A calorically perfect gas flows adiabatically (no heat exchange) through a straight…

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Viscous flow and dynamical similarity · Waves, acoustics and compressible flow

Signal Speeds in Fluids, Ch Flow through nozzles: area change, choking and thrust

Problem

A calorically perfect gas flows adiabatically (no heat exchange) through a straight duct of constant cross-sectional area and hydraulic diameter DD, with wall friction described by the Fanning friction factor fFf_{\rm F}. The governing differential relations are continuity at constant area, ρu=const\rho u = \text{const}; the momentum equation with a wall-shear term, ρu du+dp+4fFDρu22 dx=0;\rho u\,\mathrm{d}u + \mathrm{d}p + \frac{4f_{\rm F}}{D}\frac{\rho u^2}{2}\,\mathrm{d}x = 0; the adiabatic energy equation cp dT+u du=0c_p\,\mathrm{d}T + u\,\mathrm{d}u = 0; and the ideal-gas relations p=ρRTp = \rho R T, M2=u2/(γRT)M^2 = u^2/(\gamma R T). ((a)) Combine these to show that du/u\mathrm{d}u/u is proportional to M2/(1−M2)M^2/(1-M^2) times 4fF dx/D4f_{\rm F}\,\mathrm{d}x/D, and hence that dM2/M2\mathrm{d}M^2/M^2 carries the same sign as 1−M21-M^2. ((b)) State in words what this implies for a subsonic flow and for a supersonic flow entering a length of pipe with friction, and say which everyday intuition it corrects.

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