Maximum flow rate in a pipe of constant cross-sectional

7 Maximum flow rate in a pipe of constant cross-sectional area 

Consider the case of steady polytropic flow in a horizontal pipe described by equation 6.49  

If the upstream pressure Pt is kept constant and the downstream pressure P2 is gradually reduced, the flow rate will gradually increase. However, as P2— 0 the density p— 0 and consequently the mass flow rate must approach zero if the gas speed u remains finite. Thus, at some value of P2 satisfying the condition 0 P2 Pi, the mass flow rate must reach a maximum. 

Putting 3G/dP2 = 0, the conditions giving the maximum flow rate Gw satisfy the equation 

Recalling equation 6.69, it will be seen that the maximum flow rate is 

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