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Next: Supersonic Branch Up: Isothermal Flow Previous: The Entrance Limitation of   Index

# Comparison with Incompressible Flow

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.

denote

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