Non-isothermal flow of an ideal gas in a horizontal pipe

In general, where an ideal gas expands or is compressed, the relation between the pressure P and the specific volume v can be represented approximately by  

For a given upstream pressure P the maximum flow rate occurs when U2 — s/kP, the velocity of transmission of a pressure wave under these conditions (equation 4.38). 

Flow under adiabatic conditions is considered in detail in the next section, addiough an approximate result may be obtained by putting k equal to y in equation 4.66 this is only approximate because equating k to y implies revei ibility. 

The conditions existing during the adiabatic flow in a pipe may be calculated using tiie approximate expression Pi/ = a constant to give the relation between the pressure and the specific volume of the fluid. In general, however, the value of the index may not be known for an irreversible adiabatic process. An alternative approach to the problem is therefore desirable.  

For a fluid flowing under turbulent conditions in a pipe, = 0 and  

For steady flow of an ideal gas between points 1 and 2, distance L apart, in a pipe of constant cross-sectional area in which no shaft work is done, the energy equation is given by equation 6.19. For the general case of polytropic flow, from equation 6.27, the equation of state can be written as 

Equation 6.49 is the general equation for polytropic flow of an ideal gas in a horizontal pipe with no shaft work. On putting k = 1, equation 6.38 for isothermal flow is obtained. 

Putting k = y gives an approximate equation for adiabatic flow. The result is only approximate because it implies an isentropic change, ie a reversible adiabatic change, but this is not the case owing to friction. A rigorous solution for adiabatic flow is given in Section 6.5. 

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