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Chapter: Mechanical - Gas Dynamics and Jet Propulsion - Flow Through Ducts

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Adiabatic Flow of a Compressible Fluid Through a Conduit

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

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.

 

a)   Velocity



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.

 

Temperature

 

Stagnation temperature –Mach number relation

 

At critical state

 

M=1

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


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