Introduction
The absorption refrigerators can be driven by ‘low-grade’ heat energy such as waste heat in industries, solar energy and geothermal energy, and have a large potential for reducing the heat pollution for the environment. Thus, absorption refrigerators for industrial and domestic use are generating renewed interest throughout the world. In the last years, finite-time thermodynamics (or endoreversible thermodynamics, or entropy generation minimization, or thermodynamic optimization) [
1,
2,
3,
4] was applied to the performance study of absorption refrigerators, and a lot of results, which are different from those by using the classical thermodynamics, have been obtained. Yan
et al. [
6], Wijeysundera [
7,
8], Goktun [
9] and Chen
et al. [
10] analyzed the performance of the three-heat-reservoir endoreversible [
6,
7,
8] and irreversible [
9,
10] absorption refrigeration cycles with Newton’s heat transfer law. Chen
et al. [
11,
12,
13] studied the performance of the three-heat-reservoir endoreversible [
11,
12] and irreversible [
13] absorption refrigeration cycles with another linear heat transfer law, i.e., linear phenomenological law,
Q∝∆(
T−1). A three-heat-reservoir absorption refrigerator is a simplified model that the temperature of a condenser is equal to that of an absorber, but a real absorption refrigerator is not. Therefore, a four-heat-reservoir absorption refrigeration cycle model is closer to a real absorption refrigerator. The performance of the four-heat-reservoir absorption refrigeration cycle with Newton’s heat transfer law was studied by Chen [
14], Shi
et al. [
15] and Zheng
et al. [
16]. Chen [
14] deduced the maximum cooling load limit and the corresponding coefficient of performance (COP) of the endoreversible four-heat-reservoir absorption refrigeration cycle with the sole irreversibility of heat transfer, Shi
et al. [
15] deduced the fundamental optimal relation between the cooling load and the COP of the endoreversible four-heat-reservoir absorption refrigeration cycle with the sole irreversibility of heat transfer, and Zheng
et al. [
16] deduced the optimal heat transfer surface areas of the four heat exchangers the endoreversible four-heat-reservoir absorption refrigeration cycle with the sole irreversibility of heat transfer. On the basis of these research work, a four-heat-reservoir endoreversible absorption refrigeration cycle with linear phenomenological heat transfer law is established in this paper. The fundamental optimal relation between the COP and the cooling load, as well as the maximum cooling load and the corresponding COP of the cycle are derived. The results can provide the theoretical bases for the optimal design and operation of real absorption refrigerator operating between four temperature levels. The present work is different from a recent work of the authors [
17]. In Ref. [
17], an endoreversible four-heat-reservoir absorption heat-transformer with Newton’s heat transfer law was established, and the fundamental optimal relation between the COP and the heating load, as well as the maximum heating load and the corresponding COP of the cycle were derived.
Physical Model
A four-heat-reservoir endoreversible absorption refrigeration cycle that consists of a generator, an evaporator, an absorber and a condenser is shown in
Fig. 1. The flow of the working fluid in the cycle system is stable and the different parts of the working fluid exchange heat with the heat reservoirs at temperature
TH,
TL,
TO and
TM during the full time (cycle period)
τ, whereas there are thermal resistances between the working fluid and the external heat reservoirs. Therefore, the corresponding working fluid temperatures are
T1,
T2,
T3 and
T4, respectively. Work input required by the solution pump in the system is negligible relative to the energy input to the generator and is often neglected for the purpose of analysis [
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17]. It is assumed that the heat transfers between the working fluid in the heat exchangers and the external heat reservoirs are carried out under a finite temperature difference and obey linear phenomenological heat transfer law [i.e., the heat-flux
Q∝∆(
T−1) ], and these heat exchange processes are isothermal and the equations of heat transfer may be written as
where
U1,
U2,
U3 and
U4 are, respectively, the overall heat-transfer coefficients of the generator, evaporator, condenser and absorber; and
A1,
A2,
A3 and
A4 are, respectively, the heat-transfer surface areas of the generator, evaporator, condenser and absorber. The overall heat-transfer surface area
A is
Fundamental Optimal Relation
From the first law of thermodynamics, one has
From the second law of thermodynamics and the endoreversible property of the cycle, one has
Defining the parameter
a (the distribution of the total heat reject quantity between the condenser and the absorber)
From equations (6) and (7), one can obtain the COP of an endoreversible absorption refrigerator
Using equations (1) - (9), the cooling load of the refrigerator can be written as
where
.
Using equation (10) and the extremal conditions
,
and
, one can derive the temperatures of the working fluid in the generator, absorber, condenser and evaporator, which correspond the optimal cooling load for the given COP. Substituting them into equation (10) yields the fundamental optimal relation between the cooling load and the COP of the four-heat-reservoir endoreversible absorption refrigeration cycle with linear phenomenological heat transfer law as follows
Equation (11) is the major results of this paper. It can reveal the
R −
ε characteristics of a four-heat-reservoir endoreversible absorption refrigeration cycle affected by thermal resistance, and some significant results and new bounds may be derived from it.
Using equations (1)-(4), (11), and the temperatures of the working fluid in the generator, absorber, condenser and evaporator which are derived, one can obtain the optimal distribution relation of the heat-transfer surface areas as follows
From equations (5) and (12)-(14), one can find the relations between the heat-transfer surface areas of each heated exchanger and the total heat-transfer surface area
A as follows
Results and Discussion
1. When
R = 0, one can obtain the reversible COP
εr of the four-heat-reservoir absorption refrigeration cycle
It can be seen that the optimal COP of the four-heat-reservoir endoreversible absorption refrigeration cycle can’t exceed the reversible COP
εr. This shows that the real absorption refrigerators must decrease the COP level if one wants to obtain some cooling load.
2. When
ε <
εr, there exists a maximum cooling load. Using equation (11) and the extremal condition
∂R/(
∂ε) = 0, one can obtain
and
where
Rm is the maximum cooling load, and
εR is the corresponding COP.
Rm and
εR are two important parameters of the four-heat-reservoir endoreversible absorption refrigeration cycle with linear phenomenological heat transfer law, because they determine the upper bound for the cooling load and the lower bound of the COP, and provide a finite-time thermodynamic criteria for the optimal design of real absorption refrigerators, i.e., the real absorption refrigerator design must match the condition
εr > ε ≥
εR to make the refrigerator operates under the optimal conditions.
When
a = 1,
TM =
TO and
U i (
i = 1,2,3,4) =
U, the four-heat-reservoir endoreversible absorption refrigeration cycle with linear phenomenological heat transfer law becomes the three-heat-reservoir endoreversible absorption refrigeration cycle with linear phenomenological heat transfer law, and equations (20) and (21) become [
11]
3. Defining the ratio
x =
TO /
TL of the temperatures of the condenser to the evaporator, and the ratio
y =
TH /
TM of the temperatures of the generator to the absorption, one can analyze the performance numerically. In the calculation,
A = 1100
m2,
TM = 305
K,
TL = 273
K,
U1 = 458.167
kW ⋅
K /
m2,
U2 = 682.223
kW ⋅
K /
m2,
U3 = 1622.5
kW ⋅
K /
m2 and
U4 = 458.167
kW ⋅
K /
m2 are set [
5,
18].
The influence of
a on the optimal cooling load
R versus the COP
ε of the four-heat-reservoir endoreversible absorption refrigeration cycle with
TH = 403
K and
TO = 313
K is shown in
Fig. 2. The influence of
x on the optimal cooling load
R versus the COP
ε with
TH = 403
K and
a = 1.5 is shown in
Fig. 3. The influence of
y on the optimal cooling load
R versus the COP
ε with
TO = 313
K and
a = 1.5 is shown in
Fig. 4.
Figure 2,
Figure 3 and
Figure 4 show that the optimal cooling load
R versus the COP
ε is a parabolic curve, and there exists a maximum cooling load
Rm and the corresponding COP
εR. When
R <
Rm, there exist two different
ε for a fixed
R, one is larger than
εR, and another is smaller than
εR. When
ε <
εR, the COP decreases with the decrease of the cooling load, so the optimal operation range of the COP of the absorption refrigerators should be selected in
εr >
ε ≥
εR.
The influences of
x on the reversible COP
εr, the maximum cooling load
Rm and the corresponding COP
εR versus
a with
TH = 403
K and
y = 1.35 are shown in
Fig. 5,
Fig. 6 and
Fig. 7.
Figure 5,
Figure 6 and
Figure 7 show that for a fixed
a,
εr,
Rm and
εR decrease with the increase of
x. When
a is larger than one in value, the influence of the
εr,
Rm and
εR are less. There is a special point
x = 1.1172 for the cycle. When
x < 1.1172,
εr ≥ 2.0747,
εR ≥ 0.5092, both
εr and
εR decrease with the increase of
a; and when
x > 1.1172,
εr < 2.0747,
εR < 0.5092, both
εr and
εR increase with the increase of
a. These imply that
εr and
εR will reach their asymptotic values, i.e.,
εr → 2.0747 and
εR → 0.5092 when
a tends to infinity.
Rm decreases with the increase of
a, and when
a is larger than one in value, the influence of them are less significant.
The influence of
y on the reversible COP
εr, the maximum cooling load
Rm and the corresponding COP
εR versus
a with
TO = 313
K and
x = 1.14 are shown in
Fig. 8,
Fig. 9 and
Fig. 10.
Figure 8,
Figure 9 and
Figure 10 show that for a fixed
a,
εr,
Rm and
εR increase with the increase of
y. Here
x > 1.1172 holds,
Rm decreases with the increase of
a,
εr and
εR increase with the increase of
a.
4 The performance optimization can be carried out by optimizing the distribution of the heat exchanger total inventory [
19,
20]. Using
UA =
U1A1 +
U2A2 +
U3A3 +
U4A4 to replace equation (5), i.e. using the distribution of the heat conductances to replace the distribution of the heat-transfer surface areas, one can obtain the optimal distribution relation of the heat conductances as follows
From equations (24)-(26), one can obtain
The optimal relation between the cooling load and the COP in this case is as following
Using equation (31) and the extremal condition
∂R/(
∂ε) = 0 yields the maximum cooling load
Rm and the corresponding COP
εR as follows