1. Vapour Power Cycles
Power cycle is a process where devices are used to continuously produce power. A typical vapor power cycle is the steam power plant shown below:
In this system:
- the boiler receives heat from a high temperature source and converts liquid water into high pressure and high-temperature steam.
- This steam is then fed to a turbine that drives an electric generator. The steam expands through the turbine and exhausts at a low pressure. This expansion process occurs adiabatically and is nearly as reversible as possible.
- The exhaust from the turbine is sent to a condenser where cooling water is used. A pump is used to increase the pressure of the liquid condensate. A small fraction of the work obtained from the turbine is used to operate the pump.
The most efficient cycle that can operate between two
constant temperature reservoirs, TH and TC is the Carnot
engine and its thermal efficiency is given by:
Let us consider a typical nuclear power plant that has
a capacity of 750,000 kW. Steam is generated from a boiler at around 300°C and
7 MPa. The condenser operates at around 40°C and 7 kPa. The thermal efficiency
of a Carnot engine operating between these two temperatures is:
This means that, in the best-case scenario the power
plant can only convert 45% of heat received by the boiler into work while the
other 55% is discarded to the surroundings.
In practice, the efficiency of an actual power plant
will be much less than that of a Carnot engine. If we assume that η = 0.3 then:
The heat of vaporization of water is approximately
2260 kJ/kg. Therefore, the steam circulation rate is:
The specific volume of steam at 300°C and 7 MPa is 0.02947
m3/kg and a reasonable velocity for high-pressure steam in a pipe is
25 m/s. Hence, the diameter of the pipe is:
The representation of a Carnot heat engine on a T-S
diagram is:
Shaft Work in a Reversible Steady-State Process
In the analysis of a power cycle, it is necessary to calculate the work required for the pump to increase the pressure of the exit stream from the condenser to the boiler pressure, i.e., process 1 → 2. In a reversible steady-state process, the equation to calculate the shaft work is derived as follows.
The first law of thermodynamics for a closed system is
given by:
Note that equations 3 and 6 consist of only properties
and their differential changes.
These properties and their changes are state functions
and are not dependent on the path or process involved. Therefore, both
equations hold for all reversible and irreversible processes and for a change
of state in either a steady-state flow system or a closed system.
Now consider the first law of thermodynamics for a
steady-state flow system
Thus, the reversible shaft work in a steady-flow
process with negligible changes in kinetic and potential energies is given by:
The power required can be obtained by multiplying equation 12 by the mass flow rate, ṁ = Q̇ρ. The result is:
2. Rankine Cycle
Named after William J.M. Rankine (1820-1872), the
Rankine cycle is the ideal cycle for a simple power plant. As was shown in the
T-S diagram of a Carnot cycle from the previous section, the fluid at the exit
of a condenser i.e., state 1 is a mixture of liquid and vapour. Practically, it
is much easier to pump a pure liquid rather than a two-phase mixture. Thus, for
a Rankine cycle, state 1 is a saturated liquid.
The Rankine cycle can be represented on a T-S diagram
as follows:
In a Rankine cycle with superheat, the steam at the
exit of the boiler is superheated as shown:
In analyzing the Rankine cycle, it is helpful to think of the efficiency depending on the average temperature at which heat is supplied and the average temperature at which heat is rejected.
Thus, if the boiler and condenser pressures are the same, the efficiency of a Rankine cycle with superheat is greater than the efficiency of a Rankine cycle.