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Experiment and simulation of the shrinkage of falling film upon direct contact with vapor
Chemical Engineering Science 135 (2015) 52–60
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Experiment and simulation of the shrinkage of falling film upon direct contact with vapor Xiaoyu Quan, Yang Geng, Pingfang Yuan, Zheqing Huang, Chunjiang Liu n School of Chemical Engineering and Technology, State Key Laboratory of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, People's Republic of China
H I G H L I G H T S
Falling film direct contacting with saturated vapor was investigated. Experiments and a 3-D CFD model are proposed to study the phenomenon. Wetting area increases with increasing inlet temperature and flow rate. Detailed analysis explains the reason for the shrinkage. The h increases with increasing flow rate and inlet temperature.
art ic l e i nf o
a b s t r a c t
Article history: Received 29 December 2014 Received in revised form 15 June 2015 Accepted 21 June 2015 Available online 8 July 2015
An experiment is conducted to study the shrinkage of falling film in countercurrent direct contact with saturated vapor. The experiment indicates that falling film shrinks along its width. The degree of shrinkage decreases with increasing liquid inlet temperature and flow rate. To study the details of the phenomenon, a three-dimensional multiphase computational fluid dynamics (CFD) model based on the method of volume of fluid (VOF) is developed. The model considers the variation of surface tension with temperature. The results of the simulation are consistent with the experimental data. Simulations reveal the temperature, velocity and tension surface profiles. The modified Marangoni number is compared to illustrate the influence of surface tension and demonstrates that the surface tension plays an important role in this phenomenon and the shrinkage of the film causes the heat transfer to decrease. This model is expected to be applied to the design of the industrial equipment with falling film. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Falling film Shrinkage CFD Surface tension Heat transfer
1. Introduction Falling film is a common flow pattern in industrial equipment such as distillation, absorption, evaporation and water desalination systems (Huppert, 1986; Oron et al., 1997; Craster, 2009). Because of falling film's high heat and mass transfer coefficients, high heating flux and low energy consumption, it is extensively used in industrial processes. In a packing distillation column, the liquid forms falling film on the packing while the vapor, generated in the reboiler, flows upward countercurrently and directly contacts the liquid film. Vapor and liquid exchange heat and mass on the interface of falling film. Although the temperature difference between liquid and vapor is small in most of the column, it is larger in certain parts where reflux occurs, such as at the top of the column,
n
Corresponding author. Tel.: þ 86 22 23502063; fax: þ86 22 27404496. E-mail address: [email protected] (C. Liu).
http://dx.doi.org/10.1016/j.ces.2015.06.055 0009-2509/& 2015 Elsevier Ltd. All rights reserved.
bscause the temperature of the reflux is lower than the boiling point in practical production. In recent years, research has investigated the flow pattern of non-isothermal falling film. Published studies have demonstrated that the surface tension gradients caused by the non-isothermal interfacial temperature increase interfacial instabilities (Davis, 1987; Schatz and Neitzel, 2001). Correspondingly, these studies focused on the interfacial instabilities such as the surface wave and breaking of the film. Joo et al. (1996) presented a long-wave evolution equation that was used to describe the surface-wave and thermocapillary instabilities of a film on a heated plate. These researchers used this model to demonstrate a mechanism of rivulet formation through their model. Kim (1999) used the nonlinear evolution equations governing 2-D surface waves to study the film at a constant heat flux and a fixed temperature. This research showed that the film at a constant heat flux was a more stable system than that at a fixed temperature condition. Miladinova and Lebon (2005) studied the dynamics of a thin evaporating liquid film falling down an inclined plate in the cases of uniformly and nonuniformly heated plates.
X. Quan et al. / Chemical Engineering Science 135 (2015) 52–60
They found that the effect of the nonuniform heating was dominant prior to the film disappearance and that it enforced film rupture. Zhang et al. (2008a) provided a 2-D theoretical model to study the temperature distribution of falling film flowing over a vertical heated/cooled plate at a constant temperature. Zhang et al. (2008b) investigated the temperature field and flow patterns of falling film at different heating conditions. Because falling film flow is influenced by many factors, it is difficult to analyze the heat and mass transfer process accurately through experimentations. During the recent the past, with the development of computer techniques and the theory of computational fluid dynamics (CFD), CFD has become an important method for researchers to simulate the flow phenomenon of falling film. By solving the continuity, momentum, energy and species equations, CFD can predict velocity, pressure, temperature and concentration profiles in complex systems (Haelssig et al., 2010). Cherif and Daif (1999) numerically studied the evaporation of the binary liquid film streaming on the internal face of one of the two parallel plates using mixing convection. The wetted plate underwent a constant uniform heat flux whereas the other was adiabatic. They showed the importance of the film thickness and mixture composition on the mass and thermal transfers. Agunaoun et al. (1998) presented a numerical analysis of the heat and mass transfer in a binary liquid film flowing on an inclined plate. The most interesting results were obtained in mixed convection, particularly in the case of an ethylene glycolewater mixture. In fact, the results obtained by Agunaoun et al. showed that it was possible to increase the accumulated evaporation rate of water when the inlet liquid concentration of ethylene glycol was less than 40%. Hoke and Chen (1992) presented a numerical study on the evaporation of a binary liquid film on a vertical plate. These authors presented the evolution of Sherwood and Nusselt numbers. Mhetar and Slattery (1997) studied the isothermal evaporation of a binary liquid film. The reserchers measured the diffusion coefficient during the evaporation of a binary liquid in a Stefan tube. With the development of computer techniques, some 3-D models were presented to simulate falling film. Chen et al. (2009) presented a 3-D two-phase model based on the VOF method to study the hydrodynamics and mass transfer behavior of packing material (Mellapak 350Y). AlRawashdeh et al. (2008, 2012) presented a pseudo 3-D CFD model, in which the 3-D Navier–Stokes equations, governing the three Cartesian velocity components, were reduced to a single 2-D Poisson equation for the cross sectional profile of the axial velocity component; this model is used to investigate the effects of channel fabrication precision and liquid flow distribution in a microreactor. Sebastia-Saez et al. (2013) presented a small-scale 3-D CFD model for the study of the hydrodynamics and physical mass transfer in structured packing elements. They found that surface textures have a strong influence on liquid misdistribution, which had a significant influence on the interfacial absorption rate. Qi et al. (2013) developed a 3-D model for predicting the wetting factor, film thickness and flow velocity of falling film. Sebastia-Saez et al. (2014) developed a VOF-based micro-scale 3D numerical model to study the influence of several operative parameters on absorption of gas into falling liquid films. Although many simulations and experiments have already been conducted on the flow pattern of non-isothermal falling film, the temperature difference was caused by the heated plate. These experiments and simulations differ from practical processes in a distillation column in which the temperature difference is caused by high temperature vapor. In this paper, an experiment is conducted to study the flow pattern of falling film in direct contact with saturated vapor. To study the details of the phenomenon and simulate it experimentally, a 3-D CFD model is presented, in which the variation of the surface tension with temperature is considered. Through the
53
model, the temperature and surface tension profiles are revealed to illustrate the shrinkage. The modified Marangoni number is estimated and compared to illustrate the influence of surface tension. Moreover, the variation of the heat transfer behavior caused by the shrinkage is investigated.
2. Experimental apparatus and procedure To study the flow pattern of falling film, that countercurrently and directly contacts high-temperature vapor, an experiment with water and vapor was performed. The schematic diagram of the experimental setup is shown in Fig. 1. The liquid stored in the feed storage tank was sent to the falling film contactor by the centrifugal pump. A temperature-controlling device controlled its temperature, and flow rate was measured by a rotameter. The liquid entered the falling contactor at the top through the liquid distributor forming falling film on a steel plate and exited at the bottom to the liquid recovery tank. The vapor was generated by a vapor generator, entered the falling film contactor through a vapor distributor at the bottom and exited at the top of the contactor. The liquid and vapor countercurrently and directly contacted and exchanged mass and heat in the contactor. The size of the steel plate was 400 mm 100 mm 6 mm (length width thickness). The flow pattern and distribution of the temperature were recorded by an infrared camera (Ti 200, Fluke, Inc). The temperature at the inlet and outlet were measured by thermoelectric thermometers. Water and vapor were chosen as the experimental materials. The temperature of the water entering the contactor was chosen to be 30, 40, 50, 60, and 70 1C, and the flow rate was set at 60, 80, 100 and 120 L/h. The vapor flow was constant and set at a 20 m3/h. This rate can maintain the temperature of the entire contactor at the saturation temperature. The effect of the vapor rate was not considered in the study, so the rate was constant.
3. Mathematical model Falling film that countercurrently and directly contacts vapor at a high temperature is influenced by many factors. It is difficult to accurately analyze the heat and mass transfer process through experimentations. Therefore, a 3-D simulation model was presented
Fig. 1. Schematic diagram of the experimental setup, (1) Feed storage (2) centrifugal pump; (3) temperature-controlling device; (4) rotameter; (5) distributor; (6) insulating layer; (7) steel plate; (8) falling film contactor; (9) recovery tank; (10) vapor generator; (11) vortex street flowmeter; (12) distributor; (13) infrared camera.
tank; liquid liquid vapor
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to analyze the flow pattern and the heat and mass transfer phenomenon. Some important properties used in the simulation are shown in table 1. 3.1. Governing equations 3.1.1. Multiphase model Because counter-current flow of liquid and vapor occurred in the experiment, the VOF multiphase model was used to solve this problem (Nikolopoulos et al., 2007; Banerjee, 2007; Schlottke and Weigand, 2008). In CFD, the VOF method is a numerical technique for tracking and locating the free surface (or fluid-fluid interphase). It solves a single set of momentum equations shared by the fluids, and the volume fraction of each phase in the cell is tracked within the entire flow zone. Interphase tracking is thus achieved by the solution of the transport equation for the volume fraction of one of the phases. The form of the VOF conservation equation is 1 ∂ αq ρq þ ∇d αq ρq uq ¼ Sαq ð1Þ ρq ∂t The volume fraction αq takes the following values: αq ¼1 shows that the cell is entirely filled with the q phase. αq ¼0 shows that the cell is inexistence of the q phase. 0 o αq o1 shows that the cell is partially filled with the q phase. The cell exists at the interphase of different phase. The volume fraction for the primary phase is not solved because it can be computed based on the following constraint: n X
αq ¼ 1
ð2Þ
q¼1
Because there are two phases in the experiment—namely, liquid and vapor—n ¼2 and ¼V and L in the equations. The properties of the fluid that appear in the conservation equations are determined from the volume fraction of each phase. For example, density can be presented as follows: X ρ¼ α q ρq ð3Þ The VOF equation provides only a diffuse approximation of the interphase location. To identify the actual location of the interphase in a cell, the interphase has to be captured or reconstructed. The most extensively used approach for interphase reconstruction is piecewise linear interphase calculation (PLIC), which assumes that the interphase between the two phases has a linear slope within each cell, and uses this property to calculate the advection of fluid through the cell faces using a geometric reconstruction scheme. In this study, PLIC is selected to reconstruct the interphase. 3.1.2. Momentum equations In the VOF model a single momentum equation is solved within the entire flow zone, and the resulting velocity is shared among the phases. The equation can be written as follows: ∂ ðρuÞ þ ∇d ρuu ¼ ∇P þ ∇d μð∇u þ ∇uT Þ þ ρg þ F σ þ F drag ∂t
ð4Þ
For the falling film flow with a counter-current vapor phase, the driving force for the liquid phase is gravity, g, whereas the force for the vapor phase is the pressure drop. Therefore a fraction pressure drop model is considered to describe the drag force source, Fdrag, between the two phases. The pressure drop between two phases can be described as follows (Woerlee et al., 2001):
ð5Þ F drag ¼ αeff f i ρV uV uiy uV uiy
where aeff is the effective interfacial area per unit volume, which can be expressed as αef f ¼ j∇αV j ¼ j∇αL j(Nicolaiewsky and Fair (1999)). fi represents the drag force factor. According to the study of Stephan, and Mayinger (1992), the drag force factor of the falling film flow with a counter-current vapor phase on a rectangular plate can be described by the following equation: nN f i ¼ 0:079ReV 0:25 1 þ 115δ ð6Þ where N ¼ 3:95 1:8 þ 3=Dn , δn and Dn are the dimensionless thickness of the film and hydrodynamic diameter to the liquid 0:5 Laplace length δ= ρL ρV g , respectively. The surface tension source term, Fσ, is considered by the continuum surface model (CFS) proposed by Brackbill et al. (1992). The source of surface tension force can be described as follows: F σ ¼ σ ij
ρκ ∇α i i 0:5 ρi þ ρj
where σ is the surface tension coefficient. The variation of surface tension with temperature was considered in this model. The relationship between surface tension and water temperature can be expressed as follows (Tang, 2000):
σ ¼ 0:09537 2:24 10 6 T 2:56 10 7 T 2
Properties
Liquid
Vapor
Inlet temperature, 1C Density, kg/m3 μ, viscosity, mPa s Flow rate
30–70 995–965 0.8007–0.3565 60–120 L/h
100 0.5542 0.0134 20 m3/h
ð8Þ
κ is the free surface curvature, which is defined as: κ ¼ ∇ U_ n¼
1 n U ∇ jnj ð∇ U nÞ jnj jnj
ð9Þ
where _ n ¼ n=jnj,n ¼ ∇αi . The unit surface normal at the live cell adjacent to the wall is replaced by the following equation: _ _ n ¼_ n w cos γ w þ t w sin γ w
ð10Þ
_ where _ n w and t w are the unit vectors normal and tangential to the wall, respectively. The contact angle γ w is the angle between the wall and the tangent to the interphase at the wall.
3.1.3. Turbulence equation The RNG k–ε turbulence model is chosen for the turbulence calculation in this work. The equations are presented as follows: ∂ðρkÞ ∂ðρkui Þ ∂ ∂k þ ¼ αk μef f ð11Þ þ Gk ρε ∂t ∂xi ∂xi ∂xi ∂ðρεÞ ∂ðρεui Þ ∂ ∂ε ε ε2 þ ¼ αk μef f þ C 1ε Gk C n2ε ρ ∂t ∂xi ∂xi ∂xi k k
Table 1 Physical properties of vapor and liquid phases.
ð7Þ
where
η ¼Sk/ε, and Sk is a user-defined source term.
C n2ε C 2ε þ where
ð12Þ
C μ η3 ð1 η=η0 Þ 1 þ βη3
η0 ¼ 4.38, β ¼0.012, C1ε ¼1.42 and C2ε ¼ 1.68.
ð13Þ
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3.1.4. Energy equation The energy equation is shared by two phases and can be written as: X ∂ ρE þ ∇ u ρE þ p ¼ ∇ðkeff ∇T hj jj Þ þ Se ð14Þ ∂t j where Se is the energy source term and keff is the effective conductivity. Correspondingly, keff ¼k þkt, where kt is the turbulent thermal conductivity. Energy E and temperature T are mass averaged as follows: Pn q 1 α q ρq E q E ¼ Pn ð15Þ q1
Pn
q1
T ¼ Pn
α q ρq
α q ρq T q α q ρq
ð16Þ
q1
3.2. Source term Owing to the difference in temperatures between liquid and vapor, upon their directly contact, the vapor would condensate on the interphase and release latent heat. Thus, the source terms that represent the exchanged heat and mass on the interphase are calculated within the domain 0.2 o αL o0.7. The vapor-liquid phase change model presented by Sun et al. (2012) was chosen for the simulation. The model is suitable for the case in which one phase is unsaturated and the other is saturated. In our experiment and simulation, the water is considered to be unsaturated and vapor is considered to be saturated. In this model, the thermal conductivity of the vapor (saturated-phase) is assumed to be zero, which means that the vapor maintains a constant temperature and exchanges heat with the liquid only through condensation. Therefore, the energy source can be expressed as follows: SE ¼ 2kef f αef f ∇T
ð17Þ
Fig. 2. Geometry and mesh of the simulation.
In the phase change model, it is assumed that the heat is exchanged between vapour and liquid through condensation. Thus, the mass source can be expressed in the following equation: SL ¼ SV ¼
2kef f αef f ∇T ΔH
ð18Þ
3.3. Flow geometries In this work, a 3-D model is presented to simulate the experiment. Falling film is postulated to be symmetric along the vertical direction. To reduce the computational cost, a 400 mm 50 mm 6 mm geometry was used in the simulation, one side was set to be symmetric, as shown in Fig. 2. With regard to the accuracy, hexahedral grid elements are used in the entire flow domain. Compared to the vapor flow regions, the momentum and temperature gradients are larger in both the liquid film flow regions and the liquid–vapor interphase regions. Therefore, the grids in the liquid flow regions and liquid–vapor interphase regions should be denser than the vapor flow regions. In this work, mesh encryption was adopted to draw the grids, such that their density gradually increased from the vapor phase to the liquid film regions as shown in Fig. 2. The difference between the simulation results and the experimental data with mesh number is shown in Fig. 3. The figure shows that increasing the grid number improves the accuracy of the simulation results, but the improvement is not obvious once the number exceeds 576,000. Considering the accuracy and computing cost, the total mesh number of the grid is chosen to be 576,000. The boundary conditions of the simulation are shown in Table 2.
Fig. 3. Variation of the simulation results with grid number (Tin ¼30 1C, F ¼60 L/h).
4. Results and discussion 4.1. Shape of falling film To ensure that the fluid region of falling film was clearly observed, red ink was added to the water. Fig. 4 shows the shape of the falling film with vapor and without vapor at the same flow rate (80 L/h, Tin ¼30 1C). The red part represents the fluid region of the falling film. It can be observed from the figure that the fluid region of falling film with vapor is smaller than that without vapor.
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Table 2 Boundary conditions. Field
Liquid inlet
Liquid outlet
Vapor inlet
Vapor outlet
Velocity Volume fraction
uL,x ¼ 0, uL,y ¼ uL,in, uL,z ¼ 0 αL ¼ 1, αV ¼0
Pressure outlet —
uV,x ¼ 0, uV,y ¼uL,in, uV,z ¼ 0 αL ¼ 0, αV ¼ 1
Pressure outlet —
Fig. 6. Variations of width on the Y direction. Fig. 4. Shape of falling film.(For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
error between simulation and experiment is 6%, which means that the simulation is acceptable in the industry. In addition to the wetting area of falling film, the width shown in Fig. 4 is also an important factor for the shape of falling film. Fig. 6 shows the variations of the width at a flow rate of 80 L/h along the Y direction. It is shown that the width of falling film decreases from the top (at Y¼0) to the bottom (at Y¼ 0.4 m). Moreover, at the same position, the width increases with increasing inlet temperature. When the inlet temperature is high (Tin ¼50, 60, 70 1C), the variance of the width is low because the location changes from Y¼ 0.3 m to the bottom Y¼ 0.4 m. 4.2. Analysis of the simulation
Fig. 5. Wetting area of falling film with different inlet temperatures and flow rates.
In the case with vapor, the falling film obviously shrinks along the X direction. The wetting area of falling film is an important factor in industrial equipments. Fig. 5 compares the wetting area of falling film at various inlet temperatures and flow rates in the study. Simulation results are consistent with the experimental data and show the same trend. The wetting area of falling film increases with increasing inlet temperature and flow rate. This means that the larger the temperature difference between vapor and inlet liquid, the smaller the wetting area. The growth rate of the low flow rate (60, 80 L/h) within the temperature range of 50–70 1C is larger than that elicited for the range of 30–50 1C, and it changes little at higher flow rates (100, 120 L/h). The growth rate ranged from 10% to 30% within the temperature range of 30–70 1C at the same flow rate. Additionally, it decreased by 35–15% within the range of 60–120 L/h at the same inlet temperature. The maximum
Fig. 7(a) shows the temperature profile of the experiment (Tin ¼ 40 1C, F¼80 L/h) which was recorded by an infrared camera. The figure shows that the temperature on the edge is obviously higher than that in the main body. Fig. 7(b) shows the temperature distribution of the black line in Fig. 7(a). This figure indicates that the temperature on the edge of the film is approximately 10 1C higher than that of the main body. To ascertain the reason behind the shrink of falling film upon contact with saturated vapor, simulation results were used to investigate the details of the process. Fig. 8 shows the liquid profile in the X–Z plane at various Y locations (Tin ¼30 1C, F ¼60 L/h). Owing to the symmetry of the simulation model, only half of the profile is shown, and the symmetry axis is presented in the figure. It is shown that the thickness of falling film on the edge is thicker than that in the middle. The cause for this phenomenon is the liquid shrinkage on the edge, which does not provide enough time for it to spread. The width thus decreases from top to bottom (Y¼0.4 m), which corresponds to the results shown in Fig. 6. Fig. 8 (a) shows the temperature profile of the liquid. It is obvious that the temperature on the edge is higher than that on the main body, which corresponds to the experimental results shown in Fig. 7. The surface tension is strongly dependent on temperature as shown in Eq. (8). Fig. 8(b) shows the variation of surface tension. Owing to the higher temperature at the edge, the surface tension
X. Quan et al. / Chemical Engineering Science 135 (2015) 52–60
at the edge is smaller than that of the main body. Because of the distribution of surface tension, the film on the edge tends to flow toward the middle, causing shrinkage of the falling film. Fig. 8 (c) shows the velocity along the X direction of the falling film, which is clearly higher on the edge. As a result, the falling film will shrink upon direct contact with the high temperature vapor. Fig. 9 shows the liquid profile at Y¼0.2 m in the case where Tin ¼ 70 1C, and F¼ 60 L/h. Compared to Fig. 8, the variation of the temperature in Fig. 9 is smaller owing to the smaller difference between liquid and vapor. Therefore, the surface tension gradient and X-velocity are smaller, as shown in Fig. 9. Because of this, the degree of shrinkage in falling film decreases with increasing inlet temperature. Fig. 10 shows the liquid profile at Y¼ 0.2 m in the case where Tin ¼ 30 1C, and F¼ 120 L/h. Compared to Fig. 8, the film is thicker and the temperature is lower. Although the variation of the surface tension is obvious, the X-velocity is smaller than that in Fig. 8 because of the high mass flow. Therefore, the degree of falling film shrinkage decreases with increasing flow rate as shown in Fig. 5.
4.3. Modified Marangoni number analysis The surface tension gradient caused by the heterogeneous temperature distribution leads to the formation of interfacial shear stress in the tangential direction, which can initiate Marangoni convection, this phenomenon is also called the Marangoni effect. In most cases, a higher temperature difference causes a larger shear stress, which has a strong effect on the liquid film flow characteristics. In this paper, a modified Marangoni number is applied to characterize the influence of surface tension on the falling liquid film flow (Kabov and Chinnov (2001)). MaX ¼
2 L∂∂Tσ ΔT lν=L'
αμ
¼
∂σ ∂T ðT e T av Þ Ul
αμ
ν lν=L'
ð19Þ
57
where MaX represents modified Marangoni numbers in the X 0 direction, L is the width of the falling film, Te is the temperature of the film edge, and Tav is the average temperature of the falling liquid film. Correspondingly, α is the thermal diffusion coefficient, defined as α ¼ k= ρcp , k is the thermal and lv is the conductivity, 1=3 viscous length factor, defined as lν ¼ ν2 =g . Fig. 11 shows the influence of the liquid phase flow rate and the temperature differences on MaX in the X direction. For liquid film in direct contact with vapor, ΔT is positive, thereby making MaX negative. Therefore, in Fig. 11, the smaller MaX is, the larger the Marangoni effect is. Upon the increase of inlet temperature, the MaX of the falling film first increases. While Tin 450 1C, with the increase of the inlet temperature, the MaX of the falling film changes slightly. Fig. 11 also shows that MaX increases with increasing flow rate. Because the faster renewal rate induced by the increased liquid flow rate reduces the temperature difference between liquid and vapor, it also results in a larger MaX. 4.4. Heat transfer behavior Heat and mass transfer have received increased attention in industrial processes. In this study, water is the single component, such that the heat transfer can substitute for the mass transfer. In practical industrial processing, the heat and mass transfers of falling film often occurs in certain areas. Therefore, to investigate the effect of falling film shrinkage on heat transfer, the actual heat transfer coefficient hactual and overall heat transfer coefficient h are defined as follow: hactual ¼
h¼
mcp ðT out T in Þ Q overall ¼ Aactual ΔT Aactual ðT sat T in þ2T out Þ
Q overall mcp ðT out T in Þ ¼ A ΔT AðT sat T in þ2T out Þ
ð20Þ
ð21Þ
where m is the mass flow of the liquid, cp is the special heat of the
Fig. 7. Temperature profile of experiment.
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Fig. 9. Liquid profile in X–Z plane at Y¼0.2 m (Tin ¼701C, F¼ 60 L/h).
Fig. 10. Liquid profile in X–Z plane at Y ¼ 0.2 m (Tin ¼30 1C, F¼ 120 L/h).
Fig. 8. Liquid profile in the X–Z plane at various Y locations, (Tin ¼30 1C, F ¼60 L/h) (a) temperature; (b) surface tension; (c) X-velocity. Fig. 11. Influence of liquid flow rate and temperature on MaX in the X direction.
liquid, Tin and Tout are the inlet and outlet temperatures of the liquid, respectively, and Tsat is the saturated temperature of the the vapor. Aactual represents the actual wetting area of falling film shown in Fig. 5 and A ¼0.04 m2, represents the entire area of the steel plate, which corresponds to a certain area of the equipment. Fig. 12 shows that hactual decreases with the increasing the inlet temperature at the same flow rate, and with increasing flow rate at the same inlet temperature. However, in industrial processes, heat and mass transfer occur in a fixed area, such that the overall heat
transfer coefficient h is more important. Compared to Fig. 12, Fig. 13 shows that h ranges from 15% to 40% smaller than hactual. This means that the heat transfer coefficient will dramatically decrease when the shrinkage occurs. The variation trend of heat transfer is opposite such that h increases with increasing flow rate and inlet temperature. It is illustrated that the shrinkage of the falling film has an important effect on heat and mass transfer. This
X. Quan et al. / Chemical Engineering Science 135 (2015) 52–60
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coefficient h increases with increasing flow rate and inlet temperature.
Nomenclature
Fig. 12. Variations of actual heat transfer coefficient.
A aeff cp D E Eff F Fσ Fdrag fi g H j k keff m P S T u
area of the falling film, m2 effective interfacial area per unit volume, m2/m³ specific heat capacity, J/kg K diffusion coefficient, m2/s energy, J/kg mass transfer efficiency momentum source term, N/m³ surface tension source term drag force source drag force factor gravitational acceleration, m/s2 latent heat of vaporization, J/kg species flux, kg/m2s thermal conductivity, W/mK effective thermal conductivity, W/mK mass transfer rate, kg/s pressure, Pa source term temperature, K velocity, m/s
Greek letters
δ α αeff σ κ γ γw Fig. 13. Variations of overall heat transfer coefficient.
shows that when falling film directly contacts high temperature vapor, it is important that the shrinkage be considered in the design.
thickness of the liquid film volume fraction effective interfacial area per unit volume, m2/m³ surface tension coefficient, N/m free surface curvature activity coefficient contact angle
Subscripts i q V L in out
ith species qth phase vapor phase liquid phase liquid inlet liquid outlet
5. Conclusion Experiments and a 3-D CFD model have been proposed to investigate the shrinkage of falling film, that countercurrently and directly contacts high temperature vapor. In the simulation model, the variation of the surface tension with temperature is considered in the momentum equation. The following conclusions can be drawn from this work: (1) It is demonstrated that falling film will shrink upon contact with the vapor, and the area of falling film increases with increasing inlet temperature and flow rate. (2) Using the proposed model, a detailed analysis of falling film shrinkage reveals temperature and surface tension dispersion and explains the reason for the shrinkage. The modified Marangoni number is compared to illustrate the influence of surface tension. (3) It is shown that the shrinkage of falling film has an important effect on heat transfer and that the overall heat transfer
Acknowledgment The authors gratefully acknowledge the financial support by the National Natural Science Foundation of China (Project no. 21406157). References Agunaoun, A., Idrissi, A., Daif, A., Barriol, R., 1998. Etude de l'évaporation en convection mixtes d'un film liquide d'un mélange binaire s'écoulant sur un plan incliné soumis à un flux de chaleur constant. Int. J. Heat Mass Transf. 41, 197–210. Banerjee, R., 2007. A numerical study of combined heat and mass transfer in an inclined channel using the VOF multiphase model. Numer. Heat Transf. Part A 52, 163–183. Brackbill, J.U., Kothe, D.B., Zemach, C., 1992. A Continuum method for modeling surface tension. J Comput. Phys. 100, 335–354.
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Cherif, A.A., Daif, A., 1999. Etude numérique du transfert de chaleur et de masse entre deux plaques planes verticales en présence d'un film de liquide binaire ruisselant sur l'une des plaques chauffées. Int. J. Heat Mass Transf. 42, 399–418. Chen, J.B., Liu, C.J., Yuan, X.G., Yu, G.C., 2009. CFD simulation of flow and mass transfer in structured packing distillation columns. Chin. J. Chem. Eng. 17, 381–388. Craster, R.V., 2009. Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1198. Davis, S.H., 1987. Thermocapillary instabilities. Annu. Rev. Fluid Mech. 19, 403–435. Haelssig, J.B., Tremblay, A.Y., Thibault, J., Etemad, S.G., 2010. Direct numerical simulation of interphase heat and mass transfer in multicomponent vapourliquid flows. Int. J. Heat Mass Transf. 53, 3947–3960. Hoke Jr., B.C., Chen, J.C., 1992. Mass transfer in evaporating falling liquid film mixtures. AIChE J. 38, 781–787. Huppert, H.E., 1986. Flow and instability of a viscous current down a slope. Nature 300, 427–429. Joo, S.W., Davis, S.H., Bankoff, S.G., 1996. A mechanism for rivulet formation in heated falling films. J. Fluid Mech. 321, 279–298. Kabov, O.A., Chinnov, E.A., 2001. Heat transfer from a local heat source to subcooled liquid film. High Temp. 39, 703–713. Kim, H., 1999. Stability analysis of heated thin liquid-him flows with constant thermal boundary conditions. Korean J. Chem. Eng. 16, 764–773. Al-Rawashdeh., M., Hessel., V., Löb., P., Mevissen., K., Schönfeld., F., 2008. Pseudo 3D simulatin of a falling film microreactor based on realistic channel and film profiles. Chem. Eng. Sci. 63, 5149–5159. Al-Rawashdeh., M., Cantu-Perez., A., Ziegenbalg., D., Löb., P., Gavriilidis., A., Hessel., V., Schönfeld., F., 2012. Microstructure-based intensification of a falling film microreactor through optimal film setting with realistic profiles and in-channel induced mixing. Chem. Eng. J. 179, 318–329. Miladinova, S., Lebon, G., 2005. Effects of nonuniform heating and thermocapillarity in evaporating films falling down an inclined plate. Acta Mech. 174, 33–49. Mhetar, V.R., Slattery, J.C., 1997. The Stefan problem of a binary liquid mixture. Chem. Eng. Sci. 52, 1237–1242.
Nicolaiewsky, E.M.A., Fair, J.R., 1999. Liquid fow over textured surfaces. 1. contact angles. Ind. Eng. Chem. Res. 39, 284–291. Nikolopoulos, N., Theodorakakos, A., Bergeles, G., 2007. A numerical investigation of the evaporation process of a liquid droplet impinging onto a hot substrate. Int. J. Heat Mass Transf. 50, 303–319. Oron, A., Davis, S.H., Bankoff, S.G., 1997. Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980. Qi, R.H., Lu, L., Yang, H.X., Qin, F., 2013. Investigation on wetted area and film thickness for falling film liquid desiccant regeneration system. Gen. Inf. 112, 93–101. Schatz, M.F., Neitzel, G.P., 2001. Experiments on thermocapillary instabilities. Annu. Rev. Fluid Mech. 33, 93–127. Schlottke, J., Weigand, B., 2008. Direct numerical simulation of evaporating droplets. J. Comput. Phys. 227, 5215–5237. Sebastia-Saez, D., Gu, S., Ranganathan, P., Papadikis, K., 2013. 3D modeling of hydrodynamics and physical mass transfer characteristics of liquid film flows in structured packing elements. Int. J. Greenh. Gas Control 19, 492–502. Sebastia-Saez, D., Gu, S., Ranganathan, P., Papadikis, K., 2014. Micro-scale cfd study about the influence of operative parameters on physical mass transfer within structured packing elements. Int. J. Greenh. Gas Control 28, 180–188. Stephan, M., Mayinger, F., 1992. Experimental and analytical study of countercurrent flow limitation in vertical gas/liquid flows. Chem. Eng. Technol. 15, 51–62. Sun, D.L., Xu, J.L., Wang, L., 2012. A vapor-liquid phase change model for two-phase boiling and condensation. J. Xi'an Jiao Tong Univ. 46, 7–11. Tang, C.Y., 2000. The relationship between surface tension with temperature of water. J. Anqing Teach. Coll. (Nat. Sci.) 6, 73–74. Woerlee, G.F., Berends, J., Olujic, Z., Graauw, J.A., 2001. A comprehensive model for the pressure drop in vertical pipes and packed columns. Chem. Eng. J. 84, 367–379. Zhang, F., Tang, D.L., Geng, J., Zhang, Z.B., 2008a. Study on the temperature distribution of heated falling liquid films. Physica D 237, 867–872. Zhang, F., Wu, Y.T., Geng, J., Zhang, Z.B., 2008b. An investigation of falling liquid films on a vertical heated/cooled plate. Int. J. Multiph. Flow 34, 13–28.
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Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces
Experiment and simulation of the shrinkage of falling film upon direct contact with vapor Xiaoyu Quan, Yang Geng, Pingfang Yuan, Zheqing Huang, Chunjiang Liu n School of Chemical Engineering and Technology, State Key Laboratory of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, People's Republic of China
H I G H L I G H T S
Falling film direct contacting with saturated vapor was investigated. Experiments and a 3-D CFD model are proposed to study the phenomenon. Wetting area increases with increasing inlet temperature and flow rate. Detailed analysis explains the reason for the shrinkage. The h increases with increasing flow rate and inlet temperature.
art ic l e i nf o
a b s t r a c t
Article history: Received 29 December 2014 Received in revised form 15 June 2015 Accepted 21 June 2015 Available online 8 July 2015
An experiment is conducted to study the shrinkage of falling film in countercurrent direct contact with saturated vapor. The experiment indicates that falling film shrinks along its width. The degree of shrinkage decreases with increasing liquid inlet temperature and flow rate. To study the details of the phenomenon, a three-dimensional multiphase computational fluid dynamics (CFD) model based on the method of volume of fluid (VOF) is developed. The model considers the variation of surface tension with temperature. The results of the simulation are consistent with the experimental data. Simulations reveal the temperature, velocity and tension surface profiles. The modified Marangoni number is compared to illustrate the influence of surface tension and demonstrates that the surface tension plays an important role in this phenomenon and the shrinkage of the film causes the heat transfer to decrease. This model is expected to be applied to the design of the industrial equipment with falling film. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Falling film Shrinkage CFD Surface tension Heat transfer
1. Introduction Falling film is a common flow pattern in industrial equipment such as distillation, absorption, evaporation and water desalination systems (Huppert, 1986; Oron et al., 1997; Craster, 2009). Because of falling film's high heat and mass transfer coefficients, high heating flux and low energy consumption, it is extensively used in industrial processes. In a packing distillation column, the liquid forms falling film on the packing while the vapor, generated in the reboiler, flows upward countercurrently and directly contacts the liquid film. Vapor and liquid exchange heat and mass on the interface of falling film. Although the temperature difference between liquid and vapor is small in most of the column, it is larger in certain parts where reflux occurs, such as at the top of the column,
n
Corresponding author. Tel.: þ 86 22 23502063; fax: þ86 22 27404496. E-mail address: [email protected] (C. Liu).
http://dx.doi.org/10.1016/j.ces.2015.06.055 0009-2509/& 2015 Elsevier Ltd. All rights reserved.
bscause the temperature of the reflux is lower than the boiling point in practical production. In recent years, research has investigated the flow pattern of non-isothermal falling film. Published studies have demonstrated that the surface tension gradients caused by the non-isothermal interfacial temperature increase interfacial instabilities (Davis, 1987; Schatz and Neitzel, 2001). Correspondingly, these studies focused on the interfacial instabilities such as the surface wave and breaking of the film. Joo et al. (1996) presented a long-wave evolution equation that was used to describe the surface-wave and thermocapillary instabilities of a film on a heated plate. These researchers used this model to demonstrate a mechanism of rivulet formation through their model. Kim (1999) used the nonlinear evolution equations governing 2-D surface waves to study the film at a constant heat flux and a fixed temperature. This research showed that the film at a constant heat flux was a more stable system than that at a fixed temperature condition. Miladinova and Lebon (2005) studied the dynamics of a thin evaporating liquid film falling down an inclined plate in the cases of uniformly and nonuniformly heated plates.
X. Quan et al. / Chemical Engineering Science 135 (2015) 52–60
They found that the effect of the nonuniform heating was dominant prior to the film disappearance and that it enforced film rupture. Zhang et al. (2008a) provided a 2-D theoretical model to study the temperature distribution of falling film flowing over a vertical heated/cooled plate at a constant temperature. Zhang et al. (2008b) investigated the temperature field and flow patterns of falling film at different heating conditions. Because falling film flow is influenced by many factors, it is difficult to analyze the heat and mass transfer process accurately through experimentations. During the recent the past, with the development of computer techniques and the theory of computational fluid dynamics (CFD), CFD has become an important method for researchers to simulate the flow phenomenon of falling film. By solving the continuity, momentum, energy and species equations, CFD can predict velocity, pressure, temperature and concentration profiles in complex systems (Haelssig et al., 2010). Cherif and Daif (1999) numerically studied the evaporation of the binary liquid film streaming on the internal face of one of the two parallel plates using mixing convection. The wetted plate underwent a constant uniform heat flux whereas the other was adiabatic. They showed the importance of the film thickness and mixture composition on the mass and thermal transfers. Agunaoun et al. (1998) presented a numerical analysis of the heat and mass transfer in a binary liquid film flowing on an inclined plate. The most interesting results were obtained in mixed convection, particularly in the case of an ethylene glycolewater mixture. In fact, the results obtained by Agunaoun et al. showed that it was possible to increase the accumulated evaporation rate of water when the inlet liquid concentration of ethylene glycol was less than 40%. Hoke and Chen (1992) presented a numerical study on the evaporation of a binary liquid film on a vertical plate. These authors presented the evolution of Sherwood and Nusselt numbers. Mhetar and Slattery (1997) studied the isothermal evaporation of a binary liquid film. The reserchers measured the diffusion coefficient during the evaporation of a binary liquid in a Stefan tube. With the development of computer techniques, some 3-D models were presented to simulate falling film. Chen et al. (2009) presented a 3-D two-phase model based on the VOF method to study the hydrodynamics and mass transfer behavior of packing material (Mellapak 350Y). AlRawashdeh et al. (2008, 2012) presented a pseudo 3-D CFD model, in which the 3-D Navier–Stokes equations, governing the three Cartesian velocity components, were reduced to a single 2-D Poisson equation for the cross sectional profile of the axial velocity component; this model is used to investigate the effects of channel fabrication precision and liquid flow distribution in a microreactor. Sebastia-Saez et al. (2013) presented a small-scale 3-D CFD model for the study of the hydrodynamics and physical mass transfer in structured packing elements. They found that surface textures have a strong influence on liquid misdistribution, which had a significant influence on the interfacial absorption rate. Qi et al. (2013) developed a 3-D model for predicting the wetting factor, film thickness and flow velocity of falling film. Sebastia-Saez et al. (2014) developed a VOF-based micro-scale 3D numerical model to study the influence of several operative parameters on absorption of gas into falling liquid films. Although many simulations and experiments have already been conducted on the flow pattern of non-isothermal falling film, the temperature difference was caused by the heated plate. These experiments and simulations differ from practical processes in a distillation column in which the temperature difference is caused by high temperature vapor. In this paper, an experiment is conducted to study the flow pattern of falling film in direct contact with saturated vapor. To study the details of the phenomenon and simulate it experimentally, a 3-D CFD model is presented, in which the variation of the surface tension with temperature is considered. Through the
53
model, the temperature and surface tension profiles are revealed to illustrate the shrinkage. The modified Marangoni number is estimated and compared to illustrate the influence of surface tension. Moreover, the variation of the heat transfer behavior caused by the shrinkage is investigated.
2. Experimental apparatus and procedure To study the flow pattern of falling film, that countercurrently and directly contacts high-temperature vapor, an experiment with water and vapor was performed. The schematic diagram of the experimental setup is shown in Fig. 1. The liquid stored in the feed storage tank was sent to the falling film contactor by the centrifugal pump. A temperature-controlling device controlled its temperature, and flow rate was measured by a rotameter. The liquid entered the falling contactor at the top through the liquid distributor forming falling film on a steel plate and exited at the bottom to the liquid recovery tank. The vapor was generated by a vapor generator, entered the falling film contactor through a vapor distributor at the bottom and exited at the top of the contactor. The liquid and vapor countercurrently and directly contacted and exchanged mass and heat in the contactor. The size of the steel plate was 400 mm 100 mm 6 mm (length width thickness). The flow pattern and distribution of the temperature were recorded by an infrared camera (Ti 200, Fluke, Inc). The temperature at the inlet and outlet were measured by thermoelectric thermometers. Water and vapor were chosen as the experimental materials. The temperature of the water entering the contactor was chosen to be 30, 40, 50, 60, and 70 1C, and the flow rate was set at 60, 80, 100 and 120 L/h. The vapor flow was constant and set at a 20 m3/h. This rate can maintain the temperature of the entire contactor at the saturation temperature. The effect of the vapor rate was not considered in the study, so the rate was constant.
3. Mathematical model Falling film that countercurrently and directly contacts vapor at a high temperature is influenced by many factors. It is difficult to accurately analyze the heat and mass transfer process through experimentations. Therefore, a 3-D simulation model was presented
Fig. 1. Schematic diagram of the experimental setup, (1) Feed storage (2) centrifugal pump; (3) temperature-controlling device; (4) rotameter; (5) distributor; (6) insulating layer; (7) steel plate; (8) falling film contactor; (9) recovery tank; (10) vapor generator; (11) vortex street flowmeter; (12) distributor; (13) infrared camera.
tank; liquid liquid vapor
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X. Quan et al. / Chemical Engineering Science 135 (2015) 52–60
to analyze the flow pattern and the heat and mass transfer phenomenon. Some important properties used in the simulation are shown in table 1. 3.1. Governing equations 3.1.1. Multiphase model Because counter-current flow of liquid and vapor occurred in the experiment, the VOF multiphase model was used to solve this problem (Nikolopoulos et al., 2007; Banerjee, 2007; Schlottke and Weigand, 2008). In CFD, the VOF method is a numerical technique for tracking and locating the free surface (or fluid-fluid interphase). It solves a single set of momentum equations shared by the fluids, and the volume fraction of each phase in the cell is tracked within the entire flow zone. Interphase tracking is thus achieved by the solution of the transport equation for the volume fraction of one of the phases. The form of the VOF conservation equation is 1 ∂ αq ρq þ ∇d αq ρq uq ¼ Sαq ð1Þ ρq ∂t The volume fraction αq takes the following values: αq ¼1 shows that the cell is entirely filled with the q phase. αq ¼0 shows that the cell is inexistence of the q phase. 0 o αq o1 shows that the cell is partially filled with the q phase. The cell exists at the interphase of different phase. The volume fraction for the primary phase is not solved because it can be computed based on the following constraint: n X
αq ¼ 1
ð2Þ
q¼1
Because there are two phases in the experiment—namely, liquid and vapor—n ¼2 and ¼V and L in the equations. The properties of the fluid that appear in the conservation equations are determined from the volume fraction of each phase. For example, density can be presented as follows: X ρ¼ α q ρq ð3Þ The VOF equation provides only a diffuse approximation of the interphase location. To identify the actual location of the interphase in a cell, the interphase has to be captured or reconstructed. The most extensively used approach for interphase reconstruction is piecewise linear interphase calculation (PLIC), which assumes that the interphase between the two phases has a linear slope within each cell, and uses this property to calculate the advection of fluid through the cell faces using a geometric reconstruction scheme. In this study, PLIC is selected to reconstruct the interphase. 3.1.2. Momentum equations In the VOF model a single momentum equation is solved within the entire flow zone, and the resulting velocity is shared among the phases. The equation can be written as follows: ∂ ðρuÞ þ ∇d ρuu ¼ ∇P þ ∇d μð∇u þ ∇uT Þ þ ρg þ F σ þ F drag ∂t
ð4Þ
For the falling film flow with a counter-current vapor phase, the driving force for the liquid phase is gravity, g, whereas the force for the vapor phase is the pressure drop. Therefore a fraction pressure drop model is considered to describe the drag force source, Fdrag, between the two phases. The pressure drop between two phases can be described as follows (Woerlee et al., 2001):
ð5Þ F drag ¼ αeff f i ρV uV uiy uV uiy
where aeff is the effective interfacial area per unit volume, which can be expressed as αef f ¼ j∇αV j ¼ j∇αL j(Nicolaiewsky and Fair (1999)). fi represents the drag force factor. According to the study of Stephan, and Mayinger (1992), the drag force factor of the falling film flow with a counter-current vapor phase on a rectangular plate can be described by the following equation: nN f i ¼ 0:079ReV 0:25 1 þ 115δ ð6Þ where N ¼ 3:95 1:8 þ 3=Dn , δn and Dn are the dimensionless thickness of the film and hydrodynamic diameter to the liquid 0:5 Laplace length δ= ρL ρV g , respectively. The surface tension source term, Fσ, is considered by the continuum surface model (CFS) proposed by Brackbill et al. (1992). The source of surface tension force can be described as follows: F σ ¼ σ ij
ρκ ∇α i i 0:5 ρi þ ρj
where σ is the surface tension coefficient. The variation of surface tension with temperature was considered in this model. The relationship between surface tension and water temperature can be expressed as follows (Tang, 2000):
σ ¼ 0:09537 2:24 10 6 T 2:56 10 7 T 2
Properties
Liquid
Vapor
Inlet temperature, 1C Density, kg/m3 μ, viscosity, mPa s Flow rate
30–70 995–965 0.8007–0.3565 60–120 L/h
100 0.5542 0.0134 20 m3/h
ð8Þ
κ is the free surface curvature, which is defined as: κ ¼ ∇ U_ n¼
1 n U ∇ jnj ð∇ U nÞ jnj jnj
ð9Þ
where _ n ¼ n=jnj,n ¼ ∇αi . The unit surface normal at the live cell adjacent to the wall is replaced by the following equation: _ _ n ¼_ n w cos γ w þ t w sin γ w
ð10Þ
_ where _ n w and t w are the unit vectors normal and tangential to the wall, respectively. The contact angle γ w is the angle between the wall and the tangent to the interphase at the wall.
3.1.3. Turbulence equation The RNG k–ε turbulence model is chosen for the turbulence calculation in this work. The equations are presented as follows: ∂ðρkÞ ∂ðρkui Þ ∂ ∂k þ ¼ αk μef f ð11Þ þ Gk ρε ∂t ∂xi ∂xi ∂xi ∂ðρεÞ ∂ðρεui Þ ∂ ∂ε ε ε2 þ ¼ αk μef f þ C 1ε Gk C n2ε ρ ∂t ∂xi ∂xi ∂xi k k
Table 1 Physical properties of vapor and liquid phases.
ð7Þ
where
η ¼Sk/ε, and Sk is a user-defined source term.
C n2ε C 2ε þ where
ð12Þ
C μ η3 ð1 η=η0 Þ 1 þ βη3
η0 ¼ 4.38, β ¼0.012, C1ε ¼1.42 and C2ε ¼ 1.68.
ð13Þ
X. Quan et al. / Chemical Engineering Science 135 (2015) 52–60
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3.1.4. Energy equation The energy equation is shared by two phases and can be written as: X ∂ ρE þ ∇ u ρE þ p ¼ ∇ðkeff ∇T hj jj Þ þ Se ð14Þ ∂t j where Se is the energy source term and keff is the effective conductivity. Correspondingly, keff ¼k þkt, where kt is the turbulent thermal conductivity. Energy E and temperature T are mass averaged as follows: Pn q 1 α q ρq E q E ¼ Pn ð15Þ q1
Pn
q1
T ¼ Pn
α q ρq
α q ρq T q α q ρq
ð16Þ
q1
3.2. Source term Owing to the difference in temperatures between liquid and vapor, upon their directly contact, the vapor would condensate on the interphase and release latent heat. Thus, the source terms that represent the exchanged heat and mass on the interphase are calculated within the domain 0.2 o αL o0.7. The vapor-liquid phase change model presented by Sun et al. (2012) was chosen for the simulation. The model is suitable for the case in which one phase is unsaturated and the other is saturated. In our experiment and simulation, the water is considered to be unsaturated and vapor is considered to be saturated. In this model, the thermal conductivity of the vapor (saturated-phase) is assumed to be zero, which means that the vapor maintains a constant temperature and exchanges heat with the liquid only through condensation. Therefore, the energy source can be expressed as follows: SE ¼ 2kef f αef f ∇T
ð17Þ
Fig. 2. Geometry and mesh of the simulation.
In the phase change model, it is assumed that the heat is exchanged between vapour and liquid through condensation. Thus, the mass source can be expressed in the following equation: SL ¼ SV ¼
2kef f αef f ∇T ΔH
ð18Þ
3.3. Flow geometries In this work, a 3-D model is presented to simulate the experiment. Falling film is postulated to be symmetric along the vertical direction. To reduce the computational cost, a 400 mm 50 mm 6 mm geometry was used in the simulation, one side was set to be symmetric, as shown in Fig. 2. With regard to the accuracy, hexahedral grid elements are used in the entire flow domain. Compared to the vapor flow regions, the momentum and temperature gradients are larger in both the liquid film flow regions and the liquid–vapor interphase regions. Therefore, the grids in the liquid flow regions and liquid–vapor interphase regions should be denser than the vapor flow regions. In this work, mesh encryption was adopted to draw the grids, such that their density gradually increased from the vapor phase to the liquid film regions as shown in Fig. 2. The difference between the simulation results and the experimental data with mesh number is shown in Fig. 3. The figure shows that increasing the grid number improves the accuracy of the simulation results, but the improvement is not obvious once the number exceeds 576,000. Considering the accuracy and computing cost, the total mesh number of the grid is chosen to be 576,000. The boundary conditions of the simulation are shown in Table 2.
Fig. 3. Variation of the simulation results with grid number (Tin ¼30 1C, F ¼60 L/h).
4. Results and discussion 4.1. Shape of falling film To ensure that the fluid region of falling film was clearly observed, red ink was added to the water. Fig. 4 shows the shape of the falling film with vapor and without vapor at the same flow rate (80 L/h, Tin ¼30 1C). The red part represents the fluid region of the falling film. It can be observed from the figure that the fluid region of falling film with vapor is smaller than that without vapor.
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Table 2 Boundary conditions. Field
Liquid inlet
Liquid outlet
Vapor inlet
Vapor outlet
Velocity Volume fraction
uL,x ¼ 0, uL,y ¼ uL,in, uL,z ¼ 0 αL ¼ 1, αV ¼0
Pressure outlet —
uV,x ¼ 0, uV,y ¼uL,in, uV,z ¼ 0 αL ¼ 0, αV ¼ 1
Pressure outlet —
Fig. 6. Variations of width on the Y direction. Fig. 4. Shape of falling film.(For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
error between simulation and experiment is 6%, which means that the simulation is acceptable in the industry. In addition to the wetting area of falling film, the width shown in Fig. 4 is also an important factor for the shape of falling film. Fig. 6 shows the variations of the width at a flow rate of 80 L/h along the Y direction. It is shown that the width of falling film decreases from the top (at Y¼0) to the bottom (at Y¼ 0.4 m). Moreover, at the same position, the width increases with increasing inlet temperature. When the inlet temperature is high (Tin ¼50, 60, 70 1C), the variance of the width is low because the location changes from Y¼ 0.3 m to the bottom Y¼ 0.4 m. 4.2. Analysis of the simulation
Fig. 5. Wetting area of falling film with different inlet temperatures and flow rates.
In the case with vapor, the falling film obviously shrinks along the X direction. The wetting area of falling film is an important factor in industrial equipments. Fig. 5 compares the wetting area of falling film at various inlet temperatures and flow rates in the study. Simulation results are consistent with the experimental data and show the same trend. The wetting area of falling film increases with increasing inlet temperature and flow rate. This means that the larger the temperature difference between vapor and inlet liquid, the smaller the wetting area. The growth rate of the low flow rate (60, 80 L/h) within the temperature range of 50–70 1C is larger than that elicited for the range of 30–50 1C, and it changes little at higher flow rates (100, 120 L/h). The growth rate ranged from 10% to 30% within the temperature range of 30–70 1C at the same flow rate. Additionally, it decreased by 35–15% within the range of 60–120 L/h at the same inlet temperature. The maximum
Fig. 7(a) shows the temperature profile of the experiment (Tin ¼ 40 1C, F¼80 L/h) which was recorded by an infrared camera. The figure shows that the temperature on the edge is obviously higher than that in the main body. Fig. 7(b) shows the temperature distribution of the black line in Fig. 7(a). This figure indicates that the temperature on the edge of the film is approximately 10 1C higher than that of the main body. To ascertain the reason behind the shrink of falling film upon contact with saturated vapor, simulation results were used to investigate the details of the process. Fig. 8 shows the liquid profile in the X–Z plane at various Y locations (Tin ¼30 1C, F ¼60 L/h). Owing to the symmetry of the simulation model, only half of the profile is shown, and the symmetry axis is presented in the figure. It is shown that the thickness of falling film on the edge is thicker than that in the middle. The cause for this phenomenon is the liquid shrinkage on the edge, which does not provide enough time for it to spread. The width thus decreases from top to bottom (Y¼0.4 m), which corresponds to the results shown in Fig. 6. Fig. 8 (a) shows the temperature profile of the liquid. It is obvious that the temperature on the edge is higher than that on the main body, which corresponds to the experimental results shown in Fig. 7. The surface tension is strongly dependent on temperature as shown in Eq. (8). Fig. 8(b) shows the variation of surface tension. Owing to the higher temperature at the edge, the surface tension
X. Quan et al. / Chemical Engineering Science 135 (2015) 52–60
at the edge is smaller than that of the main body. Because of the distribution of surface tension, the film on the edge tends to flow toward the middle, causing shrinkage of the falling film. Fig. 8 (c) shows the velocity along the X direction of the falling film, which is clearly higher on the edge. As a result, the falling film will shrink upon direct contact with the high temperature vapor. Fig. 9 shows the liquid profile at Y¼0.2 m in the case where Tin ¼ 70 1C, and F¼ 60 L/h. Compared to Fig. 8, the variation of the temperature in Fig. 9 is smaller owing to the smaller difference between liquid and vapor. Therefore, the surface tension gradient and X-velocity are smaller, as shown in Fig. 9. Because of this, the degree of shrinkage in falling film decreases with increasing inlet temperature. Fig. 10 shows the liquid profile at Y¼ 0.2 m in the case where Tin ¼ 30 1C, and F¼ 120 L/h. Compared to Fig. 8, the film is thicker and the temperature is lower. Although the variation of the surface tension is obvious, the X-velocity is smaller than that in Fig. 8 because of the high mass flow. Therefore, the degree of falling film shrinkage decreases with increasing flow rate as shown in Fig. 5.
4.3. Modified Marangoni number analysis The surface tension gradient caused by the heterogeneous temperature distribution leads to the formation of interfacial shear stress in the tangential direction, which can initiate Marangoni convection, this phenomenon is also called the Marangoni effect. In most cases, a higher temperature difference causes a larger shear stress, which has a strong effect on the liquid film flow characteristics. In this paper, a modified Marangoni number is applied to characterize the influence of surface tension on the falling liquid film flow (Kabov and Chinnov (2001)). MaX ¼
2 L∂∂Tσ ΔT lν=L'
αμ
¼
∂σ ∂T ðT e T av Þ Ul
αμ
ν lν=L'
ð19Þ
57
where MaX represents modified Marangoni numbers in the X 0 direction, L is the width of the falling film, Te is the temperature of the film edge, and Tav is the average temperature of the falling liquid film. Correspondingly, α is the thermal diffusion coefficient, defined as α ¼ k= ρcp , k is the thermal and lv is the conductivity, 1=3 viscous length factor, defined as lν ¼ ν2 =g . Fig. 11 shows the influence of the liquid phase flow rate and the temperature differences on MaX in the X direction. For liquid film in direct contact with vapor, ΔT is positive, thereby making MaX negative. Therefore, in Fig. 11, the smaller MaX is, the larger the Marangoni effect is. Upon the increase of inlet temperature, the MaX of the falling film first increases. While Tin 450 1C, with the increase of the inlet temperature, the MaX of the falling film changes slightly. Fig. 11 also shows that MaX increases with increasing flow rate. Because the faster renewal rate induced by the increased liquid flow rate reduces the temperature difference between liquid and vapor, it also results in a larger MaX. 4.4. Heat transfer behavior Heat and mass transfer have received increased attention in industrial processes. In this study, water is the single component, such that the heat transfer can substitute for the mass transfer. In practical industrial processing, the heat and mass transfers of falling film often occurs in certain areas. Therefore, to investigate the effect of falling film shrinkage on heat transfer, the actual heat transfer coefficient hactual and overall heat transfer coefficient h are defined as follow: hactual ¼
h¼
mcp ðT out T in Þ Q overall ¼ Aactual ΔT Aactual ðT sat T in þ2T out Þ
Q overall mcp ðT out T in Þ ¼ A ΔT AðT sat T in þ2T out Þ
ð20Þ
ð21Þ
where m is the mass flow of the liquid, cp is the special heat of the
Fig. 7. Temperature profile of experiment.
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X. Quan et al. / Chemical Engineering Science 135 (2015) 52–60
Fig. 9. Liquid profile in X–Z plane at Y¼0.2 m (Tin ¼701C, F¼ 60 L/h).
Fig. 10. Liquid profile in X–Z plane at Y ¼ 0.2 m (Tin ¼30 1C, F¼ 120 L/h).
Fig. 8. Liquid profile in the X–Z plane at various Y locations, (Tin ¼30 1C, F ¼60 L/h) (a) temperature; (b) surface tension; (c) X-velocity. Fig. 11. Influence of liquid flow rate and temperature on MaX in the X direction.
liquid, Tin and Tout are the inlet and outlet temperatures of the liquid, respectively, and Tsat is the saturated temperature of the the vapor. Aactual represents the actual wetting area of falling film shown in Fig. 5 and A ¼0.04 m2, represents the entire area of the steel plate, which corresponds to a certain area of the equipment. Fig. 12 shows that hactual decreases with the increasing the inlet temperature at the same flow rate, and with increasing flow rate at the same inlet temperature. However, in industrial processes, heat and mass transfer occur in a fixed area, such that the overall heat
transfer coefficient h is more important. Compared to Fig. 12, Fig. 13 shows that h ranges from 15% to 40% smaller than hactual. This means that the heat transfer coefficient will dramatically decrease when the shrinkage occurs. The variation trend of heat transfer is opposite such that h increases with increasing flow rate and inlet temperature. It is illustrated that the shrinkage of the falling film has an important effect on heat and mass transfer. This
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coefficient h increases with increasing flow rate and inlet temperature.
Nomenclature
Fig. 12. Variations of actual heat transfer coefficient.
A aeff cp D E Eff F Fσ Fdrag fi g H j k keff m P S T u
area of the falling film, m2 effective interfacial area per unit volume, m2/m³ specific heat capacity, J/kg K diffusion coefficient, m2/s energy, J/kg mass transfer efficiency momentum source term, N/m³ surface tension source term drag force source drag force factor gravitational acceleration, m/s2 latent heat of vaporization, J/kg species flux, kg/m2s thermal conductivity, W/mK effective thermal conductivity, W/mK mass transfer rate, kg/s pressure, Pa source term temperature, K velocity, m/s
Greek letters
δ α αeff σ κ γ γw Fig. 13. Variations of overall heat transfer coefficient.
shows that when falling film directly contacts high temperature vapor, it is important that the shrinkage be considered in the design.
thickness of the liquid film volume fraction effective interfacial area per unit volume, m2/m³ surface tension coefficient, N/m free surface curvature activity coefficient contact angle
Subscripts i q V L in out
ith species qth phase vapor phase liquid phase liquid inlet liquid outlet
5. Conclusion Experiments and a 3-D CFD model have been proposed to investigate the shrinkage of falling film, that countercurrently and directly contacts high temperature vapor. In the simulation model, the variation of the surface tension with temperature is considered in the momentum equation. The following conclusions can be drawn from this work: (1) It is demonstrated that falling film will shrink upon contact with the vapor, and the area of falling film increases with increasing inlet temperature and flow rate. (2) Using the proposed model, a detailed analysis of falling film shrinkage reveals temperature and surface tension dispersion and explains the reason for the shrinkage. The modified Marangoni number is compared to illustrate the influence of surface tension. (3) It is shown that the shrinkage of falling film has an important effect on heat transfer and that the overall heat transfer
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