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The new thermo-ecological performance optimization of an irreversible three-heat-source absorption heat pump
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The new thermo-ecological performance optimization of an irreversible three-heat-source absorption heat pump Paiguy Armand Ngouateu Wouagfack a,*, Re´ne´ Tchinda b a b
L2MSP, Department of Physics, University of Dschang, PO Box 67, Dschang, Cameroon LISIE, University Institute of Technology Fotso Victor, University of Dschang, PO Box 134, Bandjoun, Cameroon
article info
abstract
Article history:
The new thermo-ecological performance optimization of an absorption heat pump oper-
Received 13 July 2011
ating between three temperature levels with the losses of heat resistance, internal irre-
Received in revised form
versibility and leakage is analyzed by taking the ecological coefficient of performance
19 September 2011
(ECOP) as an objective function. The new thermo-ecological criterion takes into account the
Accepted 22 September 2011
first and second law of thermodynamics and is defined as the heating rate per unit loss rate
Available online 1 October 2011
of availability. The ecological coefficient of performance has been expressed and maximized in terms of the temperatures of the working fluid in the main components of the
Keywords:
system. The corresponding optimal temperatures and other optimum performance
Irreversibility
parameters have been derived analytically, and the effects of the internal irreversibility,
Heat pump
the heat leakage coefficient and the source temperature ratio on the global and optimal
Absorption system
performances are discussed. The obtained results may provide a general theoretical tool
Heat source
for the ecological design of absorption heat pumps. ª 2011 Elsevier Ltd and IIR. All rights reserved.
Performance Environment
Nouvelle optimisation thermo-e´cologique de la performance d’une pompe a` chaleur irre´versible a` absorption munie de trois sources de chaleur Mots cle´s : irre´versibilite´ ; pompe a` chaleur ; syste´me a` absorption ; source de chaleur ; performance ; environnement
1.
Introduction
Energy recovery is becoming more and more important in the industry where enormous heat is wasted. Absorption heat pump is considered to be the most competitive equipment for energy saving when it uses high grade heat source to recover waste heat (Zhao et al., 2010; Fu et al., 2009). In
recent years, the thermodynamics of real absorption heatpump cycle has been performed and many significant results have been obtained (Zhao et al., 2010; Huang, 2009). A three-heat-reservoir absorption heat pump is an important theoretical model of the apparatus. It is a simplified model of an absorption heat pump in that the temperature of the absorber is equal to that of the condenser, but in a real
* Corresponding author. Tel.: þ237 77 18 58 71. E-mail address: [email protected] (P.A. Ngouateu Wouagfack). 0140-7007/$ e see front matter ª 2011 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2011.09.008
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Nomenclature
3r 2
q s T U
total heat-transfer area (m ) coefficient of performance normalized COP ecological coefficient of performance normalized ECOP internal irreversibility parameter thermal conductance (W K1) rate of heat transfer (W) specific heating load (W m2) specific entropy generation rate (W K1 m2) temperature (K) overall heat-transfer coefficient (W K1 m2)
Symbol s_ s x
entropy generation rate (W K1) source temperature ratio (TO/TG) heat leakage coefficient (W K1 m2)
A COP COP ECOP ECOP I K Q_
absorption heat pump usually it is not (Zhao et al., 2010). Some researchers optimized the performance of the threeheat-reservoir heat-pump cycle with: (i) the loss of heat resistance, (ii) the loss of heat resistance and internal irreversibility, and (iii) the loss of heat resistance, heat leakage and internal irreversibility. Chen and Yan (1989a,b), Herold (1999), Goktun (1996) and Chen (1997) analyzed the performance of the three-heat-reservoir absorption heat-pump cycle with the loss of heat resistance (Chen and Yan, 1989a,b; Herold, 1999), with losses due to heat resistance and internal irreversibility (Goktun, 1996), and with losses due to heat resistance, heat leakage and internal irreversibility (Chen, 1997) with the linear (Newtonian) heat-transfer law. Chen et al. (1997, 1999a,b) studied the performance of the endoreversible three-heat-reservoir absorption heatpump cycle with the linear phenomenological heat-transfer 1 _ Þ. Chen et al. (2005) and Qin et al. (2006, 2007) law, QfDðT performed similar optimization works to research the performance of a four-heat-source absorption heat pump with the linear (Newtonian) heat-transfer law (Chen et al., 2005), and with the generalized heat-transfer law (Qin et al., 2006, 2007). Kodal et al. (2003) and Wu et al. (2005) analyzed the thermo-economic performance of the threeheat-reservoir heat-pump cycle. None of the optimization criteria of the three-heat-source absorption heat pump defined in the above referenced works describes its performance from the thermo-ecological point of view. Yan and Lin (2000) established an ecological optimal performance of an absorption system for cooling application. They obtained that the ecological optimization criterion (E ) is more advantageous than the maximum cooling load in term of entropy production and therefore, it is beneficial for making a more rational use of energy of absorption systems so as to save energy. During the last decade, Qin et al. (2005) investigated the optimal ecological performance of endoreversible absorption heat pumps and Huang et al. (2008) performed the thermo-ecological optimization works for irreversible absorption heat pumps. In the thermo-ecological optimization studies of an irreversible absorption heat pump (Huang
coefficient of performance for reversible threeheat-source heat pump
Subscripts 1 working fluid in generator 2 working fluid in evaporator 3 working fluid in absorber and condenser A absorber C condenser E evaporator env environment conditions G generator L heat leakage max maximum O absorber and condenser Superscripts * at maximum ECOP
et al., 2008), the ecological objective function is defined as _ The first term is the heating load and the E ¼ q_ mTE s. second term is its dissipation where m is the dissipation coefficient of the heating load, TE is the temperature of evaporator and s_ is the entropy generation rate. When the ecological objective function (E ) proposed in the above referenced works is examined, one can see that the objective function (E ) may take negative values. At this condition, loss of heating load (second term of E function) is greater than the heating load (first term of E function). Such an objective function in a performance analysis can be defined mathematically, however, it needs interpretation to comprehend this situation thermodynamically (Ust et al., 2005; Ust and Sahin, 2007). In a recent study, Ust (2004) proposed a new ecological objective function called ecological coefficient of performance (ECOP) and defined as the work-energy (e.g. power of an engine, cooling rate of a refrigerator, or heating rate of a heat pump) per unit loss rate of availability which is produced by the entropy generation in the system and its surroundings. The proposed ecological objective function ECOP gives the information about to the loss rate of availability or entropy generation rate in order to produce a certain power of an engine, cooling rate of a refrigerator or heating rate of a heat pump. It should be noted that for a certain power of an engine, cooling rate of a refrigerator or heating rate of a heat pump the entropy generation is minimum at maximum ECOP. Like the ecological optimization procedure, the goal of a new thermo-ecological optimization procedure is to achieve the best compromise between the work-energy (e.g. power of an engine, cooling rate of a refrigerator, or heating rate of a heat pump) and its dissipation. The ECOP function always has positive values and it is dimensionless just like the thermal efficiency in the heat engines or the coefficient of performance in the refrigerators and heat pumps. Ust et al. (2005) established the utility of the new ecological performance coefficient. They obtained that the entropy generation rate at ECOPmax conditions is less than that at maximum ecological function (Emax). Thus ECOP criterion has a marked advantage in
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ecological perspective over the E criterion in term of entropy generation. Very recently a performance analysis and optimization based on a new thermo-ecological optimization criterion has been carried out for mechanical refrigerators (Ust and Sahin, 2007) and absorption refrigerators (Ngouateu Wouagfack and Tchinda, 2011). In the present performance optimization work, an ecological performance analysis for an irreversible three-heatsource absorption heat pump has been carried out by employing the ECOP as objective function. The optimal performance and the design parameters which maximized the ECOP criterion have been determined, and also, the effects of the major irreversibilities on the thermo-ecological performances are discussed.
2. A general irreversible absorption heatpump cycle model and thermodynamic analysis It can be observed in Fig. 1 (Huang et al., 2008) that a generator, an absorber, a condenser and an evaporator are the main components of an absorption heat-pump system. The first law of thermodynamics of the absorption heat pump shown schematically in Fig. 1 is written as: Q_ G þ Q_ E Q_ C Q_ A ¼ 0
(1)
The expression of Eq. (1) is obtained by neglecting the work input required by the solution pump. It is negligibly small compared to the energy input to the generator and, therefore, is often neglected in the analysis. It may be assumed for the absorption heat pump shown in Fig. 1 that the flow of the working fluid in the cycle system is stable and that the different parts of the working fluid exchange heat with the heat reservoirs at temperatures TG, TE and TO. In the others words, the system rejects heat from the condenser and absorber to the
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heated space at the same temperature TO. It is also assumed that the working fluid in the condenser and absorber has the same temperature T3 (Kodal et al., 2003; Wu et al. 2005). This assumption is reasonable because the working fluid in the condenser and absorber exchanges heat with the heated space at the same temperature. With the above assumptions, the absorption heat pump becomes the three-heat-source absorption heat pump and the cycle of the working fluid consists of three irreversible isothermal and three irreversible adiabatic processes. There are thermal resistances between the working fluid and the external heat reservoirs. The temperatures of the working fluid in the three isothermal processes are different from those of the external heat reservoirs so that heat is transferred under a finite temperature difference, as shown in Fig. 2 (Ngouateu Wouagfack and Tchinda, 2011). In Fig. 2, Q_ O ¼ Q_ C þ Q_ A , Q_ L is the heat leak from the heated space at temperature TO to the heat sink at temperature TE. T1 and T2 are, respectively, the temperatures of the working fluid in the generator and evaporator. Real absorption three-heat-source heat pumps are complex devices and suffer from a series of irreversibilities. Besides the irreversibility of finite-rate heat transfer, which is considered in the endoreversible cycle models and the heat leak from the heated space to the heat sink, there also exist other sources of irreversibility. For example, the internal dissipation of the working fluid will be another main source of irreversibility, which can decrease the coefficient of performance and the heating load of absorption heat pumps. According to the second law of thermodynamics, we have: I
Q_ Q_ Q_ dQ_ ¼ Gþ E O<0 T1 T2 T3 T
(2)
We can introduce an irreversibility factor: Q_ G Q_ E Q_ O þ ¼0 T1 T2 IT3
(3)
Obviously, the performance of an irreversible cycle affected by thermal resistances is directly dependent on the heattransfer law between the working fluid and the heat reservoirs. When the heat transfer between the working fluid and
Fig. 1 e Schematic diagram of an absorption heat pump (Huang et al., 2008).
Fig. 2 e The irreversible cycle model of a three-heat-source absorption heat pump (Ngouateu Wouagfack and Tchinda, 2011).
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_ the heat reservoirs obeys linear heat-transfer law ðQfDðTÞÞ, one has: Q_ G ¼ UG AG ðTG T1 Þ (4) Q_ E ¼ UE AE ðTE T2 Þ
(5)
Q_ O ¼ UO ðAA þ AC ÞðT3 TO Þ
(6)
Q_ L ¼ KL ðTO TE Þ
(7)
where AG, AE, AC and AA are, respectively, the heat-transfer areas of the generator, evaporator, condenser and absorber, KL is the heat leak coefficient, UG and UE are the overall heattransfer coefficients of the generator and evaporator, and it is assumed that the condenser and absorber have the same overall heat-transfer coefficient UO. The rate of heat leakage Q_ L given in Eq. (7) from the heated space at temperature TO to the heat sink at temperature TE was first provided by Bejan (1989). The total heat-transfer area between the cycle system and the external heat reservoirs is: A ¼ A G þ AE þ AO
(8)
Q_ O Q_ L Q_ G Q_ E Q_ L s_ TO TG TE s¼ ¼ A A 1 1 εr T1 ðT2 IT3 Þ xðTO TE Þ þ 1 ¼ TE TO IT3 ðT2 T1 Þ T1 ðIT3 T2 Þ T2 ðIT3 T1 Þ þ UG ðTG T1 ÞðT1 T2 ÞIT3 UE ðTE T2 ÞðT2 T1 ÞIT3 1 1 þ UO ðT3 TO Þ
(11)
where the parameter x ¼ KL/A represents the heat leakage coefficient and its dimension is W K1 m2 and εr ¼ (1 TE/TG) TO/(TO TE) is the coefficient of performance for reversible three-heat-source heat pump. The ecological coefficient of performance (Ust, 2004; Ust et al., 2005; Ust and Sahin, 2007; Ngouateu Wouagfack and Tchinda, 2011) of an irreversible three-heat-source absorption heat pump is defined as the ratio of heating load to the loss rate of availability, i.e.,
where AO ¼ AC þ AA.
ECOP ¼
Q_ O Q_ L 1 ¼ 1 Tenv s_ Tenv T1 O TE
1 1 εr T1 ðT2 IT3 Þ T1 ðIT3 T2 Þ T2 ðIT3 T1 Þ 1 1 þ 1 xðTO TE Þ þ þ IT3 ðT2 T1 Þ UG ðTG T1 ÞðT1 T2 ÞIT3 UE ðTE T2 ÞðT2 T1 ÞIT3 UO ðT3 TO Þ (12)
According to the standard definitions of the specific heating load (q) and the coefficient of performance (COP) of an absorption heat pump, we obtain for an irreversible threeheat-source heat pump: Q_ Q_ L q¼ O A T1 ðIT3 T2 Þ T2 ðIT3 T1 Þ þ ¼ UG ðTG T1 ÞðT1 T2 ÞIT3 UE ðTE T2 ÞðT2 T1 ÞIT3 1 1 þ xðTO TE Þ (9) UO ðT3 TO Þ
ECOP ¼
Q_ O Q_ L 1 ¼ 1 Tenv s_ Tenv T1 O TE
where Tenv conditions.
is
the
temperature
in
the
environment
3. Performance optimization of three-heatsource heat pump based on ECOP criterion For the sake of convenience, let x ¼ IT3/T1, y ¼ IT3/T2, z ¼ IT3. Then Eq. (12) may be written as:
1 1 εr ð1 yÞ xð1 yÞ yð1 xÞ 1 1 þ þ þ 1 xðTO TE Þ xy UG ðTG x zÞðx yÞ UE ðTE y zÞðy xÞ UO ðz TÞ
Q_ Q_ L Q_ O Q_ 1 L ¼ COP ¼ O Q_ G Q_ G Q_ O IT3 ðT2 T1 Þ T1 ðIT3 T2 Þ 1 xðTO TE Þ ¼ T1 ðT2 IT3 Þ UG ðTG T1 ÞðT1 T2 ÞIT3 T2 ðIT3 T1 Þ 1 þ þ UE ðTE T2 ÞðT2 T1 ÞIT3 UO ðT3 TO Þ
(10)
The specific entropy generation rate (s) for an irreversible three-heat-source heat pump with heat leak loss is
(13)
where T ¼ ITO and U ¼ UO/I. Using Eq. (13) and the extremal conditions vECOP/vx ¼ 0, vECOP/vy ¼ 0 and vECOP/vz ¼ 0, one can derive the temperature of the working fluid in the generator, evaporator, absorber and condenser which correspond to the maximum ECOP (ECOPmax) as follows: T1 ¼ TG B=½ð1 þ b1 ÞB b1 T
(14)
T2
(15)
¼ TE B=½ð1 þ b2 ÞB b2 T
T3 ¼ B=I
(16)
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where
b1 ¼
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi U=UG ; b2 ¼ U=UE ; B ¼
½UTE ðTO TE Þð1 þ b2 Þb2 xT þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i 2 ðT TE ÞU þ ðTO TE Þð1 þ b2 Þ x ðTO TE ÞTE Tx 2
UTE ðTO TE Þð1 þ b2 Þ x
(17)
Substituting Eqs. (14)e(16) into Eqs. (9)e(12) yields the maximum ECOP:
ECOPmax ¼
1 1 Tenv T1 O TE
1 1 1 1 þ Db1 ½ð1 þ b1 ÞB b1 TT1 G þ ð1 DÞb2 ½ð1 þ b2 ÞB b2 TTE 1 εr D 1 xðTO TE Þ UðB TÞ
(18)
and the optimal coefficient of performance, the optimal specific heating load and the optimal specific entropy generation rate at the maximum ECOP conditions, respectively, as: 1 1 þ Db1 ½ð1 þ b1 ÞB b1 TT1 G þ ð1 DÞb2 ½ð1 þ b2 ÞB b2 TTE COP ¼ D1 1 xðTO TE Þ UðB TÞ
q ¼
UðB TÞ 1 1 þ Db1 ½ð1 þ b1 ÞB b1 TT1 G þ ð1 DÞb2 ½ð1 þ b2 ÞB b2 TTE
pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffi UG AG þ UE AE ¼ UAO
(19)
(25)
xðTO TE Þ
1 1 s ¼ TE TO "
( xðTO TE Þ þ ðεr D 1Þ
UðB TÞ 1 1 þ Db1 ½ð1 þ b1 ÞB b1 TT1 G þ ð1 DÞb2 ½ð1 þ b2 ÞB b2 TTE
where D¼
(20)
#)
(21)
TE þ b2 T ð1 þ b2 ÞB ½ð1 þ b1 ÞB b1 TTE =TG þ b2 T ð1 þ b2 ÞB
From Eqs. (1), (4)e(6), (8) and (14)e(16), we find that, when the three-heat-source absorption heat pump is operated in the state of maximum ecological coefficient of performance, the relations between the heat-transfer areas of the heat exchangers and the total heat-transfer area are determined by: 1 AG b21 DT1 G ðB TÞf1 B=½ð1 þ b1 ÞB b1 Tg ¼ 1 A 1 þ Db1 ½ð1 þ b1 ÞB b1 TTG þ ð1 DÞb2 ½ð1 þ b2 ÞB b2 TT1 E (22) 1 AE b22 T1 E ð1 DÞðB TÞf1 B=½ð1 þ b2 ÞB b2 Tg ¼ 1 A 1 þ Db1 ½ð1 þ b1 ÞB b1 TTG þ ð1 DÞb2 ½ð1 þ b2 ÞB b2 TT1 E (23) AO 1 ¼ 1 A 1 þ Db1 ½ð1 þ b1 ÞB b1 TT1 G þ ð1 DÞb2 ½ð1 þ b2 ÞB b2 TTE (24) From Eqs. (22)e(24), we obtain a concise optimum relation for the distribution of the heat-transfer areas of three-heatsource absorption heat pump as:
4.
Optimization results and discussion
The ecological coefficient of performance (ECOP) gives information about the entropy generation rate or loss rate of availability in order to produce certain amount of heating load. The utility of the new thermo-ecological optimization criterion is that: at the maximum ECOP conditions, the absorption heat pump develops a heating load by causing lesser dissipation in the environment. Therefore the higher the ECOP is, the more the absorption heat pump works under a better compromise between the heating load and its losses which can affect the environment. In order to see the results of ECOP optimization for an irreversible three-heat-source absorption heat pump, some numerical examples are given and discussed. The numerical calculations are carried out by employing the values of the relevant parameters used by Kodal et al. (2003). Fig. 3 shows the effects of internal irreversibility (I ), heat leakage coefficient (x) and source temperature ratio (s) on the optimal temperature of the working fluid in the generator, evaporator, condenser and absorber at maximum ECOP objective function conditions. In this figure, the variations of the optimal working fluid temperatures with respect to internal irreversibility parameter and source temperature ratio have been plotted for different values of heat leakage coefficient. It can be observed that the optimal temperature of the working fluid in generator (T1*) and in evaporator (T2*) decreases with the increase of I and s. For a fixed I and s, the optimal temperature of the working fluid in generator and evaporator decreases with the increase of x. The optimal
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temperature of the working fluid in the condenser and absorber (T3*) increases with the increase of I and s and increases with the increase of x for a fixed I and s. Figs. 4 and 5 show the variation of the ECOP with the specific heating load and specific entropy generation rate for different values of I and s. By observing these figures, on can see that as I and s increase the global and optimal ECOP which correspond to the maximum ECOP decrease. In Figs. 4a and 5a, it can be seen that the global and optimal ECOP decrease significantly from an endoreversible situation (I ¼ 1) to an irreversible situation (I > 1). Fig. 6 shows the variations of the normalized ECOP ðECOP ¼ ECOP=ECOPmax Þ, normalized COP ðCOP ¼ COP=COPmax Þ and the specific entropy generation rate (s) with respect to the specific heating rate (q). One interesting observation from this figure is that maximum of the ECOP and COP coincides although their functional form is different: the coefficient of performance ðCOP ¼ Q_ O Q_ L =Q_ G Þ gives information about the necessary heat rate input in order to produce certain amount of heating load and the ecological coefficient of performance _ gives information about the entropy ðECOP ¼ Q_ O Q_ L =Tenv sÞ generation rate or loss rate of availability in order to produce certain amount of heating load. It has also been seen numerically and analytically that the performance parameters T1*, T2*, T3*, q*, s*, AG*, AE*, AO* and COP* ¼ COPmax at the maximum ECOP and maximum COP are same. Getting the same optimal conditions at the maximums of the ECOP and COP is a normal
Fig. 4 e Variation of the ECOP objective function with respect to the specific heating load for various internal irreversibility parameters (a) and for various source temperature ratios (b) (TG [ 393 K, TE [ 288 K, TO [ 313 K, Tenv [ 290 K, UG [ UE [ UO [ 500 W KL1 mL2, I [ 1.025, x [ 1.082).
Fig. 3 e Variations of the optimal working fluid temperatures at maximum ECOP conditions with respect to internal irreversibility parameter (a) and source temperature ratio (b) for different heat leakage coefficients (TG [ 393 K, TE [ 288 K, TO [ 313 K, UG [ UE [ UO [ 500 W KL1 mL2, I [ 1.025).
result. Since, for a certain heating load, the maximum COP occurs from minimum heat input required by the generator so that minimization of adverse environmental impacts in term of energy saving. The minimum environmental pollution is also achieved by maximizing the ECOP. This result has been also obtained in references Ust and Sahin (2007) and Ngouateu Wouagfack and Tchinda (2011) for compression refrigerators and absorption refrigerators respectively. Although the optimal performance conditions ECOP and COP criteria are same, their impact on the system design performance is different. The coefficient of performance is used to evaluate the performance and the efficiency of heat producing systems. This method only takes into account the first law of thermodynamics which is concerned only with the conversion of energy, and therefore, cannot show how or where irreversibilities in a system or process occur. Also, when different sources and forms of energy are involved within a system, the COP criterion of the three-heat-source heat pump doesn’t describe its performance from the view point of the energy quality involved. This factor is taken into account by the
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Fig. 5 e Variation of the ECOP objective function with respect to the specific entropy generation rate for various internal irreversibility parameters (a) and for various source temperature ratios (b) (TG [ 393 K, TE [ 288 K, TO [ 313 K, Tenv [ 290 K, UG [ UE [ UO [ 500 W KL1 mL2, I [ 1.025, x [ 1.082).
second law of thermodynamics and appears in the ECOP criterion. For this important reason, the ECOP criterion can enhance the system performance of the three-heat-source heat pump by reducing the irreversible losses in the system. A better understanding of the second law of thermodynamics reveals that the ecological coefficient of performance optimization is an important technique in achieving better operating conditions. ECOP and COP are related as: ECOP ¼
COP Tenv ð1=TO 1=TE Þðεr COPÞ
(26)
Fig. 7 shows the variations of ECOP with respect to COP for various s values. ECOP increases with the increase of COP and the ECOPmax and COPmax conditions which are end points of the curves decrease with the increase of s values.
ECOPmax ¼
1 1 Tenv T1 O TE
Fig. 8 shows the effects of I, s and x on the optimal coefficient of performance, specific heating load and specific entropy generation rate at maximum ECOP objective function conditions. The optimal specific heating rate and the optimal specific entropy generation rate at ECOPmax conditions increase with the increase of I, s and x, while the optimal coefficient of performance at ECOPmax conditions decreases. From the above performance optimization works based on ECOP criterion carried out for irreversible three-heatsource heat pumps, we can deduce important optimization results regarding endoreversible three-heat-source heat pumps and irreversible and endoreversible Carnot heat pumps with heat leak loss. For instance, when TG / N and UE ¼ UO ¼ a, Eqs. (18)e(21) can be, respectively, simplified to
1 #( pffi )1 pffiffi xðTO TE Þ I þ I Iεr TE pffiffi 1 εr þ 1 aðB ITO Þ ITO 1 þ I B "
(27)
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Fig. 6 e Variations of the normalized ECOP, normalized COP and the specific entropy generation rate with respect to the specific heating load (TG [ 393 K, TE [ 288 K, TO [ 313 K, Tenv [ 290 K, UG [ UE [ UO [ 500 W KL1 mL2, I [ 1.025, x [ 1.082).
Fig. 8 e Effects of I (a) and s (b) on the coefficient of performance, specific heating load and specific entropy generation rate at maximum ECOP objective function conditions for different heat leakage coefficients (TG [ 393 K, TE [ 288 K, TO [ 313 K, Tenv [ 290 K, UG [ UE [ UO [ 500 W KL1 mL2, I [ 1.025).
" COP ¼ 1
#1 pffi #" pffiffi xðTO TE Þ I þ I ITE pffiffi 1þ aðB ITO Þ ITO 1 þ I B
aðB ITO Þ pffiffi xðTO TE Þ q ¼ Iþ I " # pffiffi ( 1 1 Iεr TE pffi xðTO TE Þ þ εr 1 þ TE TO ITO 1 þ I B #) " aðB ITO Þ pffiffi Iþ I
(28)
(29)
s ¼ Fig. 7 e Variation of the ECOP objective function with respect to COP for various source temperature ratio values (TG [ 393 K, TE [ 288 K, Tenv [ 290 K, UG [ UE [ UO [ 500 W KL1 mL2, I [ 1.025, x [ 1.082).
(30)
where
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h pffiffi pffiffi 2 i aTE ðTO TE Þ 1 þ I x T þ I ðTO TE =IÞa þ ðTO TE Þ 1 þ I x ðTO TE ÞTE TO x B¼ pffiffi 2 aTE ðTO TE Þ 1 þ I x
(31)
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When TG / N, this means that the heat reservoir at temperature TG corresponds to a work source and therefore Eqs. (27)e(30) describe precisely the new thermo-ecological optimization performance of an irreversible Carnot heat pump. Moreover, when I ¼ 1, Eqs. (18)e(21) and Eqs. (27)e(30) are respectively, just the new thermo-ecological optimization performances of an endoreversible three-heat-source heat pump and an endoreversible Carnot heat pump with heat leak loss.
5.
Conclusion
The ecological performances of the irreversible three-heatsource absorption heat pumps have been optimized by considering the new thermo-ecological criterion as the objective function. The chosen objective function, based on the coupling between the first and the second law of thermodynamics, is the ecological coefficient of performance (ECOP) and is defined as the heating load per unit loss rate of availability. The optimum performance parameters, such as the internal working fluid temperatures, the specific heating load, the coefficient of performance, the specific entropy generation rate and the heat-transfer area distributions have been obtained analytically by maximizing the defined thermoecological objective function for an irreversible absorption heat pump with respect to the internal working fluid temperatures in the main components of the system. The effects of the internal irreversibility, heat leakage coefficient and source temperature ratio on the global and optimal ECOP have been examined and discussed by the mean of the plots. The results show that the maximum COP and the maximum ECOP occur for the same operating conditions in spite of their different meaning. The coefficient of performance takes into account only the first law of thermodynamics even though the ecological coefficient of performance takes into account both the first and second law of thermodynamics and therefore can enhance the system performances by reducing the irreversible losses occur in the system. The present optimization study may provide a basis for the design of real absorption heat pump driving in the best ecological performance conditions.
references
Bejan, A., 1989. Theory of heat transfer-irreversible refrigeration plant. Int. J. Heat Transfer 32, 1631e1639. Chen, J., 1997. Optimal performance analysis of irreversible cycles used as heat pumps and refrigerators. J. Phys. 30 (4), 582e587. Chen, J., Yan, Z., 1989a. Equivalent combined systems of threeheat-reservoir heat-pumps. J. Chem. Phys. 90 (9), 4951e4955. Chen, J., Yan, Z., 1989b. Unified description of endoreversible cycles. Phys. Rev. A 39 (8), 4140e4147. Chen, L., Bi, Y., Wu, C., 1999a. Unified description of endoreversible cycles for linear phenomenological heattransfer law. Int. J. Energy Environ. Econ. 9 (2), 77e93.
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Chen, L., Qin, X., Sun, F., 2005. Irreversible absorption heat-pump and its optimal performance. Appl. Energy 81 (1), 55e71. Chen, L., Wu, C., Sun, F., 1997. Optimal coefficient of performance and heating load relationship of a three-heat-reservoir endoreversible heat-pump. Energy Convers. Manag. 38 (8), 727e733. Chen, L., Wu, C., Sun, F., Cao, S., 1999b. Maximum profit performance of a three-heat-reservoir heat-pump. Int. J. Energy Res. 23 (7), 773e777. Fu, L., Zhao, X.L., Zhang, S.G., Jiang, Y., 2009. Laboratory research on combined cooling, heating and power (CCHP) systems. Energy Convers. Manag. 50 (4), 977e982. Goktun, S., 1996. The effect of irreversibilities on the performance of a three-heat-source heat-pump. J. Phys. D 29 (20), 2823e2825. Herold, K.E., 1999. Performance predictions of absorption cycles using an endo-reversible model. In: International Sorption Heat-pump Conference, Munich, pp. 1e6. Huang, Y.W., 2009. Local asymptotic stability of an irreversible heat pump subject to total thermal conductance constraint. Energy Convers. Manag. 50 (6), 1444e1449. Huang, Y., Sun, D., Kang, Y., 2008. Performance optimization for an irreversible four-temperature-level absorption heat pump. Int. J. Therm. Sci. 47, 479e485. Kodal, A., Sahin, B., Ekmekci, I., Yilmaz, T., 2003. Thermoeconomic optimization for irreversible absorption refrigeration and heat pumps. Energy Convers. Manag. 44 (1), 109e123. Ngouateu Wouagfack, P.A., Tchinda, R., 2011. Performance optimization of three-heat-source irreversible refrigerator based on a new thermo-ecological criterion. Int. J. Refrigeration 34, 1008e1015. Qin, X., Chen, L., Sun, F., Wu, C., 2005. Compromise optimization between heating load and entropy production rate for endoreversible absorption heat pumps. Int. J. Ambient Energy 26, 106e112. Qin, X., Chen, L., Sun, F., Wu, C., 2006. Performance of an endoreversible four-heat-reservoir absorption heat pump with a generalized heat transfer law. Int. J. Therm. Sci. 45 (6), 627e633. Qin, X., Chen, L., Sun, F., Lin, H., 2007. Model of real absorption heat pump cycle with a generalized heat transfer law and its performance. Proc. IMechE, Part A: Power Energy 221 (7), 907e916. Ust, Y., 2004. Ecological performance analysis and optimization of power-generation systems, Ph.D. thesis Progress Report, Yildiz Technical University, Istanbul. Ust, Y., Sahin, B., 2007. Performance optimization of irreversible refrigerator based on a new thermo-ecological criterion. Int. J. Refrigeration 30, 527e534. Ust, Y., Sahin, B., Sogut, O.S., 2005. Performance analysis and optimization of an irreversible dual-cycle based on an ecological coefficient of performance criterion. Appl. Energy 82, 23e39. Wu, S., Lin, G., Chen, J., 2005. Optimum thermoeconomic and thermodynamic performance characteristics of an irreversible three-heat-source heat pump. Renew. Energy 30, 2257e2271. Yan, Z., Lin, G., 2000. Ecological optimization criterion for an irreversible three-heat-source refrigerator. Appl. Energy 66, 213e224. Zhao, X., Fu, L., Zhang, S., 2010. General thermodynamic performance of irreversible absorption heat pump. Energy Convers. Manag. 52, 494e499.
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journal homepage: www.elsevier.com/locate/ijrefrig
The new thermo-ecological performance optimization of an irreversible three-heat-source absorption heat pump Paiguy Armand Ngouateu Wouagfack a,*, Re´ne´ Tchinda b a b
L2MSP, Department of Physics, University of Dschang, PO Box 67, Dschang, Cameroon LISIE, University Institute of Technology Fotso Victor, University of Dschang, PO Box 134, Bandjoun, Cameroon
article info
abstract
Article history:
The new thermo-ecological performance optimization of an absorption heat pump oper-
Received 13 July 2011
ating between three temperature levels with the losses of heat resistance, internal irre-
Received in revised form
versibility and leakage is analyzed by taking the ecological coefficient of performance
19 September 2011
(ECOP) as an objective function. The new thermo-ecological criterion takes into account the
Accepted 22 September 2011
first and second law of thermodynamics and is defined as the heating rate per unit loss rate
Available online 1 October 2011
of availability. The ecological coefficient of performance has been expressed and maximized in terms of the temperatures of the working fluid in the main components of the
Keywords:
system. The corresponding optimal temperatures and other optimum performance
Irreversibility
parameters have been derived analytically, and the effects of the internal irreversibility,
Heat pump
the heat leakage coefficient and the source temperature ratio on the global and optimal
Absorption system
performances are discussed. The obtained results may provide a general theoretical tool
Heat source
for the ecological design of absorption heat pumps. ª 2011 Elsevier Ltd and IIR. All rights reserved.
Performance Environment
Nouvelle optimisation thermo-e´cologique de la performance d’une pompe a` chaleur irre´versible a` absorption munie de trois sources de chaleur Mots cle´s : irre´versibilite´ ; pompe a` chaleur ; syste´me a` absorption ; source de chaleur ; performance ; environnement
1.
Introduction
Energy recovery is becoming more and more important in the industry where enormous heat is wasted. Absorption heat pump is considered to be the most competitive equipment for energy saving when it uses high grade heat source to recover waste heat (Zhao et al., 2010; Fu et al., 2009). In
recent years, the thermodynamics of real absorption heatpump cycle has been performed and many significant results have been obtained (Zhao et al., 2010; Huang, 2009). A three-heat-reservoir absorption heat pump is an important theoretical model of the apparatus. It is a simplified model of an absorption heat pump in that the temperature of the absorber is equal to that of the condenser, but in a real
* Corresponding author. Tel.: þ237 77 18 58 71. E-mail address: [email protected] (P.A. Ngouateu Wouagfack). 0140-7007/$ e see front matter ª 2011 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2011.09.008
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Nomenclature
3r 2
q s T U
total heat-transfer area (m ) coefficient of performance normalized COP ecological coefficient of performance normalized ECOP internal irreversibility parameter thermal conductance (W K1) rate of heat transfer (W) specific heating load (W m2) specific entropy generation rate (W K1 m2) temperature (K) overall heat-transfer coefficient (W K1 m2)
Symbol s_ s x
entropy generation rate (W K1) source temperature ratio (TO/TG) heat leakage coefficient (W K1 m2)
A COP COP ECOP ECOP I K Q_
absorption heat pump usually it is not (Zhao et al., 2010). Some researchers optimized the performance of the threeheat-reservoir heat-pump cycle with: (i) the loss of heat resistance, (ii) the loss of heat resistance and internal irreversibility, and (iii) the loss of heat resistance, heat leakage and internal irreversibility. Chen and Yan (1989a,b), Herold (1999), Goktun (1996) and Chen (1997) analyzed the performance of the three-heat-reservoir absorption heat-pump cycle with the loss of heat resistance (Chen and Yan, 1989a,b; Herold, 1999), with losses due to heat resistance and internal irreversibility (Goktun, 1996), and with losses due to heat resistance, heat leakage and internal irreversibility (Chen, 1997) with the linear (Newtonian) heat-transfer law. Chen et al. (1997, 1999a,b) studied the performance of the endoreversible three-heat-reservoir absorption heatpump cycle with the linear phenomenological heat-transfer 1 _ Þ. Chen et al. (2005) and Qin et al. (2006, 2007) law, QfDðT performed similar optimization works to research the performance of a four-heat-source absorption heat pump with the linear (Newtonian) heat-transfer law (Chen et al., 2005), and with the generalized heat-transfer law (Qin et al., 2006, 2007). Kodal et al. (2003) and Wu et al. (2005) analyzed the thermo-economic performance of the threeheat-reservoir heat-pump cycle. None of the optimization criteria of the three-heat-source absorption heat pump defined in the above referenced works describes its performance from the thermo-ecological point of view. Yan and Lin (2000) established an ecological optimal performance of an absorption system for cooling application. They obtained that the ecological optimization criterion (E ) is more advantageous than the maximum cooling load in term of entropy production and therefore, it is beneficial for making a more rational use of energy of absorption systems so as to save energy. During the last decade, Qin et al. (2005) investigated the optimal ecological performance of endoreversible absorption heat pumps and Huang et al. (2008) performed the thermo-ecological optimization works for irreversible absorption heat pumps. In the thermo-ecological optimization studies of an irreversible absorption heat pump (Huang
coefficient of performance for reversible threeheat-source heat pump
Subscripts 1 working fluid in generator 2 working fluid in evaporator 3 working fluid in absorber and condenser A absorber C condenser E evaporator env environment conditions G generator L heat leakage max maximum O absorber and condenser Superscripts * at maximum ECOP
et al., 2008), the ecological objective function is defined as _ The first term is the heating load and the E ¼ q_ mTE s. second term is its dissipation where m is the dissipation coefficient of the heating load, TE is the temperature of evaporator and s_ is the entropy generation rate. When the ecological objective function (E ) proposed in the above referenced works is examined, one can see that the objective function (E ) may take negative values. At this condition, loss of heating load (second term of E function) is greater than the heating load (first term of E function). Such an objective function in a performance analysis can be defined mathematically, however, it needs interpretation to comprehend this situation thermodynamically (Ust et al., 2005; Ust and Sahin, 2007). In a recent study, Ust (2004) proposed a new ecological objective function called ecological coefficient of performance (ECOP) and defined as the work-energy (e.g. power of an engine, cooling rate of a refrigerator, or heating rate of a heat pump) per unit loss rate of availability which is produced by the entropy generation in the system and its surroundings. The proposed ecological objective function ECOP gives the information about to the loss rate of availability or entropy generation rate in order to produce a certain power of an engine, cooling rate of a refrigerator or heating rate of a heat pump. It should be noted that for a certain power of an engine, cooling rate of a refrigerator or heating rate of a heat pump the entropy generation is minimum at maximum ECOP. Like the ecological optimization procedure, the goal of a new thermo-ecological optimization procedure is to achieve the best compromise between the work-energy (e.g. power of an engine, cooling rate of a refrigerator, or heating rate of a heat pump) and its dissipation. The ECOP function always has positive values and it is dimensionless just like the thermal efficiency in the heat engines or the coefficient of performance in the refrigerators and heat pumps. Ust et al. (2005) established the utility of the new ecological performance coefficient. They obtained that the entropy generation rate at ECOPmax conditions is less than that at maximum ecological function (Emax). Thus ECOP criterion has a marked advantage in
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 5 ( 2 0 1 2 ) 7 9 e8 7
ecological perspective over the E criterion in term of entropy generation. Very recently a performance analysis and optimization based on a new thermo-ecological optimization criterion has been carried out for mechanical refrigerators (Ust and Sahin, 2007) and absorption refrigerators (Ngouateu Wouagfack and Tchinda, 2011). In the present performance optimization work, an ecological performance analysis for an irreversible three-heatsource absorption heat pump has been carried out by employing the ECOP as objective function. The optimal performance and the design parameters which maximized the ECOP criterion have been determined, and also, the effects of the major irreversibilities on the thermo-ecological performances are discussed.
2. A general irreversible absorption heatpump cycle model and thermodynamic analysis It can be observed in Fig. 1 (Huang et al., 2008) that a generator, an absorber, a condenser and an evaporator are the main components of an absorption heat-pump system. The first law of thermodynamics of the absorption heat pump shown schematically in Fig. 1 is written as: Q_ G þ Q_ E Q_ C Q_ A ¼ 0
(1)
The expression of Eq. (1) is obtained by neglecting the work input required by the solution pump. It is negligibly small compared to the energy input to the generator and, therefore, is often neglected in the analysis. It may be assumed for the absorption heat pump shown in Fig. 1 that the flow of the working fluid in the cycle system is stable and that the different parts of the working fluid exchange heat with the heat reservoirs at temperatures TG, TE and TO. In the others words, the system rejects heat from the condenser and absorber to the
81
heated space at the same temperature TO. It is also assumed that the working fluid in the condenser and absorber has the same temperature T3 (Kodal et al., 2003; Wu et al. 2005). This assumption is reasonable because the working fluid in the condenser and absorber exchanges heat with the heated space at the same temperature. With the above assumptions, the absorption heat pump becomes the three-heat-source absorption heat pump and the cycle of the working fluid consists of three irreversible isothermal and three irreversible adiabatic processes. There are thermal resistances between the working fluid and the external heat reservoirs. The temperatures of the working fluid in the three isothermal processes are different from those of the external heat reservoirs so that heat is transferred under a finite temperature difference, as shown in Fig. 2 (Ngouateu Wouagfack and Tchinda, 2011). In Fig. 2, Q_ O ¼ Q_ C þ Q_ A , Q_ L is the heat leak from the heated space at temperature TO to the heat sink at temperature TE. T1 and T2 are, respectively, the temperatures of the working fluid in the generator and evaporator. Real absorption three-heat-source heat pumps are complex devices and suffer from a series of irreversibilities. Besides the irreversibility of finite-rate heat transfer, which is considered in the endoreversible cycle models and the heat leak from the heated space to the heat sink, there also exist other sources of irreversibility. For example, the internal dissipation of the working fluid will be another main source of irreversibility, which can decrease the coefficient of performance and the heating load of absorption heat pumps. According to the second law of thermodynamics, we have: I
Q_ Q_ Q_ dQ_ ¼ Gþ E O<0 T1 T2 T3 T
(2)
We can introduce an irreversibility factor: Q_ G Q_ E Q_ O þ ¼0 T1 T2 IT3
(3)
Obviously, the performance of an irreversible cycle affected by thermal resistances is directly dependent on the heattransfer law between the working fluid and the heat reservoirs. When the heat transfer between the working fluid and
Fig. 1 e Schematic diagram of an absorption heat pump (Huang et al., 2008).
Fig. 2 e The irreversible cycle model of a three-heat-source absorption heat pump (Ngouateu Wouagfack and Tchinda, 2011).
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_ the heat reservoirs obeys linear heat-transfer law ðQfDðTÞÞ, one has: Q_ G ¼ UG AG ðTG T1 Þ (4) Q_ E ¼ UE AE ðTE T2 Þ
(5)
Q_ O ¼ UO ðAA þ AC ÞðT3 TO Þ
(6)
Q_ L ¼ KL ðTO TE Þ
(7)
where AG, AE, AC and AA are, respectively, the heat-transfer areas of the generator, evaporator, condenser and absorber, KL is the heat leak coefficient, UG and UE are the overall heattransfer coefficients of the generator and evaporator, and it is assumed that the condenser and absorber have the same overall heat-transfer coefficient UO. The rate of heat leakage Q_ L given in Eq. (7) from the heated space at temperature TO to the heat sink at temperature TE was first provided by Bejan (1989). The total heat-transfer area between the cycle system and the external heat reservoirs is: A ¼ A G þ AE þ AO
(8)
Q_ O Q_ L Q_ G Q_ E Q_ L s_ TO TG TE s¼ ¼ A A 1 1 εr T1 ðT2 IT3 Þ xðTO TE Þ þ 1 ¼ TE TO IT3 ðT2 T1 Þ T1 ðIT3 T2 Þ T2 ðIT3 T1 Þ þ UG ðTG T1 ÞðT1 T2 ÞIT3 UE ðTE T2 ÞðT2 T1 ÞIT3 1 1 þ UO ðT3 TO Þ
(11)
where the parameter x ¼ KL/A represents the heat leakage coefficient and its dimension is W K1 m2 and εr ¼ (1 TE/TG) TO/(TO TE) is the coefficient of performance for reversible three-heat-source heat pump. The ecological coefficient of performance (Ust, 2004; Ust et al., 2005; Ust and Sahin, 2007; Ngouateu Wouagfack and Tchinda, 2011) of an irreversible three-heat-source absorption heat pump is defined as the ratio of heating load to the loss rate of availability, i.e.,
where AO ¼ AC þ AA.
ECOP ¼
Q_ O Q_ L 1 ¼ 1 Tenv s_ Tenv T1 O TE
1 1 εr T1 ðT2 IT3 Þ T1 ðIT3 T2 Þ T2 ðIT3 T1 Þ 1 1 þ 1 xðTO TE Þ þ þ IT3 ðT2 T1 Þ UG ðTG T1 ÞðT1 T2 ÞIT3 UE ðTE T2 ÞðT2 T1 ÞIT3 UO ðT3 TO Þ (12)
According to the standard definitions of the specific heating load (q) and the coefficient of performance (COP) of an absorption heat pump, we obtain for an irreversible threeheat-source heat pump: Q_ Q_ L q¼ O A T1 ðIT3 T2 Þ T2 ðIT3 T1 Þ þ ¼ UG ðTG T1 ÞðT1 T2 ÞIT3 UE ðTE T2 ÞðT2 T1 ÞIT3 1 1 þ xðTO TE Þ (9) UO ðT3 TO Þ
ECOP ¼
Q_ O Q_ L 1 ¼ 1 Tenv s_ Tenv T1 O TE
where Tenv conditions.
is
the
temperature
in
the
environment
3. Performance optimization of three-heatsource heat pump based on ECOP criterion For the sake of convenience, let x ¼ IT3/T1, y ¼ IT3/T2, z ¼ IT3. Then Eq. (12) may be written as:
1 1 εr ð1 yÞ xð1 yÞ yð1 xÞ 1 1 þ þ þ 1 xðTO TE Þ xy UG ðTG x zÞðx yÞ UE ðTE y zÞðy xÞ UO ðz TÞ
Q_ Q_ L Q_ O Q_ 1 L ¼ COP ¼ O Q_ G Q_ G Q_ O IT3 ðT2 T1 Þ T1 ðIT3 T2 Þ 1 xðTO TE Þ ¼ T1 ðT2 IT3 Þ UG ðTG T1 ÞðT1 T2 ÞIT3 T2 ðIT3 T1 Þ 1 þ þ UE ðTE T2 ÞðT2 T1 ÞIT3 UO ðT3 TO Þ
(10)
The specific entropy generation rate (s) for an irreversible three-heat-source heat pump with heat leak loss is
(13)
where T ¼ ITO and U ¼ UO/I. Using Eq. (13) and the extremal conditions vECOP/vx ¼ 0, vECOP/vy ¼ 0 and vECOP/vz ¼ 0, one can derive the temperature of the working fluid in the generator, evaporator, absorber and condenser which correspond to the maximum ECOP (ECOPmax) as follows: T1 ¼ TG B=½ð1 þ b1 ÞB b1 T
(14)
T2
(15)
¼ TE B=½ð1 þ b2 ÞB b2 T
T3 ¼ B=I
(16)
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83
where
b1 ¼
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi U=UG ; b2 ¼ U=UE ; B ¼
½UTE ðTO TE Þð1 þ b2 Þb2 xT þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i 2 ðT TE ÞU þ ðTO TE Þð1 þ b2 Þ x ðTO TE ÞTE Tx 2
UTE ðTO TE Þð1 þ b2 Þ x
(17)
Substituting Eqs. (14)e(16) into Eqs. (9)e(12) yields the maximum ECOP:
ECOPmax ¼
1 1 Tenv T1 O TE
1 1 1 1 þ Db1 ½ð1 þ b1 ÞB b1 TT1 G þ ð1 DÞb2 ½ð1 þ b2 ÞB b2 TTE 1 εr D 1 xðTO TE Þ UðB TÞ
(18)
and the optimal coefficient of performance, the optimal specific heating load and the optimal specific entropy generation rate at the maximum ECOP conditions, respectively, as: 1 1 þ Db1 ½ð1 þ b1 ÞB b1 TT1 G þ ð1 DÞb2 ½ð1 þ b2 ÞB b2 TTE COP ¼ D1 1 xðTO TE Þ UðB TÞ
q ¼
UðB TÞ 1 1 þ Db1 ½ð1 þ b1 ÞB b1 TT1 G þ ð1 DÞb2 ½ð1 þ b2 ÞB b2 TTE
pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffi UG AG þ UE AE ¼ UAO
(19)
(25)
xðTO TE Þ
1 1 s ¼ TE TO "
( xðTO TE Þ þ ðεr D 1Þ
UðB TÞ 1 1 þ Db1 ½ð1 þ b1 ÞB b1 TT1 G þ ð1 DÞb2 ½ð1 þ b2 ÞB b2 TTE
where D¼
(20)
#)
(21)
TE þ b2 T ð1 þ b2 ÞB ½ð1 þ b1 ÞB b1 TTE =TG þ b2 T ð1 þ b2 ÞB
From Eqs. (1), (4)e(6), (8) and (14)e(16), we find that, when the three-heat-source absorption heat pump is operated in the state of maximum ecological coefficient of performance, the relations between the heat-transfer areas of the heat exchangers and the total heat-transfer area are determined by: 1 AG b21 DT1 G ðB TÞf1 B=½ð1 þ b1 ÞB b1 Tg ¼ 1 A 1 þ Db1 ½ð1 þ b1 ÞB b1 TTG þ ð1 DÞb2 ½ð1 þ b2 ÞB b2 TT1 E (22) 1 AE b22 T1 E ð1 DÞðB TÞf1 B=½ð1 þ b2 ÞB b2 Tg ¼ 1 A 1 þ Db1 ½ð1 þ b1 ÞB b1 TTG þ ð1 DÞb2 ½ð1 þ b2 ÞB b2 TT1 E (23) AO 1 ¼ 1 A 1 þ Db1 ½ð1 þ b1 ÞB b1 TT1 G þ ð1 DÞb2 ½ð1 þ b2 ÞB b2 TTE (24) From Eqs. (22)e(24), we obtain a concise optimum relation for the distribution of the heat-transfer areas of three-heatsource absorption heat pump as:
4.
Optimization results and discussion
The ecological coefficient of performance (ECOP) gives information about the entropy generation rate or loss rate of availability in order to produce certain amount of heating load. The utility of the new thermo-ecological optimization criterion is that: at the maximum ECOP conditions, the absorption heat pump develops a heating load by causing lesser dissipation in the environment. Therefore the higher the ECOP is, the more the absorption heat pump works under a better compromise between the heating load and its losses which can affect the environment. In order to see the results of ECOP optimization for an irreversible three-heat-source absorption heat pump, some numerical examples are given and discussed. The numerical calculations are carried out by employing the values of the relevant parameters used by Kodal et al. (2003). Fig. 3 shows the effects of internal irreversibility (I ), heat leakage coefficient (x) and source temperature ratio (s) on the optimal temperature of the working fluid in the generator, evaporator, condenser and absorber at maximum ECOP objective function conditions. In this figure, the variations of the optimal working fluid temperatures with respect to internal irreversibility parameter and source temperature ratio have been plotted for different values of heat leakage coefficient. It can be observed that the optimal temperature of the working fluid in generator (T1*) and in evaporator (T2*) decreases with the increase of I and s. For a fixed I and s, the optimal temperature of the working fluid in generator and evaporator decreases with the increase of x. The optimal
84
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temperature of the working fluid in the condenser and absorber (T3*) increases with the increase of I and s and increases with the increase of x for a fixed I and s. Figs. 4 and 5 show the variation of the ECOP with the specific heating load and specific entropy generation rate for different values of I and s. By observing these figures, on can see that as I and s increase the global and optimal ECOP which correspond to the maximum ECOP decrease. In Figs. 4a and 5a, it can be seen that the global and optimal ECOP decrease significantly from an endoreversible situation (I ¼ 1) to an irreversible situation (I > 1). Fig. 6 shows the variations of the normalized ECOP ðECOP ¼ ECOP=ECOPmax Þ, normalized COP ðCOP ¼ COP=COPmax Þ and the specific entropy generation rate (s) with respect to the specific heating rate (q). One interesting observation from this figure is that maximum of the ECOP and COP coincides although their functional form is different: the coefficient of performance ðCOP ¼ Q_ O Q_ L =Q_ G Þ gives information about the necessary heat rate input in order to produce certain amount of heating load and the ecological coefficient of performance _ gives information about the entropy ðECOP ¼ Q_ O Q_ L =Tenv sÞ generation rate or loss rate of availability in order to produce certain amount of heating load. It has also been seen numerically and analytically that the performance parameters T1*, T2*, T3*, q*, s*, AG*, AE*, AO* and COP* ¼ COPmax at the maximum ECOP and maximum COP are same. Getting the same optimal conditions at the maximums of the ECOP and COP is a normal
Fig. 4 e Variation of the ECOP objective function with respect to the specific heating load for various internal irreversibility parameters (a) and for various source temperature ratios (b) (TG [ 393 K, TE [ 288 K, TO [ 313 K, Tenv [ 290 K, UG [ UE [ UO [ 500 W KL1 mL2, I [ 1.025, x [ 1.082).
Fig. 3 e Variations of the optimal working fluid temperatures at maximum ECOP conditions with respect to internal irreversibility parameter (a) and source temperature ratio (b) for different heat leakage coefficients (TG [ 393 K, TE [ 288 K, TO [ 313 K, UG [ UE [ UO [ 500 W KL1 mL2, I [ 1.025).
result. Since, for a certain heating load, the maximum COP occurs from minimum heat input required by the generator so that minimization of adverse environmental impacts in term of energy saving. The minimum environmental pollution is also achieved by maximizing the ECOP. This result has been also obtained in references Ust and Sahin (2007) and Ngouateu Wouagfack and Tchinda (2011) for compression refrigerators and absorption refrigerators respectively. Although the optimal performance conditions ECOP and COP criteria are same, their impact on the system design performance is different. The coefficient of performance is used to evaluate the performance and the efficiency of heat producing systems. This method only takes into account the first law of thermodynamics which is concerned only with the conversion of energy, and therefore, cannot show how or where irreversibilities in a system or process occur. Also, when different sources and forms of energy are involved within a system, the COP criterion of the three-heat-source heat pump doesn’t describe its performance from the view point of the energy quality involved. This factor is taken into account by the
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 5 ( 2 0 1 2 ) 7 9 e8 7
85
Fig. 5 e Variation of the ECOP objective function with respect to the specific entropy generation rate for various internal irreversibility parameters (a) and for various source temperature ratios (b) (TG [ 393 K, TE [ 288 K, TO [ 313 K, Tenv [ 290 K, UG [ UE [ UO [ 500 W KL1 mL2, I [ 1.025, x [ 1.082).
second law of thermodynamics and appears in the ECOP criterion. For this important reason, the ECOP criterion can enhance the system performance of the three-heat-source heat pump by reducing the irreversible losses in the system. A better understanding of the second law of thermodynamics reveals that the ecological coefficient of performance optimization is an important technique in achieving better operating conditions. ECOP and COP are related as: ECOP ¼
COP Tenv ð1=TO 1=TE Þðεr COPÞ
(26)
Fig. 7 shows the variations of ECOP with respect to COP for various s values. ECOP increases with the increase of COP and the ECOPmax and COPmax conditions which are end points of the curves decrease with the increase of s values.
ECOPmax ¼
1 1 Tenv T1 O TE
Fig. 8 shows the effects of I, s and x on the optimal coefficient of performance, specific heating load and specific entropy generation rate at maximum ECOP objective function conditions. The optimal specific heating rate and the optimal specific entropy generation rate at ECOPmax conditions increase with the increase of I, s and x, while the optimal coefficient of performance at ECOPmax conditions decreases. From the above performance optimization works based on ECOP criterion carried out for irreversible three-heatsource heat pumps, we can deduce important optimization results regarding endoreversible three-heat-source heat pumps and irreversible and endoreversible Carnot heat pumps with heat leak loss. For instance, when TG / N and UE ¼ UO ¼ a, Eqs. (18)e(21) can be, respectively, simplified to
1 #( pffi )1 pffiffi xðTO TE Þ I þ I Iεr TE pffiffi 1 εr þ 1 aðB ITO Þ ITO 1 þ I B "
(27)
86
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 5 ( 2 0 1 2 ) 7 9 e8 7
Fig. 6 e Variations of the normalized ECOP, normalized COP and the specific entropy generation rate with respect to the specific heating load (TG [ 393 K, TE [ 288 K, TO [ 313 K, Tenv [ 290 K, UG [ UE [ UO [ 500 W KL1 mL2, I [ 1.025, x [ 1.082).
Fig. 8 e Effects of I (a) and s (b) on the coefficient of performance, specific heating load and specific entropy generation rate at maximum ECOP objective function conditions for different heat leakage coefficients (TG [ 393 K, TE [ 288 K, TO [ 313 K, Tenv [ 290 K, UG [ UE [ UO [ 500 W KL1 mL2, I [ 1.025).
" COP ¼ 1
#1 pffi #" pffiffi xðTO TE Þ I þ I ITE pffiffi 1þ aðB ITO Þ ITO 1 þ I B
aðB ITO Þ pffiffi xðTO TE Þ q ¼ Iþ I " # pffiffi ( 1 1 Iεr TE pffi xðTO TE Þ þ εr 1 þ TE TO ITO 1 þ I B #) " aðB ITO Þ pffiffi Iþ I
(28)
(29)
s ¼ Fig. 7 e Variation of the ECOP objective function with respect to COP for various source temperature ratio values (TG [ 393 K, TE [ 288 K, Tenv [ 290 K, UG [ UE [ UO [ 500 W KL1 mL2, I [ 1.025, x [ 1.082).
(30)
where
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h pffiffi pffiffi 2 i aTE ðTO TE Þ 1 þ I x T þ I ðTO TE =IÞa þ ðTO TE Þ 1 þ I x ðTO TE ÞTE TO x B¼ pffiffi 2 aTE ðTO TE Þ 1 þ I x
(31)
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 5 ( 2 0 1 2 ) 7 9 e8 7
When TG / N, this means that the heat reservoir at temperature TG corresponds to a work source and therefore Eqs. (27)e(30) describe precisely the new thermo-ecological optimization performance of an irreversible Carnot heat pump. Moreover, when I ¼ 1, Eqs. (18)e(21) and Eqs. (27)e(30) are respectively, just the new thermo-ecological optimization performances of an endoreversible three-heat-source heat pump and an endoreversible Carnot heat pump with heat leak loss.
5.
Conclusion
The ecological performances of the irreversible three-heatsource absorption heat pumps have been optimized by considering the new thermo-ecological criterion as the objective function. The chosen objective function, based on the coupling between the first and the second law of thermodynamics, is the ecological coefficient of performance (ECOP) and is defined as the heating load per unit loss rate of availability. The optimum performance parameters, such as the internal working fluid temperatures, the specific heating load, the coefficient of performance, the specific entropy generation rate and the heat-transfer area distributions have been obtained analytically by maximizing the defined thermoecological objective function for an irreversible absorption heat pump with respect to the internal working fluid temperatures in the main components of the system. The effects of the internal irreversibility, heat leakage coefficient and source temperature ratio on the global and optimal ECOP have been examined and discussed by the mean of the plots. The results show that the maximum COP and the maximum ECOP occur for the same operating conditions in spite of their different meaning. The coefficient of performance takes into account only the first law of thermodynamics even though the ecological coefficient of performance takes into account both the first and second law of thermodynamics and therefore can enhance the system performances by reducing the irreversible losses occur in the system. The present optimization study may provide a basis for the design of real absorption heat pump driving in the best ecological performance conditions.
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