On the Calculation of the Process of Boiling Water Discharge Through Cylindrical Nozzles with Allowance for Hydraulic and Thermodynamic Relationships
Abstract Generated abstract
This paper addresses the calculation of boiling water discharge through cylindrical nozzles, where experiments show that flow rates are lower than for cold water but higher than predicted by equilibrium adiabatic expansion formulas. It argues that the inlet section initially behaves hydraulically as a liquid flow without vapor formation, while vapor generation develops downstream with an incomplete degree of completion determined by residence time. A successive approximation method is proposed that combines thermodynamic expansion relations on a pressure volume diagram with a hydraulic pressure drop correction at the nozzle inlet. Calculations for varying back pressures and vapor formation completion show that even small amounts of vapor formation can substantially reduce the specific flow rate.
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Toward the Calculation of the Process of Boiling Water Discharge Through Cylindrical Nozzles with Allowance for Hydraulic and Thermodynamic Relations
A. A. Turchenok
(Presented at a scientific seminar of the Department of ATsPPI)
In [1] an experimental determination was carried out of the pattern of pressure variation along cylindrical nozzles of different lengths and of the flow rate during the discharge of boiling water and, for comparison of the results, of cold water. Below, in paragraph a), information is given on experiments with cold water, and in paragraph b), with boiling water.
The pattern of pressure variation along the length of the nozzle:
a) a complete pressure drop occurs in the inlet section,
b) a complete pressure drop does not occur, and a pressure \(p_{\text{in}}\) is established (Fig. 1).
The flow rate through cylindrical nozzles:
Fig. 1
a) the experimental data were obtained in order to check the correctness of the experiments. The experimental results agree with data available in the literature;
b) the magnitude of the flow rate is smaller in comparison with the flow rate of cold water, but significantly greater than according to a thermodynamic calculation using the formulas of adiabatic expansion.
Analysis of the experimental results showed that the magnitude of the pressure drop to \(p_{\mathrm{in}}\) during the discharge of boiling water is related to the experimental value of the flow rate by the well-known hydraulics formula
\[ p_1-p_{\mathrm{in}}=\frac{c_{\mathrm{in}}^2}{2\cdot v'\cdot 10^5}. \tag{1} \]
Here: \(p_1-p_{\mathrm{in}}\) is the magnitude of the pressure drop, bar,
\(v'\) is the specific volume of boiling water, \(\mathrm{m^3/kg}\),
\(c_{\mathrm{in}}\) is the flow velocity in the inlet section, \(\mathrm{m/sec}\).
The value of the flow velocity in the inlet section can be determined from the relation
\[ c_{\mathrm{in}}=g_y\cdot v'\,10^4\ \mathrm{m/sec}, \tag{2} \]
where \(g_y\) is the experimental value of the specific mass flux, \(\mathrm{kg/cm^2\,sec}\).
The results from (1) and (2) agree with one another within the accuracy of the experimental results [1]. Consequently, despite the fact that the process of discharge of boiling water is taking place, in the inlet section the flow of boiling water still contains no vapor phase, although, because of the pressure drop, the temperature in the flow is higher than the boiling temperature at the given pressure. The degree of superheating of the liquid will be equal to
\[ \Delta t=t_{p_1}-t_{p_{\mathrm{in}}}. \tag{3} \]
To maintain a superheated state in a liquid, as is known, special conditions are required: a quiescent state of the liquid, the presence of an ideally pure liquid, etc. Under discharge conditions this is not observed, and therefore a process of vapor-phase formation must arise inside the nozzle. For the process to proceed, time is needed; therefore, in the inlet section of the nozzle the flow still contains no vapor phase. According to [1], the vapor-formation process, having arisen inside the nozzle, owing to the short residence time of the liquid particles inside short nozzles, does not have time to be fully completed. In this case the flow temperature \(t\) proves to be higher than the saturation temperature \(t_{p2}\) at the back pressure \(p_2\).
The ratio
\[ \xi=\frac{t_{p_1}-t}{t_{p_1}-t_{p_2}} \tag{4} \]
characterizes the degree of completion of the vapor-formation process.
In [2] it is shown that the process of nonequilibrium discharge of boiling water, with an incomplete degree of completion of the vapor-formation process, can be plotted on a \(P\)—\(V\) diagram (Fig. 2). By planimetering the area of the expansion process \(12_{\mathrm{d}}34\), we obtain the value of the kinetic energy in the flow on the scale of the diagram
\[ -\int_{p_1}^{p_2} v\,dp=M_f\cdot S_{12_{\mathrm{d}}34}. \tag{5} \]
Here: \(M_f\) is the area scale, \(\mathrm{kgm/cm^2}\),
\(S\) is the area, \(\mathrm{cm^2}\).
By virtue of the condition
\[ c_{\mathrm{out}}=\sqrt{2\int_{p_1}^{p_2}(-v\,dp)}\ \mathrm{m/sec}, \tag{6} \]
from (5) we obtain
\[ c_{\mathrm{out}}=\sqrt{2M_f\cdot S_{12_{\mathrm{d}}34}}\ \mathrm{m/sec}, \tag{7} \]
whence
\[ g_y=\frac{c_{\mathrm{out}}}{v_{\mathrm{out}}\cdot 10^4}\ \frac{\mathrm{kg}}{\mathrm{cm^2\cdot sec}}. \tag{8} \]
Thus, the values of velocity and discharge can be calculated for various flow regimes with an incomplete degree of completion of the vaporization process \(\xi\), using only the thermodynamic relations of the flow. In this case, on the diagram of the expansion process—
Fig. 2
Fig. 3
—the process of expansion ...
does not take into account the hydraulic (without change in volume) drop in pressure in the inlet section of the nozzle. To eliminate this error, in the present work calculations were carried out by the method of successive approximation of the process of discharge of boiling water, with additional allowance for the hydraulic relations in the inlet cross section of the nozzle section.
By virtue of the continuity of the flow, the flow rate does not change along the length. Therefore, from the values of the flow parameters found from the thermodynamic relations, the magnitude of the pressure drop in the inlet cross section can be determined (formulas 1, 2). Plotting the pressure-drop line in the initial section of the nozzle 1—2′, we construct the actual expansion process inside the nozzle in the first approximation. From the refined process curve on the \(P—Y\) diagram, by planimetry, the magnitude of the velocity (7) and of the specific flow rate (8) is again determined. The flow-rate values found will be smaller than those taken as the initial data at the beginning of the calculation. Therefore the verification calculation must be carried out again, taking as the initial data the results obtained after the first refinement. The calculation is then repeated until coincident results are obtained.
Figure 3 gives the results of calculations by the indicated method of the discharge of boiling water through cylindrical nozzles with an initial pressure \(p_1 = 9.81\) bar into a region with back pressure \(p_2 = 0.98—8.8\) bar, with different degrees of completion of the vapor-formation process: \(\xi = 0.0\%\)—discharge of cold water, \(\xi = 100\%\)—the case of equilibrium adiabatic discharge, cases of incomplete completion of the vapor-formation process—\(1.1\%\), \(3.33\%\), \(5.64\%\), \(11.8\%\).
As is seen from Fig. 3, even for small values of the degree of completion of the vapor-formation process (\(\xi = 3.33\%\)), the flow rate decreases by more than a factor of two.
Conclusion
A method has been proposed for calculating the process of discharge of boiling water, taking into account the features of the hydraulic and thermodynamic dependences in the flow during discharge of boiling water.
LITERATURE
- A. A. Gurchenok. Izv. TPI, vol. 101, 1958.
- A. A. Gurchenok. Izv. TPI, vol. 110, 1962.