DETERMINATION OF THE COMPLETION OF THE VAPORIZATION PROCESS DURING THE DISCHARGE OF BOILING WATER

A. A. GURCHENOK

Presented by Professor G. I. FUKS

In [1] it was pointed out that, during the discharge of boiling water, the vaporization process does not develop fully, since the residence time of the liquid particles inside the nozzle is too short. It is of interest to determine the degree of completion of the vaporization process inside the nozzle, since this aspect of the process of discharge of boiling water has so far scarcely been studied.

Figure 1 presents a picture of the flow form and of the change in pressure along the length of a cylindrical nozzle during the discharge of boiling water according to [1]. Characteristic of the discharge of boiling water through cylindrical nozzles is a certain pressure drop in the inlet section. Farther along the length of the nozzle the pressure decreases and in the outlet section falls to the back pressure. Calculation of the discharge capacity from the magnitude of the pressure drop at the inlet, as indicated in [1], gave results in good agreement with experiment, which confirms the correctness of the flow pattern presented in Fig. 1.

Fig. 1. Flow form and pressure change during the discharge of boiling water through cylindrical nozzles.

In the present work, on the basis of the indicated flow pattern, a method is given for determining the steam content in the outlet section of the flow.

Designations not given in [1]:

\(\nu\) — coefficient of kinematic viscosity, \(m^2/sec\);

\(\tau\) — time, sec;

\(q\) — quantity of heat, kcal/kg;

\(t_s\) — boiling temperature of the liquid, °C.

The calculation is carried out in the following order:

1. From the data on the discharge capacity of nozzles for boiling water, given in [1], we determine the discharge velocity in the inlet section of the nozzle:

\[ w_1 = \frac{G_{k.v}}{f \cdot \mu \cdot \gamma_{k.v}} \; m/sec . \tag{1} \]

1. In [1] it is shown that the throughput capacity of a cylindrical nozzle is related to the magnitude of the pressure drop at the inlet section by the following dependence:

\[ G_{k.v}=f\cdot\mu\cdot\gamma_{k.v}\sqrt{2g(p_1-p_{cc})\,v_{k.v}}\ \text{kg/sec}, \tag{2} \]

whence

\[ p_1-p_{cc}=\left(\frac{G_{k.v}}{f\cdot\mu\cdot\gamma_{k.v}}\right)^2 \frac{1}{v_{k.v}\cdot 2g}\ \text{kg}/\text{m}^2 . \tag{3} \]

1. The pressure loss for overcoming friction is calculated from the relation:

\[ \Delta p_{tr}=\lambda\frac{l}{d}\frac{w^2}{2g}\ \text{kg}/\text{m}^2, \tag{4} \]

where

\[ \lambda=\frac{1}{(1.8\log Re-1.5)^2}, \tag{5} \]

and

\[ Re=\frac{w_1\cdot d}{\nu}. \tag{6} \]

1. The velocity of the liquid in the outlet section will be equal to

\[ w_2=\varphi\sqrt{2g[(p_1-p_2)-\Delta p_{tr}]\cdot v_{k.v}}\ \text{m/sec}. \tag{7} \]

1. We determine the fraction of the section occupied by liquid in the outlet section:

\[ \varepsilon_{\ell}=\frac{w_1}{w_2}\varepsilon . \tag{8} \]

1. The fraction of the section occupied by steam will be equal to:

\[ \varepsilon_{s}=1-\varepsilon_{\ell} \tag{9} \]

1. We determine the velocity of the steam in the flow downstream of the nozzle. It will be equal to:

a) For \(p_2>p_{cr}\)

\[ c_s=91.5\sqrt{i_{out}-i_2}\ \text{m/sec}, \tag{10-a} \]

and for \(p_2<p_{cr}\),

\[ c_{s,cr}=323\sqrt{p_{out}v_{out}}\ \text{m/sec}, \tag{10-b} \]

1. The volumetric quantity of steam passing per unit time will be:

\[ V_s=\varepsilon_s\cdot f\cdot c_s\ \text{m}^3/\text{sec}, \tag{11-a} \]

whence the mass flow rate of steam is equal to:

\[ G_s=V_s\cdot\gamma_s\ \text{kg/sec}. \tag{11-b} \]

1. We determine the steam content of the flow:

\[ x=\frac{G_s}{G_{k.r}}. \tag{12} \]

1. We determine the quantity of heat that has gone into the formation of steam

\[ q_s=G_s\cdot r\ \text{kcal/sec}. \tag{13} \]

1. During the discharge of boiling water, the process of steam formation takes place without heat being supplied to the liquid, at the expense of heat removed from

of the liquid itself. The temperature of the liquid decreases by the amount:

\[ \Delta t=\frac{q_n}{G_{k.v}} \tag{14} \]

and is equal to:

\[ t=t'_s-\Delta t \tag{15} \]

The magnitude of the temperature drop shows how completely the vaporization process is completed inside the nozzle. If the vaporization process were completed entirely, then the temperature of the liquid would drop to the boiling temperature at the final pressure \(t''_s\). The ratio:

\[ \frac{t'_s-t}{t'_s-t''_s} \tag{16} \]

therefore characterizes the degree of completion of the vaporization process during the discharge of boiling water.

Tables 1 and 2 give the results of calculating the degree of steam content and the degree of completion of the vaporization process from the results of the experiments presented in [1].

Table 1

From Table 1 it is seen that, during the discharge of boiling water at the same initial pressure, the process is completed more fully the longer the nozzle.

Table 2

From Table 2 it is seen that, in one and the same nozzle, the vaporization process is completed more fully the greater the nozzle length and the lower the initial pressure, i.e., the longer the residence time of the liquid particles inside the nozzle.

The obtained data on the degree of completion of the vaporization process show that, in the experiments cited, the vaporization process under the conditions of discharge does not have time to be fully completed inside the nozzle. A comparison of the calculated data on the degree of completion of the vaporization process with the results of experiments on determining the boiling rate under steady-state conditions [2] shows the following. For

for a nozzle of length 92 mm at \(p_1 - p_2 = 2.0\ \text{kg}/\text{cm}^2\), the residence time of the liquid particles inside the nozzle is equal to:

\[ \tau=\frac{l}{w_1}=\frac{0.092}{8.45}=0.011\ \text{s}. \]

During the indicated interval of time, the vaporization process is completed by 87%. According to experimental data for steady boiling, about \(0.02\) seconds are required for its completion. The calculated values for the degree of completion of the vaporization process differ somewhat from the experimental data on the rate of vaporization obtained for the case of water boiling under steady conditions at atmospheric pressure. This can be explained by the fact that, owing to the forced motion during the discharge of boiling water, the vaporization process proceeds more intensively and, consequently, a shorter time interval is required for it to occur. Thus, the proposed method for determining the steam content in a flow of boiling water correctly characterizes the course of the vaporization process during the discharge of boiling water.

Conclusions

1. A method is given for determining the steam content in a flow from the change in pressure along the length of a nozzle during the discharge of boiling water through cylindrical nozzles.

2. Data have been obtained on the degree of completion of the vaporization process inside nozzles during the discharge of boiling water with an initial pressure of 2–8 atm through cylindrical nozzles 8–92 mm long.

References

1. A. A. Gurchenok. Investigation of the process of discharge of boiling water through diaphragms and cylindrical nozzles, 1957, Izvestiya TPI, vol. 101.

2. S. S. Kutateladze. Heat Transfer in Condensation and Boiling, Mashgiz, 1952.