ON THE KINETICS OF THE PROCESS OF STEAM CONDENSATION IN A TURBINE STAGE

S. V. POLOZHY

(Presented by Prof. Dr. I. N. Butakov)

Revealing the physical essence of the process of condensation of wet steam, and the kinetics of this process in turbines, is an important task in solving the problem of stage operation in the region of wet steam. However, modern investigations of losses due to moisture in steam turbines, and the theoretical and experimental study of this question, are not at the proper level [1, 2, 3]. This applies to a certain extent to work [4], in which an attempt is made to substantiate theoretically the kinetics of the process of steam condensation in a turbine stage on the basis of many years of experimental investigations and the use of results available in the literature from the study of phase transformations [5, 6]. This state of the question is explained by the fact that modern investigations do not take into account the phenomena of adiabatic phase formation that occur during the expansion of steam in the flow passage of a turbine.

A feature of the kinetics of the process of adiabatic phase formation is the simultaneity of mutually opposite processes: adiabatic condensation and adiabatic vaporization, complicated by surface phenomena, internal interphase heat exchange, and intensive displacement of the flow. However, the final result of the process depends on the initial state of the system. When the pressure of saturated steam is reduced, the final result is adiabatic condensation—the transformation of part of the steam into liquid. When the pressure of a heated liquid is reduced, on the contrary, steam is ultimately formed—adiabatic vaporization.

In an isoentropic pressure drop of saturated steam, for a completed process the final amount of water formed may be determined from thermodynamic relations

\[ y_s=\frac{S_1''-S_2''}{r_2}\,T_{\mathrm{н}2} =\frac{S_1''-S_2''}{S_2'-S_2''}\ \text{kg/kg}, \]

where \(S_1''\), \(S_2''\) are the tabular entropies of saturated steam at the beginning and end of the process,

\(T_{\mathrm{н}2}\), \(r_2\) are the absolute temperature and heat of vaporization at the end of the process.

In the intermediate states of the process, after the pressure of saturated steam has been reduced by an amount ensuring the nucleation of the phase

in the form of associations of molecules and microdroplets of liquid, as the pressure continues to decrease both condensation and vaporization of the previously formed liquid take place. At some moment the microdroplets of liquid prove to be superheated, while the steam is supercooled relative to the new value of the saturation temperature at the given lower pressure of the system; both phases prove to be unstable and are forced to change their state of aggregation.

The driving force of these processes is the continuous fall of pressure and the tendency of the system to come into conformity with its thermodynamic state at the new, lower pressure. This continuous restructuring of both the steam and the liquid is accompanied by internal processes of heat exchange between the phases: the release of heat by the condensing part and the absorption of heat during vaporization, and by an overall decrease in the temperature of the system in accordance with its thermodynamic potential at the lower pressure.

The quantitative balance, the predominance of one process over the other both in the intermediate and in the final state of the process, is determined by the initial state of the substance. If in the initial state there is saturated steam or wet steam with predominance of the vapor phase, then in the final result liquid is formed, and the processes of condensation predominate over the processes of vaporization; conversely, saturated liquid or wet steam with predominance of the liquid phase in the initial state leads to the formation of steam and to the predominance of vaporization processes over condensation at intermediate points of the pressure-drop process. For example, for some portion of isentropic expansion of wet steam with a pressure drop by the amount \(\Delta P = P_0 - P_1\), steam is formed from the previously precipitated liquid in the amount \(\Delta x\):

\[ \Delta x=\frac{S_0' - S_1'}{r_1}\cdot T_{\mathrm{n}1}\cdot Y_0 =\frac{S_0' - S_1'}{S_1'' - S_1'}\cdot Y_0, \]

and liquid is formed from the previously existing steam, \(\Delta y\):

\[ \Delta y=\frac{S_0'' - S_1''}{r_1}\cdot T_{\mathrm{n}1}\cdot x_0 =\frac{S_0'' - S_1''}{S_1' - S_1''}\cdot x_0, \]

where \(y_0, x_0\) are the relative moisture content and dryness of the steam at pressure \(P_0\),

\(S_0'', S_1''\) are the entropy of saturated steam at the beginning and end of the process,

\(S_0', S_1'\) are the entropy of the liquid at the beginning and end of the pressure drop.

From these relations it is seen that the formation of a new phase is proportional to the initial amount of the phase, taking into account that

\[ S_0' - S_1' \simeq S_0'' - S_1'', \quad \text{and } r_1 \text{ and } T_{\mathrm{n}1} \]

are one and the same for the vapor and liquid phases at pressure \(P_1\).

Proof that the processes of adiabatic vaporization and condensation proceed simultaneously is provided by the thermodynamic state of a two-phase system (Fig. 1). The diagram of the phase state of \(\mathrm{H_2O}\), depending on the final result of adiabatic phase formation, may be divided into two regions: adiabatic vaporization ABCA and adiabatic condensation ABDA. The boundary between them is the isentrope of equal phase formation AB—condensation and vaporization—for which any magnitude of pressure drop does not lead to a change in the ratio between the vapor and liquid phases. Such a course of the process is possible when there is equality

of the vapor and liquid phases in the initial state of the process or by reducing the pressure of a substance in the critical state (point A, Fig. 1). The isentrope of equal phase formation is good evidence of the simultaneous occurrence of the processes of adiabatic condensation and adiabatic vaporization. Here the possibility is excluded of a veiled, one-sided course of the processes, for example, only condensation of steam. This contradicts the nature of the state of the substance. A drop in pressure and temperature acts equally on both the vapor and the liquid phase, bringing them into correspondence with the new physical and thermodynamic state at the lower pressure of the system. The simultaneity of the occurrence of adiabatic condensation and vaporization is also well illustrated by the final result of a pressure reduction of a two-phase system with different ratios of the liquid and vapor phases in the initial state at

Fig. 1. Phase-state diagram of H\(_2\)O. ABCA—the region of adiabatic vaporization; ABDA—the region of adiabatic condensation; AB—the adiabat of equal phase formation; \(x\)—relative amount of saturated vapor; \(y\)—relative amount of water at the saturation temperature; \(1_{\mathrm{p}}, 2_{\mathrm{p}}, 3_{\mathrm{p}}\)—adiabatic vaporization in the range of low, high, and superhigh isentropic pressure drop; \(1_{\mathrm{k}}, 2_{\mathrm{k}}, 3_{\mathrm{k}}\)—adiabatic condensation in the range of low, high, and superhigh isentropic pressure drop.

unchanged initial and final pressures of the process. In Table 1, for the value of the isentropic pressure drop \(\Delta P = P_0 - P_1 = 10 - 0.03 = 0.97\) ata, the phase ratio at the beginning and end of the process is given for wet steam of different initial moisture content.

As can be seen from Table 1, condensation of saturated steam is not proportional to its content in the initial state, but decreases as the initial moisture content of the steam increases. If, for example, in the initial—

state dry saturated steam ($x = 1.0$), then by the end of the process 14% of this steam has turned into liquid. With an initial dryness fraction of 60% ($x = 0.60$), by the end of the process only 6.6% of this steam condenses. A further increase in the moisture content of the steam in the initial state leads not to a decrease in the saturated steam at the end of the process, but to its increase, and the processes of steam formation begin to prevail over condensation. Meanwhile, according to the modern theory of one-sided continuous condensation of wet steam, which excludes steam-formation processes, the amount of liquid formed must always be proportional to the initial amount of saturated steam at the beginning of the process.

Table 1

Influence of the initial phase ratio on adiabatic phase-transformation processes

The distinctive character of the expansion of wet steam in the flow passage of a turbine is determined in many respects by the interval, magnitude, and nature of the pressure drop as the driving forces of adiabatic phase transformations. This is explained by the difference in the physical and thermodynamic properties of steam and liquid and by their variation in time and space. In the range of an ultra-high pressure drop, the physical and thermodynamic properties of steam and liquid are such that they facilitate the processes of phase transformation (small deviations from equilibrium states during the nucleation of a new phase, insignificant sizes of nuclei of the new phase, absence of sharp differences in physical properties, etc.). Low pressures, and especially the expansion of steam under vacuum, on the contrary, complicate the conditions for the appearance of a new phase; considerable supercooling or superheating is required because of the sharp difference in the physical and thermodynamic properties of steam and liquid, etc. The magnitude and method of the pressure drop, in turn, will determine the specific character of the process for one and the same pressure interval. A steep pressure fall or one extended along the length of an element will affect differently the intensity of the process, the degree of supercooling (superheating), the place of nucleation of the new phase, and the time and rate of phase transformation. A large pressure drop in a given section of the nozzle intensifies the process in all directions: the number of phase-transformation centers increases, as do the rate of their growth and decay. A small pressure drop, on the contrary, slows the processes of phase transformation, and in the limiting case, in the absence of a pressure drop,

... in general, nucleation of a new phase without the supply of heat from outside is impossible.

A similar conclusion about the general directionality of adiabatic phase formation was possible on the basis of experimental studies of adiabatic vapor formation under vacuum, which made it possible experimentally to reveal the specific features of this process: the influence of the magnitude and character of the pressure drop, of the temperature head on the intensity of the process, of the process energy, the structure and dispersion of the flow, and the time and rate of vapor formation. To trace the change in pressure and temperature of the vapor-formation process along the length of the evaporating element, to determine the degree of perfection of the vapor-formation process and the factors influencing these characteristics of the process. The use in the experiments of transparent (organic-glass) evaporating elements (various nozzles and packings) and the carrying out of experiments under vacuum at low temperatures made it possible to organize visual observations and photography of the process satisfactorily, and conveniently to measure the temperature and pressure of the vapor-formation process along the entire length of the evaporating element. Undoubtedly, in order to establish the quantitative and qualitative characteristics of the process of expansion of wet steam (adiabatic condensation), a similar arrangement of experimental studies in the wet-steam region and theoretical investigations of adiabatic condensation from the above-mentioned standpoints are necessary.

In the existing concepts of the kinetics of the process of condensation of wet steam during its expansion in the flow passage of a turbine, as already mentioned, one proceeds from the concept of a one-sided directionality of this process, as a process only of steam condensation, of the appearance and continuous growth of droplets of liquid, similar to what takes place in stationary condensation of saturated steam on a cooling surface with heat removal through the wall and a constant pressure of the condensing steam. With such a conception of the condensation of wet steam in turbines, the specific features of adiabatic phase formations and the correct physical essence of this process are not reflected in theoretical and experimental studies. In this respect, physicists have succeeded in establishing only elements of the kinetics of phase transitions, in finding the conditions necessary for the occurrence of a new phase within the limits of the old phase. Other questions of this complex process remain unstudied up to the present time. Therefore, to construct the kinetics of the process of steam condensation in turbines by restricting oneself only to the investigations of physicists in the field of phase formation, as was done in [4], is insufficient and undesirable also in the sense that such a solution of the problem not only leads to incorrect conclusions, but also does not promote the further development of theoretical and experimental investigations of the physical essence and kinetics of condensation of wet steam in the necessary direction, which, in turn, makes it difficult to solve the problem of stage operation in the wet-steam region. This question is further complicated by the fact that modern experimental studies of losses due to moisture in experimental turbines cannot serve as a check on theoretical conclusions, since they are carried out by a method that does not take into account all the factors affecting the efficiency of such turbines during the transition from superheated to wet steam [1, 2].

The known formulas for the kinetics of condensation of wet steam that are found in the literature do not reflect the actual process of adiabatic condensation. Among the latter is the formula for

for determining the growth of liquid droplets during the expansion of saturated steam in the initial stage, when the critical radius of the droplets \(\xi\) is smaller than the mean free path of molecules \(\lambda\),

\[ \frac{d\xi}{dt} = \frac{C_p v}{r \rho_{\mathrm{н}}} \sqrt{\frac{RT}{2\pi}}\, \frac{T_{\mathrm{н}}-T}{1-\dfrac{\sigma}{\xi \rho_{\mathrm{в}}\cdot r}}, \qquad \text{or} \qquad \frac{d\xi}{dt} = \frac{3}{8}\frac{P}{r\rho_{\mathrm{н}}} \sqrt{\frac{3R}{T}}\,(T_\xi-T_1), \]

and also for the growth of large droplets in the case \(\xi \gg \lambda\), of a developed process, or the formula for determining the mass of water \(M_{\mathrm{в}}\) of the condensed steam in a given section

\[ M_{\mathrm{в}}=\int_{0}^{x} M_{\mathrm{к}}(x',x)\,J(x')\,F(x')\,dx', \]

where \(x\) is the coordinate determining the position of the section under consideration, \(x'\) is the coordinate of the section in which moisture precipitates with velocity \(J(x')\), and the area of which is \(F(x')\); \(M_{\mathrm{к}}\) is the mass of droplets in section \(x\), depending on the place of their formation.

These formulas do not take into account the simultaneity of condensation and vaporization during the expansion of wet steam in a turbine, nor the influence of the initial moisture content on the final result.

The same may be said of the formulas establishing the necessary conditions for the origination of a new phase and the rates of vaporization: Kelvin’s formula, which determines the degree of supercooling of the steam,

\[ \ln \frac{P}{P_{\infty}} = \frac{2\sigma}{\rho R T \xi^{*}}, \]

or Frenkel’s formula, which establishes the degree of supercooling of the steam,

\[ \ln \frac{T_{\infty}}{T} = \frac{2\sigma V_{\mathrm{в}}}{r \xi^{*}}, \]

as well as formulas for determining the number \(I\) of simultaneously forming centers of phase formation, at a given degree of supercooling, per unit time in an amount of substance containing \(n_1\) molecules,

\[ I= \frac{n_1}{g^{*}} \left[ \exp\left( -\frac{4\pi\sigma \xi^{2*}}{3kT} \right) \right] 2\xi^{*} \frac{P}{kT} \sqrt{ \frac{kT\ln \dfrac{P}{P_{\infty}}}{3m_n} }\, g^{*}, \]

where \(P_{\infty}, T_{\infty}\) are the pressure and temperature above a plane surface; \(P, T\) are the pressure and temperature near the embryonic droplet in the equilibrium state; \(R\) is the gas constant; \(\rho\) is the density of the liquid; \(V_{\mathrm{в}}\) is the molecular volume of the liquid phase; \(r\) is the heat of vaporization; \(\xi^{*}\) is the radius of a droplet of critical size,

\[ \xi^{*}= \frac{2\sigma T_{\infty}\cdot V_{\mathrm{в}}}{\Delta T_i \cdot r}, \]

\(g^{*}\) is the number of molecules in the embryonic droplet; \(m_n\) is the mass of a molecule; \(k\) is Boltzmann’s constant.

These kinetic formulas do not reflect the entire process of condensation and the specific conditions of adiabatic phase transformations. They do not make it possible to judge the development of the process up to the final magnitude of the pressure drop of wet steam. Along with this, it is necessary to bear in mind that,

that the numerical values obtained from these formulas cannot lay claim to sufficient accuracy, since there are no reliable data for determining the surface tension at a very small droplet radius. One cannot expect these equations to be valid for a very small droplet size, \(r < 10^{-7}\ \text{cm}\), since the Thomson–Gibbs equation was derived for a continuous medium under the assumption of a constant value of the surface tension. This circumstance may lead to large distortions in the calculation. For example, an error of 1% made in determining the surface tension leads to an error of 100% in the rate of formation of water-vapor condensation nuclei [6]. Accurate calculations of the rate of formation of condensation nuclei and of droplet growth according to the existing relations are also difficult because the details of the condensation process are unclear, and also because many thermodynamic factors used in these calculations are uncertain or unknown at present. The assumption of quasi-stationarity of the process during the very rapid expansion of wet steam is conditional; on the basis of this assumption the value \(I\) is determined. The same applies to the formulas for determining droplet growth, which are valid only in the case when the thermodynamic state of the surrounding medium does not change during droplet growth, something not observed during the expansion of wet steam in steam turbines.

From this brief survey it becomes evident that the kinetics of the process of steam condensation in a turbine stage has been poorly studied, that our knowledge of the physical essence of phase transitions is insufficient, and that there is also no proper orientation of scientific research in this field.

REFERENCES

1. S. V. Polozhii. On the question of the influence of steam moisture on the efficiency of a turbine stage. Izvestiya Vuzov SSSR—Energetika, 7, 1962.

2. S. V. Polozhii. Once again on the question of the influence of steam moisture on the efficiency of a turbine stage. Izvestiya Vuzov SSSR—Energetika, No. 3, 1963.

3. I. I. Kirillov, R. M. Yablonik. The problem of improving turbine stages operating on wet steam. Teploenergetika, No. 10, 1962.

4. I. I. Kirillov, R. M. Yablonik. Kinetics of the process of steam condensation in a turbine stage. Energomashinostroenie, No. 4, 1963.

5. Ya. I. Frenkel. Kinetic Theory of Liquids. Academy of Sciences of the USSR, 1945.

6. P. P. Vegener, L. M. Mak. Condensation in supersonic and hypersonic aerodynamic tubes. In: Problems of Mechanics, Issue 3, Moscow, 1961.