An ideal gas enters subsonically and flows isentropically

Chapter 11, Problem 11.40

(choose chapter or problem)

An ideal gas enters subsonically and flows isentropically through a choked converging–diverging duct having a circular cross-sectional area A that varies with axial distance from the throat, x, according to the formula

                    \(A=0.1+x^2\)

where A is in square feet and x is in feet. For this flow situation, sketch the side view of the duct and graph the variation of Mach number, static temperature to stagnation temperature ratio, \(T/T_0\), and static pressure to stagnation pressure ratio, \(p/p_0\), through the duct from x = -0.6 ft to x = +0.6 ft. Also show the possible fluid states at x = -0.6 ft, 0 ft, and +0.6 ft using temperature-entropy coordinates. Consider the gas as being helium (use \(0.051 \leq \mathrm{Ma} \leq 5.193\)). Sketch on your pressure variation graph the nonisentropic paths that would occur with over- and underexpanded duct exit flows (see Video V11.7) and explain when they will occur. When will isentropic supersonic duct exit flow occur?