Energy Conversion and Management 42 (2001) 1575±1605
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Simulation of a simple absorption refrigeration system Khalid A. Joudi *, Ali H. Lafta Department of Mechanical Engineering, College of Engineering, Baghdad University, Baghdad, Iraq Received 5 June 2000; accepted 15 November 2000
Abstract A steady state computer simulation model has been developed to predict the performance of an absorption refrigeration system using LiBr±H2 O as a working pair. The model is based on detailed mass and energy balances and heat and mass transfer relationships for the cycle components. A computer program has been developed to simulate the eect of various operating conditions on the performance of the individual components of the simulated system. These include an absorber, a generator, a condenser, an evaporator and a liquid heat exchanger. A new model is introduced for representing the absorber. Simultaneous heat and mass transfer has been considered in the absorber, instead of heat transfer only as in other works. The performance of absorber, generator, condenser and evaporator were simulated independently. The whole system was then simulated as a working absorption cycle under various operating conditions. Comparison between the present model results and manufacturerÕs data of the simulated system showed excellent agreement. The present simulation results were compared qualitatively with other works and were in very good general agreement. Ó 2001 Published by Elsevier Science Ltd. Keywords: Absorption system simulation; LiBr absorption system simulation; Absorption refrigeration system simulation
1. Introduction The absorption refrigeration system is one of the earliest methods of producing cold. It has most commonly been used for refrigeration and air conditioning [1]. Theoretical and experimental studies of the performance of absorption refrigeration cycles, including those using LiBr±H2 O and NH3 ±H2 O as refrigerant±absorbent combinations, have already been reported by various authors. Picher [2] tested a 1000 TR capacity LiBr±H2 O absorption refrigeration machine of a two shell type. It was shown that the machine can operate with hot water at 80°C and 120°C, and it was *
Corresponding author.
0196-8904/01/$ - see front matter Ó 2001 Published by Elsevier Science Ltd. PII: S 0 1 9 6 - 8 9 0 4 ( 0 0 ) 0 0 1 5 5 - 2
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Nomenclature A Bi cP D Ds Fc FR g Gz h Dh hfg H k km L Le Lp m_ m, n MDC N Nh Np Nu Nv P Pr q_ Q R Re RF Sh T DTm U w w x X
surface area (m2 ) Biot number Hcw ds =Kso speci®c heat capacity at constant pressure (kJ/kg °C) diameter (m) mass diusivity of solution (m2 /s) heat transfer correction factor ¯ow ratio gravitational acceleration (9.81) (m/s2 ) Graetz number xaso =Cs ds speci®c enthalpy (kJ/kg) heat of absorption hv hso
1 wo /w (kJ/kg) latent heat of vaporization (kJ/kg) heat transfer coecient (kW/m2 °C) thermal conductivity (kW/m °C] mass transfer coecient (m/s) length (m) Lewis number Ds =aso length of pass (m) mass ¯ow rate (kg/s) number manufacturer design curve number of tubes number of tubes for one pass in horizontal direction number of tube passes Nusselt number HD=k number of tubes for one pass in vertical direction pressure (kPa) Prandtl number lcP =k heat ¯ow per unit length (kW/m) total heat (kW) thermal resistance (m2 °C/kW) _ NDi l Reynolds number based on Di , 4m=p 2 fouling factor (m °C/kW) Sherwood number km ds =Ds temperature (°C) logarithmic mean temperature dierence (°C) overall heat transfer coecient (kW/m2 °C) mass fraction of water in solution (kg/kg) average mass fraction (kg/kg) coordinate along wall plate (m) LiBr concentration, percent by weight in solution (%)
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y Dy
coordinate perpendicular to wall plate (m) step size in y direction (m)
Greek c c C ds g h h K l m q r /w a
dimensionless mass fraction
w wo =
we Wo average dimensionless mass fraction _ volume flow rate per wetted length m=q (m2 /s) solution ®lm thickness
3mso Cs =g (m) heat exchanger eectiveness dimensionless temperature
T To =
Te To average dimensionless temperature dimensionless heat of absorption qso Ds Dh=
1 wo kso C1 dynamic viscosity (Pa s) kinematic viscosity l=q (m2 /s) density (kg/m3 ) surface tension (N/m) derivative of h with respect to w at constant T, oh=ow (kJ/kg) thermal diffusivity k=qcP (m2 /s)
Subscripts 1,2,... state points, or, sequence number a absorber av average c condenser cw cooling water e equilibrium, or evaporator g generator i interface, or (inside) i, j, k sequence index ia inside absorber tubes ic inside condenser tubes ie inside evaporator tubes ig inside generator tubes n number o entrance, or (outside) oa outside absorber tubes oc outside evaporator tubes og outside generator tubes p pipe r refrigerant s solution so solution properties at To and Xo v vapor w wall
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found that the cooling water connected in parallel to both absorber and condenser is more ecient than series connection. The coecient of performance (COP) was between 0.68 and 0.72. Waleed [3] developed a computer program to design heat exchangers for the generator, condenser and evaporator and to predict their performance for a 2 TR capacity lithium bromide absorption machine under working conditions dierent from the design condition. Eisa et al. [4] presented possible combinations of operating temperatures of the evaporator, condenser, absorber and generator and the concentration in the absorber and the generator up to the crystallization limit. This determines the ¯ow ratio and COP for any combination of temperatures. Flow ratio is de®ned as the ratio of the mass ¯ow rate of the solution from the absorber to the mass ¯ow rate of the refrigerant. FR
m_ a m_ r
1
Eisa et al. [5] conducted an experimental study to determine the eect of changes in operating conditions in order to optimize the performance of the LiBr±H2 O absorption cooler. It was shown that the most signi®cant parameter is the generator temperature. The higher the generator temperature, the higher is the COP. The ¯ow ratio is also an important design and optimizing parameter. An increase in ¯ow ratio reduces the required generator temperature at the expense of a reduction in the COP. Also, Eisa et al. [6] conducted more experiments on the same system of Ref. [5] to determine the eect on performance of operating the absorber and the condenser at dierent temperatures. It was demonstrated that as the temperature dierence
Tc Ta is increased, the COP and the cooling capacity are decreased. Also, the COP is more sensitive to the absorber temperature than to the condenser temperature. Mclinden and Klein [7] constructed a modular, steady state model for simulation of NH3 ±H2 O absorption heat pumps. The model was based on detailed mass and energy balances and heat and mass transfer relationships for the components of the cycle and was applied to a prototype absorption heat pump and compared with experimental data. Grossman and Michelson [8] developed a modular computer simulation program for absorption systems, which makes it possible to simulate various cycle con®gurations. The program has been tested on single and double stage absorption heat pumps and heat transformers with LiBr± H2 O and NH3 ±H2 O as the working ¯uids. The results have been compared with experimental data from tests of a LiBr±H2 O heat transformer with good agreement. The present study deals with a continuous absorption refrigeration LiBr±H2 O system. A steady state simulation is based on mass balance and heat balance equations, as well as ¯uid ¯ow, heat transfer and mass transfer correlations for each of the components. A new model is introduced for representing the absorber. Simultaneous heat and mass transfer has been considered in the absorber instead of heat transfer only, in other works. The model was applied to an actual commercial absorption refrigeration plant manufactured by Mitsubishi-York, model ES-2A4-MW working on LiBr±H2 O and using hot water as a heat source with a capacity of 211.1 kW refrigeration (60 TR) [9]. The computer model was used to simulate this system performance for a variety of operating conditions. The simulated absorption refrigeration system consists of four basic components, an absorber, a generator, a condenser and an evaporator, as shown schematically in Fig. 1. An economizer heat exchanger, normally placed between the absorber and the generator, makes the process more
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Fig. 1. Schematic diagram of the simulated absorption refrigeration system.
ecient without altering its basic operation. Low pressure water vapor from the evaporator is absorbed in the absorber by the solution. The heat generated during the absorption process is removed by cooling water. A pump circulates the weak solution, with a portion being sent to the generator through the solution heat exchanger. The other portion is mixed with the concentrated solution returning from the generator through the heat exchanger to become an intermediate solution, which returns to the absorber. In the generator, the solution coming from the absorber is boiled to release water vapor by heat addition, leaving behind a solution rich with LiBr, which is returned to the absorber via a throttling valve to maintain the pressure dierential between the high and low sides of the system. In the condenser, the water vapor coming from the generator is condensed to liquid. Then, it is passed via an expansion device to the evaporator pressure. 2. Mathematical model The simulation procedure involves the casting of mathematical models for each component making up the LiBr±H2 O absorption refrigeration system. The overall system performance may then be evaluated by combining these models under the normal sequence of operation of the simulated system. All components of the system were shell and tube exchangers of the counter ¯ow type.
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Several conditions and assumptions were incorporated in the model to simplify analysis without abscuring the basic physical situation. These conditions and assumptions were as follows: 1. The system is simulated under steady state conditions. That is, the mass ¯ow rate of water vapor generated in the generator is exactly the same as the rate of water vapor being absorbed in the absorber. 2. The pressure drop in the pipes and vessels is negligible. 3. The heat losses from the generator to the surroundings and the heat gains to the evaporator from the surroundings are negligible. 4. The expansion process of the expansion device is at constant enthalpy. The heat transfer coecient of water for turbulent ¯ow in smooth tubes was calculated according to the correlation given by Dittius and Bolter [10]. The pool boiling heat transfer coecient used for pure water and aqueous solutions (LiBr±H2 O) was that given by Charters et al. [11]. The condensation heat transfer coecient was evaluated using the Nusselt equation [10]. The evaporation heat transfer coecient employed was that by Lorenz and Yung [12] for ®lm evaporation outside tubes. Water properties were derived from the Chemical EngineersÕ Handbook [13], while the thermodynamic properties of the LiBr±H2 O solution were obtained from Refs. [1,13]. 2.1. Absorber In the absorber, the LiBr solution is sprayed over horizontal tubes cooled by water ¯owing inside. It absorbs the water vapor coming from the evaporator continuously and ¯ows in a thin ®lm around the tubes. Then, it is collected in the bottom of the lower shell. To give a description of the heat and mass transport from a horizontal tube covered with a liquid ®lm, the following simpli®cation is made. The ®lm ¯ow along one half of the tube is modeled as that along a vertical cooled wall with a length of half the tube circumference, which is a model suggested by Wassenaar [14,15]. A schematic representation of the model is shown in Fig. 2. On one side of the plate, a solution of substance A (LiBr) in substance B (water) ¯ows down as a thin laminar ®lm. At the liquid±vapor interface, the water vapor is absorbed and then transported into the bulk of the ®lm. The heat of absorption is released at the interface and transported through the ®lm and the wall to the cooling medium (water). The cooling water ¯ows on the other side of the plate in a direction perpendicular to the plane of the illustration (cross ¯ow). Therefore, the cooling water temperatures may be assumed constant over the plate height, which is equivalent to assuming a constant circumferential pipe temperature. The absorber model started with the following assumptions [14]: 1. The liquid is Newtonian and has constant physical properties. The values of the properties are based on the liquid entry conditions. 2. The ®lm ¯ow may be considered laminar and one dimensional. 3. Momentum eects and shear stress at the interface are negligible. 4. The absorbed mass ¯ow is small relative to the ®lm mass ¯ow. 5. At the interface, thermodynamic equilibrium exists between the vapor and liquid. The relation between surface temperature and mass fraction is linear with constant coecient at constant pressure.
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Fig. 2. Sketch of simpli®ed geometry used in the absorber model.
6. All the heat of absorption is released at the interface. 7. The liquid is a binary mixture and only one of the components is present in the vapor phase. 8. There is no heat transfer from the liquid to vapor and no heat transfer because of radiation, viscous dissipation, pressure gradients, concentration gradients or gravitational eects. 9. There is no diusion because of pressure gradients, temperature gradients or chemical reactions. 10. Diusion of heat and mass in the ¯ow direction is negligible relative to the diusion perpendicular to it. Under the above assumptions, the equations of momentum, energy and diusion of mass and their speci®c boundary conditions for this situation are represented in four dimensionless combined ordinary dierential equations [14,15]. These equations describe the average mass fraction of water in the solution w, the average solution temperature T , the heat transfer to the cooling
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medium across the plate wall per unit width q_ w , and the mass transfer of the water vapor to the ®lm per unit width m_ v in one in®nitesimal part of the ®lm with length dx as shown in Fig. 2. For simpli®cation, these equations are written in dimensionless form, where they describe the change of average mass fraction c, the average solution temperature h, heat transfer q_ w , and mass transfer m_ v with dimensionless length dGz. They are given in ®nal form as follows: dc a
1 dGz
h
c
2
dh b
1 dGz
h
c c
h hcw
3
dq_ w c
cP so m_ s
Te To
h hcw dGz dm_ v we wo am_ s
1 h c dGz 1 wo
4
5
where ah
Le K Nui
Sh1
i;
bh
1 1 Nui
K1Sh
i;
ch
1 1 Bi
Nu1w
i
and h and c are the dimensionless temperature and mass fraction, respectively. Te is the equilibrium solution temperature for the solution at mass fraction wo at the chosen absorber (evaporator) pressure. we is the equilibrium mass fraction of water in the solution for solution temperature To at the chosen absorber pressure. To de®ne Te and we , the relation between the solution temperature and mass fraction is formed, under assumption 5 above, by a linearization of the thermodynamic equilibrium equation of the LiBr±H2 O solution at a ®xed pressure. This equilibrium equation is expressed in the solution temperature Ts as a function of the LiBr concentration in the solution X and the vapor pressure P (or refrigerant temperature Tr ), Ts f
X ; P or Ts f
X ; Tr [1]. The relation is T s C1 w C2
6a
where C1
21:8789
0:58527Tr
C2 0:0436688 1:407Tr
6b
6c
i.e., the Te and we values can be de®ned from Eq. (6) with the evaporator temperature as the refrigerant temperature Tr as follows: Te C1 wo C2
7
and we
1
To C1
C2
8
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A good estimate for Nui , Nuw and Sh are the expressions found analytically by Ref. [14]. These numbers are Nui 2:67
9a
Nuw 1:6
9b
1 Sh Le Gz 1 Sh Gz
"
r!# 6Le Gz ; p
ln 1 ln
2 Gz Le
p 12 ; 24Le p
Gz < Gz1
Gz P Gz1
9c
9d
where Gz1
p 24Le
9e
Eqs. (2)±(5) are solved numerically for a unit width of the plate by explicit ®nite dierences [16] and lead to ci1 ci
10 a
1 hi ci dGz hi1 hi
1 dGz
hi
ci c
hi hcw
q_ wi1 q_ wi c cP so m_ si
Te To
hi hcw dGz m_ si1 m_ si we wo
1 hi ci am_ si dGz 1 wo
11
12
13
for 1 6 i 6 n
n number of parts. In the solution, the length of the plate was divided into 40 equal parts. Each part is represented by four combined ordinary dierential equations in terms of ci , hi , q_ wi and m_ si . Input variables for each part are: c, h, q_ and m_ s . The inputs of the ®rst part (i 1) are: c1 0, h1 0, q_ w1 0 and m_ s1 . The main outputs are: c2 , h2 , q_ w2 and m_ s2 . Then, these are used as new inputs to the second part and so on to the end of the plate (cn , hn , q_ wn and m_ sn ). Thereafter, the following procedures are applied to determine the solution for the whole absorber length. Its geometry is described in Fig. 3. (1) The input conditions to the absorber are the mass ¯ow rate m_ 9 , temperature T9 , and concentration X9 of the solution, as well as the evaporator temperature Te , cooling water temperature inlet to the absorber T15 and mass ¯ow rate of the cooling water m_ 15 . (2) The simulated absorber consisted of multi-passes of tubes Np a (4 in this work) with pass length Lp a (4.876 m). The four passes were simulated as one long pass with total tube length equaling 4Lp a . This simpli®cation eliminates the eect of added pressure drop due to direction changes in the original 4 pass absorber. The added pressure is thought to be very small and does
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Fig. 3. Schematic of the geometry of a cooled absorber ®lm: (a) front view section of the absorber tubes and (b) side view section for the ®rst vertical row of the tubes.
not aect the heat transfer and mass transfer calculations. The total length of each tube was divided into 40 equal sections. Each of the 40 tube sections will have a length equivalent to a plate width y as shown in Fig. 3 4Lp a
14 40 (3) The solution starts with the ®rst horizontal row of tubes. The horizontal tubes are assumed to be all in the same conditions, i.e., the simulation of one tube represents the situation of all tubes (number 1 Nh a ) of that row. The input conditions for each pipe in this row are the mass ¯ow rate of the solution m_ s per y width of the section, solution temperature To and mass fraction wo : y
m_ s
m_ 9 Np a Lp a Nh a
15
and To T9 wo 1
16 X9 100
17
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(4) Under the input conditions above, Eqs. (10)±(13) can be applied to the ®rst tube (k 1) in the ®rst vertical row at the ®rst section (j 1). The inputs start with c1 0, h1 0, q_ w1 0 and m_ s1 m_ s . The main outputs are: cn k1 , hn 1 q_ wn 1 and m_ sn 1 . Then, they are used as new inputs to the second tube (k 2) and (j 1) in the same vertical row as follows: To new T n 1 hn 1
Te
To To
wo new wn 1 cn 1
we
wo wo
m_ s1 m_ sn 1 q_ w1 q_ wn 1 and start with h1 0, c1 0 and so on to the last tube (k Nv a ). In this case, the output conditions are cn Nv a , hn Nv a , q_ wn Nv a and m_ sn Nv a . All the processes are taking place at constant cooling water temperature T15
j 1. (5) Similar results can be obtained for each of the other vertical rows of tubes at the same section
j 1 because the input conditions are the same for each vertical row (step (4)). Therefore, the rate of heat transfer to the cooling water and the mass ¯ow rate of the solution for the whole section are obtained by collecting the q_ wn Nv a and m_ sn Nv a values for each vertical row, or they can be expressed in the form: q_ wj Nh a q_ wn Nv a
18
m_ sj Nh a m_ sn Nv a
19
for 1 6 j 6 m: The cn Nv a and hn Nv a values that are obtained from step (5) are the same for the other vertical rows at same section (j 1). (6) The same procedure as in steps (5) and (6) is repeated for (j 2; 3; . . . ; m) with a new cooling water temperature at each section. This temperature can be obtained from the heat balance around the cooling water circuit as follows: Tj1
q_ wj Tj m_ 15 cPj
20
(7) The total heat of the absorber and the total mass ¯ow rate of solution leaving the absorber are computed by collecting q_ wj and m_ sj for all sections as follows: m X Qa
21 q_ wj j1
m_ 2
m X
m_ sj
22
j1
The mass ¯ow rate of the refrigerant that is absorbed in the absorber is m_ 1 m_ r m_ 2
m_ 9
23
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The cooling water temperature outlet from the absorber is T16 Tm
24
The hn Nv a and cn Nv a values that are obtained from all sections (step (6)) along the absorber length are converted to T n Nv a and wn Nv a values from: T n Nv a hn Nv a
Te
To To
25
wn Nv a cn Nv a
we
wo wo
26
These values
T n Nv a and wn Nv a are collected, and the average values as T av and wav are obtained by a numerical integral method using SimpsonÕs rule [16]. The temperature and concentration of the solution leaving the absorber T2 and X2 are T2 Ta T av X2 Xa 100
1
27 wav
28
2.2. The solution pump circuit (a) Energy balance Qp m_ 3 h3
m_ 2 h2
29
Qp is the mechanical energy required to pump the solution liquid, and it will be taken as zero in the present work because its energy is very small compared with Qg . Thus, m_ 3 h3 m_ 2 h2
30
m_ 3 h3 m_ 4 h4 m_ 8 h8
31
m_ 9 h9 m_ 8 h8 m_ 7 h7
32
h9
m_ 8 h8 m_ 7 h7 m_ 9
33
(b) Conservation of total mass m_ 2 m_ 3
34
m_ 3 m_ 4 m_ 8 2m_ 4 2m_ 8
35
m_ 9 m_ 8 m_ 7
36
(c) Conservation of absorbate m_ 2 X2 m_ 3 X3
37
From Eq. (34) X2 X3 Xa
38
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m_ 3 X3 m_ 4 X4 m_ 8 X8
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39
Substituting Eq. (35) in Eq. (39) and eliminating m_ 3 provides X3 X4 X8 Xa
40
m_ 9 X9 m_ 8 X8 m_ 7 X7
41
Substituting Eqs. (34), (35) and (40) into Eq. (41) gives m_ 9 X9 12m_ 2 X2 m_ 7 X7
42
Rearranging Eq. (42), X9
1 m_ 2 m_ 7 X2 X7 2 m9 m_ 9
43
2.3. The solution heat exchanger (a) The heat transfer process It is expressed in terms of the eectiveness of the heat exchanger. The expression for the effectiveness is given as [10] g
T6 T6
T7 T4
44
Rearranging Eq. (44), T7 T6
g
T6
T4
45
(b) Energy balance m_ 4 cP 4
T5
T4 m_ 6 cP 4
T6
T7
46
Rearranging Eq. (46), T5
m_ 6 cP 6
T6 m_ 4 cP 4
T7 T4
47
noting that m_ 4 m_ 5
1=2m_ 2 and m_ 6 m_ 7 2.4. Generator (a) Energy balance Qg Qg1 m_ 6 h6 m_ 10 h10
m_ 5 h5
48
or Qg may be expressed from the external circuit as Qg2 Qg Qg2 m_ 18 cP 18
T18
T19
49
(b) Conservation of total mass m_ 6 m_ 5
m_ 10
noting that m_ 10 m_ r
50
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(c) Conservation of absorbate m_ 5 X5 m_ 6 X6
51
Substituting Eq. (50) into Eq. (51) gives m_ 5 X5
m_ 5
m_ 10 X6
52
Rearranging Eq. (52) to obtain the ¯ow ratio (FR) FR
X6 m_ 5 m_ 10 X6 X5
53
where X6 Xg . Thus, Xg FR Xg Xa
54
(d) Heat transfer process The generator is a shell and tube heat exchanger. It is assumed that the ¯ow is counter ¯ow through multi-pass tubes. The heat transfer equation employed is [10], Qg Qg3 Ug Ag DTm g Fc
55
(e) Equilibrium equation The generator pressure Pg is in equilibrium with the solution temperature in the generator Tg T6 and its concentration Xg X6 . The generator pressure equals the condenser pressure Pc . The condenser temperature is Tc T11 , and its vapor pressure is Pc . Therefore, from the equilibrium equation of the LiBr±H2 O solution, the generator temperature Tg can be expressed as a function of its concentration Xg and the condenser temperature Tc as follows [1]: Tg f
Tc ; Xg
56
2.5. Condenser (a) Energy balance Qc Qc1 m11 hfg11
57
or Qc may be expressed from the external circuit as Qc2 Qc Qc2 m16 cP 16
T17
T16
58
(b) Heat transfer process The refrigerant vapor entering the condenser is assumed to be saturated vapor at the condenser temperature T11 (Tc ). So that Qc Qc3 Uc Ac DTm c
59
2.6. Expansion device h11 h12 hc Note that m_ 11 m_ 12 .
60
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2.7. Evaporator (a) Energy balance Qe Qe1 m_ 1
h1
h12
61
but h11 h12 . So, Qe1 m_ 1
h1
h11
62
noting that m_ 1 m_ 12 . Or Qe may be expressed from the external circuit as Qe2 Qe Qe2 m13 cP 13
T13
T14
63
(b) Heat transfer process In the evaporator, the water vapor is evaporated at the saturation temperature T1
Te , thus Qe Qe3 Ue Ae DTm e
64
2.8. The COP The overall energy balance equation for the whole cycle will be Qe Qg
Qa
Qc 0
65
The COP for the system is usually de®ned as COP
Qe Qg
66
3. Computational model The simulation of any absorption system means the representation of the actual behavior of the system mathematically. This process was done by casting mathematical models for each component making up the absorption refrigeration system. These components were an absorber, a generator, an economizer (liquid to liquid heat exchanger), a condenser and an evaporator. These models were then combined and solved to give the required information about the temperature, concentration and ¯ow rate at each state point of the system and the heat quantities at each component as well as the performance of the system. In the present work, a computer program was built to simulate the eect of various operating conditions on the performance and output of the absorption refrigeration system. 3.1. Simulation of the solution heat exchanger and the absorber Simulation of the solution heat exchanger and the absorber implies the determination of the output conditions from the solution heat exchanger and the absorber block diagram for its input conditions and physical dimensions of the absorber (Table 1). This is depicted in the information
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Table 1 Design parameters of the absorber Description
Absorber
Generator
Condenser
Evaporator
Inside diameter of tube (mm) Outside diameter of tube (mm) Total number of tubes Number of passes Number of tubes per pass Number of tubes in the vertical direction per pass Number of tubes in the horizontal direction per pass Length of the absorber (m) Fouling factor (m2 °C/kW)
13.84 15.87 484 4 121 11
13.84 15.87 200 2 100 10
13.84 15.87 121 1 121 11
13.84 15.87 256 4 64 8
11
10
11
8
4.876 8.6E 06
4.876 8.6E 06
4.876 8.6E 06
4.876 8.6E 06
Fig. 4. Information ¯ow diagram of the system.
¯ow diagram in Fig. 4a. The simulation was accomplished by starting with initial values of Ta and Xa as well as the input conditions as shown in Fig. 4a. Then T5 is computed from Eq. (47). The input conditions to the absorber, which are the parameters of the intermediate solution
m_ 9 , T9 and X9 , are then calculated. In the absorber, Ta , Xa , m_ r and Qa are computed by applying Eqs. (2)±(5) for the whole absorber length and solving them numerically by the ®nite dierence method [16]. The solution is represented in the seven steps described in Section 2. The new Ta and Xa are
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compared with the initial reference values for a given accuracy. If the required accuracy is not obtained, the new Ta and Xa are taken as new reference values, and the process is repeated. The convergence criterion was set equal to 0.0001. If the required accuracy is obtained, Ta , Xa , Qa , T16 and m_ r are obtained as shown in Fig. 4a. 3.2. Simulation of the generator This involves determination of conditions at the outlet from the generator block diagram for its input conditions and for given values of the physical dimensions of the generator (Table 1). This is depicted in the information ¯ow diagram in Fig. 4b. The output data can be obtained by solving two simultaneous nonlinear equations iteratively for two variables (Xg and T19 ) by using PowellÕs method [17], which is based on the classical Newton±Raphson technique. These equations are written in the form Q1i
i 1; 2 Q11 Qg1
Qg2
67
Q12 Qg1
Qg3
68
PowellÕs method deals with nonlinear equations as Fi and their variables xi to solve them simultaneously. The simulation starts with an initial Xg and T19 . Then, Q1i
i 1; 2 values are obtained from Eqs. (67) and (68). The initial values of Xg and T19 are converted to xi (i 1; 2). Also, Q1i
i 1; 2 values are converted to Fi
i 1; 2 values. The solution of PowellÕs method q P2 2 6 aim to reduce Fi towards zero, and it is said to converge when i1 Fi 6 10 . If the required accuracy is obtained PowellÕs method will create new xi
i 1; 2 values. These values are recreated to new Xg and T19 . These values are taken as reference values, and the process is repeated to ®nd Q1i . If the convergence is satis®ed, the output conditions are obtained (Fig. 4b). 3.3. Simulation of the condenser This implies the determination of the output data from the condenser block diagram for given values of its physical dimensions (Table 1), and for its input data as shown in Fig. 4c. The output data can be obtained by solving the energy balance equations as two simultaneous nonlinear equations iteratively for two variables (Tc and T17 ) by using PowellÕs method. These equations are written in the form Q2i
i 1; 2 Q21 Qc1
Qc2
69
Q22 Qc1
Qc3
70
The simulation starts with initial Tc and T17 values. 3.4. Simulation of the evaporator This means determination of the output conditions from the evaporator block diagram for given values of its physical dimensions (Table 1) and for the input conditions to it as shown in Fig. 4d. The output data can be obtained by solving the energy balance equations as two simultaneous
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nonlinear equations iteratively for two variables (Te and T13 ) by using Powell's method. These equations are written in the form Q3i
i 1; 2 Q31 Qe1
Qe2
71
Q32 Qe1
Qe3
72
The simulation procedure is the same as that for the condenser and generator. The simulation starts with initial Te and T13 values. 3.5. Simulation of the system Simulation of the system essentially implies prediction of the heat transfer for each of the system components, the system COP, the conditions at all state points (Fig. 1) for the given physical dimensions of the plant (Table 1), and for the input conditions to the system as shown in the information ¯ow diagram in Fig. 4. As discussed earlier, this necessitates determination of the operating conditions for which the mass and energy balances for the whole system are satis®ed, together with the performance characteristics of the individual components. The simulation procedures of Sections 3.1±3.4 can obviously be used to furnish these performance characteristics. Consequently, system simulation needs an amalgamation of these component simulation procedures so that by using the aforementioned input information, the values of the desired output variables are obtained. This is eected through the interlinking of variables which appear as output from one component and are used as input in the next component (Fig. 4). The problem essentially reduces to solving nonlinear Eqs. (67)±(72) for the variables, Te , T13 , Tc , T17 , Xg , and T19 . Variables Ta and Xa depend upon these variables, which are obtained numerically by a ®nite dierence method as shown in Section 3.1. The simulation starts with reading input data and creates initial values of Te , T13 , Tc , T17 , Xg and T19 . First the iterative procedure is done in the solution heat exchanger and the absorber as shown in the block diagram (Fig. 4a). If the required convergence is obtained, Xa , m_ r , Qa and T16 at the exit of the absorber and T5 at the inlet to the generator are determined. These are used as input in the other components as shown in Fig. 4. Then, Q11 , Q12 , Q21 , Q22 , Q31 and Q32 , respectively, are computed. These values are converted to Fi
i 1; 6 values as shown in Table 2
i 1; 6 values as shown in Table 2 with their variables xi
i 1; 6. If the with their variables xiP required accuracy, 6i1 Fi2 1=2 6 10 6 , is not obtained, PowellÕs method would create new Table 2 The nonlinear equations, which must be solved simultaneously and listed as Fi
i 1; 6 Component name
No. of equations in PowellÕs method
No. of equations in each component
Reference equation
Generator
F1 F2 F3 F4 F5 F6
Q11 Q12 Q21 Q22 Q31 Q32
Eq. Eq. Eq. Eq. Eq. Eq.
Generator Evaporator
Qg1 Qg1 Qc1 Qc1 Qe1 Qe1
Qg2 Qg3 Qc2 Qc3 Qe2 Qe3
(67) (68) (69) (70) (71) (72)
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xi
i 1; 6 values. These values are taken as reference values, and the process is repeated to ®nd new Fi
i 1; 6 values. If the convergence is satis®ed, Qa , Qc , Qg , Qe , COP and all information for each state point of Fig. 1 (temperatures, concentration, and ¯ow rates) are obtained. More comprehensive details about the simulation of the components and the system are given by Lafta [18]. 4. Results and discussion The computer model was used to simulate the systems performance for a variety of operating conditions. As a reference point for evaluation of the eect of dierent parameters, a design condition was selected which corresponds to the design point for the system. The design condition is described in Table 3. The table lists, ®rst, the following input parameters: 1. Mass ¯ow rate of the hot water and its inlet temperature, mass ¯ow rate of the cooling water and its inlet temperature and mass ¯ow rate of the chilled water and its outlet temperature. 2. Mass ¯ow rate of weak solution leaving the solution pump from the absorber and the eectiveness of the solution heat exchanger. Next, the following calculated quantities are shown: 1. The temperature, mass ¯ow rate and concentration at all the state points corresponding to Fig. 1. The concentration is the LiBr concentration, percent by weight, in the solution. 2. The heat quantities in the evaporator, condenser, absorber and generator. 3. The COP. Similar calculations were made for other selected sets of operating conditions. The performance characteristics of the individual components of the system are discussed ®rst over a wide range of operating conditions, and then, the performance of the entire system is discussed. only typical results are presented for brevity. 4.1. Individual component performance The performance of the absorber has been evaluated for various values of input conditions to the absorber. The simulation included study of the eect of varying one of the input conditions (inlet cooling water temperature to the absorber T15 , evaporator temperature Te and solution concentration outlet from the generator Xg ) while keeping the others constant. Fig. 5 shows the variation of the heat rejection to the cooling water Qa as a function of the inlet cooling water temperature T15 . The cooling water inlet temperature is varied from 24°C to 34°C as possible operating limits in this ®gure and for a design value of Xg at 56%. Te was taken at 4°C, 6°C, 8°C and 12°C, respectively. It can be seen that Qa decreases linearly as T15 is increased. The values of Qa are higher at the higher evaporator temperature. This is because when T15
Ta increases, the solution will absorb less refrigerant m_ r , which keeps the solution concentration Xa at a higher value [6]. Less refrigerant absorption means less heat liberation Qa to the cooling water [6]
Heat quantity (kW)
211.1 285 296.3 221.7
Unit
Evaporator Absorber Generator Condenser COP 0:71
Calculated parameters 13 Chilled water inlet to evaporator 14 Chilled water outlet from evaporator 1 Vapor from evaporator to absorber 2 Weak solution outlet from absorber 3 Weak solution outlet from solution pump 4 Weak solution inlet to solution heat exchanger 5 Weak solution inlet to generator 6 Strong solution outlet from generator 18 Inlet hot water to generator 19 Outlet hot water from generator 7 Strong solution outlet from heat exchanger 8 Weak solution outlet from solution pump 9 Intermediate solution inlet to absorber 10 Vapor from generator to condenser 11 Condensate from condenser to expansion device 15 Inlet cooling water to absorber 16 Outlet cooling water from absorber 17 Outlet cooling water from condenser
State points (see Fig. 1)
Solution pump Solution heat exchanger
Generator
Absorber and condenser
Input parameters Evaporator
Unit
Table 3 Design conditions for the simulated system
10.08 10.08 0.089 8.03 8.03 4.015 4.015 3.926 14.1 14.1 3.926 4.015 7.94 0.089 0.089 20.1 20.1 20.1
Mass ¯ow rate (kg/s)
Temperature (°C) 12 8 5.7 33.1 33.1 33.1 66.53 74 85 80 39.17 33.1 36.1 38 38 30 33.39 36
10.08 kg/s 8°C 20.1 kg/s 30°C 14.1 kg/s 85°C 8.03 kg/s 0.85
Value
Mass ¯ow rate of the chilled water Outlet chilled water temperature Mass ¯ow rate of the cooling water Inlet cooling water temperature Mass ¯ow rate of the hot water Inlet hot water temperature Mass ¯ow rate of the solution Eectiveness of the heat exchanger
Description
± ± ± 54.6 54.6 54.6 54.6 56 ± ± 56 54.6 55.2 ± ± ± ± ±
Concentration (%)
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Fig. 5. Variation of the heat rejection with the inlet cooling water temperature.
and vice versa. At a high Te , more refrigerant is absorbed [4], and Xa becomes lower. More refrigerant absorption means more heat liberation to the cooling water. Generator simulation includes the study of the eect of varying one of the input conditions, such as inlet hot water temperature T18 , the solution concentration at the outlet from the absorber Xa and the condenser temperature Tc (or condenser pressure), on the performance while keeping other parameters constant. Fig. 6 shows the variation of the heat supplied to the generator Qg as a function of T18 at a design solution concentration Xa of 54.6% and three condenser temperatures. It can be seen from this ®gure that when T18 increases, Qg increases. The values of this parameter are higher at the lower Tc . The reason behind this behavior is that when T18 increases, the solution temperature Tg will, of course, increase. Then, more refrigerant will be generated. This, in turn, causes an increase in the solution concentration Xg with LiBr, and then, Qg will increase as shown in Fig. 6. An important result in this ®gure is the limitation in the operating hot water temperature associated with the condenser temperature. A condenser temperature of 34°C allows a wide range of prime hot water temperature (75±95°C), whereas this range is reduced to 10°C (85±95°C) when the condenser temperature becomes 44°C. This can be attributed to the fact that the absorption cycle operates when the generator concentration Xg is greater than the absorber concentration Xa to generate refrigerant vapor [1]. Therefore, the minimum hot water temperature T18 to generate refrigerant in the generator at a condenser temperature Tc of 34°C is 75°C (Tg 67:75°C). Xg is 55%, whereas Xa is 54.6%. The other values for T18 are obtained in a similar way at other condenser temperatures. The performance of the condenser includes study of the eect of varying one of the input conditions, such as refrigerant ¯ow rate m_ r and inlet cooling water temperature T16 , on the
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Fig. 6. Variation of the heat supplied to the generator with inlet hot water temperature.
Fig. 7. Variation of the heat rejection from the condenser with the inlet condenser cooling water temperature.
performance of the condenser while keeping other factors at constant values. Fig. 7 shows the condenser heat rejection Qc to the cooling water as a function of the cooling water temperature T16 for three refrigerant mass ¯ow rates m_ r . The heat rejection rate is almost insensitive within the
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Fig. 8. Variation of the evaporator load with the outlet chilled water temperature.
cooling range of the illustration. However, the values of the heat rejection go up in steps with each m_ r , as expected. The evaporator performance is predicted by varying one of the input conditions, such as mass ¯ow rate of the refrigerant m_ r , condenser temperature Tc and outlet chilled water temperature T14 , while keeping others constant. Fig. 8 shows the variation of the cooling load Qe as a function of the outlet chilled water temperature T14 over the possible operating conditions for the chilled water temperature from 5°C to 15°C and at three values of m_ r and Tc . Qe increases slightly as T14 is increased. The cooling load takes higher values at higher refrigerant ¯ow rates and lower condenser temperatures. The rate of increase of the cooling load with refrigerant ¯ow rate in steps is quite obvious, as they are directly related. 4.2. Overall system performance The performance of the total system includes study of the eect of varying one of the input conditions, such as inlet hot water temperature T18 , inlet cooling water temperature to the absorber T15 and outlet chilled water temperature T14 , on the performance while keeping the other variables in Table 3 constant. The capacity of the system Qe is represented as a percentage of the nominal cooling capacity of 211.1 kW (60 TR) in Fig. 9. This ®gure shows that when T18 is increased, the capacity increases
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Fig. 9. Variation of the capacity with the inlet hot water temperature.
linearly. This trend is expected in the absorption refrigeration system [1]. The same trend was obtained experimentally by Pichel [2] and theoretically by Waleed [3] for other conditions. Lower T15 values mean higher capacity. Also, Fig. 9 shows a comparison between the present theoretical simulated system results and MDC. It can be seen that very good agreement is obtained between the two. The percentage dierence between the two results was within 0.13±3.64%. The theoretical predictions show a much wider range in Figs. 9±16. The dashed lines are only extensions of the computer prediction. The actual range of the system capacity is, of course, limited by the machine design limitations, which are limited by the capacity range given by the manufacturer design curve. This is shown as solid lines in the computer output in the above illustrations. Fig. 10 shows the variation of the COP of the system (Eq. (66)) with T18 . The COP increases with T18 because of the increased cooling capacity. The values of COP are higher at the lower T15 values. Eisa et al. [4] obtained a similar trend of variation of COP when Tg was increased at dierent Ta and Tc values. The system in Ref. [4] did not include a solution heat exchanger. Eisa et al. [5] investigated the variation of COP with Tg experimentally for dierent Tc values. They obtained a similar trend of results as that presented here in Fig. 10. The manufacturer's COP (MDC) of the simulated system is plotted in Fig. 10 for comparison with the present theoretical curve for COP. The percentage dierence in the results was within 0.44±1.65%. Cooling water is normally supplied to the absorber and condenser of an absorption system either in parallel or in series. Figs. 11 and 12 show the variation of the capacity and the COP, respectively, as a function of T18 for cooling water in parallel and series. The inlet cooling water temperature in parallel to the absorber T15a and to the condenser T15c was 30°C for both, while in series operation, only the inlet cooling water temperature to the absorber T15 is 30°C. These ®gures show that the capacity and COP are higher for the parallel cooling method, as would be expected, because of a
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Fig. 10. Variation of the COP with the inlet hot water temperature.
Fig. 11. Variation of the capacity with the inlet hot water temperature for cooling water in parallel and series.
lower condenser temperature in the parallel operation. Therefore, the performance improves. Pichel [2] presented experimental results of the capacity against T18 for series and parallel cooling. His results agree with the trend of results in Fig. 11 of the present work.
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Fig. 12. Variation of the COP with the inlet hot water temperature for cooling water in parallel and series.
Fig. 13. Variation of the capacity with the outlet chilled water temperature.
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Fig. 14. Variation of the COP with the outlet chilled water temperature.
Fig. 15. Variation of the capacity with the inlet cooling water temperature.
Fig. 13 shows that the capacity increases as T14 is increased, and higher values of T15 indicate lower capacity. Pichel [2] presents experimental results of the refrigeration capacity on the same
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Fig. 16. Variation of the COP with the inlet cooling water temperature.
coordinates as Fig. 13. The trend of that experimental data is the same as that in Fig. 13. MDC is included in this illustration for purposes of comparison between actual design data and the present model results. It is clear that excellent agreement is obtained between them. The percentage dierence at T15 of 30°C (which is the design value) in this comparison was less than 1%. Fig. 14 shows the system COP increasing as T14 is increased. The values of COP are higher at lower T15 . Eisa et al. [4] indicated that the COP increases as Te is increased, and that the best COP is obtained when Ta and Tc are lower. Also, Eisa et al. [5] proved experimentally that the COP increases as the evaporator temperature is increased. The results of Fig. 14 of the present model are in agreement with the results of Refs. [4,5]. Fig. 15 shows the variation of the capacity as a function of T15 . T18 was kept constant at 85°C. The capacity decreases as T15 is increased and is higher at the higher values of T14 for the reasons discussed earlier. Also, Fig. 15 illustrates a comparison between the present results and the MDC of the capacity variation with cooling water temperature. It can be seen that very good agreement is obtained between them. The percentage dierence between the results was less than 1.65%. Fig. 16 shows the variation of COP with T15 . The COP decreases when T15 takes higher values, and higher values of T14 mean higher COP. Eisa et al. [4] showed that the COP decreases as the absorber temperature Ta and condenser temperature Tc are increased. Also, Eisa et al. [5] have found experimentally that the COP decreases as Ta and Tc are decreased. These results are in agreement with the present model results of Fig. 16. The variation in the absorption cycle can now be represented on the equilibrium chart for the LiBr±H2 O solution as shown in Fig. 17. The cycle is labeled with the same numbers for state points as those of Fig. 1. In Fig. 17a, T18 increases from 80°C (state A) to 85°C (state B) at a value of T15 of 30°C and T14 of 8°C. In Fig. 17b, T15 increases from 30°C (state A) to 34°C (state B)
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Fig. 17. Absorption refrigeration cycle on P±X±T diagram of LiBr±H2 O. (a) Eect of incresing T18 on the system performance at constant T15 and T14 . (b) Eect of increaing T15 on the system performance at constant T18 and T14 . (c) Eect of increaing T14 on the system performance at constant T18 and T15 .
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at a value of T18 of 85°C and T14 of 8°C, while in Fig. 17c, T14 was changed from 8°C (state A) to 10°C (state B) at a T18 value of 85°C and T15 30°C. The above representation of the variations in the absorption cycle gives a clear picture of the cycle variations with the parameters discussed.
5. Conclusions 1. The simulation of the absorber and its representation with the present model was very successful. The validity of the simulation results was established by comparison with other works. 2. The simulation results of the overall system performance showed that the eects of varying system parameters in the simulation on system performance were typical of the LiBr absorption cycle and gave quantitative as well as qualitative results. 3. Comparison between the present model results and the manufacturer's data showed excellent agreement. References [1] ASHRAE handbook, Fundamentals, ASHRAE, New York, 1985. [2] Pichel W. Development of large capacity lithium bromide absorption refrigeration machine in USSR. ASHRAE J 1996:85±8. [3] Waleed AF. Study of the eect of design parameters on a two ton lithium bromide absorption unit. MSc Thesis, University of Technology, Baghdad, Iraq, 1983. [4] Eisa MAR, Devotta S, Holland FA. Thermodynamic design data for absorption heat pump systems operating on water±lithium bromide: Part I ± cooling. Appl Energy 1986;24:287±301. [5] Eisa MAR, Holland FA. A study of the operating parameters in a water±lithium bromide absorption cooler. Energy Res 1986;10(2):137±44. [6] Eisa MAR, Diggory PJ, Holland FA. Experimental studies to determine the eect of dierences in absorber and condenser temperatures on the performance of a water±lithium bromide absorption cooler. Energy Convers Mgmt 1987;27(2):253±9. [7] Mclinden MO, Klein SA. Steady-state modeling of absorption heat pumps with a comparison to experiments. ASHRAE Trans Part-2B 1985;91:1793±806. [8] Grossman G, Michelson E. A modular computer simulation of absorption systems. ASHRAE Trans Part-2B 1985;91:1808±26. [9] Catalogue for LiBr absorption chiller model ES-2A4.MW, 60 TR capacity. Mitsubishi Heavy Industries Ltd., Energy and Environment Research Center, Baghdad. [10] Holman JP. Heat transfer, 8th ed. New York: McGraw-Hill; 1992. [11] Charters WWS, Megler VR, Chen WD, Wang YF. Atmospheric and sub-atmospheric boiling of H2O and LiBr± H2O solutions. Int J Refrig 1982;5(2):107±14. [12] Lorenz JJ, Yung D. A note on combined boiling and evaporation liquid ®lms horizontal tubes. Trans ASME, J Heat Transf 1979;101:178±80. [13] Perry JH. Chemical engineers handbook, 3rd ed. New York: McGraw Hill; 1950. [14] Wassenaar RH. A comparison of 4 absorber models. Internal Report K-176, Delft University of Technology, Delft, Netherlands, 1992. [15] Wassenaar RH. Falling ®lm absorption: a discussion on three types of model and on the data reduction of absorption measurements. Proceedings of the 19th International Congress of Refrigeration, vol. IVa, Theme 4, 11R Commission B1, 1995. p. 34±8.
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[16] Huliquist P. Numerical methods for engineering and computer scientists. New York: Addison Wesley; 1988. [17] Rabinowitz P. Numerical methods for non-linear algebraic equations. London: Gordon & Breach; 1970. [18] Lafta AH. Preliminary simulation of a simple absorption refrigeration system. MSc Thesis, Department of Mechanical Engineering, University of Baghdad, Iraq, 1988.