Detailed thermodynamic analysis of a diffusion-absorption refrigeration cycle

Energy 115 (2016) 418e434

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Detailed thermodynamic analysis of a diffusion-absorption refrigeration cycle Ahmed Taieb*, Khalifa Mejbri, Ahmed Bellagi
U. R. Thermique et Thermodynamique des Proc
ed
es Industriels, Ecole Nationale d'ing
enieurs de Monastir (E.N.I.M.), Avenue Ibn El Jazzar, 5060 Monastir, Tunisia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 20 January 2015 Received in revised form 30 August 2016 Accepted 1 September 2016

This paper proposes an advanced simulation model for a Diffusion-Absorption Refrigerator DAR using ammonia/water/hydrogen as working fluids, and developed to describe and predict the behavior of the device under different operating conditions. The system is supposed to be cooled with ambient air and actuated with solar hot water available at 200
C. The DAR is first simulated for a set of basic data; a COP of 0.126 associated to a cooling capacity of 22.3 W are found. Basing on the obtained results an exergetic analysis of the system is performed which shows that the rectifier contribution to the exergy destruction is the most important with 34%. In a second step, the thermal capacities of all heat exchangers of the DAR are evaluated and the mathematical model so modified that the calculated capacities are now used as input data. A parametric study of the cycle is then carried out. The COP is found to exhibit a maximum when the heat supplied to the boiler or to the bubble pump is varied. Similar behavior is observed for variable submergence ratio. It is further noted that the COP is very sensitive to the ambient air temperature and to the absorber efficiency. © 2016 Elsevier Ltd. All rights reserved.

Keywords: Diffusion-absorption cycle Simulation Exergy analysis Thermal conductance COP

1. Introduction The Diffusion Absorption Refrigerator (DAR) invented by Platen and Munters in 1928 [1] has been recognized as one of the most encouraging sustainable technologies for production of cold. The cycle of the machine operates at constant total pressure and uses environment compatible working fluids: ammonia as refrigerant, water as absorbent and hydrogen or helium as non-absorbable auxiliary inert gas. This inert gas is necessary to reduce the partial pressure of the refrigerant in the evaporator to allow the process of evaporation to take place in the uniform pressure device. The main characteristic of DAR is that it has no moving parts, hence its good reliability. The circulation of the liquid solutions is driven by a bubble-pump and that of the gas loop between absorber and evaporator by gravity. This system has been now used for over 80 years [2]. The first commercial diffusioneabsorption refrigerator was introduced to the market by Electrolux Company in Sweden (also known as Dometic) [3], and since then millions of such refrigerators have been built and used mainly in domestic and niche

* Corresponding author. E-mail addresses: [email protected] (A. Taieb), [email protected] (K. Mejbri), [email protected] (A. Bellagi). http://dx.doi.org/10.1016/j.energy.2016.09.002 0360-5442/© 2016 Elsevier Ltd. All rights reserved.

applications as in camping and caravans. The DAR operates only with thermal energy, no mechanical and then no electric power is needed. This energy can be provided by fossil fuel combustion (gas, fuel, etc.), but also, for temperatures varying between 90 and 200
C, by solar thermal energy, or thermal discharges, etc. The growing concerns about worldwide energy and environmental sustainability in recent years enlarge the development of the DAR. There are many theoretical and experimental studies on the performance analyses of the DAR system operated by different energy sources [4e8]. Chen et al. [9] designed a modified generator including a heat exchanger that reuses the rejected heat in the rectifier to pre-heat the rich solution from the absorber. The new configuration of the cycle showed a significant improvement in the cooling COP, as much as 50% compared to the original system for the same cooling capacity. Zohar et al. [10] considered two configurations of a DAR, with and without refrigerant condensate sub-cooling. The results showed that the COP of the cycle without sub-cooling is higher by approximately 14e20% than that of the cycle with sub-cooling. The best system performances were obtained for ammonia mass fraction in rich solution varying in the range (0.25e0.4). The search for alternative working fluids to the standard ammonia-water system has also been the focus of investigations.

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Nomenclature A d E H h I_ M m_ n_ N P sys p* Q_ r Re S, S_ s0 T T~

DT u UA V_ We0 X,Y .x y yH2 z

Heat exchange area (m2) Diameter of bubble pump tube (m) Efficiency (absorber), effectiveness (SHX) Height (m) in Fig. 2 Height (m) in Fig. 2 Irreversibility (W) Molar mass (g mol1) Mass flow rate (kg s1) Molar flow rate (mol s1) Ratio of ammonia and hydrogen molar flow rate Total pressure (bar) partial pressure of ammonia (bar) Thermal flow rate (W) Molar flow ratio Reynolds number Entropy (J K1), entropy creation (W K1) Slip ratio Temperature (
C) Mean temperature (
C) Heat-exchanger thermal pinch (K) Velocity (m s1) Overall thermal conductance of HX (W/
C) Volume flow rate (cm3 s1) Weber number Molar ratio of ammonia in liq. and vap Ammonia mole fraction in liquid Ammonia mole fraction in vapor Hydrogen mole fraction in gas Ammonia charge in hydrogen

Greek symbols Vapor fraction in liq.-vap. Mixture Liquid-phase viscosity (kg m1 s1) Chemical potential

a h m

The objective is to reduce the temperature level of the driving heat supplied to the generator, so that solar thermal energy from flat plate or evacuated tube collectors can be used to this purpose [11e16]. Ben Ezzine et al. [17] reported that the R124eDMAC mixture could yield higher COP at lower driving heat temperatures in comparison with the NH3eH2O system. They also experimentally investigated a DAR using C4H10eC9H20 as working fluid in association with helium [18], and demonstrated the feasibility of solar cooling with this system. Maiya [19] showed that helium was more advantageous than hydrogen as inert gas, although a larger quantity of it is needed because of its larger viscosity. In the same study it was demonstrated that a higher pressure of operation decreases the COP. The recent review of diffusion cooling systems by Rodríguez~ oz and Belman-Flores [20] pointed out that the actual invesMun tigation trend is the search for alternate working fluid systems that make the DAR be driven by residual or solar heat. Srikhirin and Aphornratana [21] carried out an experimental study on an NH3eH2O DAR using helium as auxiliary gas. They developed also a mathematical model to determine the appropriate operating conditions for optimal performance, and observed that mass transfer rates in evaporator and absorber have a crucial effect on the system performance. The COP of the machine was found to

r s t f Indices A AW B C CF cv EC E gen,cv hw j L lm,i P R SR V 0

419

Density (kg m3) Surface tension (N m1) Void fraction Slope ratio of actual, tPOL, and minimum operating lines, tMOL (absorber), f ¼ tPOL =tMOL

Absorber Ammonia-water mixture Boiler Condenser Cool chamber Control volume Hot inlet Evaporator Entropy generated in control volume Hot water Component j Liquid phase Mean logarithmic temperature difference in machine component i Bubble pump Rectifier Liquid sub-cooling Vapor phase Temperature of surrounding

List of abbreviations COP Coefficient of performance, COP ¼ Q_ E =Q_ P þ Q_ B CISE Computer and Information Science and Engineering DAR Diffusion-absorption refrigerator GHX Gas heat exchanger HX Heat exchanger SHX Solution heat exchanger VLE Vapor liquid equilibrium

vary in the range 0.09e0.15. Starace and De Pascalis [22] elaborated a thermodynamic model of the DAR cycle without any assumption concerning the purity of the refrigerant exiting the rectifier. This model predicts higher performances than those found in Ref. [13]. In another study, Starace and De Pascalis [23] experimentally validated their model using a prototype with a bubble pump coupled to a domestic magnetron to reduce the starting transient of the circuit. The validation was carried out by varying the heat power supplied to the thermal pump of a commercially available DAR. Yildiz and Mustafa [24] proposed a simulation model of a DAR and evaluated the energy and exergy losses in each component of the machine. The model was then validated by comparing calculated results with measurements. It was found theoretically as well as experimentally that the largest energy and exergy losses occured in the solution heat exchanger. In the present paper a thermodynamic model for ammoniawater diffusion absorption refrigeration with hydrogen as inert gas is developed. The performances of the cycle are theoretically evaluated and analyzed. The effect of bubble pump characteristics and operating conditions (heat input, temperature, submergence ratio, etc.) is investigated. Further, an exergy analysis is performed in order to evaluate the contribution of each element of the DAR to

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the deterioration of the machine COP.

2. Cycle description The investigated configuration of the diffusion-absorption refrigeration chiller is illustrated in Fig. 1. The machine is composed mainly of a generator (bubble pump þ boiler), a rectifier, a condenser, a gas heat exchanger, an expansion chamber, an evaporator, an absorber and a solution heat exchanger. The ammonia poor liquid solution (15) returning from the generator is introduced at the top of the absorber coils. It comes out as ammonia-rich solution, after dissolving refrigerant gas, and accumulates in the solution tank located under the absorber. The rich solution (11) moves thereafter to the bubble pump via the solution heat exchanger SHX where it is preheated by the hot poor solution (14) flowing counter-currently towards the absorber. The preheated rich solution (12) arriving at the bottom of the bubble pump is further heated to boiling where heat is supplied at the rate of Q_ P . During this process, small vapor bubbles are formed and coalesce in their ascend movement to larger bubbles occupying the entire tube section. By their movement upwards these bubbles lift some liquid (13 L) to the boiler located atop and where it is further heated by the auxiliary heat Q_ B to produce supplementary vapors (16). The remaining ammonia-poor solution (14) proceeds to the top of absorber. The height difference between the compartments allows overcoming the pressure losses due the circulation through the

SHX. The water/ammonia vapors generated in the bubble pump (13 V) and in the boiler (16) mix and rise to the air-cooled rectifier where a partial condensation takes place accompanied by heat rejection of Q_ R to the surroundings. The major part of the residual water is thus removed and returns to the boiler (17). The purified ammonia vapor (1) condenses in the air-cooled condenser by rejecting the heat Q_ C to ambient air. The condensate (2) flows further by gravity to the evaporator located lower. Uncondensed gas (10 ) is fed to the reservoir via a bypass. The gas heat exchanger GHX is constituted of two coaxial tubes. The hydrogen rich gas exiting the absorber circulates in the central tube and the cold two-phase mixture of inert gas and evaporating refrigerant leaving the evaporator in the annular space. The small diameter tube transporting the liquid refrigerant from the condenser (2) to the evaporator (4) is placed in direct contact with the exterior tube of the GHX and evaporator. The pre-cooled refrigerant (4) is then injected in the evaporator inlet where it is exposed to the inert gas coming from the absorber (10). Its partial pressure is so lowered that it begins evaporating at very low temperature. The evaporation continues during the flowing of the refrigerant as a falling film in the annular space of the evaporator until arriving at the outlet (6). During this process, its partial pressure and temperature increase progressively. At the evaporator exit, the major part of the liquid refrigerant has evaporated producing in the cooling chamber the useful cold Q_ E . The liquid-vapor mixture (6) flows further in the annular space of the GHX where the

Fig. 1. Schematic diagram of diffusion absorption machine.

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421

between of 3.5e5.8 mm. 4.1. Bubble pump The performance coefficient hP of the bubble pump is defined as the ratio of volume flow rates of pumped liquid and heat generated vapor. In the literature, test results are presented with various kinds of inconsistent variables such as a mixture volume flow, mass flow, heat supply, etc. The experience shows however, that hP would vary with the vapor flow. At very low vapor flow rates, the bubbles are not organized in form of “pistons”, flow individually upwards without lifting any liquid. So there would be no pumping and hP ¼ 0. At very high vapor flow rates, the head losses in the pump tube cannot be neglected any more, and accordingly hP is reduced. Therefore, there is a maximum of hP somewhere between these extremes. In the considered configuration of the DAR, the thermal bubble pump and the boiler are assumed separated in order to be make the heat supplied to the pump Q_ P an independent variable and to investigate its effect on the performance of the refrigerator. In order to generate sufficient vapor for a given cooling capacity, _ is supplied to the boiler. supplementary heat Q B For the evaluation of the pumping capacity in the investigated DAR, the correlation developed by Behringer (cited in Ref. [27]) for a tube diameter d ¼ 4 mm is adopted, relating the pumped liquid volume flow rate V_ L (cm3/s) to the vapor flow rate V_ V (cm3/s) generated and to the submergence ratio (H/h):

Fig. 2. Bubble pump configuration/design.

rest liquid is completely evaporated (7) and superheated. This ammonia-rich gas (7) is conducted to the absorber bottom. During its ascension it meets the poor liquid solution (15) flowing countercurrently as a falling film and dissolves in the liquid. The absorption heat is rejected to the surrounding air at the rate of Q_ A . The ammonia poor gas (8) exiting at the top of the absorber contains still residual water vapor. During its further ascension in the GHX, it is cooled by the cold gas coming from the evaporator (6). The major part of water and ammonia condensates and return to the absorber (80 ). The circulation of the gas stream through the gas loop (expansion chamber, evaporator, GHX and absorber) is driven by natural convection: The evaporation of the refrigerant into the inert gas in the evaporator increases its density while the gas mixture exiting the absorber after losing the major part of its ammonia becomes lighter. This density difference drives the circulation between evaporator and absorber.

2 1:69V_ V  0:136 V_ V  0:316 V_ L ¼ 1:5
Hþ1 h

(4.1)

Fig. 3 shows the performances of the thermal bubble pump according to eq. (4.1) for different submergence ratios. As can be observed, the pumped V_ L for a given V_ V e i.e. for a thermal supply rate e decreases when the submergence ratio increases. The heat supply needed to ensure a given liquid volume flow rate increases with the submergence ratio. The void fraction tv, an important parameter in the calculation and modeling of two-phase flows, is defined as:

tv ¼

V_ V _ s V L þ V_ V 0

(4.2)

3. Thermodynamic properties of the working fluids The thermodynamic properties of ammonia/water binary system are calculated using an empirical approach based on a canonical free enthalpy model of the mixture [25]. Hydrogen gas is assumed insoluble in the liquid phase. The gas phase is considered as an ideal mixture of real gases, water/ammonia mixture on one side and hydrogen on the other side. The properties of hydrogen are evaluated using the fundamental equation of state of Younglove [26]. 4. Mathematical model of the machine As noted earlier, the diffusion-absorption refrigerator DAR uses a thermal bubble pump for the circulation of the liquids. Pump tubes used in the commercialized DAR have diameters ranging

Fig. 3. Bubble pump performance for different submergence ratio H/h (Behringer model).

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where s' represents the slip ratio, e i.e. the ratio of the superficial velocity of the vapor phase uV to that of the liquid phase uL. uV and uL are defined as follows:

V_ uV ¼ V ; tv A

Dz ¼

V_ L uL ¼ ð1  tv Þ A

(4.3)

To predict the void fraction, the slip ratio can be estimated with the CISE correlation recommended by Hewitt [28], because it takes better account of the mass flow and other physical property effects than other correlations,

s0 ¼ 1 þ E1



d  dE2 1 þ dE2

0:5 (4.4)

ε_ d¼ V 1  ε_ V

(4.4.1)

V_ V _ V V þ V_ L 

rL rV

(4.4.3) 

rL rV

0:08 (4.4.4)

with Relo and We0 are defined as:

_ md

(4.4.5)

hl m_ 2 d

(4.4.6)

srL

In these relations, m_ is the total mass flux, hL the liquid-phase viscosity, and s the surface tension. 4.2. Gas loop In a diffusion-absorption chiller, the inert gas enables the evaporation of the refrigerant in the evaporator and its further transportation to the absorber. To estimate the gas circulation rate in this loop (evaporatorGHXabsorber), one starts by calculating how much gas has to circulate to allow the evaporation of 1 mol of ammonia in the evaporator. The presence of rest water vapor can be neglected, and the gas mixture is supposed ideal. If the total pressure is noted Psys and the partial pressure of ammonia p*, the ammonia mole fraction is then yNH3 ¼ p* =Psys and that of hydrogen, yH2 ¼ ðPsys  p* Þ=Psys . The ammonia charge in hydrogen, z (mol NH3/mol H2) is given by the relation

z¼

1  yH2 p* ¼ yH2 Psys  p*

(4.5)

The ammonia charge in the gas at the inlet of the expansion chamber (10) and at the entrance of absorber (7) is respectively:

z10 ¼

1  yH2 ;10 ; yH2 ;10

yH2 ;7 yH2 ;10 yH2 ;10  yH2 ;7

(4.8)

The hydrogen flow rate corresponding to the evaporation of n_ 1 liquid refrigerant exiting the condenser is:

(4.9)

The circulation rate in the gas loop can be characterized by the molar flow rate ratio r,

yH2 ;7 n_ 10 ¼ yH2 ;10  yH2 ;7 n_ 1

(4.10)

4.3. Solution procedure

0:22

E2 ¼ 0:0273 We0 Re0:51 lo

We0 ¼

NH2 ¼

(4.4.2)

E1 and E2 are given by:

Relo ¼

(4.7)

The inverse of Dz represents the quantity of hydrogen needed for the evaporation of 1 mol of ammonia,

r¼

ε_ V is the volume flow ratio:

E1 ¼ 1:578 Re0:19 lo

yH2 ;10  yH2 ;7 yH2 ;7 yH2 ;10

_ 10 n_ H2 ¼ NH2 n_ 1 ¼ yH2 ;10 n

with d is defined as:

ε_ V ¼

The difference Dz ¼ z7  z10 represents the amount of evaporated refrigerant per mole of inert gas,

z7 ¼

1  yH2 ;7 yH2 ;7

(4.6)

To establish a steady state simulation model of the refrigerator, mass, energy and entropy balances are formulated for each of its components. Further, the chemical potential equalities for the calculation of the VLE phase equilibria are written for saturated vapor and liquid phases. All these equations together with schemes of the corresponding components are given in Table 1. Equations (1)(10) represent the model of the thermal bubble pump. For given heat flux Q_ B and submergence ratio, and an assumed value of T13, the generated molar and volume vapor rates are deduced from equations (1) and (2). Relations (3) and (4) allow to evaluate the corresponding quantity of the pumped liquid. With the resolution of the equation system (6)e(9) one can get the characteristics of the fluid at the outlet of the bubble pump (T13, x13 and y13). This procedure is iteratively repeated until convergence of temperature T13. The entropy creation in the bubble pump component is determined with eq. (10) where T~ hw represents the average temperature of the hot water used for heating of bubble pump and boiler. Mass, energy and entropy balances for the boiler are expressed by equations (14)e(17). The VLE is given by Eqs. (11)e(13). The solution heat exchanger is characterized by its thermal efficiency defined in eq. (18) which allows the calculation of T15. Two mass balances are written for each flow, and energy and entropy balances are formulated for the whole HX (Eqs. (19)e(22)). The rectifier is assumed equivalent to as a multi-stage air-cooled rectification column. As illustrated in the corresponding figure in Table 1, the top stage is constituted of a partial condenser, from which the saturated vapor (1) leaves for the refrigerator condenser while the liquid phase is refluxed. The hot mixed vapor stream from bubble pump (13 V) and boiler (16) is fed to the bottom stage of the rectifier. The liquid 17 returning to the generator boiler is saturated, and at equilibrium with the vapor noted “17vapor”. During its upward flow, this vapor will be cooled und undergoes partial condensation. The rectifier is necessary to purify the ammonia vapor (1) up to an imposed composition y1. The corresponding temperature T1 is deduced from the VLE given by eqs. (23)e(24). The water rich condensate liquid (17) returning to the boiler is supposed to be saturated at a temperature T17 ¼ T16  5; its molar composition is determined from chemical potential equalities eqs. (26)e(27) and

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423

Table 1 Mathematical model of the diffusion-absorption refrigerator. Model equations and assumptions Generator: Bubble Pump and Boiler

n_ 13V ¼ ðh

Control volume

Q_ p

13L h12 Þ

a13

(1)

þ ðh13V  h13L Þ

V_ 13V ¼ n_ 13V V 13V

(2)

2 1:69V_ 13V  0:136 V_ 13V  0:316 V_ 13L ¼ 1:5 ðH=h þ 1Þ

(3)

. n_ 13L ¼ V_ 13L V 13L

(4)

n_ 13 ¼ n_ 13V þ n_ 13L ¼ n_ 12

(5)

a13 ¼

x12  x13 n_ 13V ¼ y13  x13 n_ 13

(6)

Q_ P ¼ n_ 13 ½a13 h13V þ ð1  a13 Þh13L   n_ 12 h12 















mL;NH3 T13 ; Psys ; x13 ¼ mV;NH3 T13 ; Psys ; y13

mL;H2 O T13 ; Psys ; x13 ¼ mV;H2 O T13 ; Psys ; y13

. S_gen;P ¼ n_ 13L s13L þ n_ 13V s13V  n_ 12 s12  Q_ P T~ hw;P 















mL;NH3 T14 ; Psys ; x14 ¼ mV;NH3 T16 ; Psys ; y16 mL;H2 O T14 ; Psys ; x14 ¼ mV;H2 O T16 ; Psys ; y16

(7)

(8)

(9)

(10)

(11)

(12)

T16 ¼ T14

(13)

n_ 14 þ n_ 16 ¼ n_ 13L þ n_ 17

(14)

n_ 14 x14 þ n_ 16 y16 ¼ n_ 13L x13 þ n_ 17 x17

(15)

Q_ B ¼ n_ 14 h14 þ n_ 16 h16  ðn_ 13L h13L þ n_ 17 h17 Þ

(16)

. S_gen;B ¼ n_ 14 s14 þ n_ 16 s16  ðn_ 13L s13L þ n_ 17 s17 Þ  Q_ B T~ hw;B (17) SHX ESHX ¼ ðT14  T15 Þ=ðT14  T11 Þ

(18)

n_ 12 ¼ n_ 11 ;

x12 ¼ x11

(19)

n_ 15 ¼ n_ 14 ;

x15 ¼ x14

(20)

n_ 12 h 12 þ n_ 15 h15  ðn_ 11 h11 þ n_ 14 h14 Þ ¼ 0

(21)

S_gen;SHX ¼ n_ 12 s12 þ n_ 15 s15  ðn_ 11 s11 þ n_ 14 s14 Þ

(22) (continued on next page)

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A. Taieb et al. / Energy 115 (2016) 418e434

Table 1 (continued ) Model equations and assumptions Rectifier

Control volume

















mL;NH3 T1 ; Psys ; x1 ¼ mV;NH3 T1 ; Psys ; y1 mL;H2 O T1 ; Psys ; x1 ¼ mV;H2 O T1 ; Psys ; y1

(23)

(24)

T17 ¼ T16  5

(25)

















mL;NH3 T17 ; Psys ; x17 ¼ mV;NH3 T17 ; Psys ; y17 mL;H2 O T17 ; Psys ; x17 ¼ mV;H2 O T17 ; Psys ; y17

(26)

(27)

n_ 13V þ n_ 16 ¼ n_ 1 þ n_ 17

r ¼ n_ 10 =n_ 1 ¼ yH2 ;7

(28)

. yH2 ;10  yH2 ;7

(29)

Q_ R ¼ n_ 1 h1 þ n_ 17 h17  n_ 13V h13V  n_ 16 h16

(30)

. S_gen;R ¼ n_ 1 s1 þ n_ 17 s17  n_ 13V s13V  n_ 16 s16  Q_ R T~ air

(31)

T2 ¼ TC ¼ T~ air þ DTCair

(32)

n_ 1 ¼ n_ 2 ;

y1 ¼ x2

(33)

Q_ c ¼ n_ 2 h2  n_ 1 h1

(34)

. S_gen;C ¼ n_ 2 s2  n_ 1 s1  Q_ C T~ air

(35)

T8 ¼ T~ air þ DTAair

(36)

T80 ¼ T8

(37)

T11 ¼ T8

(38)

Condenser

Absorber/Reservoir

















* * mL;NH3 T8 ; P8min ; x15 ¼ mV;NH3 T8 ; P8min ; y*8

* * mL;H2 O T8 ; P8min ; x15 ¼ mV;H2 O T8 ; P8min ; y*8

(39)

(40)

. * Psys yH2 ;8max ¼ 1  yAW;8min ¼ 1  P8min

EABS ¼

(41)

p*7  p*8 yH2 ;8  yH2 ;7 ¼ p*7  p*8min yH2 ;8max  yH2 ;7

(42)

















mL;NH3 T11 ; P7* ; x*11max ¼ mV ;NH3 T11 ; P7* ; y*11max mL;H2 O T11 ; P7* ; x*11max ¼ mV;H2 O T11 ; P7* ; y*11max  P7* ¼ 1  yH2 ;7 Psys ;

x11max ¼ x*11max

(43)

(44)

(45)

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425

Table 1 (continued ) Model equations and assumptions

tMOL

YNH3 ;7  YNH3 ;8 ¼ ; X11;max  X15

Control volume

tPOL ¼ f tMOL

YNH3 ;7  YNH3 ;8 ¼ X11  X15 (46)

YNH3 ;j ¼ yj

. 1  yj ;

XNH3 ;j ¼ xj



 1  xj with j ¼ 7; 8 (46a)

n_ 8 þ n_ 11 ¼ n_ 7 þ n_ 15 þ n_ 80

(47)

n_ 8 y8 þ n_ 11 x11 ¼ n_ 7 x7 þ n_ 15 x15 þ n_ 80 x80

(48)

Q_ A ¼ n_ 8 h8 þ n_ 11 h11  ðn_ 15 h15 þ n_ 7 h7 þ n_ 80 h80 Þ

(49)

. S_gen;A ¼ n_ 8 s8 þ n_ 11 s11  ðn_ 15 s15 þ n_ 7 s7 þ n_ 80 s80 Þ  Q_ A T~ air (50) Expansion Chamber

















mL;NH3 T5 ; P5* ; x*5 ¼ mV;NH3 T5 ; P5* ; y*5 mL;H2 O T5 ; P5* ; x*5 ¼ mV;H2 O T5 ; P5* ; y*5  P5* ¼ 1  yH2 ;5 Psys ;

Evaporator and gas heat exchanger GHX

 x5 ¼ x*5 ; y5 ¼ 1  yH2 ;5 y*5

(51)

(52)

(53)

n_ 5L þ n_ 5V ¼ n_ 4 þ n_ 10

(54)

n_ 5L x 5L þ n_ 5V x5V ¼ n_ 4 x4 þ n_ 10 x10

(55)

n_ 5V yH2 ;5 ¼ n_ 10 yH2 ;10

(56)

n_ 5L h 5L þ n_ 5V h5V  ðn_ 4 h4 þ n_ 10 h10 Þ ¼ 0

(57)

S_gen;ExCh ¼ n_ 5L s5L þ n_ 5V s5V  ðn_ 4 s4 þ n_ 10 s10 Þ

(58)

T3 ¼ T2  DTSR3

(59)

T9 ¼ T6 þ DTGHX or

T7 ¼ T8  DTGHX

(60)

T4 ¼ T~ CF þ DT4CF

(61)

T6 ¼ T~ CF þ DT6CF

(62)

T90 ¼ T9

(63)

T10 ¼ T5 þ DT510

(64)

















mL;NH3 T6 ; P6* ; x*6 ¼ mV;NH3 T6 ; P6* ; y*6 mL;H2 O T6 ; P6* ; x*6 ¼ mV;H2 O T6 ; P6* ; y*6  P6* ¼ 1  yH2 ;6 Psys ;

x6 ¼ x*6 ;

(65)

(66)

 y6 ¼ 1  yH2 ;6 y*6 (67)

(continued on next page)

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Table 1 (continued ) Model equations and assumptions   mL;NH3 T80 ; P8*0 ; x*80 ¼ mV;NH3 T80 ; P8*0 ; y*80 





mL;H2 O T80 ; P8*0 ; x*80 ¼ mV;H2 O T80 ; P8*0 ; y*80

T80 ¼ T8 ; 















mL;H2 O T9 ; P9* ; x*9 ¼ mV;H2 O T9 ; P9* ; y*9  P9* ¼ 1  yH2 ;9 Psys ;

x90 ¼ x*9 ;

(68)



 P8*0 ¼ P8* ¼ 1  yH2 ;8 Psys ;

mL;NH3 T9 ; P9* ; x*9 ¼ mV;NH3 T9 ; P9* ; y*9

Control volume

(69)

x80 ¼ x*80

(70)

(71)

(72)

 y9 ¼ 1  yH2 ;9 y*9 (73)

















* * mL;NH3 T10 ; P10 ; x*10 ¼ mV;NH3 T10 ; P10 ; y*10

* * mL;H2 O T10 ; P10 ; x*10 ¼ mV;H2 O T10 ; P10 ; y*10

 * ¼ 1  yH2 ;10 Psys ; P10

 y10 ¼ 1  yH2 ;10 y*10

(74)

(75)

(76)

n_ 5V þ n_ 5L ¼ n_ 6V þ n_ 6L

(77)

n_ 5V y5 þ n_ 5V x5 ¼ n_ 6V y6 þ n_ 6L x6

(78)

n_ 6V yH2 ;6 ¼ n_ 5V yH2 ;5

(79)

n_ 6V þ n_ 6L ¼ n_ 7

(80)

n_ 6V y6 þ n_ 6L x6 ¼ n_ 7 y7

(81)

n_ 6V yH2 ;6 ¼ n_ 7 yH2 ;7

(82)

n_ 8 þ n_ 90 ¼ n_ 80 þ n_ 9

(83)

n_ 8 y8 þ n_ 90 x90 ¼ n_ 80 x80 þ n_ 9 y9

(84)

n_ 9 yH2 ;9 ¼ n_ 8 yH2 ;8

(85)

n_ 9 ¼ n_ 90 þ n_ 10

(86)

n_ 9 y9 ¼ n_ 90 x90 þ n_ 10 y10

(87)

n_ 9 yH2 ;9 ¼ n_ 10 yH2 ;10

(88)

n_ 3 ¼ n_ 2 ;

x3 ¼ x2 ;

n_ 4 ¼ n_ 3 ; x4 ¼ x3

(89)

Q_ E ¼ n_ 4 h4 þ n_ 6L h6L þ n_ 6V h6V þ n_ 90 h90 þ n_ 10 h10  n_ 3 h3  ðn_ 5L h5L þ n_ 5V h5V Þ  n_ 9 h9 (90)

A. Taieb et al. / Energy 115 (2016) 418e434

427

Table 1 (continued ) Model equations and assumptions

Control volume

S_gen;E ¼ n_ 4 s4 þ ðn_ 6L s6L þ n_ 6V s6V Þ þ n_ 90 s90 þ n_ 10 s10  n_ 3 s3 .  ðn_ 5L s5L þ n_ 5V s5V Þ  n_ 9 s9  Q_ E T~ CF (91) n_ 2 h2 þ ðn_ 6L h6L þ n_ 6V h6V Þ þ n_ 8 h8 þ n_ 90 h90  n_ 3 h3  n_ 7 h7  n_ 80 h80  n_ 9 h9 ¼ 0

(92)

S_gen;GHX ¼ n_ 2 s2 þ ðn_ 6L s6L þ n_ 6V s6V Þ þ n_ 8 s8 þ n_ 90 s90  n_ 3 s3  n_ 7 s7  n_ 80 s80  n_ 9 s9 (93)

its molar flow from mass balance (28). The refrigerant vapor flow rate n_ 1 is deduced from the gas molar flow n_ 10 circulating in the gas loop using relation (29). The rejected heat in the rectifier is determined from the energy balance (30). Relation (31) expresses the entropy generation in the rectifier where T~ air is the mean temperature of ambient air. The condenser is an air-cooled component, characterized by a temperature pinch between condensation and air temperature, DTCair. The refrigerant liquid at the outlet (2) is assumed saturated. Mass, energy and entropy balances of this machine element are given by Eqs. (33)e(35). The air-cooled absorption process is supposed to take place isothermally at a temperature T8 ¼ T~ air þ DTAair where DTAair is a pinch to the cooling air temperature. The analysis of the absorption operation is represented schematically in Fig. 4 and formulated mathematically by equations (36)e(50) in Table 1. In fact, the falling film absorber is a continuous contact device and usually treated using mass transfer models (film theory, penetration theory, surface renewal theory, etc.). Here however, it is simulated as a stagewise contactor, and the performance of which is characterized by a number of theoretical stages, those needed in a multi-stage contactor to reach the same separation effect under similar conditions. The McCabe-Thiele construction in Fig. 4 is based on this approach. The region of feasible operation lines lies above the equilibrium curve. This region is bounded by the distribution curve on one side and by the vertical line representing the molar ratio XNH3 ;15 of the ammonia-poor solutionXNH3 ;15 ¼ x15 =ð1  x15 Þ, and the horizontal line YNH3 ;7 ¼ ð1  yH2 ;7 Þ=yH2 ;7 ¼ p*7 =ðPsys  p*7 Þ corresponding to the molar composition ratio of the inlet gas flow. The partial pressure of the ammonia/water mixture in hydrogen at absorber outlet can never go down below the partial pressure at equilibrium represented by the point ðXNH3 ;15 ; YNH3 ;8min Þ in Fig. 4 and calculated by Eqs. (39)e(41) in Table 1. The maximum amount of refrigerant that can be absorbed in the absorber coils, corresponding to the maximum reduction of the partial pressure of ammonia/water in the gas flow, is equal to:





DyNH3 ;max ¼ yH2 ;8max  yH2 ;7 ¼ p*7  p*8min Psys

Once the design recovery of the absorber is fixed, the operating curve can be constructed by first locating the point ðx15 ; yAW;8 Þ on the diagram. The intersection of the horizontal line corresponding to the inlet gas composition yAW;7 with the equilibrium curve defines the theoretical minimum liquid-to-gas flow ratio tMOL (slope of minimum operating line in Fig. 4). The actual design value of the liquid-to-gas flow ratio tPOL (slope of the practical operating line in Fig. 4) is generally 1.2 to 1.5 times this minimum [29]. Hence, the actual design line for gas absorption will pass through the point (x15,yAW,8) and intersect the line yAW ¼ yAW,7 to the left of the equilibrium curve. The maximal molar composition of the rich solution x11max is calculated at the vapor-liquid equilibrium of ammonia/water system under partial pressure P7 and at T11 ¼ T8 via Eqs. (43)e(45). For the actual absorber, the coefficient f in Eq. (46) is fixed and accordingly the actual molar fraction of the rich solution x11 is determined. Two mass balances of the absorber are given in Eqs. (47)e(48). The energy and entropy balances are formulated by Eqs. (49)e(50). The adiabatic expansion chamber is modeled by equations (51)e(58). The VLE of the mixture ðT5 ; x5 ; y5 Þ taking place at the partial pressure P5 ¼ ð1  yH2 ;5 ÞPsys is predicted with the chemical potential equalities and the energy balance. The liquid and vapor molar flows and the molar composition of the inert gas at outlet (5) are determined from mass balances. The rate of entropy creation is estimated from the entropy balance (58).

(4.11)

This situation corresponds to an ideal absorber of very large dimension with efficiency equal to unity. In reality, the amount of refrigerant absorbed in the absorber DyNH3 ¼ yH2 ;8  yH2 ;7 ¼ ðp*7  p*8 Þ=Psys is lower than DyNH3 ;max as illustrated in Fig. 4. Consequently, the practical absorber efficiency defined with equation (42) in Table 1 is smaller than 1.

Fig. 4. Operating conditions of absorber.

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A. Taieb et al. / Energy 115 (2016) 418e434

The evaporator and the GHX are composed of three independent circuits. The GHX is isolated from the surrounding and the evaporator is submerged in the cold chamber, which is considered at an average temperatureT~ CF . In the evaporator, the temperature of the four outlet flows is determined by thermal pinches (Eqs. (61)e(64)) relatively to this temperature. The liquid refrigerant is sub-cooled in the circuit (3e4) essentially by the cold chamber CF; the temperature T4 is determined relative toT~ CF . The evaporation of refrigerant in the two-phase flow (5e6) cools the inert gas (9e10) flowing counter-currently, and produces the cooling effect in the refrigerator cabinet. Therefore, the temperatures T10 and T6 are determined through pinches relative to T5 and T~ CF , respectively. The few rest liquid droplets flowing at (90 ) are supposed to be at the same temperature as flow (9). In the GHX, the liquid refrigerant stream (3) is characterized by a sub-cooling DTSR3 and the outlet temperature of the inert gas flow T9 is fixed relatively to the inlet temperature of the two-phase flow T6. The temperature T7is evaluated from the GHX energy balance. The water rich liquid returning to the absorber (80 ) is assumed to be at the same temperature of the inert gas inlet T8. The rich inert gas flow returning from the absorber to the evaporator is sub-cooled; consequently, it is purified by condensation of the major part of the water and ammonia that return to the absorber (80 ). The vapor-liquid phase behavior of ammonia/ water mixture at partial pressure P9 at state (9e90 ) is calculated using the chemical potential equalities Eqs. (71)e(73). Similar calculation is performed for state (10) using Eqs. (74)e(76). The mass balances in all the independent circuits of the evaporator and GHX are given by equations (77)e(89). The energy and entropy balances for the two components are expressed by Eqs. (90)e(93). To simulate the diffusion-absorption refrigerator operation, a Fortran program is developed basing on the nonlinear equation routine CONLES [30]. 5. Simulation results and discussion 5.1. DAR simulation in basic operating conditions The number of degrees of freedom e or variance e of the refrigerator shown in Fig. 1, is the number of independent data needed to determine completely the state of the machine. It is also the number of input quantities and model assumptions required to run the simulations. It is evaluated to 31 in the case of the considered machine. Several of the working hypotheses are expressed within the mathematical model in Table 1. A set of basic input data for the simulations are also given in Table 2. The reported temperatures are based on experimental measurements realized in our laboratory on a commercial low capacity DAR [31]. The uniform pressure in the DAR is set to 20.7 bar. The ambient air temperature is fixed to 26
C assuming that the space containing the installation is air-conditioned. The cold chamber is supposed to be at a temperature of 5
C. The hot water used for the heating of the bubble pump and the boiler is assumed at an average temperature of T~ hw ¼ 200
C. The tube diameter of the bubble pump is set to 4 mm and the submergence ratio to 3. The supplied heat to the thermal bubble pump is set to Q_ P ¼ 47 W. To produce sufficient quantity of refrigerant, a supplementary heat of Q_ B ¼ 130 W is supplied to the boiler. Condenser and absorber are cooled by air in free convection; their characteristics pinch DTCair and DTAair are set to 25 K and 20 K, respectively. The efficiency of the absorber is fixed to EA ¼ 0.8 [32]. The effect of this parameter on the COP is investigated in the parametric analysis. The coefficient f, ratio of the slopes of the actual and minimum operating lines (Fig. 4), is set to 1.3, and the solution heat exchanger efficiency to ESHX ¼ 0.7. Further the following values are adopted for the temperature

pinches in the GHX: DTSR3 ¼ 15 K, DTGHX ¼ 12 K, DT4CF ¼ 20 K, DT6CF ¼ 10 K and DT510 ¼ 10 K. The energy performance of the DAR is evaluated by its COP:

COP ¼

Q_ E Q_ P þ Q_ B

(5.1)

Table 3 provides the most important simulation results for the considered base data set. The COP of the DAR under the specified operating conditions is evaluated to 0.126. In the bubble pump, the two-phase flow is characterized by a bond number (Bo ¼ rgD2/s) of 3.43 and it is a capillary scale regime flow (Bo < 5). The superficial velocity Reynolds number (Rej ¼ rLjD/ mL with j the superficial velocity) is equal to 2985 and a turbulent flow regime (Rej > 2500) is established in the small diameter rising tube of the bubble pump. The void fraction is estimated to 0.581 with the CISE correlation (Eqs. (4.4)e(4.4.6)) which is considered one of the most appropriate for use in the estimation of void fraction in small channels [33] and inclined narrow annular channels [34]. Recent investigation carried out by Rattner et al. [35,36] in 2015, demonstrated experimentally that intermediate scale Taylor flows (5  Bo  40) are significantly different from those at the capillary and large scales. These findings help explain why mechanistic bubble-pump models that incorporate results for capillary or high Bond number conditions often have poor predictive capabilities for liquid pumping rates. New correlations were then developed for Taylor bubble rise velocities and lengths in intermediate scale flows. Laminar falling-film results were demonstrated to apply for liquid films around Taylor bubbles in intermediate scale flows. This study thus provides models for kinematic closure of intermediate scale Taylor flows. The mechanistic models that directly account for Taylor bubble length fraction and liquid film thickness will be exploited in our future investigations. Based on the simulated results given in Table 3, the thermodynamic cycle of the air-cooled DAR is represented on the water/ ammonia Oldham diagram shown in Fig. 5. In this diagram, also called Dühring diagram [2,9], equilibrium temperature, pressure and composition relationship of ammonia-water mixtures are graphically represented. The coordinate system is transformed in such a way that the iso-ammonia-composition curves appear almost as straight lines: Ordinate-axis (pressure) is in logarithmic scale and the saturation temperature (abscissa-axis) is in inverse scale (1/T). The major processes taking place in the machine are indicated in the diagram of Fig. 5. A prototype of the envisaged machine is not yet confectioned to validate experimentally the proposed model with separated bubble pump and boiler, and not combined as it is the case in the commercialized DARs, so that the heat rates supplied to the bubble pump Q_ P and to the boiler Q_ B can be considered independent variables. However, recent investigations of a small capacity DAR in our research unit [16,31,37] show that the steady-state temperature

Table 2 Set of basic input data. Variables

Values

Variables

Values

Psys Tair Tcf T~

20.7 bar 26
C 5
C 200
C

12 20 10 10

x12 y1 DTCair DTAair

1 0.99 25
C 20
C

DTGHX DT4CF DT6CF DT510 f EABS ESHX Q_

1.3 0.8 0.7 47 W

DTSR3

15
C

Q_ B

130 W

hw

P




C C C
C



A. Taieb et al. / Energy 115 (2016) 418e434

5.2. Exergetic analysis of the DAR for basic operating conditions

Table 3 Simulation results for basic case operating conditions. Variables

Values

Variables

Values

Psys T1 T4 T5 T6 T7 T8 T9 T10 T12 T13 T14 T15 r

20.7 bar 81.8
C 25.0
C 27.4
C 5.0
C 34.0
C 46.0
C 23.6
C 17.4
C 116.0
C 140.8
C 176.7
C 85.2
C 5.05 0.343 0.903

n_ 10 n_ 11 n_ 15 n_ 17

0.012872 0.011096 0.008522 0.001900 0.086 0.581 0.33427 0.13619 0.93570 0.83456 0.03169 0.02228 0.94603 0.99659 0.99985 22.23 W

a5 a6 n_ 1 n_ 8

0.002574 mol/s

n_ 80 n_ 90

0.000733 mol/s

0.013605 mol/s 0.000017 mol/s

a13 tvoid,13 x11 x15 yH2 ;5 yH2 ;7 yNH3 ;8 yH2 O;8 yH2 ;8 yH2 ;9 yH2 ;10 Q_ E

mol/s mol/s mol/s mol/s

Q_ C Q_

51.03 W

Q_ R COP

73.46 W

A

429

74.69 W 0.126

In this section, an exergetic analysis of the DAR is performed basing on the simulation results of the previous section. The exergy destruction or irreversibility rate in the component j of the machine is evaluated applying the Gouy-Stodola [38,39] theorem:

I_j ¼ T0 S_gen; j

(5.2)

where T0 represents the (ambient) reference temperature. The availability of the heat supplied to bubble pump and boiler at temperatureT~ hw is evaluated according to the relation:

T bj ¼ Q_ j 1  0 T~ hw

! (5.3)

and the exergy gain in the evaporator,

ExRec;E ¼ Q_ E 1 

T0 T~

! (5.4)

CF

measured in various locations of the refrigerator are comparable to those calculated by the model for similar operating conditions. For example, the mean evaporator temperature predicted by the present model for a driving heat temperature of 177
C is 16
C, fairly comparable to the temperature measured: 19
C at the evaporator for 180
C in the generator. It must be further pointed out that the steady-state bubble pump model used (Eqs. (4.1)e(4.4.6)) cannot predict the transient behavior of the machine and in particular the minimal heat rate input needed to start the liquid pumping. To make this complex issue clearer, experimental tests were carried out on a 7 W cooling capacity commercial DAR. Fig. 6 represents the evaporator temperature as response to the beginning of liquid pumping by the bubble pump in the start-up phase of the machine for increasing heat input to generator (and bubble pump). As can be noted, a minimal heat rate of roughly 21 W is necessary to start activating the bubble pump. This heating power is however not sufficient to ensure a smooth functioning of the refrigerator: a heat power twice that much is needed to this purpose. The steady state of the machine is reached after 75 min.

Fig. 5. Thermodynamic cycle of the DAR in the Oldham diagram.

In the base operating conditions, the system receives 65.1 W of exergy as input in the generator (bubble pump þ boiler) and yields 1.66 W of exergy in form of cold: the exergetic efficiency of the system is very low, equal to 2.55%. The progressive destruction of exergy in the various elements of the refrigerator and the details of the exergy analysis are illustrated in Fig. 7. Roughly, one third of the total irreversibility generated in the machine is due to the components GHX, evaporator, bubble pump, condenser and expansion chamber, another third is due to the assembly boiler, SHX and absorber, and the rest, one third also, of the total irreversibility is generated in the rectifier alone. The reason is that in the latter machine element a large amount of heat, 73.7 W, at high temperature is rejected in the atmosphere. This result differs from that of reference [23] where the solution heat exchanger was identified as the machine element responsible for the maximum exergy destruction. In fact the refrigerator described in Ref. [23] is somewhat different from the machine considered in the present paper and working under different operating conditions. The combined vapor stream at the inlet of the rectifier is here at 175
C and 20.7 bar, vs. 158
C and 17.8 bar in Ref. [23]: Heat is rejected in the present work at a mean temperature of 130
C vs. 105
C in Ref. [23]

Fig. 6. Time evolution of the evaporator temperature of a small capacity DAR for various heat rate supply to generator.

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A. Taieb et al. / Energy 115 (2016) 418e434

to an ambient at roughly the same temperature. Further, because of the higher temperature and pressure in the generator, the vapor treated in the present rectifier contains more water and hence, more condensation heat is rejected. For twice the refrigeration capacity (22 W vs. 10.8 W), approximately four times more heat (73.7 W vs. 16.8 W) is rejected. The total exergy product is of 4.65 W. 5.3. Characterization of the heat transfer in the components of the machine The overall thermal conductance overall thermal conductance overall thermal conductance of a machine element characterizes the heat transfer between this component and the surroundings (ambient air) and/or between different streams inside the element. It depends on the thermo-physical properties of fluids, the nature of the flow (convective heat transfer coefficients) and the thermal conductivity and geometry of the material hardware of the machine element. For the exchanged heat rate in machine element j, Q_ j ; one can write

Q_ j ¼ ðUAÞj DTlm;j

(5.5)

where DTlm,j represents generally the mean logarithmic temperature difference between heat exchanging streams and (UA)j, the overall thermal conductance of the exchanger. In Table 4 are summarized the results of the calculations for the base case study. The thermal capacities of rectifier, condenser and absorber, all three exchanging heat with ambient air in free convection, are respectively 0.812, 1.485 and 5.161 W K1. The boiler is supposed to be heated with hot water available at 200
C. The temperature difference between heating medium and boiler is set to DT~ hw;B ¼ 10 K. The overall thermal conductance of the bubble pump is of 0.8 W/K with a DTlm,P ¼ 59.2 K. Here, the heating medium is in forced convection and the liquid solution in state of boiling. The overall heat transfer coefficient will be higher and consequently its heating surface will be relatively reduced. The boiler on the other side has a high overall thermal conductance of 3.75 W K1 with a large heat transfer potential of DTlm,B ¼ 34.7 K.

5.4. Parametric study Basing on the results of the preceding section, the DAR is now investigated for modified operating conditions, but fixed overall thermal conductance's (Table 4) of all heat exchanging elements of the machine. In the mathematical model, the DTlm expressions replace now the assumptions relative to thermal pinches. Further, the refrigerant purity y1 is now predicted by the model. The so modified model is used to perform the following parametric analyses. The evolution of the refrigerant and the inert gas molar flow rates n_ 1 and n_ 10 with the boiler heat supply is illustrated in Fig. 8 and that of the DAR performance, in Fig. 9. It can be observed in the latter that the COP increases first rapidly to a maximum of 0.127 for Q_ B values between 110 and 120 W, and starts decreasing progressively thereafter. At the beginning, the generated refrigerant vapor flow rate, and consequently the cooling capacity, increases quickly with increasing heat supply to the boiler; consequently, the COP rises rapidly. Beyond the optimal value of Q_ B , the generated vapor becomes more rich in water, which will condensate in the rectifier and return to the boiler. The supplementary supplied heat is rejected in the atmosphere without proportionally contributing to the amelioration of produced cold, and hence, the performances of the machine and the COP degrade. The evolution of the cooling capacity and COP of the DAR with respect to the bubble pump's submergence ratio in the range of 2e4 is represented in Fig. 10. COP and Q_ E exhibit maximum values at about H/h y 2.8. The effect of heat input to the bubble pump, Q_ P , on the DAR performances is investigated in the range of 10e140 W with all other parameters maintained constant. Taking into account the results for the void fraction using equations (4.2)(4.4.6), the superficial velocities of vapor and liquid phases are calculated and plotted in the flow regime map [40] of Fig. 11. It can be noted that the transition from bubbly to slug flow regime takes place at heat supply to the bubble pump of about 16.5 W. Beyond this value, the pump operates essentially in slug flow regime up to Q_ P ¼ 140 W. Fig. 12 illustrates the variation of the COP and Q_ E with respect to Q_ P . It can be observed that the COP increases rapidly and attains an optimum value of 0.127 at Q_ P ¼ 55 W. After this maximum, the COP

Fig. 7. Exergy analysis of the DAR for basic operating conditions.

A. Taieb et al. / Energy 115 (2016) 418e434

431

Table 4 Thermal capacity of the various DAR elements. Component

Expression of DTlm

Q_ (W)

DTlm (W/K)

UA (W/K)

Rectifier

 T~

73.7

90.8

0.812

51.0

34.4

1.485

74.5

14.4

5.161

63.8

49.1

1.298

47.0

59.2

0.795

130.0

34.7

3.748

3.3

27.3

0.121

24.5

19.1

1.281

2.3

25.1

0.093

15.3

17.7

0.866

24.5

19.1

1.288

R;in

  T~ air;out  T1  T~ air;in ! ln

T~ R;in T~ air;out T1 T~ air;in

T~ R;in ¼ ðn_ 13V T13 þ n_ 16 T16 Þ=ðn_ 13V þ n_ 16 Þ T~ air;out ¼ T~ air þ DT~ air;R

T~ air;in ¼ T~ air ; Condenser

  T1  T~ air;out  T2  T~ air;in ! T1 T~ air;out T2 T~ air;in

ln

T~ air;in ¼ T~ air ; Absorber

T~ air;out ¼ T~ air þ DT~ air;C

  TAbs  T~ air;out  TAbs  T~ air;in ! ln

T~ air;in ¼ T~ air ;

TAbs T~ air;out TAbs T~ air;in

T~ air;out ¼ T~ air þ DT~ air;A

(94)

(95)

(96)

(97)

(98)

(99)

(100)

SHX ðT14  T12 Þ  ðT15  T11 Þ   T12 ln TT14 15 T11 Bubble pump

 T~

hw;in;P

  T13  T~ hw;out;P  T12 ! ln

T~ hw;in;P T13 T~ hw;out;P T12

T~ hw;out;P ¼ T~ hw;in;P  DT~ hw;P Boiler

 T~

hw;in;B

  T14  T~ hw;out;B  T13 ! ln

T~ hw;in;B T14 T~ hw;out;B T13

(101)

(102)

(103)

(104)

T~ hw;out;B ¼ T~ hw;in;B  DT~ hw;B

(105)

ðT2  T7 Þ  ðT3  T6 Þ   7 ln TT23 T T6

(106)

ðT8  T7 Þ  ðT9  T6 Þ   7 ln TT89 T T6

(107)

GHX: Circuit 2e3/circuit 67

GHX: Circuit 8e9/circuit 67

Evaporator: Circuit 3e4/cold chamber

  T3  T~ CF  T4  T~ CF ! ln

(108)

T3 T~ CF T4 T~ CF

Evaporator: Circuit 9e10/circuit56 ðT9  T6 Þ  ðT10  T5 Þ   6 ln TT109 T T5 Evaporator: Circuit 5e6/cold Chamber

 T~

CF

  T6  T~ CF  T5 ! ln

T~ CF T6 T~ CF T5

(109)

(110)

432

A. Taieb et al. / Energy 115 (2016) 418e434

Fig. 10. COP and Q_ E vs. bubble pump submergence ratio H/h.

Fig. 8. Refrigerant and inert gas flow rates vs. boiler heat supply Q_ B .

begins to decrease gradually. Fig. 13 depicts the evolution of



n_11 n_1

 ; the ratio of rich solution

and refrigerant flow rates, and the ammonia composition of the generated vapor in the boiler, y16, as a function of Q_ . As Fig. 13 P

shows, _n1 as well as n_ 11 increases progressively with risingQ_ P . However, their ratio ðn_ 11 =n_ 1 Þ is not constant: it increases from 3.3 for 10 W to 4.5 for 55 W, and finally to 5.7 for 140 W. In fact, the degasification rate of the liquid solution Dx ¼ x11x14 is larger at lower values of Q_ and decreases considerably when Q_ increases P

P

(0.287 for10 W and 0.135 for 140 W). Consequently, the heat supplied to generate a refrigerant unit molar flow increases with Q_ , P

with negative effects on the COP. The molar fraction y16 increases very fast for lower values of heat power input and reaches 0.62 at the optimum value of Q_ ¼ 55 W. For higher heat rates, it tends P

progressively to become constant as can be shown in Fig. 13. Indeed, a fixed heat power Q_ B corresponds approximately to constant refrigerant flow rate generated in boiler n_ 16. The amount of pumped solution at the beginning is low, the boiling temperature very high and, consequently,y16 is very low (y16 ¼ 0.11 at Q_ ¼ 10 W). But, the quantity of pumped liquid increases with Q_ , P

purity, y16. As Q_ P becomes very large and, consequently, the flow of the pumped liquid, the boiling temperature approaches the liquid saturation temperature and starts to decrease progressively, resulting in a stabilization of y16 (y16 ¼ 0.75 for Q_ ¼ 120 W). P

Fig. 14 illustrates the DAR performance as a function of ambient air temperature Tair in the range 15e40
C. As can be noted, COP and Q_ E decrease continuously, respectively, from 0.138 to 0.103 and from 24.5 to 18.3 W. In fact, the total system pressure increases when Tair rises what makes boiling difficult. The effect of the absorber efficiency, EAbs, on the refrigeration cycle performances is portrayed in Fig. 15 for an efficiency ranging between 0.5 and 1. The COP and Q_ E concomitantly increase with the absorber efficiency, from 0.096 to 17 W for EAbs ¼ 0.5 to 0.142 and 25.1 W for EAbs ¼ 1. For lower values of EAbs, the gas stream exiting the absorber (8) still contains a significant quantity of ammonia and water. On the contrary, the amount of ammonia and water in stream (8) becomes very low and the temperature T9 is remarkably reduced when the absorber efficiency approaches unit (yH2 ;8 ¼ 0.973 and T9 ¼ 14.1
C at EAbs ¼ 0.995). 6. Conclusion

P

involving a reduction of the boiling temperature, and hence, better

_ . Fig. 9. COP and cooling capacity Q_ E vs. boiler heat supply Q B

In this work, a modified configuration of a DAR operating with ammonia/water/hydrogen as working fluid is investigated by numerical simulation. The DAR is an ambient air-cooled machine and is supposed to be activated by solar heated water available at a temperature of 200
C. To this purpose, a detailed mathematical model is developed in order to study and predict the optimum operating conditions of the system. First, the DAR is simulated under a given set of basic operating conditions where the air temperature is fixed to 26
C, the driving powers of the bubble pump and the boiler to, respectively, 47 and 130 W, and the cold chamber's temperature to 5
C. The DAR's performances under these operating conditions are COP ¼ 0.126 and Q_ E ¼ 22.2 W. Basing on these results, an exergy analysis is performed. It is found that the largest exergy destruction is taking place in the air-cooled rectifier. Further, the heat capacity of each component of the refrigerator is evaluated and used, in a second modeling approach, as constant characteristic of the corresponding machine component. The modified model is then applied to perform a parametric study of the DAR. The COP was found to exhibit a maximum in respect to the boiler heat supply. It is also established that the bubble pump operates under slug regime flow. The COP decreases

A. Taieb et al. / Energy 115 (2016) 418e434

Fig. 11. Flow regimes in bubble pump.

Fig. 14. Effect of ambient air temperature on DAR performances.

Fig. 12. COP and Q_ E vs. bubble pump heat supply, Q_ P .

Fig. 15. Effect of absorber efficiency on DAR performances.

433

0.096 for an absorber efficiency of 0.5 and rises regularly to reach 0.142 for EAbs ¼ 1. The obtained results are in good agreement with published experimental data.

References

Fig. 13. Flow rate ratio n_ 11 =n_ 1 and molar composition y16vs.Q_ P .

continuously with ambient air temperature; its value is of approximately 0.138 for 15
C and 0.103 for 40
C. The performance of the absorber affects largely the COP of the machine: its value is

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