The Approximation of the Fanno Flow by Isothermal Flow

The isothermal flow model has equations that theoreticians find easier to use and to compare to the Fanno flow model.

One must notice that the maximum temperature at the entrance is ${T_0}_1$ . When the Mach number decreases the temperature approaches the stagnation temperature ( $T\rightarrow T_0$ ). Hence, if one allows certain deviation of temperature, say about 1% that flow can be assumed to be isothermal. This tolerance requires that $(T_0 -T) /T_0 =0.99$ which requires that enough for $M_1 < 0.15$ even for large $k=1.67$ . This requirement provides that somewhere (depend) in the vicinity of $\frac{4fL}{D}= 25$ the flow can be assumed isothermal. Hence the mass flow rate is a function of $\frac{4fL}{D}$ because $M_1$ changes. Looking at the table or Figure (9.2) or the results from Potto-GDC attached to this book shows that reduction of the mass flow is very rapid.

Figure: M1 as a function of 4fL/D comparison with Isothermal Flow

As it can be seen for the Figure (9.21) the dominating parameter is $\frac{4fL}{D}$ . The results are very similar for isothermal flow. The only difference is in small dimensionless friction, $\frac{4fL}{D}$ .