Combined Cycles

Abstract

Combined cycles are the standard-bearer for modern natural gas-fired power plants. Two conventional thermodynamic heat engines (a topping cycle and a bottoming cycle) are combined in a manner that addresses thermodynamic shortcomings of both. An internal combustion engine (usually a gas turbine for power plants) serves as the topping cycle and a steam turbine (Rankine-type) cycle as the bottoming cycle. High pressure steam is generated for the Rankine cycle from the hot exhaust of the topping cycle in a heat exchanger called a Heat Recovery Steam Generator (HRSG). There is an optimal operating pressure on the H₂O side of the HRSG for thermodynamic performance, unlike a stand-alone Rankine cycle (with a high temperature external combustion source) where the thermodynamic performance increases monotonically with boiler pressure. In this chapter, thermodynamic performance of a HRSG/steam turbine combination is investigated using a pinch point analysis that demonstrates how the saturation temperature, hence H₂O-side pressure, is related to the available inlet gas temperature from the topping cycle. The condenser is also analyzed as a heat exchanger, with cooling water that must ultimately exchange heat with the ambient through a cooling tower (studied more fully in the next chapter). Finally, a series of capstone workshops are conducted to simulate the performance of an industrial scale combined cycle power plant.

Appendices

Appendix 1: Pinch Point Analysis

The heat recovery steam generator (HRSG) is viewed as a counterflow heat exchanger with three distinct sections, shown schematically in Fig. 11.23. That is, hot exhaust gases enter the HRSG at the gas turbine exit conditions and a known mass flow rate (from the right side of the schematic). Water enters in a separate stream (state d, top left) at the fixed operating pressure of the Rankine cycle. Heat is transferred (internally) and the gas turbine exhaust gases exit at ambient pressure and the stack temperature while the water exits at the turbine inlet conditions. The three main subsections of the HRSG are preheater (where compressed liquid is heated to its boiling point), boiler (where phase change occurs) and superheater (where a saturated vapor is heated to a superheated condition, suitable for the turbine). A section for pollution control is shown between the preheater and boiler (for removing nitrogen oxides, NOx), but not studied any further here, and assumed to have minimal impact on the thermodynamics of a HRSG.

Fig. 11.23

Schematic of heat recovery steam generator

The exit condition of the exhaust gas is determined by the “pinch point” that occurs between the preheater and boiler sections. The gas temperature at that point must exceed the boiling point of the H₂O at its operating pressure. For this case study in which all inlet and outlet temperatures (and mass flow rate of air) are specified, this pinch point temperature difference will be calculated to confirm it is acceptable. It is possible to specify inlet and outlet temperatures that are reasonable (exit H₂O below inlet air temperature and exit air above inlet H₂O temperature), but would result in a negative pinch point temperature (air temperature below H₂O saturation temperature at the inlet to the boiler). The spreadsheet calculations to determine the internal temperatures were detailed in Table 11.1 and plotted in Fig. 11.9 (repeated here) versus the heat transfer rate (to the H₂O) at each section. The temperature of the air decreases linearly (from right to left), a consequence of constant specific heats. The H₂O temperature increases linearly where no phase change occurs (preheater and superheater) and is constant in the boiler section. The pinch point (212.8–179.9 = 32.9 °C) occurs between the preheating and boiling sections. This value is considered reasonable. The problem could be specified with a somewhat lower exit gas temperature to generate a bit more steam, but not much.

Fig. 11.24

Schematic of HRSG plus turbine

It is clear that most of the heat transferred is needed to change the phase, and very little heat is associated with superheating. Yet superheating is vital for the Rankine cycle to maintain the steam quality of the turbine exit sufficiently high to avoid mechanical erosion.

In the next example, the exit gas temperature will not be known, but the pinch point temperature difference will be set (to 20 °C, a design specification). An energy balance between the exhaust gas inlet and this point with the pinch point constraints will yield the mass flow rate of water. Further cooling of the exhaust gases occurs in the preheater, and another constraint on the HRSG is that the stack temperature must be warm enough for the chimney to achieve sufficient draft. In this case study, the stack temperature cannot be below 50 °C. One of these two constraints will uniquely define the design operating point of the HRSG, and thereby the mass flow rate of water that circulates in the Rankine cycle (the primary loop).

Workshop: Optimizing HRSG/Turbine Performance

This workshop is similar to the HRSG/turbine case study, with an important difference to demonstrate design optimization principles. The inlet hot gas stream from an internal combustion engine is the same (mass flow rate, temperature and pressure) but the exit air temperature is not specified. Rather, it is determined by a pinch-point constraint. Also, a different value of the Rankine cycle high side pressure is assigned to different student teams. The goal of this workshop is therefore two-fold: to optimize the performance of a basic HRSG with the constraint of the pinch point for a specified Rankine cycle pressure, and determine (by parametric study) the Rankine cycle pressure that maximizes power output of the bottoming cycle driven by the exhaust of an internal combustion engine.

The schematic for this workshop (Fig. 11.24) is modified from Fig. 11.7 to indicate these two effects. A parametric study of the effect of water side pressure on the overall performance will be conducted. To properly design the HRSG, a pinch point analysis will be conducted in order to determine the lowest allowable air temperature that can be obtained (extracting the most thermal energy from the air stream) given the 2nd Law constraint that the temperature of the air stream must everywhere be greater than that of the water stream.

Fig. 11.25

Spreadsheet template for HRSG Optimization Workshop. Orange cells require input for the specific assigned Rankine high side pressure. Yellow cell represents an input gas temperature exit. The value shown is obtained using the “goal seek” feature to set pinch point temperature to a value of 20 °C

A spreadsheet with the solution for an assigned pressure of 10 bar is shown in Fig. 11.25, with the corresponding temperature versus heat transferred plot in Fig. 11.26. The structure is similar, but not identical, to Fig. 11.10. The workshop consists of modifying the spreadsheet created for the HRSG/turbine case study for the assigned pressure (orange cells) and adding a plot for the temperatures. Everything should automatically update when the outlet air temperature is changed (T_b, the yellow cell). Once working, review the plot, then change T_b to a new value, and observe the changes in the output (temperature plot, mass flow rate of water, power developed in the turbine, eXergy destruction rate, and the pinch point temperature difference). Finally, use GOAL SEEK to set the pinch point to 20 °C by changing the inlet air temperature.

Fig. 11.26

Temperature versus heat transferred plot for the case of Rankine cycle high pressure of 10 bar and a pinch point temperature difference set to 20 °C

Fig. 11.27

Temperature versus heat transferred plot for the case of Rankine cycle high pressure of 10 bar and an exit air temperature set to 120 °C. The resulting pinch point temperature difference is −11.43 °C, the negative indicating that the 2nd Law is violated at the pinch point, and an exit air temperature this low would not be possible

Results

Results from this workshop are reported and discussed in the remainder of this Appendix. First, the effect of pinch point temperature difference with the case of 10 bar is investigated. Figure 11.26 shows the air and H₂O temperatures inside the HRSG as a function of heat transferred for the design case (pinch point temperature difference set to 20 °C, with a Rankine cycle pressure of 10 bar. This case is considered to give near optimal performance, subject to a pinch point constraint. The air exit temperature of 158.9 °C shows that there is considerable thermal energy vented out the stack, a consequence of the pinch point.

Figure 11.27 shows a case where the exit temperature is set to a value (120 °C) that would appear to be reasonable for a stack temperature, but it would not be possible to design a HRSG with a single Rankine pressure because of the pinch point. Figure 11.28 shows a case where the exit air temperature is close to the inlet temperature. This case would produce minimal H₂O flow rate.

Fig. 11.28

Temperature versus heat transferred plot for the case of Rankine cycle high pressure of 10 bar and an exit air temperature set to 350 °C. The resulting pinch point temperature difference is 174.2 °C. The mass flow rate of H₂O produced is very low in this case (0.034 kg/sec)

Fig. 11.29

Effect of pinch point temperature difference on steam turbine power and exit air temperature

Figure 11.29 shows a compiled plot of the effect of pinch point temperature difference on the turbine power output and exit air temperature.

Fig. 11.30

H₂O mass flow rate produced in HRSG and exit air temperature as a function of Rankine cycle high pressure

Figure 11.30 shows the effect of Rankine cycle high pressure on the mass flow rate of H₂O produced and the exit air temperature. Figure 11.31 shows the effect on the power produced in the steam turbine and the eXergy destruction rate in the HRSG. These results are compiled numbers submitted by students from an actual course, the few outlier points are indications of students that made some error in their calculations.

Fig. 11.31

Steam turbine power produced and eXergy destruction versus Rankine cycle high pressure

Fig. 11.32

Representation of a heat exchanger with the hot fluid as an equivalent thermal reservoir source. Energy and entropy balances are shown (at steady-state, with negligible stray heat and kinetic and potential energy changes)

Appendix 2: Equivalent Reservoir Temperature of a Heat Exchanger

The objective of a heat exchanger is to transfer heat from one fluid to another across a physical wall, either to change the temperature of a stream or to change its enthalpy. Generally, the temperature of both streams changes, and neither stream behaves like a thermal reservoir. However, it may be instructive for 2nd Law analysis to treat one of the streams as a thermal reservoir with a temperature that satisfies both 1st and 2nd Laws. Figure 11.32 shows the case for which the hot side fluid is treated as a thermal reservoir. For example, in the HRSG of a combined cycle, the hot side would be the exhaust gases, and the cold side is the H₂O in the Rankine cycle. The equivalent reservoir temperature would be an intermediate temperature between the hot gas inlet and the warm gas exit. Figure 11.33 shows the case for which the cold side is treated as a thermal reservoir. For example, the condenser of a Rankine cycle, where the hot side is the condensing Rankine H₂O, and the cold side is cooling water.

Fig. 11.33

Representation of a heat exchanger with the cold fluid as an equivalent thermal reservoir sink

Combining the energy and entropy balances and equating the entropy generation rate for the case of the hot fluid as an equivalent thermal reservoir (Fig. 11.32), the equivalent thermal reservoir temperature is:

$$ {T}_{\mathrm{H},\mathrm{equiv}}=\frac{h_{\mathrm{H},\mathrm{in}}-{h}_{\mathrm{H},\mathrm{out}}}{s_{\mathrm{H},\mathrm{in}}-{s}_{\mathrm{H},\mathrm{out}}} $$

Consider the case where the hot fluid is an ideal gas:

$$ {T}_{\mathrm{H},\mathrm{equiv}}=\frac{c_p\left({T}_{\mathrm{H},\mathrm{in}}-{T}_{\mathrm{H},\mathrm{out}}\right)}{c_{\mathrm{p}}\ln \left({T}_{\mathrm{H},\mathrm{in}}/{T}_{\mathrm{H},\mathrm{out}}\right)-R\ln \left({P}_{\mathrm{H},\mathrm{in}}/{P}_{\mathrm{H},\mathrm{out}}\right)}\kern0.5em \mathrm{for}\kern0.5em \mathrm{ideal}\kern0.5em \mathrm{gases}\kern0.5em \mathrm{on}\kern0.5em \mathrm{hot}\kern0.5em \mathrm{side} $$

If the process is isobaric, with T_{H, out} = T_{H, in} − ΔT_H:

$$ {T}_{\mathrm{H},\mathrm{equiv}}=\frac{\left({T}_{\mathrm{H},\mathrm{in}}-{T}_{\mathrm{H},\mathrm{out}}\right)}{\ln \left({T}_{\mathrm{H},\mathrm{in}}/{T}_{\mathrm{H},\mathrm{out}}\right)}=\frac{\Delta {T}_{\mathrm{H}}}{-\ln \left(1-\Delta {T}_{\mathrm{H}}/{T}_{\mathrm{H},\mathrm{in}}\right)} $$

Consider a high flow limit for the hot fluid, meaning that the flow rate is sufficiently high that the change in temperature of is small. Expanding the ln term:

$$ {T}_{\mathrm{H},\mathrm{equiv}}=\frac{\Delta {T}_{\mathrm{H}}}{-\left(1-\Delta {T}_{\mathrm{H}}/{T}_{\mathrm{H},\mathrm{in}}+{\left(\Delta {T}_{\mathrm{H}}/{T}_{\mathrm{H},\mathrm{in}}\right)}^2/2-{\left(\Delta {T}_{\mathrm{H}}/{T}_{\mathrm{H},\mathrm{in}}\right)}^3/3+\dots \right)} $$

Keeping one term:

$$ {T}_{\mathrm{H},\mathrm{equiv}}\approx {T}_{\mathrm{H},\mathrm{in}}\kern0.5em \mathrm{for}\kern0.5em \mathrm{isobaric}\kern0.5em \mathrm{ideal}\kern0.5em \mathrm{gases}\kern0.5em \mathrm{and}\kern0.5em \Delta {T}_{\mathrm{H}}\ll {T}_{\mathrm{H},\mathrm{in}} $$

On the other hand, consider a low flow limit for the hot fluid, in which case it exits in thermal equilibrium with entering cold-side fluid (T_{H, out} = T_{C, in})

$$ {T}_{\mathrm{H},\mathrm{equiv}}=\frac{\left({T}_{\mathrm{H},\mathrm{in}}-{T}_{\mathrm{C},\mathrm{in}}\right)}{\ln \left({T}_{\mathrm{H},\mathrm{in}}/{T}_{\mathrm{C},\mathrm{in}}\right)} $$

For example, if the hot fluid enters at 100 °C and the cold fluid enters at 0 °C, and the hot fluid is an ideal gas (with negligible pressure drop) and in a low flow limit, the equivalent hot reservoir temperature evaluates to 47.4 °C.

A similar result holds for liquids on the hot side (with no phase change):

$$ {T}_{\mathrm{H},\mathrm{equiv}}=\frac{\left({T}_{\mathrm{H},\mathrm{in}}-{T}_{\mathrm{H},\mathrm{out}}\right)}{\ln \left(\frac{T_{\mathrm{H},\mathrm{in}}}{T_{\mathrm{H},\mathrm{out}}}\right)}\kern0.5em \mathrm{for}\kern0.5em \mathrm{liquids}\kern0.5em \mathrm{on}\kern0.5em \mathrm{hot}\kern0.5em \mathrm{side} $$

Consider a phase changing case where the hot side fluid enters as a saturated vapor and exits as a saturated liquid at the same pressure. The equivalent thermal reservoir temperature is:

$$ {T}_{\mathrm{H},\mathrm{equiv}}=\frac{h_{\mathrm{fg}}}{s_{\mathrm{fg}}}\kern0.5em \mathrm{for}\kern0.5em \mathrm{isobaric}\kern0.5em \mathrm{condensation}\kern0.5em \mathrm{on}\kern0.5em \mathrm{hot}\kern0.5em \mathrm{side} $$

This analysis is therefore a formal derivation of the fundamental relationship between saturation temperature and the enthalpy and entropy of phase change, since the hot side fluid is fixed at the saturation temperature (corresponding to the constant pressure). That is:

$$ \frac{h_{\mathrm{fg}}}{s_{\mathrm{fg}}}={T}_{\mathrm{SAT}@P} $$

A similar analysis holds for the case where an equivalent cold side thermal reservoir temperature is sought (Fig. 11.33). In that case:

$$ {T}_{\mathrm{C},\mathrm{equiv}}=\frac{h_{\mathrm{C},\mathrm{in}}-{h}_{\mathrm{C},\mathrm{out}}}{s_{\mathrm{C},\mathrm{in}}-{s}_{\mathrm{C},\mathrm{out}}} $$