Flow of gases at piping systems are often metered by chokedf

Flow of gases at piping systems are often metered by choked-flow nozzles. The mass flow rate of gas is dependent on the nozzle area, pressure p, the temperature T upstream of the meter, and the gas constant R and the specific heat C_p of the gas. Determine the independent a parameters that can be formed. Use F, L, t, T as primary dimensions. Use A, p, T, R as repeating parameters.

Solution

Mass flow rate m = f(A, p, T, R, Cp)

Dimensions of m = [MT^-1]

Dimensions of area A = [L²]

Dimensions of pressure p = [ML^-1T^-2]

Dimensions of Temperature = [theta]

Dimensions of R and Cp = [L^-2T^-2theta^-1]

There are 6 variables and 4 dimensions. Hence, we can have 6-4 = 2 dimensionless variables. Taking A, p, T, R as repeating variables we have,

Pi1 = [L²]^a1 [ML^-1T^-2]^b1 [theta]^c1 [L^-2T^-2theta^-1]^d1 [MT^-1]

Pi2 = [L²]^a2 [ML^-1T^-2]^b2 [theta]^c2 [L^-2T^-2theta^-1]^d2 [L^-2T^-2theta^-1]

The expression for Pi1 gives

b1 + 1 = 0

2*a1 - b1 - 2*d1 = 0

-2*b1 - 2*d1 - 1 = 0

c1 - d1 = 0

Solving, a = 0, b = -1, c = 1/2, d = 1/2

or Pi1 = A⁰ * P^-1 * T^1/2 R^1/2 m

or Pi1 = m* sqrt (RT) / P

The expression for Pi2 gives

b2 = 0

2*a2 - b2 - 2*d2 - 2 = 0

-2*b2 - 2*d2 - 2 = 0

c2 - d2 - 1 = 0

Solving, a2 = 0, b2 = 0, c2 = 0, d2 = -1

or Pi2 = A⁰ * P⁰ * T⁰ R^-1 Cp

or Pi2 = Cp / R