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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.

Conservation of Momentum Applied to 1 D Steady Flow

Using gauss divergence theorem:

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:

Energy equation in terms for pressure and velocity for a perfect gas

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.

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 ?

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.

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