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Next: The Entrance Limitation of Up: Isothermal Flow Previous: The Control Volume Analysis/Governing Index
It is convenient to define a hydraulic diameter
Now, the Fanning friction factor8.2 is introduced, this factor is a dimensionless friction factor sometimes referred to as the friction coefficient as
Substituting equation (8.8) into momentum equation (8.2) yields
Rearranging equation (8.9) and using the identify for perfect gas yields:
Now the pressure, as a function of the Mach number has to substitute along with velocity, .
Differentiation of equation (8.11) yields
Now it can be noticed that for isothermal process and therefore
The dimensionalization of the mass conservation equation yields
Differentiation of the isotropic (stagnation) relationship of the pressure (4.11) yields
Differentiation of equation (4.9) yields:
Notice that in an isothermal flow. There is no change in the actual temperature of the flow but the stagnation temperature increases or decreases depending on the Mach number (supersonic flow of subsonic flow). Substituting for equation (8.17) yields:
Rearranging equation (8.18) yields
By utilizing the momentum equation it is possible to obtain a relation between the pressure and density. Recalling that an isothermal flow ( ) and combining it with perfect gas model yields
From the continuity equation (see equation (8.14)) leads
The four equations momentum, continuity (mass), energy, state are described above. There are 4 unknowns ( )8.3 and with these four equations the solution is attainable. One can notice that there are two possible solutions (because of the square power). These different solutions are supersonic and subsonic solution.
The distance friction, , is selected as the choice for the independent variable. Thus, the equations need to be obtained as a function of . The density is eliminated from equation (8.15) when combined with equation (8.20) to become
After substituting the velocity (8.22) into equation (8.10), one can obtain
Equation (8.23) can be rearranged into
Similarly or by other path the stagnation pressure can be expressed as a function of
The variables in equation (8.24) can be separated to obtain integrable form as follows
It can be noticed that at the entrance for which (the initial velocity in the tube isn't zero). The term is positive for any , thus, the term on the other side has to be positive as well. To obtain this restriction . Thus, the value is the limiting case from a mathematical point of view. When Mach number larger than it makes the right hand side of the integrate negative. The physical meaning of this value is similar to choked flow which was discussed in a variable area flow in Chapter (4).
Further it can be noticed from equation (8.26) that when the value of right hand side approaches infinity ( ). Since the stagnation temperature ( ) has a finite value which means that . Heat transfer has a limited value therefore the model of the flow must be changed. A more appropriate model is an adiabatic flow model yet it can serve as a bounding boundary (or limit).
Integration of equation (8.27) yields
The definition for perfect gas yields and noticing that is used to describe the relation of the properties at . By denoting the superscript symbol for the choking condition, one can obtain that
Rearranging equation (8.29) is transfered into
Utilizing the continuity equation provides
Reusing the perfect-gas relationship
Now utilizing the relation for stagnated isotropic pressure one can obtain
Substituting for equation (8.32) and rearranging yields
And the stagnation temperature at the critical point can be expressed as
Next: The Entrance Limitation of Up: Isothermal Flow Previous: The Control Volume Analysis/Governing Index Created by:Genick Bar-Meir, Ph.D.
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