review of fundamentals -fluid mechanics - amazon s3 · pdf filereview of fundamentals -fluid...
TRANSCRIPT
Review of Fundamentals - Fluid
Mechanics
© 2015 SIM University. All rights reserved.
Introduction
• Properties of Compressible Fluid Flow
• Basics of One-Dimensional Gas Dynamics
• Nozzle Operating Characteristics
• Characteristics of Shock Wave
• A gas turbine cycle through the use of H-K Diagram
© 2015 SIM University. All rights reserved.
• Approximate Mach zones:
< M0.3 Subsonic, incompressible
M0.3 – M0.8 Subsonic, compressible
M0.8 – M1.2 Transonic, shock waves appear
M1.2 – M3 Supersonic
�M3 Hypersonic
• Normal to specify P, T, V to describe state
• In compressible flow, V is often replaced by Mach number, total pressure
and total temperature.
© 2015 SIM University. All rights reserved.
Compressible Flow Properties
Without gravity effects, the Steady Flow Energy Equation (SFEE) is
For calorically perfect (constant cp, cv) gas,
The stagnation or total enthalpy ht is defined as
The stagnation or total temperature Tt is defined as
© 2015 SIM University. All rights reserved.
Total Enthalpy / Total Temperature
• For an aircraft in flight at velocity Va, the airstream velocity at the
leading edge stagnation point is negligibly small
• Kinetic energy is brought to rest and produces a rise in temperature
(aerodynamic heating)
Watch video on aerodynamic heating:
http://www.youtube.com/watch?v=RChlt5wdqBs
Adapted: “Elements of Propulsion: Gas Turbines and
Rockets” by Jack D. Mattingly
© 2015 SIM University. All rights reserved.
Total Enthalpy / Total Temperature
Inserting the total enthalpy into the SFEE:
For calorically perfect gas,
If there is no heat transfer and no work interactions,
(i.e. q - wx = 0, or q = wx = 0), then
and, for a calorically perfect gas,
© 2015 SIM University. All rights reserved.
Total Enthalpy / Total Temperature
The total pressure Pt of a flowing gas is defined as the pressure obtained
when the gas is brought to rest isentropically (sy – s1 = 0)
© 2015 SIM University. All rights reserved.Source: Soon Kim Tat
(Note: recall that )
Total Pressure
• Stagnation temperature ratio
• Stagnation pressure ratio
• At M=1 (choked nozzle) for air (isentropic),
© 2015 SIM University. All rights reserved.
Stagnation Temperature and Pressure
© 2015 SIM University. All rights reserved. Source: Soon Kim Tat
Stagnation Temperature and Pressure
• Perfect gas is brought to stagnation (V2 = 0)
• Under adiabatic (q = 0), no-shaft-work (wx = 0)
• Same final stagnation temperature will be attained whether it is
irreversible or reversible process, i.e.
Tt2, irreversible = Tt2,reversible
• However, the final total pressure
will be lower, i.e.
Pt2, irreversible < Pt2, reversible
• Py depends on the entropy increase ,
(sy – s1) - a measure of the degree of
irreversibility
Source: p 98,“Elements of Propulsion: Gas
Turbines and Rockets” by Jack D. Mattingly
© 2015 SIM University. All rights reserved.
Total Pressure (Irreversible)
• Total pressure of air passing through an engine inlet and nozzle or a
shock wave cannot increase and must decrease because of the
irreversible effects of friction.
Schlieren Imaging of Supersonic Inlet
shock Waves
Source: http://en.wikipedia.org/wiki/Unstart
© 2015 SIM University. All rights reserved.
Total Pressure (Irreversible)
For a one-dimensional flow where q = wx = 0,
For a calorically perfect gas,
Rewrite in terms of dimensionless static enthalpy
and dimensionless kinetic energy as
We obtain
or H + K = 1
© 2015 SIM University. All rights reserved.
One-Dimensional Gas Dynamics
• Dimensionless static enthalpy (H) and dimensionless kinetic
energy (K)
• Useful for
– explaining the more complex internal flow behaviour of air
breathing engines
– visualising the operation of propulsion devices
• Not a state diagram
© 2015 SIM University. All rights reserved.
H-K Diagram
M<1 M>1
1
1
Adapted: “Elements of Propulsion: Gas Turbines and Rockets” by Jack D. Mattingly© 2015 SIM University. All rights reserved.
H-K Diagram
Key: 0 = freestream reference state.
Point c =choked condition at constant impulse.
Points u and d denote end states of normal shock.
Circled numbers denote isolines of constant property as follows:
1. Static enthalpy, static temperature
2. Kinetic energy, velocity, pressure (for frictionless heating or cooling only)
3. Isoline of constant Mach number
4. Total enthalpy, total temperature (adiabatic),
5. Post-heat release adiabatic, τ > 1
6. Impulse function / stream thrust, area (for frictionless flow with heating or
cooling only), case I = I0 ;
7. Impulse function, case φ > φ0
0
1t
t
T
Tτ ≡ =
© 2015 SIM University. All rights reserved.
H-K Diagram
Adapted: “Elements of Propulsion: Gas Turbines and Rockets” by Jack D. Mattingly
© 2015 SIM University. All rights reserved.
Scramjet H-K Diagram
Source: The Jet Engine by Rolls Royce
© 2015 SIM University. All rights reserved.
Nozzle Design
• From a large chamber, a gas flows through a nozzle with mass flow
rate ṁc
– chamber pressure Pc = Pt
– chamber temperature Tc = Tt
Graphics: Soon Kim Tat
Nozzle Gas Relationships:
© 2015 SIM University. All rights reserved.
Nozzle Design
Nozzle design – To pass a given mass flow with minimum frictional
losses between 2 regions of different pressure. (Independent
variable:– P)
Nozzle operating characteristics – Given a nozzle, determine the
mass flow rates and pressure distribution for various nozzle
pressure. (Independent variable: A)
• 4 variables: P, T, V, A
• Select one variable as independent and find the remaining
© 2015 SIM University. All rights reserved.
Nozzle Design Approaches
Design Objective: To expand exhaust gas to a target static pressure
Source: “Elements of Propulsion: Gas Turbines and Rockets” by Jack D. Mattingly
© 2015 SIM University. All rights reserved.
Nozzle Design - Example
• Consider a wind-tunnel nozzle (next slide)
• As air flows from storage chamber into evacuated receiver:
– raises the pressure in the nozzle exhaust region Pa
– decreases the nozzle pressure ratio Pn = Pc / Pa.
• 7 possible distinct nozzle pressure ratio operating conditions.
© 2015 SIM University. All rights reserved.
Nozzle Flow and Shock Waves
Nozzle Pressure Ratio
Pn=Pc/Pa
1. Underexpanded
Pn>Pṅ
2. Design expansion
Pn=Pṅ
3. Overexpanded
Pn<Pṅ
4. Normal shock at
exit
5. Normal shock
inside
6. Sonic at throat,
subsonic
elsewhere
7. Subsonic flow
everywhere (mass
flow below max)
Adapted: “Elements of Propulsion: Gas Turbines and Rockets” by Jack D. Mattingly© 2015 SIM University. All rights reserved.
Nozzle Flow and Shock Waves
Summary
• Total enthalpy, total temperature and total
pressure, and their relationships
• Cycle of an air-breathing jet engine using H-K
diagram
• Evaluating the issues related to nozzle design
© 2015 SIM University. All rights reserved.
Reflection Question
• Evaluate what happens to the gas pressure,
temperature and velocity as it passes through a
convergent nozzle and a divergent duct, if the
initial velocity is:
a. Subsonic
b. Supersonic
© 2015 SIM University. All rights reserved.