Numerical study of the evaporative convection regimes in a three-dimensional channel for different types of liquid-phase coolant
Introduction
Fluid flows with an interface under the action of co-current gas fluxes, being accompanied by evaporation or condensation have been the subject of extensive theoretical and experimental investigations in the last few decades [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. The flows under study are applied in the technological systems of the fluidic cooling or thermal control of highly efficient semiconductor devices, in membrane evaporators, distillers, thermal coating technologies, etc. Experimental researches allow one to find constructive possibilities to enhance the efficiency of the two-phase system or to modify the working setup, as well as stimulate the development of new theoretical approaches to study evaporative convection and carry out preliminary analytical investigations on the basis of adequate mathematical models. To precisely forecast the process dynamics in the two-phase systems requires a comprehensive analysis based on the mathematical modeling of the two-layer flows with evaporation. The theoretical results can contribute to the clarification of the physical aspects of evaporative convection and phenomena in the working media caused by different effects.
One of the common approaches to describe the convection accompanied by phase transition is based on using the Navier–Stokes equations or their approximations (for the review of the theoretical approaches to study evaporative convection, refer to [11]). Modeling of the real fluid flows can be performed with the help of the exact solutions of the governing equations. The solutions of the group nature are particularly valuable. Group origin of the solution implies natural properties of the space–time symmetry and symmetry of the spatial fluid motion provided by the derivation of the Navier–Stokes equations, and ensures the physical plausibility and realizability of the fluid flows described by the solutions. It allows one to obtain some evaluation characteristics and to rather effectively and rapidly forecast the outcome of experiments at the development stages [12], [13]. Finally, the exact solutions of the multiparameter problems would give information on the degree and character of the influence of various factors, taken into account in the mathematical modeling, and provide the possibility to develop a more precise mathematical model. The exact solutions obtained in the framework of different approaches which are known at present and used to study the problems of evaporative convection, are reviewed in [11].
In the present work the character and structure of the joint flows of evaporating liquid and co-current laminar gas flux are investigated on the basis of a solution of a special type of the Boussinesq approximation of the Navier–Stokes equations. This solution is a 3D analogue of the well-known Ostroumov–Birikh solution [14], [15] of the Oberbeck–Boussinesq equations. The new 3D solution is a partially invariant one and can be referred to as an “exact” solution (see the corresponding definitions concerning the exact solutions in [16]). The group nature of the Birikh type solutions [17] allows one to generalize the solutions for the problems of evaporative convection in the domains with internal interfaces admitting the phase transition [11], [18], including the non-axis-symmetrical 3D case [19]. Influence of the gravity on the structure and characteristics of the arising regimes and the applicability of different types of boundary conditions for the vapor concentration function were investigated in [19] with the help of the solution.
The invariant and partially invariant solutions are used to study the essential and secondary peculiarities of the heat and mass transfer phenomena described by the convection equations due to their group properties. We note that the asymptotic character of the Birikh type solution was proved in [20], where the thermocapillary gravitational convection in a long horizontal cavity was studied experimentally and numerically on the basis of the 2D problem. Numerical investigations, including direct numerical simulation (DNS), are preferable to analyze the behavior of the systems under particular conditions. Despite the fact that there is definitely an argument against DNS, given its time consuming nature, it is a very useful tool in real practical problems. Methodology for DNS of interphase heat and mass transfer in vapor–liquid system was presented in [21]. Scenarios of change in interface shape for the cases of wetted-wall vapor–liquid contacting and liquid contacting with own vapor along the common interface were analyzed for the binary ethanol–water system filling a narrow two-dimensional channel. Numerical simulation of dynamics of the evaporating fluid film flows taking into account the effects of contact line was performed also for the cases when the action of adhesion forces and the interfacial heat resistance were additionally considered [22]. Surface deformations, temperature and velocity distributions, flow patterns were calculated numerically on the basis of one-field model derived from the interface continuum mechanical balances for two-phase evaporative and thermocapillary flows [23]. Comparison of obtained numerical results with analytical solution corresponding to the system without evaporation and experimental data obtained for the flow in the presence of the evaporation was carried out for cases of liquid film being on structured heated substrate and locally heated substrate. Computation of characteristics for 2D two-phase flows with phase change was realized in [24] in the framework of different models to evaluate an impact of surface tension force and interface mass transfer. DNS of the dynamics for two-phase thermocapillary flows in a 3D cuvette of a rectangular-section with an applied thermal load was performed on the basis of the Navier–Stokes and energy equations [25], [26]. Dynamics of the flow regimes, evolution of the interface and stability parameters were investigated depending on the Marangoni effect and characteristics of the “liquid–vapor” phase transition therein [25], [26]. A brief review of the DNS methods based on a VOF algorithm to study multiphase flows was presented in [27].
The aim of the study is to carry out the mathematical modeling of the three-dimensional convective fluid flows with evaporation as well as to analyze the influence of the thermal load and type of the heat transfer liquid on the flow patterns and evaporation characteristics in the two-phase systems with various values of the liquid layer thickness. The study is performed in the framework of the Oberbeck–Boussinesq equations taking into account the Soret and Dufour effects in the vapor–gas phase. Parameters of heat exchange can be improved due to the appropriate choice of liquid-phase coolant in the experimental setups and industrial equipment.
Section snippets
Governing equations
Consider an infinitely long three-dimensional channel in the Cartesian coordinate system . The system is oriented so that the gravity acceleration vector is expressed as follows , where is the unit vector of . The channel has solid impermeable external boundaries and is occupied by a two-layer system of viscous incompressible fluids (liquid and gas–vapor mixture). The media fill the infinite horizontal layers and :
Generalization of the Ostroumov–Birikh solution for the 3D flows with evaporation at the interface
We construct the exact solution of Eqs. (2.1)–(2.4) in a special form. The structure of the required functions is determined by the group properties of the governing equations and presupposes the specific dependence of the velocity vector components of the th phase on the transverse coordinates . Furthermore, the functions of temperature , pressure , and vapor concentration have additive components , , which also depend on the variables . The unknown functions
Boundary conditions
The boundary conditions on the surface separating the liquid and gas phases are dictated by the conservation laws and some additional assumptions resulting from certain physical constraints [28], [29]. The conditions below will be written with respect to the form of the exact solution (3.1)–(3.4) and admitted equation of the interface (feasibility of the last assumption is substantiated in detail in [19]). In this case the normal vector coincides with the unit vector of the axis.
The
Numerical simulation
The analytical calculations for the construction of the exact solution for the three-dimensional problem of convection with evaporation at the interface is complemented by the numerical investigations. Obtaining the stationary solution (3.1)–(3.4) for the two-layer convective fluid flows is reduced to sequentially solving a chain of the two-dimensional problems.
Regimes of evaporative convection
The numerical investigations are performed for the working liquid–gas systems like ethanol–nitrogen and HFE-7100–nitrogen. HydroFluoroEther-7100 is a volatile high-performance segregated dielectric used both in thermophysical experiments in parabolic flights and in-space and in real fluidic cooling systems on the space orbital platform. The physicochemical parameters of the working fluids are given in Table 1 [12], [30], [33], [34], [35]. The choice of the working fluids is explained by
Conclusions
The quasi-steady coupled problem of the convective flows in the two-layer systems with the phase transition “liquid–vapor” at the interface has been studied. The mathematical model takes into account the influence of the thermodiffusion and thermal diffusivity effects in the gas–vapor layer and at the interface. The joint convective flows of the evaporating liquid and gas–vapor mixture in a 3D channel of a rectangular cross-section are investigated with the help of the exact solution of the
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The study was funded by the Russian Foundation for Basic Research, the government of Krasnoyarsk region and Krasnoyarsk Regional Fund of Science (grant No. 18-41-242005).
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