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Table 71 It is recommended by Powell that a ratio of 04 be used although higher values of Kgo give the designer a wider choice [3] The minimum value of Kgo is limited by the consideration of choking in the feed holes which is defined by
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Pd 2 1 = Po +1
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Choking of the feed holes causes an instability that is also known as pneumatic hammer For air, the value is given as
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Pdo Po = 0528
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Table 71 The optimum value of Kgo for different eccentricity ratios [3, 31]
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01 05 09 Optimum value of Kgo 060 040 035
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This value is applicable for diatomic gases with a ratio of specific heats of 14 For other classification of gases, the critical ratio may be different The gauge pressure ratio at which the feed holes become choked can be obtained for air from the ensuing equation and from Figure 738 For atmospheric exhaust conditions, air bearings have choked feed holes at Kgo = 04 at a supply pressure in excess of 55 lbf/in2 gauge Thus, a higher supply pressure at a higher value of Kgo is often recommended Kgo =
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0528 - Pa / Po 1 - Pa / Po
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If the consideration of choked feed holes permits, for an optimum radial load capacity, a gauge pressure ratio of 04 should be used However, if small clearances are employed for limiting gas consumption and for achieving a high stiffness, the difficulty in producing small feed holes and avoiding hole blockage become the limiting factors In the analysis of flow through feed holes, it is assumed that there is no pressure loss upstream of the throat, and the pressure immediately downstream of the jet is the static pressure at the throat of the jet For a nozzle of the same throat diameter as that of the jet, the following equation expresses the relationship between the supply pressure and the static pressure at the throat
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Gas Lubricated Bearings
Fig 738: The journal bearing design choice of the gauge pressure ratio [3]
1 2 1 Pd Po = 1 2 a o
where Pd is the static pressure at the throat, v the velocity at the throat, ao the speed of sound at the supply pressure conditions and g is the ratio of specific heats for the gas The mass flow through the jet is given as m = CD rd A where CD is the coefficient of discharge, rd the density at the throat and A is the cross-section area of the throat It is always assumed in theory that in each slot around an eccentric journal bearing, the gas flows axially from the plane of the feed hole to the end of the bearing The pressure distribution in the bearing and corresponding load capacity shown in Figure 739 takes into consideration the changes
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Pressure from each hole is p felt over a chord of width D sin N Pd5 F r5 p N W F r2 Pd2 Fr2 Pd1 Fr1 Pd5 Pa Pd3 Fr3 Pd1
Pdm5 = 2 (Pd 5 P ) + Pa a 3
Pd4 F r4 P d3 Fr3 Pd2
Pd4 Fr4
Position of rows of feed holes
Pa Pdm1 = 2 ( Pd1 Pa) + Pa 3
Fig 739: The load capacity of an aerostatic journal bearing the axial flow model [3]
in the clearance value due to different eccentricities The value calculated will be more than in practice due to the effects of dispersion and non-axial flow that distort the pressure distribution as seen in Figure 736 The effect of dispersion is clear in short bearings and bearings with fewer numbers of jets per row As the flow diverges from around the feed jet, the theoretical load is reduced by a factor known as the load dispersion coefficient, Cw, given by [32]
dn Cw = 089 D
Ln D
Pd Pa
0379 0758 D D 00505 nd nL
The effect of dispersion is clear in short bearings and in bearings with fewer numbers of jets per row In short bearings, the lubricant does not have sufficient time to fill the whole bearing before being exhausted This drawback does not exist in circumferential slot feeding Initially, it is assumed that there is no non-axial flow However, in practice, the circumferential pressure distribution causes the gas to flow around the bearing [3] This reduces both the pressure difference across the shaft and the load capacity The analysis is usually done using a computer program However, Shires has deduced a semi-empirical correction factor for non-axial flow in which the constant was derived from a series of experimental results by Robinson [33] The correction factor also makes some allowance for dispersion effect
cosh(636 l / D ) 1 L 2l sinh(636 l / D ) + tanh 636 D CL = 0315 L l C Lo D
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