On the Calculation of Flow Energy Characteristics with Account for the Critical Regime in the Discharge of Boiling Water
Abstract Generated abstract
The paper examines critical flow phenomena during the discharge of boiling water through cylindrical nozzles, where a two-phase steam-water mixture may reach the local speed of sound. It applies thermodynamic relations for sound speed in a homogeneous liquid-vapor medium, including a simplified form for incompressible liquid and adjustments for nonequilibrium superheated liquid with bubble surface-tension effects, then compares calculated sound speeds with discharge velocities over varying back pressures and degrees of vaporization completion. The calculations indicate that a critical discharge regime occurs in all considered cases, with critical pressure and velocity depending strongly on the degree of vaporization completion, and that limiting critical velocity causes losses in the conversion of thermal pressure drop into kinetic energy.
Full Text
ON THE CALCULATION OF ENERGY CHARACTERISTICS OF A FLOW WITH CONSIDERATION OF THE CRITICAL REGIME DURING THE DISCHARGE OF BOILING WATER
A. A. GURCHENOK
(Presented by the scientific seminar of the Department of ATPPP)
During the discharge of boiling water, crisis phenomena may arise in the two-phase flow that is formed, caused by the attainment of the speed of sound in the flow [1].
The speed of sound for a liquid–vapor medium in a two-phase region can be calculated by the usual formula
\[ a=\sqrt{-v^2(\Delta p/\Delta v)_s}\ \text{m/sec}. \tag{1} \]
The quantity \(\Delta p/\Delta v\) is calculated from the parameters of the vapor and liquid phases,
\[ dv/dp=x(\partial v''/\partial p)_s+(1-x)(\partial v'/\partial p)_s. \tag{2} \]
Here it is assumed that the two-phase medium under consideration has a homogeneous structure, the state parameters of which and their derivatives are continuous at any point of the volume occupied by the medium. The dimensions of the inhomogeneities must be smaller than the length of the sound wave [2].
Expression (2) can be simplified if it is assumed that the liquid in the mixture is incompressible, i.e. \(dv'/dp=0\). Taking this assumption into account, expression (1) can be transformed to the form
\[ a=a''\left(\sqrt{x}+\frac{1-x}{\sqrt{x}}\frac{v'}{v''}\right)\ \text{m/sec}, \tag{3} \]
where
\[ a=\sqrt{kpv''}\ \text{m/sec}. \tag{4} \]
It follows from formula (3) that the speed in a steam–liquid medium depends on the thermodynamic parameters of the medium: the dryness fraction \(x\), the absolute pressure \(p\), and the ratio of the specific volumes of the phases \(v'/v''\).
Using the results of [3], where the case of the discharge of boiling water through cylindrical nozzles at different degrees of completion of the vapor-formation process was investigated, below an analysis is carried out of the possibility of establishing a critical discharge regime.
The process of discharge of boiling water can be represented in a \(P\)–\(V\) diagram (Fig. 1). The area between the \(P\)-axis and the curve of the expansion process represents the magnitude of the kinetic energy of the flow. To determine the possibility of establishing a critical regime under the discharge conditions, the speed of sound in the vapor phase and in the steam–liquid mixture was determined. A feature of the calculation is the use of equa-
... of equation (3) for the conditions of a nonequilibrium state of the flow (superheated liquid) increased, with allowance for surface-tension forces, by the pressure inside the bubbles of the forming vapor phase.
Figure 2 presents graphs of the discharge velocity according to the data of [3] and of the speed of sound for cases of discharge of boiling water with a complete (100%) degree of completion of the vapor-formation process at different back pressures. The point of intersection of the curves makes it possible to determine the flow parameters at which critical discharge regimes are established. The results of calculations for all discharge regimes are summarized in Table 1.
Fig. 1
It can be seen from Table 1 that the value of the critical pressure depends on the discharge conditions. Only when the degree of completion of the vapor-formation process changes from \(\xi = 1.1\%\) to \(\xi = 100\%\) does the value of the critical pressure increase from \(P_{\mathrm{cr}} = 1.2\) bar to \(P_{\mathrm{cr}} = 4.0\) bar. In all cases a critical discharge regime occurs. The value of the critical discharge velocity is \(50\text{--}120\ \mathrm{m/sec}\), or \(1.06\text{--}0.245\) of the speed of sound in steam under the given conditions.
Fig. 2
Using the data of the calculations performed, in each case the values of the area under the expansion-process curve, the losses...
[[unclear: beginning of word]] due to the establishment in the flow of the limiting critical velocity of discharge of the steam-water mixture. For all cases, the value of the relative lost area was determined as
\[ \eta = 1 - F_{\mathrm{cr}}/F. \]
The values of \(\eta\) vary from \(0,967\) at \(\xi = 1,1\%\) to \(0,265\) at \(\xi = 100\%\). The example given shows that, during the discharge of boiling
Table 1
| \(\xi,\ \%\) | \(p_{\mathrm{cr}},\ \text{bar}\) | \(p_{\mathrm{cr}}/p_1\) | \(a_{\mathrm{mix}},\ \text{m/sec}\) | \(a_{\mathrm{mix}}/a''\) | \(a'',\ \text{m/sec}\) | \(1-\dfrac{F_{\mathrm{cr}}}{F}\) |
|---|---|---|---|---|---|---|
| 1,1 | 1,2 | 0,11 | 50 | 0,106 | 472,4 | 0,967 |
| 3,3 | 1,5 | 0,15 | 54 | 0,114 | 476,3 | 0,89 |
| 5,64 | 2,0 | 0,2 | 60 | 0,125 | 480,4 | 0,74 |
| 11,8 | 3,0 | 0,3 | 72 | 0,148 | 485,8 | 0,54 |
| 100,0 | 44,0 | 0,4 | 120 | 0,245 | 489,7 | 0,26 |
water through cylindrical nozzles, in the conversion of the thermal pressure drop into kinetic energy, energy losses will be observed owing to the establishment of the critical discharge regime.
Conclusion
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During the discharge of boiling water, the parameters of the critical regime can be determined by a series of process calculations for different values of back pressure.
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The parameters of the critical regime, under identical initial conditions of the discharge process, change when the degree of completion of vaporization changes.
References
- M. D. Vaisman. Thermodynamics of Steam-Liquid Flows. 1967.
- V. B. Dvornichenko. Thermal Power Engineering, No. 10, 1966.
- A. A. Gurchenok. The present collection.