1. Introduction
The most common configuration used in the absorber of absorption cooling systems is that of falling-film on horizontal tubes. The absorbent solution entering the absorber is distributed over the tubes via the solution distributor. A thin solution film falls over the external surface of the horizontal tubes and absorbs the refrigerant vapour. Absorption heat is released by cooling water flowing inside the tubes. Absorption cooling systems are environmentally appropriate alternatives to the vapour compression systems for space conditioning and can achieve high coefficients of performance (COP) when multi-effect configurations of the cycle are used. However, the success of this technology depends on the development of compact and high-performance heat and mass exchangers which, in turn, depend on the processes of heat and mass transfer with a phase change, and on the working fluid [
1].
Several theoretical and experimental investigations in the literature are focused on the absorption process and the development of optimal designs. Amaris et al. [
2] reviewed the experimental studies dealing with the enhancement of vapour absorption processes in absorbers employing passive techniques such as advanced surfaces, additives, and nanofluids. This review included an exhaustive and detailed scrutiny of the absorption process with different configurations of the absorber and different working fluids. The authors documented experimenting techniques, operating conditions, and the latest advances in terms of heat and mass transfer enhancement in absorbers.
However, the detailed analysis of heat and mass transfer processes in horizontal tube falling-film absorbers is complex. The operation of these absorbers depends on many parameters, such as the thermodynamic and transport properties of the working fluid, the operating conditions, the design parameters, i.e., diameter, number, and length of the tubes, and the surface wettability, among others.
The development of mathematical models that describe the absorption process in falling-film absorbers has been the objective of numerous theoretical investigations. Killion and Garimella [
3] conducted a review on mathematical models that had been used to describe the simultaneous heat and mass transfer processes in horizontal tube falling-film absorbers. In the simplest models of this type of absorbers, the film formation is assumed to be in a laminar flow regime. This flow regime is developed at Reynolds numbers below 200, while a completely turbulent regime is evidenced at Reynolds numbers above 1600 (Killion and Garimella [
3], Grossman [
4]). The results obtained by these models are not precise, due to simplified assumptions made to determine the heat and mass transfer coefficients. Moreover, a series of instabilities cause the deviation of the falling-film flow from the laminar behaviour, even at low flow rates. For this reason, many investigations contemplated a turbulent flow regime for the falling-film. Recent studies have shown various flow structures, such as droplet formation, release, and impact, and film waves, any or all of which may affect the distribution of the liquid and, consequently, heat and mass transfer (Nagavarapu and Garimella [
1], Killion and Garimella [
5], Bustamante and Garimella [
6], and Bohra et al. [
7]).
Nagavarapu and Garimella [
1] developed a heat and mass transfer model for falling-film ammonia-water absorption over horizontal microchannel tubes. The model was validated using experimental data. Three regions in the absorber were considered for vapour absorption, namely, the solution pool, evolving droplets, and falling-film. The authors concluded that vapour absorption occurs mainly in the film region, and up to 7% of the absorption rate takes place in the droplets. Heat transfer coefficients were estimated in the film region, and an empirical correlation was developed for the film Nusselt number.
In experiments, using a high-resolution video, Killion and Garimella [
5] investigated the phenomenon of droplet formation in a water/LiBr falling-film absorber. The experimental absorber was formed by a single column of nine internally cooled tubes. The authors described the characteristic droplet evolution pattern, including axial elongation along the tube, the formation of a primary droplet, trailing liquid thread, and satellite droplets, and the formation of saddle waves caused by the spreading lamellae of the primary droplet impacts. This information is valuable for modelling the absorption process in horizontal tube banks as well as for validating computer models of droplet formation.
Bustamante and Garimella [
6] studied the effect produced by the design of the flow distributor used in horizontal tube banks of falling-film heat and mass components. Flow configurations generated by the distributor, from droplet to sheet regimes, droplet surface areas and volumes, and jet diameters, were all measured using high-speed videos. The authors observed significant maldistribution (>50%) for some flow rates in all the designs tested.
Bohra et al. [
7] investigated flow patterns on a horizontal-tube falling-film absorber of an ammonia-water absorption system. High-speed videos showed the droplet model to be the dominant inter-tube flow mode at the operation conditions of their experiments. However, droplet characteristics and flow transition depend strongly on the solution flow rate.
Some investigations have focused on hydrodynamics in falling-films and incorporated the analysis of heat and mass transfer (Kirby and Perez Blanco [
8], and Jeong and Garimella [
9]).
Vertical falling-film absorbers were used in most of these falling-film absorber models (Wassenaar and Westra [
10], Patnaik and Perez-Blanco [
11], and Patnaik et al. [
12]), together with water/LiBr as the working pair.
When non-volatile absorbents, such as LiBr, are used and nonabsorbable gases are not present in the absorber, the vapour phase is formed only by the pure refrigerant, and this implies that there is no resistance to mass transfer in the vapour phase. Heat and mass transfer in a differential control volume can be described by the equations of energy and mass conservation balances applying in two spatial dimensions and one time dimension. The coupling of the equations of heat and mass transfer is conducted through the boundary conditions at the interface between the liquid and the vapour [
3]. In most cases, the researchers assume that the liquid–vapour equilibrium prevails at the interface.
The review, reported by Bohra et al. [
7], showed that the study of flow patterns and heat and mass transfer in falling films on tube bundles has been conducted mainly on individual tube column geometries in adiabatic conditions. The few studies that address the coupled mechanisms of heat and mass transfer are restricted to the working fluid water/LiBr. Different theoretical models of falling-film on horizontal tubes were developed to predict the performance of the absorber. Seewald and Perez-Blanco [
13] considered the formation of three flow regimes in falling-film absorbers and developed a spiral absorber model that took into account the droplet-formation regime.
Killion and Garimella [
3] highlighted the models that describe heat and mass transfer processes in falling-film absorbers assuming the heat and mass transfer coefficients. These semi-empirical models allow for some assumptions to be made on the hydrodynamics of the falling-film and provide reasonable results due to the appropriate representation of physical phenomenon.
Kirby and Perez-Blanco [
8] developed a model for simultaneous heat and mass transfer processes occurring in a horizontal tube falling-film absorber using water/LiBr as a working fluid. Their absorber consisted of a bundle of 6 coolant tubes per row, where the cooling water and solution circulated in a counter-current direction. The operating conditions used in the model were those of commercial absorption chillers, i.e., absorber pressure 5.75–7.00 mmHg, solution inlet concentration 60–62 wt%, and Reynolds number 13–98. The absorber was divided into three flow regimes, namely, (i) droplet-formation flow, (ii) droplet-fall flow, and (iii) falling-film flow, considering non-wavy-laminar falling-film, complete wetting of the tubes, and droplet flow mode between tubes. The model used empirical heat and mass transfer coefficients and incorporated the effect of the flow regimes that the solution experiences as it flows down through the tube bundle. The model was validated using experimental data published by Nomura et al. [
14] and was well in agreement.
Jeong and Garimella [
6] developed a model for a water/LiBr horizontal tube absorber to predict heat and mass transfer performance in falling-film and droplet mode flow regimes. The absorber configuration consisted of a bank formed by 13 horizontal tubes in counter-current flow. The model was also validated using the experimental data reported by Nomura et al. [
14]. The effect of incomplete wetting was dealt with by introducing the wetting ratio defined as the ratio between the wettened and total areas. The authors concluded that vapour was mainly absorbed in the falling-film and droplet-formation flow regimes, that heat and mass transfer in the free-fall flow regime was negligible, and that the wetting ratio had a significant effect on the absorber performance.
Juarez-Romero et al. [
15] developed a computational model to characterize heat and mass transfer in a horizontal tube falling-film absorber integrated into a heat transformer used for water purification. The absorber consisted of a bundle of 16 tubes (4 per row) internally heated by the water to be purified. The absorber was designed to supply heat at high temperature for water purification. The model was validated using data reported by Holland et al. [
16] and contemplated the flow regimes proposed by Kirby and Perez-Blanco [
8]. It correctly predicted solution and heating water temperature profiles along the absorber.
Furthermore, it is well-known that the conventional working fluid water/LiBr cannot satisfactorily reach the operating conditions required in advanced configurations of absorption cooling cycles driven by high-temperature heat sources. Hence, several investigations have been conducted to identify other possible absorbents, capable of operating at high temperatures. The ternary solutions of alkaline nitrates (LiNO
3 + KNO
3 + NaNO
3, in the mass ratio 53:28:19), known as Alkitrate and originally proposed by Davidson and Erickson [
17], have appeared as a possible substitute to the conventional working fluid water/LiBr. They make better use of high-temperature heat sources, without presenting any corrosion or thermal stability issues. Álvarez et al. [
18] investigated a triple-effect absorption cooling cycle called “the Alkitrate topping cycle” using aqueous nitrate solution as a working fluid. The coefficient of performance (COP) of this cycle was higher than that of a water/LiBr triple-effect absorption cycle at heat source temperatures over 180 °C. The Alkitrate cycle represents a relevant alternative to capitalize on the thermal potential of high-temperature heat sources. Hence, it is of great importance to investigate absorber performance at unusual operating conditions, at which conventional working fluid water/LiBr does not prove viable because it is subject to corrosion and thermal instability.
The objective of the present work is to develop a simplified model for a falling-film absorber on horizontal tubes in order to provide additional information that could help with a better understanding of the absorption process of the alkaline nitrate solution LiNO
3 + KNO
3 + NaNO
3 used as a working fluid and applied in the mass proportions 53:28:19, respectively. The absorber operating conditions are those of the last stage of a triple-effect absorption cycle powered by high-temperature heat sources. The model facilitates the study of heat and mass transfer processes, and the effect of flow regimes that characterize falling-film absorbers. Therefore, this model is a useful tool for investigating absorber performance versus solution and cooling water operating conditions as well as the geometrical parameters of the absorber. The theoretical model developed in the present work is based on the investigations conducted by Juárez-Romero et al. [
15] and Kirby and Perez-Blanco [
8] and was validated with experimental data obtained by the authors in a previous study [
19]. The governing equations, assumptions, solution methodology, and results are presented herein.
2. Operating Conditions of the Falling-Film Absorber
The water vapour (refrigerant) absorption process in a falling-film of an aqueous alkaline nitrate solution (absorbent) is investigated in the present work. The absorber configuration is based on a previously designed and built experimental prototype [
20] and is shown in
Figure 1. The design of this absorber was conducted under the operating conditions established by Álvarez et al. [
18] for a triple-effect absorption cooling cycle.
The absorber is made up of a cylindrical chamber that contains a bundle of six copper tubes in line and connected in series, a solution distributor at the inlet, and a solution collector tray at the outlet of the absorber. The alkaline nitrate solution circulates over the external surface of the tubes from the top to the bottom of the absorber, creating the falling-film, and it makes contact with the vapour coming from a water vapour generator. At the outlet of the absorber, there is a more diluted solution flow than that at the inlet because the solution has absorbed the water vapour.
Figure 2 presents a schematic diagram of the absorber and flow configurations. The falling-film formed on the external surface of the tubes of the absorber is cooled down by the water that flows through the tubes that make up the absorber, and in a counter-current direction to the solution flow. The objective of the cooling water is to dissipate the heat released during the absorption process, thus, maintaining the driving potential for mass transfer.
The operating conditions considered in the present work were obtained from those used in a previous experimental investigation carried out by the authors on the same absorber [
19]. This experimental work consisted mainly of a sensitivity study of the absorber operating variables and took into account a series of absorber efficiency parameters, such as absorption rate, thermal load, solution concentration difference between the entrance and exit of the absorber, subcooling degree of the solution leaving the absorber, and heat and mass transfer coefficients. Data obtained from the experimental work were used to validate the mathematical model for the falling-film absorber developed in the present work.
Table 1 and
Table 2 summarize, respectively, the absorber geometric parameters and the inlet operating conditions used for the absorber model. A base mass concentration of 82% for the nitrate solution and an absorber operation pressure of 30.0 kPa were selected. The cooling water temperature and the solution flow rate at the inlet of the absorber were varied in order to study their effect on the absorber performance and the validity of the model to predict outlet operating conditions.
It is important to note that, in order to study the effect of each variable, the rest of the operating conditions were maintained constant. However, some variables of the study are closely related. Therefore, certain criteria were established that permitted comparison between the different tests carried out at the same operating condition baseline. The temperature and concentration of the solution at the inlet of the absorber influence the absorption process, since they determine deviation with regard to equilibrium, and this affects the driving force for heat and mass transfer processes, as well as the thermodynamic and transport properties of the solution. It is worthy of note that when the solution is subcooled to a high degree when it enters the absorber, it enhances mass transfer at the absorber entrance.
3. Mathematical Modelling of the Absorber
In the absorber, the falling-film is formed on the external surface of each tube and this is sustained by the continuous flow of solution that comes in through the top part of the absorber and falls due to gravity. The cooling water that circulates inside the tubes of the absorber is perpendicular to the flow of the solution; therefore, the model developed is two-dimensional (2D).
Moreover, the falling-film experiences changes in the flow configuration when the solution flows from one tube to another. When droplets are formed, the surface of the exposed solution increases, whereby the available area for mass transfer increases and, consequently, improves vapour absorption [
21].
The three flow regimes that are present in the falling-film absorber on horizontal tubes (see
Figure 3) are [
8]:
Falling-film flow regime, which occurs on the external surface of each tube.
Droplet-formation flow regime, which occurs at the bottom of each tube.
Droplet-fall flow regime, extending from one tube until it reaches the next one.
The falling absorbent solution experiences each one of these flow regimes successively as it flows from one tube to the other. The mathematical model developed here integrates these different flow regimes into mass and energy balances and these are used to describe the absorption process of the absorber.
It is important to highlight that the flow regime, i.e., droplets, columns, and leaves, that describes the flow of the solution on the tubes of the absorber, is dependent on the solution mass flow rate (Γ). In most absorbers, the solution mass flow rate is controlled so as to achieve droplet formation between tubes.
The average volume of the droplets formed on the bottom part of each tube is described by the following expression [
22]:
where V
droplet is the average droplet volume, g is the gravity constant, and σ and ρ are the surface tension and the density of the solution, respectively.
Moreover, Hu and Jacobi [
23] suggested that when the solution film flows between the tubes in a droplet flow regime, the value of the Reynolds number of the film is lower than the value obtained from the following Equation:
where modified Galileo dimensionless number (Ga) is defined as:
where
is the dynamic viscosity of the solution.
In another study, Hu and Jacobi [
24] proposed the following expression for the average distance between the nucleus of solution droplet formations (D
ND) (see
Figure 3):
where d
o is the external diameter of the tubes, and ξ is the length of the capillary tube defined by the following expression:
3.1. Residence Time
Due to the differences in the transport phenomena produced during the absorption process, the residence time of the solution in each flow regime is an important parameter in absorber modelling. Firstly, the average residence time of the solution in the droplet-formation flow regime is calculated from the following Equation [
8]:
where m
droplet is the droplet mass, Γ is the solution flow divided by twice the length of the tube, and N is the number of droplets per unit of length of the tube. The number of droplets can be calculated from the following Equation [
8]:
Residence time in the droplet-fall flow regime, t
fall, is calculated using a simple freefall expression [
8]:
where S
T is the distance between the tubes, and g is the gravity constant.
In the falling-film flow regime, the residence time of the solution can be calculated from the rate of descent and the film thickness. Assuming that the falling-film flow is laminar, Nusselt’s equation is applied for the film thickness, δ:
where θ (in rad) is the angle describing the solution film in relation to the centre of the tube (see
Figure 4). The descent velocity of the film (u) is calculated with the following expression:
where δ is the film thickness.
The total residence time of the falling-film (t
film) is defined by the following Equation [
8]:
In the equation above, ro is the external radius of the absorber tubes.
3.2. Governing Equations for the Different Flow Regimes
The Equations that describe each one of the flow regimes were developed by Kirby and Perez-Blanco [
8] and Juárez-Romero et al. [
15]. Below, the governing equations for each one of the three solution flow regimes are presented.
3.2.1. Droplet-Formation Flow Regime
This flow regime takes place at the bottom of each tube. The droplet formation process is complex, and the most difficulty is encountered in defining the interface conditions of the droplet when its formation occurs. In this study, the equations describing heat and mass transfer of the droplet-formation flow regime are based on the assumption of a “fresh” surface [
25] that permits the identification of interface conditions during this regime. In this manner, it is assumed that the droplet is formed by a successive series of thin spherical layers [
8]. The fresh or new solution that joins the droplet forms a thin spherical layer on top of the bulk of the old solution. In addition, the droplet formation process is assumed to be adiabatic. Below are the expressions for the mass and energy balance equations.
Mass balance: the flow of absorbed vapour in this regime is calculated from the following expression:
where, k
form is the mass transfer coefficient for the droplet-formation flow regime, r
droplet is the radius of the droplet, X
s is the mass concentration of the solution coming from the falling-film, and X
i is the mass concentration at the interface, both in % of the absorbent. The interface conditions are calculated assuming that the temperature at the interface is the same as the temperature of the film that will form the new droplet. The interface concentration is obtained from the liquid–vapour equilibrium:
Once the vapour absorption flow rate has been defined, the final droplet mass is calculated by adding the calculated flow of absorbed vapour to the initial droplet mass.
Energy balance:
where Q is the heat transferred from the nucleus of the old and hot droplet to the new and cooler spherical layer, and q
abs is the absorption heat. The interface condition of each new layer is calculated assuming that the temperature of the droplet interface is always the same as the temperature of the fresh solution that forms the droplet. Afterwards, it is assumed that the superficial layer adheres to the droplet and forms the nucleus.
The absorption heat, q
abs, and the heat, Q, are calculated as follows (Juárez-Romero et al. [
15]):
where h
v and h
s are the specific enthalpy of saturated vapour and the specific enthalpy of the solution respectively, and λ
s is the thermal conductivity of the solution. The thermophysical properties (ρ, µ, Cp, h, λ, σ) of the alkaline nitrate solution were taken from the experimental database and empirical correlations obtained by Alvarez [
20] from the literature.
3.2.2. Droplet-Fall Flow Regime
In the droplet-fall flow regime, the heat and mass transfer equations as well as the mass and energy balances applied to a droplet [
8] are written in the same way as in the droplet-formation flow regime. In this model, the effects of any internal circulation that may occur inside the droplet are not considered. Most falling-film absorbers on horizontal tubes have a small spacing between the tubes, so the falling time of the droplets is short, and consequently, the effect of the internal circulation inside the droplet is reduced [
8]. The governing equations are described below:
Energy balance:
where k
fall is the mass transfer coefficient in the droplet-fall flow regime, X
s and X
i are the absorbent concentrations in the bulk of the solution and at the interface respectively, and q
abs is the heat absorption.
In order to calculate the interface conditions of the droplet, a linear temperature profile is assumed in the case of this flow regime, since the residence time of the droplets in the spacing between the tubes is short, and the relative importance of this flow regime does not justify a more complex model [
8]. This linear temperature profile is defined as:
Equation (19) and the concentration–temperature relation at the interface (Equation (13)) define the interface conditions in this regime.
3.2.3. Falling-Film Flow Regime
In the falling-film flow regime, the heat and mass transfer equations and mass and energy balances applied to a thin film surrounding the horizontal tube are written as follows [
8]. The energy balance is performed both on the solution side and on the cooling water side.
Energy balance (solution side), assuming a stationary state:
Energy balance (cooling water side):
where m
s is the mass flow rate of the solution, m
c is the mass flow rate of the cooling water, U is the overall heat transfer coefficient, Cp
c is the specific heat of the cooling water, and z and θ are the horizontal and angular coordinates of the absorber (see
Figure 2 and
Figure 4), respectively.
In Equation (21), the overall heat transfer coefficient, U, is calculated using the following expression:
where h
s and h
c are the heat transfer coefficients of the falling-film and the cooling water respectively, λ
Cu is the thermal conductivity of copper (396.4 W·m
−1·°C
−1), and d
o and d
i are the external and internal diameters of the tubes, respectively.
The Nusselt number and the heat transfer coefficient (h
c) on the cooling water side were calculated using the Dittus-Boelter correlation, which is appropriate for turbulent flow in tubes:
where λ
c is the thermal conductivity of water (0.67 W·m
−1·°C
−1), and Pr
c is the Prandtl number of water.
The heat transfer coefficient for the falling-film solution, h
s, was obtained by Alvarez et al. [
26] by means of the artificial neural network methodology. The resulting equation uses 6 variables, namely, absorber pressure (P
abs), solution temperature at the inlet of the absorber, cooling water temperature at the inlet (T
c,in), solution mass concentration at the inlet of the absorber (X
s,in), mass flow of cooling water (m
c), and solution mass flow rate per tube length (Γ). A root mean square error (rmse) of 1.183 was attained on prediction of the convective coefficient, h
s.
where I and J are the numbers of inputs (I = 6) and neurons in the hidden layer (J = 9); IW and LW are the weight matrices in the hidden and output layers, respectively. P
i is the input “i” normalized, and b1 and b2 are the bias vectors in the hidden and output layers, respectively.
A learning model was applied by Alvarez et al. [
26] using the performance parameters for the horizontal falling-film absorber. They obtained the best adjustment in weights (IWj,i and LWk,j) and biases (b
1j and b
2k) for the neural network structure formed by a one-layered neural network with 9 neurons in the hidden layer.
Because of the differences in hydrodynamics of each region, each one of the flow regimes considered in the model must be characterized by a different mass transfer coefficient. In this study, empirical mass transfer coefficients were not obtained for each one of the flow regimes, however, the approach reported by Jeong and Garimella [
9] was used for the calculation of these mass transfer coefficients.
Moreover, Andberg and Vliet [
27] also proposed the use of a linear temperature profile in the solution film. Therefore, a linear temperature profile is assumed (Equation (25)) to calculate the interface conditions in this flow regime.
Mass balance (solution side):
The amount of vapour absorbed at the interface is determined by the following Equation:
where k
film is the mass transfer coefficient for the falling-film regime, and X
s and X
i are the concentrations in % in the bulk of the solution and at the interface, respectively.
Jeong and Garimella [
9] detailed the investigation presented by Skelland and Minhas [
28], in which they compared mass transfer coefficients for the droplet-formation flow regime, reported by different researchers and then proposed the following generic Equation:
where D is the mass diffusivity of the solution, t
form is the residence time in the droplet-formation flow regime, and C
2 is a constant. Heertjes et al. [
29] suggested a C
2 value equal to 24/7 for the cases in which the droplet-formation is fast (t
form lower than 1.05 s). In addition, the fast formation of droplets facilitates internal circulation, and therefore, diffusion velocity is lower than droplet growth velocity [
9]. In this study, t
form is lower than 1.5 s, and that is why the C
2 constant suggested by Heertjes et al. [
29] was considered appropriate.
Jeong and Garimella [
9] also recommended the mass transfer coefficient proposed by Clift et al. [
25] for the droplet-fall flow regime:
where D is the mass diffusivity coefficient of the solution, and d
droplet is the droplet diameter.
For the falling-film flow regime, the mass transfer coefficient can be calculated using the film theory suggested by Whitman [
30] in the following expression [
9]:
where D is the mass diffusivity coefficient of the solution and δ is the thickness of the falling-film.
3.3. Assumptions
To develop the differential equations that describe the dynamic behaviour of an absorption process in a falling-film absorber on horizontal tubes, the following assumptions and equilibrium considerations were taken into account [
15]:
Absorption process in the tube bundle is described using a bi-dimensional approach (2D).
Working fluid is Newtonian and the solution flow is laminar.
Absence of non-condensable gases in the vapour phase.
Pressure drop in the absorber is insignificant.
Heat transfer from the absorbent solution to the vapour is insignificant, hence heat transfer on the vapour side is negligeable for the three flow regimes.
Absorbent solution completely covers the tube surface, that means wetting of the tubes is perfect.
Tubes have a smooth surface.
Thermodynamic and transport properties depend on the temperature and concentration conditions (T, X) at each step of the calculation sequence.
Liquid–vapour equilibrium is assumed to be present at the interface.
Mass transfer coefficients for each flow regime are obtained from the correlations proposed by Jeong and Garimella [
9].
Mass diffusivity coefficient, D, is approximated to the mass diffusivity of the LiBr aqueous solution proposed by Wike-Chang [
31].
4. Calculation Procedure
The model employed for horizontal tubes falling-film absorbers assumes that the tubes are divided into several segments over which the solution film falls (see
Figure 2). The calculation code was developed in the Matlab
® program environment and based on the calculation sequence reported by Juárez-Romero et al. [
15] and Kirby and Perez-Blanco [
8]. The resolution procedure for the ordinary differential equations, which govern the 2D behaviour of the absorber, used the Runge-Kutta fourth-order method. The equations governing each one of the three flow regimes were combined to model the whole absorber. In addition, the integration of the flow regions in the absorber was achieved by making the solution flow and the inlet and outlet conditions coincide when the solution goes from one regime to the next.
In order to solve the governing equations, the absorber was divided into different segments (see
Figure 5). All flow regimes considered, i.e., droplet-formation flow, droplet-fall flow, and falling-film flow, take place in each segment. The absorbent solution flows from the top section of each segment to the bottom, while the cooling fluid flows horizontally.
The calculation code requires the following input data: inlet temperatures of the solution and cooling water, solution mass flow rate, absorber operating pressure, inlet concentration of the solution, and certain absorber design parameters, such as tube number, tube diameter, tube length, and spacing between tubes. The equations were solved step by step for each segment of the absorber. The solution and cooling water conditions of each segment were used to connect the absorber segments to those precedent and adjacent to them.
The solution flows from row to row, combining the conditions of the absorbent solution, leaving the top segment with the solution entering into the bottom segment. When the energy and mass balances for each flow regime in each segment are performed, the whole calculation sequence is applied to the next segment. In this way, the solution properties resulting from the flow over the segments of the upper tube are used as inlet solution properties for the corresponding segments of the next tube. Additionally, the cooling water conditions are coupled with the adjacent segments. The calculations are repeated for each tube segment so that the properties of the solution at the outlet are determined. The calculation sequence of the model is illustrated in
Figure 6.
It is worthy of note that experimental data available was limited to cooling water temperature, and solution temperature and concentration at the inlet and outlet of the absorber. Therefore, it was not possible to validate local profiles of the solution temperature and concentration along the tube bundle.
6. Conclusions
The most significant conclusions, made from the present study, are summarized as follows:
The mathematical model developed integrates three flow regimes that characterize the absorption process in horizontal tube falling-film absorbers. These flow regimes are the droplet-formation flow regime, which occurs at the bottom of each tube making up the absorber, the droplet-fall flow regime, which takes place between the tubes, and the falling-film flow regime, which forms on the surface of the tubes. The mathematical model also includes the thermodynamic and transport properties of the aqueous alkaline nitrate solution. The model developed is able to assess the effect of these flow regimes on the operation and performance of an absorber.
The mathematical model predicts the temperature and concentration profiles of the alkaline nitrate solution, as well as the cooling water temperature profile, along the tubes of the absorber. Data concerning the temperature and concentration of the solution leaving the absorber and temperature at the cooling water outlet were predicted by the mathematical model. This data was then compared with experimental data obtained by Alvarez and Bourouis [
19], who had used the same absorber design in a previous experimental work. The results of the two sets of data concurred closely. The average deviations for the solution temperature, solution concentration, and cooling water temperature were 1.1%, 0.6%, and 1%, respectively.
As regards the temperature and concentration profiles of the alkaline nitrate solution and the temperature profile of the cooling water circulating along the absorber tube bundle, there is no data available in the open literature that could allow for the validation of these local profiles. However, the trends of these profiles were compared with those reported in the literature (Nomura et al. [
14], Kirby and Perez-Blanco [
8], Jeong and Garimella [
9]) using similar absorber configurations and working with the conventional fluid mixture water/LiBr. All sets of data showed similar trends to those obtained in the present work.
The total residence time of the solution in the absorber was 6.6 s. The residence time of the droplet-fall flow regime was negligible, and therefore, the main heat and mass transfer contributions were made by the falling-film and droplet-formation flow regimes. The model predicted that the solution temperature will increase by an average of 0.9 °C per tube and that the solution concentration will decrease by an average of 0.6% per tube. Additionally, the solution temperature experienced a drop of 1.8 °C in the falling-film flow regime and an increase of 1 °C, on average, in the droplet-formation flow regime. As regards the solution concentration, an average decrease of 0.21% and 0.34% per tube was observed in the falling-film flow and droplet-formation flow regimes, respectively.
The mathematical model developed also predicted the absorption rate at 4.7 g·m−2·s−1 for the absorber design and operating conditions used in the present work. This value is 22% higher than the value obtained by the authors in their previous experimental work. This deviation is attributed to the approximations made in the model, especially as regards surface wettability and the calculation of mass transfer coefficients for each flow regime.