LOSSES AND EFFICIENCY OF A STEAM TURBINE

V. T. YURINSKY

Up to the present time, our ideas about losses in a steam turbine have not received any more or less complete formulation. This question is treated with complete arbitrariness and contradiction, which does great harm to the training of heat-engineering engineers. At a time when, in the literature on the testing of steam turbines [3], the question of testing a modern multi-flow turbine is, as required, considered, in books on the theory of steam turbines, even the most recent ones [1; 2; 4], no attempt has yet been made to illuminate the question of the origin and distribution, by sources, of losses in a multistage multi-flow turbine. They are always limited to considering a single-flow turbine, i.e., the simplest type, which is not characteristic of our time, when even small turbines almost invariably have unregulated extractions for regenerative feedwater heating. It is also forgotten that, alongside the main working process in the turbine, certain other, auxiliary but inevitable processes also take place.

In the present work the author would like to bring clarity to the treatment of the question of losses and relative efficiencies of a multi-flow turbine operating under real conditions. A single-flow turbine, the simplest type, is a special case of a multi-flow turbine and therefore, like any special case, will be covered by the more comprehensive, general treatment.

Causes Giving Rise to Losses in a Steam Turbine. Classification of Losses

Losses in a steam turbine arise for the following principal reasons:

1. The conversion of thermal (potential) energy into mechanical energy in the turbine takes place imperfectly, since the isentropic expansion process is replaced by polytropic processes. As a result, the amount of mechanical energy obtained in the turbine stages will be less than the theoretically possible amount.

2. Part of the steam supplied to the turbine takes no part at all in the work of the turbine and is dissipated, bypassing the flow-through part. Another part of the steam performs work not in all stages of the turbine, but only in a definite group of stages. As a result, the power of the turbine, all other conditions being equal, will be less than that which could have been obtained in the absence of dissipation of the steam flow.

3. Part of the work obtained on the working wheels of the turbine stages and delivered to the shaft is expended in overcoming mechanical resistances (friction in the bearings, actuation of the governing system, etc.). In other words, this group of losses is determined by the presence of dissipation of the mechanical energy obtained in the turbine.

Fundamental classification of losses. Characteristic features of losses of different classes

In accordance with what has been said, one can and should distinguish:

1. Losses associated with the motion of the main (working) flow of steam through the turbine. We shall agree to call this group of losses internal losses; it might perhaps be more correct to call them losses of the working flow.

2. Losses due to the separation from the working flow, passing through the turbine, of certain quantities of steam that change their state after separation outside the flow path and outside the turbine chambers, without producing work on the rotor blades and without affecting the change in the state of the steam in the main flow.

3. Losses on the turbine shaft, which constitute an expenditure of mechanical energy. All losses belonging to the last two groups shall be called external, following the more or less established terminology, although, generally speaking, it would be more correct to call them losses from energy dissipation.

Thus, losses in a turbine are determined, on the one hand, by the imperfection of the processes of converting potential energy into mechanical work (a reduction in the steam’s capacity to do work), and, on the other hand, by the dissipation of both potential and mechanical energy.

The internal losses of a turbine may again be distinguished as losses of two categories. The main (working) steam flow, passing through the turbine, undergoes in it, in a definite sequence, a number of partial processes, namely:

1) expansion in the stages;
2) throttling (crushing) of the steam;
3) mixing of two or more steam flows which, before mixing, had undergone thermal processes differing from one another.

The indicated partial processes of throttling and mixing may occur both within the turbine stages (in its chambers) and outside the stages (before the turbine, between groups of stages, after the turbine). The expansion of steam, however, leading to the conversion of thermal energy into mechanical energy, can, naturally, occur only in the turbine stages (in the nozzles, in the passages of the rotor blades).

The internal losses accompanying the partial processes of expansion in the stages and caused by the polytropic character of the expansion processes, as well as by the presence in the stages, besides expansion processes, of processes of throttling and mixing of flows, may be called losses of the turbine stage.

The internal losses accompanying processes developing outside the stages cannot be assigned to any particular stage, and must be taken into account as losses of the turbine as a whole.

A characteristic feature of any internal loss is the possibility of representing it on thermal diagrams as a factor influencing the change in the state of the working steam flow passing through the turbine. In this case, the internal losses of a turbine stage, both those directly accompanying the expansion processes in the stages and those caused by the processes of crushing and mixing in the stages, are reflected in the change in the state of the working steam flow undergoing the partial processes in the corresponding stages and, consequently, are represented on thermal diagrams as the results of deviations of the real expansion processes in the stages from ideal isentropic (adiabatic) partial expansion processes, which leads to a reduction in the steam’s capacity to do work.

The internal losses of the turbine as a whole are necessarily localized out...

stages and are represented in heat diagrams as the occurrence of particular isenthalpic throttling processes of the working steam flow.

External losses caused by the dissipation of the steam flow supplied to the turbine, i.e., by the diversion of steam at various points of the main working process (steam extractions before the turbine or from intermediate stages, leakage through end seals, leakage past valve stems, etc.), cannot be represented in heat diagrams as factors directly affecting the change of state of the working steam flow passing through the turbine. The steam of the “branched-off” flows, of course, undergoes certain processes outside the turbine, but the changes in state of the “branched-off” flows proceed independently and are not connected with the change in state of the steam of the main flow.

External losses caused by the extraction of mechanical energy from the shaft, of course, cannot be depicted in heat diagrams as processes that have no relation to the change in the state of the steam.

Losses of a Turbine Stage. Classification. Origin of Losses and Their Manifestation

Considering more closely the processes within a turbine stage, we readily observe that for an individual stage as well it is correct to divide the losses into three groups:

1) those caused by the imperfection of the conversion of thermal energy into mechanical energy;

2) those caused by the dissipation (branching) of the steam flow within the limits of the given stage;

3) those caused by the reverse conversion into heat of part of the mechanical work obtained in the given stage.

On the other hand, the internal losses of a stage may be divided into two categories according to another criterion. Some losses inevitably arise in any turbine stage, regardless of how this stage is designed, how it is arranged relative to other stages, and under what conditions it operates. These include all losses at the rim of the stage, as well as the loss due to heat exchange with the surroundings (within the limits of the stage). Even in an ideal “adiabatic” stage having no losses in the nozzles or on the moving blades ($\psi=\varphi=1$), there still exists a loss with the exit velocity of the steam, belonging to the group of losses at the rim. Therefore the losses at the rim should be assigned to the group of inevitable stage losses. The theoretically possible case for an “adiabatic” stage, $c_{2t}=0$ (with $w_{2t}=u$), leading also to the absence of loss with exit velocity, has no relation to the turbine stages of modern designs that are of interest to us.

The second category of stage losses should include those whose occurrence is determined by the structural arrangement of the stage, the conditions of its operation, and its location in the turbine. For clarification it may be pointed out that, for example, the loss due to disk friction against steam and due to ventilation is of relatively great importance for a chamber stage with partial admission of steam, but is practically absent in a drum-type stage with full admission of steam. The loss due to operation in the region of wet steam naturally does not arise when superheated steam operates in the stage. At a certain degree of reaction and with a certain stage design, the loss due to internal steam leakage (leakage flow) may disappear.

As is known, the principal share of all internal losses of a stage consists of the losses at the rim (in the nozzles, on the moving blades, with the exit velocity).

A characteristic feature of these losses is the circumstance that they are reflected in the velocity triangles of the stage, and for this reason their magnitude is characterized by the blade efficiency \((\eta_{\mathrm{bl}})\). Consequently, the velocity coefficients \(\psi\) and \(\varphi\), the angles \(\alpha_1, \beta_2\), the circumferential velocities \(u_1, u_2\), and the degree of reaction \(\rho\) fully determine also the loss with exit velocity.

It is usually assumed that only part of the internal losses is reflected by the velocity triangles, while the other part cannot be taken into account, i.e., in the blade efficiency \((\eta_{\mathrm{bl}})\). We are therefore compelled to speak of two stage efficiencies: the blade efficiency and the relative internal efficiency of the stage \((\eta_{oi}^{\mathrm{st}})\). The latter efficiency already takes into account all stage losses without exception, regardless of the category to which they belong—unavoidable or caused by the particular operating conditions of the stage, regardless of the reasons for their occurrence.

It must be said plainly that up to the present time in the literature available to broad circles of readers we do not find sufficiently verified data for calculating the losses of a turbine stage. Even for the choice of the velocity coefficients \(\varphi\) and \(\psi\) we have only questionable curves, compiled, for example, by P. Wagner (these data first appeared in 1913 and since then have been stereotyped without exception in all books on steam turbines). Only the lack of experimental material compels us to take into account the influence of steam moisture on the stage efficiency by means of Bauman’s crudely empirical formula. Having carefully traced the mechanics of the occurrence of a number of losses in the stage (leakages, impacts of moisture droplets on the blades, impact at exit, etc.), we must come to the conclusion that they influence the elements of the velocity triangles, i.e., they also influence the value of the coefficient \(\eta_{\mathrm{bl}}\). On the other hand, as was indicated above, losses caused by the dispersion of the steam flow within the stage and by the expenditure of mechanical work in the stage can also be identified. But we find it difficult to draw up an energy balance for the stage and therefore must resort to crude empiricism.

An example of such empiricism is, in addition to the aforementioned Bauman formula, the method for calculating losses due to disk friction against steam and due to ventilation. In essence this loss is mechanical (let us recall that all empirical formulas for determining the loss express it in units of power—horsepower, kilowatts). We quite reasonably classify this loss, however, as an internal loss of the stage, since the mechanical energy taken from the wheel is expended here within the chamber of the stage and is converted at the place of consumption into heat, which ultimately is reflected in the state of the steam leaving the stage. The stage not only produces mechanical energy but also consumes part of this energy. The process occurring with the release of mechanical energy is accompanied by processes associated with the absorption of mechanical energy, with its conversion into heat. In the chamber, along with the main process of the turbine stage, which delivers mechanical energy to the shaft, there occurs a secondary process (in quantitative relations) of a compressor stage consuming mechanical energy.

Internal losses from the dispersion (branching) of the steam flow within the stage are caused by the presence of steam leakages through inter-chamber seals (diaphragms), relief holes in disks, and through clearances in the flow passage.

Within each stage there is branching of the steam flow (branching and reverse merging). While the main mass of steam performs the useful process in the nozzles and on the rotor blades, a certain quantity of steam (“branched-off” streams), instead of the desired process, ...

in the nozzles and on the blades, flows through the clearances, undergoing a process of throttling or expansion without delivering work to the turbine blades. The steam forming such “branched-off” flows either retains its initial heat content or acquires considerable kinetic energy. Therefore, when the “branched-off” flows subsequently merge with the main (working) flow, the final result of the mixing process will be an increase in the heat content of the steam compared with the heat content of the steam of the main flow before mixing. The possibility of an increase in the steam-supply velocity to the next stage is also not excluded. Thus, both the scattering of the steam flow in the stage and the expenditure of mechanical energy within the stage lead to a reduction in the delivery of mechanical energy to the turbine shaft and to an increase in the store of energy at the inlet to the next stage, in particular, the heat content of the steam increases. In other words, the presence of these losses produces the same effect as the imperfection of the processes carried out in the stage by the working flow.

As we see, in a turbine stage, transformations of energy are possible, and in fact do occur, according to a fairly complex scheme of processes. Thermal energy, after passing through a series of transformations, ultimately appears in part once again in the form of heat, but now at a lower potential, having partly lost its ability to be transformed into kinetic energy, into mechanical work.

The possibility of losses passing from one class into another

The question arises: when considering the turbine as a whole, may we not encounter similar transformations of energy? Could it not turn out that some fraction of the external losses due to the scattering of the steam flow supplied to the turbine affects the state of the steam of the main flow and thus passes into the category of internal turbine losses? This question is answered in the affirmative. During operation of the turbine, some “branched-off” flows, having separated from the main (working) flow, no longer return to the turbine. It is clear that, whatever processes are performed by such branched-off flows, these processes cannot exert a direct influence on the process performed by the main flow. Consequently, it would be meaningless to try to represent the losses caused by the branching-off of such flows on heat diagrams as factors reducing the working capacity of the main (working) flow, i.e., as internal losses.

But let us imagine that the branched-off flow, after performing a series of processes, has returned to the turbine. The processes performed by the branched-off flows outside the turbine may be different. Some of the processes (expansion in an engine) may lead to a reduction in the heat content and pressure of the steam of the branched-off flow; others (cooling in a heat exchanger), theoretically speaking, only to a reduction in heat content without a reduction in pressure.

Finally, throttling processes are possible, i.e., a lowering of pressure at unchanged heat content. Having performed some combination of these processes and returned to the turbine, the branched-off flow, mixing with the main (working) flow, will influence the change in the state of the steam of the main (working) flow. From this one may draw the correct conclusion that some external losses prove to be factors influencing the change in the state of the main flow. But it would be incorrect to conclude from this that this assertion contradicts the proposition that it is impossible to represent external losses on heat diagrams as factors influencing the change in the state of the steam of the main flow. If one considers more deeply

essentially, we shall find no contradiction. Indeed, suppose, for example, that we are considering the question of an external loss caused by scattering of the steam flow in connection with the need to supply steam to the front and rear end seals of a condensing two-casing steam turbine (Fig. 1) under no-load operation. The flow distribution in the seal pipelines under this condition is established according to arrows “б”. The steam branched off before the turbine undergoes a number of parallel throttling processes in the labyrinth end seals and flows out into the chamber of the regulating stage of the HP cylinder and into the chambers of the last stage of the LP cylinder. Mixing of the branched flows with the main (working) flow leads to a change in the state of the steam in the indicated stages,

Fig. 1

which is reflected in the course of the line of change of state in the heat diagrams. But at the very moment when the branched steam flow joined the main flow, it ceased to be a branched flow. It again became part of the main (working) flow, and therefore the change of state caused by the fact of its joining was shown in the heat diagrams as a change in the state of the main steam flow. At the same time, the quantity of steam flowing through the turbine in the main (working) flow also changed. The return of the branched flow to the principal channel leads to the fact that not all the energy removed at the point with the higher potential is scattered, but only part of it; the subsequent transformations of the energy of the returned branched flow (it may be called an admitted flow) proceed as a process of the main flow and will therefore be accompanied by the occurrence of internal losses.

We see that the occurrence and development of losses of a certain class may be accompanied by the disappearance or reduction of losses of another class. This statement may also be extended to the mechanical losses of a turbine. Let us imagine, for example, that instead of an oil pump driven by the turbine shaft (this determines the need to expend power on the pump and the occurrence of the corresponding component of mechanical loss), the turbine has received an oil pump with a steam drive. The sum of mechanical losses is now reduced, but instead a new branched flow appears, necessary for ensuring the operation of the steam-

drive of the oil pump, and, in connection with this, the dissipation of the steam flow supplied to the turbine must become greater. The introduction of a hydraulic seal into the turbine design may lead to a reduction in the dissipation of the steam flow (steam leakage through the end seals will become smaller), but in return the dissipation of mechanical energy will increase, since the hydraulic seal will expend work for its operation, borrowing it from the turbine shaft.

The dissipation of mechanical energy will also include the expenditure of it on putting auxiliary mechanisms into operation (for example, the circulating and condensate pumps) directly from the turbine shaft.

Nomenclature of the Efficiencies of a Steam Turbine and the Relation between Them

Losses in a steam turbine may be represented by means of a suitably constructed system of efficiencies having a chain relation among themselves. The above consideration of the question of the causes of occurrence and classification of losses gives grounds for proposing a scheme of distribution of losses in a steam turbine and the system of efficiencies following from this scheme (Table 1).

Explanations of this scheme are given below.

Internal Losses of the Turbine

A. Stage

Losses in the stage are associated with the course of the process of the main (working) flow. As stated, at present it is customary to speak of two stage efficiencies, $\eta_{o\ell}$ and $\eta_{oi}^{st}$.

Since the losses at the blade row are represented by the stage velocity diagrams, this group of losses is taken into account by the blade-row efficiency

\[ \eta_{o\ell} = \frac{ \dfrac{A}{g}\left(u_1 c_{1u} - u_2 c_{2u}\right) }{ h_0 + \mu_0 h_{v(n-1)} - \mu_2 h_{v(n)} }. \tag{1} \]

All stage losses without exception are taken into account by means of the relative internal efficiency of the stage

\[ \eta_{oi}^{st} = \frac{ h_i }{ h_0 + \mu_0 h_{v(n-1)} - \mu_2 h_{v(n)} }. \tag{2} \]

In formulas (1), (2):

$\mu_0$ — coefficient of utilization of the leaving loss of the preceding stage,

$\mu_2$ — the same, of the stage under consideration,

$h_0$ — available heat drop of the stage, kcal/kg,

$h_i$ — internal (utilized) heat drop of the stage, kcal/kg,

$h_{v(n-1)}$ — leaving loss of the preceding stage, “

$h_{v(n)}$ — “ “ of the stage under consideration “ “.

We shall agree to understand by the relative internal efficiency of the process of steam expansion in the turbine the ratio of the sum of the internal heat drops of the stages to the available heat drop of a real turbine operating with steam throttling before the first stage, in the intervals between groups of stages, and, finally, after the last stage.

Let us denote this efficiency by \(\eta_{oi}\). Then (see Fig. 2)

\[ \eta'_{oi} = \frac{H'_i}{H^I_o + H^{II}_o + H^{III}_o + \cdots} = \frac{\sum h_i}{H^I_o + H^{II}_o + H^{III}_o + \cdots}. \tag{3} \]

Fig. 2. Losses and efficiencies of a steam turbine

Table 1

• Throttling and mixing losses
• Throttling in the starting valve
• Throttling in the regulating valves
• Throttling in the bypass regulating valves
• Throttling at the steam outlet from the turbine
• Mixing of steam flows in receivers

\[ \eta_{\mathrm{th}} \]

Turbines as a whole
(processes outside the stages)

• Stage losses
• Loss in nozzles
• Loss on blades
• Exit loss
• Other losses in the stage

\[ \eta^{\mathrm{st}}_{oi} = \eta^{\mathrm{st}}_{ol}\cdot \eta^{\mathrm{st}}_{vs}\cdot \eta^{\mathrm{st}}_{m} \]

\[ \eta'_{oi} = \frac{\sum h_o \cdot \eta^{\mathrm{st}}_{oi}} {H^I_o + H^{II}_o + \cdots} \]

Stage turbines
(processes in stages)

• Internal losses
  (imperfection of the processes of converting the potential energy of steam into mechanical work)

\[ \eta_{oi}=\eta_{\mathrm{th}}\cdot\eta'_{oi} \]

• Steam-flow losses
• Steam extractions before the turbine
• Intermediate steam extractions
• Leakages through rod clearances
• Leakages through end seals
• Return and admission of steam flows

\[ \eta_{\mathrm{uch}} \]

Dissipation of the steam flow
(occurrence of unaccounted-for flows)

• Mechanical losses
• Friction in bearings
• Oil pump
• Regulation
• Condensate pumps driven from the turbine shaft

\[ \eta_m \]

Dissipation of mechanical energy
(expenditure of work from the turbine shaft)

• External losses
  (energy dissipation)

\[ \eta_{\mathrm{uch}}\cdot\eta_m \]

• Steam-turbine losses

\[ \eta_{oe} = \eta_{\mathrm{th}}\cdot\eta'_{oi}\cdot\eta_{\mathrm{uch}}\cdot\eta_m \]

For the particular case of a turbine in which there is no steam extraction in the intervals between groups of stages,

\[ H_0^I + H_0^{II} + \ldots = H_0, \]

\[ \eta_{0i}'=\frac{\sum h_i}{H_0}=\frac{H_i}{H_0}. \tag{4} \]

Fig. 2

Since for any stage one may write

\[ \eta_{0i}^{st}=\frac{h_i}{h_0'}, \]

where \(h_0'=h_0+\psi_0 h_{v(n-1)}-\psi_2 h_{v(n)}\), then

\[ \eta_{0i}'=\frac{\sum h_0'\eta_{0i}^{st}}{H_0^I+H_0^{II}+\ldots}. \tag{5} \]

As we see, the relative internal efficiency of the steam-expansion process in the turbine takes into account, in total, the internal losses in all stages of the turbine.

This efficiency makes it possible to relate the efficiencies of the stages to the efficiency of the turbine as a whole.

B. The Turbine as a Whole (Processes of the Working Flow Occurring Outside the Stages)

In an ideal turbine there are no processes of steam throttling in the regulating organs, receiver pipes, etc. Consequently, the process of such a turbine is carried out within the pressure variation from \(p'_0\) (pressure before the stop valve) to \(p'_2\) (pressure in the condenser). The isentropic process of such an ideal turbine will be represented by the line \(A'_0 A'_{2t}\); the available heat drop of the ideal turbine will be \(H^{\text{id}}_0\) (Fig. 2).

The presence of throttling processes in a real turbine leads to the result that its available heat drop (and the power proportional to it) is reduced.

This reduction of the available heat drop in comparison with the ideal turbine can be taken into account by means of a throttling coefficient

\[ \eta_{\text{dr}}=\frac{H'_0+H''_0+\ldots}{H^{\text{id}}_0}. \tag{6} \]

In the special case of a turbine without intermediate throttling between groups of stages,

\[ \eta_{\text{dr}}=\frac{H_0}{H^{\text{id}}_0}. \tag{7} \]

The concept of the throttling coefficient must be clarified in connection with the need to extend it to a turbine having nozzle governing (Fig. 3).

Here we may reason as follows. Let \(D\) kg/hour of steam flow through a turbine with nozzle governing. This quantity of steam is divided at each instant (when not one group valve is open, but two or several group valves are open) into two flows. One flow (I) operates with minimum throttling at admission (the initial point of the process in the governing stage is point \(A'_0\)); the other—flow (II)—is throttled more strongly (initial point \(A''_0\)). Accordingly, the available heat drops are: for flow I, \(H_{0I}=H_0\), and for flow II, \(H_{0II}\).

Taking internal losses into account, flow I undergoes in the governing stage the process \(A'_0 A_I\), and flow II the process \(A''_0 A_{II}\). After mixing in the chamber of the governing stage, both flows undergo in the remaining stages the common process \(A A_2\).

Accordingly, one may speak of a weighted-average throttling coefficient.

Fig. 3

In an ideal turbine having no steam throttling, the power obtained would be

\[ N^{\text{id}}_0=\frac{D H^{\text{id}}_0}{860}\ \text{kW}. \]

In an actual turbine with nozzle governing, owing to steam throttling at admission and at exhaust (Fig. 3), even theoretically, with an isentropic process and with no losses in the stages, such a power cannot be obtained.

The losses of available power of stream I are

\[ N_{\mathrm{I}}=\frac{D_{\mathrm{I}}\left(H_0^{\mathrm{id}}-H_{0\mathrm{I}}\right)}{860}. \]

The loss of available power of stream II is

\[ N_{\mathrm{II}}=\frac{D_{\mathrm{II}}\left(H_0^{\mathrm{id}}-H_{0\mathrm{II}}\right)}{860}. \]

Here \(D_{\mathrm{I}}, D_{\mathrm{II}}\) are the quantities of steam of streams I and II in kg/h; \(D = D_{\mathrm{I}} + D_{\mathrm{II}}\) is the total steam consumption by the turbine in kg/h.

The available power of a turbine with nozzle governing is

\[ N_0 = N_0^{\mathrm{id}}-\left(N_{\mathrm{I}}+N_{\mathrm{II}}\right); \]

\[ N_0=\frac{1}{860}\left[DH_0^{\mathrm{id}}-D_{\mathrm{I}}\left(H_0^{\mathrm{id}}-H_{0\mathrm{I}}\right)-D_{\mathrm{II}}\left(H_0^{\mathrm{id}}-H_{0\mathrm{II}}\right)\right]. \]

Further,

\[ N_0=\frac{DH_0^{\mathrm{id}}}{860}\left(1-\frac{D_{\mathrm{I}}}{D}\cdot\frac{H_0^{\mathrm{id}}-H_{0\mathrm{I}}}{H_0^{\mathrm{id}}}-\frac{D_{\mathrm{II}}}{D}\cdot\frac{H_0^{\mathrm{id}}-H_{0\mathrm{II}}}{H_0^{\mathrm{id}}}\right). \]

Denoting

\[ \frac{D_{\mathrm{I}}}{D}=x_{\mathrm{I}}; \]

\[ \frac{D_{\mathrm{II}}}{D}=x_{\mathrm{II}}, \]

we have, after simple transformations,

\[ N_0=\frac{DH_0^{\mathrm{id}}}{860}\left(1-x_{\mathrm{I}}-x_{\mathrm{II}}+x_{\mathrm{I}}\frac{H_{0\mathrm{I}}}{H_0^{\mathrm{id}}}+x_{\mathrm{II}}\frac{H_{0\mathrm{II}}}{H_0^{\mathrm{id}}}\right). \]

It may be noted that

\[ x_{\mathrm{I}}+x_{\mathrm{II}}=1 \]

and, consequently,

\[ N_0=\frac{DH_0^{\mathrm{id}}}{860}\left(x_{\mathrm{I}}\frac{H_{0\mathrm{I}}}{H_0^{\mathrm{id}}}+x_{\mathrm{II}}\frac{H_{0\mathrm{II}}}{H_0^{\mathrm{id}}}\right). \]

It is self-evident that the throttling coefficient, representing the ratio of available heat drops, can also be expressed as the ratio of available powers at the same steam consumption \(D\):

\[ \eta_{\partial p}=\frac{N_0}{N_0^{\mathrm{id}}} = \frac{ \dfrac{DH_0^{\mathrm{id}}}{860} \left( x_{\mathrm{I}}\dfrac{H_{0\mathrm{I}}}{H_0^{\mathrm{id}}} + x_{\mathrm{II}}\dfrac{H_{0\mathrm{II}}}{H_0^{\mathrm{id}}} \right) }{ \dfrac{DH_0^{\mathrm{id}}}{860} }, \]

\[ \eta_{\partial p}=x_{\mathrm{I}}\frac{H_{0\mathrm{I}}}{H_0^{\mathrm{id}}}+x_{\mathrm{II}}\frac{H_{0\mathrm{II}}}{H_0^{\mathrm{id}}}. \]

If, in the governing stage, processes were carried out simultaneously for not two but a larger number of flows, then the throttling coefficient would be

\[ \eta_{dr}=x_{1}\frac{H_{01}}{H_{0}^{id}}+x_{11}\frac{H_{011}}{H_{0}^{id}}+x_{111}\frac{H_{0111}}{H_{0}^{id}} \tag{8} \]

The last formula may be represented in the following form:

\[ \eta_{dr}=x_{1}\frac{H_{01}}{H_{0}^{id}}\left(1+\frac{x_{11}}{x_{1}}\cdot\frac{H_{011}}{H_{01}}+\ldots\right) \tag{9} \]

Let flow I be subjected to the minimum throttling (Fig. 3), \(H_{01}=H_{0}\)

\[ \eta_{dr}=x_{1}\frac{H_{0}}{H_{0}^{id}}\left(1+\frac{x_{11}}{x_{1}}\frac{H_{011}}{H_{0}}+\ldots\right) \tag{10} \]

With throttle governing, only one steam flow remains in the governing stage.

Accordingly,

\[ x_{1}=1;\quad x_{11}=x_{111}=\ldots=0, \]

as a result of which we arrive at expression (7) given above.

The internal losses of the turbine as a whole are thus taken into account by the throttling coefficient. This coefficient is of very great importance. In the process of governing the turbine, the throttling coefficient is deliberately changed.

With nozzle governing, the throttling coefficient is determined by the fractional participation of steam from the different flows in the work of the turbine and by the throttling coefficients of these flows, since formula (8) may also be written as

\[ \eta_{dr}=x_{1}\eta_{dr}^{1}+x_{11}\eta_{dr}^{11}+\ldots= \]

\[ =x_{1}\eta_{dr}^{1}\left(1+\frac{x_{11}}{x_{1}}\frac{\eta_{dr}^{11}}{\eta_{dr}^{1}}+\ldots\right), \tag{11} \]

where

\[ \eta_{dr}^{1}=\frac{H_{01}}{H_{0}^{id}},\quad \eta_{dr}^{11}=\frac{H_{011}}{H_{0}^{id}},\ldots \]

are the throttling coefficients of the individual flows.

All internal losses in the turbine can be taken into account by means of the relative internal efficiency of the turbine

\[ \eta_{oi}=\eta'_{oi}\cdot\eta_{dr}. \]

For a turbine with throttle governing without intermediate steam throttling,

\[ \eta'_{oi}=\frac{H_i}{H_0}; \]

\[ \eta_{dr}=\frac{H_0}{H_0^{id}}; \]

\[ \eta_{oi}=\frac{H_i}{H_0^{id}}=\frac{\sum h'\eta_{oi}^{em}}{H_0^{id}}. \]

For a turbine with nozzle governing,

\[ \eta_{dr}=K_{dr}\cdot\eta_{dr}^{1} \]

The quantity

\[ K_{\mathrm{dr}}=x_1\left(1+\frac{x_{\mathrm{n}}}{x_1}\frac{\eta_{\mathrm{dr}}^{\mathrm{n}}}{\eta_{\mathrm{dr}}^{1}}+\ldots\right) \]

may be called the coefficient of reduction to throttling (the simplest) control.

Thus, for any turbine

\[ \eta_{oi}=K_{\mathrm{dr}}\cdot \eta_{\mathrm{dr}}^{1}\cdot \eta'_{oi}, \]

for a turbine with throttling control

\[ K_{\mathrm{dr}}=1. \]

External losses

Losses due to dispersion of the steam flow

Losses due to dispersion of the steam flow can be represented by means of the coefficient of participation of the steam in the work of the turbine. Let us denote this coefficient by \(\eta_{uy}\).

To obtain the coefficient of participation of the steam in the work of the turbine, one may be guided by the following considerations.

Let \(D\) kg/hour of steam be supplied to the turbine at the initial parameters \(p'_0, t'_0\). If all this steam worked in the turbine in the absence of branched and rotating flows, then the internal power obtained in the turbine would be

Fig. 4

Fig. 5

\[ N'_i=\frac{D H_i}{860}\ \mathrm{kW}. \]

As a result of branching of the steam flow, i.e., the appearance of branched and rotating flows (see Figs. 4, 5, 1), a smaller power will be obtained, namely

\[ N_i=\frac{1}{860}\left\{D H_i-\left[D_{ym}H'_i+D_1H_{i1}+D_2\left(H_i-H_{i2}\right)+ +D_3H_{i3}+D_4H_{i4}+D'_{ym}\left(H_i-H_{i3}\right)\right]\right\}. \]

Coefficient of participation of the steam flow in the operation of the turbine

\[ \eta_{uch}=\frac{N_i}{N'_i} =1-\frac{D_{ut}}{D}-\frac{D'_{ut}}{D}\frac{H_i-H_{i3}}{H_i} -\frac{D_1}{D}\frac{H_{i1}}{H_i}-\ldots = \]

\[ =1-\frac{D_{ut}-D'_{ut}}{D} -\frac{D_1}{D}\frac{H_{i1}}{H_i} +\frac{D'_{ut}}{D}\frac{H_{i3}}{H_i} -\frac{D_3}{D}\frac{H_{i3}}{H_i} + \]

\[ +\frac{D_2}{D}\frac{H_{i2}}{H_i} -\frac{D_4}{D}\frac{H_{i4}}{H_i}. \]

With the complete absence of branched-off steam flows, the coefficient of participation of the steam flow in the operation of the turbine becomes equal to unity, since all the “branched-off” quantities of steam separately, and in their sum, become equal to zero.

The general expression for the coefficient of participation of the steam flow in the operation of the turbine will be

\[ \eta_{uch}=1-\sum \frac{D_{otv}}{D}\cdot\frac{H_{ix}}{H_i} +\sum \frac{D_{vozvr}}{D}\cdot\frac{H_{iy}}{H_i} +\sum \frac{D_{pris}}{D}\cdot\frac{H_{iz}}{H_i}. \]

Here

\[ \begin{aligned} D &\text{ — kg/hour, the quantity of steam supplied to the turbine,}\\ D_{otv} &\text{ — kg/hour, the quantities of steam of the branched-off flows,}\\ D_{vozvr} &\text{ — the same for the returned flows,}\\ D_{pris} &\text{ — the same for the added flows,}\\ H_{ix} &\text{ — kcal/kg, internal drops not utilized by the branched-off flows,}\\ H_{iy}, H_{iz} &\text{ — internal drops utilized by the returned and added flows,}\\ H_i &\text{ — the internal drop of the turbine.} \end{aligned} \]

It is of interest to note the following: a decrease in \(\eta_{uch}\) leads, as is evident from formula (13), to a decrease in \(\eta_{oe}\) of the unit. Consequently, from the standpoint of the economy of the turbine as a device for obtaining mechanical energy, it is disadvantageous to dissipate steam; it is disadvantageous to allow it to pass outside the flow section, wherever it may be directed. But modern turbine construction is based on the broad use of the ideas of regenerative heating of feed water, i.e., on carrying out intentional (and sufficiently large) leaks from a number of intermediate stages. At the same time, a struggle is being waged against harmful leaks through end seals, etc. We know that carrying out the regenerative cycle is advantageous because it ultimately leads to an increase in the absolute effective efficiency of the steam-power installation,

\[ \eta_e=\eta_{oe}\cdot\eta_t \]

at the expense of increasing \(\eta_t\), despite the reduction in \(\eta_{oe}\).

In the absence of flows branched off from intermediate stages, in the absence of flows returned to intermediate stages (return of flow downstream of the turbine, for example, to the condenser, plays no role), and, finally, in the absence of added flows, we can write

\[ \eta_{uch}=1-\frac{D_{ut}}{D}. \]

With the complete absence of leaks \((D_{ut}=0)\)

\[ \eta_{uch}=1. \]

Let us note that the coefficient of participation of the steam flow in the turbine takes into account the presence of steam extractions not only from intermediate points of the turbine, but also before it for auxiliary purposes (ensuring operation of the condensing unit, steam-driven oil pumps, etc.).

Mechanical Losses

Mechanical losses, as is known, are accounted for by the mechanical efficiency

\[ \eta_{\mathrm{m}}=\frac{N_e}{N_i}. \]

By \(N_e\) is meant the power at the turbine coupling, transmitted to the generator, pump, etc.

In connection with the development of gas turbines, we must remember that by \(N_e\) one should understand not the power developed by the turbine, but that transmitted to the generator, i.e., after subtracting the power consumed by the compressor. Thus gas turbines will have a mechanical efficiency immeasurably lower than that of steam turbines, which do not require the expenditure of mechanical energy to ensure the operation of a compressor. This circumstance must not be forgotten when comparing steam and gas turbines.

Relative Effective Efficiency of a Turbine

As is clear from the foregoing (see Table 1), the relative effective efficiency of a steam (or gas) turbine may be expressed by the formula:

\[ \eta_{oe}=\eta'_{oi}\cdot\eta_{dr}\cdot\eta_{uch}\cdot\eta_{\mathrm{m}}. \tag{13} \]

This expression should replace the expression usually found in the literature,

\[ \eta_{oe}=\eta_{oi}\eta_{\mathrm{m}}, \]

which is correct only under the particular assumption

\[ \eta_{uch}=1; \]

\[ \eta_{dr}=1; \]

\[ \eta'_{oi}=\eta_{oi}. \]

It is thought that a methodologically correct exposition of the question of losses and efficiencies of a steam turbine, oriented toward consideration of the general case of a multi-flow turbine, should be useful in the training of thermal engineers and should guide the study of operating turbines and the conduct of experimental investigations along the proper path.

The question arises whether, for a separately taken turbine stage as well, one may apply the same classification of efficiencies, the same system of efficiencies, as for the turbine as a whole.

From what has been said it follows with sufficient clarity that this is entirely permissible and, perhaps, expedient. In a turbine stage, just as in a multi-flow turbine, branching of flows occurs, i.e., scattering of steam, and mechanical energy taken from the rotor is expended. Therefore it will be quite correct to represent the relative internal efficiency of a stage in the following form:

\[ \eta_{oi}^{st}=\eta_{ol}\cdot\eta_{uch}^{st}\cdot\eta_{\mathrm{m}}^{st}. \]

In the present case the following reasoning may be carried out: if all the steam supplied to the stage did not branch and, flowing out of the nozzle, [[unclear: word continues on next page]]

did work on the moving blades, the power developed by the stage would be

\[ N''_{\text{int}}=\frac{3600G\cdot h_i}{860}\simeq 4.2Gh_i. \]

As a result of the scattering of the steam flow, a smaller power will be obtained:

\[ N'_{\text{int}}=\frac{3600}{860}\left[Gh_i-G_{1y}h_{i1}-G_{2y}h_{i2}-G_{3y}h_i\right]. \]

Here \(h_{i1}, h_{i2}, h_i^{1})\) are the unused internal drops, \(\dfrac{\text{kcal}}{\text{kg}}\); \(G_{1y}, G_{2y}\ldots\) are leakages in \(\text{kg}/\text{sec}\); \(G\) is the total steam flow through the stage, \(\text{kg}/\text{sec}\).

\[ N'_{\text{int}}=4.2Gh_i\left[1-\frac{G_{1y}}{G}\frac{h_{i1}}{h_i}-\frac{G_{2y}}{G}\frac{h_{i2}}{h_i}-\frac{G_{3y}}{G}\right]. \]

It is clear that the coefficient of participation of the flow in the work of the stage is

\[ \eta^{\text{st}}_{\text{part}}=\frac{N'_{\text{int}}}{N''_{\text{int}}} =1-\frac{G_{1y}}{D}\frac{h_{i1}}{h_i} -\frac{G_{2y}}{G}\frac{h_{i2}}{h_i} -\frac{G_{3y}}{G}. \]

Analogously, the mechanical efficiency of a turbine stage can be determined.

If no energy were taken from the rotor in the form of mechanical work spent on increasing the heat content of the steam leaving the stage, then the stage would develop the power \(N'_{\text{int}}\). In reality, it gives a smaller power \(N_i\).

Obviously, the mechanical efficiency of the stage is

\[ \eta^{\text{st}}_{m}=\frac{N_{i\text{int}}}{N'_{\text{int}}}, \]

but

\[ N_{i\text{int}}=N'_{\text{int}}-N_{mv}, \]

\[ \eta^{\text{st}}_{m}=\frac{N'_{\text{int}}-N_{mv}}{N'_{\text{int}}} =1-\frac{N_{mv}}{N'_{\text{int}}}. \]

LITERATURE

1. Zhiritskii G. S.—Gas Turbines, Energoizdat, Moscow–Leningrad, 1948.
2. Kirillov I. I. and Kantor S. A.—Theory and Designs of Steam Turbines, State Scientific-Technical Publishing House of Machine-Building Literature, Moscow–Leningrad, 1947.
3. Shcheglyaev A. V. and Morozov N. G.—Testing of Steam Turbines, Energoizdat, Moscow–Leningrad, 1937.
4. Shcheglyaev A. V.—Steam Turbines, Energoizdat, Moscow–Leningrad, 2nd ed., 1947.

\({}^{1})\) Steam flowing past the blades in the amount \(G_{3y}\) does no work at all, i.e., it does not utilize the drop \(h_i\).