Adiabatic Flow of a Compressible Fluid Through a Conduit
Flow through pipes in a typical plant where line lengths are short, or the pipe is well insulated can be considered adiabatic. A typical situation is a pipe into which gas enters at a given pressure and temperature and flows at a rate determined by the length and diameter of the pipe and downstream pressure. As the line gets longer friction losses increase and the following occurs:
Ø Pressure decreases
Ø Density decreases
Ø Velocity increases
Ø Enthalpy decreases
Ø Entropy increases
The question is “will the velocity continue to increasing until it crosses the sonic barrier?” The answer is NO. The maximum velocity always occurs at the end of the pipe and continues to increase as the pressure drops until reaching Mach 1. The velocity cannot cross the sonic barrier in adiabatic flow through a conduit of constant cross section. If an effort is made to decrease downstream pressure further, the velocity, pressure, temperature and density remain constant at the end of the pipe corresponding to Mach 1 conditions. The excess pressure drop is dissipated by shock waves at the pipe exit due to sudden expansion. If the line length is increased to drop the pressure further the mass flux decreases, so that Mach 1 is maintained at the end of the pipe.
The effect of friction in supersonic flow of the following parameters
a) velocity b) pressure c) temperature
Flow properties at M = M* = 1 are used as reference values for non- dimensionalizing various properties at any section of the duct.
Variation of flow properties
The flow properties (P,T,ρ,C)at M=M*=1are used as reference values for non-dimensionalizing various properties at any section of the duct.
Stagnation temperature –Mach number relation
At critical state
T0 = T0*
Variation of Mach number with duct length
The duct length required for the flow to pass from a given initial mach number M1 to a given final mach number m2can be obtained from the following expression.
Mean friction coefficient with respect to duct length is given by
The distance (L) between two section of duct where the ach numbers M1 &M2 are given by