Isentropic nozzle flow
In fluid mechanics, isentropic nozzle flow describes the movement of a fluid through a narrow opening without an increase in entropy (an isentropic process).
According to the second law of thermodynamics, whenever there is a reversible and adiabatic flow, constant value of entropy is maintained.
Isentropic is the combination of the Greek word "iso" (which means - same) and entropy.
The generation of sound waves is an isentropic process.
Since there is an increase in area, therefore we call this an isentropic expansion.
Below are nine equations commonly used when evaluating isentropic flow conditions.
[1] These assume the gas is calorically perfect; i.e. the ratio of specific heats is a constant across the temperature range.
There are both actual and the isentropic stagnation states for a typical gas or vapor.
Sometimes it is advantageous to make a distinction between the actual and the isentropic stagnation states.
The actual stagnation state is the state achieved after an actual deceleration to zero velocity (as at the nose of a body placed in a fluid stream), and there may be irreversibility associated with the deceleration process.
Also, temperature variations for compressible flows are usually significant and thus the energy equation is important.
These include the flow through a jet engine, through the nozzle of a rocket, from a broken gas line, and past the blades of a turbine.
To model such situations, consider the control volume in the changing area of the conduit of Fig.
The continuity equation between two sections an infinitesimal distance dx apart is
If only the first-order terms in a differential quantity are retained, continuity takes the form
This equation applies to a steady, uniform, isentropic flow.
So, for a supersonic flow to develop from a reservoir where the velocity is zero, the subsonic flow must first accelerate through a converging area to a throat, followed by continued acceleration through an enlarging area.
The nozzles on a rocket designed to place satellites in orbit are constructed using such converging-diverging geometry.
The energy and continuity equations can take on particularly helpful forms for the steady, uniform, isentropic flow through the nozzle.
Apply the energy equation with Q, WS = 0 between the reservoir and some location in the nozzle to obtain
Any quantity with a zero subscript refers to a stagnation point where the velocity is zero, such as in the reservoir.
Consider a converging nozzle connecting a reservoir with a receiver.
If the reservoir pressure is held constant and the receiver pressure reduced, the Mach number at the exit of the nozzle will increase until Me = 1 is reached, indicated by the left curve in figure 2.
After Me = 1 is reached at the nozzle exit for pr = 0.5283p0, the condition of choked flow occurs and the velocity throughout the nozzle cannot change with further decreases in pr.
Nature allows this by providing the streamlines of a gas the ability to make a sudden change of direction at the exit and expand to a much greater area resulting in a reduction of the pressure from pe to pr.
The case of a converging-diverging nozzle allows a supersonic flow to occur, providing the receiver pressure is sufficiently low.
If pr is slightly less than p0, the flow is subsonic throughout, with a minimum pressure at the throat, represented by curve B.
There is another receiver pressure substantially below that of curve C that also results in isentropic flow throughout the nozzle, represented by curve D; after the throat the flow is supersonic.
It may appear that the supersonic flow will tend to separate from the nozzle, but just the opposite is true.
To avoid separation in subsonic nozzles, the expansion angle should not exceed 10°.
