ON THE ISSUE OF THEORETICAL AND EXPERIMENTAL STUDIES OF THE GAS DYNAMICS OF TWO-PHASE FLOWS

S. V. Polozhii

(Presented by the scientific seminar of the Department of TPP)

The study of adiabatic phase formation, its quantitative and qualitative characteristics, is becoming exceptionally important not only for solving the problem of converting thermal power plants to adiabatic phase formation, but also for solving other technical issues connected with the use of two-phase flows, and especially with the expansion of wet steam in steam turbines.

The problem of wet steam arose with the advent of condensing steam turbines, and work in this field has been carried out continuously, but this problem has not been solved up to the present time. In the last decade, in connection with the transition of power engineering to unit thermal power plants with superhigh and supercritical steam parameters and with the development of nuclear power plants, exceptionally great attention has been devoted to this problem, especially in our country. Since 1955, studies in the field of wet steam have been conducted at BITM and at KhTEZ. Later TsKTI, LPI, and LMZ joined these works, broadly establishing studies of experimental turbines on wet steam. Extensive studies of the physical processes of the gas dynamics of two-phase flows have been undertaken at MEI. Studies of full-scale turbines were also organized at KhTGZ, VTI, TsKTI, and ORGRES. Along with experimental studies, works appeared on the theory of wet-steam turbines. The theory of phase transitions was widely used in developing methods for calculating the processes of steam condensation during the expansion of wet steam. At LPI in 1961–62 a method was developed for calculating the working process with the application of the kinetic theory of Ya. I. Frenkel. These works were continued at BITM and TsKTI as applied to the calculation of individual stages in multistage turbines. In 1968, the results of numerous experimental and theoretical studies in the field of wet steam were summarized [1, 2]. These works provide a generalization of domestic and foreign studies in the field of wet steam, extensively use the authors’ own theoretical and experimental investigations, set forth calculation methods, the kinetics and gas dynamics of two-phase media, determination of losses due to steam moisture, etc., and reflect the latest achievements in this field.

The publication of these fundamental works [1, 2] in the field of wet steam indicates the correctness of identifying this group of phenomena as adiabatic phase formation [3], of recognizing this phenomenon as a natural process of nature, and of establishing its basic laws and properties. It becomes obvious that the enormous accumulated material of experimental studies of wet steam, the exposition of the gas dynamics of two-

phase flows, the determination of losses due to moisture in experimental turbines, and the methodology of experimental and theoretical investigations and their generalization are not based on ideas about adiabatic phase formation, its properties and regularities; they are therefore to a considerable extent artificial and do not reflect the physical essence of the processes of expansion of wet steam in steam turbines [3—6].

Directionality of the Phase-Formation Process during the Expansion of Wet Steam in Steam Turbines

One of the main shortcomings of modern concepts concerning the expansion of wet steam in steam turbines, and of theoretical and experimental studies in this area, is the one-sided assessment of this process as a process only of steam condensation, of the continuous growth of liquid droplets as the wet steam expands in the turbine. Such a scheme of wet-steam condensation has been adopted as the basis for the kinetics of two-phase media and for the theory of wet-steam turbines [1, 2], and appears throughout the educational, scientific, and specialized literature connected with questions of wet steam.

The one-sided directionality of the condensation process during the expansion of wet steam without heat exchange with the external medium has its theoretical justification in the form of the concept of inversion of the aggregate state in the modern literature—the boundary of transition from one monoprocess to another, for example, from condensation to steam formation, or vice versa [7, 8]. In [7] the value of the steam content of the flow corresponding to inversion of the aggregate state under isentropic compression is determined through the ratio of the heat capacities of the vapor and liquid phases.

\[ X_{\text{inv}}=\frac{C_{\text{p}}}{C_{\text{p}}-C_{\text{l}}} \quad \text{or} \quad \frac{r \cdot dX}{dz}=-(1-X)\cdot C_{\text{l}}-X\cdot C_{\text{p}}, \]

where \((1-X)C_{\text{l}}\) is the amount of heat absorbed by the liquid when its temperature is raised by \(1^\circ\text{C}\), and \(X, C_{\text{p}}\) is the amount of heat released by the vapor phase. If dry saturated steam, under isentropic compression, releases more heat than is required to heat the liquid, then part of the liquid will evaporate. If, however, the dry saturated steam releases insufficient heat to heat the liquid to the corresponding temperature, then part of the steam condenses and the degree of dryness decreases [7]. In [8] the one-sided directionality of steam condensation under adiabatic cooling of wet steam is explained by the sign of the heat capacity. In the region where the heat capacity \(C_x\) is positive, evaporation occurs. In the region where \(C_x\) is negative, condensation of steam is observed. Since \(C'\) is positive, at small degrees of dryness \(C_x\) will be positive—evaporation. At a large degree of dryness \(C_x\) is negative—condensation of wet steam occurs.

Such a one-sided assessment of the condensation of wet steam in steam turbines is the result of observations of processes of change in the aggregate state with external heat exchange through a wall, when the supply (removal) of heat ensures the continuous transition of steam into liquid during condensation or of liquid into steam during boiling. This physical picture of the formation and growth of liquid droplets, the one-sided directionality of the condensation process with heat removal through a wall, was uncritically extended also to the process of condensation of wet steam without heat exchange in steam turbines. For the same reason, no need arose for a special study of this class of phenomena; this also explains the underestimation in thermal-engineering investigations of adiabatic phase formation,

the nonrecognition even of the very terminology of this process. Attempts by the TPTI to draw attention to the need to study adiabatic phase formation in solving questions of losses due to moisture in steam turbines [4, 5] did not have the proper effect on researchers. As a result, scientific investigations connected with solving the problem of wet steam in steam turbines, the gas dynamics of two-phase media and the theory of wet-steam turbines, as well as experimental and theoretical studies in this area, proved to have been built on an erroneous physical basis, without taking into account the basic law of adiabatic phase formation and other specific properties of this process.

Heat Transfer Between Phases During the Expansion of Wet Steam

Having accepted the concept of a one-sided direction of the condensation-formation process and of the continuous growth of droplets as the pressure of wet steam is reduced in turbine nozzles, contemporaries also made another, no less important error in assessing the physical essence of the process in the kinetics of two-phase media, explaining the formation and growth of liquid droplets by heat transfer—the transfer of heat from liquid droplets to the steam during its expansion in the turbine.

Heat transfer between liquid and steam was made the basis of theoretical conclusions and calculation dependences in the gas dynamics of two-phase flows. The theory of heat transfer, in turn, required introducing a temperature difference between the steam and the liquid droplets—an excess of the droplet temperature over the temperature of the surrounding steam—in order that, in the process of growth of liquid droplets during the expansion of wet steam, heat could be removed from the liquid. Such a scheme of heat transfer made it possible, for calculating the growth of liquid droplets during the expansion of wet steam, to provide a mathematical basis for the calculation of this process and to use the usual heat-transfer equations. Thus, for example, in [2], to determine the growth of liquid droplets, the following equation is used under the conditions of a macroscopic process:

\[ \frac{d\xi}{dt}=\frac{2\lambda'(T'-T)}{r'\rho'}, \]

where \(T'\) is the temperature of the droplet surface, \(T\) is the steam temperature, \(\lambda'\) is the coefficient of thermal conductivity, \(r'\) is the heat of phase formation, and \(\rho'\) is the density of the liquid. For the initial stage, when \(\xi<\lambda\) (the mean free path length of the molecules), the growth of droplets is determined by the equation

\[ \frac{d\xi}{dt}=\frac{3}{8}\frac{P}{r\rho}\sqrt{3\frac{R}{T}(T'-T)}. \]

The interpretation of heat transfer between phases during the expansion of wet steam and the corresponding mathematical dependences are not consistent with the basic law of adiabatic phase formation, with the physical essence of this process. The quantity of condensed liquid determined by these equations does not depend on the initial phase ratio or on the initial state. The degree of wetness will vary over the widest values up to the transition of the process from adiabatic condensation to adiabatic vapor formation when the relative humidity in the initial state is greater than one half [3, 6]. Moreover, the transfer of heat from one phase to another without external heat and energy exchange cannot change the ratio of the liquid and vapor phases:

\[ \left(X_1+\frac{Q_{1,2}}{r_1}+Y_1-\frac{Q_{1,2}}{r_1}\right)=1;\qquad (X_1+\Delta Y)+(Y_1-\Delta X)=1. \]

Temperature States during the Expansion of Wet Steam

The theory of heat exchange during the condensation of wet steam in steam turbines required the construction of a corresponding temperature scheme in the flow of wet steam during its expansion in the turbine and, in particular, of the excess of the temperature of liquid droplets \(T'\) over the steam temperature \(T''\), “...which is necessary for the removal of heat in the process of condensation” ([2], p. 110). In this connection, in the flow of wet steam one distinguishes the steam temperature, the temperature of liquid droplets, the saturation temperature \(T_{s \xi}\) of droplets for each given size, and the saturation temperature above a plane surface of liquid \(T_s\); the amount of droplet superheating \(\delta T' = T' - T_{s \xi}\), and the amount of steam subcooling \(\Delta T = T_s - T''\) relative to a droplet of radius \(\xi\). Under conditions of thermodynamic equilibrium, \(\Delta T = \delta T' = 0\). In the process of expansion this quantity differs from zero, which is explained by the conditions of heat removal from a droplet growing in the process of condensation.

One can be convinced of the artificiality of such a temperature scheme in the flow, above all, on the basis of erroneous notions about the process of heat exchange and the physical essence of condensation during the expansion of wet steam in a turbine, as already discussed above, and also in view of the inconsistency of such a temperature scheme with the fundamental law of adiabatic phase formation. In particular, these temperature differences are preserved for a given interval and magnitude of pressure drop for any degree of dryness of wet steam, and must always ensure droplet formation—steam condensation both at a very high and at a very low degree of dryness of wet steam. In reality, at a degree of dryness equal to one half, no visible phase change is observed. At a lower degree of dryness the condensation process changes into vaporization.

The pressure drop of wet steam acts equally on both the vapor and the liquid phase. Neither phase has an advantage in its physical and thermodynamic state, with the exception of quantitative differences in the initial state, which may vary in the general case from \(X = 1\) to \(y = 1\) and determine the final phase-forming effect of adiabatic phase formation.

The metastable state of the flow, the subcooling of steam during its expansion in certain types of phase-forming elements and initial states of the system, is associated with surface phenomena. The kinetic energy of the flow in this case is formed only at the expense of the vapor single-phase system, which as a result acquires a correspondingly lower temperature relative to the saturation temperature. The subsequent process of phase formation ensures the generation of kinetic energy in accordance with the law of the corresponding state, with simultaneous change in the temperature of the liquid and vapor phases, with simultaneous condensation and vaporization in quantities proportional to the weight content of each phase in the flow, with condensation predominating over vaporization for the conditions of expansion of wet steam in steam turbines. In this case the increment of the liquid phase will not remain constant as wet steam expands in the flow section of the turbine, but will decrease from stage to stage in connection with the decrease in the degree of dryness before each subsequent stage.

Growth of Droplets during the Expansion of Wet Steam

The modern theory of droplet growth during the expansion of wet steam is based on the assumption of heat exchange between phases, of the removal of heat from liquid droplets by the steam. The law of droplet growth, in turn,

derived under the assumption of an unchanged thermodynamic state of the medium, which does not change during the growth of the droplets. Having determined, under such conditions, the condensation rate and the droplet size in any section of the nozzle, we find the flow rate of the condensed vapor

\[ G'=\int_{0}^{x} M_{\mathrm{k}}(X'X)\cdot I(X')\cdot F(X')\,dX', \]

where \(X'\) is the coordinate of the section in which moisture precipitates at a rate \(I(X')\) and whose area is \(F(X')\);

\(X\) is the coordinate determining the position of the section under consideration;

\(M\) is the mass of droplets in section \(X\);

\[ I=\sqrt{\frac{2}{\gamma}}\left[\exp\left(-\frac{4\pi\sigma}{3kT}\xi_{\mathrm{cr}}^{2}\right)\right]\frac{1}{\rho}\frac{P}{kT}\sqrt{\frac{\sigma N_{a}}{\mu}}\,\frac{1}{\mathrm{kg}\cdot\mathrm{s}}; \]

\(I\) is the rate of droplet formation.

Such a determination of the amount of condensed vapor is based entirely on an incorrect conception of the physical essence of the process—on the continuous growth of droplets in the section under consideration. The amount of liquid formed does not depend on the driving forces of the process—the magnitude, nature, and interval of the pressure drop. The simultaneous occurrence of condensation and vaporization is not taken into account, nor is the influence of the phase ratio in section \(X\) on the final moisture content of the vapor. The flow rate of condensed vapor according to such a dependence is always proportional to the mass of droplets in section \(X\), whereas as the moisture content of the vapor in section \(X\) increases, the final amount of condensed vapor will continually decrease, and at \(M\) equal to 50% of the total flow, no increase in moisture will occur in section \(X'\). With a further increase in moisture content in section \(X\), the condensation process will give way to vaporization, and this entire system for determining the amount of wet vapor becomes absurd.

Limiting condensation during expansion of wet vapor

The modern theory of the growth of the condensed phase and the gas dynamics of a two-phase flow do not raise the question of the limiting possibility of vapor condensation and the influence of the initial moisture content of the vapor on the final result of the transition of vapor into liquid during expansion of wet vapor in turbines, nor the question of the magnitude of the increase in condensed liquid by turbine stages associated with the change in the phase composition of the flow along the length of the turbine flow passage. Nor is the effect of moisture removal from the flow passage of the turbine on vapor condensation in subsequent stages taken into account. The modern theory of wet-steam turbines is not capable of solving these questions. In this respect, modern theory is abstract; it is not connected with and does not reflect the whole complex of phenomena and regularities, driving forces, and properties of adiabatic phase formation, on the basis of which, for a given interval and magnitude of pressure drop, there exists its own limit of the maximum transition of vapor into liquid.

Development of the theory of wet-steam turbines

The practical tasks of developing unit thermal power plants and nuclear power stations have, over the past decade, led to a broad development of theoretical and experimental studies of two-phase flows (BITM, KhTGZ, TsKTI, LPI, MEI, LMZ, and others). At present, works have appeared that generalize and systematize research in this field—

studies that consider the problem of wet-steam turbines as a whole [1, 2]. A great deal of work has been done to solve this important problem. Numerous experimental and theoretical studies have been carried out on the thermodynamics of wet steam, the gas dynamics of a two-phase medium, and the study of the motion of liquid droplets in the flow section of a turbine; a kinetic theory of phase transformations in a turbine stage has been created. Methods have been proposed for calculating condensation processes in the elements of wet-steam turbines, for determining moisture losses in a turbine stage, and for solving problems of moisture removal in turbines. Further avenues of experimental research needed for the development of the theory of wet-steam turbines and the study of physical phenomena have been outlined; recommendations have been given for experimental investigations of wet-steam turbines.

However, all these studies, theoretical conclusions, and calculation methods are not based on a correct understanding of the physical essence of the processes during the expansion of wet steam in steam turbines, or on knowledge of the properties and laws of adiabatic phase formation.

The further development of the theory of wet-steam turbines cannot proceed without a fundamental revision of present-day views and concepts concerning the kinetics of two-phase flows in light of the basic properties and laws of adiabatic phase formation, and without studies being undertaken in this direction [3–6].

The theoretical substantiation of experimental research data related to adiabatic phase formation that are available in the literature always gives good agreement between experimental and theoretical data under the most varied conceptions of the process, flow schemes, etc. The mathematical apparatus makes it possible to bring experimental and theoretical calculations into agreement even without a correct understanding of the physical essence of the processes, and this must be borne in mind when considering the present problem.

Experimental data on determining the flow rate during the discharge of heated water are a good example of this. Each researcher of this question presented a substantiation of his experimental data with the most varied notions of the physical essence of the process and of the flow scheme.

Experimental data on moisture losses obtained during the transition of a stage from superheated to wet steam have also been theoretically substantiated by researchers, but without taking into account the decrease in stage efficiency due to changes in the thermodynamic and physical operating conditions of the stage on wet steam [4, 5, 6].

The kinetics of two-phase flows is justified by its authors by experimental and computational methods without a correct understanding of the physical essence of the process of expansion of wet steam in nozzles, without knowledge of the basic law of adiabatic phase formation and other properties of this process [3, 6, 9].

LITERATURE

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

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

3. S. V. Polozhii. Adiabatic phase formation. Information bulletin on the problem of translating TPPs to nuclear power plants No. 4, NTB TPI, Tomsk, 1969.

4. S. V. Polozhii. On the question of the influence of steam moisture on the efficiency of a turbine stage. Izv. vyssh. uchebn. zavedenii SSSR, Energetika, No. 7, 1962.

5. S. V. Polozhii. Once again on the question of the influence of steam moisture on the efficiency of a turbine stage. Izv. vyssh. uchebn. zavedenii SSSR, Energetika, No. 3, 1963.

6. S. V. Polozhii. On the question of the kinetics of the process of steam condensation in a turbine stage. Izv. TPI, vol. 137, 1965.

7. M. D. Berman. Thermodynamics of Vapor–Liquid Flows. Energiya, 1967.

8. M. P. Vukalovich, I. I. Novikov. Technical Thermodynamics. GEI, 1962.

9. S. V. Polozhii. On the question of the basic law of adiabatic phase formation. Present collection.