For a subsonic flow of air through a restriction the mass ra

For a subsonic flow of air through a restriction, the mass rate of flow is m = 1.05/root T A sqrt (p_1 - P_2)P_2 Where T = inlet temperature area of the restriction, in^2 P_1 = inlet pressure psi P_2 outlet pressure psi Determine a linear approximation for m when the inlet temperature is constant

Solution

The conservation of mass (continuity) tells us that the mass flow rate mdot through a tube is a constant and equal to the product of the density r, velocity V, and flow area A:

Eq #1

mdot = r * V * A

definition of the Mach number M, and the speed of sound a:

Eq #2:

V = M * a = M * sqrt (gam * R * T)

where gam is the specific heat ratio, R is the gas constant, and T is the temperature. Now substitute Eq #2 into Eq # 1:

Eq #3:

mdot = r * A * M * sqrt (gam * R * T)

The equation of state is:

Eq #4:

r = p / (R * T)

where p is the pressure. Substitute Eq #4 into Eq # 3:

Eq #5:

mdot = A * M * sqrt (gam * R * T) * p / (R * T)

Collect terms:

Eq #6:

mdot = A * sqrt (gam / R) * M * p / sqrt(T)

From the isentropic flow equations:

Eq #7:

p = pt * (T / Tt)^(gam/(gam-1))

where pt is the total pressure and Tt is the total temperature. Substitute Eq #7 into Eq # 6:

Eq #8:

mdot = (A * pt) / sqrt(Tt) * sqrt (gam / R) * M * (T / Tt)^((gam + 1) / (2 * (gam -1 )))

Another isentropic relation gives:

Eq #9:

T/Tt = (1 + .5 * (gam -1) * M^2) ^-1

Substitute Eq #9 into Eq # 8:

Eq #10:

mdot = (A * pt/sqrt[Tt]) * sqrt(gam/R) * M * [1 + .5 * (gam-1) * M^2 ]^-[(gam + 1)/(gam - 1)/2]