On the Methodology of Experimental Investigations of Moisture Losses in Steam Turbines

S. V. Polozhii

(Presented by the Department of Thermal Power Installations)

Quantitative data from modern investigations of moisture losses do not agree with the theoretical conception of the possible magnitude of losses from droplet moisture, based on the phenomenon of adiabatic condensation in the flow path of a steam turbine. This has served as the reason for a critical examination of the modern methodology for the experimental determination of moisture losses in steam turbines [1–3].

The shortcoming of the modern methodology of experimental and theoretical investigations of moisture losses consists in comparing the efficiency of one and the same stage when it operates successively on superheated and wet steam, without taking into account the different operating conditions of the stage on superheated and wet steam.

The absolute value of the relative internal efficiency of a stage, \(\eta_{oi}^{\mathrm{wl}}\), in the wet-steam region, determined experimentally by formula (1), reflects all losses in the stage both from droplet moisture (from steam supercooling \(\xi_{\mathrm{n}}\), from the impact of droplets at entry onto the rotor blades \(\xi_{\mathrm{ud}}\), from droplet acceleration and breakup of the film in the axial clearance \(\xi_{\mathrm{r}}\)), and from losses associated with the physical and thermodynamic features of operation of the stage in the wet-steam region (available heat drop \(H_0^{\mathrm{wl}}\), steam flow rate \(G_{\mathrm{wl}}\), disk-friction loss in steam \(\xi_{\mathrm{t}}\), operating regime \(\left(\dfrac{U}{C_0}\right)_{\mathrm{wl}}\), etc.).

\[ \eta_{oi}^{\mathrm{wl}} = K \left( \frac{M \cdot n}{G \cdot H_0} \right)_{\mathrm{wl}}, \tag{1} \]

where \(K\) is a constant coefficient;

\(M, n, G\) are the torque, rotational speed, and flow rate of the steam-conducting mixture measured in the experiment;

\(H_0\) is the available heat drop, determined from diagrams or tables according to the measured initial and final parameters \((p_0, t_0(x_0), p_2)\).

Operation of the stage on superheated steam is also fully characterized by the experimental internal efficiency of the stage (2).

\[ \eta_{oi}^{\mathrm{p}} = K \left( \frac{M \cdot n}{G \cdot H_0} \right)_{\mathrm{p}} \tag{2} \]

This is a generally known fact and is beyond doubt.

The error of the modern method of experimentally determining losses due to moisture when the same stage operates on superheated and wet steam consists in comparing these efficiencies, \(\eta_{0i}^{\mathrm{wet}}\), \(\eta_{0i}^{\mathrm{sup}}\), without taking into account the differences in their operating conditions and without ensuring such operating conditions of the stage on superheated and wet steam under which formulas (1) and (2) can be used.

When comparing the efficiency of a stage on superheated and wet steam, the loss due to the change in the stage operating conditions when the stage passes from superheated to wet steam (a decrease in efficiency due to changes in the heat drop \(H_{0}^{\mathrm{wet}}\), in the density of wet steam \(\rho^{\mathrm{wet}}\), and in the non-scalable operating conditions on wet steam \(\left(U/C_{0}\right)_{\mathrm{wet}}\)) is, according to the modern method of determining moisture losses, both in experiments and in theoretical studies, attributed entirely to the loss due to droplet moisture \((\xi_{\mathrm{p}}+\xi_{\mathrm{ud}}+\xi_{\mathrm{r}})\).

\[ \eta_{0i}^{\mathrm{wet}} = \eta_{0i}^{\mathrm{sup}} - \eta_{0i}^{\mathrm{sup}} \left[ \left(\frac{\Delta M}{M_{\mathrm{p}}}\right)_{\mathrm{p}} + \left(\frac{\Delta M}{M_{\mathrm{p}}}\right)_{\mathrm{ud}} + \left(\frac{\Delta M}{M_{\mathrm{p}}}\right)_{\mathrm{r}} \right] = \eta_{0i}^{\mathrm{sup}} - \eta_{0i}^{\mathrm{sup}} (\xi_{\mathrm{p}}+\xi_{\mathrm{ud}}+\xi_{\mathrm{r}}) \tag{3} \]

Such a method for determining losses due to steam moisture in an experimental stage would be valid only if it were possible to create identical physical and thermodynamic operating conditions for the stage when operating on superheated and wet steam, at a constant ratio of the pressures entering the stage, both for superheated and for wet steam,

\[ \varepsilon=\frac{P_{1}}{P_{0}}, \]

under which the experimental determination of the loss due to moisture is carried out.

The decrease in the efficiency of the stage when passing from superheated to wet steam at

\[ \frac{P_{1}}{P_{0}}=\mathrm{idem} \]

occurs due to all losses associated with the operation of the stage in the wet-steam region—both due to changes in the physical and thermodynamic operating conditions of the stage and due to the presence of moisture,

\[ \eta_{0i}^{\mathrm{wet}} = \eta_{0i}^{\mathrm{sup}} - \eta_{0i}^{\mathrm{sup}} \left\{ \left[ \left(\frac{\Delta M}{M_{\mathrm{p}}}\right)_{\mathrm{p}} + \left(\frac{\Delta M}{M_{\mathrm{p}}}\right)_{\mathrm{ud}} + \left(\frac{\Delta M}{M_{\mathrm{p}}}\right)_{\mathrm{r}} \right] + \left[ \left(\frac{\Delta M}{M_{\mathrm{p}}}\right)_{\mathrm{n}} + \left(\frac{\Delta M}{M_{\mathrm{p}}}\right)_{\mathrm{t}} + \left(\frac{\Delta M}{M_{\mathrm{p}}}\right)_{U/C_{0}} \right] \right\}, \tag{4} \]

where \(\left(\Delta M/M_{\mathrm{p}}\right)_{\mathrm{n}}\) is the change in torque due to the decrease in the working capacity of wet steam; \(\left(\Delta M/M_{\mathrm{p}}\right)_{\mathrm{t}}\), the same due to the increase in loss from disk friction in steam; \(\left(\Delta M/M_{\mathrm{p}}\right)_{U/C_{0}}\), due to non-scalable operating conditions of the stage on wet steam; \(\left(\Delta M/M_{\mathrm{p}}\right)_{\mathrm{p}}\), energy loss due to supercooling; \(\left(\Delta M/M_{\mathrm{p}}\right)_{\mathrm{r}}\), energy loss for breaking up the film and accelerating droplets in the axial clearance; \(\left(\Delta M/M_{\mathrm{p}}\right)_{\mathrm{ud}}\), energy loss due to the off-design angle of moisture entry onto the working blades and separation into the peripheral zone.

As an example of the error of the modern method of experimental studies for determining losses due to moisture, one may cite the studies published in [4].

In [4] an analysis is presented and recommendations are given for determining moisture losses on the basis of many years of experimental investigations by MEI using the moment characteristics of stages. This method of experimental investigations of moisture losses, in the opinion of the authors of [4], made it possible to obtain the qualitative effect of energy losses due to moisture \((\xi_p, \xi_p, \xi_{\mathrm{ud}})\) on the efficiency of steam turbines. By varying the magnitude of the torque, especially with a braked rotor

\[ \left(\frac{U}{C_0}=0\right), \]

enclosed in a floating bushing, when operating on superheated steam near the saturation line \((M_p)\) and on wet steam \((M_{\mathrm{vl}})\), in (4) the moisture loss is obtained as the ratio

\[ \frac{\Delta M}{M_p}=\frac{M_p-M_{\mathrm{vl}}}{M_p} \tag{4} \]

\[ M_{\mathrm{vl}}=M_p-\sum_{i=1}^{n}\Delta M_i \tag{5} \]

The relative internal efficiency of the stage is determined from the expression

\[ \eta_{0i}^{\mathrm{vl}}=\eta_{ci}^{p}-\eta_{0i}^{p}\left[\left(\frac{\Delta M}{M_p}\right)_p+\left(\frac{\Delta M}{M_p}\right)_r+\left(\frac{\Delta M}{M_p}\right)_{\mathrm{ud}}\right]. \tag{6} \]

The structure of these formulas and the methodology of the experimental investigations, as was shown above, are based on comparing the operation of the stage on superheated and wet steam by the total decrease in the torques and stage efficiency on wet steam, taking into account only the presence of moisture in the steam and its interaction with the steam and the structural elements of the stage, including the effect of supercooling of the wet steam \((\xi_p)\).

With such a formulation of the investigations, the main causes of the decrease in torques and in the relative internal efficiency of the stage during the transition from superheated to wet steam, which were discussed above [1, 2], are not assumed and are not taken into account. Using the same experimental data [4] (Figs. 1, 3, 6), one can show the erroneous nature of the judgments and conclusions about the essence and balance of moisture losses in [4]. The results of measurements of torques with a braked rotor and at different

\[ \frac{U}{C_0} \]

as a function of steam moisture are given in Fig. 1. The reduction of the torque of wet steam with a braked rotor

\[ \frac{U}{C_0}=0 \]

in [4] is estimated as the result of energy loss from the breakup of the film and acceleration of droplets in the axial clearance. When the rotor rotates, the decrease in torques on wet steam also takes into account the loss from droplet impact at the entrance of the flow onto the rotor blades. Such a decrease in torques during the transition from superheated to wet steam in [4] is in fact determined not only by the presence of moisture in the steam and its interaction with the flow-through part of the turbine, but also by the change in the physical and thermodynamic properties of wet steam—the reduction in the working capacity of a kilogram of steam as moisture increases at an unchanged ratio

\[ \varepsilon=\frac{P_1}{P_0} \]

in the experiments, as well as by the increase in heat losses of the wet-steam stage [1]. To illustrate the actual main causes of the decrease in torques in the experiments [4] during the transition of the stage from superheated to wet steam, taking into account only the reduction in the working capacity of wet steam, Table 1 gives the value of the theoretical heat drop \(H_0^{\mathrm{vl}}\)

from moisture at \(\varepsilon=\dfrac{P_1}{P_0}=0.85\) \((P_0=2\ \text{ata},\ P_1=1.7\ \text{at})\) and the ratio of the work capacity of wet and superheated steam \(\dfrac{H_0^{\mathrm{p}}-H_0^{\mathrm{vl}}}{H_0^{\mathrm{p}}}\). As is seen from this table, the relative decrease in the work capacity of wet steam

Fig. 1. Relative decrease in the torque on the stage shaft due to moisture as a function of the ratio \(\dfrac{U}{C_0}\) according to [1]. \(d_{\mathrm{sr}}=400\ \mathrm{mm}\), \(l=25\ \mathrm{mm}\), \(\varepsilon=0.85\)

of steam \(\dfrac{H_0^{\mathrm{p}}-H_0^{\mathrm{vl}}}{H_0^{\mathrm{p}}}\), with the total loss from droplet moisture taken as zero, \(h_{\mathrm{vl}}=0\), has the same regularity as the decrease

Table 1

of torques in the experiments [4] (Fig. 1). For the given \(y_0\) and \(\varepsilon\), the change in torques as a function of the ratio \(\dfrac{U}{C_0}\) obeys the dependence

Fig. 2a. Dependence of \(\eta_{oi}\) on \(\dfrac{U}{C_0}\) with zero loss due to droplet moisture \((Y_0=7\%)\). 1 — superheated steam, 2 — wet steam, 3 — decrease of the relative internal efficiency upon transition to wet steam at \(h_{\mathrm{wl}}=0\)

Fig. 2b. Dependence of \(\eta_{oi}\) on \(\dfrac{U}{C_0}\) with zero loss due to droplet moisture \((Y_0=2\%)\). 1 — superheated steam, 2 — wet steam, 3 — decrease of the relative internal efficiency upon transition to wet steam at \(h_{\mathrm{wl}}=0\)

$\eta_{oi}$ on $\dfrac{U}{C_0}$, caused by the usual heat losses of the stage (losses in the nozzles, on the rotor blades, due to friction, and with the exit velocity—$h_c$, $h_l$, $h_{\mathrm{tv}}$, $h_v$ [1]). Figure 2 shows a scheme of the variation of torques as functions of $\dfrac{U}{C_0}$ for constant moisture content and pressure ratio across the stage $\varepsilon = \dfrac{P_1}{P_0}$. The decrease in $\eta_{oi}$ (torques) increases with increasing steam moisture content for one and the same value of $\dfrac{U}{C_0}$ (Figs. 2a, 2b). At low moisture content, the significant influence of $\dfrac{U}{C_0}$ on the decrease in the torques of wet and superheated steam disappears (point $A$ in Figs. 2a and 2b, point $D$ in Fig. 1). This is explained by the small difference between $H_0^{t}$ and $H_0^{\mathrm{vl}}$, $h_{\mathrm{tv}}^{p}$ and $h_{\mathrm{tv}}^{\mathrm{vl}}$ at low peripheral speeds of rotation of the stage disk. The decrease in torque in the experiments of [4] at $y_0 = 0$ and operation of the stage on superheated steam (points $B$ and $C$ in Fig. 1) is apparently explained by the change in the available work and physical properties (at constant $\varepsilon = \dfrac{P_1}{P_0}$) when the stage passes from superheated to slightly superheated or saturated steam. The method of determining torques, in the form in which it is used in [4], as becomes evident, does not make it possible to detect the true loss due to droplet moisture and, still less, to compile a correct balance of losses when the stage operates in the wet-steam region. However, this method and the experimental data of [4] make it possible to reveal the principal causes of the decrease in torque during the transition from superheated to wet steam, and to establish qualitatively the influence of physical and thermodynamic factors on the decrease in efficiency of a wet-steam stage. The decrease in torque as a function of the pressure ratio across the nozzle cascade $\varepsilon = \dfrac{P_1}{P_0}$ also depends on the available work of superheated and wet steam for different $\dfrac{U}{C_0}$. For example, for $y_0 = 9\%$ at $\varepsilon = 0.9$ ($P_0 = 0.55$ ata, $P_1 = 0.5$ ata), the ratio of the theoretical heat drops of wet steam and superheated steam is $0.67 \div 0.58$, or $32$–$28\%$. The torque of the stage when operating on wet steam is reduced by the corresponding amount. At $\varepsilon = 0.4$ ($P_0 = 1.25$ ata, $P_1 = 0.5$ ata), with the same moisture content of $9\%$, the ratio of the heat drops is $0.88$–$0.80$, or $16\%$. The arrangement of the experimental curves in [4] (Fig. 3) follows this same regularity without allowance for losses due to droplet moisture. For example, for $y_0 = 9\%$ at $\varepsilon = 0.9$,

\[ \frac{\Delta M}{M_p} = 13\%, \]

whereas for $\varepsilon = 0.4$ it is only $3.5\%$.

The expenditure of energy for breaking up the film and accelerating liquid droplets in the axial clearance, at low moisture content, by its physical nature cannot be a substantial quantity. Therefore the assertion in [4] that, in the region of optimal $\dfrac{U}{C_0}$, the loss due to film breakup and droplet acceleration in the axial clearance amounts to $8$–$10\%$ at a moisture content $y_0 = 7.5\%$ (Fig. 3) is incorrect. The representation of the loss balance due to droplet moisture in Fig. 4, according to the data of [4], is erroneous.

The change in \(\eta_{oi}\) during operation of a stage on superheated steam as a function of \(\dfrac{U}{C_0}\) is determined by heat losses—in the nozzles, on the rotor blades, by disk friction, and by the exit velocity. A similar dependence of \(\eta_{oi}\) on \(\dfrac{U}{C_0}\) in experiments [4] was also obtained for wet steam (Fig. 4, curve 2). The lower

Fig. 3. Energy loss for acceleration and breakup of droplets as a function of the nozzle pressure ratio \(\left(\varepsilon=\dfrac{P_1}{P_0}\right)\) and the initial moisture content according to [4]

position of curve 2 is explained by the lower work capacity of one kilogram of wet steam at the same value of \(\varepsilon=\dfrac{P_1}{P_0}\) for superheated and wet steam.

Fig. 4. Balance of losses due to moisture as a function of \(\dfrac{U}{C_0}\), according to [4].
1 — superheated steam, 2 — wet steam. \(\varepsilon=0.85,\ y_0=7.5\%,\ d_{\mathrm{avg}}=400\ \mathrm{mm},\ l=25\ \mathrm{mm}\)

Even with the same loss in a stage for superheated and wet steam, if the loss due to droplet moisture is taken as equal-

zero, curve 2 will be located below curve 1 (Fig. 4). Thus, for example, at $\varepsilon^{\mathrm{vl}}=\varepsilon^{\mathrm{p}}=0.5$ ($P_0=1.0$ ata, $P_1=0.5$ at), the theoretical heat drop of superheated steam is $H_0^{\mathrm{p}}=28$ kcal/kg ($t_0=145^\circ\mathrm{C}$, $t_1=80^\circ\mathrm{C}$, $X_1=1.0$), while for wet steam it is 24 kcal/kg ($y_0=7\%$, $y_1=10\%$). If the heat losses are assumed equal, $h_{\text{loss}}^{\mathrm{p}}=h_{\text{loss}}^{\mathrm{vl}}=5$ kcal/kg, the relative internal efficiency of the stage for superheated steam is in this case 82%, and for wet steam, 79%. Taking into account the higher disk-friction losses in the medium of wet steam [1], the difference in the efficiencies of wet and superheated steam will increase. An increase in steam wetness acts in the same direction.

In drawing up the balance of losses from droplet moisture in [4], an error was also made in the part concerning the dependence of these losses on $\dfrac{U}{C_0}$.

The loss due to breaking up the film and accelerating the droplets in the axial clearance does not depend on $\dfrac{U}{C_0}$ for unchanged $\varepsilon=\dfrac{P_1}{P_0}$, and its magnitude was determined in [4] for a stalled rotor, whereas in Fig. 4 this loss depends on $\dfrac{U}{C_0}$ and, for a stalled rotor, is equal to zero. The same also applies to the loss due to steam supercooling $\xi_{\mathrm{p}}$, which in one case in [4] does not depend on $\dfrac{U}{C_0}$ (for a stalled rotor), while in the other case in Fig. 4 (Fig. 6 in [4]) this loss depends on $\dfrac{U}{C_0}$ and at $\dfrac{U}{C_0}=0$ also turns out to be equal to zero. The authors of [4] do not explain these contradictions in their article.

REFERENCES

1. S. V. Polozhii. On the question of the influence of steam wetness on the efficiency of a turbine stage. Izv. vuzov SSSR—Energetika, No. 7, 1962.

2. S. V. Polozhii. Once again on the question of the influence of steam wetness on the efficiency of a turbine stage. Izv. vuzov SSSR—Energetika, No. 3, 1963.

3. S. V. Polozhii. On the question of the kinetics of the process of steam condensation in a turbine stage. Izv. TPI, vol. 137, 1965.

4. M. E. Deich, G. A. Filippov, V. V. Pryakhin, O. A. Povarov. Energy losses arising during the flow of wet steam in a turbine stage. Teploenergetika, No. 12, 1966.