SELECTION OF THE NUMBER OF WATER PASSES IN SURFACE CONDENSERS

V. A. BRAGIN

(Presented by Honored Worker of Science and Technology Prof. I. P. Butakov)

The number of water passes in surface condensers determines to a considerable extent the expenditure of energy for the power plant’s own needs and the consumption of nonferrous metal for creating the cooling surface. Therefore it is very important, in each specific case, to determine correctly the required number of water passes \((z)\). At present, two-pass condensers have become the most widespread, while single-pass condensers are very rare, although the latter may in a number of cases prove more expedient \([2, 3, 6]\). The present article attempts to facilitate the problem of choosing one or two water passes in a condenser, taking into account the specific local conditions of the plant location.

When determining the number of water passes, the formula \([2, 5]\) is usually used

\[ z = \frac{A}{m \sqrt[3]{F}}, \]

where
\[ m \]
is the cooling ratio,

\[ F \]
is the condenser surface,

\[ A \]
is a quantity depending on the design characteristics of the condenser, the water velocity in the tubes, and the steam load.

With the aid of this formula, \(z\) can be determined in two ways: first, by assigning \(z\) and checking the acceptability of the condenser’s design dimensions and other quantities \([2, 4]\); second, by adopting certain design forms and, by means of complex techno-economic calculations, establishing for the specific local conditions the optimum values of \(m\) and \(z\). The second method is the most accurate, but because of its complexity it is seldom used, preference being given to the first method, which does not take into account a whole series of factors determined by the specific conditions of the power plant’s location. In this case, most often, as a precaution, \(z = 2\) is adopted, which is not always the correct decision. If there are no restrictions of any kind, then, in principle, for one and the same turbine it is always possible to use a single-pass and

two-pass condensers, and to design them in such a way that the following will be observed:

a) equality of the surfaces of condensers of both types for different amounts of cooling water, corresponding to cooling ratios \(m_1^\ell\) \((z=1)\) and \(m_2\) \((z=2)\), or, what is the same, to the temperature differences \(\Delta t_{k1}^\ell\) and \(\Delta t_{k2}\),

\[ F_1 = F_2; \tag{1} \]

b) equality of the power expenditures for supplying cooling water for different condenser surfaces,

\[ N_1 = N_2, \tag{2} \]

to which correspond the cooling ratios \(m_1^u\) and \(m_2\), or the temperature differences \(\Delta t_{k1}^u\) and \(\Delta t_{k2}\).

It is known [3] that if, for example, in the one-pass condenser being designed the selected cooling ratio \(m_1\) is less than \(m_1^\ell\) or greater than \(m_1^u\), then its use is irrational, since it will lead either to an increased surface or to an overexpenditure of energy for supplying the cooling water. Thus, condition (1) is one—the lower—boundary separating the optimal regions of use of condensers with \(z=1\) and \(z=2\), while condition (2) is the upper boundary. Therefore, if the inequality \(\Delta t_{k1}^u > \Delta t_{k1} > \Delta t_{k1}^\ell\) is satisfied, it is advantageous to use a one-pass condenser; if, however, \(\Delta t_{k1}^u < \Delta t_{k1} < \Delta t_{k1}^\ell\), then a two-pass condenser should be used. The equalities (1) and (2) written above can also be represented in the following form:

\[ k_1^\ell \Delta t_{cp1}^\ell = k_2 \Delta t_{cp2}, \tag{1'} \]

\[ W_1^u \left(H_{k1}^u - \Delta H_1^u\right) = W_2\left(H_{k2} + \Delta H_2\right). \tag{2'} \]

Here the notation is as follows:

\(k\)—heat-transfer coefficient,

\(\Delta t_{cp}\)—mean logarithmic temperature difference,

\(W\)—cooling-water flow rate,

\(H_k\)—hydraulic resistance of the condenser proper,

\(\Delta H\)—hydraulic resistance of the entire system, excluding the condenser.

The indices “\(\ell\)” and “\(u\)” indicate that the given quantity pertains to the condition of the lower or upper boundary.

Assuming that the condensers being compared are designed for the same operating conditions, that their type and principal design characteristics are identical, and also that the geometrical dimensions of the tubes are identical, after simple transformations equations (1′) and (2′) can be represented in the following form:

\[ B_\ell \cdot \frac{\left(t_k - t_1 - \Delta t_{k1}^\ell\right)^{1-x}} {\ln \dfrac{t_k - t_1}{\Delta t_{k1}^\ell}} = \frac{\left(t_k - t_1 - \Delta t_{k2}\right)^{1-x}} {\ln \dfrac{t_k - t_1}{\Delta t_{k2}}}, \tag{1''} \]

\[ \frac{B_v}{\ln^{\frac{2}{1-x}}\dfrac{t_k-t_1}{\Delta t_{k2}^{v}}} = \frac{1}{ \left[ (1+\psi)\dfrac{t_k-t_1-\Delta t_{k1}^{v}}{t_k-t_1-t_{k2}^{v}} -\psi \right] \cdot \ln^{\frac{2}{1-x}}\dfrac{t_k-t_1}{\Delta t_{k1}^{v}} }. \tag{2''} \]

The last equations give the required relations between \(\Delta t_{k1}^{n}\), \(\Delta t_{k1}^{v}\), and \(\Delta t_{k2}\), which determine the limits of the optimum application of \(z=1\) or \(z=2\).

Here the following additional notation has been introduced:

\(t_k\) is the saturation temperature of the steam corresponding to the pressure in the condenser \(P_k\);

\(t_1\) is the initial temperature of the cooling water;

\[ \psi=\Delta H_2/H_{k2}; \]

\(B_n\), \(B_v\), and \(x\) are coefficients depending on \(t_1\):

\[ B_n=0.5^x\left(1-0.1\cdot\frac{35-t_1}{35}\right), \]

\[ B_v=8\, \frac{2^{\frac{2x}{1-x}}} {\left(1-0.1\cdot\frac{35-t_1}{35}\right)^{\frac{2}{1-x}}}, \]

\[ x=0.096(1+0.15\cdot t_1). \]

In deriving the indicated equations, the well-known expression of Doctor of Technical Sciences L. D. Berman [1] was used for the heat-transfer coefficient:

\[ K=3500\cdot a\cdot \Phi_w\cdot \Phi_t\cdot \Phi_z\cdot \Phi_d, \]

and the logarithmic mean temperature difference was determined by the Grasshof formula, taking \(t_k\) as unchanged along the path of the steam [2]. In addition, it is assumed that, for the design conditions, \(\Delta H_1^v \simeq \Delta H_2\).¹

Fig. 1. Visible graph labels: vertical axis \(\delta t_k,\,^\circ\mathrm{C}\); horizontal axis \(\Delta t_{k2}^{v},\,^\circ\mathrm{C}\); \(t_1=20^\circ\mathrm{C}\); curve labels \(\psi=0\), \(\psi=1.0\), \(\psi=2.0\), \(\psi=3.0\); legend \(P_k\): \(0.04\) ata, \(0.03\), \(0.05\).

Using the equations obtained, (1″) and (2″), graphs were plotted in Figs. 1 and 2 for the most frequently encountered design values of the cooling-water temperature \(t_1 = 10;\ 15^\circ\mathrm{C}\) and for various pressures \(P_k\). In constructing them it was assumed that \(a \simeq 0.80\), which is valid for once-through water supply and for recirculating systems with sufficient blowdown, or when the water is chemically treated [1]. With their aid it is easy to find, for any specific conditions of the station location, the optimum number of water passes. To do this, having specified the recommended value \(\Delta t_{k2}\) and certain design values [2,5], one determines \(H_k\) and \(\Delta H_2\). Then, from \(\Delta t_{k2}\) and \(\psi\), the presence of an optimum zone \(z = 1\) is established from the corresponding graph. If it is absent, it is more advantageous to use \(z = 2\). If there is a zone in which \(z = 1\) is advantageous, the design value of the temperature head is selected in it,

\[ \Delta t_{k1}\bigl(\Delta t_{k1}^{v} \ge \Delta t_{k1} \ge \Delta t_{k1}^{n}\bigr) \]

for a single-pass condenser, determined by the permissible water velocity in the tubes. If the cooling water contains suspended particles that may settle on the inner surfaces of the tubes, it is recommended that the water velocity be no lower than \(1.8\text{–}2.0\ \mathrm{m/sec}\), which very greatly reduces the possibility of using single-pass condensers.

Fig. 2.

In the graphs shown, the heavy lines indicate the lower limits for using \(z = 1\), and the thin lines—the upper limits.

As is evident from the graphs, the optimum zone of use becomes larger the more: 1) the lower the initial temperature of the cooling water; 2) the higher the design steam pressure in the condenser; 3) the lower, for the two-pass condenser being replaced: a) the design temperature head \(\Delta t_{k3}\), b) the magnitude \(\psi\) (that is, the closer the station is located to the water-supply source, the lower the geometric elevation of its location, the larger the diameters of the water conduits, etc.).

The proposed method makes it possible to approach the question of selecting the number of water passes comparatively simply and soundly, taking into account the specific conditions of the power plant being designed.

REFERENCES

1. L. L. Berman. Heat-transfer coefficient of a surface condenser. Izvestiya VTI, No. 3, 1951.

2. V. P. Blyudov. Condensing Installations of Steam Turbines. GEI, 1951.

3. V. I. Buchanin. On single- and double-pass surface condensers in steam-turbine units. TsKTI, book 7, “Condensers of Steam Turbines,” 1947.

4. A. S. Feinshtein. On a method for the rapid calculation of surface condensers. Kotloturbostroenie, No. 1, 1949.

5. Heat Engineering Handbook, vol. 1, GEI, 1957.

6. E. N. Shadrin. On the question of selecting the type of surface condenser. Izvestiya TPI, vol. 70, 1950.

1. In fact, usually \(\Delta H_1^v<\Delta H_2\). Taking \(\Delta H_1^v=\Delta H_2\) somewhat lowers the upper boundary. ↩