- Email: [email protected]
A global optimization method for regenerative air refrigeration systems
Accepted Manuscript A Global Optimization Method for Regenerative Air Refrigeration Systems Jun-Hong Hao, Qun Chen, Yun-Chao Xu PII:
S1359-4311(14)00027-1
DOI:
10.1016/j.applthermaleng.2014.01.021
Reference:
ATE 5321
To appear in:
Applied Thermal Engineering
Received Date: 23 August 2013 Revised Date:
1 December 2013
Accepted Date: 15 January 2014
Please cite this article as: J.-H. Hao, Q. Chen, Y.-C. Xu, A Global Optimization Method for Regenerative Air Refrigeration Systems, Applied Thermal Engineering (2014), doi: 10.1016/ j.applthermaleng.2014.01.021. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Highlight
An optimization method is proposed for regenerative air refrigeration systems.
Heat transfer processes in heat exchangers is analyzed by the entransy theory.
A mathematical relation between design parameters and requirements is derived.
This optimization method is validated by analyzing a typical refrigeration system.
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
A Global Optimization Method for Regenerative Air
2
Refrigeration Systems
3
Jun-Hong Hao, Qun Chen *, Yun-Chao Xu
4
Key Laboratory for Thermal Science and Power Engineering of Ministry of Education,
5
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
RI PT
1
6
Abstract: The air refrigeration systems always involve such physical processes as heat transfer
8
processes in heat exchangers, compression processes in compressors and expansion processes in
9
expanders. This contribution proposes a theoretical global optimization method for regenerative
10
air refrigeration systems and introduces the entransy theory to analyze the heat transfer processes
11
in the hot-end heat exchanger, the cold-end heat exchanger and the regenerator. Integration of the
12
heat transfer analyses and the thermodynamic analyses for the compression and expansion
13
processes yields a mathematical model to describe the physical relation between the design
14
requirements and the unknown design parameters, i.e. the heat transfer area of each heat
15
exchanger and the heat capacity rate and the intermediate temperature of the air. Based on this
16
model, the optimization can be converted into a conditional extremum problem. That is, solving
17
the problem via the Lagrange multiplier method offers an optimization equation group, which
18
directly leads to the optimal values of all the unknown parameters. Finally, this optimization
19
method is validated through an optimization case to minimize the total thermal conductance of all
20
the heat exchangers in a typical regenerative air refrigeration system.
AC C
EP
TE D
M AN U
SC
7
21 22
Keywords:
23
thermodynamics, entransy theory
regenerative
air
refrigeration
systems,
optimization,
24
*
Corresponding author, Tel. & Fax: 86-10-62796332. Email: [email protected]
1
energy conservation,
ACCEPTED MANUSCRIPT 1
area, m2
F
Lagrange function
G
heat capacity rate, W/K
K
Heat transfer coefficient, W/(m2· K)
KA
the thermal conductance of exchangers, W/K
p
the pressure of the air in the cycle, Pa
Q, q
heat transfer rate, W
Rh
entransy dissipation-based thermal resistance, K/W
s
specific entropy, J/(kg· K)
T
temperature, K
γ
the adiabatic exponent of air
π
pressure ratio
λ
Lagrange multiplier
Φ
entransy dissipation rate, W K
ω
Lagrange multiplier
3 4
the air outlet of the cold-end exchanger
AC C
2
EP
Subscripts
1
M AN U
TE D
2
SC
A
RI PT
Nomenclatures
the air inlet of the compressor the air outlet of the compressor the air outlet of the hot-end exchanger
5
the air inlet of the expander
6
the air inlet of the cold-end exchanger
a
the air
h
the hot-end
i
the inlet of the exchanger 2
ACCEPTED MANUSCRIPT l
the cold-end
o
the outlet of the exchanger
r
the regenerator
AC C
EP
TE D
M AN U
SC
RI PT
1 2
3
ACCEPTED MANUSCRIPT 1
1. Introduction
The refrigeration systems not only play an important role in modern human lives but also
3
consume more and more energy. In this era with severe energy shortages, energy efficiency
4
promotion via theoretical or engineering optimization methods becomes a very hot topic. However,
5
the strong nonlinear and multivariate characteristic of the refrigeration systems is always the
6
obstacle confusing the researchers.
RI PT
2
In theory, Bejan [1] simplified several actual thermodynamic cycles as Carnot cycles and
8
found the optimal area allocation of all the heat exchangers to maximize the refrigeration
9
capacities. Thereafter, Grazzini and Rinaldi [2] optimized an irreversible inverse Rankine cycle to
10
search the maximal coefficient of performance (COP). Sahin and Kodal [3] analyzed the optimal
11
distribution of heat exchanger areas with the aim of maximizing the cooling load per total cost.
12
Ait-Ali [4] optimized both the refrigerators and heat pumps to maximize their COPs. All these
13
researches contributed a lot in developing the optimization theory for thermodynamic cycles.
14
However, simplifying actual thermodynamic cycles by Carnot cycles in these works is always too
15
general to offer the detailed information for the internal components, such as the performance of
16
compressors and expanders/throttling valves and the physical property of the working fluids.
TE D
M AN U
SC
7
In view of this limitation, for the optimization of air refrigeration cycles, scientists separately
18
analyzed the heat transfer processes in heat exchangers and the compression/expansion processes
19
in compressors and expanders, and integrated both aspects of analyses, which made the theoretical
20
optimization more reasonable and valid. For instance, Chen et al. [5] analyzed an air refrigerator
21
with non-isentropic compression and expansion processes based on the finite-time
22
thermodynamics theory, deduced the relations of the cooling load and the COP to the pressure
23
ratio, and finally maximized the COP. Chen et al. [6] applied the exergy theory to analyze the
24
multi-irreversibilities of an irreversible Brayton refrigeration cycle, and maximized the exergetic
25
efficiency with consideration of the influence of heat leak between heat reservoirs. Tu et al. [7]
26
emphasized the analysis for the influencing factors of the maximization of the cooling load and
27
the COP, i.e. the total heat exchanger inventory, the efficiency of compressor and expander, and
28
the heat capacity rates of all the fluids. Similarly, Liu et al. [8] studied the influences of the
AC C
EP
17
4
ACCEPTED MANUSCRIPT pressure ratio and the inlet temperature of compressor and expander on the optimal COP of an
2
actual air refrigeration cycle. However, because the multiple degrees of freedom of the
3
optimization problem is quite difficult to handle, they had to decrease the multiple degrees of
4
freedom of the system to only one degree via fixing the majority of the unknown parameters,
5
which can obviously make the problems simpler but hardly obtain the global optimal performance
6
of the systems.
RI PT
1
On the other hand, based on the entransy theory introduced by Guo et al. [9], Chen and his
8
colleagues proposed an entransy dissipation-based optimization method to theoretically realize the
9
global optimization of several energy utilization systems with strong multivariate and nonlinear
10
characteristic, e.g. heat exchangers networks in buildings [10,11], thermal management systems in
11
spacecrafts [12] and evaporative cooling systems [13], and found several direct mathematical
12
relations between the design requirements and all the design parameters. According to this
13
mathematical relations, the optimization problems were converted into the conditional extremum
14
problems which offer the govern equation groups for optimization. Furthermore, they extended
15
this method to optimize an air refrigeration cycle [14] and found the minimum of the sum of the
16
hot-end and cold-end heat exchangers.
TE D
M AN U
SC
7
Different from the simple air refrigeration cycles [14], the regenerative air refrigeration
18
cycles here contain two kinds of heat exchangers with different functions, i.e. the hot-end and
19
cold-end heat exchangers and the regenerator. The former transfer heat between the air flow and
20
outside fluids; while the latter recovers heat between hot and cold air flows. And the existence of
21
regenerator apparently affects the performance and optimization of the whole system. This
22
contribution intends to propose a global optimization method for regenerative air refrigeration
23
systems. Through combing the heat transfer analysis for heat exchangers based on the entransy
24
theory and the thermodynamic analysis for compression and expansion processes in compressor
25
and expander, we establish a mathematical relation between the design parameters, i.e. the thermal
26
conductance of each heat exchanger, the heat capacity rate and the intermediate temperature of the
27
air, and the design requirements, and deduce an optimization equation group by the conditional
28
extremum method. Finally, the validity of the method is proved by optimizing a typical
29
regenerative air refrigeration system.
AC C
EP
17
5
ACCEPTED MANUSCRIPT 1
2. Physical model for a regenerative air refrigeration system
Fig. 1 gives the schematic diagram of a regenerative air refrigeration system which consists
3
of a hot-end heat exchanger, a cold-end heat exchanger, a regenerator, a compressor and an
4
expander. Fig. 2 shows the T-s diagram for the refrigeration cycle. The air at the status 1 enters the
5
regenerator and absorbs heat from the hot air with the temperature promoting to the status 2. The
6
air at the the status 2 flows through compressor and leaves with higher temperature and pressure at
7
the status 3 due to compression. Then, the air flows into the hot-end heat exchanger and rejects
8
heat to the hot-end fluid with the temperature dropping to the status 4. After the heat rejection in
9
the regenerator, the air at the status 5 flows into the expander and expands to the status 6 with both
10
temperature and pressure decreasing. Finally, after the heat absorption in the cold-end heat
11
exchanger, the air returns to the the status 1.
M AN U
SC
RI PT
2
Fig. 1 The schematic diagram of a regenerative air regeneration system 12
Fig. 2 The T-s diagram of the regenerative air regeneration cycle
The regenerative air refrigeration system involves two kinds of physical processes: 1) the
14
heat transfer processes in each heat exchanger; 2) the thermodynamic processes, i.e. the
15
compression process in the compressor and the expansion process in the expander. Therefore, in
16
order to establish a physical model of the whole system, it is essential to apply the theory of heat
17
transfer and thermodynamics together to analyze the heat transfer and thermodynamic processes
18
separately.
19
3. Entransy-dissipation based heat transfer analysis
AC C
EP
TE D
13
20
Fig. 3 shows the T-q diagram of the heat transfer process of the counter-flow hot-end heat
21
exchanger in the system, where q represents the heat transfer rate between the hot-end fluid and
22
the air, and T represents the temperatures of the fluids [15]. Th stands for the temperature of the
23
hot-end fluid. T3 and T4 respectively represent the temperatures of air at the inlet and the outlet. i
24
and o separately stand for the inlet and the outlet of the heat exchanger. The total heat transfer rate 6
ACCEPTED MANUSCRIPT 1
Qh is expressed as: Qh = Ga (T3 − T4 ) = Gh (Th ,o − Th ,i ) ,
2
(1)
where Ga and Gh separately stand for the heat capacity rates of the air and the hot-end fluid. Fig. 3 The T-q diagram for the heat transfer process in the hot-end heat exchanger
4
According to the definition [9], the entransy dissipation rate of the heat transfer process in the hot-end exchanger is: 1 1 − Ga Gh
( KA)h
6
area of the shadow part in the T-q diagram [10]: T + T Th ,o + Th ,i Φ h = Qh 3 4 − . 2 2
7 8
(3)
Combining Eqs. (1) - (3) yields the expression of the thermal conductance of the hot-end heat exchanger (KA)h:
Ga ( T3 − Th , o ) 1 1 − . ln Ga (T3 − Th ,i ) − Qh Ga Gh −1
TE D
( KA)h = 9
(2)
On the other hand, the entransy dissipation rate can also be obtained through calculating the
M AN U
5
1 2 1 1 e +1 Qh − . 1 1 2 G G h ( KA )h G − G a h a e −1
SC
Φ h = Qh 2 Rh =
RI PT
3
(4)
Similarly, the thermal conductance of the cold-end heat exchanger (KA)l is expressed: G (T − T ) 1 1 ( KA)l = − ln a l ,i 1 , Ga (Tl , o − T1 ) + Ql Gl Ga −1
EP
(5)
where Gl and Tl respectively stand for the heat capacity rate and the temperature of the cold-end
11
fluid.
12
AC C
10
The total heat transfer rate of the cold-end heat exchanger, Ql, is calculated as following: Ql = Ga (T1 − T6 ) = Gl (Tl ,i − Tl ,o ) .
(6)
13
For the counter-flow regenerator, because the heat capacity rates of both the hot and the cold
14
air are the same, the temperature differences between the hot and cold air at the inlet and the outlet
15
are equivalent: T4 − T2 = T5 − T1 ,
16
Combination of Eqs. (1) and (7) offers:
7
(7)
ACCEPTED MANUSCRIPT Qh , Ga
T2 = T3 − T5 + T1 −
1 2
(8)
The thermal conductance of the regenerator (KA)r is calculated via the total heat transfer rate over the arithmetic mean temperature:
( KA)r
=
Qr , T5 − T1
(9)
where T5 is the temperature of the air at the inlet of the expander. Qr is the total heat transfer rate
4
between the hot and the cold airs in the regenerator:
RI PT
3
Qr = Ga (T4 − T5 ) .
5
(10)
( KA)r
=
Ga (T3 − T5 ) − Qh
T5 − T1
.
SC
Substituting Eqs. (1) and (10) into Eq. (9) provides:
(11)
Eqs. (4), (5) and (11) respectively give the expressions of the thermal conductance of the
7
hot-end and the cold-end heat exchangers and the regenerator. Therefore, the total thermal
8
conductance of all the heat exchangers in the system is
( KA)total = ( KA)h + ( KA)l + ( KA)r
M AN U
6
Ga (T3 − Th ,o ) Ga (Tl ,i − T1 ) G (T − T ) − Qh . 1 1 1 1 = − + − + a 3 5 ln ln T5 − T1 Ga (T3 − Th ,i ) − Qh Gl Ga Ga (Tl , o − T1 ) + Ql Ga Gh
(12)
4. Thermodynamic analysis
EP
9
−1
TE D
−1
For simplicity, the compression and the expansion processes in the compressor and the
11
expander are assumed to be adiabatic. Therefore, the temperature and the pressure of the air at the
12
inlet and the outlet of the compressor and the expander have the following relations [16]:
13
and
AC C
10
T2 p2 = T3 p3
T6 p6 = T5 p5
γ −1 γ
,
(13)
,
(14)
γ −1 γ
14
where p2 and p3 are the pressures of the air at the inlet and the outlet of the compressor; p5 and p6
15
are the pressures of the air at the inlet and the outlet of the expander; γ is the adiabatic exponent of
16
air. 8
ACCEPTED MANUSCRIPT
2
3
4
If assuming that there is no pressure loss of the fluids during flowing in all the heat exchangers, the heat transfer processes are all isobaric: p2 = p6 , p3 = p5 .
(15)
T6 T2 = . T5 T3
(16)
Combining Eqs. (13) - (15) gives:
Substituting Eqs. (6) and (8) into Eq. (16) provides that: ln
G (T + T − T ) − Qh GaT1 − Ql . = ln a 1 3 5 T5 T3
RI PT
1
(17)
Eq. (17) reveals the physical constraint which the heat capacity rate and the intermediate
6
temperature of the air have to satisfy. In this article, we want to demonstrate the core idea and
7
brief steps of the newly proposed optimization method. Therefore, we adopted the ideal
8
assumptions for the system, e.g. adiabatic compression and the heat exchangers without pressure
9
drop, to offer a clear illustration of the method. Based on the method, we can also account for the
10
compressor efficiency by replacing the adiabatic processes with polytropic processes. And the
11
pressure drop through heat exchangers can be considered to be decided by the flow rates of the
12
working fluids and the area of the heat exchanger, which can be mathematically expressed as the
13
function of thermal conductance of the heat exchanger and the heat capacity rates of the fluids.
14
Therefore, the pressure of air flow at the inlet and outlet of the heat exchanger can also be
15
expressed as the functions of the thermal conductance of the heat exchanger and the heat capacity
16
rates of the fluids. By applying similar derivation, the method can still work.
17
5. Optimization model for the regenerative air refrigeration system
AC C
EP
TE D
M AN U
SC
5
18
The performance of the regenerative air refrigeration systems is determined by all the design
19
parameters of the systems, e.g. the thermal conductance of heat exchangers, the heat capacity rate
20
and the intermediate temperatures of the air. That is, all the various optimization targets for the
21
systems, e.g. the power consumption, the fixed, operation or total cost, the thermodynamic
22
efficiency and the total heat exchanger area, can be mathematically expressed as the functions of
23
the aforementioned design parameters in the systems. When assuming the heat transfer
24
coefficients of all the heat exchangers are uniform for simplicity, minimizing the total heat transfer 9
ACCEPTED MANUSCRIPT area of all the heat exchangers is equivalent to minimizing the total thermal conductance of all the
2
heat exchangers. Here, we take the minimization of the total thermal conductance, min{(KA)total},
3
as the optimization target to demonstrate the core idea and solving steps of the method. This target
4
is described by the function about T1，T3，T5 and Ga in Eq. (12). These four parameters have to
5
satisfy not only the physical constraint in Eq. (17), but also the relation as following when the
6
thermal conductance of the regenerator is prescribed according to Eq. (11): Ga (T3 − T5 ) − Qh − ( KA) r (T5 − T1 ) = 0 .
RI PT
1
(18)
When the cooling capacity, the net input work, the heat capacity rates and the inlet
8
temperatures of hot-end and cold-end fluids are prescribed, the global optimization problem of the
9
regenerative air refrigeration system can be mathematically converted into a conditional extremum
10
problem. To solve this conditional extremum problem, we apply the Lagrange multiplier method
11
and construct the following function:
M AN U
SC
7
G (T + T − T ) − Qh G T − Ql F = ( KA )h + ( KA )l + ( KA )r + λ ln a 1 − ln a 1 3 5 T5 T3 ,
+ ω ( Ga (T3 − T5 ) − Qh − ( KA )r (T5 − T1 ) )
13
where λ and ω are Lagrange multipliers.
TE D
12
The differential of Eq. (19) with respect to Ga gives
AC C
EP
−2 −1 Ga (T3 − Th , o ) T3 − Th ,i 1 ∂F Ga 1 1 = 1 − ln + − − ∂Ga Gh Ga (T3 − Th ,i ) − Qh Ga Gh Ga Ga (T3 − Th ,i ) − Qh −2 −1 T −T Ga (Tl ,i − T1 ) Tl , o − T1 Ga 1 1 1 + 3 5 . − − 1 ln + − − Ga (Tl , o − T1 ) + Ql Gl Ga Ga Ga ( Tl ,o − T1 ) + Ql T5 − T1 Gl T1 T1 + T3 − T5 + λ − + ω ( T3 − T5 ) = 0 GaT1 − Ql Ga ( T1 + T3 − T5 ) − Qh
14
(20)
The differential of Eq. (19) with respect to T1 yields Ga 1 Ga (T3 − T5 ) − Qh − + + 2 (T5 − T1 ) Ga (Tl , o − T1 ) + Ql Tl ,i − T1 . 1 1 + λ Ga − + ω ( KA )r = 0 GaT1 − Ql Ga (T1 + T3 − T5 ) − Qh
1 1 ∂F = − − ∂T1 Gl Ga
15
(19)
−1
The differential of Eq. (19) with respect to T3 is
10
(21)
ACCEPTED MANUSCRIPT ∂F = ∂T3 −1 1 Ga 1 1 − − Ga Gh T3 − Th , o Ga ( T3 − Th ,i ) − Qh =0
(22)
The differential of Eq. (19) with respect to T5 offers 1 Ga ∂F Ga (T1 − T3 ) + Qh = + λ − + 2 ∂T5 T G T + T ( (T5 − T1 ) a 1 3 − T5 ) − Qh 5
RI PT
1
Ga Ga 1 + − λ − + ωGa . T5 − T1 Ga (T1 + T3 − T5 ) − Qh T3
− ω ( Ga + ( KA )r ) = 0 .
(23)
Eqs. (17), (18), (20) – (23) offer 6 optimization equations which contain 6 unknown
3
parameters, i.e. Ga, T1, T3, T5, λ and ω. Thereafter, the solution of these optimization equations can
4
directly lead to the minimization of the total thermal conductance of all the heat exchangers and
5
the corresponding optimal values for all the unknown parameters.
6
6. Optimization results and discussion
M AN U
SC
2
For the regenerative air refrigeration systems in Fig. 1, the inlet temperatures and the heat
8
capacity rates of the hot-end and the cold-end fluids are: Thi = 303 K, Tli = 293 K, Gh = 500 W/K,
9
Gl = 400 W/K. The cooling capacity and the net input work of the system are: Ql =1000 W,
10
W=500 W. In this situation, if the thermal conductance of the regenerator (KA)r is given as 10
11
W/K. Solving Eqs. (17), (18) and (20) – (23) provides the optimal values for all the unknown
12
parameters corresponding to the minimization of the total thermal conductance of all the heat
13
exchangers, which are listed in Table 1. The optimal pressure ratio of compressor, π, is 4.77.
AC C
EP
TE D
7
Table. 1 Optimization results 14
Furthermore, for the regenerator prescribed with any other size, the minimum of the total
15
thermal conductance of the heat exchangers and the corresponding optimal configuration of all the
16
unknown parameters can also be obtained. Fig. 4 shows the influence of the prescribed value of
17
(KA)r on the minimum of the total thermal conductance of all the heat exchangers (KA)total and that
18
of the hot-end and the cold-end heat exchangers (KA)h+(KA)l. As (KA)r increases, the minimum
19
total thermal conductance of all the heat exchangers monotonously increases as well, which
20
indicates that the total thermal conductance of all the heat exchangers has no extremum if the 11
ACCEPTED MANUSCRIPT thermal conductance of the regenerator is not fixed. Meanwhile, the section above the curve of
2
(KA)total represents the total thermal conductance in this area is physically possible to satisfy the
3
design requirements but not the minimum, while the total thermal conductance in the section
4
below the curve is impossible to satisfy the design requirements. On the other hand, in Fig. 4, as
5
(KA)r increases, the total thermal conductance of the hot-end and the cold-end heat exchangers
6
(KA)h+(KA)l reaches its maximum at about (KA)r = 20 W/K. When (KA)r < 20 W/K, a small
7
variation will lead to a sharp change of (KA)h+(KA)l, which means that the thermal conductance of
8
the regenerator should be carefully selected if there is a specific limit for the thermal conductance
9
of the hot-end and the cold-end heat exchangers during the optimization design. On the contrary,
10
when (KA)r > 20 W/K, increasing the thermal conductance of the regenerator has quite smaller
11
influence on the (KA)h+(KA)l.
M AN U
SC
RI PT
1
Fig. 4 The influence of the prescribed value of (KA)r on the minimum of the total thermal conductance of all the heat exchangers and that of the hot-end and the cold-end heat exchangers Fig. 5 shows the variation of the optimal pressure ratio of the compressor and the heat capacity
13
rate of the air versus the thermal conductance of the regenerator. The pressure ratio π equals to
14
p2/p3. When (KA)r increases, the decreasing optimal pressure ratio means that increasing the
15
thermal conductance of the regenerator can obviously reduce the pressure ratio of the compressor
16
and the expander. On the other hand, the optimal heat capacity rate of the air Ga has its minimum
17
when (KA)r = 30 W/K. That is, when (KA)r < 30 W/K, Ga drops sharply with (KA)r increasing;
18
When (KA)r > 30 W/K, Ga increases slowly. And the increasing of the area of the hot-end and
19
cold-end heat exchangers reduces the heat transfer irreversibility and the temperature difference
20
between air flow and the outside fluids [14]. While the latter recovers heat between hot and cold
21
air flows and can lead to the drop of the pressure ratio of the compressor. In other words, if we
22
don’t care about the performance of the compressor, for a refrigeration cycle, both hot-end and
23
cold-end heat exchangers are essential; while a regenerator is not. Therefore, the sum of the
24
hot-end and cold-end heat exchangers in the simple air refrigeration cycles has the minimum as
25
mentioned in the published paper. However, the total thermal conductance of all the three heat
26
exchangers in the regenerative air refrigeration system has no minimum if the area of the
27
regenerator is not prescribed.
AC C
EP
TE D
12
12
ACCEPTED MANUSCRIPT Fig. 5 The variation of the optimal pressure ratio of the compressor (π= p2/p3) and the heat capacity rate of the air versus the thermal conductance of the regenerator Fig. 6 demonstrates the variation of the optimal thermal conductance of the hot-end and the
2
cold-end heat exchangers, (KA)h and (KA)l, and their ratio, (KA)h/(KA)l, versus the thermal
3
conductance of the regenerator. As showed in Fig. 6, when (KA)r increases, both (KA)h and (KA)l
4
increase fast at first and then decrease slowly. Furthermore, the thermal conductance of the
5
hot-end heat exchanger is always larger than that of the cold-end heat exchanger, which is also
6
indicated by the fact that the ratio (KA)h/(KA)l is always above 1. Therefore, the equivalence of the
7
thermal conductance of both hot-end and cold-end heat exchangers [1] is not always essential to
8
obtain the optimal performance of the refrigeration systems.
SC
RI PT
1
M AN U
Fig. 6 The variation of the optimal thermal conductance of the hot-end and the cold-end heat exchangers and their ratio versus the thermal conductance of the regenerator Fig. 7 indicates the variation of T2 and T5 versus the thermal conductance of the regenerator.
10
Because the increase of the regenerator strengthens the heat recovery, the outlet temperature of the
11
cold air in the regenerator, T2, promotes and that of the hot air, T5, goes down. Fig. 8 illustrates the
12
variation of T1 and T6 versus the thermal conductance of the regenerator. T1 sharply increases at
13
first and then decreases slowly. While T6 decreases at first and then goes up slowly. The average
14
temperature of the air in the cold-end heat exchanger, T1-6 = (T1+T6)/2, has its maximum. Based on
15
the Eq. (6), T1and T6 has the relation as:
EP
TE D
9
T1 − T6 =
Ql . Ga
(24)
Because the heat capacity rate of the air, Ga, first decreases and then rises as (KA)r increases
17
in Fig. 5, the difference between T1 and T6 first increases and then decreases. According to the
18
variation of the thermal conductance of the cold-end heat exchanger in Fig. 4, the temperature
19
difference between the cold-end fluid and the air rises after falling. Therefore, T1-6 appears to fall
20
after rising. Fig. 9 shows the variation of T3 and T4 versus the thermal conductance of the
21
regenerator. T3 sharply first increases and then decreases slowly, while T4 first decreases and then
22
goes up slowly. The average temperature of the air in the hot-end heat exchanger T3-4 = (T3+T4)/2
23
has its minimum.
AC C
16
Fig. 7 The variation of T2 and T5 versus the thermal conductance of the regenerator 13
ACCEPTED MANUSCRIPT 1 Fig. 8 The variation of T1 and T6 versus the thermal conductance of the regenerator 2
3
RI PT
Fig. 9 The variation of T3 and T4 versus the thermal conductance of the regenerator
7. Conclusions
The regenerative air refrigeration systems involve both heat transfer and thermodynamic
5
processes. This paper introduces the entransy theory into the analysis for the heat transfer process
6
in the hot-end and the cold-end heat exchangers and the regenerator. Via combining the
7
thermodynamic analysis for the compression and the expansion processes in the compressor and
8
the expander, a mathematical model for the whole system is established to describe the physical
9
relation between the unknown design parameters and the design requirements. Based on this
10
model, the optimization for the regenerative air refrigeration systems can be converted into a
11
conditional extremum problem, which offers a theoretical optimization equation group by the
12
Lagrange multiplier method. Solving the equation group directly obtains the best performance of
13
the systems and the corresponding optimal configurations of all the unknown parameters.
TE D
M AN U
SC
4
The validity of the proposed optimization method is proved through an actual optimization
15
case to minimize the total thermal conductance of all the heat exchangers with the cooling
16
capacity, the net input work, the heat capacity rates and the inlet temperatures of hot-end and
17
cold-end fluids given. Meanwhile, the following results are obtained: 1) The minimum of the total
18
thermal conductance of all the heat exchangers in the system monotonously rises as the prescribed
19
value of the thermal conductance of the regenerator increases. If the thermal conductance of the
20
regenerator is not prescribed, the total thermal conductance of the heat exchangers in the system
21
has no extremum. 2) Increasing the thermal conductance of the regenerator decreases the optimal
22
pressure ratio of the compressor and the expander. 3) As the thermal conductance of the
23
regenerator increasing, the thermal conductance of the hot-end and the cold-end heat exchangers
24
and their total thermal conductance have their own maximums. Meanwhile, the thermal
25
conductance of the hot-end heat exchanger is always larger than that of the cold-end one, which
AC C
EP
14
14
ACCEPTED MANUSCRIPT indicates that the equivalence of the thermal conductance of both hot-end and cold-end heat
2
exchangers is not always essential to obtain the optimal performance of the refrigeration systems.
3
4) The optimal heat capacity rate of the air has its minimum when the regenerator gets larger.
4
Acknowledgement
RI PT
1
The authors would like to thank the National Natural Science Foundation of China (Grant No.
6
51006060), the National Basic Research Program of China (Grant No. 2010CB227305) and the
7
Foundation for the Author of National Excellent Doctoral Dissertation of China (Grant No.
8
201150).
9
References
M AN U
SC
5
[1] A. Bejan, Theory of heat transfer-irreversible refrigeration plants, International Journal of Heat
11
and Mass Transfer. 32 (1989) 1631-1639.
12
[2] G. Grazzini, R. Rinaldi, Thermodynamic optimal design of heat exchangers for an irreversible
13
refrigerator, International Journal of Thermal Sciences. 40 (2001) 173-180.
14
[3] B. Sahin, A. Kodal, Thermoeconomic optimization of a two stage combined refrigeration
15
system: a finite-time approach, International Journal of Refrigeration. 25 (2002) 872-877.
16
[4] M. A. AitAli, The maximum coefficient of performance of internally irreversible refrigerators
17
and heat pumps, Journal of Physics D: Applied Physics. 29 (1996) 975-980.
18
[5] L. G Chen, C. Wu, F. R. Sun, Cooling load versus COP characteristics for and irreversible air
19
refrigeration cycle, Energy Conversion and Management. 39 (1998) 117-125.
20
[6] C. K. Chen, Y. F. Su, Exergetic efficiency optimization for an irreversible Brayton refrigeration
21
cycle, International Journal of Thermal Sciences. 44 (2005) 303-310.
22
[7] Y. M. Tu, L. G. Chen, F. R. Sun, C. Wu, Cooling load and coefficient of performance
23
optimizations for real air-refrigerators, Applied Energy. 83 (2006) 1289-1306.
24
[8] S. J. Liu, Z. Y. Zhang, L. L. Tian, Thermodynamic analysis of the actual air cycle refrigeration
25
system, Systems Engineering Procedia. 1 (2011) 112-116.
AC C
EP
TE D
10
15
ACCEPTED MANUSCRIPT [9] Z. Y. Guo, H. Y. Zhu, X. G. Liang, Entransy-A physical quantity describing heat transfer ability,
2
International Journal of Heat and Mass Transfer. 50 (2007) 2545-2556.
3
[10] Q. Chen, Y. C. Xu, An entransy dissipation-based optimization principle for building central
4
chilled water systems, Energy. 37 (2012) 571-579.
5
[11] Y. C. Xu, Q. Chen, An entransy dissipation-based method for global optimization of district
6
heating networks, Energy Build. 48 (2012) 50-60.
7
[12] Y. C. Xu, Q. Chen, Minimization of mass for heat exchanger networks in spacecrafts based
8
on the entransy dissipation theory, International Journal of Heat and Mass Transfer. 55 (2012)
9
148-156.
SC
RI PT
1
[13] F. Yuan, Q. Chen, A global optimization method for evaporative cooling systems based on the
11
entransy theory, Energy. 42 (2012) 181-191.
12
[14] Q. Chen, Y. C. Xu, J. H. Hao, An optimization method for gas refrigeration cycle based on
13
the combination of both thermodynamics and entransy theory, Applied energy. submitted.
14
[15] Q. Chen, Entransy dissipation-based thermal resistance method for heat exchanger
15
performance design and optimization, International Journal Heat and Mass Transfer. 60 (2013)
16
156-162.
17
[16] M. J. Moran, H. N. Shapiro, Fundamentals of Engineering thermodynamics, fifth ed.,
18
Hoboken NJ, John Wiley & Sons, Inc., 2004.
TE D
M AN U
10
AC C
EP
19
16
ACCEPTED MANUSCRIPT 1
Figure list
Fig. 1 The schematic diagram of a regenerative air regeneration system
3
Fig. 2 The T-s diagram of the regenerative air regeneration cycle
4
Fig. 3 The T-q diagram for the heat transfer process in the hot-end heat exchanger
5
Fig. 4 The influence of the prescribed value of (KA)r on the minimum of the total thermal
6
conductance of all the heat exchangers and that of the hot-end and the cold-end heat exchangers
7
Fig. 5 The variation of the optimal pressure ratio of the compressor (π= p2/p3) and the heat
8
capacity rate of the air versus the thermal conductance of the regenerator
9
Fig. 6 The variation of the optimal thermal conductance of the hot-end and the cold-end heat
SC
RI PT
2
exchangers and their ratio versus the thermal conductance of the regenerator
11
Fig. 7 The variation of T2 and T5 versus the thermal conductance of the regenerator
12
Fig. 8 The variation of T1 and T6 versus the thermal conductance of the regenerator
13
Fig. 9 The variation of T3 and T4 versus the thermal conductance of the regenerator
M AN U
10
AC C
EP
TE D
14
17
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
1
Fig. 1 The schematic diagram of a regenerative air regeneration system
TE D
2
AC C
EP
3
18
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
1 2
Fig. 2 The T-s diagram of the regenerative air regeneration cycle
AC C
EP
TE D
3
19
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
1 2
Fig. 3 The T-q diagram for the heat transfer process in the hot-end heat exchanger
AC C
EP
TE D
3
20
ACCEPTED MANUSCRIPT 60
100
56
40
(KA)h+(KA)l
0
14
28
42
56
-1
(KA)h+(KA)l / W⋅K
70
RI PT
(KA)total
52
48 70
SC
(KA)total / W⋅K
-1
-1
130
2
Fig. 4 The influence of the prescribed value of (KA)r on the minimum of the total thermal
3
conductance of all the heat exchangers and that of the hot-end and the cold-end heat exchangers
M AN U
1
(KA)r/ W⋅K
AC C
EP
TE D
4
21
ACCEPTED MANUSCRIPT
54
π
Ga
4.8
Ga / W⋅K
34
24
14
1
28
42
M AN U
0
SC
4.6
4.4
44 -1
5.0
RI PT
π
5.2
-1
(KA)r/ W⋅K
56
14 70
2
Fig. 5 The variation of the optimal pressure ratio of the compressor (π= p2/p3) and the heat
3
capacity rate of the air versus the thermal conductance of the regenerator
AC C
EP
TE D
4
22
ACCEPTED MANUSCRIPT 1
1.10
28.6
RI PT
27.2 1.06
25.8 (KA)h
24.4
1.04
(KA)l (KA)h/(KA)l
0
14
28
42
-1
(KA)r/ W⋅K
56
M AN U
23.0 2
(KA)h/(KA)l
1.08
SC
(KA)h & (KA)l / W⋅K
-1
30.0
1.02 70
3
Fig. 6 The variation of the optimal thermal conductance of the hot-end and the cold-end heat
4
exchangers and their ratio versus the thermal conductance of the regenerator
AC C
EP
TE D
5
23
ACCEPTED MANUSCRIPT 355
T5
335
T2
RI PT
T/ K
315 295
255
0
17
34
51
SC
275
68
85
-1
(KA)r/ W⋅K
2
M AN U
1
Fig. 7 The variation of T2 and T5 versus the thermal conductance of the regenerator
AC C
EP
TE D
3
24
241
270
237
T1 T1-6
229
250
240
0
17
34
51
-1
(KA)r/ W⋅K
68
M AN U
1 2
233
RI PT
T6
260
T6 / K
280
SC
T1 & T1-6 / K
ACCEPTED MANUSCRIPT
225 85
Fig. 8 The variation of T1 and T6 versus the thermal conductance of the regenerator
AC C
EP
TE D
3
25
ACCEPTED MANUSCRIPT 355
T3-4
347
T4
339
T4 / K
T3
390
RI PT
T3 & T3- 4 / K
405
375
331
0
17
34
51
-1
323 85
SC
360
68
2
Fig. 9 The variation of T3 and T4 versus the thermal conductance of the regenerator
M AN U
1
(KA)r/ W⋅K
AC C
EP
TE D
3
26
ACCEPTED MANUSCRIPT 1
Table list
2
Table 1 Optimization results (KA)h
(KA)l
(KA)total
Ga
optimization results
29.92
27.56
66.59
21.92
3
RI PT
parameters/ W K-1
parameters/ K
T1
T2
T3
T4
T5
T6
optimization results
274.10
291.19
397.10
328.67
311.58
228.48
AC C
EP
TE D
M AN U
SC
4
27
S1359-4311(14)00027-1
DOI:
10.1016/j.applthermaleng.2014.01.021
Reference:
ATE 5321
To appear in:
Applied Thermal Engineering
Received Date: 23 August 2013 Revised Date:
1 December 2013
Accepted Date: 15 January 2014
Please cite this article as: J.-H. Hao, Q. Chen, Y.-C. Xu, A Global Optimization Method for Regenerative Air Refrigeration Systems, Applied Thermal Engineering (2014), doi: 10.1016/ j.applthermaleng.2014.01.021. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Highlight
An optimization method is proposed for regenerative air refrigeration systems.
Heat transfer processes in heat exchangers is analyzed by the entransy theory.
A mathematical relation between design parameters and requirements is derived.
This optimization method is validated by analyzing a typical refrigeration system.
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
A Global Optimization Method for Regenerative Air
2
Refrigeration Systems
3
Jun-Hong Hao, Qun Chen *, Yun-Chao Xu
4
Key Laboratory for Thermal Science and Power Engineering of Ministry of Education,
5
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
RI PT
1
6
Abstract: The air refrigeration systems always involve such physical processes as heat transfer
8
processes in heat exchangers, compression processes in compressors and expansion processes in
9
expanders. This contribution proposes a theoretical global optimization method for regenerative
10
air refrigeration systems and introduces the entransy theory to analyze the heat transfer processes
11
in the hot-end heat exchanger, the cold-end heat exchanger and the regenerator. Integration of the
12
heat transfer analyses and the thermodynamic analyses for the compression and expansion
13
processes yields a mathematical model to describe the physical relation between the design
14
requirements and the unknown design parameters, i.e. the heat transfer area of each heat
15
exchanger and the heat capacity rate and the intermediate temperature of the air. Based on this
16
model, the optimization can be converted into a conditional extremum problem. That is, solving
17
the problem via the Lagrange multiplier method offers an optimization equation group, which
18
directly leads to the optimal values of all the unknown parameters. Finally, this optimization
19
method is validated through an optimization case to minimize the total thermal conductance of all
20
the heat exchangers in a typical regenerative air refrigeration system.
AC C
EP
TE D
M AN U
SC
7
21 22
Keywords:
23
thermodynamics, entransy theory
regenerative
air
refrigeration
systems,
optimization,
24
*
Corresponding author, Tel. & Fax: 86-10-62796332. Email: [email protected]
1
energy conservation,
ACCEPTED MANUSCRIPT 1
area, m2
F
Lagrange function
G
heat capacity rate, W/K
K
Heat transfer coefficient, W/(m2· K)
KA
the thermal conductance of exchangers, W/K
p
the pressure of the air in the cycle, Pa
Q, q
heat transfer rate, W
Rh
entransy dissipation-based thermal resistance, K/W
s
specific entropy, J/(kg· K)
T
temperature, K
γ
the adiabatic exponent of air
π
pressure ratio
λ
Lagrange multiplier
Φ
entransy dissipation rate, W K
ω
Lagrange multiplier
3 4
the air outlet of the cold-end exchanger
AC C
2
EP
Subscripts
1
M AN U
TE D
2
SC
A
RI PT
Nomenclatures
the air inlet of the compressor the air outlet of the compressor the air outlet of the hot-end exchanger
5
the air inlet of the expander
6
the air inlet of the cold-end exchanger
a
the air
h
the hot-end
i
the inlet of the exchanger 2
ACCEPTED MANUSCRIPT l
the cold-end
o
the outlet of the exchanger
r
the regenerator
AC C
EP
TE D
M AN U
SC
RI PT
1 2
3
ACCEPTED MANUSCRIPT 1
1. Introduction
The refrigeration systems not only play an important role in modern human lives but also
3
consume more and more energy. In this era with severe energy shortages, energy efficiency
4
promotion via theoretical or engineering optimization methods becomes a very hot topic. However,
5
the strong nonlinear and multivariate characteristic of the refrigeration systems is always the
6
obstacle confusing the researchers.
RI PT
2
In theory, Bejan [1] simplified several actual thermodynamic cycles as Carnot cycles and
8
found the optimal area allocation of all the heat exchangers to maximize the refrigeration
9
capacities. Thereafter, Grazzini and Rinaldi [2] optimized an irreversible inverse Rankine cycle to
10
search the maximal coefficient of performance (COP). Sahin and Kodal [3] analyzed the optimal
11
distribution of heat exchanger areas with the aim of maximizing the cooling load per total cost.
12
Ait-Ali [4] optimized both the refrigerators and heat pumps to maximize their COPs. All these
13
researches contributed a lot in developing the optimization theory for thermodynamic cycles.
14
However, simplifying actual thermodynamic cycles by Carnot cycles in these works is always too
15
general to offer the detailed information for the internal components, such as the performance of
16
compressors and expanders/throttling valves and the physical property of the working fluids.
TE D
M AN U
SC
7
In view of this limitation, for the optimization of air refrigeration cycles, scientists separately
18
analyzed the heat transfer processes in heat exchangers and the compression/expansion processes
19
in compressors and expanders, and integrated both aspects of analyses, which made the theoretical
20
optimization more reasonable and valid. For instance, Chen et al. [5] analyzed an air refrigerator
21
with non-isentropic compression and expansion processes based on the finite-time
22
thermodynamics theory, deduced the relations of the cooling load and the COP to the pressure
23
ratio, and finally maximized the COP. Chen et al. [6] applied the exergy theory to analyze the
24
multi-irreversibilities of an irreversible Brayton refrigeration cycle, and maximized the exergetic
25
efficiency with consideration of the influence of heat leak between heat reservoirs. Tu et al. [7]
26
emphasized the analysis for the influencing factors of the maximization of the cooling load and
27
the COP, i.e. the total heat exchanger inventory, the efficiency of compressor and expander, and
28
the heat capacity rates of all the fluids. Similarly, Liu et al. [8] studied the influences of the
AC C
EP
17
4
ACCEPTED MANUSCRIPT pressure ratio and the inlet temperature of compressor and expander on the optimal COP of an
2
actual air refrigeration cycle. However, because the multiple degrees of freedom of the
3
optimization problem is quite difficult to handle, they had to decrease the multiple degrees of
4
freedom of the system to only one degree via fixing the majority of the unknown parameters,
5
which can obviously make the problems simpler but hardly obtain the global optimal performance
6
of the systems.
RI PT
1
On the other hand, based on the entransy theory introduced by Guo et al. [9], Chen and his
8
colleagues proposed an entransy dissipation-based optimization method to theoretically realize the
9
global optimization of several energy utilization systems with strong multivariate and nonlinear
10
characteristic, e.g. heat exchangers networks in buildings [10,11], thermal management systems in
11
spacecrafts [12] and evaporative cooling systems [13], and found several direct mathematical
12
relations between the design requirements and all the design parameters. According to this
13
mathematical relations, the optimization problems were converted into the conditional extremum
14
problems which offer the govern equation groups for optimization. Furthermore, they extended
15
this method to optimize an air refrigeration cycle [14] and found the minimum of the sum of the
16
hot-end and cold-end heat exchangers.
TE D
M AN U
SC
7
Different from the simple air refrigeration cycles [14], the regenerative air refrigeration
18
cycles here contain two kinds of heat exchangers with different functions, i.e. the hot-end and
19
cold-end heat exchangers and the regenerator. The former transfer heat between the air flow and
20
outside fluids; while the latter recovers heat between hot and cold air flows. And the existence of
21
regenerator apparently affects the performance and optimization of the whole system. This
22
contribution intends to propose a global optimization method for regenerative air refrigeration
23
systems. Through combing the heat transfer analysis for heat exchangers based on the entransy
24
theory and the thermodynamic analysis for compression and expansion processes in compressor
25
and expander, we establish a mathematical relation between the design parameters, i.e. the thermal
26
conductance of each heat exchanger, the heat capacity rate and the intermediate temperature of the
27
air, and the design requirements, and deduce an optimization equation group by the conditional
28
extremum method. Finally, the validity of the method is proved by optimizing a typical
29
regenerative air refrigeration system.
AC C
EP
17
5
ACCEPTED MANUSCRIPT 1
2. Physical model for a regenerative air refrigeration system
Fig. 1 gives the schematic diagram of a regenerative air refrigeration system which consists
3
of a hot-end heat exchanger, a cold-end heat exchanger, a regenerator, a compressor and an
4
expander. Fig. 2 shows the T-s diagram for the refrigeration cycle. The air at the status 1 enters the
5
regenerator and absorbs heat from the hot air with the temperature promoting to the status 2. The
6
air at the the status 2 flows through compressor and leaves with higher temperature and pressure at
7
the status 3 due to compression. Then, the air flows into the hot-end heat exchanger and rejects
8
heat to the hot-end fluid with the temperature dropping to the status 4. After the heat rejection in
9
the regenerator, the air at the status 5 flows into the expander and expands to the status 6 with both
10
temperature and pressure decreasing. Finally, after the heat absorption in the cold-end heat
11
exchanger, the air returns to the the status 1.
M AN U
SC
RI PT
2
Fig. 1 The schematic diagram of a regenerative air regeneration system 12
Fig. 2 The T-s diagram of the regenerative air regeneration cycle
The regenerative air refrigeration system involves two kinds of physical processes: 1) the
14
heat transfer processes in each heat exchanger; 2) the thermodynamic processes, i.e. the
15
compression process in the compressor and the expansion process in the expander. Therefore, in
16
order to establish a physical model of the whole system, it is essential to apply the theory of heat
17
transfer and thermodynamics together to analyze the heat transfer and thermodynamic processes
18
separately.
19
3. Entransy-dissipation based heat transfer analysis
AC C
EP
TE D
13
20
Fig. 3 shows the T-q diagram of the heat transfer process of the counter-flow hot-end heat
21
exchanger in the system, where q represents the heat transfer rate between the hot-end fluid and
22
the air, and T represents the temperatures of the fluids [15]. Th stands for the temperature of the
23
hot-end fluid. T3 and T4 respectively represent the temperatures of air at the inlet and the outlet. i
24
and o separately stand for the inlet and the outlet of the heat exchanger. The total heat transfer rate 6
ACCEPTED MANUSCRIPT 1
Qh is expressed as: Qh = Ga (T3 − T4 ) = Gh (Th ,o − Th ,i ) ,
2
(1)
where Ga and Gh separately stand for the heat capacity rates of the air and the hot-end fluid. Fig. 3 The T-q diagram for the heat transfer process in the hot-end heat exchanger
4
According to the definition [9], the entransy dissipation rate of the heat transfer process in the hot-end exchanger is: 1 1 − Ga Gh
( KA)h
6
area of the shadow part in the T-q diagram [10]: T + T Th ,o + Th ,i Φ h = Qh 3 4 − . 2 2
7 8
(3)
Combining Eqs. (1) - (3) yields the expression of the thermal conductance of the hot-end heat exchanger (KA)h:
Ga ( T3 − Th , o ) 1 1 − . ln Ga (T3 − Th ,i ) − Qh Ga Gh −1
TE D
( KA)h = 9
(2)
On the other hand, the entransy dissipation rate can also be obtained through calculating the
M AN U
5
1 2 1 1 e +1 Qh − . 1 1 2 G G h ( KA )h G − G a h a e −1
SC
Φ h = Qh 2 Rh =
RI PT
3
(4)
Similarly, the thermal conductance of the cold-end heat exchanger (KA)l is expressed: G (T − T ) 1 1 ( KA)l = − ln a l ,i 1 , Ga (Tl , o − T1 ) + Ql Gl Ga −1
EP
(5)
where Gl and Tl respectively stand for the heat capacity rate and the temperature of the cold-end
11
fluid.
12
AC C
10
The total heat transfer rate of the cold-end heat exchanger, Ql, is calculated as following: Ql = Ga (T1 − T6 ) = Gl (Tl ,i − Tl ,o ) .
(6)
13
For the counter-flow regenerator, because the heat capacity rates of both the hot and the cold
14
air are the same, the temperature differences between the hot and cold air at the inlet and the outlet
15
are equivalent: T4 − T2 = T5 − T1 ,
16
Combination of Eqs. (1) and (7) offers:
7
(7)
ACCEPTED MANUSCRIPT Qh , Ga
T2 = T3 − T5 + T1 −
1 2
(8)
The thermal conductance of the regenerator (KA)r is calculated via the total heat transfer rate over the arithmetic mean temperature:
( KA)r
=
Qr , T5 − T1
(9)
where T5 is the temperature of the air at the inlet of the expander. Qr is the total heat transfer rate
4
between the hot and the cold airs in the regenerator:
RI PT
3
Qr = Ga (T4 − T5 ) .
5
(10)
( KA)r
=
Ga (T3 − T5 ) − Qh
T5 − T1
.
SC
Substituting Eqs. (1) and (10) into Eq. (9) provides:
(11)
Eqs. (4), (5) and (11) respectively give the expressions of the thermal conductance of the
7
hot-end and the cold-end heat exchangers and the regenerator. Therefore, the total thermal
8
conductance of all the heat exchangers in the system is
( KA)total = ( KA)h + ( KA)l + ( KA)r
M AN U
6
Ga (T3 − Th ,o ) Ga (Tl ,i − T1 ) G (T − T ) − Qh . 1 1 1 1 = − + − + a 3 5 ln ln T5 − T1 Ga (T3 − Th ,i ) − Qh Gl Ga Ga (Tl , o − T1 ) + Ql Ga Gh
(12)
4. Thermodynamic analysis
EP
9
−1
TE D
−1
For simplicity, the compression and the expansion processes in the compressor and the
11
expander are assumed to be adiabatic. Therefore, the temperature and the pressure of the air at the
12
inlet and the outlet of the compressor and the expander have the following relations [16]:
13
and
AC C
10
T2 p2 = T3 p3
T6 p6 = T5 p5
γ −1 γ
,
(13)
,
(14)
γ −1 γ
14
where p2 and p3 are the pressures of the air at the inlet and the outlet of the compressor; p5 and p6
15
are the pressures of the air at the inlet and the outlet of the expander; γ is the adiabatic exponent of
16
air. 8
ACCEPTED MANUSCRIPT
2
3
4
If assuming that there is no pressure loss of the fluids during flowing in all the heat exchangers, the heat transfer processes are all isobaric: p2 = p6 , p3 = p5 .
(15)
T6 T2 = . T5 T3
(16)
Combining Eqs. (13) - (15) gives:
Substituting Eqs. (6) and (8) into Eq. (16) provides that: ln
G (T + T − T ) − Qh GaT1 − Ql . = ln a 1 3 5 T5 T3
RI PT
1
(17)
Eq. (17) reveals the physical constraint which the heat capacity rate and the intermediate
6
temperature of the air have to satisfy. In this article, we want to demonstrate the core idea and
7
brief steps of the newly proposed optimization method. Therefore, we adopted the ideal
8
assumptions for the system, e.g. adiabatic compression and the heat exchangers without pressure
9
drop, to offer a clear illustration of the method. Based on the method, we can also account for the
10
compressor efficiency by replacing the adiabatic processes with polytropic processes. And the
11
pressure drop through heat exchangers can be considered to be decided by the flow rates of the
12
working fluids and the area of the heat exchanger, which can be mathematically expressed as the
13
function of thermal conductance of the heat exchanger and the heat capacity rates of the fluids.
14
Therefore, the pressure of air flow at the inlet and outlet of the heat exchanger can also be
15
expressed as the functions of the thermal conductance of the heat exchanger and the heat capacity
16
rates of the fluids. By applying similar derivation, the method can still work.
17
5. Optimization model for the regenerative air refrigeration system
AC C
EP
TE D
M AN U
SC
5
18
The performance of the regenerative air refrigeration systems is determined by all the design
19
parameters of the systems, e.g. the thermal conductance of heat exchangers, the heat capacity rate
20
and the intermediate temperatures of the air. That is, all the various optimization targets for the
21
systems, e.g. the power consumption, the fixed, operation or total cost, the thermodynamic
22
efficiency and the total heat exchanger area, can be mathematically expressed as the functions of
23
the aforementioned design parameters in the systems. When assuming the heat transfer
24
coefficients of all the heat exchangers are uniform for simplicity, minimizing the total heat transfer 9
ACCEPTED MANUSCRIPT area of all the heat exchangers is equivalent to minimizing the total thermal conductance of all the
2
heat exchangers. Here, we take the minimization of the total thermal conductance, min{(KA)total},
3
as the optimization target to demonstrate the core idea and solving steps of the method. This target
4
is described by the function about T1，T3，T5 and Ga in Eq. (12). These four parameters have to
5
satisfy not only the physical constraint in Eq. (17), but also the relation as following when the
6
thermal conductance of the regenerator is prescribed according to Eq. (11): Ga (T3 − T5 ) − Qh − ( KA) r (T5 − T1 ) = 0 .
RI PT
1
(18)
When the cooling capacity, the net input work, the heat capacity rates and the inlet
8
temperatures of hot-end and cold-end fluids are prescribed, the global optimization problem of the
9
regenerative air refrigeration system can be mathematically converted into a conditional extremum
10
problem. To solve this conditional extremum problem, we apply the Lagrange multiplier method
11
and construct the following function:
M AN U
SC
7
G (T + T − T ) − Qh G T − Ql F = ( KA )h + ( KA )l + ( KA )r + λ ln a 1 − ln a 1 3 5 T5 T3 ,
+ ω ( Ga (T3 − T5 ) − Qh − ( KA )r (T5 − T1 ) )
13
where λ and ω are Lagrange multipliers.
TE D
12
The differential of Eq. (19) with respect to Ga gives
AC C
EP
−2 −1 Ga (T3 − Th , o ) T3 − Th ,i 1 ∂F Ga 1 1 = 1 − ln + − − ∂Ga Gh Ga (T3 − Th ,i ) − Qh Ga Gh Ga Ga (T3 − Th ,i ) − Qh −2 −1 T −T Ga (Tl ,i − T1 ) Tl , o − T1 Ga 1 1 1 + 3 5 . − − 1 ln + − − Ga (Tl , o − T1 ) + Ql Gl Ga Ga Ga ( Tl ,o − T1 ) + Ql T5 − T1 Gl T1 T1 + T3 − T5 + λ − + ω ( T3 − T5 ) = 0 GaT1 − Ql Ga ( T1 + T3 − T5 ) − Qh
14
(20)
The differential of Eq. (19) with respect to T1 yields Ga 1 Ga (T3 − T5 ) − Qh − + + 2 (T5 − T1 ) Ga (Tl , o − T1 ) + Ql Tl ,i − T1 . 1 1 + λ Ga − + ω ( KA )r = 0 GaT1 − Ql Ga (T1 + T3 − T5 ) − Qh
1 1 ∂F = − − ∂T1 Gl Ga
15
(19)
−1
The differential of Eq. (19) with respect to T3 is
10
(21)
ACCEPTED MANUSCRIPT ∂F = ∂T3 −1 1 Ga 1 1 − − Ga Gh T3 − Th , o Ga ( T3 − Th ,i ) − Qh =0
(22)
The differential of Eq. (19) with respect to T5 offers 1 Ga ∂F Ga (T1 − T3 ) + Qh = + λ − + 2 ∂T5 T G T + T ( (T5 − T1 ) a 1 3 − T5 ) − Qh 5
RI PT
1
Ga Ga 1 + − λ − + ωGa . T5 − T1 Ga (T1 + T3 − T5 ) − Qh T3
− ω ( Ga + ( KA )r ) = 0 .
(23)
Eqs. (17), (18), (20) – (23) offer 6 optimization equations which contain 6 unknown
3
parameters, i.e. Ga, T1, T3, T5, λ and ω. Thereafter, the solution of these optimization equations can
4
directly lead to the minimization of the total thermal conductance of all the heat exchangers and
5
the corresponding optimal values for all the unknown parameters.
6
6. Optimization results and discussion
M AN U
SC
2
For the regenerative air refrigeration systems in Fig. 1, the inlet temperatures and the heat
8
capacity rates of the hot-end and the cold-end fluids are: Thi = 303 K, Tli = 293 K, Gh = 500 W/K,
9
Gl = 400 W/K. The cooling capacity and the net input work of the system are: Ql =1000 W,
10
W=500 W. In this situation, if the thermal conductance of the regenerator (KA)r is given as 10
11
W/K. Solving Eqs. (17), (18) and (20) – (23) provides the optimal values for all the unknown
12
parameters corresponding to the minimization of the total thermal conductance of all the heat
13
exchangers, which are listed in Table 1. The optimal pressure ratio of compressor, π, is 4.77.
AC C
EP
TE D
7
Table. 1 Optimization results 14
Furthermore, for the regenerator prescribed with any other size, the minimum of the total
15
thermal conductance of the heat exchangers and the corresponding optimal configuration of all the
16
unknown parameters can also be obtained. Fig. 4 shows the influence of the prescribed value of
17
(KA)r on the minimum of the total thermal conductance of all the heat exchangers (KA)total and that
18
of the hot-end and the cold-end heat exchangers (KA)h+(KA)l. As (KA)r increases, the minimum
19
total thermal conductance of all the heat exchangers monotonously increases as well, which
20
indicates that the total thermal conductance of all the heat exchangers has no extremum if the 11
ACCEPTED MANUSCRIPT thermal conductance of the regenerator is not fixed. Meanwhile, the section above the curve of
2
(KA)total represents the total thermal conductance in this area is physically possible to satisfy the
3
design requirements but not the minimum, while the total thermal conductance in the section
4
below the curve is impossible to satisfy the design requirements. On the other hand, in Fig. 4, as
5
(KA)r increases, the total thermal conductance of the hot-end and the cold-end heat exchangers
6
(KA)h+(KA)l reaches its maximum at about (KA)r = 20 W/K. When (KA)r < 20 W/K, a small
7
variation will lead to a sharp change of (KA)h+(KA)l, which means that the thermal conductance of
8
the regenerator should be carefully selected if there is a specific limit for the thermal conductance
9
of the hot-end and the cold-end heat exchangers during the optimization design. On the contrary,
10
when (KA)r > 20 W/K, increasing the thermal conductance of the regenerator has quite smaller
11
influence on the (KA)h+(KA)l.
M AN U
SC
RI PT
1
Fig. 4 The influence of the prescribed value of (KA)r on the minimum of the total thermal conductance of all the heat exchangers and that of the hot-end and the cold-end heat exchangers Fig. 5 shows the variation of the optimal pressure ratio of the compressor and the heat capacity
13
rate of the air versus the thermal conductance of the regenerator. The pressure ratio π equals to
14
p2/p3. When (KA)r increases, the decreasing optimal pressure ratio means that increasing the
15
thermal conductance of the regenerator can obviously reduce the pressure ratio of the compressor
16
and the expander. On the other hand, the optimal heat capacity rate of the air Ga has its minimum
17
when (KA)r = 30 W/K. That is, when (KA)r < 30 W/K, Ga drops sharply with (KA)r increasing;
18
When (KA)r > 30 W/K, Ga increases slowly. And the increasing of the area of the hot-end and
19
cold-end heat exchangers reduces the heat transfer irreversibility and the temperature difference
20
between air flow and the outside fluids [14]. While the latter recovers heat between hot and cold
21
air flows and can lead to the drop of the pressure ratio of the compressor. In other words, if we
22
don’t care about the performance of the compressor, for a refrigeration cycle, both hot-end and
23
cold-end heat exchangers are essential; while a regenerator is not. Therefore, the sum of the
24
hot-end and cold-end heat exchangers in the simple air refrigeration cycles has the minimum as
25
mentioned in the published paper. However, the total thermal conductance of all the three heat
26
exchangers in the regenerative air refrigeration system has no minimum if the area of the
27
regenerator is not prescribed.
AC C
EP
TE D
12
12
ACCEPTED MANUSCRIPT Fig. 5 The variation of the optimal pressure ratio of the compressor (π= p2/p3) and the heat capacity rate of the air versus the thermal conductance of the regenerator Fig. 6 demonstrates the variation of the optimal thermal conductance of the hot-end and the
2
cold-end heat exchangers, (KA)h and (KA)l, and their ratio, (KA)h/(KA)l, versus the thermal
3
conductance of the regenerator. As showed in Fig. 6, when (KA)r increases, both (KA)h and (KA)l
4
increase fast at first and then decrease slowly. Furthermore, the thermal conductance of the
5
hot-end heat exchanger is always larger than that of the cold-end heat exchanger, which is also
6
indicated by the fact that the ratio (KA)h/(KA)l is always above 1. Therefore, the equivalence of the
7
thermal conductance of both hot-end and cold-end heat exchangers [1] is not always essential to
8
obtain the optimal performance of the refrigeration systems.
SC
RI PT
1
M AN U
Fig. 6 The variation of the optimal thermal conductance of the hot-end and the cold-end heat exchangers and their ratio versus the thermal conductance of the regenerator Fig. 7 indicates the variation of T2 and T5 versus the thermal conductance of the regenerator.
10
Because the increase of the regenerator strengthens the heat recovery, the outlet temperature of the
11
cold air in the regenerator, T2, promotes and that of the hot air, T5, goes down. Fig. 8 illustrates the
12
variation of T1 and T6 versus the thermal conductance of the regenerator. T1 sharply increases at
13
first and then decreases slowly. While T6 decreases at first and then goes up slowly. The average
14
temperature of the air in the cold-end heat exchanger, T1-6 = (T1+T6)/2, has its maximum. Based on
15
the Eq. (6), T1and T6 has the relation as:
EP
TE D
9
T1 − T6 =
Ql . Ga
(24)
Because the heat capacity rate of the air, Ga, first decreases and then rises as (KA)r increases
17
in Fig. 5, the difference between T1 and T6 first increases and then decreases. According to the
18
variation of the thermal conductance of the cold-end heat exchanger in Fig. 4, the temperature
19
difference between the cold-end fluid and the air rises after falling. Therefore, T1-6 appears to fall
20
after rising. Fig. 9 shows the variation of T3 and T4 versus the thermal conductance of the
21
regenerator. T3 sharply first increases and then decreases slowly, while T4 first decreases and then
22
goes up slowly. The average temperature of the air in the hot-end heat exchanger T3-4 = (T3+T4)/2
23
has its minimum.
AC C
16
Fig. 7 The variation of T2 and T5 versus the thermal conductance of the regenerator 13
ACCEPTED MANUSCRIPT 1 Fig. 8 The variation of T1 and T6 versus the thermal conductance of the regenerator 2
3
RI PT
Fig. 9 The variation of T3 and T4 versus the thermal conductance of the regenerator
7. Conclusions
The regenerative air refrigeration systems involve both heat transfer and thermodynamic
5
processes. This paper introduces the entransy theory into the analysis for the heat transfer process
6
in the hot-end and the cold-end heat exchangers and the regenerator. Via combining the
7
thermodynamic analysis for the compression and the expansion processes in the compressor and
8
the expander, a mathematical model for the whole system is established to describe the physical
9
relation between the unknown design parameters and the design requirements. Based on this
10
model, the optimization for the regenerative air refrigeration systems can be converted into a
11
conditional extremum problem, which offers a theoretical optimization equation group by the
12
Lagrange multiplier method. Solving the equation group directly obtains the best performance of
13
the systems and the corresponding optimal configurations of all the unknown parameters.
TE D
M AN U
SC
4
The validity of the proposed optimization method is proved through an actual optimization
15
case to minimize the total thermal conductance of all the heat exchangers with the cooling
16
capacity, the net input work, the heat capacity rates and the inlet temperatures of hot-end and
17
cold-end fluids given. Meanwhile, the following results are obtained: 1) The minimum of the total
18
thermal conductance of all the heat exchangers in the system monotonously rises as the prescribed
19
value of the thermal conductance of the regenerator increases. If the thermal conductance of the
20
regenerator is not prescribed, the total thermal conductance of the heat exchangers in the system
21
has no extremum. 2) Increasing the thermal conductance of the regenerator decreases the optimal
22
pressure ratio of the compressor and the expander. 3) As the thermal conductance of the
23
regenerator increasing, the thermal conductance of the hot-end and the cold-end heat exchangers
24
and their total thermal conductance have their own maximums. Meanwhile, the thermal
25
conductance of the hot-end heat exchanger is always larger than that of the cold-end one, which
AC C
EP
14
14
ACCEPTED MANUSCRIPT indicates that the equivalence of the thermal conductance of both hot-end and cold-end heat
2
exchangers is not always essential to obtain the optimal performance of the refrigeration systems.
3
4) The optimal heat capacity rate of the air has its minimum when the regenerator gets larger.
4
Acknowledgement
RI PT
1
The authors would like to thank the National Natural Science Foundation of China (Grant No.
6
51006060), the National Basic Research Program of China (Grant No. 2010CB227305) and the
7
Foundation for the Author of National Excellent Doctoral Dissertation of China (Grant No.
8
201150).
9
References
M AN U
SC
5
[1] A. Bejan, Theory of heat transfer-irreversible refrigeration plants, International Journal of Heat
11
and Mass Transfer. 32 (1989) 1631-1639.
12
[2] G. Grazzini, R. Rinaldi, Thermodynamic optimal design of heat exchangers for an irreversible
13
refrigerator, International Journal of Thermal Sciences. 40 (2001) 173-180.
14
[3] B. Sahin, A. Kodal, Thermoeconomic optimization of a two stage combined refrigeration
15
system: a finite-time approach, International Journal of Refrigeration. 25 (2002) 872-877.
16
[4] M. A. AitAli, The maximum coefficient of performance of internally irreversible refrigerators
17
and heat pumps, Journal of Physics D: Applied Physics. 29 (1996) 975-980.
18
[5] L. G Chen, C. Wu, F. R. Sun, Cooling load versus COP characteristics for and irreversible air
19
refrigeration cycle, Energy Conversion and Management. 39 (1998) 117-125.
20
[6] C. K. Chen, Y. F. Su, Exergetic efficiency optimization for an irreversible Brayton refrigeration
21
cycle, International Journal of Thermal Sciences. 44 (2005) 303-310.
22
[7] Y. M. Tu, L. G. Chen, F. R. Sun, C. Wu, Cooling load and coefficient of performance
23
optimizations for real air-refrigerators, Applied Energy. 83 (2006) 1289-1306.
24
[8] S. J. Liu, Z. Y. Zhang, L. L. Tian, Thermodynamic analysis of the actual air cycle refrigeration
25
system, Systems Engineering Procedia. 1 (2011) 112-116.
AC C
EP
TE D
10
15
ACCEPTED MANUSCRIPT [9] Z. Y. Guo, H. Y. Zhu, X. G. Liang, Entransy-A physical quantity describing heat transfer ability,
2
International Journal of Heat and Mass Transfer. 50 (2007) 2545-2556.
3
[10] Q. Chen, Y. C. Xu, An entransy dissipation-based optimization principle for building central
4
chilled water systems, Energy. 37 (2012) 571-579.
5
[11] Y. C. Xu, Q. Chen, An entransy dissipation-based method for global optimization of district
6
heating networks, Energy Build. 48 (2012) 50-60.
7
[12] Y. C. Xu, Q. Chen, Minimization of mass for heat exchanger networks in spacecrafts based
8
on the entransy dissipation theory, International Journal of Heat and Mass Transfer. 55 (2012)
9
148-156.
SC
RI PT
1
[13] F. Yuan, Q. Chen, A global optimization method for evaporative cooling systems based on the
11
entransy theory, Energy. 42 (2012) 181-191.
12
[14] Q. Chen, Y. C. Xu, J. H. Hao, An optimization method for gas refrigeration cycle based on
13
the combination of both thermodynamics and entransy theory, Applied energy. submitted.
14
[15] Q. Chen, Entransy dissipation-based thermal resistance method for heat exchanger
15
performance design and optimization, International Journal Heat and Mass Transfer. 60 (2013)
16
156-162.
17
[16] M. J. Moran, H. N. Shapiro, Fundamentals of Engineering thermodynamics, fifth ed.,
18
Hoboken NJ, John Wiley & Sons, Inc., 2004.
TE D
M AN U
10
AC C
EP
19
16
ACCEPTED MANUSCRIPT 1
Figure list
Fig. 1 The schematic diagram of a regenerative air regeneration system
3
Fig. 2 The T-s diagram of the regenerative air regeneration cycle
4
Fig. 3 The T-q diagram for the heat transfer process in the hot-end heat exchanger
5
Fig. 4 The influence of the prescribed value of (KA)r on the minimum of the total thermal
6
conductance of all the heat exchangers and that of the hot-end and the cold-end heat exchangers
7
Fig. 5 The variation of the optimal pressure ratio of the compressor (π= p2/p3) and the heat
8
capacity rate of the air versus the thermal conductance of the regenerator
9
Fig. 6 The variation of the optimal thermal conductance of the hot-end and the cold-end heat
SC
RI PT
2
exchangers and their ratio versus the thermal conductance of the regenerator
11
Fig. 7 The variation of T2 and T5 versus the thermal conductance of the regenerator
12
Fig. 8 The variation of T1 and T6 versus the thermal conductance of the regenerator
13
Fig. 9 The variation of T3 and T4 versus the thermal conductance of the regenerator
M AN U
10
AC C
EP
TE D
14
17
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
1
Fig. 1 The schematic diagram of a regenerative air regeneration system
TE D
2
AC C
EP
3
18
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
1 2
Fig. 2 The T-s diagram of the regenerative air regeneration cycle
AC C
EP
TE D
3
19
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
1 2
Fig. 3 The T-q diagram for the heat transfer process in the hot-end heat exchanger
AC C
EP
TE D
3
20
ACCEPTED MANUSCRIPT 60
100
56
40
(KA)h+(KA)l
0
14
28
42
56
-1
(KA)h+(KA)l / W⋅K
70
RI PT
(KA)total
52
48 70
SC
(KA)total / W⋅K
-1
-1
130
2
Fig. 4 The influence of the prescribed value of (KA)r on the minimum of the total thermal
3
conductance of all the heat exchangers and that of the hot-end and the cold-end heat exchangers
M AN U
1
(KA)r/ W⋅K
AC C
EP
TE D
4
21
ACCEPTED MANUSCRIPT
54
π
Ga
4.8
Ga / W⋅K
34
24
14
1
28
42
M AN U
0
SC
4.6
4.4
44 -1
5.0
RI PT
π
5.2
-1
(KA)r/ W⋅K
56
14 70
2
Fig. 5 The variation of the optimal pressure ratio of the compressor (π= p2/p3) and the heat
3
capacity rate of the air versus the thermal conductance of the regenerator
AC C
EP
TE D
4
22
ACCEPTED MANUSCRIPT 1
1.10
28.6
RI PT
27.2 1.06
25.8 (KA)h
24.4
1.04
(KA)l (KA)h/(KA)l
0
14
28
42
-1
(KA)r/ W⋅K
56
M AN U
23.0 2
(KA)h/(KA)l
1.08
SC
(KA)h & (KA)l / W⋅K
-1
30.0
1.02 70
3
Fig. 6 The variation of the optimal thermal conductance of the hot-end and the cold-end heat
4
exchangers and their ratio versus the thermal conductance of the regenerator
AC C
EP
TE D
5
23
ACCEPTED MANUSCRIPT 355
T5
335
T2
RI PT
T/ K
315 295
255
0
17
34
51
SC
275
68
85
-1
(KA)r/ W⋅K
2
M AN U
1
Fig. 7 The variation of T2 and T5 versus the thermal conductance of the regenerator
AC C
EP
TE D
3
24
241
270
237
T1 T1-6
229
250
240
0
17
34
51
-1
(KA)r/ W⋅K
68
M AN U
1 2
233
RI PT
T6
260
T6 / K
280
SC
T1 & T1-6 / K
ACCEPTED MANUSCRIPT
225 85
Fig. 8 The variation of T1 and T6 versus the thermal conductance of the regenerator
AC C
EP
TE D
3
25
ACCEPTED MANUSCRIPT 355
T3-4
347
T4
339
T4 / K
T3
390
RI PT
T3 & T3- 4 / K
405
375
331
0
17
34
51
-1
323 85
SC
360
68
2
Fig. 9 The variation of T3 and T4 versus the thermal conductance of the regenerator
M AN U
1
(KA)r/ W⋅K
AC C
EP
TE D
3
26
ACCEPTED MANUSCRIPT 1
Table list
2
Table 1 Optimization results (KA)h
(KA)l
(KA)total
Ga
optimization results
29.92
27.56
66.59
21.92
3
RI PT
parameters/ W K-1
parameters/ K
T1
T2
T3
T4
T5
T6
optimization results
274.10
291.19
397.10
328.67
311.58
228.48
AC C
EP
TE D
M AN U
SC
4
27