On the shape of self-sustained evaporation front in a metastable liquid

A theoretical model which takes into account new experimental data is proposed for the description of a stationary surface of the selfsustained evaporation front propagating in a layer of superheated liquid flowing over a flat heater. An approximate analytical dependence of the vapour layer thickness on the coordinate and physical parameters is obtained and found to satisfactorily agree with the experimental results. A dimensional parameter is introduced that allows the description to be presented in an invariant dimensionless form.


Introduction
Experiments on the heat exchange in boiling liquids have revealed a regime of interest for both basic science and technical applications, in which a vapor layer propagates over a cylindrical heater under conditions of liquid superheating above the boiling temperature at a given pressure [1][2][3][4]. Models proposed previously [1,[5][6][7][8][9] aimed at approximate description of the dependence of evaporation front propagation velocity on some set of physical parameters. Some of these models [5,7,9] suggest that a "frontal stagnation point" exists at the interface. However, some recent experimental data did not confirm the existence of the frontal stagnation point [2], which implies the need for developing a new approach to the description of these phenomena. In addition, the aforementioned models do not describe the shape of a stationary interface. In this context, the present work aims at developing the analytical description [10] of a stationary shape of self-sustained evaporation front propagating in a layer of superheated liquid flowing over a flat heater.

Description of the model
Reported shadow images [2] showed that the thickness of a vapor layer monotonically increases along the heater surface. In the frontal region, the interface is initially smooth. Thus the model assumptions are as follows. The flow is laminar, the flow velocity varies insignificantly, the curvature of the interface is small. Under these conditions, we can ignore the dynamic and surface pressure and assume that the liquid flows at a constant velocity over an almost flat free surface y = f (x) with velocity V (in the frame of reference connected with evaporation front). The proposed model also ignores the following factors: hydrostatic pressure; compressibility and viscosity of phases; buoyancy, wettability, and thermocapillary effects; finite thickness of the interface; heat transfer between the vapor and the heater. It is also assumed that the evaporated liquid is subsequently not condensed. In the steady-state problem formulation, we will ignore variations of heater temperature T W and heated-layer thickness δ T (within the time interval under consideration) and consider the vapor layer as semi-infinite. It is taken into account that the ratio V L U U H is small ( 1 H ), subscripts L, V refer to the liquid and vapor phases, correspondingly. Using Bernoulli integral, conditions of continuity at the interface (for material flow, energy flux, normal and tangential components of momentum flux), and also the balance between vapour production and vapour flow, the analytical solution (1) has been derived [10,11], which describes the vapor layer thickness f depending on the coordinate x and main physical parameters ( mx M , g mf ): The solution is presented in an invariant dimensionless form with the unique parameter of the model:

Discussion
The obtained analytical solution (1) doesn't take into account the curvature of the interphase surface along the transversal direction (z), which is about 1 R (R is radius of the cylindrical heater). Nevertheless, the solution (1) Fig. 1. The comparison shows that solution (1) well describes the experimental data [2], except for some initial region of the interface with vapour layer thickness 1 g . It may be assumed that the stagnation zone with the closed stream lines is formed near the forward point 0 x . This is confirmed by the results of numerical modeling of the steady-state incompressible isothermal liquid flow (Fig. 2)   The analytical solution (1) also provides good correspondence to the experimental data [12] with acetone ( 65 Fig. 3. Fig. 3. Photo of the propagating vapor cavityexperimental data [12] and the analytical solution (1) (curves) with 1 0.4 m mm.
The conditions of the experiments [13] with acetone also satisfy the inequality 1 R m ! , and the obtained solution is close to approximations of the experimental data (here 119 Fig. 4. According to [13], the thickness of the vapor cavity is proportional to (1) has the same dependence: The solution (1) noticeably exceeds experimental data in case 1 R m , when the curvature of the interphase surface in z direction can not be neglected comparatively with the curvature along x-axis.

Conclusions
The analytical dependence (1) of the vapor-layer thickness on the coordinate (derived in the framework of proposed approximate theoretical model of stationary propagation of the selfsustained evaporation front in a layer of superheated liquid flowing over a flat heater) satisfactorily agrees with the known experimental data for cylindrical heaters under the condition 1 R m !
. The governing parameter m in physical sense plays the role of characteristic curvature of the interphase surface in the direction of evaporation front propagation. This stationary solution can be used as the base to study the problem of weak perturbation stability at the liquid-vapor boundary. This work was supported by the Russian Foundation for Basic Research, project no. 15-08-01359.