ON THE CALCULATION OF FLOW CHARACTERISTICS DURING THE DISCHARGE OF BOILING WATER TAKING INTO ACCOUNT THE RELATIVE MOTION OF THE VAPOR PHASE

A. A. GURTSENOK

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

Symbols not explained in the text:
$\xi$ — degree of completion of the vapor-formation process, %,
$w$ — velocity of the steam-water mixture, m/sec,
$g$ — specific mass flow rate, kg/cm² sec,
$V$ — volumetric flow rate, m³/sec,
$\mu$ — kinematic viscosity, kg·sec/m²,
$\sigma$ — surface-tension force, kg/m.

In calculating the process of discharge of boiling water, one encounters a number of special features, the study and consideration of which make it possible to bring the results of theoretical calculations closer to experimental data [1, 2]. The available information on the velocity of the relative motion of steam in a steam-liquid mixture makes it possible to refine the method for calculating the process of discharge of boiling water [3, 4, 5].

In the present article, on the basis of the available information on the process of discharge of boiling water and on the relative motion of the vapor phase, a method is given for calculating the flow characteristics of boiling water with allowance for the indicated phenomenon.

The flow of a steam-water mixture may be conventionally divided into two volumes: through one passes water, through the other — steam. The condition is written in the form

\[ f = f_{\text{п}} + f_{\text{в}}, \tag{1} \]

where $f$ is the total cross section of the flow.

Hence the fraction of the cross section occupied by steam is

\[ \varphi_{\text{п}} = f_{\text{п}}/f, \tag{2} \]

and by water

\[ \varphi_{\text{в}} = f_{\text{в}}/f = 1 - \varphi_{\text{п}} . \tag{3} \]

The value of the reduced velocity is written as:

for the vapor phase

\[ w_0'' = V_{\text{п}}/f, \tag{4} \]

for the water part

\[ w_0' = V_{\text{в}}/f. \tag{4-a} \]

From (4) one may write

\[ V_{\text{п}} = w_0'' \cdot f, \tag{5} \]

\[ V_{\text{в}} = w_0' \cdot f. \tag{5-a} \]

The values of the actual velocity of the vapor and liquid parts of the flow are obtained in the form:

\[ w_{\mathrm{p}}^{\mathrm{d}}=w_0''/\varphi_{\mathrm{p}}, \tag{6} \]

\[ w_{\mathrm{v}}^{\mathrm{d}}=w_0'/1-\varphi_{\mathrm{p}}. \tag{6-a} \]

When the velocities of the two phases are equal, the mass and volumetric vapor contents will be identical, i.e. \(\varphi_{\mathrm{p}}=\beta\).

Hence the value of the mass vapor content can be determined from (6) and (6-a)

\[ \beta=w_0'/w_0'+w_0''. \tag{7} \]

After transforming (7), taking into account the presence of different values of the velocities of steam and water, we obtain

\[ \varphi_{\mathrm{p}}=\beta\cdot w_{\mathrm{cm}}/w_{\mathrm{p}}^{\mathrm{d}}. \tag{8} \]

This expression shows the degree of deviation of the real flow from the conventional one obtained when the phase velocities are equal. According to the data of [3], the dependence between \(\varphi_{\mathrm{p}}\) and \(\beta\) is linear in character,

\[ \varphi_{\mathrm{p}}=c\beta, \tag{9} \]

where \(c\) is the proportionality coefficient. The numerical value of \(c\) can be determined from the graph in Fig. 1.

In [4], results are given for the experimental determination of the relative velocity of steam in a steam-liquid mixture in vertical tubes. The experimental results are processed in the form of the dependence

\[ w_0=f(w_0'',\,w_0',\,v',\,v'',\,\mu',\,\sigma). \tag{10} \]

On the basis of dimensional analysis, the following criterion equation was obtained

\[ w_0'/w_0=3.0\cdot (w_0''/w_0')^{0.05}\cdot (w_0'\cdot \mu'/\sigma)^{0.1}\cdot (v'/v'')^{0.33}. \tag{11} \]

Fig. 1

The indicated results were used in the calculations presented below.

2

In [2], the case of the discharge of boiling water through cylindrical nozzles was investigated at an initial pressure \(P_1 \ne 9.8\) bar into a region with back pressure \(P_2=0.98\text{--}9.0\) bar at various degrees of completion of the vapor-formation process.

Table 1 gives the flow characteristics of a steam-liquid mixture during the discharge of boiling water under the conditions indicated above, determined without taking into account the possibility of slip of the vapor phase.

Using the data given below, the relative velocity of the vapor phase was determined. To determine the velocity and flow rate, closing equations of the kinetic-energy balance of the flow were compiled. The calculation was carried out in the following sequence.

From the data of [2] the following were determined: the value of the vapor content

\[ \varphi_{\mathrm{p}}=xv''/v_{\mathrm{cm}}, \tag{12} \]

and the volume fraction of water

\[ \varphi_{\mathrm{v}}=1-\varphi_{\mathrm{p}}. \tag{13} \]

Here \(v''\), \(v_{\mathrm{mix}}\) are, respectively, the specific volume of steam and of the steam-water mixture at the flow parameters.

The mass contents of steam and water were determined from the volumetric contents by the relations:

for steam

\[ g_{\mathrm{s}}=\varphi_{\mathrm{s}}\cdot v_{\mathrm{mix}}/v'', \tag{14} \]

for water

\[ g_{\mathrm{w}}=\varphi_{\mathrm{w}}\cdot v_{\mathrm{mix}}/v'. \tag{15} \]

Table 1

The increase in steam velocity occurs owing to its acceleration during the outflow of the stream. The kinetic-energy balance equation for the steam-liquid flow will have the form

\[ g_{\mathrm{s}}\left(\rho_{\mathrm{s}} w_{\mathrm{s}}^{2}/2-\rho_{\mathrm{s}} w_{\mathrm{mix}}^{2}/2\right) = g_{\mathrm{w}}\left(\rho_{\mathrm{w}} w_{\mathrm{mix}}^{2}/2-\rho_{\mathrm{w}} w_{\mathrm{w}}^{2}/2\right). \tag{16} \]

After transforming (16), we obtain

\[ w_{\mathrm{w}}^{2} = w_{\mathrm{mix}}^{2} - g_{\mathrm{s}}\rho_{\mathrm{s}}/g_{\mathrm{w}}\rho_{\mathrm{w}} \left(w_{\mathrm{s}}^{2}-w_{\mathrm{mix}}^{2}\right). \tag{17} \]

The value of the actual volumetric steam content will be equal to

\[ \varphi_{\mathrm{s}}^{*} = \frac{xv''/w_{\mathrm{s}}} {xv''/w_{\mathrm{s}}+(1-x)v'/w_{\mathrm{w}}}, \tag{18} \]

respectively, for water

\[ \varphi_{\mathrm{w}}^{*}=1-\varphi_{\mathrm{s}}^{*}. \tag{19} \]

Applying the flow-continuity equation, we obtain the value of the discharge in the form

\[ g^{*}=f_{\mathrm{s}}w_{\mathrm{s}}/v''+f_{\mathrm{w}}w_{\mathrm{w}}/v'\ \mathrm{kg/cm^{2}\,s}. \tag{20} \]

The results of calculations from the data of [2] are given in Table 2. From the data of Table 2 it is seen:

a) for the investigated modes of boiling-water discharge, the values of the relative velocity of steam are \(0\)--\(24.3\) m/s when determined according to [3], and \(9.6\)--\(97\) m/s according to [4];

b) the value of the actual volumetric fraction of the vapor phase, when determined with allowance for the relative velocity of its motion, turns out to be ...

turns out to be smaller by 0–25% when calculated according to [3], and smaller by 1–47% when calculated according to [4].

Table 2

c) the value of the discharge rate proves to be increased by 0–110% when calculated according to [3], and by 3–120% when calculated according to [4].

Conclusion

In investigating the process of boiling-water discharge, it is necessary to take into account the presence of the relative velocity of the vapor phase, which changes the values of the characteristics in the flow and increases the discharge values.

References

1. A. A. Gurchenok. Izv. TPI, vol. 110, 1962.
2. A. A. Gurchenok. The present collection.
3. M. A. Styrikovich. Boiler Units, 1959, pp. 212–218.
4. A. L. Bolotov. Thermal Engineering, No. 11, 1967.
5. D. A. Labuntsov. Thermal Engineering, No. 4, 1968.