On the Issue of Adiabatic Outflow of Water with Phase Formation

V. P. Matyukhin

(Presented by the scientific seminar of the Department of Theoretical and Experimental Electrical Engineering)

To date, a considerable number of experimental studies have been carried out on adiabatic outflow of water with the formation of a vapor phase. Such experiments have been conducted with nozzles of the most varied profiles and over a fairly wide range of initial pressures \((0.1\text{–}5\ \mathrm{MN}/\mathrm{m}^2)\). Interest in studying this phenomenon is stimulated by the fact that it is very often encountered in various production processes and technological cycles. However, there is still no established theory of this process that has been properly formulated mathematically. This is explained by the presence of a large number of heterogeneous factors determining this process, which are as yet insufficiently studied in themselves, and also by the absence of a successful hypothesis that would promote the emergence of such a mathematically formulated theory.

In studying the adiabatic outflow of water with vapor formation through diaphragms, short cylindrical and conoidal nozzles with a ratio of length to outlet diameter \((l/d)\) of the order of \(0.6\text{–}1.6\), and at initial pressures \(P_1 = 0.1\text{–}5\ \mathrm{MN}/\mathrm{m}^2\), a substantial deviation of the process from its equilibrium course along an isentrope was found. By contrast, for long cylindrical and conoidal nozzles with a ratio \(l/d > 10\) and the same initial parameters, a certain approach was observed to the isentropic flow of boiling water from the standpoint of the final effect—specified values of vapor content, velocity, and pressure. From this fact a number of researchers [1, 2, 6, 8] concluded that the decisive circumstance in decreasing or increasing the deviation of the process from the isentrope is the residence time of water particles in the nozzle, which, in order of magnitude, is supposedly comparable with the time required for completion of the vapor-formation process.

In other words, the residence time of water in short nozzles is small and the vapor-formation process does not have time to be completed; in long nozzles, by contrast, the residence time is sufficiently large and the vapor-formation process manages to be completed almost to full equilibrium between the phases, an indication of which could be the “ratio between the process time and the relaxation time” [1].

This point of view was first expressed quite clearly in the works of A. A. Gurchenko [5, 6]. In connection with this point of view, we would like to make a number of comments which, in our opinion, may be of value in setting up subsequent experiments or in the mathematical solution of the problem posed.

Using as examples the phenomena occurring in the adiabatic outflow of wet steam—such as the rate of droplet growth, the sizes of condensation shocks, the rate of temperature equalization over the volume of a droplet, etc.—one may conclude that, also in the case of adiabatic vaporization during the outflow of water through nozzles, the times of the processes associated with the formation of vapor bubbles are of the same order, since they are based on similar macro- and microprocesses.

An approximate estimate [by formula IV, 18, 7] makes it possible to evaluate the formation of vapor bubbles of size \(r = 10^{-3}—10^{-2}\) mm at liquid superheats \(\Delta T = 25—30^\circ\mathrm{C}\) over a time \(\tau = 10^{-6}—10^{-5}\) s, which turns out to be an order of magnitude less than the time (\(10^{-4}\) s) for the liquid to pass through short nozzles with ratio \(l/d = 0.6\) [1].

This circumstance gives reason to doubt the correctness of relying on the ratio between the process time and the vapor-formation time in studying the adiabatic outflow of boiling water. Apparently, such factors as metastability, large pressure gradients, phase slip, and also, under certain conditions, intense turbulization of the flow, which promotes fine dispersion of the flow and cavitation, and surface forces do not permit the realization, over short time intervals, of the intrinsically high rates of the vapor-formation process.

Let us turn to the schemes of processes \(a\) [1, 2, 8] and \(b\) [3] (Fig. 1), which occurred when experiments were carried out with conoidal and cylindrical nozzles. The entire process of adiabatic outflow of water with vapor formation may be divided into two zones:

Fig. 1. Nature of pressure variation in nozzles: \(a\) — conoidal; \(b\) — cylindrical

\(l_{\mathrm{m}}\) — the zone of the metastable state, where a pressure drop by the amount \(\Delta P_{\mathrm{m}}\) occurs at constant liquid temperature;

\(l_{\mathrm{p}}\) — the zone of adiabatic boiling (vapor formation), where a further pressure drop to the back pressure in the outlet section occurs, and the temperature falls in accordance with the equilibrium conditions for a curvilinear surface.

The total time \((\tau)\) for the liquid to flow through the nozzle is composed of the times for the liquid to pass through the zone of the metastable state \((\tau_{\mathrm{m}})\) and the zone of vapor formation \((\tau_{\mathrm{p}})\):

\[ \tau = \tau_{\mathrm{m}} + \tau_{\mathrm{p}} . \]

When analyzing the experiments [3] (Figs. 3, 4) with a cylindrical nozzle \(l = 70\) mm, \(d = 2.2\) mm, one can see that \(\tau_m\) is substantially greater than \(\tau_p\) and that this time belongs to the isothermal section, where pressure loss occurs due to friction. It is clear that this time is to a certain degree arbitrary and does not affect the time \(\tau_p\). Indeed, keeping \(\Delta P_{\mathrm{fr}}\) the same, but changing \(l_n\), we shall obtain different values of \(\tau_m\) at the very same flow velocity, and consequently also \(\tau\). Thus, the same magnitude of disturbance and, correspondingly, the same relaxation time will correspond to a different process time, although the final effect will be the same (the final pressure and steam content in the outlet section).

In general, one can achieve the same effect of steam formation over different times of the process, with the same ratio \(P_1/P_2\), by profiling the nozzles in different ways. From this it may be concluded that one cannot use the “relationship between the process time and the relaxation time” without introducing certain restrictions. This state of affairs forces one to turn to the search for other quantities (complexes or factors) that unambiguously and sufficiently strongly influence the process of adiabatic outflow of saturated water.

Although many researchers of this process pay attention to the influence of the magnitude \(\operatorname{grad} P\), the very nature of such influence still remains in the background. It is interesting, for example, to consider such a particular question as the influence of \(\operatorname{grad} P\) on the shape and size of a steam bubble and on the steam flows in its cavity.

Let us imagine a steam bubble in a uniform stream of boiling water (in the direction of the \(x\)-axis) (see Fig. 2). Under the action of the longitudinal \(\operatorname{grad} P\), such a steam bubble will be compressed in the direction of the \(x\)-axis; it will assume some shape different from spherical.

If, to simplify the reasoning, we assume that such a bubble, compressed in the direction of motion, has in the section by the \(xy\) plane the form of an ellipse, then it may be said that the greatest mean curvature \((c)\) will occur at \(x = 0\) and the least at \(y = 0\). In this connection, the steam pressure in the bubble space when going around the surface must be described by the Laplace equation:

\[ P_{\mathrm{p}} = P_{\mathrm{l}} + \sigma\left(\frac{1}{r_1} + \frac{1}{r_2}\right). \]

Fig. 2. Diagram of a steam bubble and steam flows in its cavity

The difference of pressures at different points of the bubble space, associated with the difference in radii of curvature, will cause steam flows \((I_{\mathrm{p}})\) in the bubble cavity, the character of whose motion will correspond to the scheme shown in Fig. 2. However, whether the deformation of the bubble under the action of \(\operatorname{grad} P\) will lead to any significant decrease in the volume of the bubble because of the presence of a pressure difference inside the volume, and also because of the influence of curvature on \(\sigma\), is still unclear. If, however, we suppose that the volume does not change substantially, then with significant deformation there will be an increase in the surface of the bubble. With a twofold and greater increase in surface, spontaneous rupture of the initial bubble into smaller bubbles is possible. This phenomenon can be described mathematically. Let:

\(V\) — volume of the steam bubble;

\(r\) — its radius before deformation;

\(S, S_{\text{comp}}\) are the surface areas before and after compression, respectively;
\(\Pi\) is the surface energy;
\(a\) and \(b\) are the major and minor semi-axes of the ellipse (Fig. 2).

Then \(V = \frac{4}{3}\pi r^3 \simeq \frac{4}{3}\pi b a^2\) and \(ba^2 = r^3\);

\[ S_{\text{comp}} \simeq 2\pi a^2 + 2\pi b^2 = 2\pi\left(\frac{r^3}{b} + b^2\right); \]

\[ \Pi = \sigma \cdot S_{\text{comp}} = 2\pi\left(\frac{r^3}{b} + b^2\right)\sigma . \]

Under mechanical equilibrium, equality holds between the forces due to \(\operatorname{grad} P\) and surface tension:

\[ \frac{\partial \Pi}{\partial b} = -\left(\frac{\partial P}{\partial X}\cdot 2b\right)\pi a^2, \]

and since

\[ \frac{\partial \Pi}{\partial b} = 2\pi\sigma\left(2b-\frac{r^3}{b^2}\right), \]

we obtain

\[ \frac{\partial P}{\partial X} = \sigma\left(\frac{1}{b^2}-2\frac{b}{r^3}\right). \tag{1} \]

The critical value of the minor semi-axis of the ellipse \((b_{\text{cr}})\), at which spontaneous rupture of the bubble may occur, is determined by the condition: \(2S = S_{\text{comp}}\). Taking into account that \(S = 4\pi r^2\), we obtain \(b_{\text{cr}} = \frac{r}{2}\). Substituting \(b_{\text{cr}}\) into formula (1), we find:

\[ \frac{\partial P}{\partial x} = \frac{3}{r^2}\cdot \sigma . \tag{2} \]

Formula (2) determines the upper value of the bubble radius for a given pressure gradient. In reality this value will be smaller if the presence of shear stresses in the flow is taken into account. It is also quite probable that \(\operatorname{grad} P\) affects the magnitude of the critical radius when a bubble arises, which would contribute to an increase in the superheating of the liquid. It is also known that the flow of a two-phase medium is characterized by the presence of a relative velocity \(W_{\text{rel}}\) between the phases, whose magnitude is also determined by \(\operatorname{grad} P\). An increase in \(\operatorname{grad} P\) leads to an increase in the relative velocity. This, in turn, leads to an increase in the flow cross sections for the liquid phase and, consequently, to a higher water flow rate through the nozzle than the flow rate obtained from the conditions of equilibrium flow along the isentrope. In addition, it must be taken into account that the presence of \(W_{\text{rel}}\) leads to an increase in the nonequilibrium of the process itself.

This is associated with the fact that a certain portion of the steam bubbles, overtaking the liquid flow, enters a region with such temperature values at which the bubble radius proves to be smaller than its critical value corresponding to the given liquid temperature; this leads to the collapse of the bubble and, under certain conditions, to the occurrence of cavitation. All these phenomena may occur in short nozzles, where \(\operatorname{grad} P\) reaches considerable values (10 bar/mm and more).

Thus, if one summarizes all the results of the analysis carried out at different times by various researchers with respect to the adiabatic outflow of water with steam formation, the impression is created that, apparently, there is no single decisive factor that would determine the process in all cases.

References

1. M. D. Vaisman. Thermodynamics of Vapor–Liquid Flows. Energiya, 1967.

2. M. E. Deich, G. A. Filippov. Gas Dynamics of Two-Phase Media. Energiya, 1968.

3. S. V. Polozhii. On the calculation of flow parameters under adiabatic vapor formation. Izv. TPI, vol. 125, 1964.

4. S. V. Polozhii. On the process of vapor formation during the outflow of heated water. Izv. vuzov SSSR, “Energetika”, No. 2, 1960.

5. A. A. Gurchenok. Investigation of the process of outflow of boiling water through diaphragms and cylindrical nozzles. Izv. TPI, vol. 101, 1958.

6. A. A. Gurchenok. Determination of the completeness of the vapor-formation process during the outflow of boiling water. Izv. TPI, vol. 101, 1958.

7. I. I. Kirillov, R. M. Yablonik. Fundamentals of the Theory of Wet-Steam Turbines. Mashinostroenie, 1968.

8. K. S. Polozhii. On the calculation of nozzles and cylindrical channels under adiabatic flow of an evaporating liquid. Trudy LPI, vol. 247, “Turbomachines,” 1965.