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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 ($ dP/P$ ) and $ dU/U$ 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 $ C_p$ and by utilizing the definition Mach number yields
$\displaystyle {dT \over T} + {1 \over \underbrace{\left( kR \over (k -1) \right)}_{C_p}} { 1 \over T} {U^2 \over U ^2} d \left( U^2 \over 2 \right) =$    
$\displaystyle \hookrightarrow {dT \over T} + {( k -1 ) \over \underbrace{kRT}_{c^2}} {U^2 \over U ^2} d \left( U^2 \over 2 \right) =$    
$\displaystyle \hookrightarrow {dT \over T} + {k -1 \over 2} M^2 {dU^2 \over U^2} = 0$ (9.20)

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, $ \frac{4fL}{D}$ , is chosen as the independent variable since the change in the dimensionless resistance, $ \frac{4fL}{D}$ , causes the change in the other variables.

Combining equations (9.19) and (9.21) when eliminating $ dT / T$ results

The term $ {d\rho \over \rho}$ can be eliminated by utilizing equation (9.18) and substituting it into equation (9.22) and rearrangement yields
The term $ dU^2/U^2$ can be eliminated by using (9.23)
The second equation for Mach number, $ M$ variable is obtained by combining equation (9.20) and (9.21) by eliminating $ dT / T$ . Then $ d\rho / \rho$ and $ U$ are eliminated by utilizing equation (9.18) and equation (9.22). The only variable that is left is $ P$ (or $ dP/P$ ) 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 $ T_0 = T (1 + (1-k)/2 M^2 )$ . Taking derivative of this expression when $ M$ remains constant yields $ dT_0 = dT (1 + (1-k)/2 M^2 ) $ 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, $ (dT_0 =0)$ . Therefore the entropy change is
Using the equation for stagnation pressure the entropy equation yields

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