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The energy equation for isentropic nozzle provides

Utilizing equation (4.27) when
leads to

For the isentropic process
when the
at any point of the flow.
The equation (7.2) becomes

The continuity equation as developed earlier (mass conservation
equation isn't effected by the gravity)

Substituting

from equation

7.3,
into equation (

7.2)
moving

to the right hand side, and diving by

yields

Rearranging equation (

7.5) yields

And further rearranging yields

Equation (

7.7) can be rearranged as

Equation (

7.8) dimensionless form by utilizing

and

is the nozzle length

And the final form of equation (

7.9) is

The term
is considered to be very small
(
%) for ``standard'' situations.
The dimensionless number,
sometimes referred as
Ozer number determines whether gravity should be considered in the
calculations.
Nevertheless, one should be aware of value of Ozer number
for large magnetic fields (astronomy) and low temperature,
In such cases, the gravity effect can be considerable.

As it was shown before the transition must occur when
.
Consequently, two zones must be treated separately.
First, here the Mach number is discussed and not the pressure as
in the previous chapter.
For
(the subsonic branch) the term
is positive and the
treads determined by gravity and the area function.

or conversely,

For the case of

(the supersonic branch) the term

is negative and therefore

For the border case

, the denominator

, is zero
either

or

And the

is indeterminate.
As it was shown in chapter (

4)
the flow is chocked (

) only when

It should be noticed that when
is zero, e.g. horizontal flow,
the equation (7.11) reduced into
that was developed previously.

The ability to manipulate the location provides a mean to
increase/decrease the flow rate.
Yet this ability since Ozer number is relatively very small.

This condition means that the critical point can occurs in
several locations that satisfies equation
(7.11).
Further, the critical point, sonic point is
If
is a positive function, the critical point happen at
converging part of the nozzle (before the throat) and if
is
a negative function the critical point is diverging part of the
throat.
For example consider the gravity,
a flow in a nozzle
vertically the critical point will be above the throat.

** Next:** Isothermal Nozzle (T=constant)
** Up:** Nozzle Flow With External
** Previous:** Nozzle Flow With External
** Index**
Created by:Genick Bar-Meir, Ph.D.

On:
2007-11-21