On the Calculation of Flow Parameters During the Discharge of Boiling Water

A. A. Gurchenok

(Presented by Prof. G. I. Fuks, Doctor of Technical Sciences)

In calculating the parameters and the flow rate during the discharge of boiling water, it is usually assumed that the expansion occurs adiabatically [1] (Fig. 1). Hence the following calculation relationships are obtained:

Fig. 1. 1–2 line of adiabatic expansion of boiling water.

steam content at the end of expansion:

\[ x = \frac{s - s'}{s'' - s'} \; \text{kg/kg}; \tag{1} \]

discharge velocity:

\[ c = 91.5 \sqrt{i_1 - i_2}\ \text{m/sec}; \tag{2} \]

specific volume at the end of expansion:

\[ v = v' + x(v'' - v')\ \text{m}^3/\text{kg}; \tag{3} \]

flow rate

\[ g=\frac{c}{v \cdot 10^4}\ \text{kg}/\text{cm}^2\text{sec}. \tag{4} \]

A study of the process of discharge of boiling water through diaphragms and nozzles showed that the flow-rate values turn out to be several times greater than those determined from relations (1, 2, 3, 4) [2]. Such a discrepancy in the results, as shown in [3], may be explained by incomplete completion of the vaporization process, owing to the excessively short residence time of the liquid particles inside the nozzle. In this case the connection between the principal parameters in the flow of the steam-water mixture is disrupted, and their values do not correspond to the states through which the flow passes during equilibrium adiabatic expansion along line 1–2, Fig. 1.

The present work shows the possibility of determining the flow rate during the discharge of boiling water using the indicated relations (1, 2, 3, 4) on the \(T\)–\(S\) or \(P\)–\(V\) diagram, provided that the specified value of the degree of completion of the vaporization process is given.

Theoretical determination of the flow rate of boiling water

Figure 2 shows the pattern of the pressure decrease inside a nozzle during the discharge of boiling water [2]. The drop in pressure immediately at the inlet section causes the flow of boiling water to become superheated. The magnitude of the superheating is then determined by the difference between the saturation temperature \(t_{s_1}\) at the initial pressure \(P_1\) and the saturation temperature \(t_{s_2}\) corresponding to the established pressure \(P_2\). To preserve the superheated state of a liquid, as is known, special conditions are required: the presence of an ideally pure liquid in which there are no centers of vaporization, and a quiescent state of the liquid. Therefore, during the discharge of boiling water, the process of formation of the vapor phase arises immediately inside the nozzle.

Fig. 2. Pattern of pressure decrease during discharge through a diaphragm: \(P_1\)—initial pressure, \(P_2\)—back pressure.

Owing to the short time interval, the vaporization process does not have time to be fully completed inside the nozzle, and in the flow at the outlet from the nozzle its temperature \(t\) turns out to be higher than the saturation temperature \(t_{s_2}\) at the established pressure \(P_2\).

The ratio

\[ \xi=\frac{t_{s_1}-t}{t_{s_1}-t_{s_2}} \tag{5} \]

characterizes the degree of completion of the vaporization process. If there is no vaporization, then

\[ t=t_{s_1}, \quad \text{whence} \quad \xi=0, \]

and if the process is completed, then

\[ t=t_{s_2}, \quad \text{whence} \quad \xi=1. \]

Representing the state in the case of an incomplete vapor-formation process by point \(2'\) (Fig. 1), we obtain

\[ x_{2'}=\frac{s_1-s'_{2'}}{s''_{2'}-s'_{2'}}\ \text{kg/kg}. \tag{6} \]

It is obvious that \(x_{2'}<x\) from (1), since

\[ t_s-t<t_{s_1}-t_s . \]

The steam forming in the flow is in the form of separate bubbles moving together with the flow. Owing to the presence of surface tension at the phase-interface line, additional forces arise, producing inside the steam bubbles a pressure higher than that of the surrounding medium. According to [4], the magnitude of the difference between the pressures inside the steam bubbles \(p\) and the pressure of the medium \(p_2\) is determined from the formula

\[ p-p_2=\Delta t\left(\frac{\partial p}{\partial t}\right)_s . \tag{7} \]

Here, \(\Delta t\) is the superheating of the liquid, °C;

\[ \left(\frac{\partial p}{\partial t}\right)_s \]

is the derivative of pressure with respect to temperature on the saturation line, equal to

\[ \left(\frac{\partial p}{\partial t}\right)_s= \frac{r\gamma'\gamma''}{A T_s(\gamma'-\gamma'')} . \tag{8} \]

where \(r\) is the heat of vaporization, kcal/kg,

\(\gamma'\), \(\gamma''\) are, respectively, the specific weight of liquid and steam, kg/m\(^3\);

\(A\) is the thermal equivalent of work, kcal/kgm,

\(T_s\) is the saturation temperature, °K.

Taking (7) into account, equation (3), for determining the value of the specific volume in the case of state \(2'\), is written in the form

\[ v_{2'g}=v'_{p_2}+x_2\left(v''_{p}-v'_{p_2}\right),\ \text{m}^3/\text{kg}. \tag{9} \]

Here, \(v'_{p_2}\) is the specific volume of boiling water at pressure \(p_2\),

\(v''_{p}\) is the specific volume of steam at pressure \(p\).

It follows from (9) that the value of the specific volume determined with allowance for the additional pressure inside the steam bubbles is smaller than when it is determined from (3) for equilibrium adiabatic expansion. Depending on the degree of completion of the vapor-formation process, the value of \(v_{2'g}\) changes from \(v'_1\) at \(\xi=0\) to \(v_2\) at \(\xi=1\).

The expansion process plotted according to the values of the actual specific volume, with allowance for the additional pressure inside the steam bubbles, is represented by the line \(1-2g\), passing to the left of the adiabatic-expansion line (Fig. 3). At \(\xi=0\) the expansion line coincides with the left limiting curve; at \(\xi=1\), with the adiabatic-expansion line. The position of point \(2g\) is determined by two parameters: the value of the specific volume \(v_{2'g}\) and the pressure at the end of expansion \(p_2\).

From the representation of the actual expansion process \(1-2g\) on the \(T-S\) diagram, one can determine the magnitude of the increase in kinetic energy in the flow (the area \(12g3\) (Fig. 3) is equal, in scale, to the indicated quantity). It should be noted that, when representing the line

of the actual expansion on the \(T-S\) diagram, the following convention is allowed: the state of the flow at the end of the expansion (point \(2g\)) is characterized by the pressure \(p_2\) and by the steam dryness fraction \(x_{2'}\), whereas the value of the temperature in the flow is higher than or equal to \(T \ge T_2\). The state of the flow at point \(2g\) is unstable, and as the vaporization process is completed there is a transition to a state to the right of point 2, and theoretically, without taking losses into account, to point 2.

Fig. 3. \(1—2g\)—line of expansion of boiling water with an incomplete degree of completion of the vaporization process.

Similarly, the process of discharge of boiling water, taking into account incomplete completion of the vaporization process, can be plotted on a \(P—V\) diagram. For this purpose it is necessary first to construct the \(P—V\) diagram of the states of boiling water and of the wet-steam region, on which to plot the line of adiabatic expansion of boiling water (\(1—2\), Fig. 4).

Fig. 4. \(1—2\) adiabatic expansion. \(1—2g\) — line of actual expansion of the flow of boiling water.

The line of actual expansion \(1—2g\) is constructed as a function of the specified degree of completion of the vaporization process, according to which

the value of the specific volume at the end of the expansion (point \(2g\)) is determined. To determine the positions of the intermediate points on the line \(1—2g\), the segments \(p_1 - p\) and \(p_1 - p_2\) are divided into equal parts. The intermediate points obtained on the line \(1—2'\) are brought down to their intersection with the corresponding point on the segment \(p_1 - p_2\). In doing so, it is assumed that the expansion line \(1—2g\) must represent a proportionally enlarged segment \(1—2'\) on the adiabatic-expansion line (point \(2'\) corresponds to the saturation pressure at the flow temperature \(t\)).

Projecting points \(1\) and \(2g\) onto the pressure axis, we obtain the figure \(12g\,3/4\), the area of which represents, to scale, the magnitude of the increase in kinetic energy during the discharge process. Analytically this dependence is written in the form

\[ -\int_{p_1}^{p_2} v\,dp = \frac{c^2 - c_H^2}{2g}. \tag{10} \]

In the case considered, the magnitude of the approach velocity \(c_H\) is small, and therefore expression (10) may be written in the form

\[ \frac{c^2}{2g} = -\int_{p_1}^{p_2} v\,dp, \tag{10a} \]

whence

\[ c = \sqrt{\,2g\int_{p_1}^{p_2}(-v\,dp)\,}\ \text{m/sec}. \tag{11} \]

Equation (11) can be solved: a) by numerically determining the value of the integral through a sum of the form

\[ \int_{p_1}^{p_2} v\,dp = \sum_{1}^{n} v_i \Delta p_i, \tag{12} \]

where \(v_i\) is the mean value of the volume over the \(i\)-th interval,

\(\Delta p_i\) is the interval of pressure change there;

b) by planimetering the area \(12g\,3/4\) by virtue of the condition

\[ -\int_{p_1}^{p_2} v\,dp = M_f \cdot S_{12g34}, \tag{13} \]

where \(M_f\) is the area scale, \(\text{kgm}/\text{cm}^2\),

\(S\) is the area, \(\text{cm}^2\).

Then from (11) and (13) we obtain

\[ c = \sqrt{2g\,M_f S_{12g34}}\ \text{m/sec}. \tag{14} \]

Table 1 gives the values of velocity and discharge obtained by the indicated method (an example of the calculation is given in the Appendix). The problem was solved by constructing the lines of actual expansion in a \(P—V\) diagram and by planimetering the area for each specified

Table 1

Determination of the flow rate during the discharge of boiling water with initial pressure
\(p_1 = 2\) ata into a space with back pressure \(p_2 = 1\) ata

values of the degree of completion of the vapor-formation process (the initial parameters of the boiling water were specified on the basis of the possibility of carrying out an experimental investigation). From Table 1 it is seen that, as the degree of completion of the vapor-formation process increases, the flow rate of boiling water decreases. The maximum flow rate is obtained for discharge without vapor formation (\(\xi = 0,0\%\)); the minimum is obtained in the case of equilibrium adiabatic expansion (\(\xi = 100\%\)).

Experimental determination of the flow rate and discussion of results

The experimental determination of the flow rate of boiling water was carried out on cylindrical nozzles of various lengths. Since the residence time of liquid particles inside a nozzle is associated with the length of the nozzle, and consequently also the degree of completion of the vapor-formation process under the same discharge conditions, the results obtained will reflect the indicated dependence.

For carrying out the experiments, the laboratory apparatus described in [2] was used. A series of cylindrical nozzles was investigated with lengths: a)—0,5 mm, b)—8,0 mm, c)—92 mm, d)—250 mm, e)—500 mm, f)—1500 mm. The initial pressure was varied within the range 0,1–1,2 ati. Discharge took place into a space with atmospheric back pressure. The experiments were performed with cold and boiling water.

The experimental results are presented in Fig. 5 in graphs a, b, c, d, e, f. From Fig. 5 it is seen that, under the same initial conditions, the flow rate of boiling water is the smaller, the longer the nozzle. The generalized

The results of the experiments at \(p_1 = 2\) ata are represented by curve 1 in Fig. 6. Curve 2 is also plotted there—the dependence of the flow rate on the degree of completion of the vaporization process according to the data of Table 1. It is clear from Fig. 6 that the pattern of change in flow rate, both on the experimental curve and on the theoretically determined one, is identical.

Fig. 5. Dependence of flow rate on pressure difference during the discharge of boiling and cold water through cylindrical nozzles of different length, according to experimental data.

Visible labels in Fig. 5:
\(g,\ \mathrm{kg}/(\mathrm{cm}^2\cdot\mathrm{s})\); pressure, \(\mathrm{kg}/\mathrm{cm}^2\).
Panels: \(a\): \(\varnothing 2.2\ \mathrm{mm},\ l=0.5\ \mathrm{mm}\); \(b\): \(\varnothing 2.2\ \mathrm{mm},\ l=8\ \mathrm{mm}\); \(c\): \(\varnothing 2.2\ \mathrm{mm},\ l=92\ \mathrm{mm}\); \(d\): \(\varnothing 3\ \mathrm{mm},\ l=250\ \mathrm{mm}\); \(e\): \(\varnothing 3\ \mathrm{mm},\ l=500\ \mathrm{mm}\); \(f\): \(\varnothing 3\ \mathrm{mm},\ l=1500\ \mathrm{mm}\).
Legend: solid line—boiling water; dashed line with circles—cold water.

From comparison of the curves, a dependence is established of the degree of completion of the vaporization process on the length of the nozzles and, consequently, on the residence time of the liquid particles in them during discharge. Thus, in the experiment with a nozzle length of \(1590\ \mathrm{mm}\), the flow rate is equal to the theoretically determined flow rate for a completely completed vaporization process. Admitting the indicated analogy, one can determine the interval of time during which, under the specified conditions, the vaporization process is fully completed. In the data of Table 1, the outlet velocity in this case is \(77\ \mathrm{m/sec}\). According to the law of continuity of flow, the flow rate is the same in any cross section of the nozzle. Therefore, for the inlet and outlet sections one may write:

\[ g=\frac{c_{\mathrm{in}}}{v'\cdot 10^4} =\frac{c_{\mathrm{out}}}{v\cdot 10^4}. \tag{15} \]

Solving (15), we obtain

\[ c_{\mathrm{in}}=1.2\ \mathrm{m/sec}. \]

A path equal to the length of the nozzle, under uniformly accelerated motion, will be traversed by the flow in a time interval equal to

\[ \tau=\frac{l}{c_{\mathrm{in}}+0.5\left(c_{\mathrm{out}}-c_{\mathrm{in}}\right)}; \qquad \tau=0.04\ \mathrm{sec}. \]

There are no data on the duration of the vapor-formation process under discharge conditions. However, there are data from high-speed filming of the growth of vapor bubbles during boiling under atmospheric conditions, according to which the growth of vapor bubbles is completed in \(0.0295—1.0\) sec [5]. The agreement of the results is a certain verification of the proposed method for determining the thermodynamic parameters in the process of discharge of boiling water.

Fig. 6. 1—experiment, 2—theoretical calculation.

Appendix

Example. The discharge of boiling water occurs at an initial pressure \(p_1 = 2\) ata into a region with atmospheric pressure \(p_2 = 1\) ata with a degree of completion of the vapor-formation process \(\xi = 16.8\%\). Determine the flow parameters at the end of expansion and the flow rate.

Solution. From tables of the thermodynamic properties of water and steam we find that the saturation temperatures are:
at \(p_1\)—\(t_{s_1} = 119.62^\circ\mathrm{C}\), \(p_2\)—\(t_{s_2} = 100^\circ\mathrm{C}\).
Then from (5)

\[ 0.168 = \frac{119.62 - t}{119.62 - 100} \]

we obtain \(t = 116.33^\circ\mathrm{C}\),
which corresponds to a saturation pressure \(P_s = 1.8\) ata. The indicated thermodynamic state is denoted by point \(2'\), lying on the adiabatic expansion line (Fig. 1).

We determine the dryness fraction of the steam at point \(2'\)

\[ x_{2'} = \frac{s_1 - s'_{2'}}{s''_{2'} - s'_{2'}} . \]

After substituting the numerical values, we obtain

\[ x_{2'} = \frac{0.3539 - 0.3554}{1.7123 - 0.3554} = 0.00527\ \text{kg/kg}. \]

The steam formed is contained in the volume of the liquid in the form of vapor bubbles. Inside the vapor bubbles there will be an elevated pressure, the magnitude of which is determined from formula (7).

For the conditions of the problem, the values of \(\left(\dfrac{\partial p}{\partial t}\right)_s\) and \(\Delta t\) are:

\[ \left(\frac{\partial p}{\partial t}\right)_s = \frac{427 \cdot 540 - 960 \cdot 0.58}{373(960 - 0.58)} = 358\ \text{kg}/\text{m}^2{}^\circ\text{C}, \]

\[ \Delta t = 116.33 - 100 = 16.33^\circ\text{C}. \]

Then from (7) we obtain

\[ \Delta p = 358 \cdot 16.33 = 5850\ \text{kg}/\text{m}^2 = 0.585\ \text{kg}/\text{cm}^2, \]

whence the total pressure is

\[ p = 1.033 + 0.585 = 1.618\ \text{kg}/\text{cm}^2. \]

We determine the value of the specific volume of the steam-water mixture, taking into account the additional pressure within the steam phase. From (9), after substituting the numerical values, we obtain

\[ v_{2g} = 0.00105 + 0.0067 \cdot 1.11 = 0.008\ \text{m}^3/\text{kg}. \]

We construct a \(P—V\) state diagram with the scale: pressure—\(1\) ata—\(20\) cm, specific volume \(0.0173\ \text{m}^3/\text{kg}\)—\(10\) cm, area \(M_f = 0.862\ \text{kg}/\text{cm}^2\). Then on the \(P—V\) diagram we plot the curve of adiabatic expansion \(1—2\). Next we construct the curve of actual expansion \(1—2g\) by transferring point \(2'\) to the pressure line \(p_2 = 1\) ata at point \(2g\) (Fig. 4) and correspondingly transferring the intermediate points. By planimetering the area between the process curve \(1—2g\) and the ordinate axis, we obtain \(S_{12g34} = 60\ \text{cm}^2\), \(M_f \cdot S_{12g34} = 52\ \text{kgm}\). The magnitude of the flow velocity at the outlet section is

\[ c = 4.3 \sqrt{52} = 31\ \text{m/sec}. \]

The flow rate of boiling water is

\[ g = \frac{31}{0.008 \cdot 10^4} = 0.39\ \text{kg}/\text{cm}^2\text{sec}. \]

LITERATURE

1. Perelman I. L. Blowdown Devices of Steam Boilers Using Superheated Water. GEI, 1956.

2. Gurchenok A. A. Investigation of the process of discharge of boiling water through diaphragms and cylindrical nozzles, Izvestiya TPI, vol. 101, 1958.

3. Gurchenok A. A. Determination of the completeness of the vapor-formation process during the discharge of boiling water, Izvestiya TPI, vol. 101, 1958.

4. Mikheev M. A. Fundamentals of Heat Transfer. GEI, 1958.

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