Question

Consider a Kantrowitz-Donaldson diffuser designed for a flight Mach number of 1.8. The entrance area equals 3.0 ft2and the ambient airtemperature and pressure are 500R and 0.8atmosphere. The flow is isentropic everywhere except across the normal shockwave. Determine:(a) the minimum cross-sectional area of the throat such that a normal shock may be stabilized at the entrance, (b) the maximum mass flow, and(c) the maximum stagnation pressure possible at the end of the diffuser (with subsonic flow only in the divergent portion).In each of these optimizations, consider the flight Mach number fixed at the design value while the final pressure (at the end of the diffuser) is allowed to adjust.

Answer 1

Solution:

Given:

Mach Number 1.8

Entrance area: A_i - 3.0 ft²

Ambient Air Temperature T₀ - 500 R

Ambient Air Pressure: p₀ - 0.8 ata

a) Minimum corss-sectional area of the throat

From the Flight Mach Number Chart For M = 1.8 A_i/A_t = 1.1

i.e 3.0/A_t = 1.1

Therefore A_t = 3.0/1.1 = 2.72 ft²

b) maximum mass flow

m_t = p₀ [A_t/ x sqrt() x (2/+1)^(+1/2x-1)]

where R = 0.00461915 g mol Rankine

= 1.4

    = 0.8[2.72/sqrt(0.004461915 x 500) x sqrt(1.4) x (2/1.4+1)^((1.4+1)/2x1.4-1))]

= 1.410 g/sec

c) maximum stagnatio pressure possible at the end of diffuser

p₁/p₀ = 1+ ((2 x )/(+1)) x (M² - 1)

p1/0.8 = 1 + ((2 x 1.4)/(1.4+1)) x (1.8² - 1)

Therefore p1 = 2.09 ata