Mass Flow Rate (Number)

One of the important engineering parameters is the mass flow rate which for ideal gas is

$$\dot{m} = \rho U A = {P \over RT} U A$$ (4.44)

This parameter is studied here, to examine the maximum flow rate and to see what is the effect of the compressibility on the flow rate. The area ratio as a function of the Mach number needed to be established, specifically and explicitly the relationship for the chocked flow. The area ratio is defined as the ratio of the cross section at any point to the throat area (the narrow area). It is convenient to rearrange the equation (4.44) to be expressed in terms of the stagnation properties as

$${\dot{m} \over A} = { P \over P_{0} } \; {P_{0} U \over \sqrt{kRT... ...rt{k \over R } \overbrace{{ P \over P_{0} } \sqrt{ {T_{0} \over T}} } ^{f(M,k)}$$ (4.45)

Expressing the temperature in terms of Mach number in equation (4.45) results in

$${\dot{m} \over A} = \left(k M P_0 \over \sqrt{kR T_{0}}\right) \left( 1 + {k -1 \over 2} M^{2} \right)^ {-{ k+ 1 \over 2(k -1)}}$$ (4.46)

It can be noted that equation (4.46) holds everywhere in the converging-diverging duct and this statement also true for the throat. The throat area can be denoted as by $A^{*}$ . It can be noticed that at the throat when the flow is chocked or in other words M=1 and that the stagnation conditions (i.e. temperature, pressure) do not change. Hence equation (4.46) obtained the form

$${\dot{m} \over A^{*}} = \left( {\sqrt{k} P_0 \over \sqrt{R T_{0}}} \right) \left( 1 + {k -1 \over 2} \right)^ {-{ k+ 1 \over 2(k -1)}}$$ (4.47)

Since the mass flow rate is constant in the duct, dividing equations (4.47) by equation (4.46) yields

$${A \over A^{*}} = { 1 \over M} \left( { 1 + {k -1 \over 2} M^{2} \over {k +1\over 2}} \right) ^ {k+ 1 \over 2 (k -1 )}$$ (4.48)

Equation (4.48) relates the Mach number at any point to the cross section area ratio.

The maximum flow rate can be expressed either by taking the derivative of equation (4.47) in with respect to M and equating to zero. Carrying this calculation results at M=1 .

$$\left(\dot{m} \over A^{*}\right)_{max} { P_0 \over \sqrt{T_0}} = \sqrt{k \over R} \left( {k +1 \over 2} \right)^ {-{ k+ 1 \over 2(k -1)}}$$ (4.49)

For specific heat ratio, $k=1.4$

$$\left(\dot{m} \over A^{*}\right)_{max} {P_0 \over \sqrt{T_0}} \sim {0.68473 \over \sqrt{R}}$$ (4.50)

The maximum flow rate for air ( $R=287 j/kg K$ ) becomes,

$${ \dot{m} \sqrt{T_0}\over A^{*} P_0 } = 0.040418$$ (4.51)

Equation (4.51) is known as Fliegner's Formula on the name of one of the first engineers who observed experimentally the choking phenomenon. It can be noticed that Fliengner's equation can lead to definition of the Fliengner's Number.

$${ \dot{m} \sqrt{T_0}\over A^{*} P_0 } = {\dot{m} \overbrace{\sqrt... ...} P_0} = \overbrace{\dot{m} c_0 \over \sqrt{R}A^{*} P_0 }^{Fn} {1\over\sqrt{k}}$$ (4.52)

The definition of Fliengner's number (Fn) is

$${Fn} \equiv {\dot{m} c_0 \over \sqrt{R} A^{*} P_0 }$$ (4.53)

Utilizing Fliengner's number definition and substituting it into equation (4.47) results in

$$Fn = kM\left( 1+ {k -1 \over 2}M^2 \right)^ {-{ k+ 1 \over 2(k -1)}}$$ (4.54)

and the maximum point for $Fn$ at M=1 is

$$Fn = k\left( {k +1 \over 2} \right)^ {-{ k+ 1 \over 2(k -1)}}$$ (4.55)