Analysis of A Disturbance in A Gas Flow

1
Analysis of A Disturbance in A Gas Flow

• P M V Subbarao
• Associate Professor
• Mechanical Engineering Department
• I I T Delhi

Search for More Physics through Mathematics .
2
Analysis of Plane Disturbance

• A control volume for this analysis is shown, and
  the gas flows from left to right.
• The conditions to the right of the disturbance
  are uniform, but different from the left side and
  vice versa.

• The thickness of disturbance is very small.
• No chemical reactions.
• There is no friction or heat loss at the
  disturbance.

3
Conservation of Mass Applied to 1 D Steady Flow
Conservation of Mass
Conservation of Mass for 1DSF
Integrate from inlet to exit
4
Gauss Divergence Theorem
If the velocity is normal to the area
5
Conservation of mass
The area of the disturbance is constant.
Conservation of momentum The momentum is the
quantity that remains constant because there are
no external forces.
6
Conservation of Momentum Applied to 1 D Steady
Flow
Using gauss divergence theorem
7
If the velocity is normal to the area
Steady, Inviscid 1-D Flow, Body Forces negligible
The area of the disturbance is constant.
8
Conservation of Energy Applied to 1 D Steady Flow
Steady flow with negligible Body Forces and no
heat transfer is an adiabatic flow
For a blissful fluid the rate of work transfer is
only due to pressure.
9
For a total change from inlet to exit
Using gauss divergence theorem
One dimensional flow normal to the area of cross
section
10
Using conservation of mass
With negligible body forces
11
(No Transcript)
12
The process is adiabatic, or nearly adiabatic,
and therefore the energy equation can be written
as
For calorically perfect gas
The equation of state for perfect gas reads
13
Solution of Simultaneous Equations

• If the conditions upstream are known, then there
  are four unknown conditions downstream.
• A system of four unknowns and four equations is
  solvable.
• There exist multiple solutions because of the
  quadratic form of equations.
• Out of these multiple solutions, some are
  physically possible and some are not.
• These Physically possible solutions refer to the
  universal law of direction of happening.
• Different Physically possible solutions will lead
  to development of different products or
  processes.
• The only tool that brings us to the right
  direction of happening is the second law of
  thermodynamics.
• This law dictates the direction of happening
  Across the disturbance the entropy can increase
  or remain constant.

14

• In mathematical terms, it can be written as
  follows

For an ideal gas

• We will not use isentropic conditions.
• Use more algebra to reduce the number of
  variables.

15
Summary of Equations
Conservation of mass
Conservation of momentum
Conservation of Energy
The equation of state for perfect gas
Constraint
16
Change in Mach Number between points x y
Dividing this equation by cx
17
Conservation of momentum
18
Dividing this through by cx2/g
19
Momentum Equation
Continuity Equation

20
Energy equation in terms for pressure and
velocity for a perfect gas
Dividing this by
21
Energy Equation
Combined Mass Momentum Equation
Combined Mass, Momentum and Energy Conservation
22
Combined Mass, Momentum and Energy Conservation
23
Nothing Happening
24
If there is something happening between x y
With a disturbance between x y,
This equation relates the downstream Mach number
to the upstream. It can be used to derive
pressure ratio, the temperature ratio, and
density ratio across the disturbance.
25
Substitute value of My
26
(No Transcript)
27
Change in Entropy Across the disturbance
28
Physically possible solution 2
Infeasible
Mx
29
The Nature of Irreversible Phenomenon
My
g constant1.4
Mx
This Strong Irreversibility is called as Normal
Shock.
30
Nature of Normal Shock

• The flow across the shock is adiabatic and the
  stagnation temperature is constant across a
  shock.
• The effect of increase in entropy across a shock
  will result in change of supersonic to subsonic
  flow.
• The severity of a shock is proportional to
  upstream Mach Number.
• Normal Shock is A severe irreversible Diffuser.
• No capital investment.
• Can we promote it ?

31
Normal Shock Past F-18
32
Reentry Interface Gas Dynamics
33
Occurrence of Normal Shock Reentry Vehicles
34
Weakening of Normal Shock
35
Brayton Cycle for Jet Propulsion
Burner
36
Gas Dynamic Analysis of Turbojet Engine
37
Jet Engine Inlet Duct

• All jet engines have an inlet to bring free
  stream air into the engine.
• The inlet sits upstream of the compressor and,
  while the inlet does no work on the flow.
• Inlet performance has a strong influence on
  engine net thrust.
• Inlets come in a variety of shapes and sizes with
  the specifics usually dictated by the speed of
  the aircraft.
• The inlet duct has two engine functions and one
  aircraft function .
• First it must be able recover as much of the
  total pressure of the free air stream as possible
  and deliver this pressure to the front of the
  engine compressor .
• Second the duct must deliver air to the
  compressor under all flight conditions with a
  little turbulence .
• Third the aircraft is concerned , the duct must
  hold to a minimum of the drag.

38

• The duct also usually has a diffusion section
  just ahead of the compressor to change the ram
  air velocity into higher static pressure at the
  face of the engine .
• This is called ram recovery .
• SUBSONIC INLETS
• A simple, straight, short inlet works quite well.
  
• On a typical subsonic inlet, the surface of the
  inlet from outside to inside is a continuous
  smooth curve with some thickness from inside to
  outside.
• The most upstream portion of the inlet is called
  the highlight, or the inlet lip.
• A subsonic aircraft has an inlet with a
  relatively thick lip.

39
Subsonic Inlet Duct
40
Axi-Symmetric Inlets
41
Rectangular Inlets
42
High Supersonic Flight
43
(No Transcript)
44
Supersoinc Intakes