A global optimization method for evaporative cooling systems based on the entransy theory

Energy 42 (2012) 181e191

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A global optimization method for evaporative cooling systems based on the entransy theory Fang Yuan, Qun Chen* Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 14 November 2011 Received in revised form 13 March 2012 Accepted 29 March 2012 Available online 26 April 2012

Evaporative cooling technique, one of the most widely used methods, is essential to both energy conservation and environment protection. This contribution introduces a global optimization method for indirect evaporative cooling systems with coupled heat and mass transfer processes based on the entransy theory to improve their energy efficiency. First, we classify the irreversible processes in the system into the heat transfer process, the coupled heat and mass transfer process and the mixing process of waters in different branches, where the irreversibility is evaluated by the entransy dissipation. Then through the total system entransy dissipation, we establish the theoretical relationship of the user demands with both the geometrical structures of each heat exchanger and the operating parameters of each fluid, and derive two optimization equation groups focusing on two typical optimization problems. Finally, an indirect evaporative cooling system is taken as an example to illustrate the applications of the newly proposed optimization method. It is concluded that there exists an optimal circulating water flow rate with the minimum total thermal conductance of the system. Furthermore, with different user demands and moist air inlet conditions, it is the global optimization, other than parametric analysis, will obtain the optimal performance of the system. Ó 2012 Elsevier Ltd. All rights reserved.

Keywords: Evaporation Energy Heat transfer Optimization Entransy dissipation Global design

1. Introduction Heat, ventilation and air conditioning systems (HVACs), commonly appeared in civil engineering, always create a comfortable indoor environment on one hand, but simultaneously increase energy consumption on the other hand. It is reported that about 20% of the total energy all over the world is consumed by HVACs, which accelerates us to develop some novel HVACs with higher energy efficiency [1]. Among all types of HVACs, the evaporative cooling techniques obtain rapid development and application because moist air is friendly to the environment and is also renewable [2]. Until now, many scientists have undertaken a large amount of researches on the optimization of evaporative cooling systems, which can be categorized into three types. One is the process design, which aims at making full use of the evaporative cooling ability of moist air by introducing several direct/indirect evaporative cooling systems [3e5]. Two is the parameter design, which optimizes the evaporative cooling performance through analyzing the impact of such operating and structural parameters as the temperature, humidity and flow rate of moist air [6,7], the temperature and flow rate of water [8], and different types of packing [8,9] on the * Corresponding author. Tel./fax: þ86 10 62781610. E-mail address: [email protected] (Q. Chen). 0360-5442/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2012.03.070

evaporative cooling performance for different types of direct/indirect evaporative coolers, including plate type [10,11], tubular type [12,13] and heat pipe [14]. In this category, scholars usually list several possible combinations of the structural and operating parameters, estimate their influences on the evaporation cooling performance, and finally choose a better, but not the best values by experiment [7e9] or numerical simulation [5]. Therefore, it is actually a “try-and-error” method. The last is the exergy method [15e18], which analyzes the exergy loss of all irreversible processes in evaporative cooling systems, and then makes some adjustments empirically to get the scheme with lower exergy loss. However, like the “try-and-error” method, it cannot ensure the optimized system is the optimal one. Furthermore, there are some scholars who questioned if the minimum exergy loss, or the minimum entropy generation, is the optimization criterion for all heat and mass transfer processes, regardless of the nature of the applications [19e22]. On the other hand, in the irreversibility analysis and optimization of heat transfer processes, Guo et al. [23] introduced the physical parameters of entransy and entransy dissipation, to respectively describe the heat transfer ability of an object or a system and evaluate the irreversibility of heat transfer processes. After that, many scholars employed the extremum of entransy dissipation as an alternative criterion to optimize all the three heat

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Nomenclature T D cp

g0 m k A (kA)i G

FG Q

temperature, K humidity, kg kg1 constant pressure specific heat, J kg1 K1 evaporation latent heat, J kg1 mass flow rate, kg s1 thermal conductivity, W m1 K1 area, m2 thermal conductance for each heat exchanger, W K1 entransy flow rate, W K entransy dissipation rate, W K heat flow rate, W

transfer processes like heat conduction [23,24], convective heat transfer [25e27], thermal radiation [28] and heat exchangers [29]. Moreover, Chen et al. extended the entransy theory to both mass transfer [30] and coupled heat and mass transfer processes [31,32], on the basis of the analogy between heat and mass transfer, and subsequently introduced the moisture entransy to represent the endothermic ability of moist air, and used its dissipation to analyze the performance of some typical evaporative cooling systems. Jiang et al. [21] and Xie et al. [33] optimized the heat and moisture transfer processes based on the entransy theory, which already had some applications in some evaporative cooling systems. However, all the aforementioned optimizations in the literature cannot globally optimize the coupled heat and mass performance in evaporative cooling systems. The reason is that we still lack a theoretical and mathematical relationship to connect the user demands with all the structural and operating parameters, i.e. the thermal conductance of each heat exchanger and the circulating water flow rate in each branch. Therefore, it is needed to advance a new method to realize the global optimization for evaporative cooling systems. This paper aims at proposing a novel global optimal design approach for evaporative cooling systems. We first make use of the concept of moisture entransy dissipation to measure the irreversibility of the whole system including heat transfer processes in single-phase heat exchangers, evaporative cooling process in airewater direct evaporative cooling tower, and non-isothermal fluid flow mixing processes, and take it as a bridge to construct the relations of structural and operating parameters of devices, user demands and inlet parameters of moist air and user’s medium. Then, we deduce two groups of optimization functions of both structural and operating parameters for two typical optimal problems by the conditional extremum method. Finally, the application of our novel optimization method in the optimal design of a typical indirect evaporative cooling system is proposed to show the applications and advantages of the newly proposed optimization approach. 2. Irreversibility in evaporative cooling systems The moist air enters the airewater direct evaporative cooling tower (2) from the air pre-cooler (1), and directly contacts with the circulating water. Taking advantage of the pressure difference between the water vapor of the moist air and the saturated water vapor over the circulating water surface, the water will evaporate and absorb heat to achieve cooling capacity. After humidification, the moist air will be discharged by the fan (4). Meanwhile, under the action of the water circulating pump (5), the cooled water divides into two branches. One flows into the air pre-cooler (1), to cool the moist air entering into the airewater direct evaporative

arrangement factor, K W1 RG entransy dissipation-based thermal resistance, K W1 l, b, h, ai,l0 , b0 , h0 , ai0 Lagrange multipliers

x

Subscript a w v sat 0 dp wb ave

air water water vapor saturated ambient dew point wet bulb arithmetic average

cooling tower (2), while the other supplies the user with a cold source. The circulating water out of the air pre-cooler (1), and the user’s heat exchanger (3), mix together and flow into the direct evaporative cooling tower (2). The whole system provides cooling capacity, which takes circulating water as a carrier and makes use of the pressure difference between the water vapor of moist air and the saturated water vapor on liquid water surface. The capitals A w K, stand for different states in the system, where the saturated moist air at the ambient temperature is taken as a benchmark, T0 and D0 represent its temperature and humidity, respectively. The whole indirect system includes three types of irreversible processes: (1) the heat transfer process in the air pre-cooler and user’s heat exchanger; (2) the mixing process of circulating water in two branches; (3) the coupled heat and mass transfer process in the airewater direct evaporative cooling tower. For simplicity, we assume that: (1) the mass flow rate of total circulating water of the system keeps constant except for in the evaporative cooling tower; (2) the property parameters of moist air. e.g. constant pressure specific heat cp,a, remain constant; (3) the saturation line of water is linear, and satisfy the following empirical relation, Dv,sat ¼ a$Ta,sat þ b; (4) the evaporation latent heat g0 keeps constant. And then it is readily to evaluate the irreversibility of the aforementioned three types of processes by entransy dissipation. 2.1. The coupled heat and mass transfer process in the direct evaporative cooling tower In the airewater direct evaporative cooling tower, the moist air with the mass flow rate, temperature and humidity of ma, Ta,B and Da,B, respectively, cools the circulating water. For clearly describing the states of moist air during the evaporative cooling process, we mark them in the enthalpy-humidity chart, shown in Fig. 2, in which the symbols j,wb and j,dp represent the wet bulb and dew point temperatures of the moist air at state j (j ¼ A, B, C), respectively. It has already been demonstrated in the literature [5] that a coupled heat and moisture transfer process in a direct evaporative cooling tower can be seemed as a “single-phase” heat transfer process between moist air at its wet bulb temperature and water, its equivalent energy conservation equation can be written as



 Q2 ¼ ma cp;ea Ta;C;wb  Ta;B;wb ¼ mw;K cp;w Tw;K  Tw;D ;

(1)

where, cp,ea ¼ cp,a þ a$g0 refers to the equivalent pressure specific heat of moist air, while the heat transfer equation of this process is

Q2 ¼

cp;ea ðkAÞ2 DTLMT ; cp;a

(2)

F. Yuan, Q. Chen / Energy 42 (2012) 181e191

183

which represents the entransy dissipation resulting from temperature difference between dry bulb temperature and dew point temperature of moist air at state B. For moist air at state j, the entransy flow rate [31] can be calculated by



2 1  1 Ga;j ¼ ma;j cp;a Ta;j  T0 þ g0 ma;j Ta;j;dp  T0 Da;j  D0 : 2 2

(8)

The dew point temperature Ta,B,dp and humidity Da,B of the moist air at state B equal to those at the state A. In addition, based on the thermal energy conservation equation in the air pre-cooler



 Q1 ¼ ma cp;a Ta;A  Ta;B ¼ mw;E cp;w Tw;G  Tw;E ;

(9)

together with the isenthalpic process from state B to state B,wb, ha,B ¼ ha,B,wb, as shown in Fig. 2, we can obtain the expression of the wet bulb temperature of moist air at state B by the inlet parameters of moist air and the heat flow rate in the air pre-cooler Fig. 1. The schematic of a typical indirect evaporative cooling system.

Ta;B;wb ¼

DTLMT

DTmax  DTmin ¼ ; DTmax ln DTmin 

(3)



cp;ea DTmax 1 1 ¼ exp  ðkAÞ2  DTmin mw;K cp;w ma;A cp;ea cp;a

 ;

(4)

where, DTLMT represents the logarithmic mean temperature of the moist air at wet bulb temperature and circulating water. At this time, it is easy to find that the equivalent thermal conductance kA, i.e. the product of heat transfer coefficient and heat transfer area, is cp,ea/cp,a times the original one. Although the thermal energy and the mass of water are both conversed in the evaporative cooling tower, the moisture entransy [31] is dissipated owing to the temperature and moisture difference between the moist air and the water. If the outlet moist air is saturated, the entransy dissipation in the direct evaporative cooling tower equals to the entransy difference of the inlet and outlet moist air and circulating water

FG2 ¼ Ga;B  Ga;C;wb þ Gw;K  Gw;D ;

FG2



 ¼ Ga;B  Ga;B;wb þ Ga;B;wb  Ga;C;wb þ Gw;K  Gw;D ;

FG2;1 ¼ Ga;B  Ga;B;wb ;

FG2;2 ¼ Ga;B;wb  Ga;C;wb þ Gw;K  Gw;D ;

(7)

(11)

which stands for the irreversibility of “single-phase” heat transfer between the moist air at its wet bulb temperature and the circulating water. For water at state j, the entransy [23] is calculated as

Gw;j ¼


2 1 m cp;w Tw;j  T0 : 2 w;j

(12)

And for saturated moist air, the expression of entransy, Eq. (8), is simplified into

Ga;j;wb ¼

 2 1 ma;j cp;ea Ta;j;wb  T0 : 2

(13)

According to the definition of entransy dissipation-based thermal resistance [34]

RG ¼

(6)

including two terms, one is

(10)

Then, substituting Eq. (8)e(10) into Eq. (7) yields the difference between the entransy values of moist air at state B and its wet bulb state, i.e. the first part of Eq. (6). The second term in the right hand of Eq. (6) is

(5)

where Gi,j stands for the entransy flow rate of working medium i at state j, the subscript i ¼ a (air) ¼ w (water). For further analysis, Eq. (5) is rewritten as

 ma;A cp;a Ta;A þ g0 Da;A þ ð273:15a  bÞg0  Q1 ma;A cp;ea

FG Q2

¼

Dtave Q

;

(14)

together with its specific expression for counter-flow and parallelflow heat exchangers [35] and substituting Eq. (1)e(4) into Eq. (14), we will obtain the entransy dissipation-based thermal resistance for the coupled heat and mass transfer in the evaporative cooling tower,



RG2;2 ¼



x2 exp ðkAÞ2 x2 cp;ea =cp;a þ 1 

; 2 exp ðkAÞ2 x2 cp;ea =cp;a  1

(15)

where, the subscript ave means the arithmetic average, x2 is a parameter relevant to arrangement of heat exchanger, defined as the arrangement factor. For counter-flow,

x2 ¼

Fig. 2. The enthalpy-humidity chart of the moist air in the evaporative cooling system.

1 1  ; mw;K cp;w ma;A cp;ea

(16)

From Eq. (15), we can see the entransy dissipation-based thermal resistance is relevant to thermal conductance and the mass flow rate, i.e. the structural and the operating parameters of the evaporative cooling tower. According to Eqs. (14) and (15), the second part of entransy dissipation in Eq. (6) can be calculated by

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F. Yuan, Q. Chen / Energy 42 (2012) 181e191

FG2;2 ¼ Q22 RG2;2

 x exp ðkAÞ2 x2 cp;ea =cp;a þ 1 

¼ Q22 2 : 2 exp ðkAÞ2 x2 cp;ea =cp;a  1

(17)

Then, the total entransy dissipation in the airewater direct evaporative cooling tower is the sum of the aforementioned two parts, i.e. Eqs. (7) and (17). 2.2. The heat transfer processes in the air pre-cooler and the user’s heat exchanger According to both the definition and the formula of entransy dissipation-based thermal resistance for “single-phase” heat exchangers with several different arrangements [35], together with the energy conservation equation, Eq. (9) and the heat transfer equation, we will obtain the expression of entransy dissipation in the air pre-cooler

FG1 ¼

Q12 RG1

¼

 exp ðkAÞ1 x1 þ 1 

; 2 exp ðkAÞ1 x1  1

x Q12 1

(18)

where, the expression of x1 is

x1 ¼

1 1  : ma;A cp;a mw;E cp;w

(19)



 Q3 ¼ mw;H cp;w Tw;H  Tw;I ¼ mw;F cp;w Tw;J  Tw;F ;

1 1  : mw;H cp;w mw;F cp;w

(21)

Then, the entransy dissipation in user’s heat exchanger is written as





x3 exp ðkAÞ3 x3 þ 1 

: 2 exp ðkAÞ3 x3  1

(22)

2.3. The mixing process of circulating waters in each branch

(23)

where, the entransy values of water at states J, G and K can be calculated from Eq. (12). That is, simultaneously taking three energy conservation equations of heat exchangers, i.e. Eqs. (1), (9) and (20), into account, Eq. (23) is rewritten as

FG;mix ¼

FG ¼ Gw;H  Gw;I þ Ga;A  Ga;C;wb ;

(26)

where Gw,H and Gw,I can be simply expressed by Eq. (12), while Ga,A is calculated by Eq. (8). Furthermore, to calculate the entransy of moist air at saturated state with the same wet bulb temperature as state C, i.e. Ga,C,wb, we should first obtain the expression of the temperature and humidity of moist air at state C,wb. Substituting the empirical relation of the saturation line, Eq. (10), and energy conservation of the system, i.e.

Q32 Q12 Q22 1 1 1 þ  : 2 mw;F cp;w 2 mw;E cp;w 2 mw;K cp;w

(27)

into Eq. (1), we will have

Ta;C;wb ¼

Q3 þ ma;A cp;a Ta;A þ ma;A g0 Da;A þ ð273:15a  bÞma;A g0 ; ma;A cp;ea (28)

Da;C;wb ¼ a$Ta;C;wb þ b;

(29)

and then substituting Eqs. (28) and (29) into Eq. (8) gives the value of Ga,C,wb. Then, from the two different calculation approaches for the total entransy dissipation of the system, Eqs. (25) and (26), we can establish the following equivalent relation as






F0 ¼ F Q ; m ; T ; m  ; Ta;A ; Da;A
  G 3
w;H w;H a;A FG1 ðkAÞ
1 ; Q1 ; mw;E  þ FG2;1
ðQ1 Þ þ FG2;2 ðkAÞ2; Q2 ; ¼ þFG3 ðkAÞ3 ; mw;F þ FG;mix Q1 ; Q2 ; mw;E ; mw;F (30)

The circulating waters out of the user’s heat exchanger at state J and the air pre-cooler at state G will mix together to state K and then flow into the airewater direct evaporative cooling tower, the entransy dissipation during the mixing process is

FG;mix ¼ Gw;J þ Gw;G  Gw;K ;

(25)

On the other hand, we can also obtain the total entransy dissipation of the system by the difference of the entransy rate flowing in and out of the system

(20)

where, Q3 is the heat flow rate, i.e. the user’s demand, and the arrangement factor, x3, is

FG3 ¼ Q32 RG3 ¼ Q32

FG ¼ FG1 þ FG2;1 þ FG2;2 þ FG3 þ FG;mix :

Q3 þ Q1 ¼ Q2 ;

Likewise, for the heat transfer in the user’s heat exchanger, the energy conservation equation is

x3 ¼

to calculate the total entransy dissipation in the system is to add all the entransy dissipation rates during these irreversible processes, i.e.

(24)

which directly connects the user’s demand Q3, the inlet parameters of user and moist air, i.e. tw,H, mw,H, ta,A, Da,A and ma,A to the structural parameters, i.e. (kA)1, (kA)2, (kA)3 and operating parameters mw,E and mw,F, which makes it possible to globally optimize all the structural and operating parameters at the same time. In addition, for optimizations of evaporative cooling systems in engineering fields, the demands of users are always fixed, and the inlet conditions of moist air often depend on external environment which are also prescribed, then Ta,C,wb and Da,C,wb are constant, so as the entransy flow rate for each state in Eq. (26). Therefore, the total entransy dissipation of the system calculated by Eq. (26) is constant, which can be represented as F0. 3. Optimal design of evaporative cooling systems

2.4. The total entransy dissipation of the evaporative cooling system From the aforementioned analysis, we obtain the expression of entransy dissipation of each irreversible process in the evaporative cooling system shown in Fig. 1, which are the functions of heat flow rate, structural and operating parameters. Then one way

Generally speaking, the cost in an evaporative cooling system can be categorized into two aspects: one is the investment cost mainly related to the area of heat exchangers, and the other is the operating cost related to the energy consumption of circulating water pump. Aim to these two different costs, there mainly exist two types of optimization problems: i) minimization of the total

F. Yuan, Q. Chen / Energy 42 (2012) 181e191

thermal conductance of all heat exchangers in the system, i.e. the investment cost for a given total circulating water flow rate; ii) minimization of the total circulating water flow rate, i.e. the operating cost for a given total thermal conductance. 3.1. Optimization with prescribed total circulating water flow rate For a given total circulating water flow rate of an evaporative cooling system,

mw;E þ mw;F ¼ mw;K ¼ const;

(31)

the total entransy dissipation rate is constant which is already concluded in Section 2.4,

FG1 þ FG2;1 þ FG2;2 þ FG3 þ FG;mix ¼ F0 ¼ const:

Tw;F ¼

1 1 Q3 þ T   Q3 RG3 : T 2 w;H 2 w;I 2mw;F cp;w

185

(37)

Then, we can construct a Lagrange function based on the conventional extremum mathematical method with the purpose of minimizing total thermal conductance. Taking the constraints of Eqs. (32)e(34) together with the energy and mass conservation equations, Eq. (27) and Eq. (31), into account, we get

2

3 3
 P ðkAÞi þ l FG1 þ FG2;1 þ FG2;2 þ FG3 þ FG;mix  F0 7 6 7
 P1 ¼ 6 6 i¼1 7; 4 þbðQ1  Q2 þ Q3 Þ þ h mw;E þ mw;F  mw;K 5

 þa1 Tw;D  Tw;E þ a2 Tw;D  Tw;F

(32)

(38)

Meanwhile, the temperature of water out of the direct evaporative cooling tower must equal to that flowing into the air precooler and the user’s heat exchanger, i.e.

where l, b, h, a1, a2 are all Lagrange multipliers. The differential of Eq. (38) with respect to the thermal conductances of each heat exchanger, i.e. (kA)1, (kA)2 and (kA)3, yield

Tw;E ¼ Tw;D ;

(33)

Tw;F ¼ Tw;D ;

(34)

Simultaneously solving the energy conservation equation and the expression of entransy dissipation for the direct evaporative cooling tower, i.e. Eqs. (1) and (11)e(14), gives

vP1 eðkAÞ1 x1 2 ¼ 1  ðlQ1 þ a1 Þ Q1 x1

2 ¼ 0; vðkAÞ1 eðkAÞ1 x1  1

(39)

vP1 eðkAÞ2 x2 cp;ea =cp;a 2 cp;ea ¼ 1  ðlQ2 þ a1 þ a2 Þ Q2 x2 ¼ 0;

vðkAÞ2 cp;a eðkAÞ2 x2 cp;ea =cp;a  12 (40)

Tw;D ¼ Ta;C;wb 

Q2 Q2  þ Q2 RG2 : 2ma;A cp;ea 2cp;w mw;K

(35)

The calculation method for the temperatures of water at states E, F is almost the same as state D. Likewise, we may obtain Tw,E and Tw,F,

Tw;E ¼ Ta;A 

Q1 Q1   Q1 RG1 ; 2ma;A cp;a 2mw;E cp;w

vP1 eðkAÞ3 x3 2 ¼ 1  ðlQ3 þ a2 Þ Q3 x3

2 ¼ 0: vðkAÞ3 eðkAÞ3 x3  1

(41)

The differential of Eq. (38) with respect to the heat transfer rates, Q1 and Q2, offer

(36)

39

 8 2 eðkAÞ1 x1 þ 1 ma;A cp;a Ta;A þ g0 Da;A þ ð273:15a  bÞg0  Q1 > > > x þ Q 7> > 6 1 1 eðkAÞ1 x1  1 > > 7> ma cp;ea > 6 > > 7> > > l6 < 7= vP1 4 Q1 Q1 5 ¼ ¼ 0; Ta;A þ þ > > vQ1 > > ma;A cp;a mw;E cp;w > > > >   > > > > 1 1 1 > > ; : þb þ a1 þ þ 2ma;A cp;a 2mw;E cp;w ðkAÞ1

8 9 ! > > > > > > > > eðkAÞ2 x2 cp;ea =cp;a þ 1 Q2 > > > > l x b  Q  > > 2 2 < = x ðkAÞ c =c p;ea p;a 2 m c 2 e 1 vP1 w;K p;w # " ¼ ¼ 0: x ðkAÞ c =c p;ea p;a 2 > vQ2 1 1 1 e 2 þ1 > > > > > > > a a x ð þ Þ þ  1 2 > > > 2ma;A cp;ea 2mw;K cp;w 2 2 eðkAÞ2 x2 cp;ea =cp;a  1 > > > : ;

(42)

(43)

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F. Yuan, Q. Chen / Energy 42 (2012) 181e191

And the differential of Eq. (38) with respect to the mass flow rates of water at states E and F in Fig. 1, i.e. mw,E and mw,F, give

vP1 vmw;E

vP1 vmw;F

9 8 > 0 1> h i > > 1 > > > > > > 2 Q12 e2ðkAÞ1 x1  1  Q12 ðkAÞ1 x1 eðkAÞ1 x1 > Q > > B2 C> 1 > >  > > l@ A 

> 2 = < 2m2w;E cp;w > m2w;E cp;w eðkAÞ1 x1  1 ¼ 0; ¼ 0 1

 > > > > ðkAÞ1 x1 1  ðkAÞ x > > Q e  Q > > 1 1A 1 1 > > > > þh þa1 @ > > 2

ðkAÞ x > > 2 c > > 1 1  1 m e > > ; : w;E p;w

(44)

9 8 > 0 1> h i > > 1 > > > > > > 2 Q32 e2ðkAÞ3 x3  1  Q32 ðkAÞ3 x3 eðkAÞ3 x3 > > Q > > B C 3 2 > > l  > > @ A 

> 2 2 = < 2mw;F cp;w > m2w;F cp;w eðkAÞ3 x3  1 ¼ ¼ 0: 0 1

 > > > > ðkAÞ3 x3 1  ðkAÞ x > > Q e  Q > > 3 3 3 3 > > þa @ Aþh > > 2 > > 2

> > > > m2w;F cp;w eðkAÞ3 x3  1 > > ; :

There are 12 unknown parameters in Eqs. (27) and (31)e(34) and (39)e(45), simultaneously solving these 12 optimization equations may obtain the values of those unknown variables, i.e. the optimal distribution of thermal conductances of each heat exchanger and circulating water mass flow rates in two branches of the system. 3.2. Optimization with prescribed total thermal conductance Similar to the optimization method used in Section 3.1, when the total thermal conductance is given,

ðkAÞ1 þ ðkAÞ2 þ ðkAÞ3 ¼ ðkAÞ0 ¼ const;

(45)

The differential of Eq. (47) with respect to the thermal conductances, (kA)1, (kA)2 and (kA)3, the heat transfer rates, Q1 and Q2, the and mass flow rates, mw,E, mw,F, yield


0  vP2 eðkAÞ1 x1 2 ¼ h0  l Q1 þ a01 Q1 x1

2 ¼ 0; vðkAÞ1 ðkAÞ e 1 x1  1
0  vP2 ¼ h0  l Q2 þ a01 þ a02 vðkAÞ2 2 cp;ea

 Q2 x 2

eðkAÞ2 x2 cp;ea =cp;a

cp;a eðkAÞ2 x2 cp;ea =cp;a

(48)

(49)

2 ¼ 0; 1

(46)

we can also establish the following Lagrange function with the constraints of Eqs. (27) and (32)e(34) with the purpose of minimizing the pump work, i.e. minimizing the total mass flow rate of


0  vP2 eðkAÞ3 x3 2 ¼ h0  l Q3 þ a02 Q3 x3

2 ¼ 0; vðkAÞ3 eðkAÞ3 x3  1

9 8 0
F0 = < mw;E þ mw;F þ l FG1 þ FG2;1 þ FG2;2 þ FG3 þ FG;mix   0 P2 ¼ þb0 ðQ ;
1  Q2 þ Q3 Þ þ 0h
ðkAÞ1 þ ðkAÞ  2 þ ðkAÞ3  ðkAÞ0 : ; 0 þa1 Tw;D  Tw;E þ a2 Tw;D  Tw;F

(50)

(47)

water at state K in Fig. 1. where l0 , b0 , h0 , a10, a20 are Lagrange multipliers.

9 8 3> > 2 > > x ðkAÞ 1 > > 1 e þ1 Q1 Q1 > > > > > >  Ta;A þ þ Q1 x1 ðkAÞ x 7 > > 6 > > m c m c 1 1  1 7 e p;a p;w > > w;E a;A 6 0 > > 7 > > l6 = <

 7 vP2 4 m a;A cp;a Ta;A þ g0 Da;A þ ð273:15a  bÞg0  Q1 5 ¼ ¼ 0; þ > > vQ1 > > m c > > a p;ea > >   > > > > 1 1 1 > > 0 > > > > þ þ þb  a01 > > > > 2ma;A cp;a 2mw;E cp;w ðkAÞ1 ; :

(51)

F. Yuan, Q. Chen / Energy 42 (2012) 181e191

8 9 # " > > > > > > x ðkAÞ c =c > > e 2 2 p;ea p;a þ 1 Q2 0 0 > > > >
 l x b   Q > > 2 2 < = x ðkAÞ c p;ea =cp;a  1 2 2 c þ m m e P v 2 p;w w;E w;F # " ¼ ¼ 0; x ðkAÞ c =c p;ea p;a 2 > vQ2 2
 1 1 1 e þ1 > > > >  a0 þ a0 > > > x þ  > 1 2 2m > > 2cp;w mw;K 2 2 eðkAÞ2 x2 cp;ea =cp;a  1 > > > a;A cp;ea : ;

vP 2 vmw;E

vP 2 vmw;F

9 8 h i > > 0 1Q 2 e2ðkAÞ1 x1  1  Q 2 ðkAÞ x eðkAÞ1 x1 1 > > > > > > 1 1 1 > > 2 1 > > > >  B C 

> > 2 > > 2 x ðkAÞ B C > > 1 1 mw;E cp;w e 1 > > > > B C > > > > B C > > > > B C 9 > > 8 > > B C > > > > > B C > > > > > > > B> C x ðkAÞ c p;ea =cp;a 2 > > > > 2 1 2e þ1 > > > B> C > > > > Q > > > > B C > > 2 ðkAÞ x2 cp;ea =cp;a > > > > > > B C 2 2 e  1 > > > > Q > > = < C 0B 1 0 > > 1 > >  a1 1þlB C > > 2 c > > B C 2m > p;w > 2 > > w;E > > B C mw;K cp;w > > > > x ðkAÞ c =c p;ea p;a 2 > > 2 > > B C c e > > p;ea > > 2 > > > > B C x ðkAÞ Q > > > > 2 

2 = < > 2 2 B> C > cp;a eðkAÞ2 x2 cp;ea =cp;a  1 > > > B C > : ¼ ¼ 0; ; B C > > B C > > > > B C > > > > B C > > > > @ A > > 2 2 > > Q Q > > > > 2 1 > > þ  > > > > 2 c 2 c > > 2m 2m > > w;K p;w w;E p;w > > > > > > 8 8 9 9 > > > > 2 x ðkAÞ c =c > > p;a > > > > 2 2 p;ea Q > > 1 e þ 1 > > > > > > 2 > > > > > > > > > > Q  > > 2 > > > > > > x ðkAÞ c > > > > p;ea =cp;a  1 2 2 2 2 > > e << = = > >
 1 > > > > 0 0 > > a a þ > > 1 2 >> 2 > > > > x ðkAÞ c =c > > c m p;a 2 2 p;ea > > > > p;w c > > e w;K p;ea > >> > > > > > > > > > x ðkAÞ þQ > > > > > > 2 2 

2 > > > > 2; > > : : ; c x ðkAÞ c =c > > p;a p;ea p;a ; : 1 e 2 2

8 9 0 1 i > > 1 2 h 2ðkAÞ1 x1 > 2 ðkAÞ1 x1 > > > x  1  Q ðkAÞ e e Q > > 1 1 1 B2 1 C > > > > B C > >  > > 

2 > > B C 2 x ðkAÞ > > 1 1  1 m c e > > B C p;w > > w;E > > B C > > > > B C > > 9 > > B8 C > > > > B C > > > > > > > B C > > > > x ðkAÞ c =c p;ea p;a 2 > > > > 2 B > 1 2e C þ 1 > > > > > > > B C > > Q > > > > > > 2 ðkAÞ x2 cp;ea =cp;a B C > > > > 2 2 e  1 > > > > B C Q > > = 0B < 3 0 > > C 1 > > l a  1 þ > > 2 B C 2 > 2mw;F cp;w > > > B C 2 > > > > c m > > > > p;w x ðkAÞ c =c B C p;ea p;a w;K > > 2 2 > > c e > > p;ea > > B> C 2 x ðkAÞ > > > > > Q > > B C 2 

> > 2 > > 2 2 < = cp;a eðkAÞ2 x2 cp;ea =cp;a  1 > B> C > > : > B C ; ¼ ¼ 0: B C > > B C > > > > B C > > > > B C > > > > B C > > > > Q22 Q12 @ A > > > > þ  > > > > 2 2 > > 2mw;K cp;w 2mw;E cp;w > > > > > > > > 9 > > 8 9 8 > > > > > > > 2 ðkAÞ2 x2 cp;ea =cp;a þ 1 > > > > > > Q 1 e > > > > > > 2 > > > > > > > >> >  Q2 ðkAÞ x c =c > > > > > > > > > > > > > > p;ea p;a 2 2 2 2 e 1 = << = > > > >
 1 > > 0 0 > > a a þ þ > > > > 1 2 >> 2 > > > > ðkAÞ2 x2 cp;ea =cp;a c m > > > > > > p;w c e > > p;ea > >> > w;K > > > > > > x ðkAÞ þQ > > > > > > 2 2 

2 > > > >: > > 2; > > > : c x ðkAÞ c =c p;a e p;a  1 > > ; 2 2 p;ea : ;

After solving Eqs. (27) and (32)e(34), (46) and (48)e(54) simultaneously, we also can obtain the optimal thermal conductances of each heat exchanger and water mass flow rates of each branch when the total circulating flow rate reaches minimum with prescribed total thermal conductance and user’s demands in the system.

187

(52)

(53)

(54)

To sum up, from the calculations of the above two groups of optimization equations, it is convenient to optimize the coupled heat and mass transfer processes in an evaporative cooling system. Different from the “try-and-error” method, we don’t need to list several possible combinations of operating and structural parameters and estimate their influences on the evaporative cooling performance

188

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so as to choose a better but not the best values of those parameters, which lead to huge calculation amount but low accuracy.

Table 1 The structural and operating parameters of the system at the optimal point. Point A

4. Entransy dissipation-based optimization of typical evaporative cooling systems 4.1. Fixed total circulating water flow rate Considering a typical indirect evaporative cooling system shown in Fig. 1, where the temperature, humidity and mass flow rate of moist air are 308 K, 0.008 kg kg1 and 1 kg s1 respectively, water flowing into the user’s heat exchanger is 299 K with the infinite heat capacity rate. The user’s demanded cooling capacity is 21.88 kW. To ensure the reliability of the optimization calculation program, we have validated the simulation program for this indirect evaporative cooling system by the results in literature [33]. Based on the validated program, Fig. 3 shows the trend of minimum total thermal conductance with different total circulating water flow rates, which indicates that increasing the total circulating water flow rate will first reduce the minimum total thermal conductance rapidly but then increase slowly. There exists an optimal mass flow rate of circulating water making the minimum total thermal conductance at the extreme point shown in Fig. 3 (point A). The structural and operating parameters at this point are listed in Table 1. It is readily to see when the given total thermal conductance is lower than that value of point A, the system will never satisfy the user’s demanded cooling capacity whatever the total circulating water flow rate is. In the optimal design for a “single-phase” heat exchanger, when the heat capacity rates of cold and hot fluids are equal, i.e. the equilibrium flow, the heat transfer performance reaches the optimum. Therefore, we further investigate the optimization results when the flow in the direct evaporative cooling tower is equilibrium, i.e.

mw;K cp;w ¼ 1 ma;A cp;ea

(55)

With the specific values list in Table 2, compared to optimal values for the optimal flow (point A), it is clear, as shown in Fig. 3, that the total thermal conductance at point A is lower than that of point B by about 6.7%. That is, the equilibrium flow is not the precondition to get the minimum total thermal conductance. Moreover, Figs. 4 and 5 give the variation trends of thermal conductances for each heat exchanger and mass flow rates in each branch with the increasing total circulating water flow rate. With the circulating water flow rate increasing, both the thermal

(kA)1/kW/K

(kA)2/kW/K

(kA)3/kW/K

mw,E/kg/s

mw,F/kg/s

1.61

4.91

6.61

0.17

0.88

conductance of the air pre-cooler and the user’s heat exchanger decrease monotonically, but that of the airewater direct evaporative cooling tower slightly decreases at first and then increases, while the mass flow rates of water in two branches both increase. All parameters at the optimal point are not the extreme value, that indicates the distributions of thermal conductances and water mass flow rates in the cycle are mutual influence, optimizing some local parameters of the system is difficult to obtain the optimal system. In addition, we investigate the outlet temperature of water from the direct evaporative cooling tower with the same user demanded cooling capacity but different inlet temperature of user’s medium, and with the same inlet temperature of user’s medium but different user demanded cooling capacity, shown in Figs. 6 and 7 respectively. It is found that the lower inlet temperature and the larger user demanded cooling capacity lead to the lower outlet temperature of water from the direct evaporative cooling tower. Therefore, in order to satisfy different user demands, we need to design different optimal systems with a certain outlet temperature of water, that is, the optimization of evaporative cooling systems by minimizing the outlet temperature of water from the evaporative cooling tower as much as possible is meaningless for some applications. Finally, Figs. 8 and 9 give the impact of different heat capacity rates of user’s medium and inlet mass flow rate of moist air on the global optimization for the evaporative cooling system, respectively. It is obvious that increasing the heat capacity rate of user’s medium and inlet mass flow rate of moist air will reduce the minimum total thermal conductance but enlarge the corresponding circulating water flow rate. This phenomenon can be explained form the following two aspects. On the one hand, when the heat capacity rate of user’s medium increasing, the heat transfer temperature difference will increase, and then the needed thermal conductance for user’s heat exchanger will decrease so as the total thermal resistance. One the other hand, with the increasing mass flow rate of moist air, the cooling capability of moist air will also increase, the total thermal conductance may decrease, so as the corresponding total mass flow rate. This conclusion points out the impact of inlet parameters of water and moist air on the optimization results. 4.2. Fixed total thermal conductance We apply the same inlet conditions of moist air and user’s medium and the same user demanded cooling capacity in Section 4.1 to the optimization with fixed total thermal conductance, Fig. 10 shows the minimum circulating water mass flow rate satisfying the user demand with different total thermal conductances, apparently, with the increasing total thermal conductance, the needed circulating water mass flow rate reduces correspondingly. As well known, the optimization results of minimum circulating water flow rate with a certain total thermal conductance should equal to the

Table 2 The comparison optimization results at equilibrium flow point and optimal point. P (kA)2/kW/K (kA)3/kW/K (kA)i/kW/K (kA)1/kW/K Fig. 3. The minimum total thermal conductance with different total circulating water flow rate.

Point B Point A

2.03 1.61

5.73 4.91

6.31 6.61

14.07 13.13

F. Yuan, Q. Chen / Energy 42 (2012) 181e191

Fig. 4. The thermal conductance for each heat exchanger with different circulating water flow rate.

189

Fig. 7. The outlet temperature of the direct evaporative cooling tower with the same inlet temperature of user’s medium but different user demanded cooling capacity (TH ¼ 299 K).

Fig. 5. Circulating water mass flow rate for each branch with different circulating water flow rate. Fig. 8. The minimum total thermal conductance versus circulating water flow rate with different heat capacity rate of user’s medium.

optimization results of minimum thermal conductance at that circulating flow rate. After optimization, we find the optimal curve of Fig. 10 is exactly the curve on the left side of the extreme point A in Fig. 3, which can also demonstrate the validity of our optimization program. Figs. 11 and 12 give the optimal distributions of thermal conductances for each heat exchanger and mass flow rates of water in each branch with different total thermal conductance.

As the total thermal conductance increasing, the thermal conductances of both the air pre-cooler and the direct evaporative cooling tower increase, but that of the user’s heat exchanger first decreases then increases, while the mass flow rates of water in two branches decrease.

Fig. 6. The outlet temperature of the direct evaporative cooling tower with the same user demanded cooling capacity but different inlet temperature of user’s medium (Qc ¼ 22 kW).

Fig. 9. The minimum total thermal conductance versus circulating water flow rate with different inlet mass flow rate of moist air.

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Fig. 10. The minimum total circulating water flow rate with different total thermal conductance of the system.

Fig. 13. The outlet water temperature of the direct evaporative cooling tower with different total thermal conductance.

flow rate even corresponds to the maximum outlet water temperature. To sum up, it is not one such characteristic parameter as the outlet water temperature of direct evaporative cooling tower, the thermal conductances of some certain heat exchangers and the mass flow rates in the system can decide the performance of whole system. For the optimal design results for the evaporative cooling system, it should reasonably distribute thermal conductances and mass flow rates of water in each branch from an overall perspective. 5. Conclusions

Fig. 11. The thermal conductance for each heat exchanger with different total thermal conductance.

Moreover, we investigate the variation trend of temperature of circulating water out of the direct evaporative cooling tower at different total thermal conductances, shown in Fig. 13. As the total thermal conductance increasing, the outlet water temperature of the airewater direct evaporative cooling tower decreases. The optimal point (point C), i.e. the minimum circulating water mass

Fig. 12. Circulating water mass flow rate for each branch with different total thermal conductance.

In this paper, after explicitly analyzing the irreversible heat and mass transfer processes in evaporative cooling systems, we calculate the entransy dissipation of all irreversible processes in the system and then give the total entransy dissipation in the system. Base on that, we establish the theoretical and mathematical relation of the thermal conductances of each heat exchanger and the mass flow rates of water in each branch with the user demands so as to deduce two groups of optimization equations via the conditional extremum mathematical method to obtain the optimal system performance. Furthermore, the newly proposed entransy dissipation-based method is employed to optimize a typical indirect evaporative cooling system. For the fixed total circulating water flow rate, there exists an optimal mass flow rate making the minimum total thermal conductance obtain the extreme value, and the total thermal conductance at optimal flow point is 6.7% lower than that at equilibrium flow point, which indicates that the equilibrium flow is not necessary, and sometimes contrary, for the optimal evaporative cooling systems. For the fixed total thermal conductance, increasing the system thermal conductance will reduce the minimum total circulating flow rate and the outlet water temperature of direct evaporative cooling tower. Moreover, when the circulating water flow rate gets the minimum value, the outlet water temperature reaches the maximum value. In addition, the results for two kinds of optimization problems both clarify that globally optimize and distribute both the heat exchanger conductances and the water mass flow rates, other than some local parameters optimization, will reach the best performance of the system. In summary, the entransy dissipation-based global optimization method develops a new approach to the global optimization of evaporative cooling systems, which is superior to the “try-anderror” method.

F. Yuan, Q. Chen / Energy 42 (2012) 181e191

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