Unknown flow rate orifice

Three classes of problems involving orifices (or other obstruction meters) that the engineer might encounter are similar to the types of problems encountered in pipe flows. These are the unknown pressure drop, unknown flow rate, and unknown orifice diameter problems. Each... 

In the case of an unknown flow rate, the pressure drop across a given orifice is measured for a fluid with known properties, and the flow rate is to be determined. 

That the cross-section of a jet of liquid from a non-circular orifice vibrates between the form of the orifice and a circle was first observed by Bidone.i This produces a series of waves. The explanation of the phenomenon as due to surface tension was given by BufF, the mathematical theory and experimental method being developed by Lord Rayleigh. Piccard and Meyer used the method for comparative measurements, refinements being introduced by Pedersen and Bohr. Rayleigh showed that for an ideal jet of radius r at its circular section, the time of oscillation is r=Ki(Qr layi where q is the density of the liquid. For an actual liquid r depends on the flow-rate and corrections are necessary. The period r is related to the directing force F and moment of inertia I by the equation t=7t(IIF) f. Since I is proportional to the mass or density and depends in an unknown way on the form of the orifice, and F is proportional to the surface tension a, it follows that r=7iA(Qla) where is a constant. Since r=l jn and where A=wave-... 

In the case of an unknown pressure drop we want to determine the pressure drop to be expected when a given fluid flows at a given rate through a given orifice. 

The Borda mouthpiece is of interest because it is one device for which the contraction coefficient can be very simply calculated. For all other orifices and tubes, there is a reduction of the pressure on the wall adjacent to the opening, but the exact pressure values are unknown. However, for the reentrant tube, the fluid is at rest on the wall around the tube hence, the pressure must be exactly that due to the depth below the surface. The only unbalanced pressure is that on an equal area opposite to the tube, and its value is whAo. The time rate of change of momentum due to the flow out of the tube is (W/g)V = wA V2lg, where A is the area of the jet. Equating force to time rate of change of momentum, whA0 - wA V2/g. Ideally, V2 = 2gh, and thus ideally Cc = A Ao = 0.5. If it is assumed that Cv = 0.98, then the actual values will be Cc = 0.52 and Cd = 0.51. 

We see that the only molecular constant involved is the molecular weight M of the gas, so that a measurement of the rate of flow of a gas through an orifice can be used to calculate M il S, the area of the orifice, is known. This formula was first verified by Knudsen and has been applied since to the measurement of molecular weights of unknown gases. 

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