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Lecture 1: Rankine Power Cycles Power Generation – Vapor Vapor Power Systems The Rankine cycle is the basic building block of the vapor power plants. The component of the power plant plant is shown in in the Figure Figure 1.1. The The whole plant plant is broken broken into four four major subsystems subsystems from A – D. D. Our main concern in these lectures will be subsystem B, where the conversion of heat into into work occurs. occurs. The other subsystems will be discussed in later lectures.
The Ideal Rankine Power Cycle Rankine Cycle is a vapor-and-liquid cycle, and accepted as the standard for steam power plants. Rankine cycle will be studied in its simplest ideal form to its more complex non-ideal form with modifications and additions that render it one of the most efficient means of generating electricity
Figure 1.1: (a) Rankine power cycle with two-phase working fluid (b) T – s diagram s diagram of Rankine cycle [Moran and Shapiro, Fundamentals of Engineering Thermodynamics]
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A schematic of the components of a Rankine cycle is shown in Figure 1.1(a). The cycle is shown on P – v, T – s and h – s coordinates in Figure 1.1(b) and 1.2. The internally reversible processes in the Rankine cycle are as follows: Process 1 – 2: Isentropic expansion of the working fluid through a turbine from saturated vapor at state 1 to the condenser pressure. The quality decreases from unity Process 2 – 3: Heat transfer from the working fluid as it flows at constant pressure through the condenser with saturated liquid at state 3. Process 3 – 4: Isentropic compression in the pump to state 4 in the compressed liquid region. Process 4 – 1: Heat transfer to the working fluid as it flows at constant pressure through the boiler to complete the cycle.
The ideal Rankine cycle also includes the possibility of superheating the vapor, as in cycle 1' – 2' – 3 – 4' – 1'. The importance of superheating is discussed later.
[ P – v coordinates] [T – s coordinates] [h – s coordinates] Figure 1.2: Rankine cycle diagram. Stations correspond to those in Figure 1.1 In the Rankine cycle, the mean temperature at which heat is supplied is less than the maximum temperature, T 2, so that the efficiency is less than that of a Carnot cycle working between the same maximum and minimum temperatures. The heat absorption takes place at constant pressure over eab, but only the part ab is isothermal. The heat rejected occurs over cd ; this is at both constant temperature and pressure.
Performance Parameters The thermal efficiency gauges the extent to which the energy input to the working fluid passing through the boiler is converted to the net work output. Using the enthalpies, heat and work quantities and expressions, the thermal efficiency of the power cycle is;
4 − ℎ3 ) = ̇ ̇ ̇ −⁄̇ ̇ ⁄̇ = (ℎ − ℎℎ2) −− (ℎ ℎ
4
The net work output equals the net heat input. Thus, the thermal efficiency can be expressed alternatively as;
2 − ℎ3 ) = ̇ ⁄̇ ̇− ⁄̇̇ ⁄̇ = 1 − ̇ ̇ ⁄⁄̇ ̇ = = 1 − (ℎ (ℎ − ℎ )
4
The heat rate is the amount of energy added by heat transfer to the cycle, usually in Btu, to produce a unit of net work output, usually in kW.h. Accordingly, the heat rate, which is inversely
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proportional to the thermal efficiency, has unit of Btu/kW.h. Another parameter used to describe power plant performance is the back work ratio, or bwr, defined as the ratio of the pump work input to the work developed by the turbine, it is written as;
̇ ⁄̇ (ℎ4 − ℎ3) = ̇ ⁄̇ = ℎ − ℎ 4 There are several features that should be noted about Figure 1.2 and the Rankine cycle in general: 1. The T – s and the h – s diagrams are not similar in shape, as they were with the perfect gas with constant specific heats. The slope of a constant pressure reversible heat addition line is,
ℎ = .
In the two-phase region, constant pressure means also constant
temperature, so the slope of the constant pressure heat addition line is constant and the line is straight. 2. The effect of irreversibilities is represented by the dashed line from b to c' . Irreversible behavior during the expansion results in a value of entropy sc' at the end state of the c' expansion that is higher than s c. The enthalpy at the end of the expansion (the turbine exit) is thus higher for the irreversible process than for the reversible process, and, as seen for the Brayton cycle, the turbine work is thus lower in the irreversible case. 3. The Rankine cycle is less efficient than the Carnot cycle for given maximum and minimum temperatures, but, as said earlier, it is more effective as a practical power production device.
Enhancements of, and Effect of Design Parameters on, Rankine Cycles The basic Rankine cycle can be enhanced through processes such as superheating and reheat. Diagrams for a Rankine cycle with superheating are given in Figure 1.3. The heat addition is continued past the point of vapor saturation, in other words the vapor is heated so that its temperature is higher than the saturation temperature associated with P a (= P b = P c = P d ) . This does several things. First, it increases the mean temperature at which heat is added, T m2 , thus increasing the efficiency of the cycle. Second is that the quality of the two-phase mixture during the expansion is higher with superheating, so that there is less moisture content in the mixture as it flows through the turbine. (The moisture content at e is less than that at e' .) This is an advantage in terms of decreasing the mechanical deterioration of the blades.
[ P – v coordinates] [T – s coordinates] [h – s coordinates] Figure 1.3: Rankine cycle with superheating
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The heat exchanges in the superheated cycle are:
Along abcd , which is a constant pressure (isobaric) process: q 2 = hd - ha . Along ef : q1 = hf – he (< 0).
The thermal efficiency of the ideal Rankine cycle with superheating is
= (ℎ − ℎℎ ) −− ℎ(ℎ − ℎ )
This can be expressed explicitly in terms of turbine work and compression (pump) work as
=
(ℎ − ℎ ) − (ℎ − ℎ ) ℎ − ℎ
Compared to the basic cycle, superheating has increased the turbine work, increased the mean temperature at which heat is received, T m2, and increased the cycle efficiency.
Figure 1.4: Comparison of Rankine cycle with superheating and Carnot cycle
Figure 1.5: Rankine cycle with superheating and reheat for space power application
A comparison of the Carnot cycle and the Rankine cycle with superheating is given in Figure 1.4. The maximum and minimum temperatures are the same, but the average temperature at which
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heat is absorbed is lower for the Rankine cycle. To alleviate the problem of having moisture in the turbine, one can heat again after an initial expansion in a turbine, as shown in Figure 1.5, which gives a schematic of a Rankine cycle for space power application. This process is known as reheat. The main practical advantage of reheat (and of superheating) is the decrease in moisture content in the turbine because most of the heat addition in the cycle occurs in the vaporization part of the heat addition process.
Figure 1.6: Effect of exit pressure on Rankine c ycle efficiency We can also examine the effect of variations in design parameters on the Rankine cycle. Consider first the changes in cycle output due to a decrease in exit pressure. In terms of the cycle shown in Figure 1.6, the exit pressure would be decreased from P 4 to P4'. The original cycle is 1 – 2 – 3 – 4 – 1, and the modified cycle is 1' – 2' – 3 – 4' – 1'. The consequences are that the cycle work, which is the integral of Tds around the cycle, is increased. In addition, as drawn, although the levels of the mean temperature at which the heat is absorbed and rejected both decrease, the largest change is the mean temperature of the heat rejection, so that the thermal efficiency increases.
Figure 1.7: Effect of maximum boiler pressure on Rankine cycle efficiency
Another design parameter is the maximum cycle pressure. Figure 1.7 shows a comparison of two cycles with different maximum pressure but the same maximum temperature, which is set by material properties. The average temperature at which the heat is supplied for the cycle with a higher maximum pressure is increased over the original cycle, so that the efficiency increases.
