SOME ISSUES IN OPTIMIZING THE PARAMETERS OF THE THEORETICAL CYCLE OF STEAM-TURBINE PLANTS WITH ADIABATIC PHASE FORMATION

V. I. Bespalov

(Presented by the scientific seminar of the Department of TES)

The cycle of a steam-turbine plant with adiabatic phase formation was proposed and developed at Tomsk Polytechnic Institute [1].

Figure 1 shows the simplest schematic of a steam-turbine plant with adiabatic phase formation (APF). The plant consists of a once-through supercritical water-heating boiler unit BK, an evaporator-generator IG, structurally designed as a single-disk turbine in whose evaporating elements steam is produced and the kinetic energy of the flow is generated, a moisture separator S, a steam superheater PP, feedwater and condensate pumps, and a steam turbine. In the general case the plant has regenerative heating of the feedwater.

The theoretical cycle of such a steam-turbine plant with APF is conventionally shown in Fig. 2 on a \(TS\)-diagram. In the diagram, 1 corresponds to the state of the working fluid before the evaporating elements; 2—at the outlet from the IG; 3—to the state of dry saturated steam at the outlet from the moisture separator; 4—to the state of superheated steam before the steam turbine; 7 and 9—respectively to the state of the condensate and feedwater downstream of the condensate and feedwater pumps.

Fig. 1

Fig. 2

Considering the proposed cycle, it may be noted that, at a specified pressure \(P_{1v}\) and temperature of the superheated water \(T_{1v}\), and with unchanged heat-rejection temperature in the condenser \(T_k\) and steam temperature before the

... by the steam turbine \(T_{\mathrm{p}}\)—an increase in the pressure in the separator leads to a supercritical-pressure cycle and to an inadmissibly high final moisture content of the steam in the last stages of the turbine. On the other hand, a decrease in the pressure in the separator leads to degradation of the cycle of the steam-turbine plant.

Consequently, there exists a certain optimum value of the pressure in the separator \(P_c\) and a corresponding temperature \(T_c\), at which, for the prescribed pressure and temperature of the superheated water before the evaporator-generator and the pressure in the condenser, the useful work of the cycle will be maximal.

In the general case, the efficiency of a steam-turbine cycle with adiabatic phase formation is determined from the expression

\[ \eta_i = \frac{i_{1\mathrm{v}} - i_c + x_c\left(i_{\mathrm{p}} - i_{\mathrm{k}}\right)} {i_{1\mathrm{v}} - i_{\mathrm{pv}} + x_c\left(i_{\mathrm{p}} - i_{\mathrm{k}1} - r_c\right)} . \tag{1} \]

Here:
\(i_{1\mathrm{v}}\)—enthalpy of the superheated water before the evaporator-generator, kJ/kg;
\(i_c\)—enthalpy of the steam-water mixture at the outlet from the evaporator-generator, kJ/kg;
\(i_{\mathrm{p}}\)—enthalpy of the superheated steam before the steam turbine, kJ/kg;
\(i_{\mathrm{k}}\)—enthalpy of the exhaust steam, kJ/kg;
\(i_{\mathrm{pv}}\)—enthalpy of the water downstream of the feed pump, kJ/kg;
\(i'_{\mathrm{k}}, i_{\mathrm{k}1}\)—respectively, the enthalpies of the condensate before and after the condensate pump, kJ/kg;
\(r_c = i''_c - i'_c\)—latent heat of vaporization at the pressure in the separator, kJ/kg;
\(x_c\)—dryness fraction of the wet steam at the outlet from the evaporator-generator.

The maximum value of the cycle efficiency is determined from the condition:

\[ \frac{\partial \eta_i}{\partial T_c}=0; \qquad \frac{\partial^2 \eta_i}{\partial T^2}<0. \]

We shall determine the optimum pressure in the separator for the theoretical cycle of a steam-turbine plant with adiabatic phase formation, neglecting, for purposes of simplification, the work of the feed and condensate pumps.

For this case:

\[ i_{\mathrm{pv}} = i'_c; \quad i_{\mathrm{k}1}=i'_{\mathrm{k}}; \]

\[ i_c=i_{ct}; \quad i_{\mathrm{k}}=i_{\mathrm{kt}}; \quad x_c=x_{ct}. \]

We solve the problem under the conditions:

\[ P_{1\mathrm{v}}=\mathrm{const}, \quad t_{\mathrm{p}}=\mathrm{const}, \]

\[ t_{1\mathrm{v}}=\mathrm{const}, \quad t_{\mathrm{k}}=\mathrm{const}, \quad P_c=\mathrm{var}. \]

Differentiating expression (1) with respect to \(T_c\), we obtain:

\[ \eta_i^{\max} = \frac{ -\dfrac{\partial i_{ct}}{\partial T_c} + \dfrac{\partial x_{ct}}{\partial T_c}\left(i_{\mathrm{p}} - i_{\mathrm{kt}}\right) + x_{ct}\left( \dfrac{\partial i_{\mathrm{p}}}{\partial T_c} - \dfrac{\partial i_{\mathrm{kt}}}{\partial T_c} \right) }{ -\dfrac{\partial i'_c}{\partial T_c} + \dfrac{\partial x_{ct}}{\partial T_c}\left(i_{\mathrm{p}} - i'_{\mathrm{k}} - r_c\right) + x_{ct}\left( \dfrac{\partial i_{\mathrm{p}}}{\partial T_c} - \dfrac{\partial i'_{\mathrm{k}}}{\partial T_c} - \dfrac{\partial r_c}{\partial T_c} \right) }. \]

Using the known thermodynamic relations [2, 3], as well as those obtained later in the works of other authors [4, 5, 6], we find the values of the partial derivatives:

\[ \left( \frac{\partial i_{ct}}{\partial T_c} \right)_{r_{\mathrm{p}},\,T_{\mathrm{p}},\,S} = V_{ct}\frac{\Delta S}{\Delta V} = \left(x_{ct}\Delta V + V'_c\right)\frac{\Delta S}{\Delta V}. \]

Here: \(\Delta V = V_c'' - V_c'\), \(\Delta S = S_c'' - S_c'\).

\[ \left(\frac{\partial x_{ct}}{\partial T_c}\right)_{S} = \frac{C_x}{r_c}, \]

where \(C_x\) is the heat capacity of wet steam in the process \(x=\mathrm{const}\).

\[ \left(\frac{\partial i_p}{\partial T_c}\right)_{T_p} = V_p\frac{\Delta S}{\Delta V}(1-\alpha_p), \]

where \(V_p\) is the specific volume of superheated steam before the turbine,

\[ \alpha_p=\frac{T_p}{V_p}\left(\frac{\partial V_p}{\partial T_p}\right)_{P} \]

is the coefficient of isobaric expansion of steam, which, with sufficient accuracy, can be determined from the approximate relation:

\[ \alpha_p=\frac{T_p}{V_p}\left(\frac{\Delta V_p}{\Delta T_p}\right)_{P}, \]

\[ \left(\frac{\partial i_{kt}}{\partial T_c}\right)_{T_p,T_k,S} = -\alpha_p V_p \frac{\Delta S}{\Delta V}\cdot \frac{T_k}{T_p}, \]

\[ \left(\frac{\partial i_c'}{\partial T_c}\right)_{x} = C_x' - V_c'\frac{\Delta S}{\Delta V}, \]

where \(C_x'\) is the heat capacity of water on the saturation line under the condition \(x=\mathrm{const}\);

\[ \left(\frac{\partial i_k'}{\partial T_c}\right)_{T_k} = 0, \]

\[ \left(\frac{\partial r_c}{\partial T_c}\right) = r_c' = \Delta C-\Delta S, \qquad \Delta C=C_x''-C_x'. \]

Here \(C_x''\) is the heat capacity of dry saturated steam on the saturation line at \(x=\mathrm{const}\). In practice it is more convenient to use the graphical dependence \(r_c''=f(T_c)\), constructed from the approximate equality:

\[ r_c''=\frac{\Delta r_c}{\Delta T_c}. \]

Substituting the values of the partial derivatives into the expression for \(\eta_t^{\max}\), making the necessary transformations, and solving it with respect to \(T_c\), we obtain an expression for determining the optimum value \(T_c^{\mathrm{opt}}\):

\[ T_c^{\mathrm{opt}} = \frac{r_c'}{\lambda} \cdot \frac{ \frac{V_c'}{V_p}(\eta_t^{\max}+1) + x_{ct}\left[\alpha_p(\eta_{tk}-\eta_t^{\max})-(1-\eta_t^{\max})+\lambda\right] }{ \frac{C_x}{r_c}\left[(i_p-i_{kt}')-\eta_t^{\max}(i_p-i_k'-r_c)\right] + \eta_t^{\max}(C_x'+x_{ct}r_c') } \;^\circ\mathrm{K}. \]

Here:

\[ \lambda=\frac{\Delta V}{V_p}; \qquad \eta_{tk}=1-\frac{T_k}{T_p}. \]

The resulting equation can be solved by the method of successive approximations.

Taking into account that equations obtained from differential relations are highly sensitive even to a slight deviation from the extremum, it is advisable first to estimate its position on the basis of variant calculations. Figure 3 shows the dependences of the thermal efficiency \((\eta_t)\) of a steam-turbine cycle with adiabatic phase formation on \(T_c\), constructed according to equation (1).

For the specified parameters \(t_{1v}=400^\circ\mathrm{C}\), \(P_{1v}=290\) bar, \(P_k=0.05\) bar, and \(t_p=565^\circ\mathrm{C}\), the optimum temperature of moisture separation in the separator was \(t_c^{\mathrm{opt}}=283^\circ\mathrm{C}\), which corresponds to the saturation pressure \(P_c^{\mathrm{opt}}=67.2\) bar. The theoretical efficiency of such a cycle is \(\eta_t=0.462\). In this case, the dryness fraction of the steam-water mixture at the outlet from the evapor-

the turbine-generator is 56%, and at the outlet from the steam turbine it is 82.7%.

At a superheated-water pressure before the evaporating elements of \(P_{\mathrm{v}}=295\) bar, the optimum separation temperature is \(t_c^{\mathrm{opt}}=307^\circ\mathrm{C}\), which corresponds to a saturation pressure of \(P_c^{\mathrm{opt}}=94.7\) bar. The dryness fraction of the steam that has passed through the steam turbine decreases to 80%.

Fig. 3

From the calculations presented it is evident that the initial pressure of the superheated water has a substantial effect on the magnitude of the optimum separation temperature \(T_c^{\mathrm{opt}}\).

To determine the effect of the work of the feed and condensate pumps on the value of the optimum temperature of moisture separation in the separator, it is necessary to substitute the following values into expression (1):

\[ i_{\mathrm{pv}}=i_c' + V_c'(P_{1\mathrm{v}}-P_c), \]

\[ i_{k1}=i_k' + V_k'(P_c-P_k). \]

Differentiating (1) with respect to \(T_c\) at \(i_{1\mathrm{v}}=\mathrm{const}\), we obtain:

\[ \eta_t^{\max}= \frac{ -\dfrac{\partial i_{ct}}{\partial T_c} +\dfrac{\partial x_{ct}}{\partial T_c}(i_{\mathrm{p}}-i_{kt}) +x_{ct}\left(\dfrac{\partial i_{\mathrm{p}}}{\partial T_c} -\dfrac{\partial i_{kt}}{\partial T_c}\right) }{ -\dfrac{\partial i_{\mathrm{pv}}}{\partial T_c} +\dfrac{\partial x_{ct}}{\partial T_c}(i_{\mathrm{p}}-i_{k1}-r_c) +x_{ct}\left(\dfrac{\partial i_{\mathrm{p}}}{\partial T_c} -\dfrac{\partial i_{k1}}{\partial T_c} -\dfrac{\partial r_c}{\partial T_c}\right) }. \]

In this expression, in addition to the partial derivatives found earlier, there are new partial derivatives:

\[ \left(\frac{\partial i_{k1}}{\partial T_c}\right)_S = V_k'\frac{\Delta S}{\Delta V}, \qquad \left(\frac{\partial i_{\mathrm{pv}}}{\partial T_c}\right)_S = C_x' - 2V_c'\frac{\Delta S}{\Delta V}. \]

Substituting the values of the derivatives into the expression for \(\eta_t^{\max}\), carrying out the necessary transformations, and solving it with respect to \(T_c^{\mathrm{opt}}\), we obtain:

\[ T_c^{\mathrm{opt}} = \frac{r_c}{\lambda}\, \frac{ \dfrac{V_c'}{V_{\mathrm{p}}'}\left[ 2-x_{ct}\dfrac{V_k'}{V_c'}\eta_t^{\max}+1 \right] +x_{ct}\left[ a_{\mathrm{p}}(\eta_{tk}-\eta_t^{\max})-(1-\eta_t^{\max})-\lambda \right] }{ \dfrac{c_x}{r_c} \left[ (i_{\mathrm{p}}-i_{kt})-\eta_t^{\max}(i_{\mathrm{p}}-i_{k1}-r_c) \right] +\eta_t^{\max}(C_x' + x_{ct}r_c') } K. \]

Calculations made using this expression showed that taking into account the operation of the feed and condensate pumps leads to a reduction in the optimal value of \(T_c^{\mathrm{opt}}\) by several degrees. At \(P_{1b} = 290\) bar and \(t_{1b} = 400^\circ\mathrm{C}\), the optimal temperature decreased from \(283^\circ\mathrm{C}\) to \(277^\circ\mathrm{C}\).

Conclusions

1. The optimal pressure for separating the steam–water mixture downstream of the evaporator-generator in a steam-turbine plant with adiabatic phase formation can be determined analytically.

2. The value of the optimal moisture-separation pressure is substantially influenced by the parameters of the superheated water ahead of the evaporator-generator.

3. Taking into account the operation of the feed and condensate pumps leads to a reduction in the optimal value of the moisture-separation pressure.

References

1. S. V. Polozhii. Steam-power plants with adiabatic steam generation. Izv. vuzov SSSR, Energetika, No. 1, 1965.

2. V. A. Kirillin et al. Technical Thermodynamics. Energiya, Moscow, 1968.

3. A. M. Litvin. Technical Thermodynamics. Gosenergoizdat, 1963.

4. A. I. Andryushchenko. Thermodynamic Calculations of Optimal Parameters of Thermal Power Stations. Vysshaya shkola, Moscow, 1963.

5. Problems of optimization and calculation of high-capacity steam-turbine units. Scientific Proceedings of SPI, issue No. 39, Saratov, 1969.

6. Optimization of parameters and rational use of fuel in power installations. Reports of the regional conference of young scientists. Saratov, 1969.