Chapter 2 Propulsion Gas Dynamics

Lecture by Clark M. (Mike) Butler, P.E.

ME 4953.007 & 5013.006 Aircraft Propulsion

LECTURE 1 Air-Standard Analysis, Irreversibility, Gas

Turbines for Aircraft Propulsion

Ideal Gas Model Review Question: What is the difference between an Ideal gas and a Perfect gas? A perfect gas is an ideal gas which is also chemically non-reactive and is always in thermodynamic equilibrium.

u(T) and h(T) are evaluated either from tables or equations on either a mass or molar basis.

Ideal Gas Model Review

►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. ►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. ►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

Relating the Ideal Otto and Brayton Thermodynamic Cycles

Ideal Otto Cycle

Ideal Brayton Cycle

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 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-Standard Brayton Cycle

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

Air-Standard Brayton Cycle

►Later we show that 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

Heat addition

Heat rejection

Air-Standard Brayton Cycle

►The thermal efficiency is

►The back work ratio is

►Since the above equations 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.

Air-Standard Brayton Cycle

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 compressor 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.

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.

►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 (environmental heat sink).

Ideal Air-Standard Brayton Cycle

►Since the ideal Brayton cycle involves internally reversible processes, results from entropy review 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.

Ideal Air-Standard Brayton Cycle

►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.

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

Ideal Air-Standard Brayton Cycle

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.

Effects of Compressor Pressure Ratio on Brayton Cycle Performance

►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. ►Accordingly, the Brayton cycle thermal efficiency increases as the compressor pressure ratio increases.

60

η th (

%)

2 4 6 8 10

Compressor Pressure Ratio

Effects of Compressor Pressure Ratio on Brayton Cycle Performance

►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. ►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′.

Effects of Compressor Pressure Ratio on Brayton Cycle Performance

►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.

Effects of Compressor Pressure Ratio on Brayton Cycle Performance

►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.

Effects of Compressor Pressure Ratio on Brayton Cycle Performance

►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.

Effects of Compressor Pressure Ratio on Brayton Cycle Performance

Gas Turbine Power Plant Irreversibility

►The most significant irreversibility by far is the irreversibility of combustion. ►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.

mW

mW

mW

ctnet −=

marked decrease in net work of the power plant

irreversibilites decrease turbine work

irreversiblities increase compressor work

)()(

)/()/(

s43

43

st

tt hh

hhmWmW

−−

==

η

►Isentropic turbine efficiency 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

hhmWmW

−−

==

η

►Isentropic compressor efficiency 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 Irreversibility

►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. ►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.

Gas Turbines for Aircraft Propulsion

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.

Gas Turbines for Aircraft Propulsion

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.

►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:

Gas Turbines for Aircraft Propulsion

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 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:

Gas Turbines for Aircraft Propulsion

►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

)(022

cvcv eiei

ei zzghhmWQ

Review: Nozzle and Diffuser Modeling

.0cv =W

−+−=

2VV

)(022ei

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

2V2

aa1 += hh

ha Va

h1 V1 ≈ 0

a

1

−+−=

2VV

)(022ei

ei hh

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

−+−=

2VV

)(02

12a

1a hh

Gas Turbines for Aircraft Propulsion

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

−+−=

2VV

)(025

24

54 hh

)(2V2

V545

25

54 hhhh −=→+=

h4 V4 ≈ 0

4

5

h5 V5

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

−+−=

2VV

)(022ei

ei hh

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

Gas Turbines for Aircraft Propulsion

►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.

Gas Turbines for Aircraft Propulsion