ch09_pt2

Chapter 9

Gas Power Systems

Learning Outcomes

►Conduct air-standard analyses of internal combustion engines based on the Otto, Diesel, and dual cycles, including the ability to►sketch p-v and T-s diagrams and evaluate

property data at principal states.►apply energy, entropy, and exergy balances.►determine net power output, thermal

efficiency, and mean effective pressure.

Learning Outcomes►Conduct air-standard analyses of gas turbine

power plants based on the Brayton cycle and its modifications, including the ability to►sketch T-s diagrams and evaluate property data

at principal states.►applying mass, energy, entropy, and exergy

balances.►determine net power output, thermal efficiency,

back work ratio, and the effects of compressor pressure ratio on performance.

Learning Outcomes

►Analyze subsonic and supersonic flows through nozzles and diffusers, including the ability to►describe the effects of area change on flow

properties and the effects of back pressure on mass flow rate.

►explain the occurrence of choking and normal shocks.

►analyze the flow of ideal gases with constant specific heats.

Considering Gas Turbine Power Plants►Gas turbine power plants are more quickly constructed, less costly, and more compact than the vapor power plants considered in Chapter 8.►Gas turbines are suited for stationary power generation as well as for powering vehicles, including aircraft propulsion and marine power plants.►Gas turbines are

►increasingly used for large-scale power generation, and ►for such applications fueled primarily by natural gas, which is relatively abundant today.

Considering Gas Turbine Power Plants►Gas turbines may operate on an open or closed basis, as shown in the figures.►The open gas turbine is more commonly used and is the main focus of our study of gas turbines.►Study of the individual components of these configurations requires the control volume forms of the mass, energy, and entropy balances.

Open to the atmosphere Closed

Considering Gas Turbine Power Plants►The open mode gas turbine is an internal combustion power plant.

►Air is continuously drawn into the compressor where it is compressed to a high pressure.

►Combustion products exit at elevated temperature and pressure. ►Combustion products expand through the turbine and then are discharged to the surroundings.

►Air then enters the combustion chamber (combustor) where it mixes with fuel and combustion occurs.

The remainder is available as net work output to drive an electric generator, to propel a vehicle, or for other uses.

Part of the turbine work is used to drive the compressor.

Considering Gas Turbine Power Plants►The closed gas turbine operates as follows:

►A gas circulates through four components: turbine, compressor, and two heat exchangers at higher and lower operating temperatures, respectively.►The turbine and compressor play the same roles as in the open gas turbine. ►As the gas passes through the higher-temperature heat exchanger, it receives energy by heat transfer from an external source.►The thermodynamic cycle is completed by heat transfer to the surroundings as the gas passes through the lower-temperature heat exchanger.

Considering Gas Turbine Power Plants

►The heat transfer associated with the higher-temperature heat exchanger of the closed gas turbine originates from an external source, which may include

►External combustion of biomass, municipal solid waste, fossil fuels such as natural gas, and other combustibles.►Waste heat from industrial processes.►Solar thermal energy.►A gas-cooled nuclear reactor.

►To conduct elementary analyses of open gas turbine power plants, simplifications are required. Although highly idealized, an air-standard analysis can provide insights and qualitative information about actual performance.►An air-standard analysis has the following elements:

►The working fluid is air which behaves as an ideal gas. Ideal gas relations are reviewed in Table 9.1.►The temperature rise that would be brought about by combustion is accomplished by heat transfer from an external source. ►With an air-standard analysis, we avoid the complexities of the combustion process and the change in composition during combustion, which simplifies the analysis considerably. Combustion is studied in Chapter 13.►In a cold air-standard analysis, the specific heats are assumed constant at their ambient temperature values.

Air-Standard Analysis of Open Gas Turbine Power Plants

Air-Standard Brayton Cycle►The schematic of a simple open air-standard gas turbine power plant is shown in the figure.►The energy transfers by heat and work are in the directions of the arrows.►Air circulates through the components:

►Process 1-2: the air is compressed from state 1 to state 2.►Process 2-3: The temperature rise that would be achieved in the actual power plant with combustion is realized here by heat transfer,

►At state 1, air is drawn into the compressor from the surroundings.

.inQ

Air-Standard Brayton Cycle

►Air returns to the surroundings at state 4 with a temperature typically much greater than at state 1.►After interacting with the surroundings, each unit of mass returns to the same condition as the air entering at state 1, thereby completing a thermodynamic cycle.

►Process 3-4: The high-pressure, high-temperature air expands through the turbine. The turbine drives the compressor and develops net power, .cycleW


►Air returns to the surroundings at state 4 with a temperature typically much greater than at state 1.►After interacting with the surroundings, each unit of mass returns to the same condition as the air entering at state 1, thereby completing a thermodynamic cycle.

►Process 3-4: The high-pressure, high-temperature air expands through the turbine from state 3 to state 4. The turbine drives the compressor and develops net power, .cycleW

►We imagine process 4-1 being achieved by a heat exchanger, as shown by the dashed line in the figure.


►Cycle 1-2-3-4-1 is called the Brayton cycle.►The compressor pressure ratio, p2/p1, is a key

Brayton cycle operating parameter.


►Analyzing each component as a control volume at steady state, assuming the compressor and turbine operate adiabatically, and neglecting kinetic and potential energy effects, we get the following expressions for the principal work and heat transfers, which are positive in accord with our convention for cycle analysis.

Turbine

Compressor

(Eq. 9.15)

(Eq. 9.16)

(Eq. 9.17)

(Eq. 9.18)

Heat addition

Heat rejection

http://www.wiley.com/college/moran/0470495901/ig/Ch9/pages/eq_09_16.htm




Air-Standard Brayton Cycle►The thermal efficiency is

(Eq. 9.19)

►The back work ratio is

(Eq. 9.20)

►Since Eqs. 9.15 through 9.20 have been developed from mass and energy balances, they apply equally when irreversibilities are present and in the absence of irreversibilities.

Note: A relatively large portion of the work developed by the turbine is required to drive the compressor. For gas turbines, back work ratios range from 20% to 80% compared to only 1-2% for vapor power plants.



Ideal Air-Standard Brayton Cycle►The ideal air-standard Brayton cycle provides an especially simple setting for study of gas turbine power plant performance. The ideal cycle adheres to additional modeling assumptions:

►Frictional pressure drops are absent during flows through the heat exchangers. These processes occur at constant pressure. These processes are isobaric.►Flows through the turbine and pump occur adiabatically and without irreversibility. These processes are isentropic.►Accordingly, the ideal Brayton cycle consists of two isentropic processes alternated with two isobaric processes. In this respect, the ideal Brayton cycle is in harmony with the ideal Rankine cycle, which also consists of two isentropic processes alternated with two isobaric processes (Sec. 8.2.2).

Process1-2: Isentropic compression of air flowing through the compressor.Process 2-3: Heat transfer to the air as it flows at constant pressure through the higher-temperature heat exchanger.

Ideal Air-Standard Brayton Cycle►The ideal air-standard Brayton cycle consists of four internally reversible processes:

Process 3-4: Isentropic expansion of the air through the turbine.Process 4-1: Heat transfer from the air as it flows at constant pressure through the lower-temperature heat exchanger.

Ideal Air-Standard Brayton Cycle

►Since the ideal Brayton cycle involves internally reversible processes, results from Sec. 6.13 apply.►On the p-v diagram, the work per unit of mass flowing is –∫vdp. Thus on a per unit of mass flowing basis,

►Area 1-2-a-b-1 represents the compressor work input.►Area 3-4-b-a-3 represents the turbine work output.►Enclosed area 1-2-3-4-1 represents the net work developed.

►Area 2-3-a-b-2 represents the heat added.►Area 4-1-b-a-4 represents the heat rejected.►Enclosed area 1-2-3-4-1 represents the net heat added or equivalently, the net work developed.

Ideal Air-Standard Brayton Cycle

►On the T-s diagram, the heat transfer per unit of mass flowing is ∫Tds. Thus, on a per unit of mass flowing basis,

Effects of Compressor Pressure Ratio on Brayton Cycle Performance

►That the compressor pressure ratio, p2/p1, is an important operating parameter for gas turbines is brought out simply by the following discussions centering on the T-s diagram:


►Increasing the compressor pressure ratio from p2/p1 to p2′/p1 changes the cycle from 1-2-3-4-1 to 1-2′-3′-4-1.

►Since the average temperature of heat addition is greater in cycle 1-2′-3′-4-1, and both cycles have the same heat rejection process, cycle 1-2′-3′-4-1 has the greater thermal efficiency.



►Since the average temperature of heat addition is greater in cycle 1-2′-3′-4-1, and both cycles have the same heat rejection process, cycle 1-2′-3′-4-1 has the greater thermal efficiency.►Accordingly, the Brayton cycle thermal efficiency increases as the compressor pressure ratio increases.

60

th (

%)

2 4 6 8 10

CompressorPressure Ratio



►Since the average temperature of heat addition is greater in cycle 1-2′-3′-4-1, and both cycles have the same heat rejection process, cycle 1-2′-3′-4-1 has the greater thermal efficiency.►Accordingly, the Brayton cycle thermal efficiency increases as the compressor pressure ratio increases.►The turbine inlet temperature also increases with increasing compressor ratio – from T3 to T3′.


►However, there is a limit on the maximum temperature at the turbine inlet imposed by metallurgical considerations of the turbine blades. ►Let’s consider the effect of increasing compressor pressure ratio on Brayton cycle performance when the turbine inlet temperature is held constant.►This is investigated using the T-s diagram as presented next.


►The figure shows the T-s diagrams of two ideal Brayton cycles having the same turbine inlet temperature but different compressor pressure ratios.

►Cycle A has the greater compressor pressure ratio and thus the greater thermal efficiency.

►Cycle B has the larger enclosed area and thus the greater net work developed per unit of mass flow.

►For Cycle A to develop the same net power as Cycle B, a larger mass flow rate would be required and this might dictate a larger system.


►Accordingly, for turbine-powered vehicles, where size and weight are constrained, it may be desirable to operate near the compressor pressure ratio for greater net work per unit of mass flow and not the pressure ratio for greater thermal efficiency.

Gas Turbine Power Plant Irreversibility►The most significant irreversibility by far is the irreversibility of combustion. This type of irreversibility is considered in Chap. 13, where combustion fundamentals are developed.►Irreversibilities related to flow through the turbine and compressor also significantly impact gas turbine performance. They act to

►decrease the work developed by the turbine and►increase the work required by the compressor, ►thereby decreasing the net work of the power plant.

m

W

m

W

m

W

ctnet

marked decrease in net work of the power plant

irreversibilites decreaseturbine work

irreversiblities increasecompressor work

Gas Turbine Power Plant Irreversibility

)(

)(

)/(

)/(

s43

43

st

tt hh

hh

mW

mW

►Isentropic turbine efficiency, introduced in Sec. 6.12.1, accounts for the effects of irreversibilities within the turbine in terms of actual and isentropic turbine work, each per unit of mass flowing through the turbine.

work developed in the actual expansion from turbine inlet state

to the turbine exit pressure

work developed in an isentropic expansion from turbine inlet

state to exit pressure

Gas Turbine Power Plant Irreversibility

)(

)(

)/(

)/(

12

1s2

c

scc hh

hh

mW

mW

►Isentropic compressor efficiency, introduced in Sec. 6.12.3, accounts for the effects of irreversibilities within the compressor in terms of actual and isentropic compressor work input, each per unit of mass flowing through the compressor.

work input for the actual process from compressor inlet state to the compressor exit pressure

work input for an isentropic process from compressor inlet state to exit pressure

Gas Turbine Power Plant Loss

►The exhaust gas temperature of a simple gas turbine is typically well above the ambient temperature. Thus, the exhaust gas has considerable thermodynamic utility (exergy) that would be irrevocably lost were the gas discharged directly to the ambient.►Regenerative gas turbines (Sec. 9.7) and gas turbine-based combined cycles (Sec. 9.9) aim to avoid such a significant loss by using the hot exhaust gas cost-effectively.

►The regenerator allows air exiting the compressor to be preheated, process 2-x, as the turbine exhaust gas cools, process 4-y.►Preheating reduces the heat added per unit of mass flowing (and thus the amount of fuel that must be burned):

Regenerative Gas Turbines

)( x3in hhm

Q

►The hot turbine exhaust can be utilized with a preheater called a regenerator.

)( 23in hhm

Q

►The net work per unit of mass flowing is not altered with the inclusion of a regenerator. Accordingly, since the heat added is reduced, thermal efficiency increases.

With Regeneration Without Regeneration

►Since a finite temperature difference must exist between the two streams of the regenerator for heat transfer to take place between the streams, the cold-side exiting temperature, Tx, must be less than the hot-side entering temperature, T4.

Regenerator Effectiveness

►As the stream-to-stream temperature difference becomes small Tx approaches T4, but cannot exceed it. Accordingly, Tx ≤ T4.

►As the enthalpy of the air varies only with temperature, we also have hx ≤ h4.

T4

►The regenerator effectiveness is defined as the ratio of the actual enthalpy increase of the air flowing through the cold side of the regenerator, hx – h2, to the maximum theoretical enthalpy increase, h4 – h2.


(Eq. 9.27)


►In practice, regenerator effectiveness values range from 60-80%, approximately. Thus, the temperature Tx at the combustor inlet is invariably below the temperature T4 at the turbine exit.

►Selection of a regenerator is largely an economic decision.


►With regeneration less fuel is consumed by the combustor but another component, the regenerator, is required.►When considering use of a regenerator, the trade-off between fuel savings and regenerator cost must be weighed.

►A modification of the Brayton cycle that increases the net work developed is multistage expansion with reheat.►The figure shows a cycle with two turbine stages and a reheat combustor between the stages.

Gas Turbines with Reheat and Regeneration

Cycle with reheat

►The ideal Brayton cycle with reheat is 1-2-3-a-b-4-1. The ideal Brayton cycle without reheat is 1-2-3-4′-1.►The reheat cycle has a larger enclosed area than the cycle without reheat and thus a greater net work developed per unit of mass flowing, which is the aim.


Cycle without reheat

►The figure also shows that the temperature at the exit of the second-stage turbine, state 4, is greater than at the exit of the single turbine of the cycle without reheat, state 4′. Accordingly, with reheat the potential for regeneration is also enhanced.►When reheat and regeneration are used together, the thermal efficiency can increase significantly over that for the cycle without reheat.


T4′

T4

►Another modification of the Brayton cycle that increases the net work developed is compression with intercooling.►The figure shows two compressor stages and an intercooler between the stages.

Gas Turbines with Intercooling and Regeneration

►The accompanying p-v diagram shows the processes for internally reversible operation:

Gas Turbines withIntercooling and Regeneration

►Isentropic compression without intercooling is represented by process 1-c-2′.

►Process 1-c. Isentropic compression from state 1, where pressure is p1, to state c, where pressure is pi.►Process c-d. Constant-pressure cooling from temperature Tc to temperature Td.►Process d-2. Isentropic compression to state 2, where pressure is p2.

►Recalling that for such internally reversible processes the work input per unit of mass flowing is given by ∫vdp, the following area interpretations apply, each per unit of mass flowing:


►With intercooling, area 1-c-d-2-a-b-1 represents the work input.►Without intercooling, area 1-2′-a-b-1 represents the work input.►The cross-hatched area c-d-2-2′-c represents the reduction in work achieved with intercooling.

►If the total turbine work remains the same, a reduction in compressor work results in an increase in the net work developed, which is the aim.

►While compression with and without intercooling each bring the air to the same final pressure, p2, the final temperature with intercooling, T2, is lower than the final temperature without intercooling, T2′.


►Comparing states 2 and 2′ on the T-s diagram, T2 < T2′.

►The lower temperature at the compressor exit with intercooling enhances the potential for regeneration.

T2

T2′


►When compression with intercooling is used together with regeneration, the thermal efficiency can increase significantly over that for the cycle without intercooling.►The T-s diagram also shows that for cooling to the surroundings the temperature Td at the intercooler exit cannot be less than T1, the temperature of the air entering the compressor from the surroundings: Td ≥ T1.

T1

Td

►Shown here is a regenerative gas turbine that incorporates reheat and intercooling.►With these modifications to the basic Brayton cycle:

Regenerative Gas Turbinewith Reheat and Intercooling

►The net work output is increased.►The thermal efficiency is increased.

►Applying mass and energy rate balances at steady state, we obtain the following expressions, each per unit of mass flowing:


►Total turbine work:

(h6 – h7) + (h8 – h9) = t1(h6 – h7s) + t2(h8 – h9s)m

W

t

=

where t1 and t2 denote the isentropic efficiencies of turbines 1 and 2, respectively.

►Total compressor work:

(h2 – h1) + (h4 – h3) = (h2s – h1)/c1 + (h4s – h3)/c2m

W

c

=

where c1 and c2 denote the isentropic efficiencies of compressors 1 and 2, respectively.

►Applying mass and energy rate balances at steady state, we obtain the following expressions, each per unit of mass flowing:


►Total heat added:

(h6 – h5) + (h8 – h7)m

Q

in

=

►In this application, the regenerator effectiveness is:

(h5 – h4)/(h9 – h4)reg =

►For cooling to the surroundings, the temperature at the exit of the intercooler, T3, cannot be less than the temperature of the air entering the compressor from the surroundings: T3 ≥ T1.

►The exhaust temperature of the simple gas turbine is typically well above the ambient temperature, and thus the hot gas exiting the turbine has significant thermodynamic utility (exergy) that can be used cost-effectively.►Ways to utilize this potential include:

►The regenerative cycle previously considered.►A combined cycle – namely, a cycle that couples two power cycles such that the energy discharged by heat transfer from the higher-temperature cycle is used as a heat input for the lower-temperature cycle.

Gas Turbine-Based Combined Cycle

►Illustrated here is a combined cycle involving gas and vapor power cycles:

►The cycles are combined using an interconnecting heat-recovery steam generator that serves as the boiler for the vapor power cycle.

►The combined cycle has the gas turbine’s high average temperature of heat addition and the vapor power cycle’s low average temperature of heat rejection.

►Thermal efficiency is greater than either cycle would have individually.

Combined Gas Turbine-Vapor Power Cycle

►Increasingly, combined gas turbine-vapor power plants are being used world-wide for electric power generation.

Combined Gas Turbine-Vapor Power Cycle►The net power developed by the combined cycle is the sum of the net power developed by each cycle.►The thermal efficiency of the combined cycle is the net power output divided by the rate of heat addition.

►For an adiabatic heat recovery steam generator, mass and energy rate

(Eq. 9.28)

balances reduce to give the following relationship involving the mass flow rates of the two cycles:

(Eq. 9.29)

Combined-cycle District Heating

►Alternatively, steam exiting the turbine may be sent directly to the community while its condensate returns to the pump, thereby eliminating the condenser.

►Shown here is a combined gas turbine-vapor power cycle applied for district heating. District heating plants are located within communities to deliver steam or hot water

together with electricity for domestic, commercial, and industrial use.

►Because of their favorable power-to-weight ratio, gas turbines are well suited for aircraft propulsion. The turbojet engine is commonly used for this purpose.►The figure provides the schematic of a turbojet engine.

Gas Turbines for Aircraft Propulsion

Va V5

Va V5

►The increase in velocity from diffuser inlet, Va, to nozzle exit, V5, gives rise to the thrust developed by the engine in accord with Newton’s second law of motion (Eq. 9.31).►In harmony with air-standard analysis, we assume air modeled as an ideal gas flows through the engine shown in the schematic and the temperature rise that would be obtained with combustion is achieved by heat transfer from an external source.


Va V5

►If the air flows through the components of the turbojet engine without irreversibilities and stray heat transfer, air undergoes the five processes shown on the T-s diagram:


►Process a-1: Air at velocity Va enters the diffuser and decelerates isentropically, while experiencing an increase in pressure.►Process 1-2: The air experiences a further increase in pressure isentropically, owing to work done by the compressor.

Va V5


►Process 2-3: The temperature of the air increases at constant pressure as it receives a heat transfer from an external source.►Process 3-4: The high-pressure, high-temperature air expands isentropically through the turbine, driving the compressor.


Va V5


►Process 4-5: The air continues to expand isentropically through the nozzle, achieving a velocity, V5, at the engine exit much greater than the velocity, Va, at the engine inlet, and thereby developing thrust.


►If the change in potential energy from inlet to exit is negligible, g(zi – ze) drops out.

►If the heat transfer with surroundings is negligible, drops out.

)(

2

)V(V)(0

22

cvcv eiei

ei zzghhmWQ

Review: Nozzle and Diffuser Modeling

.0cv W

2

VV)(0

22ei

ei hh

cvQ

►

►The one-inlet, one-exit energy rate balance at steady state reads:

►For a control volume enclosing a nozzle or diffuser,

►For the diffuser, i = a and e = 1. Then,

►The energy rate balance applicable to the diffuser takes the form


2

V2a

a1 hh

ha

Va

h1

V1 ≈ 0

a

1

2

VV)(0

22ei

ei hh

►Since exit velocity is negligible, the energy rate balance reduces to

2

VV)(0

21

2a

1a hh

►The energy rate balance applicable to the nozzle takes the form


2

VV)(0

25

24

54 hh

)(2V2

V545

25

54 hhhh

h4

V4 ≈ 0

4

5

h5

V5

►For the nozzle, i = 4 and e = 5. Then,

2

VV)(0

22ei

ei hh

►Since inlet velocity is negligible, the energy rate balance reduces to

►Since the final expressions obtained for the diffuser and nozzle are deduced from mass and energy rate balances, they apply equally when irreversibilities are present and in the absence of irreversibilities.


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