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Next: The Mechanics and Why Up: Fanno Flow Previous: Model   Index

# Non-Dimensionalization of the Equations

Before solving the above equation a dimensionless process is applied. By utilizing the definition of the sound speed to produce the following identities for perfect gas

Utilizing the definition of the perfect gas results in

Using the identity in equation (9.14) and substituting it into equation (9.11) and after some rearrangement yields

By further rearranging equation (9.16) results in

It is convenient to relate expressions of ( ) and in terms of the Mach number and substituting it into equation (9.17). Derivative of mass conservation ((9.2)) results in

The derivation of the equation of state (9.5) and dividing the results by equation of state (9.5) results

Derivation of the Mach identity equation (9.14) and dividing by equation (9.14) yields

Dividing the energy equation (9.4) by and by utilizing the definition Mach number yields

Equations (9.17), (9.18), (9.19), (9.20), and (9.21) need to be solved. These equations are separable so one variable is a function of only single variable (the chosen as the independent variable). Explicit explanation is provided for only two variables, the rest variables can be done in a similar fashion. The dimensionless friction, , is chosen as the independent variable since the change in the dimensionless resistance, , causes the change in the other variables.

Combining equations (9.19) and (9.21) when eliminating results

The term can be eliminated by utilizing equation (9.18) and substituting it into equation (9.22) and rearrangement yields

The term can be eliminated by using (9.23)

The second equation for Mach number, variable is obtained by combining equation (9.20) and (9.21) by eliminating . Then and are eliminated by utilizing equation (9.18) and equation (9.22). The only variable that is left is (or ) which can be eliminated by utilizing equation (9.24) and results in

Rearranging equation (9.25) results in

After similar mathematical manipulation one can get the relationship for the velocity to read

and the relationship for the temperature is

density is obtained by utilizing equations (9.27) and (9.18) to obtain

The stagnation pressure is similarly obtained as

The stagnation temperature expresses as . Taking derivative of this expression when remains constant yields and thus when these equations are divided they yield

In similar fashion the relationship between the stagnation pressure and the pressure can be substituted into the entropy equation and result in

The first law requires that the stagnation temperature remains constant, . Therefore the entropy change is

Using the equation for stagnation pressure the entropy equation yields

Next: The Mechanics and Why Up: Fanno Flow Previous: Model   Index
Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21