

Problem 5.08: Jet Pump 

The device connected between compartments A and B is a simplified version of a jet pump. A jet (or ejector) pump is a device which uses a small, very highspeed jet with relatively low volume flow rate to move fluid at much larger volume flow rates against a pressure differential D p, as shown in the figure. The pump in the figure consists of a contoured inlet section leading to a pipe segment of constant area A_{2}. A small jet draws fluid from compartment A and ejects it at high velocity V_{j} and area A_{j} at the entrance plane (1) of the constantarea pipe segment. Between (1) and (2), the jet (the "primary" stream) and the secondary fluid flow which is drawn in from compartment A via the contoured inlet section mix in a viscous, turbulent fashion and eventually, at station (2), emerge as an essentially uniformvelocity stream. The pump operates in steady state. To simplify the analysis, we make several physical assumptions that are not unreasonable. We assume
We also make two assumption about operating conditions that are also reasonable and considerably simplify the mathematics involved in the analysis: (a) Derive an expression for Dp as a function of the total volume flow rate Q from compartment A to compartment B. The given quantities are A_{1}, A_{2}, r and V_{j}. Indicate the volume flow rate Q_{o} when Dp = 0 (the "shortcircuit" volume flow rate) and the pressure Dp_{0} at which Q = 0. Write the pressurevolume flow rate relationship in universal dimensionless form as Dp/Dp_{0} vs Q/Q_{0} and sketch it for positive values of pressure This is the “pump curve” in dimensionless form. Show that for A_{j} << A_{2}, Q_{0} >> V_{j}A_{j}. 
(b) Sketch the pressure distributions along the line ab for the cases Dp = 0 and Dp > 0.  
(c) Is your formulation in (a) valid when Q=0, i.e. when the total volume flow rate from A to B is zero? Explain. What is the minimum value Q_{min} of Q for which your formulation in (a) is valid?  