Velocity and temperature profiles extending over the liquid and gas phases of two-phase flow falling down vertical plates
Introduction
A film flow falling along vertical plates can be often observed in many engineering devices such as evaporators, condensers and absorbers. The heat and fluid flow behavior of the film flow plays an important role in determining the heat and mass transfer performance. A better understanding of the heat and fluid flow characteristics allows us to predict heat and mass transfer performance more accurately and to develop more effective and optimal devices to improve the performance under the same operating conditions. Therefore, the characteristics of the thin falling liquid film have been of interest to engineers and have been extensively investigated experimentally, analytically and numerically [1], [2], [3], [4]. Basic and comprehensive review of falling film flow on the flat plate have been done by Kapitza and Kapitza [5] and Alekseenko et al. [6].
Practically, however, it is difficult to measure accurately velocity and temperature profiles inside the film flow, because the film thickness is very thin. Therefore, the details inside the film flow are greatly indebted to numerical calculations. In a calculation of such two-phase flow that the different phases of liquid and gas exist, it is necessary to solve a moving boundary problem where the interface between the liquid and gas phases is varying as the calculation goes on. Complicated calculations are required in order to capture the interface and to set the boundary conditions there. As efficient numerical methods to analyze the moving boundary problem, the Volume of Fluid (VOF) Method [7] or the Level-Set (LS) Method [8] have been used, where the mass conservation can be preserved in the VOF method and the unit normal vector at the interface can be accurately calculated in the LS method. Very recently, a Coupled Level-Set and Volume of Fluid (CLSVOF) method [9], [10], [11] is noticed as one of the high degree of completeness, because it adopts the merits of the VOF and LS methods and cancels their demerits.
On the other hand, a theoretical solution can be derived for steady and fully developed laminar film flow in the liquid phase on the flat plate, from a simple balance between the gravitational body force and the shear force at the solid wall. The film thickness remains constant, and the velocity profile across the film becomes parabolic in the fully developed region. Although there are many variety of treatments for the temperature at the interface, the temperature profile is a linear heat conduction state in the liquid phase because the film thickness is very thin. This solution is called a Nusselt solution. The velocity and temperature profiles have been widely used as a benchmark solution to check the validity of numerical calculations and as a boundary condition at the inlet. However, it is an important issue that the gas phase is not taking into consideration in the theory.
In a recent computation of the two-phase flow, the gas phase as well as the liquid phase can be calculated using one fluid equation model. Then the conservation equations for each phase are coupled through jump conditions at the interface. In order to satisfy the jump conditions, there are two models. One model is that a smooth and narrow transition layer is set across over the interface [12], where physical quantities such as density, viscosity, surface tension etc. are changed smoothly across the interface. The other one is more realistic and useful for practical engineering because such physical quantities change sharply at the interface. An efficient method to handle the sharp change at the interface is a Ghost Fluid Method (GFM) [13], [14], [15]. In either case, spurious vortices appear near the inlet region in numerical simulations if the velocity and temperature profiles from the Nusselt solution are applied as an inlet boundary condition because the gas phase is not taken into consideration. Furthermore, it is inconvenient to use the solution without the gas phase in comparison with the obtained simulation results of the two-phase flow. Therefore, it is necessary to derive a two-phase version of the Nusselt solution which considers the velocity and temperature profiles in the gas phase as well as in the liquid phase and the jump conditions at the interface. However, such extended analytical velocity and temperature profiles extending over the liquid and gas phases are not derided in the present circumstances.
In this study, we analytically derive the velocity and temperature profiles extending over the liquid and gas phases of two-phase flow falling down vertical plates as a simple model of plate-type absorber and therefore we do not consider the phase change in this paper. The profiles are expected to be used as a simple and efficient benchmark solution to check the validity of numerical calculations for solving the model of plate-type absorber and be applied as a boundary condition at the inlet. In addition, the heat and fluid flow characteristics extending over the two-phases are investigated for the effect of the viscosity and thermal conductivity ratios.
Section snippets
Governing equations
We consider a two-phase flow along a smooth flat plate as shown in Fig. 1. The flow is assumed to be two-dimensional and incompressible. Then the governing equations for the velocity, pressure and temperature are written in nondimensional forms as
All the variables have been nondimensionalized using a characteristic length , a film surface velocity , the density of the liquid phase , the temperature on the plate
Analytical treatment of the two-phase flow
We analytically derive velocity and temperature profiles in the two-phase flow falling down vertical plates. The flow and temperature fields are assumed to be steady state, fully developed and unchanged in the x direction, and therefore the film thickness of the liquid phase is assumed to be constant and unity as indicated by the dashed line shown in Fig. 1. Then the velocity and temperature profiles can be expressed as u = (u(y),0) and T = T(y). Substituting the velocity and temperature
Conclusions
We have analytically derived the velocity and temperature profiles extending over the liquid and gas phases of two-phase flow falling down vertical plates. As previously mentioned, the gas phase as well as the liquid phase can be calculated using one fluid equation model in a recent computation of the two-phase flow, where the conservation equations for each phase are coupled through jump conditions at the interface. Satisfying the jump conditions by using the GFM, we can obtain realistic
Acknowledgements
The computation in this work has been done using the facilities of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo.
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