For this type of flow, the stagnation temperature is constant, then A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 72 For this type of flow, the stagnation temperature is constant, then Taking logarithmic of this expression and then differentiating gives Substitute for dT⁄T The last term can be manipulated to be: Substitute this expression into last equation and rearrange gives:

 For supersonic the value of fLmax lies between O and M=1 and A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 73 Integration of this equation gives For Fanno flow, the integration limits are This equation relates friction factor f, , to M directly. For air γ=1.4, then;  For supersonic the value of fLmax lies between O and M=1 and O.8215 at M=∞.  For subsonic the value of fLmax becomes very large as M becomes very small. 5.3 Reference state and Fanno Flow Table The equations developed in this chapter are the means of computing the properties at one location in terms of those given at some other location. The key to problem solution is predicting the Mach number at the new location through the use of last equation. The solution of this equation for the unknown M2 presents a messy task, as no explicit relation is possible between M2 and M1. In reference case we imagine that we continue by Fanno flow (i.e., more duct is added) until the velocity reaches M=1. Figure (5.2) shows a physical

A course in Gas Dynamics…………………………………. …. …Lecturer: Dr A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 74 system together with its T –s diagram for a subsonic Fanno flow. We know that if we continue along the Fanno line (remember that we always move to the right), we will eventually reach the limiting point where sonic velocity exists. The dashed lines show assumed elongation duct of sufficient length to enable the flow to traverse the remaining portion of the upper branch and reach the limit point. This is the (*) reference point for Fanno flow. Figure 5.2 The * reference for Fanno flow We now rewrite the working equations in terms of the Fanno flow ∗ reference condition Where T* is the static temperature at M=1, and

Substitute from continuity equation for constant area duct A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 75 Figure 5.3 Fanno line Substitute from continuity equation for constant area duct

A course in Gas Dynamics…………………………………. …. …Lecturer: Dr A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 76 Figure 5.4

Figure 5.6 mass flow reduction. A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 77 Suppose now that the duct is long enough for a flow initially subsonic to reach Mach 1, and an additional length is added, as shown in Figure (5.5). The flow Mach number for the given mass flow cannot go past 1 without decreasing entropy. This is impossible from the second law. Hence the additional length brings about a reduction in mass flow. The flow jumps to another Fanno Line (see Figure 5.6). Essentially, the duct is choked due to friction. Corresponding to a given inlet subsonic Mach number, there is a certain maximum duct length Lmax beyond which a flow reduction occurs. Figure 5.5 Figure 5.6 mass flow reduction.

Table 1 Absolute Roughness of common materials A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 78 Now suppose the inlet flow is supersonic and the duct length is made greater than Lmax to produce Mach =1. With the supersonic flow unable to sense changes in duct length occurring ahead of it, the flow adjusts to the additional length by means of a normal shock rather than a flow reduction. The location of the shock in the duct is determined by the back pressure imposed on the duct. 5.4 Friction factor Dimensional analysis of the fluid flow in fluid mechanics shows that the friction factor can be expressed as = f(Re, cfDe ) . Where cfDe is the relative roughness. The relationship among, Re, and cfDe is determined experimentally and plotted on a chart called a Moody chart or a Moody diagram. Typical values of , the absolute roughness are shown in the following table. Table 1 Absolute Roughness of common materials If the flow rate is known together with the duct size and material, the Reynolds number and relative roughness can easily be calculated and the value of the friction factor is taken from the diagram. The curve in the laminar flow region can be represented by f = 64fRe

A course in Gas Dynamics…………………………………. …. …Lecturer: Dr A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 79 For noncircular cross sections the equivalent diameter can be used. 4A This equivalent diameter may be used in the determination of relative roughness and Reynolds number, and hence the friction factor. However, care must be taken to work with the actual average velocity in all computations. Experience has shown that the use of an equivalent diameter works quite well in the turbulent zone. In the laminar flow region this concept is not sufficient and consideration must also be given to the aspect ratio of the duct. In some problems the flow rate is not known and thus a trial-and-error solution results. As long as the duct size is given, the problem is not too difficult; an excellent approximation to the friction factor can be made by taking the value corresponding to where the ε/D curve begins to level off. This converges rapidly to the final answer, as most engineering problems are well into the turbulent range. Figure 5.7 Moody diagram for friction factor in circular ducts De =

Table 5.1 Fluid Property Variation for Fanno Flow A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 8O 5.5 APPLICATIONS The following steps are recommended to develop good problem-solving technique: 1. Sketch the physical situation (including the hypothetical ∗ reference point). 2. Label sections where conditions are known or desired. 3. List all given information with units. 4. Compute the equivalent diameter, relative roughness, and Reynolds number. 5. Find the friction factor from the Moody diagram. 6. Determine the unknown Mach number. 7. Calculate the additional properties desired. Table 5.1 Fluid Property Variation for Fanno Flow 5.6 CORRELATION WITH SHOCKS Now we can imagine a supersonic Fanno flow leading into a normal shock. If this is followed by additional duct, subsonic Fanno flow would

of the same Fanno line. [See Figure 5.8b]. A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 81 occur. Such a situation is shown in Figure 5.8a. Note that the shock merely causes the flow to jump from the supersonic branch to the subsonic branch of the same Fanno line. [See Figure 5.8b]. Figure 5.8a Combination of Fanno flow and normal shock (physical system). Figure 5.8b Combination of Fanno flow and normal shock. Example A large chamber contains air at a temperature of 300 K and a pressure of 8 bar abs (Figure E5.6). The air enters a converging–diverging nozzle with an area ratio of 2.4. A constant-area duct is attached to the nozzle and a normal shock stands at the exit plane. Receiver pressure is 3 bar abs. Assume the entire system to be adiabatic and neglect friction in the nozzle. Compute the f Δx/D for the duct.

A3 p3 A2 p03 We proceed to compute p5 pS pS p01 p03 p3 3 1 A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 82 Figure E5.6 A3 p3 A2 p03 We proceed to compute p5 pS pS p01 p03 p3 3 1 01 03 3 For = 2.4, M3 = 2.4, and = 0.0684 p* = p = ( ) (1) ( ) (0.3111) = 1.7056 p* 8 0.0684 p* p p