It is convenient to take the sonic state (NMa = 1) as the reference state for application of these equations. Thus, if the upstream Mach number is NMa, the length of pipe through which this gas must flow to reach the speed of sound (NMa = 1) will be L*. This can be found by integrating Eq. (9-57)

from (L - 0, NMa) to (L - L*, NMa - 1). The result is

where f is the average friction factor over the pipe length L*. Because the mass velocity is constant along the pipe, the Reynolds number (and hencef) will vary only as a result of variation in the viscosity, which is usually small. If AL - L - L* — L* is the pipe length over which the Mach number changes from NMa1 to NMa2, then

4fAL /4fL*

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