Bulletin of the Tomsk Polytechnic Institute, Order of the Red Banner of Labor, named after S. M. Kirov

Vol. 150　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1968

Determination of the Cooling Ratio in Steam-Turbine Units in the Design of Thermal Power Plants

E. N. Shadrin, G. F. Shilin

(Presented by Prof. Dr. I. N. Butakov)

As is known, most modern turbines are designed and manufactured for a calculated final pressure of \(0.03\text{--}0.035\) ata and a cooling ratio of \(55 \div 60\). In practice, turbines may operate under various conditions both with respect to fuel and with respect to water supply. However, when designing a thermal power plant, the circulating-water-supply system is designed for the calculated amount of cooling water. Such a decision leads either to unjustified capital expenditures, or to excessive consumption of electric power for pumping cooling water, or, finally, to excessive fuel consumption for the generation of \(1\) kWh of electric power. Indeed, for stations using cheap grades of fuel, the use of high cooling ratios and the considerable capital expenditures on the circulating system associated with this may not be repaid by the savings obtained thereby through reduced fuel consumption due to operation of the unit at a deep vacuum. Conversely, at stations using expensive grades of fuel, an underestimated cooling ratio may cause unjustified overconsumption of fuel because of the poor vacuum resulting from a small amount of cooling water, although the capital expenditures on the circulating system will here be smaller than at stations with high cooling-ratio values.

Thus, the cooling ratio is a techno-economic quantity affecting both the capital expenditures at the station and the operating costs. It is recommended that the optimal cooling ratio \(m^{\mathrm{opt}}\) be determined from the condition of minimum total production costs for generating a specified amount of electric power, including in this sum only those components of these costs that vary as a function of \(m\). These components should be taken to include the fuel costs \(S_{\mathrm{t}}^{\mathrm{e}}\) incurred in generating \(E_{\mathrm{year}}\), the annual amount of electric power, the costs of pumping cooling water \(S_{\mathrm{t}}^{\mathrm{p}}\), and the calculated costs depending on capital expenditures in water conduits \(S_{\mathrm{k}}^{\mathrm{w}}\) and in the pumping station \(S_{\mathrm{k}}^{\mathrm{p}}\).

Thus, the sum of the variable components of the annual production costs for generating \(E_{\mathrm{year}}\) kWh of electric power is

\[ \sum S = S_{\mathrm{t}}^{\mathrm{e}} + S_{\mathrm{t}}^{\mathrm{p}} + S_{\mathrm{k}}^{\mathrm{w}} + S_{\mathrm{k}}^{\mathrm{p}} . \tag{1} \]

To find the optimal cooling ratio, it is necessary to solve the equation:

\[ \frac{d \Sigma S}{dm} = \frac{dS_{\mathrm{T}}^{\mathrm{e}}}{dm} + \frac{dS_{\mathrm{T}}^{\mathrm{p}}}{dm} + \frac{dS_{\mathrm{k}}^{\mathrm{v}}}{dm} + \frac{dS_{\mathrm{k}}^{\mathrm{n}}}{dm}. \tag{2} \]

We proceed to expand the differential relations of equation (2).

The overall specific fuel consumption in kg for the generation of 1 kWh of electric energy is

\[ b_{\mathrm{e}} = \frac{\varepsilon_{\mathrm{p}}(i_0-i_{\mathrm{feed}})} {H_0 \eta_{\mathrm{oe}}\eta_{\mathrm{ku}}\eta_{\mathrm{loss}}\eta_{\mathrm{aux}} Q_{\mathrm{p}}^{\mathrm{n}}}. \tag{3} \]

Here \(i_0\) is the initial heat content of the steam, kJ/kg;
\(i_{\mathrm{feed}}\) is the heat content of the feedwater, kJ/kg;
\(H_0\) is the available heat drop in the turbine, kJ/kg;
\(Q_{\mathrm{p}}^{\mathrm{n}}\) is the lower calorific value of the fuel, kJ/kg;
\(\eta_{\mathrm{oe}}, \eta_{\mathrm{ku}}, \eta_{\mathrm{loss}}, \eta_{\mathrm{aux}}\) are the efficiencies, respectively, the relative electrical efficiency, boiler-installation efficiency, flow efficiency, and auxiliary-plant efficiency;
\(\varepsilon_{\mathrm{p}}\) is a coefficient accounting for the increase in steam consumption by the turbine due to regenerative extractions.

When designing a thermal power plant, the initial steam parameters, as well as the turbine type, are known. Therefore \(i_0\), \(i_{\mathrm{feed}}\), \(\eta_{\mathrm{oe}}\), \(\eta_{\mathrm{ku}}\), \(\eta_{\mathrm{loss}}\), and \(\eta_{\mathrm{aux}}\) may be considered constant, independent of the final parameters and, consequently, of the cooling ratio \(m\). The exception is the relative internal efficiency \(\eta_{oi}\), whose value will vary with changes in the final pressure \(p_k\). But since \(p_k\), within the range of variation of \(m\), will fluctuate within rather narrow limits, the change in \(\eta_{oi}\) may be neglected in these calculations.

The value \(Q_{\mathrm{p}}^{\mathrm{n}}\) depends on the grade of fuel burned and is determined by specific local conditions.

The annual fuel costs incurred for generating \(E_{\mathrm{yr}}\) kWh of electric energy are determined from the expression:

\[ S_{\mathrm{T}}^{\mathrm{e}} = \frac{3.6\,\varepsilon_{\mathrm{p}}(i_0-i_{\mathrm{feed}})R_{\mathrm{T}}AN_{\mathrm{e}}} {H_0\eta_{\mathrm{oe}}\eta_{\mathrm{ku}}\eta_{\mathrm{loss}}\eta_{\mathrm{aux}}Q_{\mathrm{p}}^{\mathrm{n}}}, \tag{4} \]

where \(R_{\mathrm{T}}\) is the cost of coal, rub/t;
\(A\) is the number of hours of utilization of installed capacity, h/yr;
\(N_{\mathrm{e}}\) is the turbine capacity, kW.

In formula (4), \(H_0=f(m)\). Therefore

\[ \frac{dS_{\mathrm{T}}^{\mathrm{e}}}{dm} = \frac{\partial S_{\mathrm{T}}^{\mathrm{e}}}{\partial H_0}\cdot\frac{dH_0}{dm} = - \frac{3.6\,\varepsilon_{\mathrm{p}}(i_0-i_{\mathrm{feed}})R_{\mathrm{T}}AN_{\mathrm{e}}} {H_0^2\eta_{\mathrm{oe}}\eta_{\mathrm{ku}}\eta_{\mathrm{loss}}\eta_{\mathrm{aux}}Q_{\mathrm{p}}^{\mathrm{n}}} \cdot \frac{dH_0}{dm}. \tag{5} \]

The available heat drop in the turbine may approximately be expressed as a function of the cooling ratio as

\[ H_0 = a-bt_s = a-b\left(t_1+\Delta t_k+\frac{i_k-i'_k}{4.19\,m}\right). \tag{6} \]

Here \(a\) and \(b\) are coefficients depending on the initial steam parameters. Their value may be taken in accordance with Table 1;
\(t_s\) is the temperature of the exhaust steam, °C;

\(t_1\) — temperature of the circulating water at the inlet to the condenser, °C;

\(\Delta t_k\) — temperature head in the condenser, °C;

\(i_k - i'_k\) — amount of heat transferred in the condenser to the cooling water during condensation of 1 kg of steam, kJ/kg.

Table 1

If the value of \(\dfrac{dH_0}{dm}\) is substituted into equation (5), then after simple transformations we obtain

\[ \frac{dS_T^{e}}{dm} = - \frac{M_1 h}{\kappa m^2 + lm + h^2}. \tag{7} \]

In equation (7)

\[ M_1 = \frac{3.6\,\varepsilon_p (i_0 - i_{\mathrm{pit}})\,\Delta N_e R_T} {\eta_{oe}\eta_{ku}\eta_{hoz}\eta_{pot} Q_p^{n}}, \]

\[ \kappa = [a - b(t_1 + \Delta t_k)]^2, \]

\[ l = [0.477 b^2(t_1 + \Delta t_k)(i_k - i'_k) - 0.477ab(i_k - i'_k)], \]

\[ h = 0.238\,b(i_k - i'_k). \]

The total annual expenditure in rubles for pumping circulating water, under the fuel-cost item, may be determined as

\[ S_T^{p} = \frac{0.001\,E_{uv} R_T}{\eta_k Q_p^{n}}, \tag{8} \]

where \(E_{uv}\) is the annual expenditure of electric power, in kJ, for pumping circulating water.

Since the expenditure of electric power for pumping circulating water depends on the cooling ratio, then

\[ \frac{dS_T^{p}}{dm} = \frac{\partial S_T^{p}}{\partial E_{uv}} \cdot \frac{dE_{uv}}{dm} = \frac{0.001\,R_T}{\eta_k Q_p^{n}} \cdot \frac{dE_{uv}}{dm}. \tag{9} \]

The expenditure of electric power for pumping circulating water, kJ per year, is

\[ E_{uv} = \frac{3.6\,mD_k H_{cn} A\varphi}{\rho \eta_n}. \tag{10} \]

Here \(H_{cn}\) is the total head overcome by the circulating pumps, N/m²;

\(D_k\) — steam flow rate to the condenser, kg/s;

\(\eta_n\) — pump efficiency;

\(\varphi = 1.04 \div 1.08\) — coefficient allowing for the additional water flow for the oil and air coolers.

The total pump head consists of the head required to overcome the resistance of the condenser \(H_k\), the circulating water conduits \(H_v\), and also the head \(H_g\) caused by the difference between the geodetic elevations of the condenser and the water level in the source.

The resistance of the condenser as a function of the cooling ratio may be written as

\[ H_k = Z \psi \left( \frac{m}{m_p} \right)^2 w_{\mathrm{cr}}^{\,2}, \tag{11} \]

where \(Z\) is the number of condenser passes;

\(\psi\) is the geometric characteristic of the condenser. Its value may be found from Fig. 1 from the known length and diameter of the condenser tubes.

\(m_p\) is the design cooling ratio;

\(w_{\mathrm{cr}}\) is the design water velocity in the condenser tubes, m/sec.

Fig. 1. Dependence of the coefficient \(\psi\) on the diameter and length of condenser tubes.

The resistance of the water conduit may be determined from the known formula

\[ H_v = \lambda_p \frac{\beta L_0}{d_v}\cdot \frac{w_v^2}{2}\ \text{N/m}^2 . \tag{12} \]

Here \(L_0\) is the actual length of the water conduit, m;

\(d_v\) is the diameter of the water conduit, m. Its value, depending on the cooling ratio for a single water conduit, may be found from the formula

\[ d_v = 0.0357\, m^{0.5} D_k^{0.5} w_v^{-0.5} \varphi^{0.5}\ \text{m}. \tag{13} \]

For a double water conduit,

\[ d_v = 0.0252\, m^{0.5} D_k^{0.5} w_v^{-0.5} \varphi^{0.5}\ \text{m}. \tag{14} \]

The velocity of water in the circulating water conduit is, in turn, a techno-economic quantity and may be determined from the formula [1]

\[ w_v = \sqrt[3]{\frac{48.2}{\dfrac{AR}{\eta_k Q_t^{\mathrm{n}}} + 7.8}} . \]

If, in equation (12), instead of \(d_v\) its value from (13) and (14) is substituted, then the expression for the resistance of a double water conduit will have the form

\[ H_v = 596\, \beta L_0 w_v^{2.5} m^{-0.5} D_k^{-0.5} \varphi^{-0.5}\ \text{N/m}. \tag{15} \]

Then the total head of the circulating pumps for a double water conduit, as a function of the cooling ratio, is

\[ H_{\mathrm{circ.p}} = Z \psi \left( \frac{w_{\mathrm{cr}}}{m_p} \right)^2 m^2 + 596\, \beta L_0 w_v^{2.5} m^{-0.5} D_k^{-0.5} \varphi^{-0.5} + 9807\, H_g . \tag{16} \]

If in equation (9) we substitute its value for \(H_{\text{sh}}\), take the derivative \(\dfrac{dЭ_{\text{sh}}}{dm}\), and make some transformations, then, after substitution into expression (9), for a double water conduit we obtain

\[ \frac{dS_{\text{t}}^{\text{n}}}{dm}=C\left(Tm^2+B\right), \tag{17} \]

where

\[ C=\frac{0.0036\,R_{\text{t}}A\varphi}{\eta_{\text{n}}\eta_{\text{k}}Q_{\text{r}}^{\text{n}}\rho}; \]

\[ T=3z\psi\beta\,\frac{w_{\text{kr}}^2}{W_{\text{r}}^2}\,D_{\text{k}}^3;\quad B=37.25\,\beta L_0 w_{\text{v}}^{2.5}D_{\text{k}}^{0.5}\varphi^{-0.5} +9807D_{\text{k}}H_2. \]

For a single water conduit, expression (17) is obtained analogously, with the only difference that the coefficient \(B\) will have the form:

\[ B=26.6\,\beta L_0 w_{\text{v}}^{2.5}D_{\text{k}}^{0.5}\varphi^{-0.5} +9807D_{\text{k}}H_2, \]

We proceed to determine the production costs that depend on the capital expenditures for the circulating water conduits and the pumping station. The cost of one linear meter of water conduit in place [1] is

\[ k_{\text{v}}=a'd_{\text{v}}\ \text{rubles/linear m}. \tag{18} \]

Here \(a'\) is a coefficient whose value depends on the number of water-conduit lines and may be taken as 110 rubles/linear m for a single water conduit, and 22 rubles/linear m for a double one.

If, instead of \(d_{\text{v}}\), its value from equations (13) and (14) is substituted, then the cost of 1 linear m for a single water conduit is

\[ k_{\text{v}}=0.0357\,m^{0.5}D_{\text{k}}^{0.5}w_{\text{v}}^{-0.5}\varphi^{0.5}a_1'\ \text{rubles/linear m}. \tag{19} \]

Then, for a double water conduit,

\[ k_{\text{v}}=0.0252\,m^{0.5}D_{\text{k}}^{0.5}w_{\text{v}}^{-0.5}\varphi^{0.5}a_2'\ \text{rubles/linear m}. \tag{20} \]

The calculated costs for the double water conduit are

\[ S_{\text{k}}^{\text{v}}=0.000252\,(u_{\text{v}}+\delta)L_0m^{0.5}D_{\text{k}}^{0.5}w_{\text{v}}^{-0.5}\varphi^{0.5}a_1'. \tag{21} \]

Then, after differentiation, for a double water conduit we obtain

\[ \frac{dS_{\text{k}}^{\text{v}}}{dm} = 1.57\cdot10^{-5}(u_{\text{v}}+\delta)L_0D_{\text{k}}^{0.5}w_{\text{v}}^{-0.5}\varphi^{0.5}a_2' = M_2'. \tag{22} \]

For a single water conduit,

\[ \frac{dS_{\text{k}}^{\text{v}}}{dm} = 2.23\cdot10^{-5}(u_{\text{v}}+\delta)L_0D_{\text{k}}^{0.5}w_{\text{v}}^{-0.5}a_1'\varphi^{0.5} = M_2'. \tag{23} \]

The cost of the pumping station \(K_{\text{n}}\) may be taken as proportional to its capacity \(N_{\text{n}}\) and represented as the sum of the cost of the civil-engineering part and of the equipment:

\[ K_{\text{n}}=k_{\text{s}}N_{\text{n}}+k_0N_{\text{n}}, \tag{24} \]

where the capacity of the pumping station may be expressed by the formula

\[ N_{\text{n}} = \frac{0.001\,mD_{\text{k}}}{\rho} \left[ z\psi\left(\frac{w_{\text{kr}}}{m_{\text{r}}}\right)^2m^2 + 596\beta L_0w_{\text{v}}^{2.5}D_{\text{k}}^{-0.5}\varphi^{-0.5} + 9807H_2 \right]. \tag{25} \]

Then the calculated costs with respect to the capitalization factor for the pump will be written as

\[ S_{\mathrm{k}}^{\mathrm{n}}=[0.01(u_{\mathrm{c}}+\delta)\kappa_{\mathrm{c}}+0.01(u_{0}+\delta)\kappa_{0}]N_{\mathrm{n}}. \tag{26} \]

Here the pump power \(N_{\mathrm{n}}=f(m)\). Therefore

\[ \frac{dS_{\mathrm{k}}^{\mathrm{n}}}{dm} = \frac{\partial S_{\mathrm{k}}^{\mathrm{n}}}{\partial N_{\mathrm{n}}}\cdot \frac{dN_{\mathrm{n}}}{dm}. \tag{27} \]

If we take the derivative and substitute the expression obtained into formula (27), then after simple transformations we obtain

\[ \frac{dS_{\mathrm{k}}^{\mathrm{n}}}{dm}=P(Tm^{2}+B), \tag{28} \]

where the expression for \(P\) will have the form:

\[ P=\frac{10^{-8}}{\eta_{\mathrm{n}}}\,[(u_{\mathrm{c}}+\delta)\kappa_{\mathrm{c}}+(u_{0}+\delta)\kappa_{0}]. \]

If, further, the values of the quantities entering into expression (2) are substituted from (7), (17), (22), and (28), then after a number of transformations we obtain the equation

\[ A_{1}m^{4}+A_{2}m^{3}+A_{3}m^{2}+A_{4}m+A_{5}=0. \tag{29} \]

Here

\[ A_{1}=Tk(C+P);\qquad A_{2}=Tl(C+P); \]

\[ A_{3}=T(C+P)+BK(C+P)+M_{2}k; \]

\[ A_{4}=l(CB+BP+M_{2});\qquad A_{5}=(CB+BP+M_{2})h^{2}-M_{1}h. \]

It is expedient to represent equation (29) in the following form:

\[ m^{2}= \frac{-\dfrac{A_{5}}{m}-A_{4}} {\dfrac{A_{3}}{m}+A_{1}m+A_{2}}. \tag{30} \]

Equation (30) can be solved either by trial selection or graphically. For this purpose one may write

\[ y_{1}=m^{2};\qquad y_{2}= \frac{-\dfrac{A_{5}}{m}-A_{4}} {\dfrac{A_{3}}{m}+A_{1}m+A_{2}}. \]

The first expression is the equation of a parabola, which can be plotted once and for all. The position of the curve \(y_{2}=f(m)\) is determined by the local specific conditions. At the intersection of the curves \(y_{1}=f(m)\) and \(y_{2}=f(m)\) we find the optimum velocity, as shown in Fig. 2.

In Fig. 3, by the method set forth above, the dependence \(m^{\mathrm{opt}}=f(R_{\mathrm{t}})\) is constructed for the particular case when

\[ L_{0}=450\ \text{m};\quad H_{2}=7.5\ \text{m};\quad \beta=1.15;\quad z=2; \]

\[ m_{\mathrm{p}}=50;\quad \eta_{\mathrm{k}}=0.312;\quad \varphi=1.06, \]

for various fuel costs and for various capacities.

It is evident from Fig. 3:

Fig. 2. Determination of the optimal cooling ratio.

Fig. 3. Dependence of the optimal cooling ratio on fuel cost.

Fig. 4. Dependence of the optimal final pressure on fuel cost.

1) as the cost of fuel decreases, lower values of the cooling ratio should be taken; conversely, if the plant is designed to operate on expensive fuel, the cooling ratio is taken to be higher;

2) the temperature of the cooling water has little effect on the design cooling ratio;

3) as turbine capacity increases, the design cooling ratio increases, and the increase is greater the higher the cost of fuel.

In Fig. 4, for the data of the same example, we construct a graph of the dependence of the optimum final pressure \(p_k^{\mathrm{opt}}\) on the cost of fuel. The pressure \(p_k^{\mathrm{opt}}\) was found from the temperature of the exhaust steam

\[ t_s = t_1 + \Delta t_k + \frac{i_k - i_k'}{4.19\,m^{\mathrm{opt}}}, \]

where \(m^{\mathrm{opt}}\) was determined from Fig. 3. From Fig. 4 it is evident that, at low fuel costs of \(1 \div 2\) rubles/t, the design final pressure in turbine condensers should be higher, \(0.45 \div 0.05\) ata,

REFERENCES

1. E. N. Shadrin, Selection of the optimum water velocity in circulating water conduits. Izvestiya vuzov SSSR—Energetika, No. 5, 1963.