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The Mach number of a bubbly secondary flow at the throat entrance is (see Reference 5):

When this Mach number is 1.0, the LJGL pump has reached limiting flow; that is, a reduction in back pressure Pd no longer causes an increase in the bubbly secondary-flow rate.

Pump Efficiencies The LJGL pump produces two useful work results:

• Static-pressure increase of the liquid component of the secondary flow stream.

• If a gas is entrained in this liquid stream, isothermal compression of the gas component.

With W as the work rate, ft-lb/s, (power) WL = Q2 (Pd — Ps) is the work rate on the liquid component, and WG = pGsQGsRTln(Pd — Ps) is the work rate on the gas component. The energy rate input is Ein = Q1(Pi — Ps). The LJGL pump mechanical efficiency is the total work rate divided by the energy rate in

The Jet Loss Jet pumps in practical applications have nozzle-to-throat spacings sp/Dth of one or more mixing-throat diameters. The power jet traverses from a static pressure at or near Ps down to Po, with no useful work recognized in the one-dimensional theory. Thus a "jet loss" occurs, which is in addition to the frictional and mixing losses (see Reference 9).

In the LJL and LJGL (but not the LJG) pumps, throat-inlet pressure drops—and hence jet losses—are significant (Ref. 6).

Pump Efficiency, Incorporating Jet Loss In Eq. (9), (Pi — Pd) is expanded: (Pi — Pd) = (Pi — Ps) — (Pd — Ps), and (Pi — Ps) = (Pi — Po) — (Ps — Po) = Z(1 + Kn) — j(Ps — Po), where j = 1 for a fully inserted nozzle, no jet loss: and j = 0 for the usual case of retracted nozzle, which produces full jet loss. Eq. 9 now becomes

Eq. (10) is recommended for predicting liquid-jet pump efficiencies as follows: Use j = 0 for pumps with normally-retracted nozzles (full jet loss); use j = 1 for no-jet-loss pumps (thin-walled nozzle tip fully inserted so sp = 0). The pressure in Eq. (10) should be calculated from the one-dimensional theory using Eqs (1), (3), (5), and (7). (See below for the LJL jet pump.)

Computer Programs for LJGL and LJG Models Solutions for the compressible flow cases are generated using computer spreadsheet or Fortran programs. Values for Z, b, Ps, Ts, R, p1, S and the four K coefficients are fixed/assumed for each pump and operating conditions. Eqs. (3), (5), and (7) are then solved for each step increase in flow-ratio M, with fs held constant. Alternatively, M may be held constant and the equations solved for step increase in fs. Eqs. (3), (5), and (7) are interdependent: solution of Eq. (5) requires Po values from Eq. (3) and solution of Eq. (7) requires Pt values from Eq. (5). The program outputs at each flow-ratio step are static pressure Po, PtPd, and the three pump efficiencies defined by Eq. 10.

LJGL FLOW CUT-OFF Compressible-flow choking of the secondary stream at the throat entrance will occur at MNo = 1. The flow ratio at which this will occur can be predicted from critical-flow theory. For further details, see Reference 5.

Performance of the gas compressor (LJG) can be calculated from Eqs. (1), (3), (5), and (7) by setting M = 0. Although simplified, the equations are still coupled as in the LJGL case. The one-dimensional theory predicts actual performance quite well (see References 6 and 7) provided the mixing is completed within the length of the mixing throat. Theory-experiment agreement fails and the gas compressor efficiency declines—mixing is allowed to extend into the diffuser.

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