Analysis of A Disturbance in A Gas Flow

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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 .…. Analysis of Plane Disturbance. A control volume for this analysis is shown, and the gas flows from left to right.

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### 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 .…

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.
Conservation of Mass Applied to 1 D Steady Flow

Conservation of Mass:

Conservation of Mass for 1DSF:

Integrate from inlet to exit :

Gauss Divergence Theorem

If the velocity is normal to the area :

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.

If the velocity is normal to the area :

Steady, Inviscid 1-D Flow, Body Forces negligible

The area of the disturbance is constant.

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.

For a total change from inlet to exit :

Using gauss divergence theorem:

One dimensional flow normal to the area of cross section

Using conservation of mass

With negligible body forces:

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

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.
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.
Summary of Equations

Conservation of mass:

Conservation of momentum:

Conservation of Energy:

The equation of state for perfect gas

Constraint:

Momentum Equation :

Continuity Equation :

&

Dividing this by

Energy Equation :

Combined Mass & Momentum Equation :

Combined Mass, Momentum and Energy Conservation :

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.

Physically possible solution 2

Solution - 1

Infeasible

Mx

The Nature of Irreversible Phenomenon

My

g = constant=1.4

Mx

This Strong Irreversibility is called as Normal Shock.

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 ?
Turbofan

Turbine +

Nozzle

Compressor

4

3

1

2

1'-2"

1'-11"

10"

A

B

C

D

Brayton Cycle for Jet Propulsion

Burner

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.
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.