On the Calculation of Heat-Transfer Intensity in the Process of Flashing During the Outflow of Boiling Water

A. A. Gurtienok

(Presented by the seminar of the Department of Automation of Thermal-Power Processes of Industrial Enterprises)

The process of outflow of boiling water proceeds, as is known, nonequilibrially. In a thermodynamic description of the process, therefore, the introduction of empirical corrections is necessary. In the present work an attempt is made to apply the method of heat-transfer theory to describe the process. A problem has been formulated and solved, by means of the numerical grid method, concerning the transfer of heat from the liquid to individual bubbles of the arising vapor phase [1].

During the outflow of boiling water inside a nozzle, a pressure drop occurs. The liquid becomes superheated, and its flashing occurs. The vapor phase arises throughout the entire volume in the form of individual vapor bubbles. The heat necessary for growth is taken from the liquid, whose temperature is higher than the saturation temperature of the forming vapor phase. The intensity of heat supply to the centers of growth of the vapor phase determines the rate of their growth. The growth process will continue until thermodynamic equilibrium is established. Owing to the short time interval, the vapor-formation process does not have time to be completed fully in short nozzles, and therefore the temperature of the liquid at the nozzle exit proves to be higher than the saturation temperature of the vapor phase,

\[ \xi = \frac{t_{s1} - t}{t_{s1} - t_{s2}} . \]

This ratio characterizes the degree of completion of the vapor-formation process [2]. To construct a calculation scheme, let us assume that the volume of liquid is divided into separate cells, within each of which a single vapor bubble has arisen and grows, removing heat from the neighboring liquid layers. The amount of heat that can be transferred from the liquid is determined from the heat balance according to the values of the flow parameters at the beginning and at the end of the process. Taking into account the magnitude of the degree of completion of the process, the following can be determined:

the volume occupied by the vapor phase

\[ V_{\mathrm{v}} = x \cdot v_{\mathrm{v}} \cdot G; \tag{2} \]

the volume of the liquid phase

\[ V_{\mathrm{l}} = (1 - x) \cdot v \cdot G; \tag{3} \]

the total volume of the cell

\[ V = V_{\mathrm{v}} + V_{\mathrm{l}} . \tag{4} \]

Here \(x\) is the dryness fraction of the two-phase medium, kg/kg;
\(v_{\text{п}}\) is the specific volume of the vapor inside a vapor bubble, m³/kg.

On the basis of the data on the process of outflow of boiling water given in [2], the geometric parameters of the cell were calculated from (2), (3), and (4), proceeding from a spherical model. As a result, the geometric structure of the cell was obtained for each stage of the vapor-formation process by bulk boiling during the outflow of boiling water (Table 1).

Table 1

The resulting picture of the growth of the vapor phase in the cell is idealized. However, it is clear from Table 1 that, as the vapor phase grows, its volume increases so much that the liquid can occupy the volume only in the form of thin spherical shells. It is obvious that, in the process of the change in shape, strong deformation and good mixing of the layers occur, and consequently the temperature is equalized over the entire volume. The equation of unsteady heat conduction in polar grids, if at the initial instant the temperature distribution is a function of a single radius \(r\), has the form

\[ \frac{\partial T}{\partial \tau} = a \frac{\partial^2 T}{\partial r^2} + \frac{1}{r}\frac{\partial T}{\partial r} \tag{5} \]

when the initial and boundary conditions are satisfied,

\[ T(r,0)=f(r), \tag{6} \]

\[ T(R,\tau)=F(\tau). \tag{7} \]

Equation (5) can be represented as a system of finite-difference equations

\[ \frac{\partial T_{i,k}}{\partial \tau} = \frac{T_{i,k+1}-T_{i,k}}{l}, \tag{8} \]

\[ \frac{\partial^2 T_{i,k}}{\partial r^2} = \frac{T_{i-1,k}-2T_{i,k}+T_{i+1,k}}{h^2}, \tag{9} \]

\[ \frac{\partial T_{i,k}}{\partial r} = \frac{T_{i+1,k}-T_{i-1,k}}{2h}. \tag{10} \]

Substituting (8), (9), and (10) into (5), we obtain the computational arithmetic relation.

\[ T_{i,\kappa+1}=\left[1-\frac{2al}{h^2}\right]T_{i,\kappa} +\frac{al}{h^2}\left[T_{i+1,\kappa}+T_{i-1,\kappa}\right]+ \]

\[ +\frac{al}{2hr_i}\left[T_{i+1,\kappa}-T_{i-1,\kappa}\right]. \tag{11} \]

Application of relation (11) makes it possible to determine the value of the temperature \(T\) at the grid nodes in the \(r-\tau\) plane.

The calculations are carried out as follows. We divide the radius of the cell sphere into \(n\) equal parts:

\[ r=0,\quad r=h,\ldots,\quad r=ih,\ldots,\quad r=n\cdot h=R \]

The time value is plotted along the ordinate axis. In order to obtain stable difference solutions, the size of the time step must satisfy the relation

\[ l\leq \frac{h^2}{4a}. \tag{12} \]

Taking the condition

\[ l=\frac{h^2}{4a}, \tag{13} \]

from (13) we obtain, for the \(i\)-th row,

\[ T_{i,\kappa+1}=\frac{1}{2}T_{i,\kappa} +\frac{1}{4}\left[T_{i-1,\kappa}+T_{i+1,\kappa}\right] +\frac{1}{8i}\left[T_{i+1,\kappa}-T_{i-1,\kappa}\right]. \tag{14} \]

The relations obtained from (14) were used as the basis for calculating unsteady heat conduction in the cell. The calculation is carried out step by step. In the first step we construct the grid region (Fig. 1). We enter on the grid the temperature values at the initial instant of time \((\tau=h)\). The dimensions of the grid step are chosen on the basis of conditions (12) and (13). Then, from (14), we calculate the temperature values for the given value of the sphere radius after the time interval \(l\). The quantity of heat transferred from the liquid will be determined by calculation from the relation

\[ \Delta Q=V\cdot c\cdot \rho\left[T_{i,\kappa}-T_{i,\kappa+1}\right]. \tag{15} \]

Fig. 1.

The calculation of the first iteration step is complete. Next, for the new conditions obtained, we again carry out an analogous calculation. The calculations are carried out until the final parameters have been obtained. The results of the calculation performed are presented in Table 2.

A model of the cell in the form of a sphere with a diameter of \(1.0\) m was calculated. As a result of the calculation, data were obtained on the duration in time and the interval of temperature decrease for each iteration step. To pass from the calculation on the model to the actual process, use was made of ...

was used, obtained from the similarity condition by equality of the Fourier criterion,

\[ \frac{a\tau_{n}}{L_{n}^{2}}=\frac{a\tau_{m}}{L_{m}^{2}} . \tag{16} \]

Since the properties of the liquid in the model calculation were taken unchanged, in (16) the quantities \((a)\) on the left- and right-hand sides of the equation

Table 2

will be identical. Then equation (16) can be rewritten in the form

\[ \tau_{n}=\tau_{m}\left[\frac{L_{n}}{L_{m}}\right]^{2}. \tag{16a} \]

Condition (16a) makes it possible to pass to other model dimensions and to the full-scale process. Thus, calculation from the indicated condition showed that the size of the liquid cell for the conditions of the process of the outflow of boiling water in [2], [3] is equal to \(0.75 \times 10^{-3}\ \text{m}\). The time during which the vapor-formation process is completed is \(0.04\) s. The data obtained agree with the results of high-speed filming of the growth of vapor bubbles during boiling under atmospheric conditions [4]).

REFERENCES

1. P. P. Yushkov. Approximate solution of problems of unsteady heat conduction by the finite-difference method. Proceedings of the Institute of Power Engineering, issue 6, Academy of Sciences of the BSSR, 1958.

2. A. A. Gurchenok. On the calculation of parameters in a flow during the outflow of boiling water. Izv. TPI, vol. 110, 1962.

3. A. A. Gurchenok. On the intensity of the vapor-formation process during flashing in the outflow of boiling water. Izv. TPI, vol. 245.

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