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Next: The Mechanics and Why Up: Fanno Flow Previous: Model Index Non-Dimensionalization of the EquationsBefore solving the above equation a dimensionless process is applied. By utilizing the definition of the sound speed to produce the following identities for perfect gasUtilizing 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 second law reads 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 |