- Email: [email protected]
A mathematical model with experiments of single effect absorption heat pump using LiBr–H2O
Applied Thermal Engineering 30 (2010) 2753e2762
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
A mathematical model with experiments of single effect absorption heat pump using LiBreH2O Jian Sun*, Lin Fu, Shigang Zhang, Wei Hou Department of Building Science, Tsinghua University, Beijing, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 May 2010 Accepted 27 July 2010 Available online 11 August 2010
A mathematical model of a single effect, LiBreH2O absorption heat pump operated at steady conditions is presented. This model took into consideration of crosscurrent flow of fluids for heat and mass exchangers, two-dimensional distribution of temperature and concentration fields, local values of heat and mass transfer coefficients, thermal parameter dependent physical properties of working fluids and operation limits due to the danger of the LiBr aqueous solution hydrates and crystallization. Improvements of the calculation method make this simulation much more convenient and efficient. An improved absorber experiment set-up and a complete absorption heat pump were built and tested for further study. It was found that the mass flux of vapor increased with the increase of absorber pressure, coolant flow rate, spray density of LiBr solution and decrease of coolant and input temperature of solution. And the vapor mass flux increased almost linearly with the increase of absorber pressure. Results derived from this model show agreement within 7% with experimental values. Crown Copyright Ó 2010 Published by Elsevier Ltd. All rights reserved.
Keywords: Absorption heat pump Model Lithium bromide Experiment
1. Introduction Energy recovery is becoming more and more important in the industry where enormous heat is wasted. Among heat driven devices, absorption heat pump can use low grade heat in various industrial processes. Besides, absorption heat pump can benefit the atmosphere by reducing the emission of carbon dioxide and adopting environment friendly working pair. There are many mathematical models developed for H2OeLiBr absorption cycles basing on mass and energy balances [1e5], which neglected the process of mass transfer by diffusion. The models of falling films absorption were also reviewed [6]. Some improved models of falling film absorption were given in recent works [7e24]. However, studies including modeling of diffusion are fairly limited yet. Heat transfer formulas were commonly applied as simplified, dimensionless equations in most of given models. In fact, heat transfer is a three-dimensional problem. Besides, such a method will impose the necessity of integration of differential equations at strictly specified boundary conditions. However, large area of heat and mass transfer with low thickness of falling film is extremely complicated. A simplified numerical analysis of vapor absorption in the absorber was presented. Similarity solutions were used to solve * Corresponding author. Tel.: þ86 10 62773885; fax: þ86 10 62770544. E-mail address: [email protected] (J. Sun).
momentum and energy transport in their research. Simulation results got showed reasonable accuracy when compared with experimental results [7]. Eddy diffusivity correlations were used to describe the heat and mass transfer near the wall and the interface in the absorber model. And energy and diffusion equations were solved simultaneously. Two cases (a constant-temperature and an adiabatic wall) were especially investigated [8]. A theoretical absorption heat pump model was given to predict the transient operating characteristics when recovering waste heat of 30e40 C, which indicated that a higher capacity was obtained with an increase in driving steam temperature, waste heat temperature, and mass flow rate of hot water and waste water. Influences of heat and mass transfer were not considered in their work [9]. A single coupled heat and mass transfer model was developed to extract the transfer coefficients for falling-films from the measurements on a tubular absorber. An absorber experiment was also given to estimate their model. However, the error between the predictions and their results for the heat transfer coefficient was not satisfying. And temperature and concentration fields could not be calculated in their simple model, which was important to realize heat and mass transfer principles in absorbers [10]. Besides, with higher precision, our absorber experiment set-up is much more convenient for operation, and less time was needed to reach a steady state.
1359-4311/$ e see front matter Crown Copyright Ó 2010 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2010.07.032
2754
J. Sun et al. / Applied Thermal Engineering 30 (2010) 2753e2762
A theoretical model was used to analyze the simultaneous heat and mass transfer processes involved in a flat plate absorber. They found that the absorption heat and mass fluxes, the total heat and mass transfer rates and the heat and mass transfer coefficients could reach high values at the inlet region but decreased at the outside of the inlet region. An analysis for the constant wall temperature condition has also been carried out to investigate the reliability of the present numerical method through, which was compared with previous investigations [11]. An analysis of heat and mass transfer characteristics of a horizontal tube absorber using H2OeLiBr was given. The results showed that heat transfer coefficient did not vary with the concentration of LiBr solution but slightly with the film Reynolds number. And mass transfer coefficient varied with the concentration of LiBr solution and the position in the absorber but not significantly with the film Reynolds number [12]. However, specific model was not given, and heat and mass transfer coefficients were just correlated and analyzed. A method to investigate multistage absorption cycles when COP of single stage heat pumps was known was discussed. Different cycles of heat pumps, refrigerators, heat transformers and heat pump transformers were compared, which adopted LiBreH2O and NH3eH2O as working fluids. The major irreversibilities of the given advanced cycles were taken into account by using real efficiencies of the single stage cycles. Besides, their model could be applied to calculate different kinds of absorption cycles much faster for choosing an appropriate cycle for each application [13]. Absorption heat pump research trends in solar driven air conditioning were also reviewed [14]. A computational model was developed for energy and exergy analysis of single effect and series flow double effect LiBreH2O absorption systems. A series of new efficient property equations for LiBr solution were developed in this model. They found that COP of single effect system was in the range of 0.6e0.75, which was 1e1.28 of the series flow double effect systems. Besides, irreversibility was highest in the absorber in both systems [15,16]. Three domestic refrigerators (vapor compression, thermoelectric and absorption refrigeration) were compared with aspects of the energy efficiency, noise produced and cost. They found that the vapor compression refrigerator consumed the least energy, was least costly and noisiest. And the absorption refrigerator was the quietest but most expensive. At the same time, the thermoelectric refrigerator was the costliest and less energy efficient than the absorption refrigerator [17]. An air-cooled adiabatic absorption refrigeration system with parallel flow type was developed to solve the problems occurring in conventional air-cooled absorption refrigeration systems with falling film absorber. Results they got indicated that solution distribution ratio reduction helped to make the cycle operate in a vacuum state and to obtain higher COP when outdoor air temperature are higher than that in normal operation conditions. Besides, outdoor air temperature had strong influences on cooling capacity and COP [18]. However, most of experimental studies were done with vertical plates and round tubes to investigate the heat and mass transfer coefficients and the absorption rates. Complete absorption heat pump experimental results were rarely reported. Experiments have been performed for vapor absorption into 50 and 60 wt% LiBr solution films flowing down a vertical surface to investigate the effects of liquid diffusivity values, molecular properties of the concentrated solutions and non-absorbable gases. It was interesting that the base case with 1% non-absorbable in the bulk vapor could lead to about 50% less mass transfer than if no air were present in the vapor [19]. Dependence of the heat and mass transfer coefficients in the absorber under different conditions were studied, and a visualized model was compared with experimental
results [20]. A comprehensive, flow mechanism-based model of vapor absorption with LiBr aqueous solution was presented. The effect of incomplete wetting was considered by the concept of wetting ratio, which was an improvement than previous models. The governing equations were solved using a differential algebraic equation solver (LSODI). Results indicated that tubes near the top absorbed more vapor than tubes at the bottom, and more vapor was absorbed in the falling film region than other regimes [21]. Experimental research was given about heat and mass transfer in a helical absorber, the absorber was simplified to a series of vertical falling films with mixing conditions in between. A mathematical formulation with partial differential equations was based on very general assumptions, and measurements were obtained at tube spacings: 0, 3, 15 and 24 mm and with sheet, droplet and jet flow. They found that an increase in tube spacing from 15 to 24 mm had no significant effect and surface tension gradients could lead to incomplete wetting of the tubes [22]. An experimental investigation was done to explore the correlations for the coupled heat and mass transfer processes, and a numerical model with three flow regimes was given when the absorbent mixture traveled from one pipe to another: drop formation at the tube, free droplet fall at the bottom of the tube, and falling film between tubes [23]. Besides, tests on a smooth tube and on several other tube geometries were presented to determine the possible enhancements achievable through film mixing, LiBreH2O and LiCleH2O were simulated as working fluids and compared with experimental data. Results indicated that the absorption increased the mass flow rate by up to 2% for Reynolds number up to 150 and flow length up to 1 m [24]. This paper established a new model of absorption heat pump including two-dimensional heat and mass transfer fields, which was aimed to beyond the existing restricts of modeling and improve the previous models. The improvement of numerical approach of calculation of two dimensional heat and mass transfer of the absorber was deeply considered. Time-consuming continual procedures of numerical integration for each operation point was substituted. The calculated results were approximated for each part in the absorption heat pump by the mean of multivariable method for a given geometry. Therefore, it was possible to replace differential equations with algebraic equations, especially convenient for absorber performance simulation. Parameters of heat and mass transfer for each exchanger were simulated, which were important for the feasibility analysis for both rated and part-load exploitation conditions. 2. Mathematical model of absorption of absorption heat pump This complete model was calculated by two steps: zero dimensional energy and mass conservation and two dimensional heat and mass transfer fluxes for the absorber. Compared with other models, the main benefit of this model is the improvement of arithmetic for concentration and temperature field calculation of the absorber. The time-consuming numerical integration of many differential equations for each operation point is instead by means of much easier calculations of units in the absorption heat pump with input data. For a given geometry, this result could be approximated by means of multivariable second-order polynomials. So it is possible to replace complicated differential equations with algebraic equations, which is much more expedient for non-linear equations of LiBr solution. The first step was aiming to give two-dimensional models for each heat and mass exchanger in the absorption heat pump, which was only executed once for a given absorption heat pump geometry. The second step was promoting the model of zero-dimension for energy and mass balancing, which was continually performed
J. Sun et al. / Applied Thermal Engineering 30 (2010) 2753e2762
later at each operating point shown in Fig. 1. Because of this, this simulation was limited strictly to the solution of the algebraic nonlinear equations system. The schematic diagram was demonstrated in Fig. 2. The model of LiBreH2O absorption heat pump was combined by typical industrial single stage units shown in Fig. 2. The fundamental simplifications assumed for the model were as follows: Steady state of the absorption heat pump; No radiation heat transfer; Pressure loss only occurred in throttle valve, evaporator and absorber nozzles; Saturated vapor in the absorber; No heat loss through each tube jacket; The input data included geometry and operation information of each unit: Number, material, arrangement, surface structure, internal and external diameter, length of tubes in generator, condenser, evaporator, and absorber; Operation parameters of the solution pump; Mass flow rate M1, temperature T1 and pressure P1 values of input hot water; Mass flow rate M10, temperature T10 and pressure P10 values of input heated water; Mass flow rate M14, temperature T14 and pressure P14 values of input low grade water; Circulation ratio of the solution pump (M3/M18); And the total unknown parameters in this model were listed as follows:
Fig. 1. Schematic diagram of the total system simulation.
2755
: Absorption pressure P9 ¼ P14 ¼ P19, generation pressure P5 ¼ P6 ¼ P3 ¼ P4 ¼ P7 ¼ P8 ¼ P18 ¼ P15; : Input mass flow rate M19 ¼ M8, input temperature T19 ¼ T8, input LiBr concentration CL;19 ¼ CL;8 of the absorber; : Output mass flow rate M5 ¼ M9, output temperature T5 ¼ T6 ¼ T9, output LiBr concentration CL;3 ¼ CL;5 ¼ CL;6 ¼ CL;9 of the absorber; : Output mass flow rate M4 ¼ M7, output temperature T4, output LiBr concentration CL;4 ¼ CL;7 of the generator; : Input mass flow rate M3, output temperature T3 of the generator; : Input mass flow rate of weak solution of the solution mixer M6; : Output temperature of strong solution of the solution heat exchanger T7; : Input temperature of water vapor into the condenser T18; : Output temperature of heated water through the absorber T16; : Output temperature of heated water through the condenser T17; : Output temperature of hot water through the generator T2; : Output temperature of low-grade water through the evaporator T13; : Mass flow rate of refrigerant M11 ¼ M14 ¼ M15 ¼ M18; Simulation results were got when all following algebraic equations were solved. In absorber, overall mass balance and LiBr mass balance were described as follows:
M8 þ M18 ¼ M9
(1)
M8 CL;8 ¼ M9 CL;9
(2)
Fig. 2. Calculation diagram of absorption heat pump.
2756
J. Sun et al. / Applied Thermal Engineering 30 (2010) 2753e2762
In generator, overall mass balance and LiBr mass balance were described as follows:
M4 þ M18 ¼ M3
(3)
M3 CL;9 ¼ M4 CL;4
(4)
In the solution mixer, overall mass balance and LiBr mass balance were described as follows:
M6 þ M7 ¼ M8
(6)
M8 Hsol T8 ; CL;8 þ M18 Hvsv ðP9 Þ þ M10 Hw ðP10 ; T10 Þ ¼ M9 Hsol T9 ; CL;9 þ M10 Hw ðP10 ; T16 Þ
ð7Þ
ð8Þ
¼ M10 Hw ðP10 ; T16 Þ M10 Hw ðP10 ; T10 Þ Z
dMdif ¼ Mdif;abs P9 ; M8 ; T8 ; CL;8 ; M10 ; T10 ¼ M18
ð14Þ (15)
Z
dQ ¼ Qgen P5 ; M3 ; T3 ; CL;9 ; M1 ; T1
Agen
¼ M1 Hw ðP1 ; T1 Þ M1 Hw ðP1 ; T2 Þ Z
dMdif ¼ Mdif;gen P5 ; M3 ; T3 ; CL;9 ; M1 ; T1 ¼ M18
ð16Þ (17)
Mean temperature of vapor in the generator was derived from the mean value of an area integral.
Z
M18 Hv ðP5 ; T18 Þ þ M10 Hw ðP10 ; T16 Þ sl ðP5 Þ þ M10 Hw P10; T17 ¼ M18 Hw
ð9Þ
Tv dA Agen
Agen
sl M18 Hw ðP5 Þ þ M14 Hw ðP14 ; T14 Þ ¼ M18 Hvsl ðP9 Þ þ M14 Hw P14; T13
ð10Þ
M6 Hsol T9 ; CL;9 þ M4 Hsol CL;4 ; T7 ¼ M8 Hsol T8 ; CL;8
(11)
¼ Tv;gen P5 ; M3 ; T3 ; CL;9 ; M1 ; T1 ¼ T18
(18)
Similarly, heat transfer rates in the condenser and evaporator were derived from the numerically integrated differential heat transfer equations.
Z dQ ¼ Qcon ðP5 ; T18 ; M10 ; T16 Þ Acon
ð12Þ
Energy balance of the solution pump was given as follows:
M9 ðP P9 Þ hs hem rsol ðT9 ; C9 Þ 5
Aabs
Agen
M4 Hsol T4 ; CL;4 þ M1 Hw ðP1 ; T2 Þ þ M18 Hv ðP5 ; T18 Þ ¼ M3 Hsol T3 ; CL;9 þ M1 Hw ðP1 ; T1 Þ
Nel ¼
dQ ¼ Qabs P9 ; M8 ; T8 ; CL;8 ; M10 ; T10
Aabs
Energy balances in the absorber, generator, condenser, evaporator, solution mixer, and solution heat exchanger were listed as follows one by one.
M3 Hsol T9 ; CL;9 þ M4 Hsol CL;4 ; T4 ¼ M3 Hsol T3 ; CL;9 þ M4 Hsol T7 ; CL;4
Z
(5)
M8 CL;8 ¼ M6 CL;9 þ M7 CL;4
derived by means of an approximation procedure from the numerically integrated differential heat and mass transfer equations.
¼ M10 Hw ðP10 ; T17 Þ M10 Hw ðP10 ; T16 Þ
ð19Þ
Z dQ ¼ Qeva ðP5 ; M14 ; T14 Þ
(13)
Heat and mass transfer rates were given as follows, which were
Fig. 3. Calculation diagram for a single tube in the horizontal falling film absorber.
Aeva
¼ M14 Hw ðP14 ; T14 Þ M14 Hw ðP14 ; T13 Þ
(20)
Fig. 4. Temperature and mass ratio profiles distribution along the tube radius.
J. Sun et al. / Applied Thermal Engineering 30 (2010) 2753e2762
2757
Table 1 Results comparison with Ref. [21].
Mass flow rate of refrigerant Output concentration of LiBr solution in absorber Input concentration of LiBr solution in absorber Output concentration of LiBr solution in generator Output temperature of LiBr solution from solution pump Output temperature of LiBr solution in absorber Temperature of strong solution at the exit of solution exchanger Temperature of vapor in the generator Output temperature of hot water Output temperature of heated water Input temperature of chilled water Output temperature of chilled water Load of evaporator (recycled heat) COP (coefficient of performance)
Simulation results
Experimental results [21]
Error
0.089 kg/s 54.6 wt% (mass fraction of lithium bromide) 56 wt% (mass fraction of lithium bromide) 56 wt% (mass fraction of lithium bromide) 66.53 C
0.086 kg/s 54.4 wt% (mass fraction of lithium bromide) 55.8 wt% (mass fraction of lithium bromide) 55.8 wt% (mass fraction of lithium bromide) 65.1 C
3.37% 0.37%
33.1 C 39.17 C
33.8 C 39.6 C
38 C 80 C 36 C 12 C 8 C 211.1 kW 0.71
38.5 C 80.7 C 35.7 C 11.8 C 8.0 C 209.6 kW 0.70
Heat transfer rate for the solution heat exchanger was given as follows (counter flowing):
Qshx M3 ; T9 ; CL;9 ; M4 ; T4 ; CL;4 ¼ M4 Hsol T4 ; CL;4 M4 Hsol T7 ; CL;4
(21)
The numerical integration of differential heat and mass transfer formula in the horizontal falling film absorber was introduced later. The approaches used for other falling film exchangers were analogous bearing in mind with individual conditions of heat and mass transfer. The non-linear equations were solved by a modified Newton’s method in MATLAB [25]. 3. Modeling of heat and mass transfer in the horizontal falling film absorber
0.36% 0.36% 2.12% 2.18% 1.17% 1.34% 0.88% 0.67% 1.42% 0.00% 0.69% 1.41%
Due to well mixing of water vapor in the absorber, the geometry arrangement was not the issue for mass transfer. So the pressure of water vapor was assumed to be unique in the absorber [26]. Equations for the absorber were established on classic Nusselt solution for the plate type heat exchanger, which were based on these assumptions: Unmixed fluids of each tube; Heat flux perpendicular to the heat exchange area; One dimensional heat transfer given by one dimensional conduction and heat transfer coefficients for convection; The schematic model for the absorber was given on the cylindrical coordinate system, and heat transfer and mass transfer were attached by one dimensional mass transfer coefficients. Convective heat transfer between the vapor and solution was neglected due to small values of the convective heat transfer coefficient [26]; The falling film thickness was given according to Nusselt solution assumption [12]:
In the absorption heat pump, vapor is absorbed into lithium bromide solution in the absorber. And absorbers are often consisted of a bank of horizontal tubes. The concentrated solution, driven by gravity and water vapor, flows down outside horizontal tubes. The heat of absorption is set free predominantly.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 v2 d¼ Resol 4 g sin q
Fig. 5. Distribution of calculated interface concentration of falling film in the absorber.
Fig. 6. Distribution of calculated interface temperature of falling film in the absorber.
(22)
2758
J. Sun et al. / Applied Thermal Engineering 30 (2010) 2753e2762
Table 2 Input data of parameter distribution investigation of the falling film.
Table 3 Range and uncertainty of parameters.
Parameters
Values
Parameter
Range
Experimental error
Uncertainty
Coolant flow rate Coolant inlet temperature Strong solution flow rate Strong solution inlet temperature Strong solution inlet concentration
0.065 kg/s 29.1 C 0.0132 kg/s 39.5 C 59.3 wt% (mass fraction of lithium bromide) 2130 Pa 19 mm 0.8 mm 25
Msol Csol Tsol Thw Pabs
0.036e0.072 kg s1 60e62 wt% 39.0e49.6 C 26.0e35.3 C 930e3130 Pa
1.1 104 kg/s 0.1 wt% 0.1 C 0.1 C 3.3 Pa
0.20% 0.16% 0.23% 0.33% 0.16%
Vapor pressure Outer diameter of the copper pipe Thickness of the copper pipe Number of copper tubes
Identical solution flow rates for each tube as well as uniform flow distribution of cooling water inside each tube; Flows of all fluids were halved vertically and the model equations were formulated for half part of each tube; Basing on Nusselt formulation, the second-order differentials of heat transfer and diffusive mass flow rate were given as follows:
d2 Qabs ¼ 2
d Mdif;abs
U1 Tcw dldq T 2p sol;es X ¼ 1 Weq W dldq 2p
(23)
(24)
The driving force for heat transfer DT ¼ Tsol;es Tcw was defined. Radial distribution of the cooling water temperature was assumed uniform except the boundary layer [26]. Besides, heat transfer coefficient was also given [2]. The driving force for mass transfer DW ¼ Weq W was defined as a mass ratio difference between the external surface of the solution film and interior. And radial distribution of LiBr concentration was assumed to be uniform except the mass transfer layer [26], which was modeled by using the mass
transfer coefficient. The external of the falling film was considered in the saturated state. So we could establish this equation:
Weq ¼ f ðP; TÞ
(25)
For each small element in the falling film in Fig. 3, the mass and energy balances of heat and mass transfer were formulated as follows: Solution mass balance:
M vW d2 Mdif;abs ¼ dML dW l¼idem ¼ L dldq L vq
(26)
Solution energy balance:
d2 Qabs ¼ dML d½ðW þ 1ÞHsol l¼idem þ d2 Mdif;abs Hvsv M vW vT ¼ L S1 þ S2 sol dldq L vq vq
(27)
where:
vH S1 ¼ Hvsv Hsol ðW þ 1Þ sol vW
(28)
vH S2 ¼ ðW þ 1Þ sol vTsol
(29)
Fig. 7. Schematic diagram of absorber experiment system.
J. Sun et al. / Applied Thermal Engineering 30 (2010) 2753e2762
Fig. 10. Relation between input temperature of strong solution and mass flux of vapor.
Fig. 8. Relation between absorber pressure and mass flux of vapor.
Heated water energy balance:
M C vT d2 Qabs ¼ dMhw dHw q¼idem ¼ hw hw hw dldq 3 vl Through mathematical transformation, differentials could be eliminated as follows:
the
2759
(30) second-order
J1 Weq W L vW ¼ 2pML vq
(31)
U1 Tsol;es Thw 3 vThw ¼ vl 2pMhw Chw
(32)
Solution sprinkling density was uniform along the tube length; Each column of tubes was irrespective to the nearby column tubes; Heated water flowed identically in each tube; Because of this, the falling film temperature boundary conditions for each tube in each column were affected by the temperature distribution of upper tube. The mean values of the output LiBr concentration were calculated by integration.
1 W ¼ L
ZL Wi¼m ðl; q ¼ pÞdl
(34)
0
vTsol 1 L ¼ S1 J1 Weq W U1 Tsol;es Thw S2 2pML vq
(33)
The distribution of solution temperature along the radial coordinate was based on the assumption of the heat transfer regime in the laminar film with pure conductive heat transfer. This simplification allowed the extraction of value Tsol;es . Temperature and mass ratio profiles along the tube radius were demonstrated in Fig. 4. Physical and thermal properties of LiBr solution could be found [27]. In a traditional horizontal falling film absorber, such assumptions were given:
Fig. 9. Relation between input temperature of cooling water and mass flux of vapor.
T hw;i ¼
1
p
Zp Thw;i ðl ¼ L; qÞdq i ¼ 1;2;.mðpipe row indexÞ
(35)
0
Overall exchanged heat and mass transfer rates could be calculated with the input and output values of heated water temperature and lithium bromide aqueous solution concentration. Due to the crosscurrent flow regime for other heat and mass exchangers, an analogous approach was employed for each component of the model, except the solution heat exchanger which
Fig. 11. Relation between spray density and mass flux of vapor in the absorber.
2760
J. Sun et al. / Applied Thermal Engineering 30 (2010) 2753e2762 Table 4 Input data of the model.
Fig. 12. A solar driven absorption heat pump system.
was considered count flowing. Obviously, individual values of heat and mass transfer coefficient were calculated for a particular exchanger with the specific character of heat and mass transfer conditions. 4. Model analysis and comparison Before this work, several attempts have been made to improve models which could join the computational effectiveness of procedures with the non-uniform distribution of driving forces for heat and mass transfer in each absorption exchanger, which resulted in procedures for calculations of one-dimensional [28] or pseudo two dimensional [26] fields of thermal parameters for the heat and mass exchange surface area. A series of numerical simulations for a model of an open absorption cycle was presented [28]. In their work, the dehumidifier operating as an absorber and regenerator working as a generator were all modeled as one dimensional counter-current exchangers. A model was developed for a LiBreH2O absorption chiller where the absorption process was modeled as the pseudo crosscurrent heat and mass transfer on a horizontal tube bank [26]. The heat and mass transfer procedures on the surface of pipes were simplified to be a vertical isothermal plate, while the mechanism of the mass diffusion was modeled with the mass transfer coefficients determined from the dimensionless analogy mass transfer numbers. Multivariable operating characteristics of absorption cycles could also be analyzed with this model. Simultaneous heat and mass transfer in absorber was deeply considered, however, performance of absorber, generator, condenser and evaporator was simulated independently [25], which was not so convenient for calculation. However, whether in absorption heat pump or in absorption chiller, this presented model is both available. Equations given in this model could also be applied for absorption chillers. With same input data supplied in their work, comparison was brought forth in Table 1. This simulation model could be employed for calculation of the overall performance of absorption heat pumps as well as to 2D distribution of physical parameters of the working fluids. Figs. 5 and 6 illustrated concentration and temperature distribution of the interface of falling film in the absorber with input data in Table 2. Interface concentration field across the copper tubes was shown in Fig. 5. Because of the large vapor pressure difference between the incoming equilibrium solution and incoming vapor, there was rapid absorption of vapor in the interface of the falling film. And the falling film interface, assumed in the saturated equilibrium state,
Parameter
Value
Pressure of hot water Pressure of heated water Pressure of low-grade heat Input temperature of hot water Input temperature of heated water Input temperature of low-grade water Mass flow rate of hot water Mass flow rate of heated water Mass flow rate of low-grade water Circulation ratio
856 kPa 486 kPa 532 kPa 98.1 C 31.9 C 9.8 C 48.8 kg/s 139.5 kg/s 58.1 kg/s 13
decreased in the direction of solution flow. The interface temperature field of the falling film was demonstrated in Fig. 6. Temperatures rose rapidly due to vapor absorption, which released lots of absorption heat. Besides, the temperatures varied almost in a linear pattern. 5. Experimental investigations For further study of this absorption procedure, two experimental systems were built. The first experimental set-up was shown in Fig. 7, which was employed for further research for the absorber model. And the second experimental system shown in Fig. 11 was a complete solar driven absorption heat pump manufactured by us, which was used to recycle the low grade industrial waste heat. 5.1. Absorber experiment A schematic diagram of the experimental set-up was shown in Fig. 7. The absorber was designed and constructed followed closely those of real absorbers in absorption heat pump systems. It was consisted by a series of 15 horizontal copper tubes with nominal outer diameter 19 mm, wall thickness 1 mm and effective length 120 mm. The solution of LiBr and water in the evaporator was heated with an electric heater to boil off the refrigerant, which in this case was vapor. The vapor traveled up a tube connecting the evaporator to the absorber unit located above. The strong solution was pumped on to the top of the absorber tubes through the heat exchanger and sprayed using a solution distributor. The solution absorbed vapor as it flowed as a falling film over the absorber tubes. The weak solution was returned to the refrigerant evaporator by gravity. The hot water returning from the absorber was cooled in a heat exchanger and passed through a temperature-controlled water bath. A series of copper-constantan thermocouples were installed in the solution and cooling water flow circuits to measure temperatures. All thermocouples were calibrated using a master thermometer. Two conductivity probes were used to measure the concentration of LiBr solution at the inlet and exit of the absorber. These were calibrated over a range of mass concentrations and temperatures using samples of lithium bromide of known concentrations before they were installed in the experimental setup. The output of the thermocouples and the conductivity meters were recorded continuously in a data acquisition system. Range and uncertainty of parameters of this experiment system were shown in Table 3. The effect of absorber pressure, inlet temperature and flow rate of coolant, inlet temperature and flow rate of solution were experimented. The effect of the absorber pressure on the vapor mass flux was shown in Fig. 8. Obviously, the absorber was operated at a slightly lower pressure than the vapor pressure in the
J. Sun et al. / Applied Thermal Engineering 30 (2010) 2753e2762
2761
Table 5 Experimental and simulation results comparison.
Pressure in the absorber Pressure in the generator Mass flow rate of refrigerant Input concentration of LiBr solution in absorber Output concentration of LiBr solution in absorber Output concentration of LiBr solution in generator Input temperature of LiBr solution in absorber Output temperature of LiBr solution in absorber Temperature of strong solution at the exit of solution exchanger Temperature of vapor in the generator Output temperature of hot water Output temperature of heated water Output temperature of low-grade water Load of evaporator (recycled heat) COP (coefficient of performance)
Experimental results
Simulation results
Simulation deviation
Experimental uncertainty
803 Pa 6163 Pa 0.33 kg/s 60.7 wt% 58.1 wt% 63.3 wt% 45.1 C 33.9 C 57.3 C
784 Pa 6397 Pa 0.31 kg/s 61.1 wt% 58.4 wt% 63.8 wt% 44.4 C 33.5 C 57.8 C
2.37% 3.80% 6.06% 0.66% 0.52% 0.79% 1.55% 1.18% 0.87%
0.41% 0.54% 0.33% 0.16% 0.17% 0.16% 0.22% 0.29% 0.17%
86.7 C 95.2 C 33.3 C 6.4 C 759 kW 0.68
87.9 C 94.1 C 33.6 C 6.6 C 769 kW 0.70
1.38% 1.16% 0.90% 3.12% 1.32% 2.94%
0.12% 0.11% 0.30% 1.60% 1.96% 1.62%
evaporator to ensure the necessary pressure difference for the vapor to flow from the evaporator to the absorber. It was found that the vapor mass flux increased almost linearly with the increase of absorber pressure. The vapor pressure was changed by varying the evaporator temperature from 54 to 77 C. Higher vapor pressure increased the equilibrium concentration at the film interface which led to a higher vapor absorption rate. The effect of the coolant inlet temperature and coolant flow rate on the vapor absorption rate was shown in Fig. 9. It was noticed that vapor mass flux decreased both with the increasing in inlet temperature and the decreasing in the flow rate of the coolant because these variations caused the absorber surface temperature and the bulk solution temperature to rise. The variation of the vapor absorption rate with solution inlet temperature and mass flow rate were shown in Figs. 10 and 11, respectively. Lower inlet temperature of the solution and coolant could maintain low temperature of the falling film. Therefore, film interface absorbs more water vapor to reach equilibrium state. As expected the absorption rate is higher for larger solution mass flow rates. For small solution flow rates, the concentration of the bulk solution decreases rapidly following the absorption of a small amount of water vapor in the entrance region of the absorber. As a result, the difference between film interface concentration and solution bulk concentration decreases thus hindering subsequent vapor diffusion into the film. Moreover, higher solution flow rates result in a more uniform film covering a large surface area of the absorber which enhances vapor absorption. 5.2. Absorption heat pump system experiment In recent studies, authors used special setups, which were consisted of a generator/absorber unit and so on. They did not use an actual absorption heat pump system where coupling between the components may create additional effects. The feedback of the other components may yield boundary conditions for the absorber which differ from the typical setting in stand-alone experiments. So there is a need for additional research into this topic. An actual absorption heat pump shown in Fig. 12 was designed and built. This solar driven absorption heat pump using LiBreH2O as working fluid was manufactured in 2009 and installed in Beijing. A series of measurements were taken to observe the performance of this system. Temperatures, concentrations and flowing rates of LiBr solution were measured by internal measurement devices, which could be easily read from the control interface. Input and output data got in the experiment and simulation results for this absorption heat pump at a steady state were compared in Tables 4 and 5,
respectively. Results derived from this model show agreement within 7% with experimental values. 6. Conclusion and discussion A new mathematical model of a single effect absorption heat pump using LiBreH2O was presented, which took into account: crosscurrent flow of fluids for heat and mass exchangers; twodimensional distributions of temperature and concentration fields; local values of heat and mass transfer coefficients; calculation efficiency of total simulation. The main practical advantage of this model is the possibility of assessing the influence of both the geometry parameters and operation parameters on thermal performance (temperature and concentration value in different places, COP, etc.). The effect of absorber pressure, inlet temperature and flow rate of coolant, inlet temperature and flow rate of solution were deeply investigated. It was found that the vapor mass flux increased almost linearly with the increase of absorber pressure. Besides, the difference between film interface concentration and solution bulk concentration decreases thus hindering subsequent vapor diffusion into the film. Moreover, higher solution flow rates could result in a more uniform film covering a large surface area of the absorber which enhances vapor absorption. Experimental results showed that the mass flux of vapor increased with the increase of absorber pressure, coolant flow rate, spray density of LiBr solution and decrease of coolant and input temperature of solution. Calculated results showed satisfying agreement within 7% with results got by our experiments and other researchers’ models. Acknowledgements The authors gratefully acknowledge financial support from the national science and technology support plan of Peoples Republic of China (No.: 2007BAB23B01) and the key projects of the Beijing municipal science and technology plan (No.: D07040600560701). Thanks are also due to two anonymous referees for their helpful remarks. Nomenclature A, surface area of tubes, m2 c, specific heat capacity, J/(kg K) C, mass fraction of lithium bromide in solution M, mass flow rate, kg/s g, gravitational acceleration, m/s2
2762
J. Sun et al. / Applied Thermal Engineering 30 (2010) 2753e2762
h, specific enthalpy, J/kg Ul, total heat transfer coefficient in cylindrical coordinates, W/(m K) L, tube length, m l, linear coordinate, m N, solution pump power, W P, pressure, Pa Q, heat transfer rate, W T, temperature, C W, mass ratio of water to the mass of pure lithium bromide d, laminar film thickness, m z, dimensionless linear coordinate h, solution pump efficiency y, kinematic viscosity, m2/s 3, total angle of a gravitational flow along tube perimeter, rad q, angular coordinate, rad r, density, kg/m3 J1, total mass transfer coefficient in cylindrical coordinates, kg/(m s) j, dimensionless angular coordinate, j ¼ q=p Re, Reynolds number Subscripts abs, absorber con, condenser gen, generator eva, evaporator SHX, solution heat exchanger hw, heated water dif, diffusional transfer el, electric em, electromechanical eq, equilibrium state es, external surface L, lithium bromide sol, solution s, isentropic v, vapor w, water sl, saturated liquid sv, saturated vapor References [1] A. Bula, H. Diane, F.N. Luis, A.C. Lesme, Thermodynamics Simulation of Solar Absorption Refrigeration Systems, Generator Heat Exchanger. AMSE, France, 2000. [2] G.A. Florides, S.A. Kalogirou, S.A. Tassou, L.C. Wrobel, Design and construction of a LiBrewater absorption machine, Energy Conversion and Management 44 (2003) 2483e2508. [3] F. Assilzadeh, S.A. Kalogirou, Y. Ali, K. Sopian, Simulation and optimization of a LiBr solar absorption cooling system with evacuated tube collectors, Renewable Energy 30 (2005) 1143e1159. [4] W. Wang, R.Z. Wang, Y.X. Xu, Investigation on adsorption refrigeration with a single adsorbent bed, International Journal of Energy Research 22 (1998) 1157e1163.
[5] A. Yokozeki, Theoretical performances of various refrigeranteabsorbent pairs in a vapor-absorption refrigeration cycle by the use of equations of state, Applied Energy 80 (2005) 383e399. [6] D.K. Jesse, G. Srinias, A critical review of models of coupled heat and mass transfer in falling film absorption, International Journal of Refrigeration 24 (2001) 755e797. [7] J.W. Andberg, G.C. Vliet, A simplified model for absorption of vapors into liquid films flowing over cooled horizontal tubes, ASHRAE Transactions 93 (1987) 2454e2466. [8] G. Grossman, M.T. Heath, Simultaneous heat and mass transfer in absorption of gases in turbulent liquid films, International Journal of Heat and Mass Transfer 27 (1984) 2365e2376. [9] S. Jeong, B.H. Kang, S.W. Karng, Dynamic simulation of an absorption heat pump for recovering low grade waste heat, Applied Thermal Engineering 18 (1998) 1e12. [10] M.R. Islam, N.E. Wijeysundera, J.C. Ho, Evaluation of heat and mass transfer coefficients for falling-films on tubular absorbers, International Journal of Refrigeration 26 (2003) 197e204. [11] J.I. Yoon, T.T. Phan, C.G. Moon, P. Bansal, Numerical study on heat and mass transfer characteristic of plate absorber, Applied Thermal Engineering 25 (2005) 2219e2235. [12] D. Bredow, P. Jain, A. Wohlfeil, F. Ziegler, Heat and mass transfer characteristics of a horizontal tube absorber in a semi-commercial absorption chiller, International Journal of Refrigeration 31 (2008) 1273e1281. [13] F. Ziegler, G. Alefeld, Coefficient of performance of multistage absorption cycles, International Journal of Refrigeration 10 (1987) 285e295. [14] G. Grossman, Solar-powered systems for cooling, denhumidification and airconditioning, Solar Energy 72 (2002) 53e62. [15] S.C. Kaushik, A. Arora, Energy and exergy analysis of single effect and series flow double effect waterelithium bromide absorption refrigeration systems, International Journal of Refrigeration 32 (2009) 1247e1258. [16] A. Arora, S.C. Kaushik, Theoretical analysis of LiBr/H2O absorption refrigeration systems, International Journal of Energy Research 33 (2009) 1321e1340. [17] P.K. Bansal, A. Martin, Comparative study of vapour compression, thermoelectric and absorption refrigerators, International Journal of Energy Research 24 (2000) 93e107. [18] G.M. Chen, L. Wang, Q. Wang, M. Zhong, Thermodynamic performance analysis of gas-fired air-cooled adiabatic absorption refrigeration systems, Applied Thermal Engineering 27 (2007) 1642e1652. [19] K.J. Kim, N.S. Berman, S.C. Chaud, Absorption of water vapour into falling films of aqueous lithium bromide, International Journal of Refrigeration 18 (1995) 486e494. [20] T. Nomura, N. Nishimura, S. Wei, Heat and mass transfer mechanism in the absorber of water/LiBr conventional absorption refrigerator: experimental examination by visualized model, in: International Sorption Heat Pump Conference. ASME, New York, 1993, pp. 203e208. [21] S.Y. Jeong, S. Garimella, Falling film and droplet mode heat and mass transfer in a horizontal tube LiBr/water absorber, International Journal of Heat and Mass Transfer 45 (2002) 1445e1458. [22] R.H. Wassenaar, Measured and predicted effect of flow rate and tube spacing on horizontal tube absorber performance, International Journal of Refrigeration 19 (1996) 347e355. [23] M.J. Kirby, B.H. Perez, A design model for horizontal tube water/lithium bromide absorbers, ASME heat pump and refrigeration systems design, Annals of Applied Probability 32 (1994) 1e10. [24] R. Yang, B.D. Wood, A numerical modeling of an absorption process on a liquid falling film, Solar Energy 48 (1992) 195e198. [25] M.J. Powell, A hybrid method for nonlinear equations. in: P. Rabinowitz (Ed.), Numerical Methods for Nonlinear Algebraic Equations. Gordon and Breach, London, 1970, pp. 87e114. [26] K.A. Joudi, A.H. Lafta, Simulation of a simple absorption refrigeration system, Energy Conversion and Management 42 (2001) 1575e1605. [27] H.T. Chua, H.T. Toh, Improved thermodynamic property fields of LiBreH2O solution, International Journal of Refrigeration 23 (2000) 412e429. [28] H.M. Hellmann, G. Grossman, Simulation and analysis of an open-cycle dehumidifiereevaporatoreregenerator (DER) absorption chiller for low grade heat utilization, International Journal of Refrigeration 18 (1995) 177e189.
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
A mathematical model with experiments of single effect absorption heat pump using LiBreH2O Jian Sun*, Lin Fu, Shigang Zhang, Wei Hou Department of Building Science, Tsinghua University, Beijing, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 May 2010 Accepted 27 July 2010 Available online 11 August 2010
A mathematical model of a single effect, LiBreH2O absorption heat pump operated at steady conditions is presented. This model took into consideration of crosscurrent flow of fluids for heat and mass exchangers, two-dimensional distribution of temperature and concentration fields, local values of heat and mass transfer coefficients, thermal parameter dependent physical properties of working fluids and operation limits due to the danger of the LiBr aqueous solution hydrates and crystallization. Improvements of the calculation method make this simulation much more convenient and efficient. An improved absorber experiment set-up and a complete absorption heat pump were built and tested for further study. It was found that the mass flux of vapor increased with the increase of absorber pressure, coolant flow rate, spray density of LiBr solution and decrease of coolant and input temperature of solution. And the vapor mass flux increased almost linearly with the increase of absorber pressure. Results derived from this model show agreement within 7% with experimental values. Crown Copyright Ó 2010 Published by Elsevier Ltd. All rights reserved.
Keywords: Absorption heat pump Model Lithium bromide Experiment
1. Introduction Energy recovery is becoming more and more important in the industry where enormous heat is wasted. Among heat driven devices, absorption heat pump can use low grade heat in various industrial processes. Besides, absorption heat pump can benefit the atmosphere by reducing the emission of carbon dioxide and adopting environment friendly working pair. There are many mathematical models developed for H2OeLiBr absorption cycles basing on mass and energy balances [1e5], which neglected the process of mass transfer by diffusion. The models of falling films absorption were also reviewed [6]. Some improved models of falling film absorption were given in recent works [7e24]. However, studies including modeling of diffusion are fairly limited yet. Heat transfer formulas were commonly applied as simplified, dimensionless equations in most of given models. In fact, heat transfer is a three-dimensional problem. Besides, such a method will impose the necessity of integration of differential equations at strictly specified boundary conditions. However, large area of heat and mass transfer with low thickness of falling film is extremely complicated. A simplified numerical analysis of vapor absorption in the absorber was presented. Similarity solutions were used to solve * Corresponding author. Tel.: þ86 10 62773885; fax: þ86 10 62770544. E-mail address: [email protected] (J. Sun).
momentum and energy transport in their research. Simulation results got showed reasonable accuracy when compared with experimental results [7]. Eddy diffusivity correlations were used to describe the heat and mass transfer near the wall and the interface in the absorber model. And energy and diffusion equations were solved simultaneously. Two cases (a constant-temperature and an adiabatic wall) were especially investigated [8]. A theoretical absorption heat pump model was given to predict the transient operating characteristics when recovering waste heat of 30e40 C, which indicated that a higher capacity was obtained with an increase in driving steam temperature, waste heat temperature, and mass flow rate of hot water and waste water. Influences of heat and mass transfer were not considered in their work [9]. A single coupled heat and mass transfer model was developed to extract the transfer coefficients for falling-films from the measurements on a tubular absorber. An absorber experiment was also given to estimate their model. However, the error between the predictions and their results for the heat transfer coefficient was not satisfying. And temperature and concentration fields could not be calculated in their simple model, which was important to realize heat and mass transfer principles in absorbers [10]. Besides, with higher precision, our absorber experiment set-up is much more convenient for operation, and less time was needed to reach a steady state.
1359-4311/$ e see front matter Crown Copyright Ó 2010 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2010.07.032
2754
J. Sun et al. / Applied Thermal Engineering 30 (2010) 2753e2762
A theoretical model was used to analyze the simultaneous heat and mass transfer processes involved in a flat plate absorber. They found that the absorption heat and mass fluxes, the total heat and mass transfer rates and the heat and mass transfer coefficients could reach high values at the inlet region but decreased at the outside of the inlet region. An analysis for the constant wall temperature condition has also been carried out to investigate the reliability of the present numerical method through, which was compared with previous investigations [11]. An analysis of heat and mass transfer characteristics of a horizontal tube absorber using H2OeLiBr was given. The results showed that heat transfer coefficient did not vary with the concentration of LiBr solution but slightly with the film Reynolds number. And mass transfer coefficient varied with the concentration of LiBr solution and the position in the absorber but not significantly with the film Reynolds number [12]. However, specific model was not given, and heat and mass transfer coefficients were just correlated and analyzed. A method to investigate multistage absorption cycles when COP of single stage heat pumps was known was discussed. Different cycles of heat pumps, refrigerators, heat transformers and heat pump transformers were compared, which adopted LiBreH2O and NH3eH2O as working fluids. The major irreversibilities of the given advanced cycles were taken into account by using real efficiencies of the single stage cycles. Besides, their model could be applied to calculate different kinds of absorption cycles much faster for choosing an appropriate cycle for each application [13]. Absorption heat pump research trends in solar driven air conditioning were also reviewed [14]. A computational model was developed for energy and exergy analysis of single effect and series flow double effect LiBreH2O absorption systems. A series of new efficient property equations for LiBr solution were developed in this model. They found that COP of single effect system was in the range of 0.6e0.75, which was 1e1.28 of the series flow double effect systems. Besides, irreversibility was highest in the absorber in both systems [15,16]. Three domestic refrigerators (vapor compression, thermoelectric and absorption refrigeration) were compared with aspects of the energy efficiency, noise produced and cost. They found that the vapor compression refrigerator consumed the least energy, was least costly and noisiest. And the absorption refrigerator was the quietest but most expensive. At the same time, the thermoelectric refrigerator was the costliest and less energy efficient than the absorption refrigerator [17]. An air-cooled adiabatic absorption refrigeration system with parallel flow type was developed to solve the problems occurring in conventional air-cooled absorption refrigeration systems with falling film absorber. Results they got indicated that solution distribution ratio reduction helped to make the cycle operate in a vacuum state and to obtain higher COP when outdoor air temperature are higher than that in normal operation conditions. Besides, outdoor air temperature had strong influences on cooling capacity and COP [18]. However, most of experimental studies were done with vertical plates and round tubes to investigate the heat and mass transfer coefficients and the absorption rates. Complete absorption heat pump experimental results were rarely reported. Experiments have been performed for vapor absorption into 50 and 60 wt% LiBr solution films flowing down a vertical surface to investigate the effects of liquid diffusivity values, molecular properties of the concentrated solutions and non-absorbable gases. It was interesting that the base case with 1% non-absorbable in the bulk vapor could lead to about 50% less mass transfer than if no air were present in the vapor [19]. Dependence of the heat and mass transfer coefficients in the absorber under different conditions were studied, and a visualized model was compared with experimental
results [20]. A comprehensive, flow mechanism-based model of vapor absorption with LiBr aqueous solution was presented. The effect of incomplete wetting was considered by the concept of wetting ratio, which was an improvement than previous models. The governing equations were solved using a differential algebraic equation solver (LSODI). Results indicated that tubes near the top absorbed more vapor than tubes at the bottom, and more vapor was absorbed in the falling film region than other regimes [21]. Experimental research was given about heat and mass transfer in a helical absorber, the absorber was simplified to a series of vertical falling films with mixing conditions in between. A mathematical formulation with partial differential equations was based on very general assumptions, and measurements were obtained at tube spacings: 0, 3, 15 and 24 mm and with sheet, droplet and jet flow. They found that an increase in tube spacing from 15 to 24 mm had no significant effect and surface tension gradients could lead to incomplete wetting of the tubes [22]. An experimental investigation was done to explore the correlations for the coupled heat and mass transfer processes, and a numerical model with three flow regimes was given when the absorbent mixture traveled from one pipe to another: drop formation at the tube, free droplet fall at the bottom of the tube, and falling film between tubes [23]. Besides, tests on a smooth tube and on several other tube geometries were presented to determine the possible enhancements achievable through film mixing, LiBreH2O and LiCleH2O were simulated as working fluids and compared with experimental data. Results indicated that the absorption increased the mass flow rate by up to 2% for Reynolds number up to 150 and flow length up to 1 m [24]. This paper established a new model of absorption heat pump including two-dimensional heat and mass transfer fields, which was aimed to beyond the existing restricts of modeling and improve the previous models. The improvement of numerical approach of calculation of two dimensional heat and mass transfer of the absorber was deeply considered. Time-consuming continual procedures of numerical integration for each operation point was substituted. The calculated results were approximated for each part in the absorption heat pump by the mean of multivariable method for a given geometry. Therefore, it was possible to replace differential equations with algebraic equations, especially convenient for absorber performance simulation. Parameters of heat and mass transfer for each exchanger were simulated, which were important for the feasibility analysis for both rated and part-load exploitation conditions. 2. Mathematical model of absorption of absorption heat pump This complete model was calculated by two steps: zero dimensional energy and mass conservation and two dimensional heat and mass transfer fluxes for the absorber. Compared with other models, the main benefit of this model is the improvement of arithmetic for concentration and temperature field calculation of the absorber. The time-consuming numerical integration of many differential equations for each operation point is instead by means of much easier calculations of units in the absorption heat pump with input data. For a given geometry, this result could be approximated by means of multivariable second-order polynomials. So it is possible to replace complicated differential equations with algebraic equations, which is much more expedient for non-linear equations of LiBr solution. The first step was aiming to give two-dimensional models for each heat and mass exchanger in the absorption heat pump, which was only executed once for a given absorption heat pump geometry. The second step was promoting the model of zero-dimension for energy and mass balancing, which was continually performed
J. Sun et al. / Applied Thermal Engineering 30 (2010) 2753e2762
later at each operating point shown in Fig. 1. Because of this, this simulation was limited strictly to the solution of the algebraic nonlinear equations system. The schematic diagram was demonstrated in Fig. 2. The model of LiBreH2O absorption heat pump was combined by typical industrial single stage units shown in Fig. 2. The fundamental simplifications assumed for the model were as follows: Steady state of the absorption heat pump; No radiation heat transfer; Pressure loss only occurred in throttle valve, evaporator and absorber nozzles; Saturated vapor in the absorber; No heat loss through each tube jacket; The input data included geometry and operation information of each unit: Number, material, arrangement, surface structure, internal and external diameter, length of tubes in generator, condenser, evaporator, and absorber; Operation parameters of the solution pump; Mass flow rate M1, temperature T1 and pressure P1 values of input hot water; Mass flow rate M10, temperature T10 and pressure P10 values of input heated water; Mass flow rate M14, temperature T14 and pressure P14 values of input low grade water; Circulation ratio of the solution pump (M3/M18); And the total unknown parameters in this model were listed as follows:
Fig. 1. Schematic diagram of the total system simulation.
2755
: Absorption pressure P9 ¼ P14 ¼ P19, generation pressure P5 ¼ P6 ¼ P3 ¼ P4 ¼ P7 ¼ P8 ¼ P18 ¼ P15; : Input mass flow rate M19 ¼ M8, input temperature T19 ¼ T8, input LiBr concentration CL;19 ¼ CL;8 of the absorber; : Output mass flow rate M5 ¼ M9, output temperature T5 ¼ T6 ¼ T9, output LiBr concentration CL;3 ¼ CL;5 ¼ CL;6 ¼ CL;9 of the absorber; : Output mass flow rate M4 ¼ M7, output temperature T4, output LiBr concentration CL;4 ¼ CL;7 of the generator; : Input mass flow rate M3, output temperature T3 of the generator; : Input mass flow rate of weak solution of the solution mixer M6; : Output temperature of strong solution of the solution heat exchanger T7; : Input temperature of water vapor into the condenser T18; : Output temperature of heated water through the absorber T16; : Output temperature of heated water through the condenser T17; : Output temperature of hot water through the generator T2; : Output temperature of low-grade water through the evaporator T13; : Mass flow rate of refrigerant M11 ¼ M14 ¼ M15 ¼ M18; Simulation results were got when all following algebraic equations were solved. In absorber, overall mass balance and LiBr mass balance were described as follows:
M8 þ M18 ¼ M9
(1)
M8 CL;8 ¼ M9 CL;9
(2)
Fig. 2. Calculation diagram of absorption heat pump.
2756
J. Sun et al. / Applied Thermal Engineering 30 (2010) 2753e2762
In generator, overall mass balance and LiBr mass balance were described as follows:
M4 þ M18 ¼ M3
(3)
M3 CL;9 ¼ M4 CL;4
(4)
In the solution mixer, overall mass balance and LiBr mass balance were described as follows:
M6 þ M7 ¼ M8
(6)
M8 Hsol T8 ; CL;8 þ M18 Hvsv ðP9 Þ þ M10 Hw ðP10 ; T10 Þ ¼ M9 Hsol T9 ; CL;9 þ M10 Hw ðP10 ; T16 Þ
ð7Þ
ð8Þ
¼ M10 Hw ðP10 ; T16 Þ M10 Hw ðP10 ; T10 Þ Z
dMdif ¼ Mdif;abs P9 ; M8 ; T8 ; CL;8 ; M10 ; T10 ¼ M18
ð14Þ (15)
Z
dQ ¼ Qgen P5 ; M3 ; T3 ; CL;9 ; M1 ; T1
Agen
¼ M1 Hw ðP1 ; T1 Þ M1 Hw ðP1 ; T2 Þ Z
dMdif ¼ Mdif;gen P5 ; M3 ; T3 ; CL;9 ; M1 ; T1 ¼ M18
ð16Þ (17)
Mean temperature of vapor in the generator was derived from the mean value of an area integral.
Z
M18 Hv ðP5 ; T18 Þ þ M10 Hw ðP10 ; T16 Þ sl ðP5 Þ þ M10 Hw P10; T17 ¼ M18 Hw
ð9Þ
Tv dA Agen
Agen
sl M18 Hw ðP5 Þ þ M14 Hw ðP14 ; T14 Þ ¼ M18 Hvsl ðP9 Þ þ M14 Hw P14; T13
ð10Þ
M6 Hsol T9 ; CL;9 þ M4 Hsol CL;4 ; T7 ¼ M8 Hsol T8 ; CL;8
(11)
¼ Tv;gen P5 ; M3 ; T3 ; CL;9 ; M1 ; T1 ¼ T18
(18)
Similarly, heat transfer rates in the condenser and evaporator were derived from the numerically integrated differential heat transfer equations.
Z dQ ¼ Qcon ðP5 ; T18 ; M10 ; T16 Þ Acon
ð12Þ
Energy balance of the solution pump was given as follows:
M9 ðP P9 Þ hs hem rsol ðT9 ; C9 Þ 5
Aabs
Agen
M4 Hsol T4 ; CL;4 þ M1 Hw ðP1 ; T2 Þ þ M18 Hv ðP5 ; T18 Þ ¼ M3 Hsol T3 ; CL;9 þ M1 Hw ðP1 ; T1 Þ
Nel ¼
dQ ¼ Qabs P9 ; M8 ; T8 ; CL;8 ; M10 ; T10
Aabs
Energy balances in the absorber, generator, condenser, evaporator, solution mixer, and solution heat exchanger were listed as follows one by one.
M3 Hsol T9 ; CL;9 þ M4 Hsol CL;4 ; T4 ¼ M3 Hsol T3 ; CL;9 þ M4 Hsol T7 ; CL;4
Z
(5)
M8 CL;8 ¼ M6 CL;9 þ M7 CL;4
derived by means of an approximation procedure from the numerically integrated differential heat and mass transfer equations.
¼ M10 Hw ðP10 ; T17 Þ M10 Hw ðP10 ; T16 Þ
ð19Þ
Z dQ ¼ Qeva ðP5 ; M14 ; T14 Þ
(13)
Heat and mass transfer rates were given as follows, which were
Fig. 3. Calculation diagram for a single tube in the horizontal falling film absorber.
Aeva
¼ M14 Hw ðP14 ; T14 Þ M14 Hw ðP14 ; T13 Þ
(20)
Fig. 4. Temperature and mass ratio profiles distribution along the tube radius.
J. Sun et al. / Applied Thermal Engineering 30 (2010) 2753e2762
2757
Table 1 Results comparison with Ref. [21].
Mass flow rate of refrigerant Output concentration of LiBr solution in absorber Input concentration of LiBr solution in absorber Output concentration of LiBr solution in generator Output temperature of LiBr solution from solution pump Output temperature of LiBr solution in absorber Temperature of strong solution at the exit of solution exchanger Temperature of vapor in the generator Output temperature of hot water Output temperature of heated water Input temperature of chilled water Output temperature of chilled water Load of evaporator (recycled heat) COP (coefficient of performance)
Simulation results
Experimental results [21]
Error
0.089 kg/s 54.6 wt% (mass fraction of lithium bromide) 56 wt% (mass fraction of lithium bromide) 56 wt% (mass fraction of lithium bromide) 66.53 C
0.086 kg/s 54.4 wt% (mass fraction of lithium bromide) 55.8 wt% (mass fraction of lithium bromide) 55.8 wt% (mass fraction of lithium bromide) 65.1 C
3.37% 0.37%
33.1 C 39.17 C
33.8 C 39.6 C
38 C 80 C 36 C 12 C 8 C 211.1 kW 0.71
38.5 C 80.7 C 35.7 C 11.8 C 8.0 C 209.6 kW 0.70
Heat transfer rate for the solution heat exchanger was given as follows (counter flowing):
Qshx M3 ; T9 ; CL;9 ; M4 ; T4 ; CL;4 ¼ M4 Hsol T4 ; CL;4 M4 Hsol T7 ; CL;4
(21)
The numerical integration of differential heat and mass transfer formula in the horizontal falling film absorber was introduced later. The approaches used for other falling film exchangers were analogous bearing in mind with individual conditions of heat and mass transfer. The non-linear equations were solved by a modified Newton’s method in MATLAB [25]. 3. Modeling of heat and mass transfer in the horizontal falling film absorber
0.36% 0.36% 2.12% 2.18% 1.17% 1.34% 0.88% 0.67% 1.42% 0.00% 0.69% 1.41%
Due to well mixing of water vapor in the absorber, the geometry arrangement was not the issue for mass transfer. So the pressure of water vapor was assumed to be unique in the absorber [26]. Equations for the absorber were established on classic Nusselt solution for the plate type heat exchanger, which were based on these assumptions: Unmixed fluids of each tube; Heat flux perpendicular to the heat exchange area; One dimensional heat transfer given by one dimensional conduction and heat transfer coefficients for convection; The schematic model for the absorber was given on the cylindrical coordinate system, and heat transfer and mass transfer were attached by one dimensional mass transfer coefficients. Convective heat transfer between the vapor and solution was neglected due to small values of the convective heat transfer coefficient [26]; The falling film thickness was given according to Nusselt solution assumption [12]:
In the absorption heat pump, vapor is absorbed into lithium bromide solution in the absorber. And absorbers are often consisted of a bank of horizontal tubes. The concentrated solution, driven by gravity and water vapor, flows down outside horizontal tubes. The heat of absorption is set free predominantly.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 v2 d¼ Resol 4 g sin q
Fig. 5. Distribution of calculated interface concentration of falling film in the absorber.
Fig. 6. Distribution of calculated interface temperature of falling film in the absorber.
(22)
2758
J. Sun et al. / Applied Thermal Engineering 30 (2010) 2753e2762
Table 2 Input data of parameter distribution investigation of the falling film.
Table 3 Range and uncertainty of parameters.
Parameters
Values
Parameter
Range
Experimental error
Uncertainty
Coolant flow rate Coolant inlet temperature Strong solution flow rate Strong solution inlet temperature Strong solution inlet concentration
0.065 kg/s 29.1 C 0.0132 kg/s 39.5 C 59.3 wt% (mass fraction of lithium bromide) 2130 Pa 19 mm 0.8 mm 25
Msol Csol Tsol Thw Pabs
0.036e0.072 kg s1 60e62 wt% 39.0e49.6 C 26.0e35.3 C 930e3130 Pa
1.1 104 kg/s 0.1 wt% 0.1 C 0.1 C 3.3 Pa
0.20% 0.16% 0.23% 0.33% 0.16%
Vapor pressure Outer diameter of the copper pipe Thickness of the copper pipe Number of copper tubes
Identical solution flow rates for each tube as well as uniform flow distribution of cooling water inside each tube; Flows of all fluids were halved vertically and the model equations were formulated for half part of each tube; Basing on Nusselt formulation, the second-order differentials of heat transfer and diffusive mass flow rate were given as follows:
d2 Qabs ¼ 2
d Mdif;abs
U1 Tcw dldq T 2p sol;es X ¼ 1 Weq W dldq 2p
(23)
(24)
The driving force for heat transfer DT ¼ Tsol;es Tcw was defined. Radial distribution of the cooling water temperature was assumed uniform except the boundary layer [26]. Besides, heat transfer coefficient was also given [2]. The driving force for mass transfer DW ¼ Weq W was defined as a mass ratio difference between the external surface of the solution film and interior. And radial distribution of LiBr concentration was assumed to be uniform except the mass transfer layer [26], which was modeled by using the mass
transfer coefficient. The external of the falling film was considered in the saturated state. So we could establish this equation:
Weq ¼ f ðP; TÞ
(25)
For each small element in the falling film in Fig. 3, the mass and energy balances of heat and mass transfer were formulated as follows: Solution mass balance:
M vW d2 Mdif;abs ¼ dML dW l¼idem ¼ L dldq L vq
(26)
Solution energy balance:
d2 Qabs ¼ dML d½ðW þ 1ÞHsol l¼idem þ d2 Mdif;abs Hvsv M vW vT ¼ L S1 þ S2 sol dldq L vq vq
(27)
where:
vH S1 ¼ Hvsv Hsol ðW þ 1Þ sol vW
(28)
vH S2 ¼ ðW þ 1Þ sol vTsol
(29)
Fig. 7. Schematic diagram of absorber experiment system.
J. Sun et al. / Applied Thermal Engineering 30 (2010) 2753e2762
Fig. 10. Relation between input temperature of strong solution and mass flux of vapor.
Fig. 8. Relation between absorber pressure and mass flux of vapor.
Heated water energy balance:
M C vT d2 Qabs ¼ dMhw dHw q¼idem ¼ hw hw hw dldq 3 vl Through mathematical transformation, differentials could be eliminated as follows:
the
2759
(30) second-order
J1 Weq W L vW ¼ 2pML vq
(31)
U1 Tsol;es Thw 3 vThw ¼ vl 2pMhw Chw
(32)
Solution sprinkling density was uniform along the tube length; Each column of tubes was irrespective to the nearby column tubes; Heated water flowed identically in each tube; Because of this, the falling film temperature boundary conditions for each tube in each column were affected by the temperature distribution of upper tube. The mean values of the output LiBr concentration were calculated by integration.
1 W ¼ L
ZL Wi¼m ðl; q ¼ pÞdl
(34)
0
vTsol 1 L ¼ S1 J1 Weq W U1 Tsol;es Thw S2 2pML vq
(33)
The distribution of solution temperature along the radial coordinate was based on the assumption of the heat transfer regime in the laminar film with pure conductive heat transfer. This simplification allowed the extraction of value Tsol;es . Temperature and mass ratio profiles along the tube radius were demonstrated in Fig. 4. Physical and thermal properties of LiBr solution could be found [27]. In a traditional horizontal falling film absorber, such assumptions were given:
Fig. 9. Relation between input temperature of cooling water and mass flux of vapor.
T hw;i ¼
1
p
Zp Thw;i ðl ¼ L; qÞdq i ¼ 1;2;.mðpipe row indexÞ
(35)
0
Overall exchanged heat and mass transfer rates could be calculated with the input and output values of heated water temperature and lithium bromide aqueous solution concentration. Due to the crosscurrent flow regime for other heat and mass exchangers, an analogous approach was employed for each component of the model, except the solution heat exchanger which
Fig. 11. Relation between spray density and mass flux of vapor in the absorber.
2760
J. Sun et al. / Applied Thermal Engineering 30 (2010) 2753e2762 Table 4 Input data of the model.
Fig. 12. A solar driven absorption heat pump system.
was considered count flowing. Obviously, individual values of heat and mass transfer coefficient were calculated for a particular exchanger with the specific character of heat and mass transfer conditions. 4. Model analysis and comparison Before this work, several attempts have been made to improve models which could join the computational effectiveness of procedures with the non-uniform distribution of driving forces for heat and mass transfer in each absorption exchanger, which resulted in procedures for calculations of one-dimensional [28] or pseudo two dimensional [26] fields of thermal parameters for the heat and mass exchange surface area. A series of numerical simulations for a model of an open absorption cycle was presented [28]. In their work, the dehumidifier operating as an absorber and regenerator working as a generator were all modeled as one dimensional counter-current exchangers. A model was developed for a LiBreH2O absorption chiller where the absorption process was modeled as the pseudo crosscurrent heat and mass transfer on a horizontal tube bank [26]. The heat and mass transfer procedures on the surface of pipes were simplified to be a vertical isothermal plate, while the mechanism of the mass diffusion was modeled with the mass transfer coefficients determined from the dimensionless analogy mass transfer numbers. Multivariable operating characteristics of absorption cycles could also be analyzed with this model. Simultaneous heat and mass transfer in absorber was deeply considered, however, performance of absorber, generator, condenser and evaporator was simulated independently [25], which was not so convenient for calculation. However, whether in absorption heat pump or in absorption chiller, this presented model is both available. Equations given in this model could also be applied for absorption chillers. With same input data supplied in their work, comparison was brought forth in Table 1. This simulation model could be employed for calculation of the overall performance of absorption heat pumps as well as to 2D distribution of physical parameters of the working fluids. Figs. 5 and 6 illustrated concentration and temperature distribution of the interface of falling film in the absorber with input data in Table 2. Interface concentration field across the copper tubes was shown in Fig. 5. Because of the large vapor pressure difference between the incoming equilibrium solution and incoming vapor, there was rapid absorption of vapor in the interface of the falling film. And the falling film interface, assumed in the saturated equilibrium state,
Parameter
Value
Pressure of hot water Pressure of heated water Pressure of low-grade heat Input temperature of hot water Input temperature of heated water Input temperature of low-grade water Mass flow rate of hot water Mass flow rate of heated water Mass flow rate of low-grade water Circulation ratio
856 kPa 486 kPa 532 kPa 98.1 C 31.9 C 9.8 C 48.8 kg/s 139.5 kg/s 58.1 kg/s 13
decreased in the direction of solution flow. The interface temperature field of the falling film was demonstrated in Fig. 6. Temperatures rose rapidly due to vapor absorption, which released lots of absorption heat. Besides, the temperatures varied almost in a linear pattern. 5. Experimental investigations For further study of this absorption procedure, two experimental systems were built. The first experimental set-up was shown in Fig. 7, which was employed for further research for the absorber model. And the second experimental system shown in Fig. 11 was a complete solar driven absorption heat pump manufactured by us, which was used to recycle the low grade industrial waste heat. 5.1. Absorber experiment A schematic diagram of the experimental set-up was shown in Fig. 7. The absorber was designed and constructed followed closely those of real absorbers in absorption heat pump systems. It was consisted by a series of 15 horizontal copper tubes with nominal outer diameter 19 mm, wall thickness 1 mm and effective length 120 mm. The solution of LiBr and water in the evaporator was heated with an electric heater to boil off the refrigerant, which in this case was vapor. The vapor traveled up a tube connecting the evaporator to the absorber unit located above. The strong solution was pumped on to the top of the absorber tubes through the heat exchanger and sprayed using a solution distributor. The solution absorbed vapor as it flowed as a falling film over the absorber tubes. The weak solution was returned to the refrigerant evaporator by gravity. The hot water returning from the absorber was cooled in a heat exchanger and passed through a temperature-controlled water bath. A series of copper-constantan thermocouples were installed in the solution and cooling water flow circuits to measure temperatures. All thermocouples were calibrated using a master thermometer. Two conductivity probes were used to measure the concentration of LiBr solution at the inlet and exit of the absorber. These were calibrated over a range of mass concentrations and temperatures using samples of lithium bromide of known concentrations before they were installed in the experimental setup. The output of the thermocouples and the conductivity meters were recorded continuously in a data acquisition system. Range and uncertainty of parameters of this experiment system were shown in Table 3. The effect of absorber pressure, inlet temperature and flow rate of coolant, inlet temperature and flow rate of solution were experimented. The effect of the absorber pressure on the vapor mass flux was shown in Fig. 8. Obviously, the absorber was operated at a slightly lower pressure than the vapor pressure in the
J. Sun et al. / Applied Thermal Engineering 30 (2010) 2753e2762
2761
Table 5 Experimental and simulation results comparison.
Pressure in the absorber Pressure in the generator Mass flow rate of refrigerant Input concentration of LiBr solution in absorber Output concentration of LiBr solution in absorber Output concentration of LiBr solution in generator Input temperature of LiBr solution in absorber Output temperature of LiBr solution in absorber Temperature of strong solution at the exit of solution exchanger Temperature of vapor in the generator Output temperature of hot water Output temperature of heated water Output temperature of low-grade water Load of evaporator (recycled heat) COP (coefficient of performance)
Experimental results
Simulation results
Simulation deviation
Experimental uncertainty
803 Pa 6163 Pa 0.33 kg/s 60.7 wt% 58.1 wt% 63.3 wt% 45.1 C 33.9 C 57.3 C
784 Pa 6397 Pa 0.31 kg/s 61.1 wt% 58.4 wt% 63.8 wt% 44.4 C 33.5 C 57.8 C
2.37% 3.80% 6.06% 0.66% 0.52% 0.79% 1.55% 1.18% 0.87%
0.41% 0.54% 0.33% 0.16% 0.17% 0.16% 0.22% 0.29% 0.17%
86.7 C 95.2 C 33.3 C 6.4 C 759 kW 0.68
87.9 C 94.1 C 33.6 C 6.6 C 769 kW 0.70
1.38% 1.16% 0.90% 3.12% 1.32% 2.94%
0.12% 0.11% 0.30% 1.60% 1.96% 1.62%
evaporator to ensure the necessary pressure difference for the vapor to flow from the evaporator to the absorber. It was found that the vapor mass flux increased almost linearly with the increase of absorber pressure. The vapor pressure was changed by varying the evaporator temperature from 54 to 77 C. Higher vapor pressure increased the equilibrium concentration at the film interface which led to a higher vapor absorption rate. The effect of the coolant inlet temperature and coolant flow rate on the vapor absorption rate was shown in Fig. 9. It was noticed that vapor mass flux decreased both with the increasing in inlet temperature and the decreasing in the flow rate of the coolant because these variations caused the absorber surface temperature and the bulk solution temperature to rise. The variation of the vapor absorption rate with solution inlet temperature and mass flow rate were shown in Figs. 10 and 11, respectively. Lower inlet temperature of the solution and coolant could maintain low temperature of the falling film. Therefore, film interface absorbs more water vapor to reach equilibrium state. As expected the absorption rate is higher for larger solution mass flow rates. For small solution flow rates, the concentration of the bulk solution decreases rapidly following the absorption of a small amount of water vapor in the entrance region of the absorber. As a result, the difference between film interface concentration and solution bulk concentration decreases thus hindering subsequent vapor diffusion into the film. Moreover, higher solution flow rates result in a more uniform film covering a large surface area of the absorber which enhances vapor absorption. 5.2. Absorption heat pump system experiment In recent studies, authors used special setups, which were consisted of a generator/absorber unit and so on. They did not use an actual absorption heat pump system where coupling between the components may create additional effects. The feedback of the other components may yield boundary conditions for the absorber which differ from the typical setting in stand-alone experiments. So there is a need for additional research into this topic. An actual absorption heat pump shown in Fig. 12 was designed and built. This solar driven absorption heat pump using LiBreH2O as working fluid was manufactured in 2009 and installed in Beijing. A series of measurements were taken to observe the performance of this system. Temperatures, concentrations and flowing rates of LiBr solution were measured by internal measurement devices, which could be easily read from the control interface. Input and output data got in the experiment and simulation results for this absorption heat pump at a steady state were compared in Tables 4 and 5,
respectively. Results derived from this model show agreement within 7% with experimental values. 6. Conclusion and discussion A new mathematical model of a single effect absorption heat pump using LiBreH2O was presented, which took into account: crosscurrent flow of fluids for heat and mass exchangers; twodimensional distributions of temperature and concentration fields; local values of heat and mass transfer coefficients; calculation efficiency of total simulation. The main practical advantage of this model is the possibility of assessing the influence of both the geometry parameters and operation parameters on thermal performance (temperature and concentration value in different places, COP, etc.). The effect of absorber pressure, inlet temperature and flow rate of coolant, inlet temperature and flow rate of solution were deeply investigated. It was found that the vapor mass flux increased almost linearly with the increase of absorber pressure. Besides, the difference between film interface concentration and solution bulk concentration decreases thus hindering subsequent vapor diffusion into the film. Moreover, higher solution flow rates could result in a more uniform film covering a large surface area of the absorber which enhances vapor absorption. Experimental results showed that the mass flux of vapor increased with the increase of absorber pressure, coolant flow rate, spray density of LiBr solution and decrease of coolant and input temperature of solution. Calculated results showed satisfying agreement within 7% with results got by our experiments and other researchers’ models. Acknowledgements The authors gratefully acknowledge financial support from the national science and technology support plan of Peoples Republic of China (No.: 2007BAB23B01) and the key projects of the Beijing municipal science and technology plan (No.: D07040600560701). Thanks are also due to two anonymous referees for their helpful remarks. Nomenclature A, surface area of tubes, m2 c, specific heat capacity, J/(kg K) C, mass fraction of lithium bromide in solution M, mass flow rate, kg/s g, gravitational acceleration, m/s2
2762
J. Sun et al. / Applied Thermal Engineering 30 (2010) 2753e2762
h, specific enthalpy, J/kg Ul, total heat transfer coefficient in cylindrical coordinates, W/(m K) L, tube length, m l, linear coordinate, m N, solution pump power, W P, pressure, Pa Q, heat transfer rate, W T, temperature, C W, mass ratio of water to the mass of pure lithium bromide d, laminar film thickness, m z, dimensionless linear coordinate h, solution pump efficiency y, kinematic viscosity, m2/s 3, total angle of a gravitational flow along tube perimeter, rad q, angular coordinate, rad r, density, kg/m3 J1, total mass transfer coefficient in cylindrical coordinates, kg/(m s) j, dimensionless angular coordinate, j ¼ q=p Re, Reynolds number Subscripts abs, absorber con, condenser gen, generator eva, evaporator SHX, solution heat exchanger hw, heated water dif, diffusional transfer el, electric em, electromechanical eq, equilibrium state es, external surface L, lithium bromide sol, solution s, isentropic v, vapor w, water sl, saturated liquid sv, saturated vapor References [1] A. Bula, H. Diane, F.N. Luis, A.C. Lesme, Thermodynamics Simulation of Solar Absorption Refrigeration Systems, Generator Heat Exchanger. AMSE, France, 2000. [2] G.A. Florides, S.A. Kalogirou, S.A. Tassou, L.C. Wrobel, Design and construction of a LiBrewater absorption machine, Energy Conversion and Management 44 (2003) 2483e2508. [3] F. Assilzadeh, S.A. Kalogirou, Y. Ali, K. Sopian, Simulation and optimization of a LiBr solar absorption cooling system with evacuated tube collectors, Renewable Energy 30 (2005) 1143e1159. [4] W. Wang, R.Z. Wang, Y.X. Xu, Investigation on adsorption refrigeration with a single adsorbent bed, International Journal of Energy Research 22 (1998) 1157e1163.
[5] A. Yokozeki, Theoretical performances of various refrigeranteabsorbent pairs in a vapor-absorption refrigeration cycle by the use of equations of state, Applied Energy 80 (2005) 383e399. [6] D.K. Jesse, G. Srinias, A critical review of models of coupled heat and mass transfer in falling film absorption, International Journal of Refrigeration 24 (2001) 755e797. [7] J.W. Andberg, G.C. Vliet, A simplified model for absorption of vapors into liquid films flowing over cooled horizontal tubes, ASHRAE Transactions 93 (1987) 2454e2466. [8] G. Grossman, M.T. Heath, Simultaneous heat and mass transfer in absorption of gases in turbulent liquid films, International Journal of Heat and Mass Transfer 27 (1984) 2365e2376. [9] S. Jeong, B.H. Kang, S.W. Karng, Dynamic simulation of an absorption heat pump for recovering low grade waste heat, Applied Thermal Engineering 18 (1998) 1e12. [10] M.R. Islam, N.E. Wijeysundera, J.C. Ho, Evaluation of heat and mass transfer coefficients for falling-films on tubular absorbers, International Journal of Refrigeration 26 (2003) 197e204. [11] J.I. Yoon, T.T. Phan, C.G. Moon, P. Bansal, Numerical study on heat and mass transfer characteristic of plate absorber, Applied Thermal Engineering 25 (2005) 2219e2235. [12] D. Bredow, P. Jain, A. Wohlfeil, F. Ziegler, Heat and mass transfer characteristics of a horizontal tube absorber in a semi-commercial absorption chiller, International Journal of Refrigeration 31 (2008) 1273e1281. [13] F. Ziegler, G. Alefeld, Coefficient of performance of multistage absorption cycles, International Journal of Refrigeration 10 (1987) 285e295. [14] G. Grossman, Solar-powered systems for cooling, denhumidification and airconditioning, Solar Energy 72 (2002) 53e62. [15] S.C. Kaushik, A. Arora, Energy and exergy analysis of single effect and series flow double effect waterelithium bromide absorption refrigeration systems, International Journal of Refrigeration 32 (2009) 1247e1258. [16] A. Arora, S.C. Kaushik, Theoretical analysis of LiBr/H2O absorption refrigeration systems, International Journal of Energy Research 33 (2009) 1321e1340. [17] P.K. Bansal, A. Martin, Comparative study of vapour compression, thermoelectric and absorption refrigerators, International Journal of Energy Research 24 (2000) 93e107. [18] G.M. Chen, L. Wang, Q. Wang, M. Zhong, Thermodynamic performance analysis of gas-fired air-cooled adiabatic absorption refrigeration systems, Applied Thermal Engineering 27 (2007) 1642e1652. [19] K.J. Kim, N.S. Berman, S.C. Chaud, Absorption of water vapour into falling films of aqueous lithium bromide, International Journal of Refrigeration 18 (1995) 486e494. [20] T. Nomura, N. Nishimura, S. Wei, Heat and mass transfer mechanism in the absorber of water/LiBr conventional absorption refrigerator: experimental examination by visualized model, in: International Sorption Heat Pump Conference. ASME, New York, 1993, pp. 203e208. [21] S.Y. Jeong, S. Garimella, Falling film and droplet mode heat and mass transfer in a horizontal tube LiBr/water absorber, International Journal of Heat and Mass Transfer 45 (2002) 1445e1458. [22] R.H. Wassenaar, Measured and predicted effect of flow rate and tube spacing on horizontal tube absorber performance, International Journal of Refrigeration 19 (1996) 347e355. [23] M.J. Kirby, B.H. Perez, A design model for horizontal tube water/lithium bromide absorbers, ASME heat pump and refrigeration systems design, Annals of Applied Probability 32 (1994) 1e10. [24] R. Yang, B.D. Wood, A numerical modeling of an absorption process on a liquid falling film, Solar Energy 48 (1992) 195e198. [25] M.J. Powell, A hybrid method for nonlinear equations. in: P. Rabinowitz (Ed.), Numerical Methods for Nonlinear Algebraic Equations. Gordon and Breach, London, 1970, pp. 87e114. [26] K.A. Joudi, A.H. Lafta, Simulation of a simple absorption refrigeration system, Energy Conversion and Management 42 (2001) 1575e1605. [27] H.T. Chua, H.T. Toh, Improved thermodynamic property fields of LiBreH2O solution, International Journal of Refrigeration 23 (2000) 412e429. [28] H.M. Hellmann, G. Grossman, Simulation and analysis of an open-cycle dehumidifiereevaporatoreregenerator (DER) absorption chiller for low grade heat utilization, International Journal of Refrigeration 18 (1995) 177e189.