Adiabatic flow of an ideal gas in a horizontal pipe

Equation 6.19 is the basic equation relating the pressure drop to the flow rate. The difficulty that arises in the case of adiabatic flow is that the equation of state is unknown. The relationship, PVy = constant, is valid for a reversible adiabatic change but flow with friction is irreversible. Thus a difficulty arises in determining the integral in equation 6.19 an alternative method of finding an expression for dPIV is sought. 

For adiabatic flow in a horizontal pipe with no shaft work, equation 6.7 reduces to 

The differential form of equation 6.50 with u expressed in terms of G and V is 

The enthalpy change can be found from the following two fundamental thermodynamic relationships which, in the case of ideal gases, are valid for irreversible processes as well as reversible ones  

Integrating equation 6.59 gives the desired relationship between P and V  

Calculations for adiabatic flow require the use of equations 6.63 and 6.64. For example, if the upstream conditions P and V are known, and G and di are specified, C can be calculated from equation 6.64 then V2 from equation 6.63. Substituting this value of V2 in equation 6.64 gives P2. If the logarithmic term is not negligible, an iterative calculation will be needed to determine V2 from equation 6.63. 

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An ideal gas flows in steady state adiabatic flow along a horizontal pipe of inside diameter d, = 0.02 m. The pressure and density at a point are P = 20000 Pa and p = 200 kg/m3 respectively. The density drops from 200 kg/m3 to 100 kg/m3 in a 5 m length. Calculate the mass flux assuming that the Fanning friction factor /= 9.0 x 10 3 and the ratio of heat capacities at constant pressure and constant volume y = 1.40. 

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