11.2 The Rankine cycle

These shortcomings inherent in a steam power plant realizing the Carnot cycle with wet steam can be partly remedied if heat is rejected from the wet steam in the condenser before the entire steam condenses. Thus, not wet steam with a small density but water is compressed from pressure p2 to pressure p1 Compared to the specific volumes of wet steam at point 3 (see the      T-s diagram in Fig. 11.2), the specific volume of water is small. Its compressibility is also negligible compared with that of wet steam. Water is displaced from the condenser into the boiler and its pressure is simultaneously increased not with the aid of compressors but pumps of compact design, simple arrangement and actuated by a rather low-power drive.

Such a cycle was suggested in the 1850's almost simultaneously by the Scottish engineer and physicist W. Rankine and by R. Clausius, and the cycle is usually called the Rankine cycle. The schematic layout of a steam power plant operating on a Rankine cycle is similar to the schematic layout depicted in Fig. 11.1, the only difference being that in the Rankine cycle the 5 denotes a water pump and not a wet-steam compressor.

 

Fig.11.4.jpg

 

Fig. 11.4

 

Figure 11.4 shows the Rankine cycle on a T-s diagram. In the condenser wet steam condenses completely along the isobar p2 = const (point 3 in Fig. 11.4). Water is then compressed in the pump from the pressure p2 to the pressure p1; this adiabatic process is represented on the T-s diagram by the vertical line 3-5.

The length of line 3-5 is small; as was already mentioned in Chapter 6, in the liquid region the isobars are plotted on the T-s diagram very close to one another. Thus, when water at a temperature of 25 °C and a saturation pressure of 3.1 kPa (0.032 kgf/cm2) is compressed isentropically to a pressure of 29 400 kPa (300 kgf/cm2), water temperature rises by less than 1 °C, and we can assume with a good degree of accuracy that in the liquid region the isobars for water virtually coincide with the left boundary curve. Therefore, in plotting the Rankine cycle on a T-s diagram, in the water region the isobars are shown as merging with the left boundary curve.

The small length of adiabat 3-5 is evidence of the small amount of work expended by the pump to compress the water.

It will be recalled that, as was shown in Sec. 7.9, the work spent to com­press the gas from pressure p2 to pressure p1 in a compressor is determined from relationship (7.188), which for      1 kg of working medium takes the fol­lowing form:

 

 

[since v (p1) < v (p2), we have l21 < 0], and the work expended to realize the entire compressor cycle (the mechanical work of compression) is determined from the formula

 

 

It will also be recalled that for adiabatic compression, in accordance with Eq. (8.15),

 

 

where iout and iin are the enthalpies of the gas at the compressor outlet (pressure p1) and before compression (pressure p2), respectively.

Equations (7.188) and (7.195a) do not depend on the substance compres­sed; of course, the equations also hold for the compression of liquid with the aid of a pump.

As a first approximation, quite sufficient for technical calculations, water can be assumed as virtually incompressible (vw = const, i.e. dvw = 0) and, consequently,

 

 

As to the mechanical work done by a pump, taking in Eq. (7.195a) the quantity vw outside the integral sign, we obtain:

                                                                                       (11.1)

 

(the minus sign indicates that work must be transferred to the pump from an external source of work).

The mechanical work done by a pump to compress water is also small. For instance, if water is compressed from a pressure p2 = 3.1 kPa (0.032 kgf/cm2) to a pressure p1 = 49 030 kPa (500 kgf/cm2), according to Eq. (11.1), the work of the pump is

 

                                                      [1]

 

The same result can be obtained, using equation (8.15). For this purpose, with the aid of an i-s diagram or Steam Tables, we find the difference between the enthalpies of water on a given isentrop at pressures p1 and p2.

The feed pump delivers water at a pressure pl to the boiler in which heat is added isobarically at p1 = const. In the boiler water is first heated to the boiling point (section 5-4 of the isobar  p1 = const in Fig. 11.4) and then vapourized (section 4-1 of the isobar p1 = const in Fig. 11.4). The dry saturated vapour, generated in the boiler, passes into the turbine; the process of steam expansion in the turbine is represented by the adiabat 1-2. The waste wet steam is exhausted into the condenser and the vapour cycle closes.

From the viewpoint of thermal efficiency the Rankine cycle seems to be less expedient than the reversible Carnot cycle depicted in Fig. 11.2, inasmuch as the area ratio (just as the average temperature of heat addition) is less for the former. However, allowing for the practical conditions under which the cycle is to be realized and also for the considerably smaller effect of the irreversibility of the process of water compression, compared with the compression of wet vapour, on the overall efficiency of a cycle, the Rankine cycle is more economical than the corresponding Carnot cycle for wet steam. At the same time the replacement of the cumbersome compressor, ensuring compression of the wet steam with a compact feed water pump permits a substantial reduction in the costs involved in building a steam power plant, and a simplification of its maintenance.

 

Fig.11.5.jpg

 

Fig. 11.5

 

Fig.11.6.jpg

 

Fig. 11.6

 

Thus, the internal absolute efficiencies of the two cycles will be approx­imately the same.

The thermal efficiency of the Rankine cycle is increased by superheating the steam in a special element of the steam boiler, the steam superheater (denoted SH in Fig. 11.5) in which steam is heated to a temperature exceed­ing the saturation temperature at the given pressure p1. The T-s diagram of the Rankine cycle with superheated steam is shown in Fig. 11.6. With superheating the mean temperature of heat addition increases compared with the temperature at which heat is added in a cycle without superheat. Consequently the thermal efficiency of the cycle increases too.

It can be seen from Fig. 11.6 that in the Rankine cycle with superheat the process of steam expansion in the turbine 1-2, realized up to the same pressure as before, p2, ends inside the two-phase region at a dryness fraction higher than in the cycle depicted in Fig. 11.4. Because of this, the turbine blading operates under lighter conditions and, consequently, there is an increase in the internal relative efficiency of the turbine and in the internal efficiency of the cycle ; for a cycle with superheat the efficiency increases both because of an increase in  and in the internal relative efficiency .

The Rankine cycle with superheat is the basic cycle for thermopower plants with application in up-to-date heat and power engineering.

The quantity of heat added to the working medium in the cycle, q1, is represented on the        T-s diagram shown in Fig. 11.6 by the area a-3-5-4-6-l-b-a. The heat rejected in the cycle, q2, is equivalent to area a-3-2-b-a, and the work output of the cycle, to area 3-5-4-6-1-2-3.

Since in the Rankine cycle the processes of heat addition and rejection are isobaric, and in an isobaric process the quantity of heat added (rejected) is equal to the difference between the enthalpies of the working medium at the beginning and end of the cycle, as applied to the Rankine cycle, we can then write

                                                                                                                          (11.2)

                                                                                                                          (11.3)

 

 (the subscripts for i correspond to the notations of the state of the working medium used in Fig. 11.6).

Here, i1 is the enthalpy of superheated water vapour (steam) at the exit of the boiler[2]  (at a pressure p1 and a temperature T1); i5 is the enthalpy of water at the boiler inlet, i.e. at the pump outlet (at a pressure p1 and a tem­perature T6); i2 is the enthalpy of wet steam at the turbine exit (exhaust steam), i.e. at the condenser inlet (at a pressure p2 and with a dryness frac­tion x); and i3 is the enthalpy of water at the condenser outlet (equal to the enthalpy of water on the saturation line, i', at the saturation temperature T2 determined directly by pressure p2).

Taking the above relationship into account, from the general expression for thermal efficiency of a cycle,

 

applied to the reversible Rankine cycle we have:

                                                                                                                 (11.4)

 

This equation can be presented in the following form:

                                                                                                               (11.4a)

 

The difference   represents the available enthalpy drop converted into the kinetic energy of flow and then, into work in the turbine. In accor­dance with Eq. (8.15), the difference  represents the mechanical work of the pump. Thus, the work of the cycle can be considered as the diffe­rence between the work done in the turbine and the work expended to drive the pump.

If we introduce the following notations:

                                                                                                                     (11.5)

and

                                                                                                                    (11.6)

then

                                                                                                              (11.7)

 

the superscripts "theor" and "r" indicate that these quantities pertain to a theoretical reversible cycle, not accounting for losses due to the irrevers­ibility of real processes.  

 

Fig.11.7.jpg

 

Fig. 11.7

 

The quantity  should not be confused with the work of expansion, and  with the work of compression in a cycle. The Rankine cycle is represented on the p-v diagram in          Fig. 11.7 (the notations are the same as in Fig. 11.6). On this diagram the isobar 5-4-6-1           (p1 = const) represents the addition of heat in the cycle, line 1-2 shows adiabatic expansion of steam in the turbine, line 2-3 is the isobar (p2 = const) along which heat is rejected in the condenser, and line 3-5 represents the adiabatic compression of water in the pump (due to the small compressibility of water, this adiabat practically coincides with an isochor). As can be seen from this diagram, the work of expansion is equal to the area c-5-l-2-d-c, the work of compres­sion to the area c-3-2-d-c, and the work output of the cycle is represented by area 1-2-3-5-1.

The quantities   and    are represented on the p-v diagram in the following manner. In accordance with Eq. (8.15),   is represented by area 1-2-m-n-l. Equation (7.195a) indicates that the difference   is represented by area 5-3-m-n-5. It follows that the work output of the cycle, equal to the difference   is represented by area 1-2-3-5-1.

 

Taking into account Eq. (11.1) for the mechanical work performed by the pump,

 

                                                                                                          (11.8)

 

for relationship (11.4a) we have:

 

                                                                                                           (11.9)

 

Equations (11.4a) or (11.9) make it possible to determine with the aid of an i-s diagram or Steam Tables the thermal efficiency of the reversible Rankine cycle in terms of the known initial parameters of the steam (i.e. the steam pressure p1 and temperature T1 at the turbine inlet) and the steam pressure p2 in the condenser.

Thus, if the initial steam conditions are pressure p1 = 16 670 kPa (170 kgf/cm2) and temperature T1 = 550 °C, and condenser pressure is maintained equal to p2 = 4 kPa                           (0.04 kgf/cm2), the magnitude of the thermal efficiency η is calculated in the following way. From the Steam Tables[3] we find that at a pressure of 16 670 kPa (170 kgf/cm2) and a temperature of 550 °C the enthalpy of steam is i1 = 3438 kj/kg (821.2 kcal/kg), the entropy of steam s1 = 64 619 kJ/(kg-K) [15 434 kcal/(kg-K)]. Now an i-s diagram is used to find the enthalpy of wet steam i2 at a pressure p2 = 4 kPa (0.04 kgf/cm2) and the same as at point 1, value of entropy (in a reversible process the expansion adiabat coincides with an isentrop). This enthalpy is i2 = 1945 kj/kg (464.5 kcal/kg).

The enthalpy of water on the saturation line at a pressure p2 = 4 kPa (0.04 kgf/cm2) is              i3 = 120 kj/kg (28.7 kcal/kg). The entropy of water in this state is equal to 0.4178 kJ/kg-K [0.0998 kcal/(kg-K)]. From the Steam Tables we find the value of the enthalpy of water at point 5 (the pump exit) at a pressure 16 670 kPa (170 kgf/cm2) and at the same value of the entropy as at point 3: i5 = 137 kj/kg (32.7 kcal/kg); the tempera­ture of water T5 = 29 °C.

Thus,  = 1493 kJ/kg (356.7 kcal/kg);  = 17 kJ/kg (4.0 kcal/kg);                             = 3301 kJ/kg (788.5 kcal/kg). Substituting these values into Eq. (11.4a), we obtain the thermal efficiency of this reversible Rankine cycle, η = 0.46. It will be indi­cated for the sake of comparison that the thermal efficiency of a reversible Carnot cycle realized in the same temperature interval (550 °C to 28.6 °C) is = 0.63, much grea­ter than the thermal efficiency of the reversible Rankine cycle calculated above.

Figure 11.8 shows the Rankine cycle on an i-s diagram (the notations are the same as on the  T-s and p-v diagrams shown in Figs. 11.6 and 11.7). It is clear, in accordance with Eq. (11.4a), that on this diagram the distance along the i-axis between points 1 and 2 corresponds to the work done by the turbine, the distance between points 5 and 3 represents the work expended in the pump, the distance between points 1 and 5 represents the heat q1 added in the cycle, and the distance between points 2 and 3 shows the amount of heat q2 rejected in the cycle.

If the work done by the pump, , is negligible compared with the drop in enthalpy in the turbine, , i.e. if we consider that i3 = i5, then Eq. (11.4a) can be presented in the following form:

 

                                                                                                                          (11.10)

 

Fig.11.8.jpg

 

Fig. 11.8

 

Fig.11.9.jpg

 

Fig. 11.9

This relationship is quite suitable for estimating calculations of low-pressure steam power cycles. When dealing with high-pressure steam power plants the work of the pump cannot be ignored.

Let us find the dependence of the thermal efficiency of the Rankine cycle on the initial conditions of the steam.

Under the same initial steam conditions (p1 and T1) a decrease in con­denser pressure p2 leads to a higher thermal efficiency: inasmuch as in the two-phase region pressure is directly related with temperature, a decrease in p2 means a decrease of the temperature at which heat is rejected in the cycle, T2. Thus, the temperature interval of the cycle widens and the thermal efficiency rises.

The nature of the dependence of thermal efficiency η on condenser pressure p2 is illustrated graphically in Fig. 11.9. This graph is plotted for the above cycle realized with initial steam conditions p1 =16670 kPa (170 kgf/cm2) and T1=550°C; the values of the thermal efficiency are calculated with the aid of Eq. (11.4a).

In modern steam power plants condenser pressure p2, usually predeter­mined by the temperature of the condenser cooling water, is 3.5 to 4.0 kPa (0.035 to 0.040 kgf/cm2); a pressure of 4.0 kPa (0.04 kgf/cm2) corresponds to a saturation temperature T2 =28.6°C. A further reduction of condenser pressure is inexpedient. First, a greater rarefaction (vacuum) causes the specific volume of the exhaust steam flowing into the condenser from the turbine to increase, requiring a larger condenser and much longer blades in the last turbine stages. Second, a greater rarefaction causes the tempe­rature of the wet steam in the condenser to decrease (for a pressure of 3.0 kPa the water saturation temperature is 23.8°C, and for a pressure of 2.0 kPa it is 17.2 °C), resulting in a very small difference between the temperatures of the condensing steam and the condenser cooling water[4] to which the con­denser's external surfaces are exposed, requiring a larger condenser.

 

Fig.11.10.jpg

 

Fig. 11.10

 

Fig.11.11.jpg

 

Fig. 11.11

However, the thermal efficiency of the Rankine cycle depends above all on the initial steam condition,  p1 and T1 .With a rise in superheat tempera­ture T1 at the same pressure the thermal efficiency of the cycle increases, since there is a higher mean temperature of heat addition in the cycle, as illustrated in Fig. 11.10. To illustrate, Fig. 11.11 shows a graph on which the thermal efficiency is plotted against T1 for a Rankine cycle in which the initial steam pressure                p1 = 16 670 kPa (170 kgf/cm2), and the pressure of steam in the condenser p2 = 4.0 kPa (0.04 kgf/cm2). If T1 is constant, an increase in the pressure p1 also leads to a rise of the thermal efficiency of the cycle; the higher the p1 the greater the cycle areas ratio and the higher the mean temperature of heat addition (Fig. 11.12).

 

Fig.11.12.jpg

 

Fig. 11.12

 

Fig.11.13.jpg

 

Fig. 11.13

 

Fig.11.14.jpg

 

Fig. 11.14

However, with rising p1 at the same superheat temperature, the wetness of exhaust steam (at the turbine exit) increases involving a drop in turbine relative internal efficiency. Therefore, when raising the initial steam pres­sure, it is also desirable to increase the throttle steam temperature. In Fig. 11.13 the thermal efficiency of the Rankine cycle is plotted against p1 at a superheat temperature T1 = 550 °C and p2 = 4.0 kPa (0.04 kgf/cm2).

It is clear that the higher the steam pressure p1 and temperature T1, the higher the thermal efficiency of the Rankine cycle. In Fig. 11.14 the thermal efficiency η of the reversible Rankine cycle is plotted against p1.

Thus, to raise the thermal efficiency of a Rankine cycle, in principle an attempt should be made to raise the initial steam conditions.

At present the basic initial steam conditions practiced in Russian electric power sta­tions are p1 = 23 500 kPa (240 kgf/cm2) and T1 = 565 °C. Pilot plants are being operated with steam conditions p1 = 29 400 kPa (300 kgf/cm2) and a throttle steam temperature up to T1 = 650 °C.

A further increase in the initial steam conditions is restricted by the properties of the construction materials presently available: at high pressures and temperatures the strength of pearlitic grades of steel deteriorates, and they must be replaced with considerably more expensive austenitic steels. Although such a change permits operation at higher p1 and T1, resulting in a somewhat higher thermal efficiency of the cycle, investments increase. In other words, although fuel is saved, more expensive metals are consumed. Considering the problem from this viewpoint, a further increase in initial steam conditions is inexpedient, especially where cheap grades of fuel are available. This problem is solved on the basis of a comprehensive technical and economic analysis.

 

 


[1] It will be noted for comparison that to compress an ideal gas in the same pres­sure interval several times more work must be expended [for instance, for an ideal gas with k = 1.4 and R = 490 J/(kg-K) the work of compression will be 5380 kj/kg (1285 kcal/kg)]

 

[2] It is assumed for the sake of simplicity that on the way from the boiler to the turbine steam pressure and temperature do not change. Actually, due to the resistance to steam flow offered by the steam pipeline and the inevitable heat losses, the pressure and temperature of steam drop somewhat.

 

[3] This example pertains to the cycle of an actual steam-turbine plant operating under the following initial conditions: p1 = 170 kgf/cm2 and T-i = 550 °C, at a condenser pressure p2 = 0.04 kgf/cm2. The values of i and s used in the calculation are taken from Steam Tables compiled on the basis of rounded values of pressures, expressed in kgf/cm2, while in up-to-date Steam Tables the values of i and s are given in kcal/kg and kcal/(kg-K), respectively, and then converted into units of the SI system. This also pertains to the example considered in Sec. 11.3.

 

[4] Cooling water is delivered into condensers from rivers, lakes or from water-cooling towers; it is clear that the temperature of the water depends on climatic conditions and varies during the year: it may vary from 0 to 30 °C.