In this article we will discuss about:- 1. Carnot Cycle 2. Rankine Cycle 3. Modified Rankine Cycle.
For some type of heat engines (gases) heat exchange at constant temperature is impracticable and hence, Carnot cycle cannot be practicable for gas power plants. Heat exchange at constant temperature for saturated vapour also takes place at constant pressure in some devices such as condenser or steam generator and hence, Carnot cycle could be used as the basis for a vapour power cycle. We shall consider in our discussion steam as a working substance.
Carnot Cycle:
A Carnot cycle employing steam as a working substance is drawn in fig. 3-25 on pressure-volume and temperature-entropy diagrams, ab represents isothermal operation at higher temperature T1 and heat is absorbed at that temperature.
A frictionless reversible isentropic expansion takes place along bc. cd represents isothermal operation at lower temperature T2 at which heat is removed. The wet vapour is compressed isentropically from d until the initial state a has been reached.
The area abfe of T-ɸ diagram represents the heat absorbed at a constant temperature T1. The heat absorbed during the cycle equals (ɸc – ɸa) T2.
The area cdef represents the heat rejected at a constant temperature T2. The heat rejected during the cycle is equal to (ɸc – ɸd) T2.
This cycle is practicable upto certain points. The isothermal expansion of steam at a constant temperature is practicable in a boiler and the isentropic expansion of steam in the engine or turbine is reasonable. The impracticable part is in the handling of the steam in the condenser and feed pump. In the condenser the steam is to be partially condensed upto the point which is very difficult to achieve. Also the feed pump must handle both wet steam and water.
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A slight modification of this cycle will produce a cycle which is more practical, though of slightly lower thermal efficiency. This practical cycle is known as Rankine cycle and is usually accepted as ideal cycle for steam plants.
Another reason for non-acceptance of Carnot cycle as the power cycle is its lower work ratio. The factor known as the work ratio is very useful for expressing the performance of the cycles. The work ratio is defined as the network delivered to the surroundings to the work delivered by the prime mover.
For a Carnot cycle more work is required by the feed pump as it handles wet vapour. A cycle with a low work ratio suffers more from irreversibilities than with a high work ratio. A cycle designed with a high work ratio would be more desirable than one with a lower ratio, even though the thermal efficiency of the former would be low, because in actual plants shaft work and compression work both are affected adversely.
Rankine Cycle:
The conversion of the heat energy of the fuel into the mechanical energy or power with aid of steam is carried out in steam power plants. The schematic lay out of the simplest power plant is shown in fig. 3-26 and the changes in the substance (steam) are represented on temperature-entropy diagram in fig. 3-27.
In Rankine cycle, the isothermal operations of Carnot cycle are replaced by constant pressure processes. Fig. 3-26 shows the flow diagram for a Rankine cycle. There are four events of the cycle which occurs in boiler, engine or turbine, condenser and feed pump.
(1) Constant pressure heating in boiler
(2) Isentropic expansion in engine or turbine
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(3) Constant pressure condensation in condenser
(4) Isentropic compression in feed pump.
The water at a lower pressure P2 from the condenser enters the feed pump where the water is compressed isentropically from P2 to P1. The feed pump discharges the water at a higher pressure P1 into the boiler where it is evaporated into steam.
The steam leaving the boiler enters the engine or turbine and after performing the work leaves the prime mover at a lower pressure P2 and enters the condenser. In the condenser the steam is condensed and leaves it as saturated liquid and flows back to the pump.
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The Rankine cycle is sketched in fig. 3-27 on temperature-entropy diagram. The water at a pressure P2 is considered to be in a saturated condition represented by the point a. As the pumping process is considered as reversible isentropic it is represented by a vertical line 3-4. As the temperature of the liquid leaving the pump is lower than the saturation temperature corresponding to pressure P1 it is known as a sub-cooled or compressed liquid.
The line bc represents the heating of water at pressure P1 in a boiler until the saturation temperature represented by the point 5 is reached. After the saturation point is reached, the saturated liquid is evaporated, the final condition being represented by the point 1 (1-2). ab, a’ – b’, a” – b” and a'” – b”‘ represents the frictionless isentropic expansion in the engine.
During the frictionless isentropic operation the pressure of vapour falls from P1 to P2. The vapour leaving the prime mover is condensed at constant pressure P2 in a condenser. The working substance leaving the condenser is a saturated liquid at a lower pressure. The operation on T-ɸ diagram is represented by the horizontal line 2-3.
The evaporation process may be continued upto the point a’ or a”. The former state represents the condition of the vapour to be dry and saturated while the latter state represents the condition of the vapour to be superheated.
The process 3-4-5 is greatly exaggerated to explain the working of the cycle. In actual plotting it is very difficult to differentiate between the path 3-4-5 and 3-5 and hence, the process 3-4-5 is assumed to take place along 3-5.
The heat is added in a boiler, and superheater if any, at constant pressure and its value is (H1 – h2) where H1 is the enthalpy of unit mass of steam leaving the boiler unit and h2 is the enthalpy of unit mass of water leaving the pump and entering the boiler.
The work is performed by the vapour as it flows through the engine. If we neglect the kinetic energy, potential energy and heat loss terms, the work done by vapour per kg as it flows through the turbine is equal to change in enthalpy between the vapour as it enters and leaves the turbine.
∴ Work done in turbine = (H1 – H2)
where H2 is the enthalpy of unit mass of steam leaving the engine.
Heat is rejected in the condenser at a constant pressure. The work must be done on the water as it passes through the pump. If the liquid is considered incompressible, even though it is slightly compressible, there has been no change in volume during the process. Under this condition, pump work Wp= Vf (P1 – P2) where Vf is the specific volume of water at condenser pressure, P1 is the pressure of steam in a boiler and P2 is the pressure in the condenser.
Comparison of Rankine and Carnot Cycles on Temperature- Entropy Diagram (fig. 3-28):
The area abcda represents work of Rankine cycle which is larger than the work of Carnot cycle represented by the area abe.
The unavailable energy for the Rankine cycle is larger than that for the Carnot cycle by an amount equal to area aegf. The fact that rise in unavailable energy is greater than the gain in work output which means that the efficiency for the Rankine cycle is less than that for the Carnot cycle.
Work Done During Rankine Cycle on Pressure-Volume Diagram (fig. 3-29):
Fig. 3-29 represents pressure-volume diagram for a Rankine cycle when work done by the feed pump is neglected and the expansion of steam in the engine or turbine is expressed in the form PVn = constant.
Modified Rankine Cycle:
In steam engine plants the steam is not expanded down to condenser pressure. It is released at a higher pressure and then there is a pressure drop at constant volume down to condenser pressure. This early release causes reduction in efficiency because the work for the cycle is reduced while the heat supplied per cycle remains the same.
This cycle is known as modified Rankine cycle. The reason for the early release is that at the lower pressure the specific volume of steam is high. In order to accommodate such rapidly expanding steam, large cylinder volume is necessary and the extra work obtained is very small.
Thus in order to limit the size of the reciprocating engine, the toe of the Rankine diagram is cut off. This is shown in fig. 3-30. The steam is expanded in prime move upto 2′ and released to condenser therefore pressure reduces at constant volume to condenser pressure.
Advantages of Modified Rankine Cycle:
The modified Rankine cycle provides number of advantages as shown below:
(1) The toe of a Rankine cycle has been cut off which results in minor reduction in work and the fuel consumption is slightly increased.
(2) The cylinder volume required for a Rankine cycle engine is reduced because full expansion is not obtained.
(3) This saves high initial cost.
(4) The length of engine cylinder is reduced.
(5) The weight of the engine is drastically reduced with minor loss of power.
(6) The power to weight ratio of engine is increased.
(7) This results in high fuel economy.