shown in Fig. 1(a), a variable-temperature heat reservoir irreversible AHP cycle includes one absorber, one condenser, one evaporator and one generator. All four external heat reservoir heat capacities are finite and the internal cycle working substance is working in four different temperature levels. Heat is exchanged through four heat exchangers between the external heat reservoir and the internal working substance, respectively. The inlet and outlet temperatures of external heat reservoirs in absorber, condenser, evaporator and generator are Tain and Taout , Tcin and Tcout , Tein and Teout , as well as Tgin and Tgout , respectively, as shown in Fig. 1(b). Absorber, condenser, evaporator and generator heat reservoir heat capacities are Ca , Cc , Ce and Cg , respectively. Heat reservoirs exchange heat with internal working substance at temperature Ta′ , Tc′ , Te′ and
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Tg′ in different heat exchangers. Absorber, condenser, evaporator and generator heat transfer surface area, heat exchanger effectiveness, heat transfer coefficient and heat exchange rate are Aa , Ea , Ua and Qa′ , Ac , Ec , Uc and Qc′ , Ae , Ee , Ue and Qe′ , as well as Ag , Eg , Ug and Qg′ , respectively. In four external heat exchangers, the internal working substance temperature is invariable and the external heat reservoir temperature is variable, and one has
6
Qg′ = Cg Eg (Tgin − Tg′ )
(1)
7
Qa′ = Ca Ea (Ta′ − Tain )
(2)
8
Qc = Cc Ec (Tc −
)
(3)
− Te )
(4)
9
10 11 12 13 14
′
′
′
Qe =
(
Ce Ee Tein
Tcin ′
where Eg = 1 − exp(−Ug Ag /Cg ), Ea = 1 − exp(−Ua Aa /Ca ), Ec = 1 − exp(−Uc Ac /Cc ) and Ee = 1 − exp(−Ue Ae /Ce ). Since the heat reservoir temperature can be more than or less than the surrounding temperature, the heat transfer (the heat leakage) losses between surrounding and the heat reservoir are taken into account in the model. It is assumed that the generator, absorber, condenser and evaporator heat reservoir heat leakage rates Qgl , Qal , Qcl and Qel are related to the difference between the external heat reservoir inlet temperature and surrounding temperature. Thus,
15
Qgl = Kgl (Tgin − Ts )
(5)
16
Qal = Kal (Tain − Ts )
(6)
17
Qcl
(
− Ts )
(7)
18
Qel = Kel (Ts − Tein )
(8)
19 20 21 22 23 24 25 26
27
=
Kcl
Tcin
where Kal , Kcl , Kel and Kgl are absorber, condenser, evaporator and generator heat reservoir heat leakage coefficients, respectively; and Ts is the surrounding temperature. In the generator, the external heat reservoir releases heat to the internal working substance and synchronously releases heat to surroundings. In the evaporator, the external heat reservoir releases heat to the internal working substance and synchronously absorbs heat leakage from surroundings. In the absorber and condenser, the external heat reservoir absorbs heat from the internal working substance and synchronously releases heat leakage to surroundings. Thus, absorber and condenser heat reservoir heat addition rates (Qa and Qc ), and generator and evaporator heat reservoir heat rejection rates (Qg and Qe ) can be written as Qg = Qg′ + Qgl ′
(9)
28
Qa = Qa −
Qal
(10)
29
Qc = Qc′ − Qcl
(11)
30
31 32 33
34
35 36 37 38
39
40 41
′
Qe = Qe −
Qel
.
(12)
As a result of the internal dissipation in the internal working substance cycle, there also exists another source of irreversibility, named as internal irreversibility. One can assume that the internal irreversibility can be represented by an irreversibility factor I [32–34,37] I = (Qa′ /Ta′ + Qc′ /Tc′ )/(Qg′ /Tg′ + Qe′ /Te′ ).
(13)
When internal irreversibility can be ignored, irreversibility factor I = 1. When internal irreversibility cannot be ignored, irreversibility factor I > 1. The internal working substance rejects heat to the external heat reservoir in the absorber and condenser. One can assume that the distribution ratio can be represented by a parameter a a = Qc′ /(Qa′ + Qc′ ).
(14)
COP (ψ ) and Heating load (Π ) are two key parameters for AHP. According to the model described above, ψ and Π can be written as
42
ψ = (Qa + Qc )/Qg
(15)
43
Π = Qa + Qc .
(16)
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3. General COP and heating load characteristic
1
3.1. COP and heating load relation equation
2
Aimed at the internal working substance cycle in the model described above, one has
3
Qa + Qc = Qg + Qe . ′
′
′
′
(17)
Using Eqs. (9)–(12) and (14)–(17), one obtains
5
Qg′ = Π /ψ − Qgl Qa′ = (1 − a)(Π + Qal + Qcl ) Qc = a(Π + ′
Qal
+
Qcl
)
Qe = Π (ψ − 1)/ψ + ′
Qal
+
Qcl
+
Qgl
.
(18)
6
(19)
7
(20)
8
(21)
9
According to Eqs. (1) and (18), (2) and (19), (3) and (20), and (4) and (21), one obtains Tg′ = Tgin − Ta′ = Tain + Tc′ = Tcin + Te′ = Tein −
10
Π − Kgl (Tgin − Ts )ψ Cg Eg ψ
(1 − a)[Π + Kal (Tain − Ts ) + Kcl (Tcin − Ts )] Ca Ea a[Π + Kal (Tain − Ts ) + Kcl (Tcin − Ts )] Cc Ec
Π (ψ − 1) + [Kal (Tain − Ts ) + Kcl (Tcin − Ts ) + Kgl (Tgin − Ts )]ψ Ce Ee ψ
.
(22)
11
(23)
12
(24)
13
(25)
14
Substituting Eqs. (18)–(21) and (22)–(25) into Eq. (13) yields
Tgin
−
−1
+
Tgin
1
Tein
+
Tcin
a[ Π +
Kal
(
Tain
− Ts ) +
Kcl
(
Tcin
15
Π − ( − Ts )ψ Cg Eg ψ [Π + ( − T s ) + −1 ψ Tain 1 − + I (1 − a)[Π + Kal (Tain − Ts ) + Kcl (Tcin − Ts )] Ca Ea −1 Kgl
− Ts )]
4
Kal
+
1
Cc Ec
Tain
= 0.
Kcl
(
Tcin
− Ts ) + Kgl (Tgin − Ts )]ψ − Π
−
1
−1 16
Ce Ee ψ
17
(26)
Eq. (26) is a general equation including the COP, the heating load and some key irreversibility parameters of variabletemperature heat reservoir irreversible AHP coupled to a FTL cycle. Eq. (26) may be used directly to analyze the general characteristic and the effect of key irreversibility parameters on cycle characteristic of a practical AHP. 3.2. COP and heating load characteristic To analyze the performance of AHP, one assumes that a = 0.45, Tain = 325 K, Tcin = 330 K, Tein = 300 K, Tgin = 380 K,
Ts = 300 K, Ua = Ug = 0.5 kw m−2 K−1 , Uc = Ue = 1.0 kw m−2 K−1 , Ca = Cc = Ce = Cg = 50.0 kW K−1 , Aa = Ac = Ae = Ag = 100 m2 , Ea = Eg = 0.6321 and Ec = Ee = 0.8647. Using Eq. (26), the characteristic curves among the heating load, the COP and some key irreversibility parameters are shown in Fig. 2. When heat leakages and internal irreversibility are ignored, the characteristic is curve 1 in Fig. 2. When heat leakages are ignored, the characteristic is curve 2 in Fig. 2. When internal irreversibility is ignored, the characteristic is curve 3 in Fig. 2. When internal irreversibility and heat leakages with different values are considered, the characteristics are curves 4–6 in Fig. 2. One can see that the COP monotonically decreases as the heating load increases, according to the curves 1 and 2 in Fig. 2. At these situations, the irreversible factors considered include heat resistance and internal irreversibility, but do not include heat leakage. From the curves 3–6 in Fig. 2, one can see that COP does not monotonically decrease as the heating load increases. At these situations, the irreversible factors considered include heat leakage and other factors. Thus, the heat leakage should be considered in an irreversible AHP cycle model. From the curve 4 in Fig. 2, one can see that a maximum COP (ψmax ) exists in the characteristic curves of AHP cycles. This reflects the true performance characteristic of a practical
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Fig. 2. The heating load, the COP and key irreversibility parameters characteristic curves.
Fig. 3. Characteristic curves before and after optimizing distribution A and UA.
2
AHP cycle. From the curves 1–6 in Fig. 2, one can also see that the performance of AHP decreases as any irreversibility factor increases.
3
4. Optimal characteristic
4
4.1. Optimal distribution of A
1
5 6 7 8 9 10 11 12 13 14 15 16 17 18
Absorber, condenser, evaporator and generator heat transfer surface areas Aa , Ac , Ae and Ag are key factors to decide the cost of AHP. The total area A is A = Aa + Ac + Ae + Ag .
(27)
If A = constant, optimizing the distribution of Aa , Ac , Ae and Ag , one can find the optimal cycle performance. When the heating load is selected, one can take COP as the optimization objective function. When COP is selected, one can take the heating load as the optimization objective function. With the constraint A = constant, using the Powell search method [38] and numerical algorithm, Fig. 3 shows the characteristic relationship curves before and after optimizing distribution A of an AHP cycle. In calculations, the parameters are the same as Section 3.2, and I = 1.01 and Kgl = Kal = Kcl = Kel = 0.1 kW K−1 are set. From Fig. 3, one can see that performances after optimizing distribution A are better than before optimizing. Heat capacities Ca , Cc , Ce and Cg can affect cycle performances. In order to discuss influences of heat capacities, the optimal Π versus Ca , Cc , Ce and Cg curves when ψ = 1.42 are shown in Fig. 4, the optimal ψ versus Ca , Cc , Ce and Cg curves when Π = 710 kW are shown in Fig. 5. From Figs. 4 and 5, one can see that the optimal performances increase as Cg , Ca , Cc and Ce increase, and the increasing degree decreases as Cg , Ca , Cc and Ce increase. These show that the cycle can
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Fig. 4. Relationship curves of the optimal heating load versus heat capacities.
Fig. 5. Relationship curves of the optimal COP versus heat capacities.
be ameliorated by increasing any heat reservoir heat capacities, but one also should not pursue the maximum performance and utilize quite large heat capacity excessively.
1 2
4.2. Optimal distribution of UA Absorber, condenser, evaporator and generator heat exchanger inventories Ua Aa , Uc Ac , Ue Ae and Ug Ag are key factors to decide the performance of AHP. If the total heat exchanger inventory (UA = Ug Ag + Ua Aa + Uc Ac + Ue Ae ) is fixed, optimizing distribution UA can also be carried out [39–43]. Using UA = constant constraint to replace A = constant constraint, the optimal performance of AHP after optimizing distribution of UA can be obtained. The optimal performance after optimizing distribution of UA is shown in Fig. 3, which shows the performances after optimizing distribution of UA are better than before optimizing. 5. Special cases The finite-time-thermodynamics model of the heat pump cycle includes two heat reservoir, three heat reservoir and fourtemperature-level endoreversible and irreversible cycle models, coupled to infinite and finite heat capacity heat reservoirs. Eq. (26) is universal, which includes the results of all above heat pump cycle models.
3
Q2
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2
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5.1. Case 1: I = 1, Kgl = Kal = Kcl = Kel = 0 If internal irreversibility and heat leakage losses are ignored, i.e., I = 1, Kgl = Kal = Kcl = Kel = 0, Eq. (26) becomes
3
Tgin
−
Π
1
−1
+
Cg Eg ψ
Tein
−
Π (ψ − 1)
−1
1 Ce Ee ψ
−ψ
Tain
(1 − a)Π
+
1
−1 +
Ca Ea
Tcin
aΠ
+
1
−1
Cc Ec
= 0.
(28)
5
Eq. (28) is the expression including the COP, heating load and key characteristic parameters based on a variabletemperature heat reservoir FTL endoreversible AHP cycle model.
6
5.2. Case 2: Cg → ∞, Ca → ∞, Cc → ∞, Ce → ∞
4
7 8
When heat capacities of heat reservoirs are infinite (Cg → ∞, Ca → ∞, Cc → ∞ and Ce → ∞), heat reservoirs can be assumed as constant-temperature reservoirs, Eq. (26) becomes
10
14 15
16
17
19
20 21
a[Π + Kal (Ta − Ts ) + Kcl (Tc − Ts )]
Tg
1
−
Π
26
29 30
+
1
−1
= 0.
Uc Ac
(29)
+
Te
Π (ψ − 1)
−
−1
1 Ue Ae ψ
−ψ
Ta
(1 − a)Π
+
1
−1
Ua Aa
+
Tc aΠ
+
1
−1
Uc Ac
= 0.
(30)
In this case, the absorber and condenser can be simplified to one heat exchanger. The AHP cycle can be modeled by a THR model, then Eq. (26) becomes Tgin
Π − Kgl (Tgin − Ts )ψ
−
ψ
[Π +
l Kac
+
Cg Eg ψ
in Tac
I
−
−1
1
(Tacin − Ts )]
+
1
Tein
[Π + Kacl (Tacin − Ts ) + Kgl (Tgin − Ts )]ψ − Π −1
Cac Eac
= 0.
−
1
−1
Ce Ee ψ (31)
Eq. (31) is the expression including the COP, heating load and key characteristic parameters based on a variabletemperature THR irreversible AHP cycle model. Moreover, when internal irreversibility and heat leakage losses are ignored (I = 1 and Kgl = Kal = Kcl = Kel = 0), Eq. (31) becomes
28
−1
in l 5.3. Case 3: a = 1/2, Tain = Tcin = Tac , Ea = Ec = Eac , Ca = Cc = Cac /2, Kal = Kcl = Kac /2
23
27
1
Eq. (30) is the expression including the COP, heating load and key characteristic parameters based on a constanttemperature heat reservoir FTL endoreversible AHP cycle model [33,34].
22
24
−1
Ug Ag ψ
25
Te
Eq. (29) is the expression including the COP, heating load and key characteristic parameters based on a constanttemperature heat source FTL irreversible AHP cycle model [34]. Moreover, when internal irreversibility and heat leakage losses are ignored (I = 1 and Kgl = Kal = Kcl = Kel = 0), Eq. (29) becomes
18
Tc
+
11
12
−1
1
− + − Π − Kgl (Tg − Ts )ψ Ug Ag ψ [Π + Kal (Ta − Ts ) + Kcl (Tc − Ts ) + Kgl (Tg − Ts )]ψ − Π Ue Ae ψ −1 1 ψ Ta − + I (1 − a)[Π + Kal (Ta − Ts ) + Kcl (Tc − Ts )] Ua Aa
9
13
Tg
Tgin
Π
−
1 Cg Eg ψ
−1
+
Tein
Π (ψ − 1)
−
1 Ce Ee ψ
−1
−ψ
in Tac
Π
+
1 Cac Eac
−1
= 0.
(32)
Eq. (32) is the expression including the COP, heating load and key characteristic parameters based on a variabletemperature three heat reservoir THR endoreversible AHP cycle.
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in 5.4. Case 4: Tain = Tcin = Tac , Cg → ∞, Ca → ∞, Cc → ∞, Ce → ∞, Uac = Ua = Uc , Aac = 2Aa = 2Ac , a = 1/2
1
In this case, heat capacities of heat reservoirs are infinite (Cg → ∞, Ca → ∞, Cc → ∞ and Ce → ∞), heat reservoirs can be assumed as a constant-temperature reservoir. The absorber and condenser can be simplified to one heat exchanger, the AHP cycle can be modeled by THR cycle model. Eq. (26) becomes
1
Tg
−1
Te
1
− + − Π − Kgl (Tg − Ts )ψ Ug Ag ψ [Π + Kacl (Tac − Ts ) + Kgl (Tg − Ts )]ψ − Π Ue Ae ψ −1 Tac 1 ψ + = 0. − I [Π + Kacl (Tac − Ts )] Uac Aac
Tg
Π
−
1 Ug Ag ψ
−1
+
Te
Π (ψ − 1)
−
1 Ue Ae ψ
−1
−ψ
Tac
Π
+
1
−1
Uac Aac
= 0.
3 4
−1 5
(33)
Eq. (33) is the expression including the COP, heating load and key characteristic parameters based on a constanttemperature THR irreversible AHP cycle model [34]. Moreover, when internal irreversibility and heat leakage losses are ignored (I = 1 and Kgl = Kal = Kcl = Kel = 0), Eq. (33) becomes
2
(34)
Eq. (34) is the expression including the COP, heating load and key characteristic parameters based on a constanttemperature THR endoreversible AHP cycle model [33,34]. in 5.5. Case 5: Tgin → ∞, a = 1/2, Tain = Tcin = Tac , Ea = Ec = Eac , Ca = Cc = Cac /2
In this case, the model becomes a variable-temperature two heat reservoir (Carnot) irreversible heat pump model. Moreover, if I = 1 and Kgl = Kal = Kcl = Kel = 0, the model becomes a variable-temperature two heat reservoir (Carnot) endoreversible heat pump model. in 5.6. Case 6: Tgin → ∞, a = 1/2, Tain = Tcin = Tac , Cg → ∞, Ca → ∞, Cc → ∞, Ce → ∞, Uac = Ua = Uc , Aac = 2Aa = 2Ac
In this case, the model becomes a constant-temperature two heat reservoir (Carnot) irreversible heat pump model. Moreover, if I = 1 and Kgl = Kal = Kcl = Kel = 0, the model becomes a constant-temperature two heat reservoir endoreversible heat pump model. 6. Conclusion A variable-temperature heat reservoir model is closer to a practical AHP cycle, in which irreversibility losses include heat resistance losses, internal working substance dissipation losses, and heat leakage losses between heat reservoir and surrounding. In accordance with this irreversible AHP cycle model, the general characteristic about COP versus heating load has a maximum COP value (ψmax ) with the corresponding heating load (Πψ ), and this reflects the true performance characteristic of a practical AHP. The finite heat capacity of heat reservoirs, heat resistance losses, internal irreversibility loss and heat leakage losses must be considered in modeling and quantitative analysis of AHP cycles. COP and heating load can be increased by optimizing distribution heat exchanger surface area or heat exchanger inventory. When any heat reservoir heat capacity increases, the cycle performance is improved, but the increment decreases as any heat reservoir heat capacity increases. Eq. (26) of this paper is universal, which includes the results of all heat pump cycles (two, three, and four heat reservoir; endoreversible and irreversible; infinite and finite heat capacity heat reservoirs). Therefore, the results obtained herein are useful and universal, and can supply some guidelines for the design and operation of a practical AHP, whether the heat reservoir is variable-temperature or not. Acknowledgments This paper is supported by The National Natural Science Foundation of P. R. China (Project No. 10905093). The authors wish to thank the reviewers for their careful, unbiased and constructive suggestions, which led to this revised manuscript.
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37 38 39
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References [1] [2] [3] [4]
7
[5] [6]
8
[7]
9
14
[8] [9] [10] [11] [12] [13]
15
[14]
16
[15]
17
[16]
18
21
[17] [18] [19] [20]
22
[21]
23
[22]
24
[23]
25
[24]
26 27
[25] [26]
28
[27]
29
[28]
30
[29]
31
[30]
32
[31]
33
[32]
34
[33]
35 37
[34] [35] [36]
38
[37]
39
[38] [39]
6
10 11 12 13
19 20
36
40 41 42 43 44
X. Qin et al. / Physica A xx (xxxx) xxx–xxx
[40] [41] [42] [43]
F. Ziegler, State of the art in sorption heat pumping and cooling technologies, Int. J. Refrig. 25 (4) (2002) 450–459. D.S. Kim, C.A.I. Ferreira, Solar refrigeration options—a state-of-the-art review, Int. J. Refrig. 31 (1) (2008) 3–15. M.U. Siddiqui, S.A.M. Said, A review of solar powered absorption systems, Renewable Sustainable Energy Rev. 42 (2015) 93–115. A. Bejan, Entropy generation minimization: The new thermodynamics of finite-size device and finite-time processes, J. Appl. Phys. 79 (3) (1996) 1191–1218. R.S. Berry, V.A. Kazakov, S. Sieniutycz, Z. Szwast, A.M. Tsirlin, Thermodynamic Optimization of Finite Time Processes, Wiley, Chichester, 1999. L.G. Chen, C. Wu, F.R. Sun, Finite time thermodynamic optimization or entropy generation minimization of energy systems, J. Non-Equilib. Thermodyn. 24 (4) (1999) 327–359. A. Durmayaz, O.S. Sogut, B. Sahin, H. Yavuz, Optimization of thermal systems based on finite-time thermodynamics and thermoeconomics, Prog. Energy Combust. Sci. 30 (2) (2004) 175–217. L.G. Chen, F.R. Sun, Advances in Finite Time Thermodynamics: Analysis and Optimization, Nova Science Publishers, New York, 2004. L.G. Chen, Finite Time Thermodynamic Analysis of Irreversible Processes and Cycles, Higher Education Press, Beijing, 2005, (in Chinese). M. Feidt, Optimal use of energy systems and processes, Int. J. Exergy 5 (5–6) (2008) 500–531. M. Feidt, Thermodynamics applied to reverse cycle machines, a review, Int. J. Refrig. 33 (7) (2010) 1327–1342. B. Andresen, Current trends in finite-time thermodynamic, Angew. Chem. Int. Ed. 50 (12) (2011) 2690–2704. M. Feidt, Evolution of thermodynamic modelling for three and four heat reservoirs reverse cycle machines: A review and new trends, Int. J. Refrig. 36 (1) (2013) 8–23. X.Y. Qin, L.G. Chen, Y.L. Ge, F.R. Sun, Finite time thermodynamic studies on absorption thermodynamic cycles: A state of the arts review, Arab. J. Sci. Eng. 38 (3) (2013) 405–419. P.A. Ngouateu Wouagfack, R. Tchinda, Finite-time thermodynamics optimization of absorption refrigeration systems: A review, Renewable Sustainable Energy Rev. 21 (2013) 524–536. E. Açıkkalp, Modified thermo-ecological optimization for refrigeration systems and an application for irreversible four-temperature-level absorption refrigerator, Int. J. Energy Environ. Eng. 4 (1) (2013) 20. E. Açıkkalp, Models for optimum thermo-ecological criteria of actual thermal cycles, Therm. Sci. 17 (3) (2013) 915–930. E. Açıkkalp, H. Yamik, Limits and optimization of power input or output of actual thermal cycles, Entropy 15 (8) (2013) 3309–3338. S. Sieniutycz, J. Jezowski, Energy Optimization in Process Systems and Fuel Cells, Elsevier, Oxford, UK, 2013. E. Açıkkalp, Entransy analysis of irreversible heat pump using Newton and Dulong–Petit heat transfer laws and relations with its performance, Energy Convers. Manage. 86 (2014) 792–800. K.H. Hoffmann, B. Andresen, P. Salamon, Finite-time thermodynamics tools to analyze dissipative processes, in: A.R. Dinner (Ed.), Proceedings of The 240 Conference: Science’s Great Challences, in: Advances in Chemical Physics, vol. 157, Wiley, 2015, pp. 57–67. E. Açıkkalp, N. Caner, Determining performance of an irreversible nano scale dual cycle operating with Maxwell–Boltzmann gas, Physica A 424 (2015) 342–349. E. Açıkkalp, H. Yamik, Modeling and optimization of maximum available work for irreversible gas power cycles with temperature dependent specific heat, J. Non-Equilib. Thermodyn. 40 (2015) 25–39. J. Chen, B. Andresen, Optimal analysis of primary performance parameters for an endoreversible absorption heat pump, Heat Recovery Syst. CHP 15 (8) (1995) 723–731. S. Goktun, The effect of irreversibilities on the performance of a three-heat-source heat pump, J. Phys. D: Appl. Phys. 29 (20) (1996) 2823–2825. G. Lin, Z. Yan, Optimization of erergy loss rate in an irreversible three-heat-source heat pump, Acta Energiae Solaris Sin. 22 (1) (2001) 107–110. (in Chinese). S. Wu, G. Lin, J. Chen, Optimum thermoeconomic and thermodynamic performance characteristics of an irreversible three-heat-source heat pump, Renew. Energy 30 (15) (2005) 2257–2271. P.A. Ngouateu Wouagfack, R. Tchinda, The new thermo-ecological performance optimization of an irreversible three-heat-source absorption heat pump, Int. J. Refrig. 35 (1) (2012) 79–87. G. Su, Z. Yan, The optimal ecological performance of three-heat-source heat pump with linear phenomenological heat transfer law, Acta Energiae Solaris Sin. 20 (2) (1999) 204–208. (in Chinese). X.Y. Qin, L.G. Chen, F.R. Sun, C. Wu, Compromise optimization between heating load and entropy production rate for endoreversible absorption heat pumps, Int. J. Ambient Energy 26 (2) (2005) 106–112. P.A. Ngouateu Wouagfack, R. Tchinda, Optimal ecological performance of a four-temperature-level absorption heat pump, Int. J. Therm. Sci. 54 (2012) 209–219. J. Chen, The general performance characteristics of an irreversible absorption heat pump operating between four temperature levels, J. Phys. D: Appl. Phys. 32 (12) (1999) 1428–1433. Y. Huang, D. Sun, Y. Kang, Performance optimization for an irreversible four-temperature-level absorption heat pump, Int. J. Therm. Sci. 47 (4) (2008) 479–485. L.G. Chen, X.Y. Qin, F.R. Sun, C. Wu, Irreversible absorption heat-pump and its optimal performance, Appl. Energy 81 (1) (2005) 55–71. X. Zhao, L. Fu, S. Zhang, General thermodynamic performance of irreversible absorption heat pump, Energy Convers. Manage. 52 (1) (2011) 494–499. X.Y. Qin, L.G. Chen, F.R. Sun, C. Wu, Performance of an endoreversible four-heat-reservoir absorption heat pump with a generalized heat transfer law, Int. J. Therm. Sci. 45 (6) (2006) 627–633. X.Y. Qin, L.G. Chen, F.R. Sun, C. Wu, Model of real absorption heat pump cycle with a generalized heat transfer law and its performance, Proc. IMechE Part A: Power Energy 221 (7) (2007) 907–916. M.J.D. Powell, Nonlinear Optimization, Academic Press, New York, 1982. A. Bejan, J.V.C. Vargas, Optimal allocation of a heat exchanger inventory in heat driven refrigerators, Int. J. Heat Mass Transfer 38 (16) (1995) 2997–3004. M. Sokolov, J.V.C. Vargas, A. Bejan, Thermodynamic optimization of solar-driven refrigerators, Trans. ASME J. Sol. Energy Eng. 118 (2) (1996) 130–135. S.A. Klein, D.T. Reindl, The relationship of optimum heat exchanger allocation and minimum entropy generation rate for refrigeration cycles, Trans. ASME J. Energy Res. Technol. 120 (2) (1998) 172–178. M. Feidt, M. Costea, C. Petre, S. Petrescu, Optimization of direct Carnot cycle, Appl. Therm. Eng. 27 (5–6) (2007) 829–839. J. Sarkar, S. Bhattacharyya, Overall conductance and heat transfer area minimization of refrigerators and heat pumps with finite heat sources, Energy Convers. Manage. 48 (3) (2007) 803–808.
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Thermodynamic modeling and performance analysis of the variable-temperature heat reservoir absorption heat pump cycle Q1
Xiaoyong Qin, Lingen Chen, Yanlin Ge ∗ , Fengrui Sun Institute of Thermal Science and Power Engineering, Naval University of Engineering, Wuhan 430033, PR China Military Key Laboratory for Naval Ship Power Engineering, Naval University of Engineering, Wuhan 430033, PR China College of Power Engineering, Naval University of Engineering, Wuhan 430033, PR China
highlights • • • •
An irreversible absorption heat pump cycle model is established. It is a variable-temperature heat reservoir four-temperature-level cycle. General relationship between heating load and COP is derived. Optimal performance characteristics between heating load and COP are obtained.
article
info
Article history: Received 31 January 2015 Received in revised form 27 April 2015 Available online xxxx Keywords: Finite-time-thermodynamics Absorption heat pump Variable-temperature heat reservoir Four-temperature-level
abstract For practical absorption heat pump (AHP) plants, not all external heat reservoir heat capacities are infinite. External heat reservoir heat capacity should be an effect factor in modeling and performance analysis of AHP cycles. A variable-temperature heat reservoir AHP cycle is modeled, in which internal working substance is working in four temperature levels and all irreversibility factors are considered. The irreversibility includes heat transfer irreversibility, internal dissipation irreversibility and heat leakage irreversibility. The general equations among coefficient of performance (COP), heating load and some key characteristic parameters are obtained. The general and optimal characteristics are obtained by using numerical calculations. Besides, the influences of heat capacities of heat reservoirs, internal dissipation irreversibility, and heat leakage irreversibility on cycle performance are analyzed. The conclusions can offer some guidelines for design and operation of AHP plants. © 2015 Published by Elsevier B.V.
1. Introduction Many low grade heats, for example geothermal energy, solar energy, discharged heat from various enterprises, etc., exist in our surroundings. AHP (the type I absorption heat pump) can utilize these heats, and at the same time, AHP can use environment friendly working substance. Thus, AHPs have active function for decreasing the environment pollution introduced by cycle working substance. Recent 20 years, many scholars have developed many researches about AHP for industrial uses [1–3].
∗ Corresponding author at: College of Power Engineering, Naval University of Engineering, Wuhan 430033, PR China. Tel.: +86 27 83615046; fax: +86 27 83638709. E-mail addresses: [email protected], [email protected] (Y. Ge). http://dx.doi.org/10.1016/j.physa.2015.05.081 0378-4371/© 2015 Published by Elsevier B.V.
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Nomenclature A total heat transfer surface area of all heat exchangers, m2 Ai (i = a, c , e, g ) heat transfer surface area of heat exchanger, m2 a the distribution ratio of heat rejection rate Ci (i = a, c , e, g ) heat capacity of heat reservoir, kW K−1 Ei (i = a, c , e, g ) effectiveness of heat exchanger I internal irreversibility factor Kil (i = a, c , e, g ) heat leakage coefficient of heat reservoir, kW K−1 Qi (i = a, c ) heat addition rate of heat reservoir, kW Qi (i = e, g ) heat rejection rate of heat reservoir, kW Qil (i = a, c , e, g ) heat leakage rate between heat reservoir and surrounding, kW Qi′ (i = a, c , e, g ) heat exchange rate between heat reservoir and internal working substance, kW Tiin (i = a, c , e, g ) inlet temperature of external heat reservoir, K Tiout (i = a, c , e, g ) outlet temperature of external heat reservoir, K Ti′ (i = a, c , e, g ) internal working substance temperature in heat exchanger, K Ui (i = a, c , e, g ) heat transfer coefficient, kW m−2 K−1 UA total heat exchanger inventory of all heat exchangers, kW K−1 Greek symbols
Π ψ
heating load, kW COP
Subscripts a c e g max s
ψ
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
absorber condenser evaporator generator maximum surrounding at maximum COP
The performance of AHP can be analyzed by classical thermodynamic and finite time thermodynamics [4–23]. Some new results have been obtained by using finite time thermodynamics analysis, which are not obtained or inconsistent with the results by using classical thermodynamic analysis. In accordance with endoreversible three-heat-reservoir (THR) model, Chen and Andresen [24] discussed performances of AHP cycles on Newtonian heat transfer law (Q ∝ (1T )). In accordance with irreversible THR model, Goktun [25], Lin and Yan [26], Wu et al. [27], and Ngouateu and Wouagfack [28] discussed characteristics of AHP cycles on Newtonian heat transfer law (Q ∝ (1T )). In accordance with endoreversible THR model, Su and Yan [29] discussed characteristics of AHP cycles on heat transfer law of Q ∝ (1T −1 ). The THR cycle model assumes that the external heat reservoir temperature and internal working substance temperature are the same in the condenser and absorber. But, internal working substance temperature cannot be the same or the external heat reservoir temperature cannot be the same in the condenser and absorber, in fact. Therefore, a model assumed that the absorber working substance temperature can be different to the condenser working substance is closer to an actual AHP cycle, which is called four-temperature-level (FTL) AHP cycle model [30–37]. In accordance with a FTL endoreversible cycle model, Qin et al. [30] and Ngouateu and Tchinda [31] analyzed characteristics of AHP cycles based on Newtonian heat transfer law. In accordance with a FTL irreversible cycle model, Chen [32], Huang et al. [33], Chen et al. [34], and Zhao et al. [35] analyzed performances of AHP cycles based on Newtonian heat transfer law. In accordance with four-temperature-level cycle models, Qin et al. [36,37] analyzed characteristics of AHP cycles based on heat transfer law of Q ∝ ∆(T n ). For many thermal energy systems, the heat reservoir heat capacities cannot all be infinite and the heat reservoir temperatures cannot all be constants (variable-temperature heat reservoirs). But, almost all of these studies [24–37] on AHP mentioned above were assumed that all heat reservoir heat capacities are infinite (constant-temperature heat reservoirs). Not all heat reservoir heat capacities of practical AHP plants are finite, too. The effects of heat capacities of heat reservoirs should be taken into account in finite time thermodynamic modeling and performance analyses for AHP cycles. Based on these achievements mentioned above [24–37], a variable-temperature heat reservoir irreversible AHP cycle model coupled to FTL will be established in this paper. The irreversibility induced by finite-rate heat transfer lied in the external heat reservoir and the internal working substance, the irreversibility induced by heat leakage losses between surroundings and the external heat reservoirs, and the irreversibility induced by internal working substance dissipation will be considered. The general equations including coefficient of performance (COP), heating load and some key irreversibility parameters of
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3
Fig. 1. A variable-temperature heat reservoir irreversible AHP model.
AHP cycles will be deduced based on this model. Using numerical calculations, the general and optimal performances among the COP, the heating load and key irreversibility parameters will also be gained. Furthermore, the influences of heat reservoir heat capacity, heat leakage irreversibility and internal irreversibility on cycle performance will be analyzed. 2. Physical model As shown in Fig. 1(a), a variable-temperature heat reservoir irreversible AHP cycle includes one absorber, one condenser, one evaporator and one generator. All four external heat reservoir heat capacities are finite and the internal cycle working substance is working in four different temperature levels. Heat is exchanged through four heat exchangers between the external heat reservoir and the internal working substance, respectively. The inlet and outlet temperatures of external heat reservoirs in absorber, condenser, evaporator and generator are Tain and Taout , Tcin and Tcout , Tein and Teout , as well as Tgin and Tgout , respectively, as shown in Fig. 1(b). Absorber, condenser, evaporator and generator heat reservoir heat capacities are Ca , Cc , Ce and Cg , respectively. Heat reservoirs exchange heat with internal working substance at temperature Ta′ , Tc′ , Te′ and
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4
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Tg′ in different heat exchangers. Absorber, condenser, evaporator and generator heat transfer surface area, heat exchanger effectiveness, heat transfer coefficient and heat exchange rate are Aa , Ea , Ua and Qa′ , Ac , Ec , Uc and Qc′ , Ae , Ee , Ue and Qe′ , as well as Ag , Eg , Ug and Qg′ , respectively. In four external heat exchangers, the internal working substance temperature is invariable and the external heat reservoir temperature is variable, and one has
6
Qg′ = Cg Eg (Tgin − Tg′ )
(1)
7
Qa′ = Ca Ea (Ta′ − Tain )
(2)
8
Qc = Cc Ec (Tc −
)
(3)
− Te )
(4)
9
10 11 12 13 14
′
′
′
Qe =
(
Ce Ee Tein
Tcin ′
where Eg = 1 − exp(−Ug Ag /Cg ), Ea = 1 − exp(−Ua Aa /Ca ), Ec = 1 − exp(−Uc Ac /Cc ) and Ee = 1 − exp(−Ue Ae /Ce ). Since the heat reservoir temperature can be more than or less than the surrounding temperature, the heat transfer (the heat leakage) losses between surrounding and the heat reservoir are taken into account in the model. It is assumed that the generator, absorber, condenser and evaporator heat reservoir heat leakage rates Qgl , Qal , Qcl and Qel are related to the difference between the external heat reservoir inlet temperature and surrounding temperature. Thus,
15
Qgl = Kgl (Tgin − Ts )
(5)
16
Qal = Kal (Tain − Ts )
(6)
17
Qcl
(
− Ts )
(7)
18
Qel = Kel (Ts − Tein )
(8)
19 20 21 22 23 24 25 26
27
=
Kcl
Tcin
where Kal , Kcl , Kel and Kgl are absorber, condenser, evaporator and generator heat reservoir heat leakage coefficients, respectively; and Ts is the surrounding temperature. In the generator, the external heat reservoir releases heat to the internal working substance and synchronously releases heat to surroundings. In the evaporator, the external heat reservoir releases heat to the internal working substance and synchronously absorbs heat leakage from surroundings. In the absorber and condenser, the external heat reservoir absorbs heat from the internal working substance and synchronously releases heat leakage to surroundings. Thus, absorber and condenser heat reservoir heat addition rates (Qa and Qc ), and generator and evaporator heat reservoir heat rejection rates (Qg and Qe ) can be written as Qg = Qg′ + Qgl ′
(9)
28
Qa = Qa −
Qal
(10)
29
Qc = Qc′ − Qcl
(11)
30
31 32 33
34
35 36 37 38
39
40 41
′
Qe = Qe −
Qel
.
(12)
As a result of the internal dissipation in the internal working substance cycle, there also exists another source of irreversibility, named as internal irreversibility. One can assume that the internal irreversibility can be represented by an irreversibility factor I [32–34,37] I = (Qa′ /Ta′ + Qc′ /Tc′ )/(Qg′ /Tg′ + Qe′ /Te′ ).
(13)
When internal irreversibility can be ignored, irreversibility factor I = 1. When internal irreversibility cannot be ignored, irreversibility factor I > 1. The internal working substance rejects heat to the external heat reservoir in the absorber and condenser. One can assume that the distribution ratio can be represented by a parameter a a = Qc′ /(Qa′ + Qc′ ).
(14)
COP (ψ ) and Heating load (Π ) are two key parameters for AHP. According to the model described above, ψ and Π can be written as
42
ψ = (Qa + Qc )/Qg
(15)
43
Π = Qa + Qc .
(16)
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3. General COP and heating load characteristic
1
3.1. COP and heating load relation equation
2
Aimed at the internal working substance cycle in the model described above, one has
3
Qa + Qc = Qg + Qe . ′
′
′
′
(17)
Using Eqs. (9)–(12) and (14)–(17), one obtains
5
Qg′ = Π /ψ − Qgl Qa′ = (1 − a)(Π + Qal + Qcl ) Qc = a(Π + ′
Qal
+
Qcl
)
Qe = Π (ψ − 1)/ψ + ′
Qal
+
Qcl
+
Qgl
.
(18)
6
(19)
7
(20)
8
(21)
9
According to Eqs. (1) and (18), (2) and (19), (3) and (20), and (4) and (21), one obtains Tg′ = Tgin − Ta′ = Tain + Tc′ = Tcin + Te′ = Tein −
10
Π − Kgl (Tgin − Ts )ψ Cg Eg ψ
(1 − a)[Π + Kal (Tain − Ts ) + Kcl (Tcin − Ts )] Ca Ea a[Π + Kal (Tain − Ts ) + Kcl (Tcin − Ts )] Cc Ec
Π (ψ − 1) + [Kal (Tain − Ts ) + Kcl (Tcin − Ts ) + Kgl (Tgin − Ts )]ψ Ce Ee ψ
.
(22)
11
(23)
12
(24)
13
(25)
14
Substituting Eqs. (18)–(21) and (22)–(25) into Eq. (13) yields
Tgin
−
−1
+
Tgin
1
Tein
+
Tcin
a[ Π +
Kal
(
Tain
− Ts ) +
Kcl
(
Tcin
15
Π − ( − Ts )ψ Cg Eg ψ [Π + ( − T s ) + −1 ψ Tain 1 − + I (1 − a)[Π + Kal (Tain − Ts ) + Kcl (Tcin − Ts )] Ca Ea −1 Kgl
− Ts )]
4
Kal
+
1
Cc Ec
Tain
= 0.
Kcl
(
Tcin
− Ts ) + Kgl (Tgin − Ts )]ψ − Π
−
1
−1 16
Ce Ee ψ
17
(26)
Eq. (26) is a general equation including the COP, the heating load and some key irreversibility parameters of variabletemperature heat reservoir irreversible AHP coupled to a FTL cycle. Eq. (26) may be used directly to analyze the general characteristic and the effect of key irreversibility parameters on cycle characteristic of a practical AHP. 3.2. COP and heating load characteristic To analyze the performance of AHP, one assumes that a = 0.45, Tain = 325 K, Tcin = 330 K, Tein = 300 K, Tgin = 380 K,
Ts = 300 K, Ua = Ug = 0.5 kw m−2 K−1 , Uc = Ue = 1.0 kw m−2 K−1 , Ca = Cc = Ce = Cg = 50.0 kW K−1 , Aa = Ac = Ae = Ag = 100 m2 , Ea = Eg = 0.6321 and Ec = Ee = 0.8647. Using Eq. (26), the characteristic curves among the heating load, the COP and some key irreversibility parameters are shown in Fig. 2. When heat leakages and internal irreversibility are ignored, the characteristic is curve 1 in Fig. 2. When heat leakages are ignored, the characteristic is curve 2 in Fig. 2. When internal irreversibility is ignored, the characteristic is curve 3 in Fig. 2. When internal irreversibility and heat leakages with different values are considered, the characteristics are curves 4–6 in Fig. 2. One can see that the COP monotonically decreases as the heating load increases, according to the curves 1 and 2 in Fig. 2. At these situations, the irreversible factors considered include heat resistance and internal irreversibility, but do not include heat leakage. From the curves 3–6 in Fig. 2, one can see that COP does not monotonically decrease as the heating load increases. At these situations, the irreversible factors considered include heat leakage and other factors. Thus, the heat leakage should be considered in an irreversible AHP cycle model. From the curve 4 in Fig. 2, one can see that a maximum COP (ψmax ) exists in the characteristic curves of AHP cycles. This reflects the true performance characteristic of a practical
18
19 20 21
22
23 24 25 26 27 28 29 30 31 32 33 34
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Fig. 2. The heating load, the COP and key irreversibility parameters characteristic curves.
Fig. 3. Characteristic curves before and after optimizing distribution A and UA.
2
AHP cycle. From the curves 1–6 in Fig. 2, one can also see that the performance of AHP decreases as any irreversibility factor increases.
3
4. Optimal characteristic
4
4.1. Optimal distribution of A
1
5 6 7 8 9 10 11 12 13 14 15 16 17 18
Absorber, condenser, evaporator and generator heat transfer surface areas Aa , Ac , Ae and Ag are key factors to decide the cost of AHP. The total area A is A = Aa + Ac + Ae + Ag .
(27)
If A = constant, optimizing the distribution of Aa , Ac , Ae and Ag , one can find the optimal cycle performance. When the heating load is selected, one can take COP as the optimization objective function. When COP is selected, one can take the heating load as the optimization objective function. With the constraint A = constant, using the Powell search method [38] and numerical algorithm, Fig. 3 shows the characteristic relationship curves before and after optimizing distribution A of an AHP cycle. In calculations, the parameters are the same as Section 3.2, and I = 1.01 and Kgl = Kal = Kcl = Kel = 0.1 kW K−1 are set. From Fig. 3, one can see that performances after optimizing distribution A are better than before optimizing. Heat capacities Ca , Cc , Ce and Cg can affect cycle performances. In order to discuss influences of heat capacities, the optimal Π versus Ca , Cc , Ce and Cg curves when ψ = 1.42 are shown in Fig. 4, the optimal ψ versus Ca , Cc , Ce and Cg curves when Π = 710 kW are shown in Fig. 5. From Figs. 4 and 5, one can see that the optimal performances increase as Cg , Ca , Cc and Ce increase, and the increasing degree decreases as Cg , Ca , Cc and Ce increase. These show that the cycle can
X. Qin et al. / Physica A xx (xxxx) xxx–xxx
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Fig. 4. Relationship curves of the optimal heating load versus heat capacities.
Fig. 5. Relationship curves of the optimal COP versus heat capacities.
be ameliorated by increasing any heat reservoir heat capacities, but one also should not pursue the maximum performance and utilize quite large heat capacity excessively.
1 2
4.2. Optimal distribution of UA Absorber, condenser, evaporator and generator heat exchanger inventories Ua Aa , Uc Ac , Ue Ae and Ug Ag are key factors to decide the performance of AHP. If the total heat exchanger inventory (UA = Ug Ag + Ua Aa + Uc Ac + Ue Ae ) is fixed, optimizing distribution UA can also be carried out [39–43]. Using UA = constant constraint to replace A = constant constraint, the optimal performance of AHP after optimizing distribution of UA can be obtained. The optimal performance after optimizing distribution of UA is shown in Fig. 3, which shows the performances after optimizing distribution of UA are better than before optimizing. 5. Special cases The finite-time-thermodynamics model of the heat pump cycle includes two heat reservoir, three heat reservoir and fourtemperature-level endoreversible and irreversible cycle models, coupled to infinite and finite heat capacity heat reservoirs. Eq. (26) is universal, which includes the results of all above heat pump cycle models.
3
Q2
4 5 6 7 8 9
10
11 12 13
8
1
2
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5.1. Case 1: I = 1, Kgl = Kal = Kcl = Kel = 0 If internal irreversibility and heat leakage losses are ignored, i.e., I = 1, Kgl = Kal = Kcl = Kel = 0, Eq. (26) becomes
3
Tgin
−
Π
1
−1
+
Cg Eg ψ
Tein
−
Π (ψ − 1)
−1
1 Ce Ee ψ
−ψ
Tain
(1 − a)Π
+
1
−1 +
Ca Ea
Tcin
aΠ
+
1
−1
Cc Ec
= 0.
(28)
5
Eq. (28) is the expression including the COP, heating load and key characteristic parameters based on a variabletemperature heat reservoir FTL endoreversible AHP cycle model.
6
5.2. Case 2: Cg → ∞, Ca → ∞, Cc → ∞, Ce → ∞
4
7 8
When heat capacities of heat reservoirs are infinite (Cg → ∞, Ca → ∞, Cc → ∞ and Ce → ∞), heat reservoirs can be assumed as constant-temperature reservoirs, Eq. (26) becomes
10
14 15
16
17
19
20 21
a[Π + Kal (Ta − Ts ) + Kcl (Tc − Ts )]
Tg
1
−
Π
26
29 30
+
1
−1
= 0.
Uc Ac
(29)
+
Te
Π (ψ − 1)
−
−1
1 Ue Ae ψ
−ψ
Ta
(1 − a)Π
+
1
−1
Ua Aa
+
Tc aΠ
+
1
−1
Uc Ac
= 0.
(30)
In this case, the absorber and condenser can be simplified to one heat exchanger. The AHP cycle can be modeled by a THR model, then Eq. (26) becomes Tgin
Π − Kgl (Tgin − Ts )ψ
−
ψ
[Π +
l Kac
+
Cg Eg ψ
in Tac
I
−
−1
1
(Tacin − Ts )]
+
1
Tein
[Π + Kacl (Tacin − Ts ) + Kgl (Tgin − Ts )]ψ − Π −1
Cac Eac
= 0.
−
1
−1
Ce Ee ψ (31)
Eq. (31) is the expression including the COP, heating load and key characteristic parameters based on a variabletemperature THR irreversible AHP cycle model. Moreover, when internal irreversibility and heat leakage losses are ignored (I = 1 and Kgl = Kal = Kcl = Kel = 0), Eq. (31) becomes
28
−1
in l 5.3. Case 3: a = 1/2, Tain = Tcin = Tac , Ea = Ec = Eac , Ca = Cc = Cac /2, Kal = Kcl = Kac /2
23
27
1
Eq. (30) is the expression including the COP, heating load and key characteristic parameters based on a constanttemperature heat reservoir FTL endoreversible AHP cycle model [33,34].
22
24
−1
Ug Ag ψ
25
Te
Eq. (29) is the expression including the COP, heating load and key characteristic parameters based on a constanttemperature heat source FTL irreversible AHP cycle model [34]. Moreover, when internal irreversibility and heat leakage losses are ignored (I = 1 and Kgl = Kal = Kcl = Kel = 0), Eq. (29) becomes
18
Tc
+
11
12
−1
1
− + − Π − Kgl (Tg − Ts )ψ Ug Ag ψ [Π + Kal (Ta − Ts ) + Kcl (Tc − Ts ) + Kgl (Tg − Ts )]ψ − Π Ue Ae ψ −1 1 ψ Ta − + I (1 − a)[Π + Kal (Ta − Ts ) + Kcl (Tc − Ts )] Ua Aa
9
13
Tg
Tgin
Π
−
1 Cg Eg ψ
−1
+
Tein
Π (ψ − 1)
−
1 Ce Ee ψ
−1
−ψ
in Tac
Π
+
1 Cac Eac
−1
= 0.
(32)
Eq. (32) is the expression including the COP, heating load and key characteristic parameters based on a variabletemperature three heat reservoir THR endoreversible AHP cycle.
X. Qin et al. / Physica A xx (xxxx) xxx–xxx
9
in 5.4. Case 4: Tain = Tcin = Tac , Cg → ∞, Ca → ∞, Cc → ∞, Ce → ∞, Uac = Ua = Uc , Aac = 2Aa = 2Ac , a = 1/2
1
In this case, heat capacities of heat reservoirs are infinite (Cg → ∞, Ca → ∞, Cc → ∞ and Ce → ∞), heat reservoirs can be assumed as a constant-temperature reservoir. The absorber and condenser can be simplified to one heat exchanger, the AHP cycle can be modeled by THR cycle model. Eq. (26) becomes
1
Tg
−1
Te
1
− + − Π − Kgl (Tg − Ts )ψ Ug Ag ψ [Π + Kacl (Tac − Ts ) + Kgl (Tg − Ts )]ψ − Π Ue Ae ψ −1 Tac 1 ψ + = 0. − I [Π + Kacl (Tac − Ts )] Uac Aac
Tg
Π
−
1 Ug Ag ψ
−1
+
Te
Π (ψ − 1)
−
1 Ue Ae ψ
−1
−ψ
Tac
Π
+
1
−1
Uac Aac
= 0.
3 4
−1 5
(33)
Eq. (33) is the expression including the COP, heating load and key characteristic parameters based on a constanttemperature THR irreversible AHP cycle model [34]. Moreover, when internal irreversibility and heat leakage losses are ignored (I = 1 and Kgl = Kal = Kcl = Kel = 0), Eq. (33) becomes
2
(34)
Eq. (34) is the expression including the COP, heating load and key characteristic parameters based on a constanttemperature THR endoreversible AHP cycle model [33,34]. in 5.5. Case 5: Tgin → ∞, a = 1/2, Tain = Tcin = Tac , Ea = Ec = Eac , Ca = Cc = Cac /2
In this case, the model becomes a variable-temperature two heat reservoir (Carnot) irreversible heat pump model. Moreover, if I = 1 and Kgl = Kal = Kcl = Kel = 0, the model becomes a variable-temperature two heat reservoir (Carnot) endoreversible heat pump model. in 5.6. Case 6: Tgin → ∞, a = 1/2, Tain = Tcin = Tac , Cg → ∞, Ca → ∞, Cc → ∞, Ce → ∞, Uac = Ua = Uc , Aac = 2Aa = 2Ac
In this case, the model becomes a constant-temperature two heat reservoir (Carnot) irreversible heat pump model. Moreover, if I = 1 and Kgl = Kal = Kcl = Kel = 0, the model becomes a constant-temperature two heat reservoir endoreversible heat pump model. 6. Conclusion A variable-temperature heat reservoir model is closer to a practical AHP cycle, in which irreversibility losses include heat resistance losses, internal working substance dissipation losses, and heat leakage losses between heat reservoir and surrounding. In accordance with this irreversible AHP cycle model, the general characteristic about COP versus heating load has a maximum COP value (ψmax ) with the corresponding heating load (Πψ ), and this reflects the true performance characteristic of a practical AHP. The finite heat capacity of heat reservoirs, heat resistance losses, internal irreversibility loss and heat leakage losses must be considered in modeling and quantitative analysis of AHP cycles. COP and heating load can be increased by optimizing distribution heat exchanger surface area or heat exchanger inventory. When any heat reservoir heat capacity increases, the cycle performance is improved, but the increment decreases as any heat reservoir heat capacity increases. Eq. (26) of this paper is universal, which includes the results of all heat pump cycles (two, three, and four heat reservoir; endoreversible and irreversible; infinite and finite heat capacity heat reservoirs). Therefore, the results obtained herein are useful and universal, and can supply some guidelines for the design and operation of a practical AHP, whether the heat reservoir is variable-temperature or not. Acknowledgments This paper is supported by The National Natural Science Foundation of P. R. China (Project No. 10905093). The authors wish to thank the reviewers for their careful, unbiased and constructive suggestions, which led to this revised manuscript.
6
7 8 9 10
11
12 13
14
15 16 17
18
19 20 21
22
23 24 25 26 27 28 29 30 31 32 33 34 35
36
37 38 39
10
1
2 3 4 5
References [1] [2] [3] [4]
7
[5] [6]
8
[7]
9
14
[8] [9] [10] [11] [12] [13]
15
[14]
16
[15]
17
[16]
18
21
[17] [18] [19] [20]
22
[21]
23
[22]
24
[23]
25
[24]
26 27
[25] [26]
28
[27]
29
[28]
30
[29]
31
[30]
32
[31]
33
[32]
34
[33]
35 37
[34] [35] [36]
38
[37]
39
[38] [39]
6
10 11 12 13
19 20
36
40 41 42 43 44
X. Qin et al. / Physica A xx (xxxx) xxx–xxx
[40] [41] [42] [43]
F. Ziegler, State of the art in sorption heat pumping and cooling technologies, Int. J. Refrig. 25 (4) (2002) 450–459. D.S. Kim, C.A.I. Ferreira, Solar refrigeration options—a state-of-the-art review, Int. J. Refrig. 31 (1) (2008) 3–15. M.U. Siddiqui, S.A.M. Said, A review of solar powered absorption systems, Renewable Sustainable Energy Rev. 42 (2015) 93–115. A. Bejan, Entropy generation minimization: The new thermodynamics of finite-size device and finite-time processes, J. Appl. Phys. 79 (3) (1996) 1191–1218. R.S. Berry, V.A. Kazakov, S. Sieniutycz, Z. Szwast, A.M. Tsirlin, Thermodynamic Optimization of Finite Time Processes, Wiley, Chichester, 1999. L.G. Chen, C. Wu, F.R. Sun, Finite time thermodynamic optimization or entropy generation minimization of energy systems, J. Non-Equilib. Thermodyn. 24 (4) (1999) 327–359. A. Durmayaz, O.S. Sogut, B. Sahin, H. Yavuz, Optimization of thermal systems based on finite-time thermodynamics and thermoeconomics, Prog. Energy Combust. Sci. 30 (2) (2004) 175–217. L.G. Chen, F.R. Sun, Advances in Finite Time Thermodynamics: Analysis and Optimization, Nova Science Publishers, New York, 2004. L.G. Chen, Finite Time Thermodynamic Analysis of Irreversible Processes and Cycles, Higher Education Press, Beijing, 2005, (in Chinese). M. Feidt, Optimal use of energy systems and processes, Int. J. Exergy 5 (5–6) (2008) 500–531. M. Feidt, Thermodynamics applied to reverse cycle machines, a review, Int. J. Refrig. 33 (7) (2010) 1327–1342. B. Andresen, Current trends in finite-time thermodynamic, Angew. Chem. Int. Ed. 50 (12) (2011) 2690–2704. M. Feidt, Evolution of thermodynamic modelling for three and four heat reservoirs reverse cycle machines: A review and new trends, Int. J. Refrig. 36 (1) (2013) 8–23. X.Y. Qin, L.G. Chen, Y.L. Ge, F.R. Sun, Finite time thermodynamic studies on absorption thermodynamic cycles: A state of the arts review, Arab. J. Sci. Eng. 38 (3) (2013) 405–419. P.A. Ngouateu Wouagfack, R. Tchinda, Finite-time thermodynamics optimization of absorption refrigeration systems: A review, Renewable Sustainable Energy Rev. 21 (2013) 524–536. E. Açıkkalp, Modified thermo-ecological optimization for refrigeration systems and an application for irreversible four-temperature-level absorption refrigerator, Int. J. Energy Environ. Eng. 4 (1) (2013) 20. E. Açıkkalp, Models for optimum thermo-ecological criteria of actual thermal cycles, Therm. Sci. 17 (3) (2013) 915–930. E. Açıkkalp, H. Yamik, Limits and optimization of power input or output of actual thermal cycles, Entropy 15 (8) (2013) 3309–3338. S. Sieniutycz, J. Jezowski, Energy Optimization in Process Systems and Fuel Cells, Elsevier, Oxford, UK, 2013. E. Açıkkalp, Entransy analysis of irreversible heat pump using Newton and Dulong–Petit heat transfer laws and relations with its performance, Energy Convers. Manage. 86 (2014) 792–800. K.H. Hoffmann, B. Andresen, P. Salamon, Finite-time thermodynamics tools to analyze dissipative processes, in: A.R. Dinner (Ed.), Proceedings of The 240 Conference: Science’s Great Challences, in: Advances in Chemical Physics, vol. 157, Wiley, 2015, pp. 57–67. E. Açıkkalp, N. Caner, Determining performance of an irreversible nano scale dual cycle operating with Maxwell–Boltzmann gas, Physica A 424 (2015) 342–349. E. Açıkkalp, H. Yamik, Modeling and optimization of maximum available work for irreversible gas power cycles with temperature dependent specific heat, J. Non-Equilib. Thermodyn. 40 (2015) 25–39. J. Chen, B. Andresen, Optimal analysis of primary performance parameters for an endoreversible absorption heat pump, Heat Recovery Syst. CHP 15 (8) (1995) 723–731. S. Goktun, The effect of irreversibilities on the performance of a three-heat-source heat pump, J. Phys. D: Appl. Phys. 29 (20) (1996) 2823–2825. G. Lin, Z. Yan, Optimization of erergy loss rate in an irreversible three-heat-source heat pump, Acta Energiae Solaris Sin. 22 (1) (2001) 107–110. (in Chinese). S. Wu, G. Lin, J. Chen, Optimum thermoeconomic and thermodynamic performance characteristics of an irreversible three-heat-source heat pump, Renew. Energy 30 (15) (2005) 2257–2271. P.A. Ngouateu Wouagfack, R. Tchinda, The new thermo-ecological performance optimization of an irreversible three-heat-source absorption heat pump, Int. J. Refrig. 35 (1) (2012) 79–87. G. Su, Z. Yan, The optimal ecological performance of three-heat-source heat pump with linear phenomenological heat transfer law, Acta Energiae Solaris Sin. 20 (2) (1999) 204–208. (in Chinese). X.Y. Qin, L.G. Chen, F.R. Sun, C. Wu, Compromise optimization between heating load and entropy production rate for endoreversible absorption heat pumps, Int. J. Ambient Energy 26 (2) (2005) 106–112. P.A. Ngouateu Wouagfack, R. Tchinda, Optimal ecological performance of a four-temperature-level absorption heat pump, Int. J. Therm. Sci. 54 (2012) 209–219. J. Chen, The general performance characteristics of an irreversible absorption heat pump operating between four temperature levels, J. Phys. D: Appl. Phys. 32 (12) (1999) 1428–1433. Y. Huang, D. Sun, Y. Kang, Performance optimization for an irreversible four-temperature-level absorption heat pump, Int. J. Therm. Sci. 47 (4) (2008) 479–485. L.G. Chen, X.Y. Qin, F.R. Sun, C. Wu, Irreversible absorption heat-pump and its optimal performance, Appl. Energy 81 (1) (2005) 55–71. X. Zhao, L. Fu, S. Zhang, General thermodynamic performance of irreversible absorption heat pump, Energy Convers. Manage. 52 (1) (2011) 494–499. X.Y. Qin, L.G. Chen, F.R. Sun, C. Wu, Performance of an endoreversible four-heat-reservoir absorption heat pump with a generalized heat transfer law, Int. J. Therm. Sci. 45 (6) (2006) 627–633. X.Y. Qin, L.G. Chen, F.R. Sun, C. Wu, Model of real absorption heat pump cycle with a generalized heat transfer law and its performance, Proc. IMechE Part A: Power Energy 221 (7) (2007) 907–916. M.J.D. Powell, Nonlinear Optimization, Academic Press, New York, 1982. A. Bejan, J.V.C. Vargas, Optimal allocation of a heat exchanger inventory in heat driven refrigerators, Int. J. Heat Mass Transfer 38 (16) (1995) 2997–3004. M. Sokolov, J.V.C. Vargas, A. Bejan, Thermodynamic optimization of solar-driven refrigerators, Trans. ASME J. Sol. Energy Eng. 118 (2) (1996) 130–135. S.A. Klein, D.T. Reindl, The relationship of optimum heat exchanger allocation and minimum entropy generation rate for refrigeration cycles, Trans. ASME J. Energy Res. Technol. 120 (2) (1998) 172–178. M. Feidt, M. Costea, C. Petre, S. Petrescu, Optimization of direct Carnot cycle, Appl. Therm. Eng. 27 (5–6) (2007) 829–839. J. Sarkar, S. Bhattacharyya, Overall conductance and heat transfer area minimization of refrigerators and heat pumps with finite heat sources, Energy Convers. Manage. 48 (3) (2007) 803–808.