Tribology an Design 95 A new ientification metho of the supply hole ischarge coefficient of gas bearings G. Belforte, F. Colombo, T. Raparelli, A. Trivella & V. Viktorov Department of Mechanics, Politecnico i Torino, Italy Abstract The paper shows a new methoology to ientify the orifice ischarge coefficient of pneumatic externally pressurize bearings. A common metho of ientification is to measure the experimental air flow, the supply pressure an the local maximum of the pressure istribution in the bearing clearance near the supply hole. This metho however involves some practical ifficulties especially with small supply hole iameters ue to the high pressure an velocity graients near the supply holes. The propose new metho requires both theoretical an experimental approaches. It iffers from the previous metho because the experimental activity involves measuring only the pressure in corresponence of one specifie point uner the pa far from supply holes. The pressure istribution uner the pa is calculate by solving the Reynols equation. The pressure bounary conition near the supply holes an consequently the pressure istribution in the pa clearance epen on the supply hole ischarge coefficient. By using the bisection metho the ischarge coefficient value is change iteratively in orer to fin the convergence of the experimental pressure value with the numerical one. The ischarge coefficient is obtaine when the experimental pressure measure in the selecte point correspons with the calculate one. This new metho is simpler to realize experimentally an it is less sensitive to the position of the measuring hole. Air pas with ifferent supply hole iameters are teste. The pressure istribution uner the pas is etermine experimentally using a test bench at the purpose realize. The tests are carrie out with ifferent air clearances an supply pressures. Keywors: pneumostatic air pa, ischarge flow coefficient, gas lubrication, gas bearing. oi:10.495/td100091
96 Tribology an Design NOMENCLATURE b Critical pressure ratio c Discharge coefficient of supply orifices Supply orifice iameter 0 Insert iameter k T Temperature coefficient 0 T / T l Supply orifice length h Air clearance q Inlet mass flow rate per unit area r Raial coorinate N Number of supply orifices P Absolute pressure P a Absolute ambient pressure P s Absolute supply pressure P c Absolute supply orifice ownstream pressure R 0 Gas constant = 87.6 m /s K T 0 Absolute temperature in normal conition T Orifice upstream absolute temperature Re Reynols number S Supply orifice cross section Circumferential coorinate e Pa external iameter i Pa internal iameter Air viscosity = 17.89 10-6 Pa s 1 Introuction Gas bearings are use for those applications in which high precision an repeatability in positioning are require. The absence of stick an slip an the very low friction makes these components very appropriate in precision linear guies. Externally pressurize air bearings are use to support measuring an testing machines an their static an ynamic performance is the object of continuous improvement in the esign process. While the equations that escribe in goo approximation the air flow in the clearance uner the pa are well known, a problem not yet completely solve is the efinition of the ischarge coefficient of the supply holes. The researchers have ealt with this problem both experimentally an by means of numerical methos. In Al-Bauer an Van Brussel [1] an experimental activity on a centrally fe circular air pa is shown an compare with simulate results obtaine with a metho, calle separation of variables. A common approach consists in measuring empirically the ischarge coefficient that relates the theoretical mass flow rate to the experimental one with the following formula Gexp c G th, (1)
Tribology an Design 97 where the theoretical mass flow rate is calculate consiering an isentropic expansion. Some papers present a single number, e.g. Lun [] an Bryant et al. [3], while others propose empirical formulas, e.g. Elro an Glanfiel [4] an Kazimierski an Trojnarski [5]. Other authors like Renn an Hsiao [6], Belforte et al. [7] an Ly an Ding [8] solve the complete flow problem by using commercial CFD coes. Belforte et al. [9] propose a formula to calculate the ischarge coefficient on the base of experimental measurements of mass flow rates an of pressure istributions near the supply holes. However this metho presents some practical ifficulties when the supply holes have a iameter smaller than 0.1 mm. In the present paper is propose another ientification metho that nees to measure experimentally the pressure only in corresponence of one point uner the pa. This metho is simpler to be realize an less time consuming because it oesn t nee to measure the pressure istribution uner the pa near the supply holes, where the pressure graients are very high. Pas uner test Four annular stainless steel pneumatic pas are examine. They have the same outer iameter of 40 mm an inner iameter of 10 mm. The central pocket is at ambient pressure. Figure 1 shows the sketch of the pas an figure their photo. In corresponence of the supply circumference of iameter 5 mm, N equispace supply holes of iameter an length l=0.4 mm are realize. The nominal iameter values are 0. mm, 0.3 mm, 0.4 mm an 0. mm for pas 1,, 3 an 4 respectively. The holes are rille on brass inserts of iameter 0 = 4 mm. Table 1: Geometric parameters. Pa type Number of holes N 1 3 3 3 3 4 6 Orifice Measure orifice iameter m [mm] 1 0.37 0.11 3 0.19 1 0.31 0.313 3 0.315 1 0.41 0.415 3 0.44 1 0.8 0.30 3 0.1 4 0.19 5 0.18 6 0.17 Mean iameter value [mm] Roughness Ra [m] 0. 0.45 0.313 0.46 0.40 0.61 0.1 0.38
98 Tribology an Design In Table 1 are shown the principal geometric parameters of the pas. Each supply hole was measure by a microscope along two perpenicular irections an the mean value m is reporte in the table with also the measure mean iameter value of all the orifices. In the table are reporte also the relative errors with respect to the nominal values an the roughness of the pas. 3 Pas 1,,3 r 1 Pa 4 4 3 5 r 6 1 Ø l 0 Figure 1: 5 0 1.5 Sketch of the pas. Figure : Photo of the pas. 3 Numerical moel The well known Reynols equation () for compressible fluis is solve to calculate the pressure istribution uner the pa. 1 3 P 1 3 P 0 0 G Ph rh h R T r r r r 4 4 () rr t
Tribology an Design 99 Mass flow rate G is calculate in accorance with the isentropic expansion equation (3), where P c is the supply port ownstream absolute pressure. G c G th G c k1 k P k P k s c c c P s 4 k 1 P s P if P c 0. 58, s RT Ps G th k s 1 k c Ps if Pc 0. 58. (3) 4 k 1 k 1 RT P The ischarge coefficient c can be ientifie experimentally. In paper [9] the following formula is propose s c 0.85 1 e h 8. 1 e 0.001Re (4) where Reynols number is calculate with eqn (5) 4G Re (5) If formula (4) is use coefficient c is calculate on the base of the mass flow rate G of the previous iteration. Eqn () is iscretize with finite ifference metho consiering central erivatives. The surface uner the pa is meshe with n an m noes along the raial an the circumferential irections respectively. The position of the noes is chosen in orer to make some of them coincie with the supply holes. The Reynols equation is solve with explicit Euler metho with the initial conition of ambient pressure in corresponence of all noes. The following bounary conitions are consiere: P=P a at the outer an inner eges of the pa coincience of pressure for =0 an =. The loa carrying capacity of the pa is calculate integrating the relative pressure uner the pa with eqn (6): r e 3.1 Discharge flow coefficient ientification F P P r r (6) 0 ri In Belforte et al. [9] the ischarge coefficient c was calculate with eqn (1) as the ratio between the experimental an the theoretical mass flow rate, obtaine a
100 Tribology an Design consiering the experimental supply holes ownstream pressure level P c. This metho however requires to measure the pressure istribution near the supply holes an iniviuate with goo precision the ownstream pressure level P c. This coul be ifficult especially with small hole iameters because of the high pressure graients. Moreover the measuring hole machine on the crossmember must have a smaller iameter than the supply hole in orer to measure correctly the pressure istribution. This limits the iameter of the supply holes for the pas that can be teste correctly on the test bench. Figure 3: Block iagram of the ientification algorithm. A new metho of ientification of the supply holes ischarge coefficient is propose hereuner. The metho consists in measuring the pressure P exp in corresponence of a etermine point uner the pa for ifferent clearance values. This pressure is compare with the numerical value P num that one obtains with a ischarge coefficient c of first approximation. The c value is then moifie an the relative pressure is calculate numerically; ifferent iterations are performe using the bisection metho until convergence is reache. The final numerical ischarge coefficient correspons to a numerical pressure value equal to the experimental one. In figure 3 is shown the block iagram of the algorithm use. The number of iterations performe is iter=8, to which correspons a convergence error err<0.%, where err=(p exp P num )/P exp.
Tribology an Design 101 4 Experimental set-up 4.1 Test rig escription The test rig is compose of a rigi structure an a moving screw with which it is possible to impose ifferent air clearance heights between the pa an the stationary member. The air gap is measure by means of two micrometric inuctive probes with sensitivity 0.1 µm. The loa carrying capacity of the pa is measure with a loa cell with kn full range. The pressure uner the pa is measure with a piezoresistive sensor with resolution 10-4 MPa, connecte to the measuring hole rille in the stationary pa (see figure 4). The latter can be move raially or circumferentially in orer to measure the pressure value in r insert pa 1.5 h 0/ stationary member pressure measuring hole Ø 0. Figure 4: Sketch of the measuring system. Figure 5: Test rig.
10 Tribology an Design each point uner the pa. A potentiometric isplacement transucer measures the raial position of the stationary member with respect to the center of the pa. The air flow rate is measure by a flowmeter situate upstream the pa. High efficiency air filters (97% an 99% efficiency with particles of iameter 0.1 m) are use to avoi problems of clogging of the supply holes. Figure 5 shows the test rig. 4. Test proceure The loa carrying capacity an the air consumption are strongly influence by the air gap. For this reason a test proceure was efine in orer to obtain a goo repeatability an precision of the measurements. The pa is initially cleane an the supply holes are checke with a microscope to exclue an eventual clogging. The pa is then supplie with compresse air at a efine pressure an an increasing ownwars vertical force is applie on it until the air clearance is equal to zero. In this conition the isplacement transucers are set to zero. Higher loas are avoie in orer not to eform the pa an the stationary member. In this way the reaing of the isplacement transucers coincies with the air clearance. The eformation of the pa an of the stationary member are negligible. The force applie to the pa is then ecrease an the pressure uner the pa, the loa capacity an the air flow rate are measure for ifferent air gaps. The supply pressure is maintaine constant for each clearance value up to h=0 m. The measuring hole is positione far away the supply holes where the pressure graients are low. In particular it was positione in corresponence of the supply circumference in the mile of two supply holes. 4.3 Experimental results an iscussion The pressure in corresponence of the measuring point was measure at ifferent clearance values with P s =0.6 MPa. In figure 6 the measure pressures uner the Pressure [Pa] 3.5 x 105 3.5 1.5 Pa 1 Pa Pa 3 Pa 4 1 0 10 0 30 Air clearance h [m] Figure 6: Experimental pressure in the measuring point uner the pa vs air clearance; comparison between ifferent pas.
Tribology an Design 103 four pas are compare vs the air clearance. As one can see, the pressure increases both with the iameter of supply holes (pas 1 to 3) an with the number of holes (pas 1 an 4). The secon effect is preominant. In figure 7 the air mass flow rate is reporte vs the air clearance. The air consumption of pa 4 is almost twice than that of pa 1, being the number of holes twice an the supply hole iameter the same. In figure 8 the experimental loa capacity of the pas are compare with the numerical ones. The numerical solution unerestimates the experimental loa capacity. In figure 9 the values of the ischarge coefficients obtaine with the propose metho are reporte. The values are compare with the ones obtaine with the empirical formula (4). Consiering that the value of c is very sensitive to the clearance value h, the comparison is quite goo. The error between the formula an the experimental value is comparable with the tolerance amitte. x 10-4 Mass flow rate [kg/s] 1.5 1 0.5 Pa 1 Pa Pa 3 Pa 4 0 0 10 0 30 Air clearance h [m] Figure 7: Experimental air consumption vs air clearance; comparison between ifferent pas. F [N] 50 00 150 100 pa 1, exp pa 1, num pa, exp pa, num pa 3, exp pa 3, num pa 4, exp pa 4, num 50 0 5 10 15 0 5 30 Air clearance [m] Figure 8: Loa capacity vs air clearance; comparison between experimental an numerical results.
104 Tribology an Design ischarge coefficient 0.8 0.7 0.6 0.5 0.4 0.3 formula pa 1 pa pa 3 pa 4 0. 0.1 0.0 0.04 0.06 0.08 0.1 0.1 0.14 h/ Figure 9: Discharge coefficient c vs air clearance; comparison between the values obtaine with the ientification metho an with eqn (4). 5 Conclusions A metho of ientification of the supply holes ischarge coefficient for gas bearings is presente. The metho allows the etermination of the coefficient c without the nee of measuring the complete pressure istribution uner the supply holes. This involves some practical avantages that are more evient when the supply hole iameter is smaller than 0.1 mm. The iameter of the measuring hole is not neee to be smaller than the supply hole iameter because the pressure is measure far-off the supply holes, where the pressure graients are not high. As the air consumption an the pressure uner the pa are very sensitive to the air clearance, the experimental measures must be carrie on with carefulness in orer to have a goo estimation of the ischarge coefficient. References [1] Al-Bauer, F., Van Brussel H., Symmetric raial laminar channel flow with particular reference to aerostatic bearings, Journal of Tribology, 114, pp. 630-636, 199. [] Lun, J.W., The hyrostatic gas journal bearing with journal rotation an vibration, Journal of Basic Engineering, 86, pp. 38-336, 1964. [3] Bryant, M.R., Velinsky, S.A., Beachley, N.H., Froncza, K.F.T., A esign methoology for obtaining infinite stiffness in an aerostatic thrust bearing, Journal of Mechanisms, Transmissions an Automation in Design, 108, pp. 448-456, 1986. [4] Elro, H.G., Glanfiel, G.H., Computer proceures for the esign of flexibly mounte, externally pressurize, gas lubricate journal bearing, Gas bearing Symposium, University of Southampton, pp..1-.37, 1971.
Tribology an Design 105 [5] Kazimierski, Z., Trojnarski, J., Investigation of externally pressurize gas bearing with ifferent feeing systems, Journal of Lubrication Technology, 10, pp. 59-64, 1980. [6] Renn., J., Hsiao, C., Experimental an CFD stuy on the mass flow-rate characteristic of gas through orifice-type restrictor in aerostatic bearings, Tribology International, 37, pp. 309-315, 004. [7] Belforte, G., Raparelli, T., Trivella, A., Viktorov, V., Visconte, C., Numerical analysis on the supply hole ischarge coefficient in aerostatic bearings, International Conference on Tribology AITC-AIT 006, 0- September 006, Parma, Italy. [8] Ly, Y., Ding, H., Influences of the geometrical parameters of aerostatic thrust bearing with pockete orifice-type restrictor on its performance, Tribology International, 40, pp. 110-116, 007. [9] Belforte, G., Raparelli, T., Viktorov, V., Trivella, A., Discharge coefficients of orifice-type restrictor for aerostatic bearings, Tribology International, 40, pp. 51-51, 008.