On the Question of Boiling of a Liquid with Heat Supplied Through a Wall

S. V. Polozhii

Solving the problem of intensifying heating surfaces—the principal direction in the development of modern steam generators—is closely connected with the study of the process of steam formation. Heat transfer during a change in the state of aggregation is the object of intensive study by whole groups of scientific workers in many countries, and especially in our Soviet Union. Significant successes in this field have been achieved by the Soviet school of thermal engineers [3; 4; 8], which extended the theory of similarity to this class of phenomena. However, when these questions are presented in the literature, a sufficiently consistent physical treatment is not always given of the boiling process [6; 7; 9; 13] and of many phenomena connected with it.

As is known, surface phenomena play an essential role in processes associated with phase transformations, such as, for example, the transition from the liquid state to the gaseous state—steam formation or boiling—and the reverse transition from the gaseous state to the liquid state—condensation.

The dependence between the surface tension of a liquid, the radius of curvature of a bubble, and the excess of the boiling temperature of the liquid over the tabulated saturation temperature is determined by an equation of the form:

\[ \Delta t_s=\frac{2\sigma}{P'\rho}\frac{\gamma'}{\gamma'-\gamma''}, \tag{1} \]

\(\Delta t_s\) — excess of the boiling temperature of the liquid over the tabulated saturation temperature;

where \(\sigma\) is the coefficient of surface tension of the liquid, kg/m;

\[ P'=\left(\frac{dp}{dt}\right)_s=\frac{r\gamma'\gamma''}{AT_s(\gamma'-\gamma'')} \]

— the dependence of the pressure of saturated steam on the saturation temperature;

\(\rho\) — minimum radius of curvature of the surface (of the bubble), m;

\(\gamma', \gamma''\) — specific weight of the liquid and of the steam, kg/m³;

\(T_s, r\) — absolute saturation temperature and heat of vaporization;

\(A\) — thermal equivalent of work.

For a given external pressure, formula (1) has the following final form

\[ \Delta t_s=\frac{2\sigma}{\rho}\frac{AT_s}{r\gamma''}=\frac{B}{\rho}, \tag{2} \]

where

\[ B=\frac{2\sigma AT_s}{r\gamma''}. \]

In this formula \(r, \gamma'', \sigma\) have the values at the saturation temperature over a plane surface.

The value of the surface tension decreases with increasing temperature and, at the critical temperature, becomes equal to zero (Fig. 1). The quantity \(P'=\left(\dfrac{dp}{dt}\right)_s\) increases with increasing pressure (Fig. 2); \(\gamma''\) increases with increasing external pressure, while \(r\) decreases.

Fig. 1.

Fig. 2.

Therefore, for one and the same minimum radius of the germinal body, the excess of the initial temperature over the tabulated saturation temperature \((t_s)\) depends on the pressure above the liquid, decreasing as the pressure increases. At the critical pressure \(\sigma=0\) and \(\Delta t_s=0\). Under vacuum, the increase of \(\sigma\), \(\gamma'\), and the decrease of \(P\) causes a considerable value of \(\Delta t_s\). Accordingly, the onset of nucleation of the vapor phase at high pressures is possible at a small value of \(\Delta t_s\) and at very small \(\rho\). Under rarefaction, considerable initial dimensions of the germinal bodies are required, or, in their absence, a high value of \(\Delta t_s\).

The influence of germinal bodies on the superheat temperature of a liquid is well illustrated by heating a degassed liquid in a glass vessel. In this way, for example, it has been possible to bring the temperature of a liquid to \(200^\circ\mathrm{C}\) at atmospheric pressure. Such a state is customarily called metastable, unstable.

At a quite definite value of the heat flux in the process of vapor formation at a given external pressure, the initial dimensions of vapor bubbles are determined by the microroughness of the heating surface of the vessel with the liquid, to which there corresponds a quite definite temperature for the onset of phase formation (boiling). However, a change in the heat flux, all other conditions being equal, affects both the value of \(\Delta t_s\) and the minimum size of the germinal bodies. An increase in the heat flux raises the superheat temperature of the liquid; its magnitude is established by the entry into action of new, additional centers of vapor formation with smaller dimensions of the germinal bodies. This new stage already proceeds at a higher superheat temperature and at the corresponding magnitude and number of germinal centers of vapor formation. At low heat fluxes, the centers of vapor formation are the largest individual roughnesses with considerable \(\rho\) and small \(\Delta t_s\) established in this process.

The quantities \(\Delta t_c\) and \(\rho\), at one and the same heat flux, in turn depend on the external pressure, decreasing as the external pressure increases, as was already considered above (formula 1).

In all cases of vapor formation in the region of subcritical external pressures, the presence of a superheat temperature of the liquid is obligatory; it is determined by the kinetics of the phase transition.

Numerous experiments on the boiling of various liquids confirm the theoretical propositions of the kinetics of phase transitions [10].

In all experiments on the boiling of liquids, the existence was established of a difference between the temperature of the liquid at the heating surface and the saturation temperature above the liquid. This difference, ultimately due to surface tension, has received the name of the temperature head, or superheat, of the liquid.

Therefore the term “isothermal” or “isobaric” process of vapor formation characterizes not the entire process of phase transition, but only its final stage—the emergence of vapor bubbles at the plane surface of phase separation and the vapor phase above the liquid, which is under a constant external pressure with the tabulated saturation temperature. And only in this sense may these terms be used in educational and technical literature. Numerous experiments also confirm the nonconstancy of the temperature of the initial phase of vapor formation for one and the same external pressure (the final boiling temperature) and its dependence on the conditions under which the process proceeds. In this case, the temperature of the initial stage of boiling with heat supplied through a wall is established in the volume of liquid adjacent to the heating surface, and without special measurements it was not possible to determine it. The usual measurement of the temperature of a boiling liquid in everyday life and in technology did not reveal the essence of the process and was the cause of an incorrect view of this process. Without taking hydrostatic pressure into account, the temperature at which boiling of a liquid begins is determined from the equation:

\[ t = t_s + \Delta t_c, \tag{3} \]

where \(t_s\) is the saturation temperature above a plane surface of phase separation (the temperature of the final stage of boiling),

\(\Delta t_c\) is the temperature difference due to the surface tension of the liquid.

When

\[ \Delta t_c = 0 \qquad t = t_s, \]

i.e., the temperature at which boiling begins is equal to the temperature of saturated vapor above the plane of phase separation—there is equilibrium of the liquid and vapor phases. The process of vapor formation is absent. In fact, when \(\Delta t_c = 0\), the heat flux \(q = c \cdot \Delta t_c G = 0\), and the amount of vapor formed

\[ D = \frac{q}{r} \]

is also equal to zero,

where

\(q\) is the heat flux in \(\text{kcal}/\text{m}^2\) hour,

\(r\) is the heat of vaporization in \(\text{kcal}/\text{kg}\),

\(c\) is the heat capacity of the liquid.

To illustrate the subsequent arguments, we shall use the experimental data on boiling with heat supplied by Jakob and Fritz [12; 14], who measured the boiling temperature and the growth rate of vapor bubbles in the volume at atmospheric pressure by means of high-speed cinematography.

Figure 3 graphically shows the change in the temperature of boiling water with depth when it is heated from below. Figure 4 shows the growth rate of a vapor bubble, obtained by means of high-speed cinematography of the boiling process.

Fig. 3.

Fig. 4.

As is seen from Fig. 3, the temperature of the boiling water at the surface of the free level is somewhat higher than the saturation temperature \(t_s\). In the direction from the free level toward the heating surface, the temperature of the boiling liquid is at first almost constant and only in a small layer adjacent to the heating surface does it rise sharply and substantially exceed the saturation temperature above the free surface of the liquid.

In both figures we have shaded the zone of rapid temperature drop (Fig. 3) and the corresponding zone of bubble formation and rapid bubble growth (Fig. 4).

Accordingly, the entire process of vapor formation during boiling of a liquid in bulk may be divided into two zones:

I — the intensive zone, or the zone of boiling proper (the shaded region directly adjacent to the heating surface);

II — the nonintensive boiling zone—the overlying region of the liquid. In this region there occurs an extremely slow decrease in the temperature of the boiling liquid and slow growth of the bubble.

Table 1 gives the main characteristics of zones I and II of boiling, according to the data of this experiment in our treatment.

As is seen from Table 1, the numerical data characterizing the boiling zones differ sharply from one another. The duration of the process (bubble growth) in zone I is 8 times less than the duration of the process in zone II. At the same time, the bubble growth rate and the temperature drop in zone I many times exceed those in zone II.

Nevertheless, in the nonintensive zone considerable vapor formation may occur, mainly owing to the large amount of liquid and the long residence time of the bubble in this zone. If, however, one calculates the total amount of heat expended on vapor formation in zones I and II without taking into account time and the volume of liquid, as is usually done [9, 13], then both in experiments [12, 14] and in other experiments the heat transferred to vapor formation in zone II proves to be greater than in zone I (in the experiments of Jakob and Fritz, the heat transferred in zone I amounted to 5–10% of the total heat flux). This circumstance leads to a reassessment of the role of the height ...

Table 1

layer of liquid above the heating surface [9, 12] and to incorrect conclusions.

With an increase in the height of the liquid level above the heating surface, the amount of steam obtained in zone II will increase.

Thus, the process of vapor formation at constant external pressure and saturation temperature above a plane phase-separation surface proceeds at a variable temperature, caused by surface phenomena of the liquid. In this respect, according to the change and the absolute magnitude of the liquid temperature, two boiling zones are observed:

1. The intensive zone—the zone of actual boiling of the liquid. It characterizes the basic process of phase formation. In this zone the vapor phase is nucleated and grows rapidly; moreover, as the vapor bubble grows, the capillary pressure and the corresponding vapor temperature in the bubble and in the surrounding liquid decrease sharply. The height of this zone is relatively small, for example, 3–5 mm for boiling water under atmospheric conditions, and depends on the departure diameter of the vapor bubble, decreasing with increasing external pressure.

2. The non-intensive zone—the final stage of vapor formation. It is characterized by a slow fall in the temperature of the liquid and vapor as the detached bubble moves toward the free surface of the liquid. Bubble growth in this zone occurs slowly. The height of the liquid layer in this zone is an arbitrary quantity, determined by the specific conditions of the process—the height of the liquid when it boils in the bulk.

Despite the existence of a developed theory of phase transitions [10] and numerous experimental data on the boiling of various liquids, the literature nevertheless contains an incorrect interpretation of the boiling temperature and of the process of vapor formation as a whole.

Thus, for example, Mlodzeevskii (Molecular Physics, 1941) writes: “The boiling temperature of any liquid is that temperature at which the pressure of its saturated vapor becomes equal to the external pressure.”

Pavlov (Course of Physics, vol. 1, 1949) assesses the boiling temperature in exactly the same way: “The boiling point or boiling temperature is called that-”

boiling temperature at which the elasticity of the saturated vapor of the liquid becomes equal to the external pressure.”

Kashin (Course of Physics, vol. 1, 1948) gives two laws of boiling:

“1. Boiling begins when the temperature is reached at which the pressure of the saturated vapor of the liquid is equal to the external pressure.

1. Throughout the entire process this temperature remains constant.”

Frisch and Timoreva (Course of Physics, vol. 1, 1952) define boiling as vigorous evaporation, which begins when the liquid is heated to such a temperature that the elasticity of its saturated vapors becomes equal to the external pressure.

In the physics textbook for higher technical schools and universities edited by Academician Papaleksi (vol. 1, 1948) it is written: “Boiling is not the birth of a two-phase system, but the transition of a two-phase system from a quiet existence to a vigorous one, in which bubbles grow rapidly and float up.”

Strauf (Molecular Physics, 1949) gives a formulation of the boiling process that in principle does not differ from the preceding ones:

“For boiling it is necessary: 1. That the pressure of the saturated vapor at least reach the value of the external pressure. 2. That heat be supplied to the liquid not through the free surface.”

However, Strauf takes into account the influence on the boiling temperature of a liquid of other factors as well—in particular surface tension—but considers it accidental and therefore does not include it in the above formulation.

Similar formulations occur even in some specialized literature. Thus, for example, Vukalovich and Novikov (Technical Thermodynamics, 1952) define boiling of a liquid as “...a process of intensive evaporation throughout the entire volume, characterized by equality of the vapor pressure inside the vapor bubbles and the external pressure on the liquid and by constancy of temperature during the process.” And further: “the boiling temperature of a liquid at a given external pressure on the liquid is equal to the temperature at which the pressure of the saturated vapor of the liquid (above its plane surface) is equal to the external pressure.”

However, having given such a definition of the boiling of a liquid, the authors subsequently substantiate the presence of a difference between the temperature of the liquid \(T_\rho\), at which boiling of the liquid begins at a given external pressure \(P_\rho\), and the temperature \(T_s\), at which the pressure of the saturated vapors of the liquid is equal to the external pressure \(P_0\), calling it the degree of superheating of the liquid

\[ \Delta T = T_\rho - T_s = \frac{T_s v''}{r}\,\frac{2\sigma}{\rho}, \]

and also consider the condition for the formation of nuclei of the new phase

\[ \rho_{cr} = \frac{2\sigma v''}{(v'' - v')(P_\rho - P_s)} . \]

But they regard the superheating of a boiling liquid not as a regular phenomenon, but only as an accidental one, when the liquid does not contain foreign gaseous impurities. They completely ignore the presence of “superheating” in the volume of the liquid near the heating surface, where the new phase arises and is nucleated.

In the formulations cited, the boiling temperature of a liquid is reduced entirely to the temperature of saturated vapor beneath a plane phase-separation surface at a given external pressure, and the whole process of phase transition is reduced to its final stage. The influence of external pressure on the saturation temperature has been well studied. The temperature of saturated vapor above a plane phase-separation surface corresponds strictly to the external pressure.

To regard this temperature as the boiling temperature of the liquid, as is usually done, is erroneous and follows from an incorrect conception of the phase change of the aggregate state of a substance. What is considered is not the dialectics of this process, not the course of the phase transformation in time, but only the final result, the “equilibrium” between the new and old phases, on the assumption that the new phase has reached its full development and that the interface between the phases is plane. In reality, at a given external pressure this temperature is not the temperature at which the transition of the old state into the new actually begins, but that at which the process stops, when the coexistence of the new and old phases in equilibrium is possible for an indefinitely long time.

Thus, the generally accepted formulations of the process and of the boiling temperature of a liquid do not characterize this process exhaustively. Therefore it is necessary to broaden the understanding of the law of boiling of a liquid so that it embraces all aspects of this complex process and correctly reflects it. We consider it possible to give the following formulation of the law of boiling of a liquid.

1. Boiling is a process in which vapor formation occurs not only from the surface, but also within the liquid.

2. The boiling temperature of any liquid is that temperature at which the pressure of its saturated vapor is equal to the pressure above the liquid plus the additional pressure within it.

3. The initial boiling temperature of a liquid at a given external pressure is a variable quantity; it depends on the properties of the liquid and on the conditions under which the process proceeds.

From the preceding investigation of formula (1) it follows that, at high external pressure, capillary pressure plays a smaller role in the process of vapor formation, and at the critical pressure it disappears altogether.

Already at subcritical pressures, even for small \(\Delta t_s\), the existence of sharply expressed bubble (nucleate) boiling is impossible. Any smooth surface will at the same time be rough. The most insignificant microroughnesses will serve as centers of vapor formation. Boiling is possible at very small \(\Delta t_s\). Under rarefaction, however, the role of surface tension in the process of vapor formation increases. The possibility of the appearance of centers (bubbles) of vapor formation is sharply reduced. Boiling is possible at considerable \(\Delta t_s\), or in the presence of large germ bodies. Only sharply expressed nucleate boiling is possible. At small \(\Delta t_s\), the size of the germ bodies reaches considerable dimensions, and for the boiling process to take place it is necessary to have macroroughness or the preliminary formation of large bubbles of dissolved gases.

Thus, for example, at \(\Delta t_s = 2^\circ\text{C}\) and an external pressure \(P_s = 0.05\) ata, boiling is possible only in the presence of germ bubbles of size \(R_{\min} = 0.3\) mm, which practically leads to the cessation of boiling.

In all cases, at subcritical pressures we have only nucleate vapor formation. The term “film” boiling has a conventional character when the formation of bubbles occurs in such a way that, in the process of growth, they merge into a vapor phase before detaching from the heating surface [14].

Figure 5 shows a diagram of “film” boiling at high external pressure and under rarefaction. Under rarefaction the process proceeds in such a way that the coalescence of rare vapor bubbles on the heating surface occurs even at small \(\Delta t_s\) and, correspondingly, small thermal stresses \((q)\). This conclusion is confirmed by experimental data [13]. At boi-

of boiling of water and reduction of the pressure from atmospheric to 0.09 atm, the critical temperature head remains practically constant.

Visual observations are known of the boiling of water [15] with heat supplied through a wall at various pressures, when even at \(\Delta t_s\) of \(60 \div 65^\circ\mathrm{C}\) there were no signs of “film” boiling. At high temperature differences, the formation of steam was so rapid that the surface of the tube was more or less constantly covered with bubbles.

Fig. 5.

Our visual observations of vapor formation under vacuum with and without heat supplied from outside also confirm the proposition stated.

The boiling of water under vacuum is characterized by the appearance of large bubbles and their rapid growth. The coalescence of bubbles during adiabatic boiling occurs in the form of formation of a bubbly layer of water under the liquid (during boiling in the bulk).

The size of the bubbles increased sharply with decreasing pressure above the liquid and, correspondingly, with the growth of the specific volume of the steam.

Fig. 6.

A decrease in \(\Delta t_s\) in the experiments sharply reduced the number of simultaneously arising bubbles, but increased their size. Thus, for example, at \(p = 0.03\) atm and small \(\Delta t_s\) \((1.5 \div 2^\circ)\), the size of individual bubbles on the free surface of the water reached \(100\text{--}200\) mm in cross section. The growth of the bubbles then occurred in the horizontal direction, as shown in Fig. 6a.

During boiling with heat supplied (boiling on the surface of a nichrome coil \((d = 1\ \mathrm{mm})\) under rarefaction \((p = 0.03\ \mathrm{ata})\)), the influence of external pressure on the growth and size of a bubble is illustrated especially clearly. At the initial moment of heating, the process was characterized by the instantaneous appearance of individual large bubbles at different places on the coil and by their rapid disappearance as a result of heat transfer to the colder surrounding water. The process was accompanied by a sharp metallic sound. The very moment of appearance and disappearance of a bubble was observed as a dull white flash. With further heating, individual spherical bubbles of large diameter appeared at certain places on the coil (12–15 mm—the exact measurement of the detachment diameter of the bubbles could not be made), Fig. 6b.

The mechanism of vapor formation described above provides a physically simple explanation of the process of boiling of a liquid with high surface tension, or generally of nonwetting liquids, such as mercury. For this, as a first approximation, it is sufficient to compare certain physical properties of a wetting and a nonwetting liquid, for example, water and mercury at the same external pressure.

Table 2

From consideration of formula (1) it follows that, for one and the same temperature difference \(\Delta t_c\), the magnitude of the minimum nucleation bubble in mercury is greater than for water by more than 13 times (for external pressure \(p = 1\ \mathrm{ata}\)), which follows from the relation

\[ \frac{\left(\dfrac{\sigma}{p'}\right)_{pm}} {\left(\dfrac{\sigma}{p'}\right)_{vod}} = \frac{0.000275}{0.0000207} = 13.5. \]

Therefore, for example, at one and the same \(\Delta t_c = 10^\circ\), the minimum radius of nucleation bodies required for mercury boiling is 13 times larger than for water. As a result, mercury must exhibit more sharply pronounced nucleate boiling than water.

If, however, the minimum radius of the nucleation bodies for mercury and water is taken to be equal (at \(\Delta t_c = 10^\circ\mathrm{C}\)), then mercury will begin to boil at a \(\Delta t_c\) 13.5 times greater than for water. Owing to the high thermal conductivity of mercury, the growth of a bubble will be such that it will instantaneously encompass a large zone of liquid. The growth of the bubble, because of the nonwetting property of the mercury on the heating surface, will occur along the heating surface. The bubble will be “squeezed out” of the liquid mercury. Separation of the liquid mercury from the heating surface leads to the coalescence of individual bubbles and iso-

mercury from the surface. The high thermal conductivity of liquid mercury promotes the rapid growth of individual bubbles.

It is therefore impossible to agree with the assertion of Mikheev [9] and Kryzhilin [5] that the process of vapor formation in nonwetting liquids occurs over the entire heating surface. Nor can one agree with the existing notion of the detachment of a mercury bubble while leaving a “stalk” on the surface (Fig. 7), advanced by Jakob and Fritz and preserved up to the present time.

Fig. 7.

Such a conception of the mechanism of boiling of mercury is fully confirmed by experimental data. (The presence of a considerable temperature difference \(\Delta t_s\), with large coefficients of thermal conductivity of liquid mercury, the high efficiency of heat transfer from the wall to the liquid mercury, and the motion of mercury bubbles along the heating surface [16].) Experiments on the boiling of water on a nonwetting surface fully confirm the mechanism of vapor formation in nonwetting liquids [15]. In this case individual water bubbles formed as in the boiling of nonwetting liquids, i.e., tending to flow together on the metal surface (the bubbles were “squeezed out” of the liquid), so that a relatively large area of the heating surface was covered with vapor. With an increase in \(\Delta t_s\) (about \(34^\circ\)C), the number of vapor-formation centers increased, and the tendency toward coalescence of bubbles on the heating surface was even more sharply expressed. With a further increase (\(\Delta t_s\) about \(45^\circ\)) in the heat flux, the tube became covered with a continuous layer of vapor, i.e., the coalescence of individual bubbles occurred throughout the entire tube, without the possibility of detecting individual vapor-formation centers. In these experiments \(\alpha\) decreased by as much as 10 times. When water boiled as a wetting liquid under the same conditions, coalescence of bubbles (“film” boiling) was not observed even at \(65^\circ\) and a considerably greater heat flux.

By violating the wettability property of the surface (by adding sodium carbonate to the water), the authors in these experiments achieved the reverse transition of “film” boiling into nucleate boiling.

Thus, the formation of the vapor phase of any liquids, both wetting and nonwetting, up to the critical pressure, can only be nucleate. In wetting liquids the vapor bubbles that form are evacuated through the liquid layer. At large heat loads and, consequently, large values of \(\Delta t_s\), the number of vapor-formation centers increases, and individual bubbles begin to coalesce at the heating surface, insulating the liquid layer from the heating surface. At high pressures (owing to micronucleate boiling), coalescence of vapor bubbles on the heating surface, as a result of the enormous number of vapor-formation centers, occurs at low \(\Delta t_s\) and heat fluxes. Under rarefaction, coalescence of vapor bubbles may occur with an insignificant number of vapor-

formation as a consequence of the larger departure diameter of the bubbles.

In non-wetting liquids (or when a liquid boils on a non-wetted surface), vapor bubbles are removed not through the liquid layer, but between the liquid and the heating surface. The emergence of vapor bubbles at the heating surface and their coalescence on the heating surface lead to insulation of the liquid layer from the heating surface and to deterioration of the heat-transfer coefficient. The expression “boiling of non-wetting liquids over the entire surface” contradicts the nature of the phase transition and should not be used in our educational and scientific literature.

In many specialized works the question of heat transfer during boiling is considered, and a generalization of experimental data is given. The most complete generalization of experiments, with a physical basis for wetting liquids under conditions of boiling in a large volume and natural convection, is given by Kruzhilin [4]. Kruzhilin’s data were included in M. A. Mikheev’s textbook on heat transfer [9].

Without going into all the details of the criterion dependences for the intensity of heat exchange during boiling of a liquid, let us consider only the correctness of the initial premise—the influence of the mechanism of vapor formation on heat transfer.

In the mathematical description of the process of heat transfer to a liquid during boiling in a volume, in addition to the usual differential equations, equations are introduced for the motion of the vapor bubble and for the increase in the volume of the bubble as a result of evaporation of liquid from its surface. In addition, the mutual influence of the centers of vapor formation is taken into account.

As a result of processing these equations in criterion form and on the basis of an analysis of the generalization of experimental data, Kruzhilin obtained the following equations:

For the heat-transfer coefficient,

\[ N_u = 0.075\, P_r^{0.5}\cdot K_q^{0.7}\cdot K_u^{1/3}, \]

and for the critical heat load,

\[ K_q = 99.5\, P_r^{0.5}\cdot K_u^{-0.66}\cdot A_r^{1/3}; \]

where

\[ N_u=\frac{\alpha\delta}{\lambda} \quad \text{— Nusselt criterion;} \]

\[ P_r=\frac{\nu}{a} \quad \text{— Prandtl criterion;} \]

\[ A_r=\frac{g\delta^3}{\nu^2}\,\frac{\gamma'-\gamma''}{\gamma'} \quad \text{— Archimedes criterion;} \]

\[ K_q=\frac{p'\delta^2}{\lambda\sigma}\,\frac{\gamma'-\gamma''}{\gamma'} \quad \text{— criterion determining the number of active centers of vapor formation;} \]

\[ K_u=\frac{\gamma'\,c\,\sigma}{\gamma'' r\,p'\delta}\,\frac{\gamma'}{\gamma'-\gamma''} \quad \text{— criterion determining the frequency of bubble departure;} \]

\[ \alpha \text{ — heat-transfer coefficient,} \]

\[ \lambda \text{ — thermal-conductivity coefficient,} \]

\[ \nu \text{ — coefficient of kinematic viscosity,} \]

$a$ — thermal diffusivity coefficient,
$g$ — acceleration due to gravity,
$c$ — heat capacity,
$q$ — heat flux.

These equations are extended by Kruzhilin to boiling at any pressures, up to the critical pressure. Here, in all dependences there appears the quantity “$\delta$”—the determining dimension,

\[ \delta=\sqrt{\frac{\sigma}{\gamma'-\gamma''}}, \]

proportional to the departure diameter of vapor bubbles $d_0$ ($\delta$ differs from the departure diameter $d_0$ by the absence of the factor $\Theta$). With $d_0$ is associated the frequency of bubble departure

\[ u=\frac{1}{d_0} \]

The heat-transfer coefficient $\alpha$ is related to $u$ and $z$ (the centers of vapor formation). The higher the bubble-departure frequency $u$ and the greater the number of active vapor-formation centers $z$, the more intense the heat transfer. And since the values of $u$ and $z$, for one and the same liquid at a given pressure, depend only on $\Delta t_6$, then $\alpha$ also depends on $\Delta t_6$ and $q$ ($\Delta t_6=f(q)$, for a given surface).

From this the significance of the departure diameter of bubbles $d_0$ in the heat exchange of a boiling liquid is already evident.

The magnitude of the departure diameter of vapor bubbles is determined by Kruzhilin according to Fritz’s formula [11], obtained on the basis of processing experimental data on the boiling of water under atmospheric conditions, namely:

\[ d_0=20\cdot \Theta \sqrt{\frac{\sigma}{\gamma'-\gamma''}} \ \text{mm} \]

or, for water, taking into account

\[ \sigma=0.00714\left(\frac{\gamma'-\gamma''}{1000}\right)^4 \frac{\text{kg}}{\text{m}}, \]

\[ d_0=2.67\sqrt{\left(\frac{\gamma'-\gamma''}{1000}\right)^3}\ \text{mm}, \tag{4} \]

where $\Theta$—the limiting wetting angle—is taken for water as constant, equal to $50^\circ$.

Extending Fritz’s formula to pressures different from atmospheric is erroneous, since it incorrectly reflects the mechanism of vapor formation at other pressures. According to formula (4), $d_0$ for water changes insignificantly with pressure; $d_0$ decreases as the external pressure increases. Under rarefaction, $d_0$ tends toward a maximum equal to $2.67$ mm. The change of $d_0$ from $p_s=1$ ata to $p_s=0$ according to Fritz’s formula is shown in Fig. 8, curve $d_0$.

In the same figure is shown the change of the specific volume of steam with pressure (curve $v''$). From comparison of these two quantities, the unsuitability of Fritz’s formula for other pressures is apparent.

Let us imagine what occurs in the process of evaporation of water at different external pressures $p_1=1$ ata and $p_2=0.025$ ata, in such a way that, per unit time, from a unit surface, $1$ kg of water evaporates. In the first case it is necessary to supply heat $q_1=r_1=539.4$ kcal. In the second case it is neces-

it is necessary to supply \(q_2=r_2=585.6\) kcal; in this case the volume of steam in the first case is \(v_1=v_1''=1.72\ \text{m}^3\), and in the second case \(v_2=v_2''=55.28\ \text{m}^3\).

\[ \text{Ratio}\quad \frac{q_2}{q_1}=\frac{r_2}{r_1}=1.085, \]

\[ \text{ratio}\quad \frac{v_2}{v_1}=32.1. \]

Fig. 8.

According to formula (4), the departure diameter of the bubble increases by 1.04, i.e., \(d_0\) remains practically unchanged when the pressure is lowered toward vacuum,

\[ \frac{d_{02}}{d_{01}}=\frac{2.6}{2.5}=1.04\ \text{times}. \]

The frequency of bubble formation will decrease by

\[ \frac{u_2}{u_1} = \frac{\dfrac{1}{d_{02}}}{\dfrac{1}{d_{01}}} = \frac{0.384}{0.4} = 0.96. \]

At the same time, according to equation (1), with decreasing pressure \(p'\) falls sharply and the minimum radius of the embryonic body increases greatly, as a result of which the number of active centers of steam formation falls sharply. Thus, for example, at \(\Delta t_s=10^\circ\mathrm{C}\), lowering the pressure from \(p_1=1\ \text{ata}\) to \(p_2=0.025\ \text{ata}\)

increases the radius of the embryonic bodies by 75 times

\[ \left( \frac{p_1'}{p_2'}= \frac{0.000278}{0.00000368} =75 \right) \]

and causes a corresponding decrease in the centers of steam formation. Proceeding from such a definition of the equivalent departure diameter of a bubble, it remains unclear how, in the boiling process, with almost one and the same bubble diameter, with a reduced formation frequency \((u)\), and with a significant reduction in the centers of steam formation, a sharply increased volumetric quantity of steam is realized.

The above-mentioned theoretical inconsistency of formula (4) for the departure diameter of a bubble at pressures other than atmospheric is confirmed experimentally, as was already described earlier. Under the conditions of our experiments, the departure diameter of the bubble reached a value 5–6 times greater than that determined by formula (4). Therefore, the extension

of the above-presented criterion equations for the boiling of wetting liquids in a volume at any pressures cannot be considered sufficiently rigorous in that dependence of the determining dimension on the physical properties of the liquid which is described by equation (4), adopted by Kruzhilin.

The erroneousness of extending Fritsch’s formula to excess pressures has also been proved experimentally by the work of Zysin [1], who studied the process of formation and growth of an individual vapor bubble in the range of pressure variation \(1\)—\(5\ \mathrm{kg/cm^2}\). In Fig. 9 is presented the dependence of the relative change in the departure diameter of vapor bubbles \(d_i\) for water with change in pressure. The experimental data are denoted by the solid line; the dashed line represents the same dependence obtained from formula (4).

Fig. 9.

The influence of pressure on the change in the number of active centers of vapor formation, according to the same experiments for water, is shown in Fig. 10.

Fig. 10.

As is seen from Fig. 10, the rate of increase of the number \(z\) is greater than the rate of increase of pressure. With an increase in heat load, the number of centers of vapor formation also increases. Here we have experimental proof of the influence of pressure and heat load on the minimum radius of embryonic bodies. An increase in pressure decreases \(\sigma\) and increases \(p'\), and, consequently, for one and the same \(\Delta t_c\) decreases \(\rho\)

\[ \left(\rho = \frac{2\sigma}{\Delta t_c\,p'} \frac{\gamma'}{\gamma-\gamma'}\right). \]

An increase in \(q\) at one and the same external pressure raises \(\Delta t_c\) and decreases \(\rho\); that is, intensification of the vapor-formation process proceeds mainly through an increase in the centers of vapor formation at unchanged external pressure, that is, through an increase in \(q\) and the correspond-

…its value \(\Delta t_c\). Accordingly, the higher \(z\), the more intense the heat transfer, that is, the higher the heat-transfer coefficient.

With variable external pressure, an increase in pressure at unchanged heat flux causes an increase in the centers of vapor formation and in the magnitude of the heat-transfer coefficient. However, the number of vapor bubbles arising (\(z\)) still does not completely determine the intensity of heat transfer. A second factor in the influence of the mechanism of vapor formation is the frequency of bubble detachment (\(u\)).

The higher the frequency of bubble detachment, the higher \(\alpha\) will be. The frequency of bubble formation at unchanged external pressure depends on the thermal stress of the heating surface. As \(q\) increases, the temperature head (\(\Delta t_c\)) increases; as a result, the rate of preparation of the liquid phase increases, as does the rise in its temperature caused by the minimum radius of the embryonic body. Thus the time required to establish the necessary superheating of the liquid is shortened, and the frequency of bubble formation increases. It must be assumed that there also occurs some shortening of the growth time of the bubble that has arisen, owing to the change in certain physical properties with increasing temperature. However, the development of the vapor phase occurs at a small difference between the temperature of the liquid and the temperature of the saturated vapor inside the bubble, determined by the capillary pressure. An increase in the size of the bubble leads to a decrease in the pressure and temperature inside the bubble and to the development of the vapor phase.

The increase in the frequency of bubble detachment with growth of the thermal stress of the heating surface proceeds considerably more slowly than the emergence of new centers of vapor formation. Evidence of this is the relatively rapid attainment of the maximum heat flux, at which there occurs such a mass growth of individual centers of vapor formation that it leads to the merging of bubbles in the vapor phase—to “film” boiling.

The interaction of \(d_0\), \(u\), and \(z\) as functions of \(q\) and \(p\) ultimately determines the heat-transfer coefficient in the part governed by the mechanism of vapor formation. By the relation among \(d_0\), \(u\), and \(z\), it seems to us, one can physically explain the dependence of the magnitude of the maximum heat flux on pressure both for boiling of water in a large volume and for boiling of water in a tube (solid curve) [2], Fig. 11.

Fig. 11.

The initial increase of the curves with rising pressure is characterized by the fact that the factor of increase in the centers of vapor formation proves to be more…

a favorable influence on \(q\) than the factor of bubble detachment frequency. At an external pressure from 10 to 70 ata during boiling in a tube and from 50 to 90 ata during pool boiling, there is a maximum value of the critical heat flux—the most favorable ratio of the factors \(u\) and \(z\). With a further increase in pressure, \(q_{\mathrm{cr}}\) again falls, and at a pressure close to the critical one, \(q_{\max}\) is several times smaller than at atmospheric pressure. The transition to “micro”-bubble boiling (a high value of \(z\)) at superhigh pressures causes bubbles to arise everywhere—over the entire surface, with the possibility of their coalescence before detaching from the surface, despite the small detachment diameter of the bubbles. A less sharp increase in the frequency of bubble detachment at superhigh pressures contributes to this possibility of coalescence of vapor bubbles and, consequently, to a reduction in the critical heat flux.

The proposition stated above concerning the different quantitative influence of increasing pressure on \(u\) and \(z\) is confirmed by experiment. An increase in pressure has a sharper effect on the growth in the number of vapor-generation centers than on changes in the frequency of bubble detachment. Thus, for example, using the experimental data of Zysina [1], it may be established that a change in pressure from 1 to 3 kg/cm\(^2\) increases the growth of vapor-generation centers for water by a factor of 5, while the detachment frequency over this same interval of pressure change increases only by a factor of 2. This explains the rapid transition to “film” boiling at high and superhigh pressures even when the thermal stresses of the heating surface are not high.

It seems to us that experiments to determine the dependence of \(u\) and \(z\) at superhigh pressures are absolutely necessary here.

When a liquid boils under vacuum, as the rarefaction increases, there will occur a considerable increase in the detachment diameter \((d_0)\) and a decrease in the frequency of detachment of vapor bubbles \((u)\) and in the number of active vapor-generation centers \(z\) (strongly expressed “macro”-bubble boiling of the liquid). However, with increasing vacuum the relation between the factors \(d_0\), \(u\), and \(z\) may lead to coalescence of the vapor phase of separate “rare” bubbles, to deterioration of turbulence, and to a decrease in the maximum flux. The relations among \(d_0\), \(u\), and \(z\) during boiling of water under vacuum are of such a character, as experiments [13] show, that the critical temperature head remains practically constant.

The principal contradiction of the process of vapor formation with heat supplied through a wall is the limited possibility of increasing the intensity of this process. An increase in heat flux is possible only so long as purely “nucleate” boiling takes place, that is, without coalescence of vapor bubbles at the heating surface. Quantitative changes in this process lead to a new qualitative state—to coalescence of vapor bubbles at the heating surface—to “film” boiling, to insulation of the liquid layer from the heating surface by vapor, to a sharp reduction in the heat-transfer coefficient, to an increase in wall temperature, and to a reduction in the permissible heat flux. The present-day tendency toward transferring the steam-power cycle to superhigh pressure may encounter on its way a difficult-to-overcome obstacle associated with the low permissible thermal stresses at superhigh pressures, caused by the transition to “film” boiling at small temperature heads.

The thick-walled design and the use of special high-alloy steels at superhigh pressures will require steam generators with high thermal stresses of the heating surfaces. However, the physical nature of the process of vapor formation with heat supplied through a wall may become a substantial obstacle in solving this problem. The use of adiabatic vapor formation removes these difficulties.

References

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