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increases the slope of the theoretical N(M) curve at low M values. At operating M values, however, this effect is negligible.

Finding b as a Function of Nop or Mop Figure 4 presents Nop(b) and Mop(b) for the recommended K friction coefficients of .05, 0, and .20. The Figure 4 curves are cross-plots of Mop values (=§ Mmep) and associated Nop values. (To find Mmep value at b, Eqs. (17)—(18) spreadsheets of h(M) and N(M) were prepared for b values covering the range .05—.6; these numerical values are provided in Table 3.) Graphical representation of the LJL pump characteristics as shown in Figure 4 is useful in estimating b values when the jet pump application fixes N or M, and hence b. Example 3 illustrates the utility of Figure 4.

Straight-Line N(M) Approximation Early investigators (see Reference 2) noted a nearly linear behavior of N versus M. A "straight-line" approximation for N(M) based on the axis intercepts No and Mo would, of course, be useful. A true parabolic efficiency curve would result so Mmep = 1 Mo, and Mo is easily found. Unfortunately, the degree of this linearity varies with area ratio b. At about b = .12, the N curve is in fact a straight line. But as b is increased, the N(M) curve is increasingly concave down so the h = MN "parabolic" curve is increasingly skewed to the left. The converse is true at very small b values. Accordingly, the straight-line approximation is not recommended for use in designing LJL jet pumps. Instead, the Eqs. (17) and (18) theoretical model should be solved and charted for each b-value pump under consideration. Again, the recommended operating flow ratio Mop = § Mmep.

Longitudinal Dimensions of the LJL Jet Pump One-dimensional theory includes dimensions perpendicular to flow, but provides no guidance for longitudinal shapes/profiles. Two hardware dimensions of key importance must be taken from the literature or determined experimentally: nozzle-to-throat spacing "sp" and mixing throat length "L." Both are expressed in terms of throat diameters; sp/Dth and L/Dh Recommended values are given as follows.

nozzle-throat spacing Many experimental searches for optimum sp have been reported. Sanger and Vogel both found (see References 10 and 12) that maximum efficiency (slightly above 40% for Reference 10) is obtained with sp/Dth = 0, using nozzles with an external concave tapering, leading to a thin lip at the outlet. This configuration matches the theoretical model: The jet discharges to pressure Po at the throat inlet, and thus the normal jet loss was eliminated. But a zero spacing promotes cavitation, even with a thin-lipped nozzle tip, and certainly with the rounded-nose exterior profile used in Reference 4. Sanger found that retracting nozzles a distance of about one diameter provided good cavitation resistance and at only a small loss in efficiency. Other investigators have found that performance is insensitive in the range of sp/D^ = 0.5-2 diameters, and that small spacings do promote cavitation. (see Reference 1). It is recommended that LJL jet pumps be designed with sp/Djjj = 1.

mixing-throat length The parallel-walled throat should be long enough to allow complete mixing, but throat lengths should be as short as possible to minimize frictional losses. L/Dth values as well as length (a venturi shape) ranging 1.0 to 10 have been reported for LJL pumps. Several factors affect optimum throat length:

a. When sp/D^ is finite—true for most jet pumps—mixing takes place in the distance (sp plus part or all of L, the mixing-throat length); thus, sp and L are interrelated. Sanger (see Reference 10) found that optimum sp/D^ increased from 0 to 2.3, for pumps with L/Djjj = 7.5 and 3.5 respectively.

b. The primary-flow nozzle affects required L: Long tapered nozzles promote boundary-layer build-up producing jets that delay mixing, increasing required L. Multi-hole nozzles and swirl-inducing nozzles promote mixing and reduce required length L, but pump efficiency suffers because of increased nozzle-flow losses. (See "Primary-Flow Nozzle Design.")

c. Pump area-ratio b can affect optimum throat length. Small-b pumps operate with high flow ratios and throat lengths of L/Dth = 8 were required. For pumps with larger b values, throat lengths of four diameters sufficed (see Reference 13).

d. The gas compressor jet pump (LJG) requires longer throats, as high as 10-30 throat diameters (see Reference 7).

L/Dth = 6 is recommended for general LJL design use. Efficiency of the proposed pump may subsequently be improved by optimizing L (experimentally) for the given pump and duty.

primary-flow nozzle design A short-entry internally-convex profile similar to the ASME metering flow nozzle, is recommended for the LJL pumps. Avoid long conical nozzles.

Liquid-jet flow from a sharp-edged orifice mixes readily and is recommended for the LJG gas compressor (see Reference 7). The annular-nozzle liquid-jet pump has been investigated. In this configuration the secondary flow is axial, surrounded by the primary flow at the throat entrance. This arrangement is advantageous in pumping sticky secondary fluids because it prevents wall contact of the sticky fluid at the throat entry (see Reference 1). An obvious disadvantage is the increased nozzle frictional loss caused by flow of the primary fluid over the comparatively large surface area of the annular nozzle.

throat-inlet contour Many jet pumps reported in the early literature had long (small-convergence-angle) conical sections connecting the suction chamber to the throat, and usually including a sudden wall angle change at the throat entry. Later developments led to the short (large-convergence-angle) entry, well rounded at the throat. The long (small-angle) conical entry is wrong because its proximity to the nozzle exterior throttles the secondary flow and because it promotes cavitation. Secondly, a long approach section increases wall friction (reflected in a high Ken if it is measured). A short entry to the throat and a well-rounded profile connecting the suction chamber and throat is recommended.

Laboratory Flow Tests Performance testing of LJL pumps requires a facility with appropriate instrumentation, pumps, flow meters, and control valves. The flow rates Q1 and Q2, and at least three static pressures, Pi, Ps, and Pd, must be recorded at each test point. In addition, it is recommended that the throat section(s) contain static pressure taps for measurement of Po and Pt. Jet pump test data will then permit measurement of Ken, Kth and Kdi. For on-design operation, these Ks vary little, if at all, with change in flow ratio M. Departures of K(M) from nominal levels serve to reveal otherwise hidden problems. one example is that Kth will change if an increase in flow ratio causes mixing to extend/persist from the throat section into the diffuser (inadequate throat length). Another example is that a sudden rise in Kdi may indicate diffuser-wall separation.

LIQUID-JET LIQUID (LJL) PUMP DESIGN EXAMPLES_

Equations (1), (3), (5) and (7) will model jet pumps for LJL, LJG and LJGL configurations. Equations (1), (11—19) apply specifically to the LJL jet pump. The three numerical examples provided here are for this widely used pump only. LJG and LJGL liquid-jet pumps are less common: please see References 5, 6 and 7.

Example 1 Design a jet pump to handle 50 gpm (11.36 m3/h) of water at 80°F (26.7°C) from a suction at 14.7 psia atmospheric pressure (101.325 kPa) to discharge at 40 psi (275.8kPa). Determine the required primary flow rate, jet nozzle pressure and dimensions of the jet pump.

Solution: For best efficiency, select b = .25. From Table 2, adopt Kn = 0.5, Ktd = .20, and Ken = 0. Eq. (17) then produces N(M). A computer spreadsheet table for Eqs. (17) and (18) showing output values based on increments of M is recommended, for example, see Table 1. As shown by the bottom line in Table 1, the Mmep = 1.104 (found by successive approximations using the spreadsheet program). The operating flow ratio is Mop = §Mmep = .676. The spreadsheet program at this Mop produces the Nop (.428) and h(28.934%) values shown in Table 1, second line from the bottom. From Eq. 19 (Pd - Ps)/Pi - Ps) = N/(N + 1), so and Pi - Ps = (1.428/.4828)40 = 1.3346 psi (920.212 kPa). Thus Pi = 1.3346 + 14.7 = 148.16 psia (1021.56kPa). Liquid jet pumps—with very few exceptions—operate with the nozzle tip withdrawn from the throat entry by one throat-diameter or more. The nozzle tip experiences a discharge pressure close to Ps, not Po, and "jet loss" thus occurs. For this jet loss condition, Eq. (1) changes to (Pi - Ps) = Z(1 + Kn). The jet velocity head Z = 1.3346/1.05 = 1.2710 psi (876.35kPa).

CAVITATION-LIMITED flow ratio ml The flow ratio ML is now evaluated using Eq. (20) to answer the question: will Mop = .676 avoid pump cavitation? The area ratio c = (1 - .25)/

.25 = 3. For the secondary fluid (water at 80°F, 26.7°C),Pn = .506 psia (3.49 kPa).The conservative value of s = 1.35 is used in Eq. (20) as follows:

Because Mop (.676) < ML (.863), this pump will not cavitate at the specified operating condition.

JET PUMP dimensions Q1 = Q2/Mop = 50/.676 - 73.96 gpm (16.8 m3/h) = 73.96/(7.481 X 60) = .165 ft3/s. Longitudinal dimensions: sp/Dth = 1.0, thus sp = .937 in (23.81 mm). For L/Dth = 6, L = 6 X .937 = 5.62 in (142.75 mm). A diffuser with an included angle of 5° (conservative) and area ratio a = .224 would have length of approximately 12 in (304.8 mm). A shorter diffuser may be desirable, but kinetic-energy leaving losses will be higher.

example 1 using other area ratios An infinite number of different jet pumps can be designed to handle the Example 1 duty of pumping 50 gpm at 40 psi discharge pressure. The previous numerical example (b = .25) was repeated using b = .1, .4, and .6. Table 4 compares the results of these four b-ratio pumps. In each case the design is based on Mop = §Mmep; the assumed K values and calculation procedure are similar for the four pumps.

The expression CR% in Table 4 indicates the (ML — Mop)/Mop % separation of operating flow ratio Mop and the limiting flow ratio ML; the larger this number, the better. For this constant-duty example (Q2 = 50 gpm and Pd = 40 psi), Table 4 shows that cavitation resistance can be improved by using a larger b ratio, i.e., the (ML — Mop)/Mop % figure increases with b. Note that at the smallest b value (.1), the cavitation value (sixth column) is CR = — 5.3%: this indicates that Mop > ML, and this pump would encounter cavitation: b = .1 should not be used at these Mop and s values. A redesign of the pump suction chamber (nozzle-external profile and throat-inlet profile) leading to an improvement in cavitation coefficient s could render the b = .1 pump usable (the value s = 1.35 used previously is conservative). For example, an improvement to s = 1.0 would raise ML to 2.0. With this change, a b = .1 pump could be used, cavitation-free.

Table 4 shows two other important facts: (1) The b = .25 pump provides the highest efficiency, even though all four are designed to operate at §Mmep, i.e., each at a similar position on the efficiency curve specific to that pump. (2) Small b-value pumps operate at large nozzle pressure-drops ((P; — Ps)) and with small Q1 primary-values. Conversely, high b-value pumps operate with small nozzle pressure-drops, but use a large flow rate Q1. The pumps have in common similar expenditures of energy (Q1 X pressure change, and allowing for efficiency differences) to handle the duty, which is the same in all four cases. In some design problems the nozzle-pressure—or possibly the primary-fluid flow volume— might outweigh the importance of mechanical efficiency. Table 4 shows how an adjustment in b might be used to achieve improved cavitation, albeit with efficiency sacrifices.

reducing THE flow ratio MOP TO cope with cavitation For a given b-value LJL jet pump, a reduction in Mop offers a way to improve cavitation resistance. The b = .1 pump used

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