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CFD simulation of water vapor absorption in laminar falling film solution of water-LiBr ─ Drop and jet modes
Accepted Manuscript CFD simulation of water vapor absorption in laminar falling film solution of water-LiBr ─ Drop and Jet modes Seyed Mojtaba Hosseinnia, Mohammad Naghashzadegan, Ramin Kouhikamali PII: DOI: Reference:
S1359-4311(17)30145-X http://dx.doi.org/10.1016/j.applthermaleng.2017.01.022 ATE 9786
To appear in:
Applied Thermal Engineering
Received Date: Revised Date: Accepted Date:
5 July 2016 16 November 2016 8 January 2017
Please cite this article as: S. Mojtaba Hosseinnia, M. Naghashzadegan, R. Kouhikamali, CFD simulation of water vapor absorption in laminar falling film solution of water-LiBr ─ Drop and Jet modes, Applied Thermal Engineering (2017), doi: http://dx.doi.org/10.1016/j.applthermaleng.2017.01.022
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CFD simulation of water vapor absorption in laminar falling film solution of water-LiBr ─ Drop and Jet modes
Seyed Mojtaba Hosseinnia, Mohammad Naghashzadegan, Ramin Kouhikamali Department of Mechanical Engineering, Faculty of Engineering, University of Guilan, Rasht, Iran
Abstract: The rate of absorbed water vapor over a horizontal tube bank in a water-cooled falling film absorber depends on several parameters and working conditions. The flow regime between the tubes in falling film absorbers is one of the most influential parameters in the amount of the absorbed vapor. Among different falling film regimes, in this study, the drop and jet modes are simulated numerically. The full Navier-Stokes equations are solved and the well-known volume of fluid (VOF) method is used to capture the gas/liquid interface. In addition, the energy and diffusion equations are solved in this 3Dsimulation with aid of an in-house CFD code. Adaptive mesh refinement, in accordance with the magnitude of volume fraction gradient, has strongly improved the interface capturing and therefore increased the accuracy of simulation. The simulation results reveal that by changing just the regime from drops to jets, the rate of average vapor mass flux decreases one order of magnitude, from to . Keywords: Numerical simulation, Falling film absorber, Drop mode, Jet mode, VOF 1. Introduction Application of absorption machines which are capable of using a low-grade heat source, are growing in the recent years [1-4]. The absorption technology is less harmful to the environment specially by using water as refrigerant and aqueous LiBr or LiCl as the absorbing media in the domestic air conditioning applications [5]. Absorber heat exchanger is the most important component in the absorption machine which has a great effect on the whole cycle efficiency and initial system costs [6]. In a conventional falling film absorber the binary solution, reach in a salt, falls by gravity on several horizontal cooling tubes. Because of working on the same pressure, the absorber and the evaporator are placed in one chamber. Therefore the vapor of refrigerant, coming from the
Corresponding author. Tel.: +09813 33690273; fax: +98 13 33690273. E-mail address: [email protected] (R. Kouhikamali)
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evaporator, is allowed to move to the absorber side freely. Thus the entire absorber chamber is filled with the vapor of refrigerant in its saturated state, approximately [7]. The vapor is absorbed on the interface of aqueous LiBr and releases the heat of absorption. A reliable absorption process takes place by removing the heat of absorption. Hence the cooling media, generally liquid water, flows inside the horizontal tubes and cools the solution. Several important parameters play roles in whole absorption process inside the absorber. Some of them are related to geometry and affect the hydrodynamic of falling film such as tube diameter and spacing [8]. Some others modify both hydrodynamic and heat/mass transfer characteristics with respect to thermophysical properties such as density, surface tension, and viscosity [9-10]. In addition, the rate of pouring binary solution which falls on the horizontal tubes is the other important component. A compromise among mentioned parameters dictates the flow regime between the tubes; including drop mode, jet mode and sheet mode with respect to increase in the binary solution mass flow rate. Hence, in fixed absorber geometry, with certain working pair – in this study water/LiBr– the only parameter which specifies the flow regime is the solution mass flow rate per the length of the horizontal tubes. Modeling of absorption process over horizontal-cooling tubes is a very interesting area for researchers [9-11]. Two important aspects of this modeling can be classified as hydrodynamics of the falling film and the heat/mass transfer. There are several simplifying assumptions in both domains. From hydrodynamic point of view, the simplifications are using Nusselt’s velocity profile [12] around the tube in Refs. [13-16], which gets infinite film thickness at both top and bottom of the tube and this velocity profile, cannot predict the laminar waves generated from impact of drops on the tube surface. Moreover, there are some assumptions about drop formation and fall; under and between tubes, by assuming a 3D phenomenon as a 2D one. On the other hand, from the heat/mass transfer aspects, employing correlations for the heat/mass transfer coefficients make other simplifications [17-19]. Many researchers have worked on the hydrodynamic of the falling film both experimentally and numerically. Hu and Jacobi [20-22] have studied the different aspects of the falling film mode transitions from dropwise to sheet mode experimentally. They showed that the transition mode are related to modified Galileo number (Ga) and film Reynolds number (Ref) in the form of Ref=AGaB where the constants (A and B) are adjusted with respect to experimental data. Roques et al. [23-24] have investigated the transition modes in the hydrodynamics of the falling film on horizontal low-finned tubes with different fin spacing and configuration. In their experimental study, they showed the beginning of transition becomes advanced in comparison to plain tubes. Even recently Wang and Jacobi [25] showed a thermodynamics-based analysis for prediction of the different transitions in the flow regime. They assumed each flow mode is a thermodynamic system which can be in equilibrium with the other modes by 2
considering several simplifying assumptions. These assumptions presuppose that the system is in isothermal and adiabatic conditions which mean no heat or mass transfer. However in considering just hydrodynamics, using a thermodynamic-based analysis is a simple and quick prediction. In addition to experimental and even theoretical investigations, the hydrodynamics of falling film has been studied numerically in different approaches. Killion and Garimella [26] presented a comprehensive review of numerical history in modeling two phase gravity driven falling film with VOF model and fluent commercial CFD package. In their transient simulation, the droplet creation below a horizontal tube, its growth and fall and the wave generated by its impact on the next tube is illustrated. The 3-D numerical result is validated perfectly with respect to experimental results [27]. Chen et al. [28] simulated numerically the falling film transition mode on the horizontal tubes by means of finite volume solver, Fluent. They used volume of fluid technique to capture the phase’s interface and hydrodynamics of falling film. By comparison with test data, they showed a good agreement between the numerical results and experimental data. Thus far, the hydrodynamic aspect of falling film is briefly reviewed; the heat/mass transfer point of view will be presented in the followings. As it was remarked earlier, heat and mass transfer in absorption process are coupled together. The driving force in the mass transfer is the difference between bulk concentration in the falling film and concentration of a very thin layer between gas and liquid phase; the interface. In other words, water vapor absorbs at the interface immediately and because of a very low mass diffusivity characteristic of the vapor in the binary solution, , water makes a low concentration area at the interface. The concentration at interface is an unknown boundary condition in absorption problem. The absorbed vapor brings the heat of absorption with itself which is the difference between the enthalpy of the vapor phase and the enthalpy of water in the binary solution. Hassanvand and Hashemabadi [29] used the VOF model to simulate the interphase mass transfer process in stratified gas-liquid flow between two parallel plates. Their results were validated by comparison with two benchmark mass transfer mechanisms and they concluded there is a precise agreement. Subramaniam and Garimella [30] simulated the drop mode falling film absorption numerically with VOF technique. They considered a section of tube with constant surface temperature. By assuming the chemical potential equality at the interface between water vapor and water in the aqueous LiBr, they computed the interface concentration in an iterative manner using Newton-Rophson algorithm. Moreover they considered the effect of laminar waves produced in drop impact on the pipe and showed this laminar wave mixing effect in the absorption process. This study focuses on numerical simulation of combined heat and mass transfer processes in an absorber in the drop and jet modes falling film pattern between horizontal tubes. To have a comparison between the obtained numerical results and the experimental 3
ones, the computational domain such as diameter of pipes and pipe spacing has been chosen in accordance with Kyung et al. [16] research. 2. Problem geometry and mesh design Fig.1 shows a cross-section view of a typical horizontal-tube water cooled absorber. The hot and strong solution of water and LiBr enters to the absorber via several feeders and falls on a column of horizontal cooling tubes. The end of each cooling tube may be connected to the next tube beneath it to make a serpentine tube (not showed in Fig.1) or to a vertical pipe which distribute the cooling water. The cooling water flows through these pipes from the bottom. At the same time, water vapor flows steadily to the absorber chamber in its saturated condition (approximately). For simulating in the current study, three cooling tubes below the feeder are chosen as the computational domain. Fig. 2 shows the computation domain. Each tube has an outer diameter of 19.05 mm and the spacing between them is 24.5 mm. In this simulation, a section of 0.02 m of each of the three tubes is determined to be the tube length in –z direction. This value is chosen according to the relation proposed by Young et al [31]; Eq. (1). Where is the space between drops or jets falling between horizontal tubes, is the surface tension coefficient at vapor/aqueous LiBr, and is the density of liquid. n is considered to be 2 for water vapor and aqueous LiBr [31].
A square of 1 1 mm2 at the top center of the tubes (2 mm above the first tube) plays the role of the feeder or distributer as a perforated tube or plate [8]. The entire computational domain is meshed with hexahedral structured cells. Fig. 2 illustrates a detail view of designed base meshes; cells in area of falling solution around the tubes perimeter are denser in contrast to the other area. This is mainly because the heat transfer between the binary solution around the wall of the pipes takes place in this area. The minimum cell size is around the wall of the cooling pipes. This minimum cell size (as base-mesh size) has selected after a grid independency analysis was done. The shape of interface has not changed significantly by the change in the grid size. But the film thickness around the tube is very small (around 0.2 mm) and having more grids in the falling film around the tubes will enhance more accurate results. Therefore, the bulk values of temperature and concentration for the first tube at and in drop mode with respect to variations in gird sizes are presented in Table 1. Due to very low mass diffusivity coefficient of the water vapor in aqueous LiBr, the cells at the interface of liquid/vapor needs to be refined. This is done by using an adaptive mesh refinement algorithm. In brief, interface area has a changing value of volume fraction gradients. Therefore, by setting a proper value of volume fraction gradient for 4
beginning the adaption process at the proper time steps, the cells at the interface area divided to finer cells. This adaption is known as h-refinement [32-34]. Each cell (at the interface) splits in eight cells by adding additional nodes at the centers of each face, in the middle of each edge and at the center of the original cell. However, this refinement should be limited by the minimum cell volume. In this way, the adaption process stops whenever each cell at the interface faces a volume which is lower than this minimum volume.
Fig. 1. Schematic view of a horizontal-tube water cooled absorber
Fig. 2. Geometry of the modeled case and the base mesh design (center plane view) Fig. 3 shows two detail views of the base mesh changes after adaption algorithm. The base mesh can be seen on the left side. Although at some adapted small cells Courant number is more than unity, the interface is well calculated by using an implicit approach in time stepping for the interface capturing method [34]. 5
Table 1. Bulk values of Temperature and concentration at (first tube) o Temperature ( C) Concentration (% wt LiBr) Minimum gird size ( ) 42.02 59.98 41.35 59.92 40.94 59.90 40.90 59.90
Fig. 3. Base mesh structure after adaption at the interface (center plane view) 3. Governing equations and boundary conditions In order to simulate the gravity driven two-phase flow, a powerful multiphase model which is capable of capturing the interface between the phases, is the volume of fluid model. This model has been widely used in simulation of two-phase problem including heat and mass transfer as it was mentioned in the introduction. A brief description of this model and the energy and concentration equations are presented here. 3.1. Hydrodynamic of flow The LiBr/water solution distributes over the horizontal tubes from top and this solution covers the surface of tubes and flow between them by gravity. To capture the 6
interface of solution and water vapor the VOF method is used. In this multiphase method, a variable, named volume fraction ( ), is used to identified a cell which is in the gas phase, or belong to the liquid phase. Therefore the value of the volume fraction between zero (gas phase) and one (liquid phase) presents the interface. This method is proposed by Hirt and Nichols [35]. The thermophysical variables such as density and viscosity and so on are computed by linear interpolation of pure values using volume fraction. The equations governs the hydrodynamics of the problem are consist of continuity, momentum and volume fraction. Eqs. (2) to (4) show these equations, respectively. is the velocity vector.
The last term on the right hand side of Eq. (3) is the volumetric surface tension force between two phases. The surface tension force acts in a finite layer between two phases or the interface. To turn the surface tension force from a two dimensional force into the three dimensional one, Brackbill et al. [36] proposed the continuous surface force (CSF) model which relates the surface tension force to the volume fraction gradient, Eq. (5).
In the equation (5), is the constant surface tension coefficient, and is the curvature of the interface. The last term in the left hand side of (3), , is the source term which is added (subtracted) to (from) the liquid phase (the gas phase) in the absorption process. In the case of using a surfactant, the mass transfer due to the variation of surface tension or Marangoni effect can be included by providing the appropriate function of surface tension to the Eq. (5). However the variations of surface tension in the absence of additives are not significant [37] and considering a constant surface tension is reasonable. The property in cells (in general ), such as density, is determined by multiplying to the property of each phase
3.2. Heat and mass transfer The energy and concentration equations are presented in (7) and (8), respectively.
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The second term in the parentheses in the right hand side of Eq. (7) is the interdiffusion term, the amount of heat flux which is brought about by mass flux J. The last term on the right hand side of the energy equation is the amount of heat of absorption from vapor to the solution on the interface. This volumetric energy source can be calculated from multiplying the amount of absorbed mass with the heat of absorption. The amount of mass flux which is absorbed through the interface is calculated by Fick’s law; equation (9).
The normal direction of the solution interface is changing with location and time simulation. To have a nomal direction, the gradiant of volume fraction is used here.
is the unique nomal vector through the gas-liquid interface. Therefore by dot product of concentration gradient at the inteface cells to the normal vector,
will be
achived. As it was remarked earlier, the same mass from the gas phase will absorbed by the liquid phase, therfore the volumetric mass source term in Eq. (4), , is the mass flux product by the normal projected cell area with positive sign in the liquid phase and with negative sign in the gas phase. Therefore, the normal heat flux (W m-2) at the interface, , is governed by Eq. (11). (11) Where the heat of absoption (Habs) is assumed to be constant in this study. 3.3. Boundary conditions and numerical aspects Fig. 4 shows the boundary conditions of the computational domain. The strong solution is continuously feeding from the feeder and with constant inlet bulk temperature and bulk concentration. As it can be seen from the Fig. 4, the geometry of problem is 8
symmetry from all sides. All walls of the pipes are impermeable and have convection boundary condition, Eq. (12). (12) In Eq. (12), is the cooling water temperature, is the tube surface temperature, and is the cooling water side heat transfer coefficient which is evaluated by DittusBoelter relation. Initial values haves been presented in Table 2. Table 2. Absorber working conditions [16] Pressure (Pa) Initial solution temperature (K) Degree of subcooling (K) Initial solution concentration (%wt LiBr) Strong solution mass flow rate (kg s-1 m-1) Cooling water temperature (K)
1080 320.15 0 60 0.03 (for drop mode), 0.07 (for in-line jet mode) 303.15
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Fig. 4: Boundary conditions In addition, the interface concentration of LiBr is an unknown boundary condition which should be adjusted at the interface. In the volume of fluid model, the interface consists of the cells which the volume fraction value is neither 1 nor 0. Therefore the interface is a thin layer with volume fraction value between 0 and 1. On the other hand, the vapor pressure equilibrium assumption is used widely to set the interface concentration value. Hosseinnia et al. [38] evaluate the interface equilibrium concentration with the assumption of equality of the chemical potential of water in solution with water vapor at interface pressure and temperature. The same assumption is used here to set the interface boundary condition. The thermophysical properties of LiBr solution and water vapor are assumed to be constant in the range of working conditions. These properties are shown in Table 3.
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Table 3. Thermophysical properties of the LiBr solution and water vapor Density (kg m-3) 1750 -1 -1 Thermal conductivity (W m K ) 0.4 2 -1 Solution Mass diffusivity (m s ) 1.25 10-9 cp (J kg-1 K-1) 1900 -1 -1 Viscosity (kg m s ) 0.005775 -3 Density (kg m ) 0.00774 -1 -1 Thermal conductivity (W m K ) 0.01745 Vapor -1 -1 Viscosity (kg m s ) 9.385 10-6 Heat of absorption (kJ kg-1) 2.74 103 Interface surface tension (N m-1) 0.088 Equations (2), (3), (4), (7), and (8) are discretized with second order upwind scheme. The SIMPLE algorithm is used for velocity-pressure coupling and the GeoReconstruction algorithm by Youngs [39] is used for reconstructing the interface shape. Main assumptions in this simulation are presented as follows: - Flow is laminar and Newtonian - The flow is assumed to be divided perfectly on each side of the cooling tubes (the perfect bifurcation) - Non-equilibrium cross effects such as Soret and Dufour effects are not taken into account - Both phases share a velocity at the interface - The vapor pressure equilibrium is assumed at the interface - Temperature of the Cooling water, flowing inside the tubes, is assumed to be constant (30 ) - All thermophysical properties are considered constants in the range of working temperature and concentrations 4. Result and discussion 4.1. Validation of simulation The well-known Nusselt’s solution [12] is widely used for hydrodynamic around the tube in the literature. Eq. (13) shows the Nusselt’s thickness for solution around the pipe as a function of . Fig. 5 illustrates the coordinates’ definition (A) and the film thickness around the tube obtained from this study (around the first tube) and Nusselt’s solution (B and C). Two film thicknesses are in a very good agreement with an error of around 0.01 percent. The liquid film of our simulation is slightly thicker because of the vapor absorption at the interface. Nevertheless, the film thickness around the first tube deviates from Nusselt’s solution at the top of the tube, where the solution is distributed and at the bottom of tube where the solution is forming the sessile and then the pendant shape and the falling droplets. 11
For the other two pipes below the first one, the film thickness varies due to the laminar waves produced by the falling droplets and their following satellite drops. The laminar waves cannot be predicted in a relatively simple velocity profile as Nusselt’s one. Therefore, absorption simulations which manipulate Nusselt’s velocity profile will underestimate the absorbed vapor mass flux at the wavy interface over horizontal tubes. Velocity profiles at the normal and tangential coordinates (Fig. 5-A) are shown in Eqs. (14) and (15). Fig. 6 compares the velocity profiles at different angles. Although velocity profiles match very well with Nusselt’s ones, they deviate from each other by increasing the magnitude of . Nusselt’s velocity profiles have a symmetric pattern with respect to . As it can be seen from Fig. 6, Nusselt’s profiles coincide at and . However, the falling film around the pipe makes an unsymmetrical pattern (with respect to the top of tube) due to shaping a droplet below the tube. Therefore this drop formation affects the falling film shape, and as a result, the velocity profile magnitude.
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Fig. 5. Mid-plane cross section; (A) coordinates of the first pipe, (B) Film thickness comparison, (C) detail view of two film thickness near
Fig. 6. Velocity profile at the mid-plane of the first pipe at different angles The results obtained by present CFD simulation are compared with the model by Kyung et al. [16] in the drop mode. The temperature and concentration at liquid/gas interface, at the wall of the first tube and the solution bulk are illustrated in the Fig. 7 and Fig. 8. These values are for the center plane and for the first tube. As it can be seen from these figures, the present simulation agrees very well with the model proposed by Kyung et al. [16]. The variation of angle around the tube is selected in the range of degree where the drop formation begins.
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up to around 150
Fig. 7. Temperature variations around the firs tube at the center plane
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Fig. 8. Concentration variations around the first tube at the center plane 4.2. Effect of solution mass flow rate on hydrodynamics 4.2.1. Drop mode The drop mode between tubes is related to the amount of the mass flow rate over the first tube from the feeder. The transition from drop mode to the drop-jet pattern is predicted by Hu and Jacobi [20] experimentally. The critical value of the film Reynolds number is related to the modified Galilio number according to Eq. (16). (16) In the drop mode simulation, the mass flow rate per unit length of tube has been set to 0.03 kg s-1m-1, corresponding to Ref = 20.8. By considering Eq. (16), the maximum mass flow rate is around 0.0323 kg s-1 m-1, i.e. Ref = 22.4. To have a better understanding in the drop formation stages under the upper tube, a comparison has been made between present study and available experimental results [40].
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Fig. 9 shows different stages of the drop formation. The CFD results show a good agreement with experimental results.
Fig. 9. Comparison of different stages of drop formation under the first tube, Rights: experimental results [40], Lefts: present study (CFD) The induced circulation due to the drop formation at the bottom of the pipe is illustrated in Fig. 10. This internal circulation plays an important role in the absorption process. The internal circulation inside the sessile stage in the drop formation is shown in Fig. 10. However the strength of this circulation becomes weak by marching in time and going to the pendant stage.
Fig. 10. Velocity vectors in the sessile drop at the center plane, under the first tube Fig. 11 presents the different stages of drop formation, growth, fall and impact on the horizontal cooling tubes. The satellite drops are also generated from the liquid bridge after the drop impact on the lower tube. As it can be seen from Fig. 11, the impact waves have interactions on the third tube and make more mixing effect on the falling film around the tube.
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Fig. 11. 3-D contours of the interface, drops formation and fall under the horizontal tubes, 1-10: every 10ms, 11-20 every 5ms Fig. 12 shows the velocity contours at the different stages of the drop mode falling film. The velocity of the falling drops increase from the below of the cooling tubes gradually. The intertube spacing is 24.5 mm, therefore the approximate velocity of the falling drops over the tube below, is expected to be around 0.7 m/s. However, the bridge of water following the falling drop reduces its velocity to around 0.55 m/s on the impact over the top of the next tube. Then the impact of droplet produces laminar waves over the surface of the tube. These velocity profiles can be seen in Fig. 12.
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Fig. 12. Velocity contours (m s-1) at the interface, every 30 ms (approximately) from left to right 4.2.2. Jet mode The maximum value of Re can be predicted by considering the transition between jet mode to jet-sheet. Expression (17) is obtained by Hu and Jacobi [20] for passing from the jet mode toward the sheet mode. (17) This expression gives the maximum value of Re f = 108.4; i.e. = 0.1565 kg s-1 m-1 for the water LiBr mixture. Thus any value of Re between 22.4 and 108.4 maps the falling film mode onto jet mode. The jet mode falling film between tubes is achieved by increasing the solution mass flow rate to 0.0615 kg s-1 m-1 ; i.e. Ref = 42.6. It has been observed that the jet mode can be in different sub-modes in experimental investigations [20]. These modes are: in-line jets, staggered jets and unsteady jets. In this study, we focus on the in-line jets. Fig. 13 presents the captured contours of phase and velocity for the in-line jets at the solution interface. The jet mode between the horizontal tubes is achieved by increasing the solution mass flow rate. The jet mode, in contrast to the drop mode, reaches to a steady state form in its hydrodynamic. As it can be seen from Fig. 13, the interface shape varies around each cooling tube, slightly. It is mainly due to different distribution of solution over each cooling tube. Addition to the solution mass flow rate in the jet mode causes two important changes in the hydrodynamic of the falling film. As the first change, it thickens the film around the tubes uniformly and as the second change, it increases the film velocity around and between the tubes. A reduction in the contact time between the water vapor and the falling solution is the important result of the latter. 18
Fig. 13. In-line jets, Left: 3-D contour of the interface, Right: velocity magnitude; Ref=42.6. 4.3. Heat and mass transfer 4.3.1. Drop mode Falling film around the tube is cooled by the cooling water flowing inside the pipe. However, the cooling effect of the tube reduces gradually during the formation of drops in the bottom of the tubes. On the other hand, the absorption of vapor releases the heat of absorption at the interface and causes a heat flux at the interface, thus increases the temperature of the solution. Fig. 14 shows the interface temperature and concentration counters at different drop formation steps. The binary solution becomes cooler around the cooling tubes and both temperature and concentration go down in this region, however during the formation of the drops beneath the tube, the solution experiences higher temperature and higher equilibrium concentration at the interface consequently. Fig. 15 shows the instantaneous absorbed mass flux of the water vapor at the interface, coincident with the contours of Fig. 14. The solution is moving toward the below part of the tube because of the effect of gravity and makes an extremely thin layer of solution (around 0.1 mm) on top portion of second and third tubes. This fine and very slow region experiences the lower temperature and consequently, the lower equilibrium concentration at the interface, resulting to a higher value of concentration gradient. The minimum temperature at the interface occurs at the top of cooling tubes where the falling drop is expected to fall on. This temperature is near 35 . The corresponding concentration is near 54 % LiBr, therefore a large concentration gradient exists at this point and for a very
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short time (20 ms approximately) the value of local mass flux reaches to .
Fig. 14. Temperature and concentration contours at different drop formation stages at the interface, every 30 ms from left to right
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The local mass flux varies by the droplet impact on the lower pipe (Fig. 15 the last column on the right). The average values of the local mass flux at each stage are , , , and from left to right, respectively.
Fig. 15. Local mass flux (kg s-1 m-2) at the interface during drop formation stages 4.3.2. Jet mode As it was stated before, the average velocity of the falling film goes up in the jet mode; therefore, it makes a reduction in the residence time of the falling solution in contact with the water vapor. In addition, the thickness of the falling film around the tube becomes uniform along the length of the tubes. Fig. 16 presents the temperature and concentration contours for the in-line jets. By considering the temperature contour of the in-line jets at the interface, the minimum temperature is near 39 (4 degrees hotter than the drop mode), and the corresponding equilibrium concentration is 56.5% LiBr. Despite the drop mode, the jet mode can reach a steady shape in the hydrodynamic with no very thin region on top of the second or third tube. The water vapor is absorbed at the interface of jets and this absorption process causes higher temperature on jets than the film around the cooling tubes. The temperature shows lower values between two jets around the cooling tube, where the velocity of the two neighboring jet mixed with each other.
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Fig. 16. Temperature ( ) and concentration contours at the interface in in-line jets Fig. 17 presents the local mass flux at the interface for the in-line jet regime. The local mass flux at the jet regime shows lower value with respect to the drop regime.
Fig. 17. Local mass flux (kg s-1 m-2) at the interface of in-line jet mode In spite of the drop mode, the average value of local mass flux is in the jet mode. It indicates that the local mass flux in the drop mode is around ten times of the jet mode. The main reason of this fact can be found in the 22
high absorption rate on the very thin region on the top of the second and third tubes in the drop regime. Considering this fact, and the fact that the impact of drops on the cooling tubes makes a laminar wave, which has a mixing effect, the mass transfer in the drop mode should be higher than the jet mode. 5. Conclusion Three-dimensional simulation of water vapor absorption process into water-LiBr solution has been numerically performed inside a water cooled absorber chamber, consists of a horizontal tube bank (three tubes). The multiphase volume of fluid model has been utilized to capture the interface shape. Two different flow patterns between the horizontal tubes, namely drop mode and jet mode, have been studied. The heat and mass transfer equations have been solved with the hydrodynamics of fluid flow simultaneously. The absorption of vapor takes place at the interface, therefore the adaptive mesh refinement is used to increase the accuracy of the interface capturing as well as modeling the transport equations boundary layers. Results have been compared with available data on the literature and a good agreement is obtained. The simulation results reveal the superior vapor mass flux in the drop mode with respect to the jet mode. The main reason is in the unsteady nature of the drop regime with respect to the jet one. Drops form below the tubes and then they pinch off and fall over the lower tube and this continuous process makes a transient film thickness around the tubes. In the other words, the film thickness around the tubes in the drop regime becomes thinner in some spots and as a result, cooler than the jet regime. The cooler the film temperature is, the higher the concentration gradient will be at the interface and therefore the higher local mass flux is achieved. Moreover, the impact of the droplets on the cooling pipes makes a laminar wave which accompanies the mixing effect. This mixing effect is very weak (if not missing) in the jet regime.
Nomenclature cp specific heat of solution (J kg-1K-1) C concentration of LiBr ( kg LiBr/kg solution) D solution diffusivity (m2 s-1) gravity acceleration (m s-2) - along the fall direction Ga modified Galileo number ( ) -1 h enthalpy (J kg ) Habs heat of absorption (J kg-1) conductivity coefficient (W m-1 K-1) absorbed mass flux (kg s-1m-2) normal direction vector at the interface P pressure (Pa) 23
r R Re
normal distance from the pipe surface (m) pipe radious film Reynolds number (4 -1) absorbed mass source (kg m-3) time (s) temperature (K) velocity vector (m/s) heat flux at interface (W m-2) energy source term (Wm-3)
Greek symbols volume fraction (of liquid phase) mass flow rate per tube length, (kg s-1 m-1) film thickness (m) angle (shown in Fig. 5-A) interface curvature (m) spacing between neighboring jets or drops (m) kinematic viscosity (kg/m.s) density (kg/m3) surface tension coefficient(N/m) any property in the computation domain such as density, viscosity and so on Subscripts abs absorbed vapor cw cooling water eq equilibrium condition gen source term i inner inlet inlet condition int interface j each of liquid water or LiBr o outer volumetric w wall 0 pipe radious 1,2 liquid and gas phase References [1] P. Srikhirin, S. Aphornratana, S. Chungpaibulpatana, A review of absorption refrigeration technologies, Renew Sustain Energy Rev 5 (2001) 343–372 24
[2] G.A. Florides, S.A. Tassou, S.A. Kalogirou, L.C. Wrobel, Review of solar and low energy cooling technologies for buildings, Renew Sustain Energy Rev 6 (2002) 557–572 [3] X. Wang, H. T. Chua, Absorption Cooling: A Review of Lithium Bromide-Water Chiller Technologies, Recent Patents on Mechanical Engineering 2 (2009) 193-213 [4] X.Q. Zhai, M. Qu, Yue. Li, R.Z. Wang, A review for research and new design options of solar absorption cooling systems, Renew Sustain Energy Rev 15 (2011) 4416– 4423 [5] J. Sun, L. Fu, S. Zhang, A review of working fluids of absorption cycles, Renew Sustain Energy Rev 16 (2012) 1899– 1906 [6] R. T. Wassenaar, Measured and predicted effect of flow rate and tube spacing on horizontal tube absorber performance, Int. J. Refrig. 5 (1996) 347-355 [7] ASHRAE Handbook of fundamentals, 1997 [8] J. Fernandez-Seara, A. A. Pardinas, Refrigerant falling film evaporation review: Description, fluid dynamics and heat transfer, Appl. Therm. Eng. 64 (2014) 155-171 [9] V.D. Papaefthimiou, D.C. Karampinos, E.D. Rogdakis, A detailed analysis of watervapor absorption in LiBr–H2O solution on a cooled horizontal tube, Appl. Therm. Eng. 26 (2006) 2095–2102 [10] S. J. F. Lin, Zh. Shigang, Experimental study on vertical vapor absorption into LiBr solution with and without additive, Appl. Therm. Eng. 31 (2011) 2850-2854 [11] J. D. Killion, S. Garimella, A critical review of models of coupled heat and mass transfer in falling-film absorption. Int. J. Refrig. 24 (2001) 755–797 [12] W. Nusselt, Die oberflaechenkondensateion des wasserdampfes, Zeitschrift des Vereines Deutscher Ingenieur 60 (27) (1916) 541–546 [13] S. K. Choudhury, D. Hisajima, T. Ohuchi, A. Nishiguchi, T. Fukushima, S. Sakaguchi, Absorption of vapors into liquid films flowing over cooled horizontal tubes, ASHRAE Transaction Research 99 (Part 2) (1993) 81-89. [14] Z. Lu, D. Li, S. Li, B. Yu-Chi, A semi-empirical model of the falling film absorption outside horizontal tubes, Proce. Int. Ab-soption Heat Pump Conf. 2 (1996) 473-480 [15] S. Jeong, S. Garimella, Falling-film and droplet mode heat and mass transfer in a horizontal tube LiBr/water absorber, Int. J. Heat Mass Trans. 45 (2002) 1445–1458 [16] I. Kyung, K. E. Herold, Y. T. Kang, Model for absorption of water vapor into aqueous LiBr flowing over a horizontal smooth tube, Int. J. Refrig. 30 (2007) 591-600 [17] I. Kyung, K. E. Herold, Y. T. Kang, Experimental verification of H2O/LiBr absorber bundle performance with smooth horizontal tubes, Int. J. Refrig. 30 (2007) 582-590
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[18] M.J. Kirby, P. Perez-Blanco, A design model for horizontal tube water/lithium bromide absorbers, in: Heat Pump and Refrigeration Systems Design, Analysis and Applications, ASME 32 (1994) 1-10 [19] R. Clift, J.R. Grace, M.E. Weber, Bubbles, Drops, and Particles, Academic Press, New York, 1978 [20] X. Hu, A.M. Jacobi, The Intertube Falling Film: part 1- Flow Characteristics, Mode Transitions, and Hysteresis, ASME J. Heat Trans. 118 (1996) 616-625. [21] X. Hu, A.M. Jacobi, Flow Characteristics of Liquid Droplets and Jets Falling Between Horizontal Circular Tubes, Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, 2 (1997) 1295-1302. [22] X. Hu, A.M. Jacobi, Departure-site spacing for liquid droplets and jets falling between horizontal circular tubes, Experimental Therm. Fluid Sci. 16 (1998) 322-331. [23] J. F. Roques, V. Dupont, J. R. Thome, Falling film transitions on plain and enhanced tubes, ASME transactions, journal of heat transfer, 124 (2002) 491-499 [24] J. F. Roques, J. R. Thome, Falling film transitions between droplet, column. And sheet flow modes on a vertical array of horizontal 19FPI and 40FPI low-finned tubes, Heat Trans. Eng. 24(2003) 40–45 [25] X. Wang, A.M. Jacobi, A Thermodynamic Basis for Predicting Falling-Film Mode Transitions, Int. J. Refrig. 43 (2014) 123-132 [26] J. D. Killion, S. Garimella, Simulation of pendant droplets and falling films in horizontal tube absorbers, J. Heat Transfer, 126 (6) (2004) 1003-1013 [27] J. D. Killion, S. Garimella, Pendant droplet motion for absorption on horizontal tube banks, Int. J. Heat and Mass Transfer 47 (2004) 4403–4414 [28] J. Chen, R. Zhang, R. Niu, Numerical simulation of horizontal tube bundle falling film flow pattern transformation, Renewable Energy 73 (2015) 62-68 [29] A. Hassanvand, S.H. Hashemabadi, Direct numerical simulation of interphase mass transfer in gas–liquid multiphase systems, Int. Commu. Heat Mass Trans. 38 (2011) 943– 950 [30] V. Subramaniam, S. Garimella, Numerical study of heat and mass transfer in Lithium Bromide-water falling films and droplets, Int. J. Refrig. 40 (2014) 211-226 [31] D. Yung, J.J. Lorentz, E.N. Ganic, Vapor/liquid interaction and entrainment in falling film evaporators, J. Heat Transfer 102 (1980) 20-25 [32] H. Jasak, A.D. Gosman, Automatic resolution control for the finite-volume method, Part 2: Adaptive mesh refinement and coarsening. Numerical Heat Transf., B38 (2000) 257 –271 26
[33] J. J. Cooke, L. M. Armstrong, K.H. Luo, S. Gu, Adaptive mesh refinement of gas– liquid flow on an inclined plane, Computers Chem. Eng. 60 (2014) 297– 306 [34] Z. Xie, D. Pavlidis, J. R. Percival, J. L. M. A. Gomes, C.C. Pain, O. K. Matar, Adaptive unstructured mesh modeling of multiphase flows, Int. J. Multiphase Flow 67 (2014) 104–110 [35] C.W. Hirt and B.D. Nichols, Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries, J. Comput. Phys. 39 (1981) 201-225 [36] J. U. Brackbill, D. B. Kothe, C. Zemach, A continuum method for modeling surface tension, J. Comput. Phys. 100 (1992) 335-354 [37] E. Hihara, T. Saito, Effect of surfactant on falling film absorption, Int. J. Refrig. 16 (1999) 339-346 [38] S. M. Hosseinnia, M. Naghashzadegan, R. Kouhikamali, CFD simulation of adiabatic water vapor absorption in large drops of water-LiBr solution, Appl. Therm. Eng. 102 (2016) 17-29 [39] D. L. Youngs, Time-dependent multi-material flow with large fluid distortion, In Numerical methods for fluid dynamics, edited by K. W. Morton and M. J. Baines, pp 273-285. Academic Press, 1982 [40] J. D. Killion, S. Garimella, Gravity-driven flow of liquid film and droplets in horizontal tube banks, Int. J. Refrig. 26 (2003) 516-526
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Highlights
Different stages of drop formation under horizontal tubes are numerically simulated
Hydrodynamic of in-line jet mode between tubes are captured numerically
Transient nature of drop mode makes different film thickness around the tubes
Average local absorbed mass flux in drop mode is ten times larger than the jet mode
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S1359-4311(17)30145-X http://dx.doi.org/10.1016/j.applthermaleng.2017.01.022 ATE 9786
To appear in:
Applied Thermal Engineering
Received Date: Revised Date: Accepted Date:
5 July 2016 16 November 2016 8 January 2017
Please cite this article as: S. Mojtaba Hosseinnia, M. Naghashzadegan, R. Kouhikamali, CFD simulation of water vapor absorption in laminar falling film solution of water-LiBr ─ Drop and Jet modes, Applied Thermal Engineering (2017), doi: http://dx.doi.org/10.1016/j.applthermaleng.2017.01.022
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CFD simulation of water vapor absorption in laminar falling film solution of water-LiBr ─ Drop and Jet modes
Seyed Mojtaba Hosseinnia, Mohammad Naghashzadegan, Ramin Kouhikamali Department of Mechanical Engineering, Faculty of Engineering, University of Guilan, Rasht, Iran
Abstract: The rate of absorbed water vapor over a horizontal tube bank in a water-cooled falling film absorber depends on several parameters and working conditions. The flow regime between the tubes in falling film absorbers is one of the most influential parameters in the amount of the absorbed vapor. Among different falling film regimes, in this study, the drop and jet modes are simulated numerically. The full Navier-Stokes equations are solved and the well-known volume of fluid (VOF) method is used to capture the gas/liquid interface. In addition, the energy and diffusion equations are solved in this 3Dsimulation with aid of an in-house CFD code. Adaptive mesh refinement, in accordance with the magnitude of volume fraction gradient, has strongly improved the interface capturing and therefore increased the accuracy of simulation. The simulation results reveal that by changing just the regime from drops to jets, the rate of average vapor mass flux decreases one order of magnitude, from to . Keywords: Numerical simulation, Falling film absorber, Drop mode, Jet mode, VOF 1. Introduction Application of absorption machines which are capable of using a low-grade heat source, are growing in the recent years [1-4]. The absorption technology is less harmful to the environment specially by using water as refrigerant and aqueous LiBr or LiCl as the absorbing media in the domestic air conditioning applications [5]. Absorber heat exchanger is the most important component in the absorption machine which has a great effect on the whole cycle efficiency and initial system costs [6]. In a conventional falling film absorber the binary solution, reach in a salt, falls by gravity on several horizontal cooling tubes. Because of working on the same pressure, the absorber and the evaporator are placed in one chamber. Therefore the vapor of refrigerant, coming from the
Corresponding author. Tel.: +09813 33690273; fax: +98 13 33690273. E-mail address: [email protected] (R. Kouhikamali)
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evaporator, is allowed to move to the absorber side freely. Thus the entire absorber chamber is filled with the vapor of refrigerant in its saturated state, approximately [7]. The vapor is absorbed on the interface of aqueous LiBr and releases the heat of absorption. A reliable absorption process takes place by removing the heat of absorption. Hence the cooling media, generally liquid water, flows inside the horizontal tubes and cools the solution. Several important parameters play roles in whole absorption process inside the absorber. Some of them are related to geometry and affect the hydrodynamic of falling film such as tube diameter and spacing [8]. Some others modify both hydrodynamic and heat/mass transfer characteristics with respect to thermophysical properties such as density, surface tension, and viscosity [9-10]. In addition, the rate of pouring binary solution which falls on the horizontal tubes is the other important component. A compromise among mentioned parameters dictates the flow regime between the tubes; including drop mode, jet mode and sheet mode with respect to increase in the binary solution mass flow rate. Hence, in fixed absorber geometry, with certain working pair – in this study water/LiBr– the only parameter which specifies the flow regime is the solution mass flow rate per the length of the horizontal tubes. Modeling of absorption process over horizontal-cooling tubes is a very interesting area for researchers [9-11]. Two important aspects of this modeling can be classified as hydrodynamics of the falling film and the heat/mass transfer. There are several simplifying assumptions in both domains. From hydrodynamic point of view, the simplifications are using Nusselt’s velocity profile [12] around the tube in Refs. [13-16], which gets infinite film thickness at both top and bottom of the tube and this velocity profile, cannot predict the laminar waves generated from impact of drops on the tube surface. Moreover, there are some assumptions about drop formation and fall; under and between tubes, by assuming a 3D phenomenon as a 2D one. On the other hand, from the heat/mass transfer aspects, employing correlations for the heat/mass transfer coefficients make other simplifications [17-19]. Many researchers have worked on the hydrodynamic of the falling film both experimentally and numerically. Hu and Jacobi [20-22] have studied the different aspects of the falling film mode transitions from dropwise to sheet mode experimentally. They showed that the transition mode are related to modified Galileo number (Ga) and film Reynolds number (Ref) in the form of Ref=AGaB where the constants (A and B) are adjusted with respect to experimental data. Roques et al. [23-24] have investigated the transition modes in the hydrodynamics of the falling film on horizontal low-finned tubes with different fin spacing and configuration. In their experimental study, they showed the beginning of transition becomes advanced in comparison to plain tubes. Even recently Wang and Jacobi [25] showed a thermodynamics-based analysis for prediction of the different transitions in the flow regime. They assumed each flow mode is a thermodynamic system which can be in equilibrium with the other modes by 2
considering several simplifying assumptions. These assumptions presuppose that the system is in isothermal and adiabatic conditions which mean no heat or mass transfer. However in considering just hydrodynamics, using a thermodynamic-based analysis is a simple and quick prediction. In addition to experimental and even theoretical investigations, the hydrodynamics of falling film has been studied numerically in different approaches. Killion and Garimella [26] presented a comprehensive review of numerical history in modeling two phase gravity driven falling film with VOF model and fluent commercial CFD package. In their transient simulation, the droplet creation below a horizontal tube, its growth and fall and the wave generated by its impact on the next tube is illustrated. The 3-D numerical result is validated perfectly with respect to experimental results [27]. Chen et al. [28] simulated numerically the falling film transition mode on the horizontal tubes by means of finite volume solver, Fluent. They used volume of fluid technique to capture the phase’s interface and hydrodynamics of falling film. By comparison with test data, they showed a good agreement between the numerical results and experimental data. Thus far, the hydrodynamic aspect of falling film is briefly reviewed; the heat/mass transfer point of view will be presented in the followings. As it was remarked earlier, heat and mass transfer in absorption process are coupled together. The driving force in the mass transfer is the difference between bulk concentration in the falling film and concentration of a very thin layer between gas and liquid phase; the interface. In other words, water vapor absorbs at the interface immediately and because of a very low mass diffusivity characteristic of the vapor in the binary solution, , water makes a low concentration area at the interface. The concentration at interface is an unknown boundary condition in absorption problem. The absorbed vapor brings the heat of absorption with itself which is the difference between the enthalpy of the vapor phase and the enthalpy of water in the binary solution. Hassanvand and Hashemabadi [29] used the VOF model to simulate the interphase mass transfer process in stratified gas-liquid flow between two parallel plates. Their results were validated by comparison with two benchmark mass transfer mechanisms and they concluded there is a precise agreement. Subramaniam and Garimella [30] simulated the drop mode falling film absorption numerically with VOF technique. They considered a section of tube with constant surface temperature. By assuming the chemical potential equality at the interface between water vapor and water in the aqueous LiBr, they computed the interface concentration in an iterative manner using Newton-Rophson algorithm. Moreover they considered the effect of laminar waves produced in drop impact on the pipe and showed this laminar wave mixing effect in the absorption process. This study focuses on numerical simulation of combined heat and mass transfer processes in an absorber in the drop and jet modes falling film pattern between horizontal tubes. To have a comparison between the obtained numerical results and the experimental 3
ones, the computational domain such as diameter of pipes and pipe spacing has been chosen in accordance with Kyung et al. [16] research. 2. Problem geometry and mesh design Fig.1 shows a cross-section view of a typical horizontal-tube water cooled absorber. The hot and strong solution of water and LiBr enters to the absorber via several feeders and falls on a column of horizontal cooling tubes. The end of each cooling tube may be connected to the next tube beneath it to make a serpentine tube (not showed in Fig.1) or to a vertical pipe which distribute the cooling water. The cooling water flows through these pipes from the bottom. At the same time, water vapor flows steadily to the absorber chamber in its saturated condition (approximately). For simulating in the current study, three cooling tubes below the feeder are chosen as the computational domain. Fig. 2 shows the computation domain. Each tube has an outer diameter of 19.05 mm and the spacing between them is 24.5 mm. In this simulation, a section of 0.02 m of each of the three tubes is determined to be the tube length in –z direction. This value is chosen according to the relation proposed by Young et al [31]; Eq. (1). Where is the space between drops or jets falling between horizontal tubes, is the surface tension coefficient at vapor/aqueous LiBr, and is the density of liquid. n is considered to be 2 for water vapor and aqueous LiBr [31].
A square of 1 1 mm2 at the top center of the tubes (2 mm above the first tube) plays the role of the feeder or distributer as a perforated tube or plate [8]. The entire computational domain is meshed with hexahedral structured cells. Fig. 2 illustrates a detail view of designed base meshes; cells in area of falling solution around the tubes perimeter are denser in contrast to the other area. This is mainly because the heat transfer between the binary solution around the wall of the pipes takes place in this area. The minimum cell size is around the wall of the cooling pipes. This minimum cell size (as base-mesh size) has selected after a grid independency analysis was done. The shape of interface has not changed significantly by the change in the grid size. But the film thickness around the tube is very small (around 0.2 mm) and having more grids in the falling film around the tubes will enhance more accurate results. Therefore, the bulk values of temperature and concentration for the first tube at and in drop mode with respect to variations in gird sizes are presented in Table 1. Due to very low mass diffusivity coefficient of the water vapor in aqueous LiBr, the cells at the interface of liquid/vapor needs to be refined. This is done by using an adaptive mesh refinement algorithm. In brief, interface area has a changing value of volume fraction gradients. Therefore, by setting a proper value of volume fraction gradient for 4
beginning the adaption process at the proper time steps, the cells at the interface area divided to finer cells. This adaption is known as h-refinement [32-34]. Each cell (at the interface) splits in eight cells by adding additional nodes at the centers of each face, in the middle of each edge and at the center of the original cell. However, this refinement should be limited by the minimum cell volume. In this way, the adaption process stops whenever each cell at the interface faces a volume which is lower than this minimum volume.
Fig. 1. Schematic view of a horizontal-tube water cooled absorber
Fig. 2. Geometry of the modeled case and the base mesh design (center plane view) Fig. 3 shows two detail views of the base mesh changes after adaption algorithm. The base mesh can be seen on the left side. Although at some adapted small cells Courant number is more than unity, the interface is well calculated by using an implicit approach in time stepping for the interface capturing method [34]. 5
Table 1. Bulk values of Temperature and concentration at (first tube) o Temperature ( C) Concentration (% wt LiBr) Minimum gird size ( ) 42.02 59.98 41.35 59.92 40.94 59.90 40.90 59.90
Fig. 3. Base mesh structure after adaption at the interface (center plane view) 3. Governing equations and boundary conditions In order to simulate the gravity driven two-phase flow, a powerful multiphase model which is capable of capturing the interface between the phases, is the volume of fluid model. This model has been widely used in simulation of two-phase problem including heat and mass transfer as it was mentioned in the introduction. A brief description of this model and the energy and concentration equations are presented here. 3.1. Hydrodynamic of flow The LiBr/water solution distributes over the horizontal tubes from top and this solution covers the surface of tubes and flow between them by gravity. To capture the 6
interface of solution and water vapor the VOF method is used. In this multiphase method, a variable, named volume fraction ( ), is used to identified a cell which is in the gas phase, or belong to the liquid phase. Therefore the value of the volume fraction between zero (gas phase) and one (liquid phase) presents the interface. This method is proposed by Hirt and Nichols [35]. The thermophysical variables such as density and viscosity and so on are computed by linear interpolation of pure values using volume fraction. The equations governs the hydrodynamics of the problem are consist of continuity, momentum and volume fraction. Eqs. (2) to (4) show these equations, respectively. is the velocity vector.
The last term on the right hand side of Eq. (3) is the volumetric surface tension force between two phases. The surface tension force acts in a finite layer between two phases or the interface. To turn the surface tension force from a two dimensional force into the three dimensional one, Brackbill et al. [36] proposed the continuous surface force (CSF) model which relates the surface tension force to the volume fraction gradient, Eq. (5).
In the equation (5), is the constant surface tension coefficient, and is the curvature of the interface. The last term in the left hand side of (3), , is the source term which is added (subtracted) to (from) the liquid phase (the gas phase) in the absorption process. In the case of using a surfactant, the mass transfer due to the variation of surface tension or Marangoni effect can be included by providing the appropriate function of surface tension to the Eq. (5). However the variations of surface tension in the absence of additives are not significant [37] and considering a constant surface tension is reasonable. The property in cells (in general ), such as density, is determined by multiplying to the property of each phase
3.2. Heat and mass transfer The energy and concentration equations are presented in (7) and (8), respectively.
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The second term in the parentheses in the right hand side of Eq. (7) is the interdiffusion term, the amount of heat flux which is brought about by mass flux J. The last term on the right hand side of the energy equation is the amount of heat of absorption from vapor to the solution on the interface. This volumetric energy source can be calculated from multiplying the amount of absorbed mass with the heat of absorption. The amount of mass flux which is absorbed through the interface is calculated by Fick’s law; equation (9).
The normal direction of the solution interface is changing with location and time simulation. To have a nomal direction, the gradiant of volume fraction is used here.
is the unique nomal vector through the gas-liquid interface. Therefore by dot product of concentration gradient at the inteface cells to the normal vector,
will be
achived. As it was remarked earlier, the same mass from the gas phase will absorbed by the liquid phase, therfore the volumetric mass source term in Eq. (4), , is the mass flux product by the normal projected cell area with positive sign in the liquid phase and with negative sign in the gas phase. Therefore, the normal heat flux (W m-2) at the interface, , is governed by Eq. (11). (11) Where the heat of absoption (Habs) is assumed to be constant in this study. 3.3. Boundary conditions and numerical aspects Fig. 4 shows the boundary conditions of the computational domain. The strong solution is continuously feeding from the feeder and with constant inlet bulk temperature and bulk concentration. As it can be seen from the Fig. 4, the geometry of problem is 8
symmetry from all sides. All walls of the pipes are impermeable and have convection boundary condition, Eq. (12). (12) In Eq. (12), is the cooling water temperature, is the tube surface temperature, and is the cooling water side heat transfer coefficient which is evaluated by DittusBoelter relation. Initial values haves been presented in Table 2. Table 2. Absorber working conditions [16] Pressure (Pa) Initial solution temperature (K) Degree of subcooling (K) Initial solution concentration (%wt LiBr) Strong solution mass flow rate (kg s-1 m-1) Cooling water temperature (K)
1080 320.15 0 60 0.03 (for drop mode), 0.07 (for in-line jet mode) 303.15
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Fig. 4: Boundary conditions In addition, the interface concentration of LiBr is an unknown boundary condition which should be adjusted at the interface. In the volume of fluid model, the interface consists of the cells which the volume fraction value is neither 1 nor 0. Therefore the interface is a thin layer with volume fraction value between 0 and 1. On the other hand, the vapor pressure equilibrium assumption is used widely to set the interface concentration value. Hosseinnia et al. [38] evaluate the interface equilibrium concentration with the assumption of equality of the chemical potential of water in solution with water vapor at interface pressure and temperature. The same assumption is used here to set the interface boundary condition. The thermophysical properties of LiBr solution and water vapor are assumed to be constant in the range of working conditions. These properties are shown in Table 3.
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Table 3. Thermophysical properties of the LiBr solution and water vapor Density (kg m-3) 1750 -1 -1 Thermal conductivity (W m K ) 0.4 2 -1 Solution Mass diffusivity (m s ) 1.25 10-9 cp (J kg-1 K-1) 1900 -1 -1 Viscosity (kg m s ) 0.005775 -3 Density (kg m ) 0.00774 -1 -1 Thermal conductivity (W m K ) 0.01745 Vapor -1 -1 Viscosity (kg m s ) 9.385 10-6 Heat of absorption (kJ kg-1) 2.74 103 Interface surface tension (N m-1) 0.088 Equations (2), (3), (4), (7), and (8) are discretized with second order upwind scheme. The SIMPLE algorithm is used for velocity-pressure coupling and the GeoReconstruction algorithm by Youngs [39] is used for reconstructing the interface shape. Main assumptions in this simulation are presented as follows: - Flow is laminar and Newtonian - The flow is assumed to be divided perfectly on each side of the cooling tubes (the perfect bifurcation) - Non-equilibrium cross effects such as Soret and Dufour effects are not taken into account - Both phases share a velocity at the interface - The vapor pressure equilibrium is assumed at the interface - Temperature of the Cooling water, flowing inside the tubes, is assumed to be constant (30 ) - All thermophysical properties are considered constants in the range of working temperature and concentrations 4. Result and discussion 4.1. Validation of simulation The well-known Nusselt’s solution [12] is widely used for hydrodynamic around the tube in the literature. Eq. (13) shows the Nusselt’s thickness for solution around the pipe as a function of . Fig. 5 illustrates the coordinates’ definition (A) and the film thickness around the tube obtained from this study (around the first tube) and Nusselt’s solution (B and C). Two film thicknesses are in a very good agreement with an error of around 0.01 percent. The liquid film of our simulation is slightly thicker because of the vapor absorption at the interface. Nevertheless, the film thickness around the first tube deviates from Nusselt’s solution at the top of the tube, where the solution is distributed and at the bottom of tube where the solution is forming the sessile and then the pendant shape and the falling droplets. 11
For the other two pipes below the first one, the film thickness varies due to the laminar waves produced by the falling droplets and their following satellite drops. The laminar waves cannot be predicted in a relatively simple velocity profile as Nusselt’s one. Therefore, absorption simulations which manipulate Nusselt’s velocity profile will underestimate the absorbed vapor mass flux at the wavy interface over horizontal tubes. Velocity profiles at the normal and tangential coordinates (Fig. 5-A) are shown in Eqs. (14) and (15). Fig. 6 compares the velocity profiles at different angles. Although velocity profiles match very well with Nusselt’s ones, they deviate from each other by increasing the magnitude of . Nusselt’s velocity profiles have a symmetric pattern with respect to . As it can be seen from Fig. 6, Nusselt’s profiles coincide at and . However, the falling film around the pipe makes an unsymmetrical pattern (with respect to the top of tube) due to shaping a droplet below the tube. Therefore this drop formation affects the falling film shape, and as a result, the velocity profile magnitude.
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Fig. 5. Mid-plane cross section; (A) coordinates of the first pipe, (B) Film thickness comparison, (C) detail view of two film thickness near
Fig. 6. Velocity profile at the mid-plane of the first pipe at different angles The results obtained by present CFD simulation are compared with the model by Kyung et al. [16] in the drop mode. The temperature and concentration at liquid/gas interface, at the wall of the first tube and the solution bulk are illustrated in the Fig. 7 and Fig. 8. These values are for the center plane and for the first tube. As it can be seen from these figures, the present simulation agrees very well with the model proposed by Kyung et al. [16]. The variation of angle around the tube is selected in the range of degree where the drop formation begins.
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up to around 150
Fig. 7. Temperature variations around the firs tube at the center plane
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Fig. 8. Concentration variations around the first tube at the center plane 4.2. Effect of solution mass flow rate on hydrodynamics 4.2.1. Drop mode The drop mode between tubes is related to the amount of the mass flow rate over the first tube from the feeder. The transition from drop mode to the drop-jet pattern is predicted by Hu and Jacobi [20] experimentally. The critical value of the film Reynolds number is related to the modified Galilio number according to Eq. (16). (16) In the drop mode simulation, the mass flow rate per unit length of tube has been set to 0.03 kg s-1m-1, corresponding to Ref = 20.8. By considering Eq. (16), the maximum mass flow rate is around 0.0323 kg s-1 m-1, i.e. Ref = 22.4. To have a better understanding in the drop formation stages under the upper tube, a comparison has been made between present study and available experimental results [40].
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Fig. 9 shows different stages of the drop formation. The CFD results show a good agreement with experimental results.
Fig. 9. Comparison of different stages of drop formation under the first tube, Rights: experimental results [40], Lefts: present study (CFD) The induced circulation due to the drop formation at the bottom of the pipe is illustrated in Fig. 10. This internal circulation plays an important role in the absorption process. The internal circulation inside the sessile stage in the drop formation is shown in Fig. 10. However the strength of this circulation becomes weak by marching in time and going to the pendant stage.
Fig. 10. Velocity vectors in the sessile drop at the center plane, under the first tube Fig. 11 presents the different stages of drop formation, growth, fall and impact on the horizontal cooling tubes. The satellite drops are also generated from the liquid bridge after the drop impact on the lower tube. As it can be seen from Fig. 11, the impact waves have interactions on the third tube and make more mixing effect on the falling film around the tube.
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Fig. 11. 3-D contours of the interface, drops formation and fall under the horizontal tubes, 1-10: every 10ms, 11-20 every 5ms Fig. 12 shows the velocity contours at the different stages of the drop mode falling film. The velocity of the falling drops increase from the below of the cooling tubes gradually. The intertube spacing is 24.5 mm, therefore the approximate velocity of the falling drops over the tube below, is expected to be around 0.7 m/s. However, the bridge of water following the falling drop reduces its velocity to around 0.55 m/s on the impact over the top of the next tube. Then the impact of droplet produces laminar waves over the surface of the tube. These velocity profiles can be seen in Fig. 12.
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Fig. 12. Velocity contours (m s-1) at the interface, every 30 ms (approximately) from left to right 4.2.2. Jet mode The maximum value of Re can be predicted by considering the transition between jet mode to jet-sheet. Expression (17) is obtained by Hu and Jacobi [20] for passing from the jet mode toward the sheet mode. (17) This expression gives the maximum value of Re f = 108.4; i.e. = 0.1565 kg s-1 m-1 for the water LiBr mixture. Thus any value of Re between 22.4 and 108.4 maps the falling film mode onto jet mode. The jet mode falling film between tubes is achieved by increasing the solution mass flow rate to 0.0615 kg s-1 m-1 ; i.e. Ref = 42.6. It has been observed that the jet mode can be in different sub-modes in experimental investigations [20]. These modes are: in-line jets, staggered jets and unsteady jets. In this study, we focus on the in-line jets. Fig. 13 presents the captured contours of phase and velocity for the in-line jets at the solution interface. The jet mode between the horizontal tubes is achieved by increasing the solution mass flow rate. The jet mode, in contrast to the drop mode, reaches to a steady state form in its hydrodynamic. As it can be seen from Fig. 13, the interface shape varies around each cooling tube, slightly. It is mainly due to different distribution of solution over each cooling tube. Addition to the solution mass flow rate in the jet mode causes two important changes in the hydrodynamic of the falling film. As the first change, it thickens the film around the tubes uniformly and as the second change, it increases the film velocity around and between the tubes. A reduction in the contact time between the water vapor and the falling solution is the important result of the latter. 18
Fig. 13. In-line jets, Left: 3-D contour of the interface, Right: velocity magnitude; Ref=42.6. 4.3. Heat and mass transfer 4.3.1. Drop mode Falling film around the tube is cooled by the cooling water flowing inside the pipe. However, the cooling effect of the tube reduces gradually during the formation of drops in the bottom of the tubes. On the other hand, the absorption of vapor releases the heat of absorption at the interface and causes a heat flux at the interface, thus increases the temperature of the solution. Fig. 14 shows the interface temperature and concentration counters at different drop formation steps. The binary solution becomes cooler around the cooling tubes and both temperature and concentration go down in this region, however during the formation of the drops beneath the tube, the solution experiences higher temperature and higher equilibrium concentration at the interface consequently. Fig. 15 shows the instantaneous absorbed mass flux of the water vapor at the interface, coincident with the contours of Fig. 14. The solution is moving toward the below part of the tube because of the effect of gravity and makes an extremely thin layer of solution (around 0.1 mm) on top portion of second and third tubes. This fine and very slow region experiences the lower temperature and consequently, the lower equilibrium concentration at the interface, resulting to a higher value of concentration gradient. The minimum temperature at the interface occurs at the top of cooling tubes where the falling drop is expected to fall on. This temperature is near 35 . The corresponding concentration is near 54 % LiBr, therefore a large concentration gradient exists at this point and for a very
19
short time (20 ms approximately) the value of local mass flux reaches to .
Fig. 14. Temperature and concentration contours at different drop formation stages at the interface, every 30 ms from left to right
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The local mass flux varies by the droplet impact on the lower pipe (Fig. 15 the last column on the right). The average values of the local mass flux at each stage are , , , and from left to right, respectively.
Fig. 15. Local mass flux (kg s-1 m-2) at the interface during drop formation stages 4.3.2. Jet mode As it was stated before, the average velocity of the falling film goes up in the jet mode; therefore, it makes a reduction in the residence time of the falling solution in contact with the water vapor. In addition, the thickness of the falling film around the tube becomes uniform along the length of the tubes. Fig. 16 presents the temperature and concentration contours for the in-line jets. By considering the temperature contour of the in-line jets at the interface, the minimum temperature is near 39 (4 degrees hotter than the drop mode), and the corresponding equilibrium concentration is 56.5% LiBr. Despite the drop mode, the jet mode can reach a steady shape in the hydrodynamic with no very thin region on top of the second or third tube. The water vapor is absorbed at the interface of jets and this absorption process causes higher temperature on jets than the film around the cooling tubes. The temperature shows lower values between two jets around the cooling tube, where the velocity of the two neighboring jet mixed with each other.
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Fig. 16. Temperature ( ) and concentration contours at the interface in in-line jets Fig. 17 presents the local mass flux at the interface for the in-line jet regime. The local mass flux at the jet regime shows lower value with respect to the drop regime.
Fig. 17. Local mass flux (kg s-1 m-2) at the interface of in-line jet mode In spite of the drop mode, the average value of local mass flux is in the jet mode. It indicates that the local mass flux in the drop mode is around ten times of the jet mode. The main reason of this fact can be found in the 22
high absorption rate on the very thin region on the top of the second and third tubes in the drop regime. Considering this fact, and the fact that the impact of drops on the cooling tubes makes a laminar wave, which has a mixing effect, the mass transfer in the drop mode should be higher than the jet mode. 5. Conclusion Three-dimensional simulation of water vapor absorption process into water-LiBr solution has been numerically performed inside a water cooled absorber chamber, consists of a horizontal tube bank (three tubes). The multiphase volume of fluid model has been utilized to capture the interface shape. Two different flow patterns between the horizontal tubes, namely drop mode and jet mode, have been studied. The heat and mass transfer equations have been solved with the hydrodynamics of fluid flow simultaneously. The absorption of vapor takes place at the interface, therefore the adaptive mesh refinement is used to increase the accuracy of the interface capturing as well as modeling the transport equations boundary layers. Results have been compared with available data on the literature and a good agreement is obtained. The simulation results reveal the superior vapor mass flux in the drop mode with respect to the jet mode. The main reason is in the unsteady nature of the drop regime with respect to the jet one. Drops form below the tubes and then they pinch off and fall over the lower tube and this continuous process makes a transient film thickness around the tubes. In the other words, the film thickness around the tubes in the drop regime becomes thinner in some spots and as a result, cooler than the jet regime. The cooler the film temperature is, the higher the concentration gradient will be at the interface and therefore the higher local mass flux is achieved. Moreover, the impact of the droplets on the cooling pipes makes a laminar wave which accompanies the mixing effect. This mixing effect is very weak (if not missing) in the jet regime.
Nomenclature cp specific heat of solution (J kg-1K-1) C concentration of LiBr ( kg LiBr/kg solution) D solution diffusivity (m2 s-1) gravity acceleration (m s-2) - along the fall direction Ga modified Galileo number ( ) -1 h enthalpy (J kg ) Habs heat of absorption (J kg-1) conductivity coefficient (W m-1 K-1) absorbed mass flux (kg s-1m-2) normal direction vector at the interface P pressure (Pa) 23
r R Re
normal distance from the pipe surface (m) pipe radious film Reynolds number (4 -1) absorbed mass source (kg m-3) time (s) temperature (K) velocity vector (m/s) heat flux at interface (W m-2) energy source term (Wm-3)
Greek symbols volume fraction (of liquid phase) mass flow rate per tube length, (kg s-1 m-1) film thickness (m) angle (shown in Fig. 5-A) interface curvature (m) spacing between neighboring jets or drops (m) kinematic viscosity (kg/m.s) density (kg/m3) surface tension coefficient(N/m) any property in the computation domain such as density, viscosity and so on Subscripts abs absorbed vapor cw cooling water eq equilibrium condition gen source term i inner inlet inlet condition int interface j each of liquid water or LiBr o outer volumetric w wall 0 pipe radious 1,2 liquid and gas phase References [1] P. Srikhirin, S. Aphornratana, S. Chungpaibulpatana, A review of absorption refrigeration technologies, Renew Sustain Energy Rev 5 (2001) 343–372 24
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Highlights
Different stages of drop formation under horizontal tubes are numerically simulated
Hydrodynamic of in-line jet mode between tubes are captured numerically
Transient nature of drop mode makes different film thickness around the tubes
Average local absorbed mass flux in drop mode is ten times larger than the jet mode
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