ON THE QUESTION OF SELECTING THE HEATING ZONE IN POWER-PLANT CONDENSERS AT A SPECIFIED VACUUM

I. N. Butakov

As is known, the temperature of the steam at the inlet to the condenser is

\[ t_k = t_1 + (\delta t) + (t_k - t_2), \]

where \(t_1\) is the temperature of the circulating water at the inlet;

\[ \delta t = t_2 - t_1 \]

is the temperature difference of this water at the outlet \(t_2\) and at the inlet, while the difference \((t_k - t_2)\) may be selected from Gebhardt’s table. This table is given in graphical form in Fig. 1. From Fig. 1 it is seen that \((t_k - t_2)\) increases as \(t_1\) decreases. Gebhardt’s formula for the overall heat-transfer coefficient in condensers,

\[ k = 460 \sqrt{t_1 + 17.8}\cdot \sqrt{c_w}, \]

shows that \(k\) decreases as \(t_1\) decreases; therefore the difference \((t_k - t_2)\) must increase in order to provide the cooling action of the water to the same—

Fig. 1

of the cold end as the thermal resistance of the condenser tubes increases. The overall heat-transfer coefficient according to a formula that reflects reality only roughly (Gey and Winstenley),

$k = 935 \sqrt{c_w}\cdot \sqrt[4]{t_b + 17.8}$,

depends on $t_b$, the mean temperature of the water in the condenser tubes, decreasing with this quantity. The same is indicated by Eagle and Ferguson’s curves for $\alpha_b$—the coefficient of heat transfer from the tube wall to the water [1]. The mean temperature of the water in the condenser tubes decreases together with the decrease in $t_1$. From what has been said it follows that a decrease in $t_1$ must inevitably increase the thermal resistance of the condenser tubes and necessitate an increase in $(t_k - t_2)$ at a specified vacuum. This is also confirmed by the VTI graph, Fig. 2, given in the Guidelines for the Operation of Spray Ponds (Gey, Moscow,

Fig. 2

1944, p. 44), and here this “temperature difference” proves to be even more substantial than in Gebhardt, since for steam turbines of medium and large capacities the specific steam load is usually taken to be from 50 to 60 kg of steam per hour per 1 sq. m of cooling surface [2].

The determination of the value $\delta t = t_2 - t_1$ presents much greater difficulties, as will be seen from what follows. The essential question here is how many passes the condenser should have: the value $\delta t$ increases with the number $z$ of passes. This, in turn, is connected with the cooling ratio, the expenditure of power for supplying circulating water, and also with the capital investments in the condensing installation.

The amount of circulating water supplied to the condenser is determined by the well-known formula, in kg/hour,

\[ W=\frac{i_2-t_k}{t_2-t_1}\cdot D=\frac{\Delta i}{\delta t}\cdot D. \tag{1} \]

From formula (1) it is seen that the flow rate of circulating water, while depending on \(\Delta i=i_2-t_k\) and on \(D\)—the hourly steam flow rate in kg, as a function of vacuum—depends at the same time also on \(\delta t\). For each specified vacuum there must be its own most advantageous temperature difference \(\delta t\), to the determination of which we now turn.

The resistance of the condenser on the water side may be determined by the customary formula

\[ H_k=z\left(0.031\cdot \frac{L}{d_1}\cdot \frac{c_w^2}{2g}\beta+1.4\cdot \frac{c_w^2}{2g}\right). \tag{2} \]

Here the first term characterizes the frictional resistances inside the condenser tubes, and the second—the inlet and outlet resistances; \(\beta\) is a coefficient taking into account the influence of the mean temperature and velocity of the water. From the equation of continuity of motion we determine

\[ c_w=\frac{W}{9\cdot 10^5\cdot \pi d_1^2 n_1} =\frac{D\cdot \Delta i}{9\cdot 10^5\pi\cdot d_1^2\cdot n_1\cdot \delta t}, \]

where \(n_1\) is the number of tubes in one pass. We shall take \(c_w=\text{const}\) in all variants of the calculations, in order to come close to keeping the heat-transfer coefficient unchanged.

Then

\[ n_1=\frac{n}{z}=\frac{D\cdot \Delta i}{9\cdot 10^5\cdot \pi d_1^2\cdot c_w\cdot \delta t}. \tag{3} \]

Since the cooling surface of the condenser is

\[ F=\pi\cdot d_1\cdot L\cdot n_1\cdot z =\frac{W}{k}\cdot \ln \frac{t_s-t_1}{t_s-t_2}, \]

then

\[ n_1=\frac{F}{\pi d_1 Lz} =\frac{D\cdot \Delta i}{\delta t\cdot k\cdot \pi d_1\cdot L\cdot z} \ln \frac{\Delta t_1}{\Delta t_1-\delta t}, \]

where \(t_s\) is the temperature of saturated steam at condenser pressure \(p_k\). Equating (3) and (4), we have

\[ \frac{D\cdot \Delta i}{9\cdot 10^5\pi d_1^2 c_w\cdot \delta t} = \frac{D\cdot \Delta i}{\delta t\cdot k\cdot \pi d_1 Lz} \ln \frac{\Delta t_1}{\Delta t_1-\delta t}, \]

whence, after the necessary cancellations, we determine

\[ Lz=\frac{9\cdot 10^5\cdot d_1\cdot c_w}{k} \cdot \ln \frac{\Delta t_1}{\Delta t_1-\delta t} =\beta_1\cdot \ln \frac{\Delta t_1}{\Delta t_1-\delta t}, \]

where

\[ \rho_1=\frac{9\cdot 10^5\cdot d_1\cdot c_w}{k}, \]

and, consequently,

\[ z=\frac{\rho_1}{L}\ln\frac{\Delta t_1}{\Delta t_1-\delta t}. \]

Substituting the values of \(z\) and \(Lz\) into formula (2), we obtain

\[ H_k=\frac{c_w^3}{2gk}\cdot \rho_2\cdot \ln\frac{\Delta t_1}{\Delta t_1-\delta t}, \]

where

\[ \rho_2=9\cdot 10^5\left(0.031\cdot \beta+\frac{1.4\cdot d_1}{L}\right). \]

The number of kWh expended per year in overcoming the resistance of the condenser will be

\[ \mathcal{E}_1=A\cdot N_1=\frac{W\cdot H_k\cdot A}{3600\cdot 75\cdot 1.36\cdot \eta_k}= \]

\[ =\frac{\Delta i\,\rho_2\,c_w^3\cdot \mathcal{E}\cdot d_e}{3600\cdot 75\cdot 1.36\cdot 2g\cdot \eta_k\cdot k\cdot \delta t}\cdot \ln\frac{\Delta t_1}{\Delta t_1-\delta t}, \tag{5} \]

where \(\mathcal{E}\) is the annual output of kWh by the station, \(\eta_k\) is the efficiency of the pumping unit, and \(A\) is the equipment utilization factor, h/year. The conclusion just made belongs to the former associate professor of the Department of Thermal Power Installations of the Tomsk Polytechnic Institute, N. I. Anastasieva.

But the circulating pumps overcome not only the resistances of the condenser as such, but also those of the pipelines connected to the condenser and to the circulating pumps, including the loss of head when the water enters the water chamber of the condenser. The power in kW expended in overcoming all these latter resistances is determined by

\[ N_2=\frac{W_1H}{3600\cdot 75\cdot 1.36\cdot \eta_k}, \]

where \(H\) represents the total head overcome by the circulating pumps, less the above-mentioned condenser resistances. Therefore the annual expenditure of kWh under this item could be estimated as

\[ \mathcal{E}_2=\frac{W\cdot H\cdot A}{3600\cdot 75\cdot 1.36\cdot \eta_k} =\frac{\Delta i\cdot H\cdot \mathcal{E}\cdot d_e}{3600\cdot 75\cdot 1.36\cdot \eta_k\cdot \delta t}. \tag{6} \]

The circulating water must be supplied from the water-supply source to the condensers, for which it is often necessary to install a central pumping station, which at the same time usually supplies the water required for cooling the generators and oil coolers in an amount of from 4 to 8% of the circulating-water flow for the condensers as such. Therefore the number of annual kWh expended on pumping the water will be

\[ \mathcal{E}_2=\frac{1.08\cdot W\cdot H\cdot A}{3600\cdot 75\cdot 1.36\cdot \eta_k}, \]

where \(H\), in m water column, is the total above-mentioned head to be overcome, depending both on the elevation height of the water supply from the water-supply source—

to the condenser, as well as on the frictional and local resistances of the water conduit. Therefore, if \(H_1\) is the geodetic head and \(H_2\) is the head due to the resistance of the water conduit, then

\[ \mathcal{E}_2 = \frac{1.08\, W (H_1+H_2) A}{3600 \cdot 75 \cdot 1.36 \cdot \eta_n} = \frac{1.08\, H_1 \cdot A \cdot W}{3600 \cdot 75 \cdot 1.36 \cdot \eta_n} + \frac{1.08 \cdot A \cdot \lambda \cdot L_e \cdot c_{\mathrm{vod}}^{\,2} \cdot W}{3600 \cdot 75 \cdot 1.36 \cdot \eta_n \cdot 2g \cdot d}, \tag{7} \]

where \(L_e\) is the equivalent length of the water conduit in m, \(d\) is the diameter of the water conduit in m, and \(c_{\mathrm{vod}}\) is the velocity of the water in it in m/s; roughly, for the sake of simplicity, \(\lambda = 0.03\) is adopted. Assuming in calculating all variants that \(c_{\mathrm{vod}} = \mathrm{const}\), we determine from the continuity equation

\[ d=\sqrt{\frac{4 \cdot 1.08 \cdot W}{2 \cdot 3600 \cdot 1000 \cdot \pi \cdot c_{\mathrm{vod}}}}, \]

taking into account that, in order to ensure the reliability of the water supply, two lines of the water conduit will be in operation. Substituting this value into expression (7), we obtain

\[ \mathcal{E}_2 = \frac{1.08}{3600 \cdot 75 \cdot 1.36 \cdot \eta_n} \left[ \frac{\Delta i \cdot d_e \cdot \mathcal{E} \cdot H_1}{\delta t} + \frac{3.5 \cdot A^{0.5} L_e \cdot c_{\mathrm{vod}}^{2.5}(\Delta i \cdot d_e \cdot \mathcal{E})^{0.5}}{(\delta t)^{0.5}} \right]. \tag{8} \]

Thus, the total annual cost in rubles caused by water pumping, under the fuel-cost item, will be expressed as

\[ R_{\mathrm{tsv}} = \frac{860(\mathcal{E}_1+\mathcal{E}_2)}{\eta_k \cdot Q_r^{\,n}} \left(1+\frac{\sum \Delta}{100}\right) \frac{R_t}{1000}, \tag{9} \]

where \(R_t\) is the cost of one ton of fuel in rubles, and \(\left(1+\dfrac{\sum \Delta}{100}\right)\) is the plant coefficient.

When considering the variants, it is necessary to take into account the monetary costs specifically under the fuel item, since the other operating costs may be regarded as unchanged.

We now proceed to estimate the costs depending on fixed assets. Since

\[ F=\frac{W}{k}\ln\frac{t_s-t_1}{t_s-t_2} = \frac{\Delta i \cdot D}{k \cdot \delta t} \ln\frac{\Delta t_1}{\Delta t_1-\delta t}, \]

then \(F=f(\delta t)\). The annual charges under the fixed-assets item for the condenser will be

\[ R_k^{\,k}=\frac{F \cdot u}{100}\left(\frac{S_k}{F}\right), \]

where \(\dfrac{S_k}{F}\) is the cost per sq. m of condenser cooling surface, rubles/m\(^2\), and

\(u\) is the charges in percent, so that

\[ R_k^{\,k} = \frac{\Delta i \cdot D \cdot u}{100 \cdot k \cdot \delta t} \left(\frac{S_k}{F}\right) \ln\frac{\Delta t_1}{\Delta t_1-\delta t} = \]

\[ = \frac{\Delta i \cdot E \cdot u \cdot d_e}{100 \cdot A \cdot k \cdot \delta t} \left(\frac{S_k}{F}\right)\ln \frac{\Delta t_1}{\Delta t_1-\delta t}. \tag{10} \]

On the other hand, for the water conduit we have the cost of one linear meter laid, in rubles,

\[ S_{vod}=a+b\cdot d=a+b'\sqrt{\frac{D\cdot \Delta i}{c_{vod}\cdot \delta t}}, \]

where \(a\), \(b\), and \(b'\) are quantities that may be taken as constant within known limits of variation of \(d\), and which take into account the laying of a double water conduit. The sum of the annual deductions for the water conduit is

\[ R_k^{vod}=\frac{S_{vod}\cdot u\cdot L_0}{100} =\frac{u\cdot L_0}{100}\left(a+b'\sqrt{\frac{\Delta i\cdot d_e\cdot E}{c_{vod}\cdot \delta t\cdot A}}\right), \]

where \(L_0\) is the actual length of the water conduit in m. The cost of the pumping station, \(S_{nas}=f\cdot N_{nas}\), is taken as proportional to its capacity \(N_{nas}\), where \(N_{nas}\) is in hp, and \(f\) is a numerical coefficient representing the cost of machines per 1 hp with complete installation, foundations, and also the cost of the premises.

\[ R_k^{nas}=\frac{S_{nas}u}{100} =\frac{1.36\cdot u\cdot f}{100}(N_1+N_2)= \]

\[ =\frac{1.36\cdot u\cdot f}{100}\left[ \frac{\Delta i\cdot \rho_2 c_w^3\cdot E\cdot d_e}{3600\cdot 75\cdot 2g\cdot \eta_n\cdot k\cdot A\cdot \delta t} \ln \frac{\Delta t_1}{\Delta t_1-\delta t} +\right. \]

\[ \left. +\frac{1.08\cdot H_1\cdot \Delta i\cdot E\cdot d_e}{3600\cdot 75\cdot \eta_n\cdot A\cdot \delta t} +\frac{1.08\cdot 3.5\cdot L_e\cdot c_{vod}^{2.5}(\Delta i\cdot d_e\cdot E)^{0.5}} {3600\cdot 75\cdot \eta_n\cdot (A\cdot \delta t)^{0.5}} \right]. \tag{11} \]

Thus, the sum of the annual monetary expenditures connected with supplying water for the station, as a function of \((\delta t)\), will be expressed as

\[ R=R_{tsv}+R_k^k+R_k^{vod}+R_k^{nas}. \tag{12} \]

To determine the temperature difference \(\delta t=t_2-t_1\) at which the annual monetary expenditures will be minimal, we take the first derivative

\[ \frac{dR}{d(\delta t)} \]

and set it equal to zero. Let us denote

\[ C_1=\frac{860}{\eta_k\cdot Q_p^n}\left(1+\frac{\Sigma \Delta}{100}\right)\cdot \frac{R_t}{1000}; \]

\[ C_2=\frac{\Delta i\cdot \rho_2\cdot c_w\cdot E\cdot d_e}{3600\cdot 102\cdot 2g\cdot \eta_n\cdot k}; \]

\[ C_3=\frac{1.08\cdot H_1\cdot \Delta i\cdot d_e\cdot E}{3600\cdot 102\cdot \eta_n}; \]

\[ C_4=\frac{1.08\cdot 3.5\cdot A^{0.5}L_e\cdot c_{vod}^{2.5}(\Delta i\cdot d_e\cdot E)^{0.5}} {3600\cdot 102\cdot \eta_n}; \]

\[ C_5=\frac{\Delta i \cdot E \cdot u \cdot d_e}{100 \cdot A \cdot k}\left(\frac{S_k}{F}\right); \]

\[ C_6=\frac{u \cdot L_0}{100}\cdot a; \]

\[ C_7=\frac{u \cdot L_0}{100}\cdot b' \cdot \sqrt{\frac{\Delta i \cdot d_e \cdot E}{c_{\mathrm{water}}\cdot A}}; \]

\[ C_8=\frac{1.36\cdot u \cdot f}{100}\cdot \frac{C_2}{A}; \]

\[ C_9=\frac{1.36\cdot u \cdot f}{100}\cdot \frac{C_3}{A};\qquad C_{10}=\frac{1.36\cdot u \cdot f}{100}\cdot \frac{C_4}{A}. \]

Introducing these designations, from (12) we obtain

\[ R=C_6+\frac{1}{(\delta t)^{0.5}}\left[C_7+C_{10}+C_1\cdot C_4\right]+ \frac{1}{\delta t}\left[C_9+C_1 C_3\right]+\left[C_1 C_2+C_5+C_8\right]\cdot \frac{1}{\delta t}\ln\frac{\Delta t_1}{\Delta t_1-\delta t}. \tag{13} \]

We shall further denote

\[ E_1=C_7+C_{10}+C_1\cdot C_4; \]

\[ E_2=C_9+C_1\cdot C_3; \]

\[ E_3=C_1\cdot C_2+C_5+C_8. \]

Then

\[ R=C_6+E_1\frac{1}{(\delta t)^{0.5}}+E_2\cdot\frac{1}{\delta t} +E_3\cdot\frac{1}{\delta t}\ln\frac{\Delta t_1}{\Delta t_1-\delta t}. \tag{14} \]

Carrying out the differentiation and equating the first derivative to zero, we obtain

\[ -\frac{1}{\delta t}\ln\left(1-\frac{\delta t}{\Delta t_1}\right) =\frac{1}{\Delta t_1-\delta t} -\frac{0.5\cdot E_1}{E_3(\delta t)^{0.5}} -\frac{E_2}{E_3(\delta t)}. \tag{15} \]

In order to get rid of \(\ln\left(1-\frac{\delta t}{\Delta t_1}\right)\), we expand in a series

\[ -\frac{1}{\delta t}\ln\left(1-\frac{\delta t}{\Delta t_1}\right) =\frac{1}{\Delta t_1}+0.5\frac{\delta t}{(\Delta t_1)^2} +\frac{1}{3}\cdot \frac{(\delta t)^2}{(\Delta t_1)^3}+\cdots; \]

It is possible to stop at the second term of the expansion, whereby we obtain

\[ \frac{1}{\Delta t_1}+0.5\frac{\delta t}{(\Delta t_1)^2} =\frac{1}{\Delta t_1-\delta t} -\frac{E_2}{E_3(\delta t)} -\frac{0.5\cdot E_1}{E_3(\delta t)^{0.5}}. \tag{16} \]

Bearing in mind that \((\delta t)^{0.5}=1+0.5\ln(\delta t)+0.5\cdot0.25[\ln(\delta t)]\) and that \((\delta t)\) most often varies within the limits \(5^\circ\)–\(12^\circ\mathrm{C}\), we find that it may be taken approximately as \((\delta t)^{0.5}\simeq 3\). Then, after reducing (16) to a common denominator and collecting like terms, we obtain

\[ (\delta t)^3=-\Delta t_1\left[1+0.33\cdot\frac{E_1}{E_3}(\Delta t_1)\right](\delta t)^2- \]

\[ -(\Delta t_1)^2\left[\frac{2E_2}{E_3}-\frac{0.33\,E_1}{E_3}(\Delta t_1)\right]\delta t +\frac{2E_2}{E_3}(\Delta t_1)^3, \tag{17} \]

or

\[ (\delta t)^3=M_1-M_2(\delta t)-M_3(\delta t)^2 . \tag{18} \]

Equation (18) is conveniently solved graphically. Let us denote \(y_1=(\delta t)^3\) and \(y_2=M_1-M_2(\delta t)-M_3(\delta t)^2\). The curve \(y_1=(\delta t)^3\) does not depend on any local conditions and can be constructed once and for all. As for the curve of the second order, \(y_2=M_1-M_2(\delta t)-M_3(\delta t)^2\), it is a parabola whose ordinates decrease as \((\delta t)\) increases, and whose position on the drawing depends on the values of the coefficients \(M_1\), \(M_2\), and \(M_3\). The magnitudes of these coefficients are determined by particular concrete conditions. The intersection of the cubic curve \(y_1\) with the parabola \(y_2\) determines the point whose abscissa gives us the most advantageous temperature difference \((\delta t)\).

Equation (17) can be represented in the following form:

\[ (\delta t)^3 = -\Delta t_1(\delta t)^2 +\frac{2E_2}{E_3}\left[(\Delta t_1)^3-(\Delta t_1)^2\delta t\right] + \frac{0.33\,E_1}{E_3}\left[(\Delta t_1)^3\delta t-(\Delta t_1)^2(\delta t)^2\right]. \tag{19} \]

The quantities \(\left[(\Delta t_1)^3-(\Delta t_1)^2\delta t\right]\) and \(\left[(\Delta t_1)^3\delta t-(\Delta t_1)^2(\delta t)^2\right]\) are always positive, and therefore the ordinates of the parabola \(y_2\) increase together with \(\dfrac{2E_2}{E_3}\) and \(\dfrac{0.33\,E_1}{E_3}\), while \((\delta t)\) will also increase. This increase depends substantially, as we see, on the ratios \(\dfrac{E_2}{E_3}\) and \(\dfrac{E_1}{E_3}\). We have for

\[ \frac{E_2}{E_3} = \frac{21\,H_1\,k}{\rho_2\,c_w^3} \cdot \frac{ 86\,\dfrac{A}{u}\left(1+\dfrac{\Sigma\Delta}{100}\right)R_t +1.36\,f\,\eta_k\,Q_p^n }{ 86\,\dfrac{A}{u}\left(1+\dfrac{\Sigma\Delta}{100}\right)R_t + \left[ 7\cdot 10^6\left(\dfrac{S_k}{F}\right)\cdot \frac{\eta_k}{\rho_2\,c_w^3} +1.36\,f \right]\eta_k Q_p^n } = \]

\[ = \frac{21\,H_1\,k}{\rho_2\,c_w^3} \cdot \frac{ 1+ \dfrac{ 1.36\,f\,\eta_k\,Q_p^n\cdot\dfrac{u}{A} }{ 86\left(1+\dfrac{\Sigma\Delta}{100}\right)R_t } }{ 1+ \dfrac{ 1.36\,f\,\eta_k\,Q_p^n\cdot\dfrac{u}{A} }{ 86\left(1+\dfrac{\Sigma\Delta}{100}\right)R_t } + \dfrac{ 7\cdot 10^6\left(\dfrac{S_k}{F}\right)\eta_k^2 Q_p^n\cdot\dfrac{u}{A} }{ \rho_2\,c_w^3\,86\left(1+\dfrac{\Sigma\Delta}{100}\right)R_t } }, \tag{20} \]

which can be represented in the form

\[ \frac{E_2}{E_3}=\frac{Z_1}{Z_2+Z_3}. \tag{20a} \]

The values \(Z_1\), \(Z_2\), and \(Z_3\) can be determined from the nomogram in Fig. 3, constructed by the heat-engineering student of the Tomsk Polytechnic Institute E. N. Shadrin, now already an engineer, with \(\eta_k=0.68\) adopted.

Nomogram No. 1
for determining the water-heating zone \(\delta t\)
in condensers

\[ \frac{F_{\mathrm{з}}}{F_{\mathrm{в}}} = \frac{z}{z_{1}\cdot z_{2}} \]

Fig. 3

And further we have, for

\[ \frac{E_1}{E_3} = \frac{k \cdot L_0}{(\Delta i \cdot d_{\mathrm{e}} \cdot N_{\mathrm{e}})^{0.5}} \cdot \frac{ 3.7 \cdot \alpha \cdot c_{\mathrm{wod}}^{2.5} + \dfrac{ \eta_{\mathrm{n}} \cdot b' \cdot \dfrac{36.8}{c_{\mathrm{wod}}^{2.5}} }{ \dfrac{86}{\eta_{\mathrm{k}} \cdot Q_p^{\mathrm{n}}} \left(1+\dfrac{\sum \Delta}{100}\right) R_t \cdot \dfrac{A}{u} +1.36 f } }{ 0.051 \cdot \rho_2 \cdot c_w^3 + \dfrac{ 36.8 \cdot \eta_{\mathrm{n}} \left(s_{\mathrm{k}}/F\right) }{ \dfrac{86}{\eta_{\mathrm{k}} \cdot Q_p^{\mathrm{n}}} \left(1+\dfrac{\sum \Delta}{100}\right) R_t \cdot \dfrac{A}{u} +1.36 f } }, \tag{21} \]

which can be represented in the following form for \(\eta_{\mathrm{n}}=0.68\)

\[ \frac{E_1}{E_3} = \frac{k \cdot L_0}{(\Delta i \cdot d_{\mathrm{e}} \cdot N_{\mathrm{e}})^{0.5}} \cdot \frac{72.5 \cdot \alpha \cdot c_{\mathrm{wod}}^{2.5}}{\rho_2 \cdot c_w^3} \cdot \frac{ \left[ Z_2+ \dfrac{ 25 \cdot 10^4 \cdot b' \cdot \eta_{\mathrm{k}} \cdot Q_p^{\mathrm{n}} \cdot u/A }{ 3.7 \alpha c_{\mathrm{wod}}^3 \left(1+\dfrac{\sum \Delta}{100}\right) R_t } \right] }{ Z_2+Z_3 }, \tag{22} \]

or, taking

\[ Z_4= \frac{ 6.76 \cdot 10^4 \cdot b' \cdot \eta_{\mathrm{k}} \cdot Q_p^{\mathrm{n}} u/A }{ \alpha \cdot c_{\mathrm{wod}}^3 \left(1+\dfrac{\sum \Delta}{100}\right) R_t }, \]

we have

\[ \frac{E_1}{E_3} = Z_5 \cdot \frac{Z_2+Z_4}{Z_2+Z_3}, \qquad \text{where} \qquad Z_5= \frac{k \cdot L_0}{(\Delta i \cdot d_{\mathrm{e}} \cdot N_{\mathrm{e}})^{0.5}} \cdot \frac{72.5 \cdot \alpha \cdot c_{\mathrm{wod}}^{2.5}}{\rho_2 \cdot c_w^3}. \]

To determine the values of \(Z_4\) and \(Z_5\), one may use the nomograms in Figs. 4 and 5, also constructed by E. N. Shadrin.

Fig. 4

From expressions (20) and (21) it is evident that the temperature difference \((\delta t)\) in the condenser will decrease together with a decrease in the leveling height—

\(H_1\), the length of the water conduit \(L_0\), with increasing plant capacity \(N_{\mathrm{e}}\). It is precisely these circumstances that provide grounds in the USA for selecting high cooling ratios \((m = 100\text{--}120)\) at small values of \((\delta t)\) and single-pass condensers, since many American plants, at large capacities,

Fig. 5.

are built directly on the shore of powerful water-supply sources and have small values of \(H_1\) and \(L_0\).

The regime of the rivers of the USSR is characterized by strong fluctuations in water levels depending on the season and in different years as a result of uneven-

nonuniformity of precipitation over the years. Therefore, in order to avoid flooding, power plants have to be built on steep banks, at a considerable distance from rivers, when \(H_1\) and \(L_0\) acquire values large in comparison with those in the USA, as a result of which we do not have to orient ourselves toward small differences \((\delta t)\). In addition, energy policy in the USSR is directed toward CHP plants with their reduced capacities of the condensing sections, and also toward the construction of stations of small and medium capacity, which should likewise act in the direction of increasing \((\delta t)\). Further, from expression (20) it follows that with an increase in \(Q_p^n\), \(R_t\), \(\eta_k\), and \(\dfrac{A}{u}\), \(E_2/E_3\) will decrease, and consequently \((\delta t)\) will also decrease. In the USA, power plants in the majority of cases operate on high-calorific fuel.

Statistical data show that at American stations, many of which, like Soviet ones, are at the modern level of operation and operate with high initial steam parameters, the number of hours of utilization is considerably smaller than in the USSR. Depreciation charges on fixed assets \(u\) are greater in the USA, since interest on capital is also included there, which we do not have. All the indicated circumstances existing in the USA should have induced the Americans to use large cooling ratios in condensers. In the USSR, the orientation toward the use of low-grade fuels and the large number of hours of utilization, with a comparatively small percentage of depreciation charges, create conditions for using large differences \((\delta t)\) in the condenser.

Equation (17) of the parabola mentioned above may be represented in the following form

\[ y_2 = (\Delta t_1)^2 \left\{ \frac{2E_2}{E_3}\left[(\Delta t_1)-(\delta t)\right] + \frac{0.33\,E_1}{E_3}\left[(\Delta t_1)(\delta t)-(\delta t)^2\right] - \frac{(\delta t)^2}{\Delta t_1} \right\}. \tag{23} \]

The greater \(\Delta t_1 = t_s - t_1\), the greater the ordinates \(y_2\), and the greater \(\delta t\). At a specified vacuum, and hence also at a specified temperature \(t_s\), increasing \(t_1\) reduces the ordinates \(y_2\), and reduces the difference \((\delta t)\). The mean annual temperature \(t_1\) in the USA is higher than in the USSR; therefore the difference \(t_s - t_1 = \Delta t_1\) is greater in the USSR, and consequently the difference \((\delta t)\) in our country must also be greater.

Using the method set forth above, let us determine the most advantageous temperature difference \(\delta t = t_2 - t_1\), for varying vacuums, taking as the initial data for a particular example the following: \(k = 3000\) cal/kW.m. °C hour; \(H_1 = 10\) m; \(L_e = 1100\) m; \(A = 5000\) hours/year; \(\eta_n = 0.68\); \(u = 10\%\); \(S_k/F = 120\) rub/kW.m; \(Q_n^p = 5000\) cal/kg; \(R_t = 30\) rub/t; \(\sum A = 20\%\); \(p_1 = 29\) ata; \(t_{\mathrm{per}} = 400^\circ\mathrm{C}\); \(t_1 = 10^\circ\mathrm{C}\); \(c_w = 2\) m/sec; \(c_{\mathrm{vod}} = 1.5\) m/sec; \(E = 250 \cdot 10^6\) kWh; \(d_1 = 19\) mm; \(L = 6.0\) m; \(\beta = 1\); \(f = 200\) rub/h.p.; \(a = 25\); \(b = 400\).

Below are given the results of calculations carried out by the heat-engineering student of the Tomsk Polytechnic Institute, Comrade Kudoyarov, now already an engineer, who is an active participant in this work of ours:

\[ p_2 = 9 \cdot 10^5\left(0.031 + \frac{1.4 \cdot 0.19}{6}\right)=31820; \]

\[ C_1 = \frac{860 \cdot 1.2 \cdot 30}{\eta_k \cdot 5000 \cdot 1000} = \frac{0.00619}{\eta_k}; \]

\[ C_2 = \frac{31820 \cdot 8 \cdot 250 \cdot 10^6 \cdot \Delta i \cdot d_e} {3600 \cdot 102 \cdot 2 \cdot 9.81 \cdot 0.68 \cdot 3000} = 4330 \cdot \Delta i \cdot d_e; \]

\[ C_3 = \frac{1.08 \cdot 10 \cdot 250 \cdot 10^6 \cdot \Delta i \cdot d_e} {3600 \cdot 102 \cdot 0.68} = 10800 \cdot \Delta i \cdot d_e; \]

\[ C_4=\frac{1.08\cdot 3.5\cdot 15800\cdot 1100\cdot 70.6\cdot 2.76\sqrt{\Delta i.d_z}}{3600\cdot 102\cdot 0.68} =51200\sqrt{\Delta i.d_z}; \]

\[ C_5=\frac{250\cdot 10^6\cdot 120\cdot \Delta i.d_z}{100\cdot 5000\cdot 3000} =200\cdot \Delta i.d_z; \]

\[ C_6=\frac{10\cdot 1000\cdot 25}{100}=2500; \]

\[ C_7=\frac{10\cdot 1000\cdot 400}{100} \sqrt{\frac{250\cdot 10^6\cdot 4\cdot 1.08\cdot \Delta i.d_z}{3600\cdot 2\cdot 1000\cdot 5000\cdot \pi\cdot 1.5}} =3020\sqrt{\Delta i.d_z}; \]

\[ C_8=\frac{1.36\cdot 10\cdot 200\cdot 4330.\Delta i.d_z}{100\cdot 5000} =23.6\Delta i.d_z; \]

\[ C_9=\frac{1.36\cdot 10\cdot 200\cdot 10800.\Delta i.d_z}{100\cdot 5000} =58.8\Delta i.d_z; \]

\[ C_{10}=\frac{1.36\cdot 10\cdot 200\cdot 51200\cdot \sqrt{\Delta i.d_z}}{100\cdot 5000} =278.5\sqrt{\Delta i.d_z}; \]

\[ E_1=(3298.5+317/\eta_k)\sqrt{\Delta i.d_z}; \]

\[ E_2=(58.8+67/\eta_k)\Delta i.d_z; \]

\[ E_3=(223.6+27/\eta_k)\Delta i.d_z. \]

The product \(\Delta i.d_z\) varies together with the change in vacuum. Taking \(\eta_{ku}=0.85\); \(\eta_{pop}=0.94\); \(\eta_{oi}=0.83\) for the conditions of normal vacuum \(p_2=-0.04\) ata, and also a change of \(\eta_{oi}\) of \(0.3\%\) for a change in moisture content of \(1\%\); \(\eta_{gen}=0.96\); \(\eta_{mech}=0.995\); \(t_s-t_k=1\), we obtain, rounded:

The equation

\[ (\delta t)^3=M_1-M_2(\delta t)-M_3(\delta t)^2 \]

can be represented in the following form

\[ (\delta t)^2(\delta t+M_3)=M_1-M_2(\delta t), \]

whence

\[ \delta t=\frac{M_1/\delta t-M_2}{\delta t+M_3}, \]

so that \(\delta t\) can also be determined quite simply by selection from the last equality for given \(M_1, M_2\), and \(M_3\). As a result of the selection

and the optimal values \((\delta t)\) for different vacuums were determined; these are given at the end of the above table; Fig. 6 gives the graphical dependence of \((\delta t)\) on the vacuum for the particular example analyzed above.

Figure 6 shows the changes in \((\delta t)\) for the case when the geodetic head \(H_1\) and the equivalent distance \(L_\mathrm{e}\) of the stations from the water-supply source have doubled, reaching values at which the use of once-through water supply already becomes unprofitable. On the other hand, Fig. 6 shows the case when \(H_1\) is reduced by half, and \(L_\mathrm{e}=100\ \text{m}\).

Fig. 6

Fig. 7

From the graph it is evident that the difference in the values of \((\delta t)\) for these two cases at \(p_2=0.03\ \text{ata}\) will be \(\sim 2.5^\circ\text{C}\), and at \(p_2=0.04\ \text{ata}\) even \(\sim 3.5^\circ\text{C}\).

Figure 7 establishes an interesting point: at small geodetic heads \(H_1\) and short distances \(L_\mathrm{e}\), the differences \((t_k-t_2)=t_k-(\delta t)-t_1\), obtained as a result of the present study and according to Gebhardt, turn out to be nearly coincident, which also occurs at deep vacuums of the order of \(0.03\)–\(0.04\ \text{ata}\), both when the temperature of the circulating water at the inlet is \(t_1=10^\circ\text{C}\) and when \(t_1=15^\circ\text{C}\). The situation is different when the geodetic head increases \((H_1>5\ \text{m})\) and the distance \(L_\mathrm{e}\) increases: the differences \((t_k-t_2)\) become insufficient in comparison with the empirically established norms (according to Gebhardt), particularly in the region of deep vacuums. The lower curve in Fig. 7 clearly shows the impracticability of once-through water supply at \(H_1=20\ \text{m}\) and \(L_\mathrm{e}=2200\ \text{m}\), despite the water temperature \(t_1=10^\circ\text{C}\); in this case the differences \((t_k-t_2)\) come into sharp contradiction with the data of experience and are clearly unacceptable as being very small.

From all that has been said, a number of conclusions follow:

1. The attainment of deep vacuums under the conditions of the USSR is in many cases difficult because of the characteristics of our rivers. Wide

the use of CHPPs significantly reduces their condensing capacity, which, as we have seen, adversely affects the possibility of applying deeper vacuums. The decisions of the XVIII Party Congress concerning the construction of medium- and low-capacity thermal power plants also act in the same direction.

1. The use of low-grade fuels, and the higher values of \(A/u\) that occur in the USSR, tend toward an increase of \((\delta t)\) and the use of less deep vacuums at our power stations.

2. The study has established a strong dependence of \((\delta t)\) on \((\Delta t_1)\), so that a transition from \(t_1 = 15^\circ\mathrm{C}\) to \(t_1 = 10^\circ\mathrm{C}\), at small geodetic heights \(H_1\) and distances \(L_z\), does not produce a substantial effect with respect to deepening the vacuum, if one takes into account that \((t_k - t_2)\) increases in this case.

3. It is advisable to make use of the gift of our severe climate—the low temperatures of circulating water in winter (\(1\text{–}2^\circ\mathrm{C}\))—by installing utilization plants of the Claude and Boucherot type, as described in the author’s article “The Use of the Heat of Circulating Water at Power Stations for the Production of Electric Power” [4].

4. In addition to the advantages of such a utilization plant indicated in the above-mentioned article, one may note one further possibility for saving fuel, namely by deepening the vacuum of the main turbines at \(t_1 = 15^\circ\mathrm{C}\). In a number of cases the utilization plant may be located at a comparatively small distance from the main turbines, with a sometimes considerable reduction of the geodetic height, as a result of which it will become advisable to use single-pass condensers at the main turbines, with a small difference between the water temperatures at outlet and inlet. In this case, at a vacuum \(p_2 = 0.035\) ata, it is possible to have a steam temperature in the condenser of about \(27^\circ\mathrm{C}\). The circulating water at the outlet from the condenser will then have a temperature \(t_2 = 21\text{–}22^\circ\mathrm{C}\). The cooling ratio will be about 90, which gives a 1.5-fold increase in the quantity of circulating water as compared with the case examined in the article cited above. The temperature difference in the evaporator, however, will decrease from \(10^\circ\mathrm{C}\) to \(7^\circ\mathrm{C}\), i.e., by a factor of 1.43, so that the quantity of steam obtained in the evaporator and the capacity of the utilization plant will remain almost unchanged. In the main turbines, however, the capacity increases as a result of the deepening of the vacuum. In this case one will have to face the possible necessity of abandoning single-casing turbines in the low-pressure part, which will inevitably make the turbine-generators more expensive and more complicated.

5. The decisions of the Technical Council of the NKES, in the part concerning the transition to a deep vacuum with a cooling ratio of 50–60, at a temperature at the inlet to the condenser \(t_1 = 10^\circ\mathrm{C}\), should be regarded as requiring reconsideration. The reduction of geodetic height recommended by the Technical Council of the MES (Elektricheskie stantsii, 1945, No. 4–5) by lowering the floor level of the condensing room is hardly advisable, since, besides the difficulty in a number of cases of justifying this economically, it will deprive this room of natural light; water may enter the basement during heavy rains, snowmelt, and floods; and, if the recommended bringing of the main building closer to the water source is adopted, difficulties will arise with the laying of railway tracks inside the machine hall.

References

1. Anatoliev F. A.—Calculation in the auxiliary devices of steam-power installations, ONTI, 1936, p. 26, and also Fig. 21.

2. Radtsig A. A. Theory and Calculation of Condensing Installations, GEI, 1934, p. 107.

3. Bulanin V. I.—On the method of calculating surface condensers, Soviet Boiler and Turbine Construction, No. 2, 1938.

4. Izvestiya of the Tomsk Polytechnic Institute named after S. M. Kirov, Order of the Red Banner of Labor, vol. 63, 1944.