APPLICATION OF THE METHOD OF THE THERMAL QUASI-STEADY APPROXIMATION TO THE PROBLEM OF MICROLAYER EVAPORATION

V. V. Salomatov, A. D. Gorbunov, A. V. Kuzmin

(Presented by the Department of Thermophysics and Nuclear Power Engineering)

The microlayer hypothesis proposed by Snyder [1] has recently found wide application. A number of works report results of indirect thermometric and cinematographic proof of the existence of a microlayer by means of highly sensitive thermocouples of various designs [1, 2, 4, 6], thin-film resistance thermometers [5, 7], and also by analysis of the deposition of radioactive calcium sulfate [3]. Direct measurement of the microlayer thickness was carried out by means of an optical interferometer [9]. In [10], by the same method, certain details of the microlayer geometry were refined, with simultaneous photography of bubble growth.

The results of the experimental works cited above have shown the influence of microlayer evaporation on bubble growth and on heat transfer during boiling.

On the path toward a complete analytical description of the process of rapid growth of a vapor bubble on a liquid–solid interface, the main difficulty is determining the law of the dynamics of microlayer evaporation. Therefore, the analysis given in [13] is restricted to the case of uniform initial superheating of the liquid and solid, having identical thermal activities, for which an exact solution is available [11].

In [14] a numerical solution is given for one-dimensional heat-conduction equations, which is then used to determine the volume of vapor formed as a result of microlayer evaporation.

The aim of the present work is to obtain an analytical solution of the problem of microlayer evaporation, necessary for analyzing its participation in the growth of a vapor bubble and in the mechanism of heat transfer, as well as for interpreting experimental data on wall-temperature fluctuations.

The experimentally determined values of the microlayer thickness \(\delta_0\), lying in the range \(2\)—\(20\ \mu\), are several orders of magnitude smaller than the current and departure radii of the bubble, which provides grounds for adopting a one-dimensional model. The results of [12] make it possible, in a first approximation, to regard the microlayer as immobile.

By the end of the rapid initial growth of the bubble, on a flat wall of thickness \(L\) there forms a superthin liquid layer of thickness \(\eta=1\). A flux \(K_i(F_0)\) acts on the outer surface of the wall. At the initial instant the temperature of the interface between the solid and the liquid is
\(\Theta_2(0,0)=1\).

It is assumed that the temperature at the vapor–liquid interface is constant in time and equal to \(\Theta_2(\eta(F_0),F_0)=0\). The temperature gradient in the vapor is zero. At the microlayer–wall interface, the temperature of the liquid and the solid and the corresponding heat fluxes are equal. The initial conditions are specified by the temperature distribution in the liquid \(u_2(Z_2)\) and in the solid \(u_1(Z_1)\).

Taking into account the initial assumptions, the problem under consideration is characterized by the following system of equations:

the differential energy equation in the solid

\[ k_\varepsilon^2\frac{\partial\Theta_1(Z_1,F_0)}{\partial F_0} = \frac{\partial^2\Theta_1(Z_1,F_0)}{\partial Z_1^2}, \qquad F_0>0,\quad 0<Z_1<L; \tag{1} \]

the differential energy equation in the liquid microlayer

\[ \frac{\partial\Theta_2(Z_2,F_0)}{\partial F_0} = \frac{\partial^2\Theta_2(Z_2,F_0)}{\partial Z_2^2}, \qquad F_0>0,\quad 0<Z_2<\eta(F_0); \tag{2} \]

boundary conditions at the liquid–solid interface

\[ \Theta_1(0,F_0)=\Theta_2(0,F_0), \tag{3} \]

\[ \frac{\partial\Theta_1(0,F_0)}{\partial Z_1} = -\frac{\partial\Theta_2(0,F_0)}{\partial Z_2} = q(F_0); \tag{4} \]

boundary conditions at the moving boundary

\[ \Theta_2(\eta(F_0),F_0)=0 \tag{5} \]

\[ \frac{\partial\Theta_2(\eta(F_0),F_0)}{\partial Z_2} = M\frac{d\eta(F_0)}{dF_0}; \tag{6} \]

On the outer surface of the wall a boundary condition of the second kind is prescribed,

\[ \frac{\partial\Theta_1(L,F_0)}{\partial Z_1} = K_i(F_0); \tag{7} \]

Initial conditions

\[ \Theta_1(Z_1,0)=u_1(Z_1), \tag{8} \]

\[ \Theta_2(Z_2,0)=u_2(Z_2), \tag{9} \]

\[ \eta(0)=1. \tag{10} \]

The problem under consideration consists in determining the temperature field and the dynamics of formation of the vapor phase. Note that the Stefan condition (6) makes the problem essentially nonlinear. For the solution of this...

of the nonlinear heat-conduction problem with a moving boundary, we use the method of the thermal quasi-steady approximation, which received a physical justification and development in [15, 16] and which makes it possible to cope with Stefan-type nonlinearity.

Taking into account the noted physical features of the course of the thermal process, we shall solve the problem formulated by the system of equations (1)—(10) with the aid of the Laplace integral transform. Freeing ourselves from differential operations with respect to time and solving the transformed problem, then returning to the domain of originals, we shall subsequently isolate the quasi-steady asymptotic form of the temperature field.

Numerical estimates and comparisons of the results obtained with exact solutions show that for most nonlinear problems with moving boundaries the use of two terms of the expansion guarantees an accuracy sufficient for engineering calculations.

Then the temperature field for the wall, with allowance for two terms of the expansion, takes the form

\[ \Theta_1(Z_1,F_0)= \frac{1}{k_\varepsilon^2 L}\int_0^{F_0}(K_i-q)\,dF_0 +qL\left(\frac{1-3\left(\dfrac{Z_1-L}{L}\right)^2}{6}\right) - \]

\[ {}-K_iL\left(\frac{1-3\left(\dfrac{Z_1}{L}\right)^2}{6}\right) +\frac{1}{L}\int_0^L u_1(Z_1)\,dZ_1 . \tag{11} \]

Using the procedure given above, we obtain the solution for the microlayer

\[ \Theta_2(Z_2,F_0)= \frac{1}{\eta}\int_0^{F_0}(V+q)\,dF_0 - V\eta\left(\frac{1-3\left(\dfrac{Z_2}{\eta}\right)^2}{6}\right) - \]

\[ {}-q\eta\left(\frac{1-3\left(\dfrac{Z_2-\eta}{\eta}\right)^2}{6}\right) +\frac{1}{\eta}\int_0^\eta u_2(Z_2)\,dZ_2, \tag{12} \]

where

\[ V(F_0)=M\dot{\eta}(F_0) \]

(the dot denotes differentiation with respect to time). In the solution (11)—(12), the law of motion of the phase-separation boundary \(\eta(F_0)\) and the function of the heat flux at the liquid—solid interface \(q(F_0)\) still remain unknown. Satisfying, respectively, conditions (3) and (5), we obtain a system of integro-differential equations for determining them

\[ \frac{1}{k_\varepsilon^2 L}\int_0^{F_0}(K_i-q)\,dF_0 -\frac{qL}{3} -\frac{K_iL}{6} +v_2 = \frac{1}{\eta}\int_0^{F_0}(V+q)\,dF_0 -\frac{V\eta}{6} +\frac{q\eta}{3} +v_2, \tag{13} \]

\[ \frac{1}{\eta}\int_0^{F_0}(V+q)\,dF_0 = \frac{q\eta}{6} -\frac{V\eta}{3} -v_2, \tag{14} \]

where

\[ v_1=\frac{1}{L}\int_0^L u_1(Z_1)\,dZ_1;\qquad v_2=\frac{1}{\eta}\int_0^\eta u_2(Z_2)\,dZ_2. \tag{15} \]

Methods for solving integro-differential equations are developed much less fully than those for differential or integral equations. The difficulties,

arising in the solution of integro-differential equations can be avoided by reducing them to a Volterra integral equation of the second kind. To do this, let us satisfy the heat-conduction equation (2) at \(Z_2=\eta\). We obtain a quadratic equation for \(V\), whose solution has the form

\[ V=-\frac{M}{2\eta}\left(1-\sqrt{1-\frac{q\eta}{M}}\right). \tag{15} \]

Taking expression (15) into account, the solution of equation (13), which describes the law of motion of the phase-interface boundary, is written as

\[ \widetilde{F}_0=N(F_0)(1-\eta(F_0))-\frac{\nu_2^*}{\operatorname{Ki}(F_0)} +\frac{k_\varepsilon^2 L^2}{6} -\frac{k_\varepsilon^2 L\nu_1}{\operatorname{Ki}(F_0)} +\frac{q(F_0)}{\operatorname{Ki}(F_0)} \left( \frac{\eta^2(F_0)}{6} +\right. \]

\[ \left. +\frac{k_\varepsilon^2 L^2}{3} +\frac{\eta(F_0)k_\varepsilon^2 L}{2} \right) -\frac{V(F_0)}{\operatorname{Ki}(F_0)} \left( \frac{\eta(F_0)}{3} +\frac{\eta(F_0)k_\varepsilon^2 L}{2} \right), \tag{16} \]

where the function \(q(F_0)\) is determined from the solution of the equation

\[ \int_0^{F_0} q(F_0)\,dF_0 = \frac{q\eta^2}{6} -\frac{V\eta^2}{3} -M(\eta-1)-\nu_2^*. \tag{17} \]

Here and below

\[ \nu_2^*=\int_0^1 u_2(Z_2)\,dZ_2. \]

The integral equation (17) is quite complicated for computing the function \(\eta(F_0)\), but in certain limiting cases, which are of considerable interest in practical applications, it can be substantially simplified. Below we shall consider the principal ones.

It should be noted that, in order to analyze the role of the evaporating microlayer in the dynamics of a vapor bubble, it is desirable to obtain as simple as possible analytical expressions for the law of motion of the phase-transition boundary \(\eta(F_0)\), so as to simplify the general analysis of the participation of the evaporating microlayer in the mechanism of heat transfer and in the dynamics of a vapor bubble growing on the heating surface.

The expressions for finding the temperature field can be written in a simpler form. Using the equation of the process dynamics (13), (14), we rewrite expressions (11), (12) in the simpler form

\[ \Theta_1(Z_1,F_0)= \frac{\eta}{2}(q-V) +\frac{qL}{2} \left[ 1-\left(\frac{Z_1-L}{L}\right)^2 \right] +\frac{\operatorname{Ki}}{2}\frac{Z_1^2}{L}, \tag{18} \]

\[ \Theta_2(Z_2,F_0)= \frac{q}{2\eta}(Z_2-\eta)^2 +\frac{V}{2\eta}Z_2^2 -\frac{V\eta}{2}. \tag{19} \]

Case \(k_\varepsilon^2\to 0\). The equality \(k_\varepsilon^2=0\) means that the thermal activity of the liquid predominates over the thermal activity of the wall material. The solid wall is, as it were, “transparent” to the heat flux; in other words, \(q(F_0)=K_1(F_0)\). Indeed, multiplying the first integro-differential equation (13) by \(k_\varepsilon^2=0\), we obtain

\[ \frac{1}{L}\int_0^{F_0}\left(\operatorname{Ki}(F_0)-q(F_0)\right)dF_0=0, \]

that is,

\[ K_i(F_0)=q(F_0) \]

A practical example may be the boiling of nonmetallic liquids on materials that, in their thermophysical properties, are close to insulators.

Then the law of motion of the phase-transition boundary is obtained from (16):

\[ \widetilde F_0 = N(1-\eta(F_0)) - \frac{v_2^*}{K_i(F_0)} + \frac{\eta^2(F_0)}{6} - \frac{v(F_0)}{K_i(F_0)}\frac{\eta^2(F_0)}{3}. \tag{20} \]

Of definite interest is the total evaporation time of the microlayer, which is determined from (20) at \(\eta(\widetilde F_{0k})=0\):

\[ \widetilde F_{0k}=N(F_0)-\frac{v_2^*}{K_i(F_0)} . \tag{21} \]

To determine the temperature fluctuation of the heating surface under a growing vapor bubble, one may use either of equations (18) and (19):

\[ \Theta_1(0,F_0)=\frac{\eta(F_0)}{2}\bigl(K_i(F_0)-V(F_0)\bigr), \tag{22} \]

In the case

\[ \frac{4K_i(F_0)\eta(F_0)}{M}\ll 1 \]

expressions (15), (20), (22) can be simplified further, without fear of loss of accuracy, by expanding the radical expression in (15) in a series and retaining the first term of the series:

\[ \left. \begin{aligned} V(F_0)&\simeq K_i(F_0),\\ \widetilde F_0&\simeq N(F_0)(1-\eta(F_0))-\frac{v_2^*}{K_i(F_0)}+\frac{\eta^2(F_0)}{2},\\ \Theta_1(0,F_0)&\simeq \eta(F_0)\cdot K_i(F_0). \end{aligned} \right\} \tag{23} \]

Case \(k_\varepsilon^2\to\infty\). This case corresponds, for example, to boiling of organic liquids on highly thermally conductive metallic heating surfaces. It means that the thermal activity of the wall material is much greater than the thermal activity of the boiling liquid. A characteristic feature of the process at \(K_\varepsilon^2\to\infty\) is the constancy of the temperature of the heating surface throughout the entire process of evaporation of the microlayer.

Writing equation (13) in the form

\[ \frac{1}{\eta(F_0)}\int_0^{F_0}(V+q)\,dF_0 - \frac{V\eta(F_0)}{6} + \frac{q\eta(F_0)}{3} + v_2 =1 \tag{24} \]

and solving the system (14), (24) with the additional relation (15), for initial distributions close to linear ones, we obtain the following equation for the dynamics of the evaporation process:

\[ 1-\eta^2(F_0)=2\left(1-\sqrt{\frac{M-2}{M}}\right)F_0. \tag{25} \]

The expression for determining the time at which the evaporation process ends will take the following form

\[ \mathrm{Fo}_{\mathrm{end}}= \frac{1}{2\left(1-\sqrt{\frac{M-2}{M}}\right)} . \tag{26} \]

It is interesting to note that from equations (25), (26), for the case \(k_z^2 \to \infty\), the range of solutions of the obtained relations is \(M>2\).

For heat fluxes that vary only weakly with time, with \(\mathrm{Ki}(\mathrm{Fo}) \cong \mathrm{const}\), the modified Fourier number \(\widetilde{\mathrm{Fo}}\) becomes \(\mathrm{Fo}\).

It follows from an analysis of the dynamics of the evaporation process that the principal role is played by the first term of expression (20), i.e., the criterion \(N(\mathrm{Fo})\). It is evident that the regularization of the process kinetics will proceed the more rapidly, the larger the relative evaporation criterion \(N(\mathrm{Fo})\) is.

The authors of [14] carried out numerical calculations on a computer for constant heat fluxes and a linear initial distribution of the temperature field of the form: for the wall, \(u_1=1+\mathrm{Ki}Z_1\); for the liquid microlayer,

\[ u_2=1-\mathrm{Ki}Z_2 . \]

Unfortunately, the results of calculations are presented only for two particular values of the criterion \(\mathrm{Ki}\), equal to 0.03 and 0.3.

Comparison with the numerical calculations [14] shows that, as \(\mathrm{Ki}\) increases and for sufficiently large values of the criterion \(M\), expressions (20) and (25) will respectively give the upper and lower estimates of the process dynamics, which is convenient for analyzing the participation of the microlayer in bubble growth and heat transfer.

From comparison of the solutions and analysis of the obtained analytical expressions, it can be seen that, for both cases \(k_z^2\), the solutions become self-similar with respect to the dimensionless plate thickness \(L\) and practically coincide with the results of numerical calculations for the system of a bounded and a semi-bounded body, both for the dynamics of microlayer evaporation and for the fluctuation of the temperature of the heating surface in the quasi-stationary part of the process.

The initial stage affects the dynamics of the evaporation process at small values of the phase-transition criterion \(M\), but this influence is not significant. Since at \(M=5\) only about 20% of the microlayer evaporates during this time interval, at values \(M>100\) the amount of the liquid layer that has evaporated will be less than 1%.

If necessary, the initial stage of the process can be taken into account by introducing the “depth of penetration of the thermal disturbance” from the side of the phase-transition boundary [16].

Conclusions

1. Using the method of the thermal quasi-stationary approximation, a general solution has been obtained for the problem of microlayer evaporation for arbitrary initial temperature distributions in the wall and in the liquid microlayer and for an arbitrary heat flux on the heating wall.

2. The analysis carried out made it possible to extract from the general, rather complicated solution asymptotically exact and approximate calculation relations for the dynamics of microlayer evaporation, the fluctuation of the heating surface, and the time of complete evaporation of the microlayer at limiting values of the regime parameters.

1. Comparison of the obtained solutions with some partial results of numerical calculations shows good agreement in the quasi-steady part of the microlayer evaporation process.

NOTATION

\[ \Theta=\frac{T-T_s}{T_{10}-T_s} \]
— dimensionless temperature;

\[ \eta(F_0)=\frac{\delta(\tau)}{\delta_0}; \qquad L=\frac{l}{\delta_0}\frac{\lambda_1}{\lambda_2} \]
— dimensionless thicknesses of the microlayer and the wall;

\[ Z_2=\frac{Z_1}{\delta_0}; \qquad Z_1=\frac{Z_1\lambda_1}{\delta_0\lambda_2} \]
— dimensionless coordinates;

\[ F_0=\frac{a_2\tau}{\delta_0^2} \]
— Fourier number, counted from the beginning of the establishment of the wall temperature equal to \(T_{10}\);

\[ M=\frac{r}{Cp_2(T_{10}-T_s)} \]
— phase-transition criterion;

\[ \operatorname{Ki}(F_0)=\frac{q_1(\tau)\delta_0}{\lambda_2(T_{10}-T_s)} \]
— Kirpichev criterion;

\[ k_{\xi}^{2}=\frac{\lambda_1\rho_1 Cp_1}{\lambda_2\rho_2 Cp_2} \]
— criterion characterizing the thermal activity of the wall material relative to the liquid;

\[ a_1,\ \lambda_1,\ \rho_1,\ Cp_1,\ l \]
— thermal diffusivity, thermal conductivity, density, heat capacity, and wall thickness;

\[ a_2,\ \lambda_2,\ \rho_2,\ Cp_2,\ \delta_0 \]
— the same for the liquid microlayer;

\[ \widetilde{F}_0=\frac{1}{\operatorname{Ki}(F_0)} \int_{0}^{F_0}\operatorname{Ki}(F_0)\,dF_0 \]
— modified Fourier number;

\[ N(F_0)=\frac{M}{\operatorname{Ki}(F_0)} \]
— relative criterion of the evaporation process;

\[ T_s \]
— phase-transition temperature;

\[ q_1(\tau) \]
— heat flux on the outer side of the wall;

\[ \tau \]
— time.

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