On the Question of Choosing the Type of Surface Condenser

E. N. Shadrin

The cooling surface of a condenser, in \(m^2\), is

\[ F=\frac{W}{K}\cdot \ln \frac{t_s-t_1}{t_s-t_2}, \quad \text{where} \tag{1} \]

\(W\) — the quantity of cooling water supplied to the condenser, kg/hour;
\(t_s\) — the temperature of the steam entering the condenser, °C;
\(t_1\) and \(t_2\) — the temperatures of the cooling water at the inlet to the condenser and at the outlet from it, °C;
\(K\) — the overall heat-transfer coefficient, kcal/\(m^2\) hour, °C.

Its value depends on many factors, including the velocity \(c_w\) of the water in the condenser tubes, the temperature \(t_1\) of the cooling water, the steam load \(\dfrac{D_k}{F}\) of the condenser, and also the quantity \(G_k\) of air entering the condenser. Thus one may write

\[ K=a\cdot c_w^n\cdot t_1^p; \tag{2} \]

where \(a\), \(n\), and \(p\) are constants for a particular operating regime of the condenser.

In this case,

\[ n=f_1\left(\frac{D_k}{F},\, t_1,\, G_v\right),\quad p=f_2\left(\frac{D_k}{F},\, G_v\right). \]

Since the heat-transfer coefficient \(K\) decreases with decreasing velocity \(c_w\), it follows from formulas 1 and 2 that, for equal cooling surfaces \(F_1\) and \(F_2\) of single-pass and two-pass condensers, the amount of water \(W_1\) in the case of using a single-pass condenser will be greater.

If the influence of the mean temperature of the cooling water on the heat-transfer coefficient is neglected and \(n=0.5\) is adopted, which is valid for condensers with large steam loads, then from the condition \(F_1=F_2\) we obtain:

\[ m_1 \ln^2 \frac{t_s-t_1}{t_s-t_1-\delta t_1} = \frac{m_2}{2}\ln^2 \frac{t_s-t_1}{t_s-t_1-\delta t_2}. \tag{3} \]

Here \(m_1\) — cooling ratio in the single-pass condenser;
\(m_2\) — the same in the two-pass condenser;
\(\delta t_1\) and \(\delta t_2\) — water-heating ranges in the single-pass and two-pass condensers, °C.

Figure 1 gives, in graphical interpretation, the solution of the transcendental equation (3) for \(t_1=10^\circ\text{C}\) and \(t_1=15^\circ\text{C}\) for the particular example when

\[ p_2=0.04\ \text{ata},\quad \text{and}\quad t_2-t_k=532\ \frac{\text{kcal}}{\text{kg}}. \]

Fig. 2 shows the change in the ratio \(\varphi = \dfrac{W_1}{W_2}\) as a function of \(t_1\) and \(m_2\).

Fig. 1

From Figs. 1 and 2 it can be seen that the use of higher cooling ratios, as well as lower design temperatures, affects

Fig. 2

the value of \(\varphi\) in the direction of increasing it. In constructing the curves of Figs. 1 and 2, we neglected the change in \(n\) in passing from \(t_1 = 15^\circ\mathrm{C}\) to \(t_1 = 10^\circ\mathrm{C}\). If we denote by

$H_1$ is the total head overcome by the circulating pumps with a single-pass condenser, m of water column;

$H_2$ is the same for a two-pass condenser; then, for certain values of the difference between the elevations $H_n$ of the condenser and the water level in the source, and for a quite definite distance $L_0$ of the station from the latter, it may turn out that

\[ W_1 H_1 = W_2 H_2 . \tag{4} \]

Equation 4 expresses the equality of the work expended in overcoming the resistances of the circulating circuit and the head due to the geodetic height, for single-pass and two-pass condensers.

Let us write this equation in another form.

The resistance of the condenser is

\[ H_k = Z\left( 0.031 \frac{L_k}{d_k}\frac{c^2 W}{2g}\,\beta + 1.4\frac{c^2 W}{2g} \right), \quad \text{where} \tag{5} \]

$L_k$ is the length of the condenser tubes, m;

$d_k$ is their internal diameter, mm;

$Z$ is the number of water passes;

$\beta$ is a coefficient taking into account the influence of the mean temperature and the velocity of the water.

In formula 5 the first term characterizes the frictional resistances inside the condenser tubes, and the second—the inlet and outlet resistance.

The resistance of the supply water conduits is

\[ H_v = \lambda \cdot \frac{L}{d_v}\cdot \frac{c_v^2}{2g}, \tag{6} \]

$L_e$ is the equivalent length of the water conduit, m;

$c_v$ is the velocity of the water in the conduits, m/sec;

$d_v$ is the diameter of the water conduit, m.

Then, for identical geometric dimensions of the condensers, equation (4) of equality of work can be written as

\[ \left( 0.031 \frac{L}{d_k}\cdot \frac{\beta}{2g} + \frac{1.4}{2g} \right) \left( z_2 W_2 c_{w1}^2 - z_1 W_1 c_{w2}^2 \right) + \]

\[ + \frac{\lambda L_0 \varphi}{2g d_v} \left( W_2 c_{v2}^2 - W_1 c_{v1}^2 \right) + H_n\left(W_2-W_1\right) = 0. \tag{7} \]

Here the coefficient

\[ \alpha = \frac{L_e}{L_0}. \]

If, further, we denote

\[ \psi = 0.031 \frac{L_k}{d_k}\frac{\beta}{2g} + \frac{1.4}{2g} \]

and take into account that

\[ \frac{c_{w1}}{c_{w2}} = 0.5\varphi, \qquad \frac{c_{v1}}{c_{v2}} = \varphi \quad \text{for } d_v=\text{const}, \]

then equation 7 can be written as

\[ 2\psi c_{w2}^2\left(1-\frac{\varphi^3}{8}\right) + \frac{\lambda L_0 \alpha c_{v2}^2}{2g d_v} (1-\varphi^3) + H_n(1-\varphi) = 0. \tag{8} \]

In the axes \(L_0, H_n\), the given expression represents the equation of a straight line, which is easy to construct by setting \(L_0\) and \(H_n\) alternately equal to zero.

Thus, for \(L_0 = 0\),

\[ H_n = \frac{ 2\psi \cdot t^2 w_2 \left(1 - \frac{\varphi^3}{8}\right) }{ \varphi - 1 }. \tag{9} \]

If, as a rough approximation, we take \(\lambda = 0.03\), then for \(H_n = 0\),

\[ L_0 = \frac{ 1305\psi d_v c^2 w_2 \left(1 - \frac{\varphi^3}{8}\right) }{ a \cdot c_{v2}^{\,2}(\varphi^3 - 1) }. \tag{10} \]

In Fig. 3, according to equations 9 and 10, a straight line is constructed for the particular case when \(c_{w2} = 2.5\ \text{m/sec};\ c_{v} = 1.5\ \text{m/sec};\ \psi = 0.5;\ \varphi = 1.4;\ a = 1.2;\ d_v = 1\ \text{m}\).

Fig. 3

From Fig. 3 it follows:

1) in many cases it is possible to reduce the energy expenditures for pumping cooling water in recirculating cooling systems by replacing a two-pass condenser with a single-pass condenser;

2) replacing a two-pass condenser with a single-pass condenser in once-through systems will make it possible to reduce the head in the circulating water supply, if, at a small geodetic height, the station is located in immediate proximity to the water-supply source. This will reduce the expenditure of energy for the thermal power station’s own needs.

REFERENCES USED

1. Anatoliev F. A.—Calculation of Auxiliary Devices of Steam-Power Installations, ONTI, NKTP, 1936.

2. Berman L. D.—The Influence of the Velocity and Temperature of Cooling Water on the Heat-Transfer Coefficient, Journal “Izvestiya VTI” 12, 1948.

3. Bulanin V. I.—On Single- and Two-Pass Surface Condensers of Steam-Turbine Installations, TsKTI, book 8, 1947.