# 2.25 Problem Sets - Section 5

## Problem 5.8: Jet Pump

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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 high-speed 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*and area

_{j}*A*at the entrance plane (1) of the constant-area 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 uniform-velocity stream. The pump operates in steady state.

_{j}To simplify the analysis, we make several physical assumptions that are not unreasonable. We assume

- that the flow is incompressible
- that the flow from compartment A to station (1) is inviscid,
- that, although viscous forces cause the turbulent mixing process
between (1) and (2), the shear force exerted on the walls between
those stations is small compared with
*D**pA*_{2}, - that gravitational effects are negligible, the flow being horizontal.

We also make two assumption about operating conditions that are also reasonable and considerably simplify the mathematics involved in the analysis:

*A*and

_{j}<< A_{2}*V*

_{j}A_{j}<< V_{2}A_{2}(a) Derive an expression for *D**p*
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

*D*

*p*

*= 0*(the "short-circuit" volume flow rate) and the pressure

*D*

*p*at which

_{0}*Q = 0*. Write the pressure-volume flow rate relationship in universal dimensionless form as

*D*

*p*/

*D*

*p*

_{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 a-b for the cases Dp
= 0 and Dp > 0. |
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(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
of _{min}Q for which your formulation in (a) is valid? |
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