CHAPTER FIVE FANNO FLOW 5.1 Introduction A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 66 CHAPTER FIVE FANNO FLOW 5.1 Introduction We have mentioned that area changes, friction, and heat transfer are the most important factors affecting the properties in a flow system. Up to this Chapter we have considered only one of these factors, that of variations in area. We now wish to take a look at the subject of friction losses. To study only the effects of friction, we analyze flow in a constant- area duct without heat transfer. As before, the table permits rapid solutions to many problems of this type, which are called Fanno flow. 5.2 Working Relations for Fanno Flow Consider one-dimensional steady flow of perfect gas with constant specific heats through constant area duct. In case of adiabatic, no work exchange, the flow is Fanno flow where friction effect is considered. The basic equations of continuity, energy, and momentum under the following assumptions, are derived: The stagnation temperature will be proved to be constant along the duct while the stagnation pressure will suffer from losses due to friction. The entropy is expected to increase. • State equ.

velocity” or “mass flux”. • Energy A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 67 • Continuity The flow area is constant. G is a constant, which is referred to as the “mass velocity” or “mass flux”. • Energy We start with S.F.E.E. For adiabatic and no work, this becomes If we neglect the potential term, this means that

From continuity equation A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 68 From continuity equation • Entropy Substitute for Temperature and pressure ratio, from above:

work flow of a perfect gas, we start from the following thermodynamic A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 69 To derive an expression for stagnation pressure ratio for adiabatic, no- work flow of a perfect gas, we start from the following thermodynamic relation for stagnation (total) properties Since for adiabatic flow and from energy equation, then • Momentum

Friction factor, f, is four times friction coefficient, Cf. A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 70 The external forces that act on the element are the pressure and shear forces as shown in figure (5.1) Figure 5.1 Control volume for isolated, constant area duct with frictional flow τ is shear stress due to wall friction and Asur duct surrounding surface area. The hydraulic diameter; Surface area is Friction factor, f, is four times friction coefficient, Cf.

Substitute for τ and Asur A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 71 Substitute for τ and Asur From state equation and the definition of Mach number Taking logarithmic of this expression and then differentiating gives Substitute for dp⁄p and dV⁄V gives Then