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Next: The Entrance Limitation of Up: Isothermal Flow Previous: The Control Volume Analysis/Governing   Index

Dimensionless Representation

In this section the equations are transformed into the dimensionless form and presented as such. First it must be recalled that the temperature is constant and therefore, equation of state reads
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 $ M^{2} = \rho U^{2} / k P$ yields:
Now the pressure, $ P$ as a function of the Mach number has to substitute along with velocity, $ U$ .
Differentiation of equation (8.11) yields

Now it can be noticed that $ dT = 0$ 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 $ dT_{0} \neq 0$ 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 $ T$ 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 ($ T=0$ ) 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 ( $ M, T, P, \rho$ )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, $ \frac{4fL}{D}$ , is selected as the choice for the independent variable. Thus, the equations need to be obtained as a function of $ \frac{4fL}{D}$ . 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 $ \frac{4fL}{D}$

The variables in equation (8.24) can be separated to obtain integrable form as follows
It can be noticed that at the entrance $ (x = 0)$ for which $ M = M_{x=0}$ (the initial velocity in the tube isn't zero). The term $ \frac{4fL}{D}$ is positive for any $ x$ , thus, the term on the other side has to be positive as well. To obtain this restriction $ 1= kM^{2}$ . Thus, the value $ M= {1\over \sqrt{k}}$ is the limiting case from a mathematical point of view. When Mach number larger than $ M > {1\over \sqrt{k}}$ it makes the right hand side of the integrate negative. The physical meaning of this value is similar to $ M=1$ choked flow which was discussed in a variable area flow in Chapter (4).

Further it can be noticed from equation (8.26) that when $ M \rightarrow {1\over \sqrt{k}}$ the value of right hand side approaches infinity ($ \infty$ ). Since the stagnation temperature ($ T_{0}$ ) has a finite value which means that $ dT_{0} \rightarrow \infty$ . 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 $ M^{2} = { U^{2} / kRT}$ and noticing that $ T=constant$ is used to describe the relation of the properties at $ M = 1 /\sqrt{k}$ . 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
$\displaystyle \rho U = \rho^{*} U^{*} ; \Longrightarrow {\rho \over \rho^{*}} = { 1 \over \sqrt{k} M} %\label{eq:}
$ (8.31)

Reusing the perfect-gas relationship
Now utilizing the relation for stagnated isotropic pressure one can obtain
$\displaystyle {P_{0} \over P_{0}^{*}} = {P \over P^{*}} \left[ {1 + { k -1 \ove...
... ^ {2} \over { 1 + {k -1 \over 2k} } } \right] ^ { k \over k -1 } %\label{eq:}
$ (8.33)

Substituting for $ {P \over P^{*}}$ equation (8.32) and rearranging yields
$\displaystyle {P_{0} \over P_{0}^{*}} = {1 \over \sqrt{k}} \left( {2k \over 3k-...
...left( 1 + {k -1 \over 2} M ^{2}\right)^{k \over k-1} { 1 \over M} %\label{eq:}
$ (8.34)

And the stagnation temperature at the critical point can be expressed as

These equations (8.30)-(8.35) are presented on in Figure (8.2)

Figure: Description of the pressure, temperature relationships as a function of the Mach number for isothermal flow
\begin{figure}\centerline{\includegraphics {calculations/isothermal}}

next up previous index
Next: The Entrance Limitation of Up: Isothermal Flow Previous: The Control Volume Analysis/Governing   Index
Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21