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Next: Supersonic Branch Up: Isothermal Flow Previous: The Entrance Limitation of Index
The Mach number of the flow in some instances is relatively small. In these cases, one should expect that the isothermal flow should have similar characteristics as incompressible flow. For incompressible flow, the pressure loss is expressed as follows
Now note that for incompressible flow and represent the ratio of the traditional . To obtain a similar expression for isothermal flow, a relationship between and and pressures has to be derived. From equation (8.39) one can obtained that
Substituting this expression into (8.40) yields
Because is always positive there is only one solution to the above equation even though M2.
Expanding the solution for small pressure ratio drop, , by some mathematics.
Now equation (8.41) can be transformed into
now we have to expand into a series around and remember that
and for example the first derivative of
similarly it can be shown that equation (8.45) now can be approximated as
rearranging equation (8.48) yields
and further rearrangement yields
in cases that is small
The pressure difference can be plotted as a function of the for given value of . Equation (8.51) can be solved explicitly to produce a solution for
A few observations can be made about equation (8.52).
Next: Supersonic Branch Up: Isothermal Flow Previous: The Entrance Limitation of Index Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21 include("aboutPottoProject.php"); ?>