FLOW OF A VISCOUS LIQUID ON A ROTATING NSK Thesis for flu Degree 0‘ pk. D. MICHIGAN STATE UNIVERSITY Tony Leonard Kaminski £965 mass ' . ' “““"“*'“”" {4 w munHumnimrnnummzmmmnnuuumlj farm ' 293 20125 1026 ‘dxelmnn «j . 3‘: UIi He]: HY v? 135”” "" This is to certify that the thesis entitled Flow of a Viscous Liquid on a Rotating Disk presented by Tony L. Kaminski has been accepted towards fulfillment of the requirements for Ph.D. degree in Agricultural Engineering QLKJZM WWW Major professor Date 94’037- C J, 0-169 ABSTRACT FLOW OF A VISCOUS LIQUID ON A ROTATING DISK by Tony Leonard Kaminski There are inherent advantages to the rotating disk for the atomization or distribution of liquids especially liquids which cannot be subjected to high pressure or which contain solid particles. An investigation was con— ducted to obtain a more thorough understanding of the flow of a liquid on a flat rotating disk and to determine the feasibility of using theoretical considerations to study this flow situation. In this study an exact solution to the Navier—Stokes equations, for the case of flow around a disk rotating in a fluid at rest, was applied to the present flow situation. The flow of the liquid on the disk was characterized by means of streamline projections in both the horizontal (r—e) plane and the vertical (r-z) plane. An experimental apparatus consisting of a 6—inch diameter disk powered by a variable speed motor was used to verify the theoretical results. Two mineral oils were selected for the tests: one had a kinematic viscosity of .087 inz/sec and the other had a kinematic viscosity of .194 inz/sec. The oil was introduced axially symmetrical Tony Leonard Kaminski onto the disk through a plexi—glass tube designed to reduce the axial velocity of the liquid. Effects of rotational speeds of 1000 rpm and 2&00 rpm and flow rates of 5.0 gpm and 10.2 gpm of the two oils were considered. The thickness of the oil layer on the disk was measured with a point gage and the streamline patterns in the horizontal plane were photographed. The theoretical and experimental curves were compared and the deviations between some of the curves were explained. The following conclusions derived from the Navier— Stokes equations were verified experimentally for the laminar flow of a Newtonian fluid on a rotating disk. 1. The thickness of the liquid flowing on the disk is increased by: (a) increased liquid viscosity (b) increased flow rate and (c) slower rotational speed. 2. The angular displacement of the liquid from inlet to outlet is increased by: (a) increased liquid viscosity (b) decreased flow rate and (c) higher rotational speed. 3. The angle relative to the radius with which a liquid particle leaves the disk increases with (a) increased angular displacement (b) increased radial distance and (c) decreased height above the disk surface. This angle has a maximum value of 90 degrees on the surface of the disk implying that a particle at this height has only tangential velocity. As the distance above the disk increases the flow becomes more in a radial direction. Tony Leonard Kaminski Preliminary experiments were made injecting fluid through special nozzles on a section of the disk. These tests showed that approximately twovthirds of the liquid could be made to flow off of one—fifth of the circumference Approved §;:;~L~(;-<;fEE:A/+waa Major Professor gnu. 2M Department Chairman Of the disk. FLOW OF A VISCOUS LIQUID ON A ROTATING DISK By Tony Leonard Kaminski A THESIS Submitted to Michigan State University in partial fulfillment of the requirements» for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1965 ACKNOWLEDGMENTS The author sincerely appreciates the assistance of all who have helped in this study. He is extremely appre— ciative for the council and guidance provided by Dr. Sverker Persson (Agricultural Engineering) during his research work. To the other members of the guidance committee, Professor H. F. McColly (Agricultural Engineering), Dr. N. L. Hills (Mathematics) and Dr. G. E. Mass (Metallurgy, Mechanics, and Material Science) the author expresses his gratitude for their time, professional interests, and con- structive criticisms. Dr. M. Krzywoblocki (Mechanical Engineering) made many helpful suggestions during the theoretical considerations in this study. Professor C. M. Hansen (Agricultural Engineering) was instrumental in obtaining the financial grant, from the American Oil Company, Whiting, Indiana, for the assistantship which made this work possible. Mr. James Cawood and his staff in the Agricultural Engineering Research Laboratory assisted in the development of the experimental apparatus. This dissertation is dedicated to my wife, Zena, who consistently gave her encouragement and support to this study. ii TABLE OF CONTENTS ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES LIST OF APPENDICES TABLES. ABBREVIATIONS AND SYMBOLS Chapter I. II. III. IV. VI. INTRODUCTION REVIEW OF LITERATURE THEORETICAL CONSIDERATIONS 3.1 The Flow Functions for a Semi-infinite Fluid and an Infinite Disk. 3.2 Description of the Flow Pattern EXPERIMENTAL CONSIDERATIONS 4.1 Description of the Apparatus . 4.2 Selection of a Viscous Liquid. 4.3 Experimental Procedure . . . PRESENTATION AND DISCUSSION OF RESULTS 5.1 Streamlines in the r-z Plane 5.2 Streamlines in the r-e Plane 5.3 Flow on Selected Areas of the Rotating Disk SUMMARY AND CONCLUSIONS 6.1 Summary 6.2 Conclusions. SUGGESTIONS FOR FURTHER STUDY REFERENCES. APPENDICES. iii Page ii iv viii ll ll 19 27 27 33 38 38 52 62 71, 71 73 76 77 80 LIST OF TABLES Functions for the velocity and pressure distribution as calculated by the digital computer using equations 3.29 - 3.34 Values of specific gravity and kinematic viscosity obtained from samples of the oils used in the laboratory tests. Tests conducted to study the flow of a viscous liquid on a rotating disk. Comparison of experimental and theoretical axial velocities at the tube outlet, located .300 inches above the surface of the disk, with the experimental resultant velocity at the edge of the disk . . Values of Reynolds numbers present in the laboratory tests. Comparison of the experimental and theoretical surface streamlines using angular displace— ments and angles relative to the radius at the edge of the disk Comparison of the experimental and theoretical average radial and average tangential velocities at the edge of the disk. iv Page 18 36 38 44 47 63 64 Figure 10. 11. LIST OF FIGURES Flow near a disk rotating in a fluid at rest. Velocity components: u—radial, v—circum— ferential and w—axial Example of radial and tangential velocity profiles above a rotating disk . Motion of a particle on a rotating disk Example of how the ratio of tangential velocity to radial velocity varies with radial position on the disk . . The testing apparatus (MSU Photo No. 65772-10). The point gage and plexi-glass tube mounted above the disk (MSU photo No. 65772-11) Equipment used for flow measurement (MSU Photo No. 65772—12). Comparison of experimental and theoretical streamlines in the r— 2 plane for hydraulic oil (0 = .194 inz/sec) flowing on a disk rotating at 1000 rpm . . Comparison of experimental and theoretical streamlines in the r— 2 plane for hydraulic oil (v = .194 inZ/sec) flowing on a disk rotating at 2400 rpm. . . . . Comparison of experimental and theoretical streamlines in the r— 2 plane for mineral oil (v = .087 in2/sec) flowing on a disk rotating at 1000 rpm. . . . Comparison of experimental and theoretical streamlines in the r— 2 plane for mineral oil (v = .087 in2/sec) flowing on a disk rotating at 2400 rpm. . . Page 12 21 24 25 28 29 31 39 4O 41 42 Figure Page 12. Comparison of experimental and theoretical radial velocity profiles on a disk rotat— ing at 2100 rpm in air (obtained by Gregory et al. 1955) Curve 1, theoretical laminar profile; 2, 3, 4 experimental profiles: 2, laminar region; 3, insta— bility region; 4, turbulent region. . . . 49 13. Comparison of experimental and theoretical tangential velocity profiles on a disk rotating at 2100 rpm in air (obtained by Gregory et a1. 1955) Curve 1, theoretical laminar profile; 2, 3, 4 experimental profiles: 2, laminar region; 3 instability region; 4, turbulent region . . . . . . 50 14. Comparison of experimental and theoretical streamlines in the r— 9 plane for 10. 2 gpm flow of hydraulic oil (v = .194 inZ/sec) onto a disk rotating at 1000 rpm (clockwise) (MSU Photo No. 65772-2) . . . . . . . 53 15. Comparison of experimental and theoretical streamlinesij1the r-e plane for 10.2 gpm flow of hydraulic oil (0 = .194 inz/sec) onto a disk rotating at 2400 rpm (clockwise) (MSU Photo No. 65772—3) . . . . . . . 54 16. Comparison of experimental and theoretical streamlines in the r—e plane for 5.0 gpm flow of hydraulic oil (v = .194 in2 /sec) onto a disk rotating at 1000 rpm (clockwise) (MSU Photo No. 65772— 4) . . . . 55 17. Comparison of experimental and theoretical streamlines in the r—e plane for 5.0 gpm flow of hydraulic oil (v = .194 inZ/sec) onto a disk rotating at 2400 rpm (clockwise) (MSU Photo No. 65772—1) . . . . . . . 56 18. Comparison of experimental and theoretical streamlines in the r—e plane for 10.2 gpm flow of mineral oil (v = .087 inZ/sec) onto a disk rotating at 1000 rpm (clockwise) (MSU Photo No. 65772-5) . . . . . . . 57 vi Figure 19. 20. 21. 23. 24. 25. 26. 27. Page Comparison of experimental and theoretical streamlines in the r—e plane for 10.2 flow of mineral oil (0 = .087 inz/sec) onto a disk rotating at 2400 rpm (clockwise) (MSU Photo No. 65772-6) . . . . . . . 58 Comparison of experimental and theoretical streamlines in the r—e plane for 5.0 gpm flow of mineral oil (0 = .087 inZ/sec) onto a disk rotating at 1000 rpm (clockwise) (MSU Photo No. 65772—7) . . . . . . . 59 Comparison of experimental and theoretical streamlines in the r-e plane for 5.0 gpm flow of mineral oil (v = .087 inz/sec) onto a disk rotating at 2400 rpm (clockwise) (MSU Photo No. 65772—8) . . . . . . . 6O Closed-bottom nozzle (left) and Open-bottom nozzle (right) with parrallelogram-shaped Openings (MSU Photo No. 65772—9) . . . . 66 Diagram of a parallelogram—shaped nozzle opening (MSU Photo No. 65772—9). . . . . 66 Flow pattern from the closed-bottom nozzle (N = 1000 rpm, o = 4.4 gpm, t = .194 1h3seo) (MSU Photo No. 65772-13) 69 Streamlines in the r—e plane for the closed— bottom nozzle (N = 1000 rpm, Q = 4.4 gpm, v = .194 ihZ/seo) (MSU Photo No. 65772—14) . 69 Flow pattern from the open—bottom nozzle (N = 1000 rpm, o = 5.4 gpm,v = .194 inZ/sec) (MSU Photo No. 65772—15) . . . . . . . 70 Streamlines in the r-e plane for the open- bottom nozzle (N = 1000 rpm, Q = 5.4 gpm, v = .194 inzsec) (MSU Photo No. 65772-16) . 7o Negatives of photographs are filed by Information Services, Michigan State University vii Appendix A A1. A2. A3. A4. A5. A6. A7. A8. A9. A10. LIST OF APPENDICES TABLES Dimensionless radial (F) and tangential (G) velocity components for hydraulic oil (0 = .194 inz/sec )flowing on a disk rotating at both 1000 rpm and 2400 rpm. . . . . . Radial velocity components for hydraulic oil (v = .194 in /sec.) flowing on a disk rotating at 1000 rpm Tangential velocity components for hydrau- lic oil (v = .194 in /sec.) flowing on a disk rotating at 1000 rpm Radial velocity components for hydraulic oil (v = .194 in /sec.) flowing on a disk rotating at 2400 rpm Tangential velocity components for hydrau— lic oil (v = .194 in /sec.) flowing on a disk rotating at 2400 rpm Dimensionless radial (F) and tangential (G) velocity2 components for mineral oil (v = .087 in2 /sec. ) flowing on a disk rotating at both 1000 rpm and 2400 rpm. Radial velocity2 components for mineral oil (v = .087 in2 /sec. ) flowing on a disk rotating at 1000 rpm. . . Tangential velocity2 components for mineral oil (v = .087 in2 /sec. ) flowing on a disk rotating at 1000 rpm . . . Radial velocity2 components for mineral oil (v = .087 in2 /sec. ) flowing on a disk rotating at 2400 rpm. . . . Tangential velocity2 components for mineral oil (v = .087 in2 /sec. ) flowing on a disk rotating at 2400 rpm. viii Page 81 82 83 84 85 86 87 88 89 9O Appendix B Page B1. Determination of streamlines in r—e plane for 10.2 gpm flow of hydraulic oil (0 = .194) onto a disk rotating at 1000 rpm . . . . 92 B2. Determination of streamlines in r-e plane for 10. 2 gpm flow of hydraulic oil (0 = .194) onto a dis k rotating at 2400 rpm . . . . . . 93 B3. Determination of streamlines in r—e plane for 5. 0 gpm flow of hydraulic oil (v = .194) onto a disk rotating at 1000 rpm . . . . . . . . 94 B4. Determination of streamlines in r—e plane for 5. 0 gpm flow of hydraulic oil (v = .194) onto a disk rotating at 2400 rpm . . . . . . . . 95 B5. Determination of streamlines in r—e plane for 10. 2 gpm flow of mineral oil (0 = .087) onto a disk rotating at 1000 rpm . . . . . . . 96 B6. Determination of streamlines in r—e plane for 10. 2 gpm flow of mineral oil (v = .087) onto a disk rotating at 2400 rpm . . . . . . . 97 B7. Determination of streamlines in r—e plane for 5.0 gpm flow of mineral oil (v = .087) onto a disk rotating at 1000 rpm . . . . . . . . . . 98 B8. Determination of streamlines in r—G plane for 5.0 gpm flow of mineral oil (0 = .087) onto a disk rotating at 2400 rpm . . . . . . . . . . 99 ix cm eq in in2 ips 1b In No. "U Re rpm SEC CtD¢N€ - Fr I 3r + u (avg + r 8r 2 2 u + l a u 2 av + 8 u (3.1) ‘ ‘2 ’2 _—2 ‘ "2 gg ‘—’2 r r 86 r 82 l 2 11 3;: 19.1 91 2.1: __32 3v 9 (at + u3r + r as + r + waz) Fe r 86 + u (org 1 l 2 2 2 8V V 8 V Bu 8 V + F ‘F — —7 + —2 ——7 + —7 as + ——7 (3.2) r r 80 r oz 2 l 3 3E 31 Y 3i 3E = _ 32 3 W _ _E 9 <3 + uar + r 36 W8z> F2 82 + u (arz + r or 1 2 a w 3 v +—z—-z+——‘2 (3.3) r 30 az Continuity equation: on u 1 av 8w _ E? + r + r e + 32 _ 0 (3'4) Where p = mass density t = time p = pressure F = body force u = absolute viscosity u, v, w = velocity components By neglecting the body forces and assuming steady rotationally symmetric flow of a viscous liquid the Navier— 2 Stokes equations can be written as u 33 - K + wi—Ll ar r 82 1 ap azu 1 Eu u azu =__ —_+_.___.__. ._____.> . 0 3r + v 2 r r 2 + (3 5) 8r r 823’ 2 2 av uv 8v (3 v 1 av v a v (3.6) u _ 'l' —— ‘l‘ W —— = \) —§' + -- '— .— -—-§- + j) dr r z 3r r r r az aw 3w 1 gap 32w 1 8w 32w 4 —— + W —“ = - — + V (“—2 + r *“+ ‘_E (3.7) 3r 2 p 32 or r r 3z 3! 2 3w _ 3r + r + 33 _ o (3.8) Where v = E = kinematic viscosity The no-slip condition on the disk gives the following boundary conditions z = 0 : u = 0, v = rm w = 0 (3.9) and the condition of a fluid at rest at infinity gives z = w : u = 0, v = 0 . . . . (3.10) The equations of motion and the continuity equation are satisfied by the following substitutions: u = rf(z), v = rg(z), w = h(z), p = p(z). . .(3.ll) These.assumptions can be used only in laminar flow since in turbulent flow the velocity components are not prOpor— tional to the radius but more complicated functions of the radius. Assuming laminar flow, the Navier-Stokes equations become 2 2 2 df d f _ f - g + h d? _ v 2 - 0 (3.12) dz 2 2fg + h g5 - v9—5 = o (3 13) z 2 dz 2 9.2122 dh- hdz + 9 dz - v 2 - 0 (3.14) 15 2f + —— = 0 (3.15) with the following boundary conditions 2 = 0 : f = 0, g = w, h = o, p = p (3.16) O o (3.17) z = w : f = 0, g = O, h = —c, p In order to integrate the system of equations (3.12—3.15) it is convenient to introduce a dimensionless distance from the disk c = z % (3.18) The flow equations are also changed into non—dimen— sional form by substituting f=wF(C), s=wG(c), h= NET HM), p=vaP1(C) (3.19) where P1 = P—PO Thus the following assumptions are made for the velocity components and the pressure: u=wr F(Q), v=rwG(;), w=W H(§), p= prP1 (3.20) Inserting these assumptions (3.20) into equations 3.5—3.8 Q_ = Q_ . BE dz dc dz Of four simultaneous, ordinary, non-linear differential and noting that yields the following system equations for the functions F, G, H, and P1 2 2 F - G + g3 — 9—3 = o (3.21) c d: 2 2FG + H g% - é_§ = o (3.22) 16 dP1 2 dH d H ~--.——- + H —e - —-—— = o .2 dH _ 2F + a? — o (3.24) The boundary conditions calculated from equations (3.9) and (3.10).are: C = O : F = O, G = 1, H = 0, P1 = O (3.25) C = w : F = O, G = O (3.26) As indicated earlier the first solution of this system of equations (3.21 - 3.24) was given by von Karman (1921) by an approximate integral method. Later Cochran (1934) calculated more accurate values, for a limited number of values of the independent variable, by a method Of numerical integration. Cochran's solution was Obtained by assuming a power series near C = O and an asymptotic series for large values Of C. By trial and error he connected the two series which yielded the following boundary conditions dF dG , for d? and a: . _ dF = dG = _ In order to use an existing computer program and solve the equations by a digital computer the dependent variables F, G, H, and P1 were redefined as follows: dF dG YI = H: y2 = F: y3 = a?) y4 = G) y5 = 6:: y6 = P1 = P-P (3.28) 17 The system of flow equations (3.21 - 3.24) were then written as a system of six first-order linear differential equations for the functions y1, y2, y3, yq, y5, and ye dYI 8—?— = -2y2 (3°29) dyz 52. = y3 (3.30) d 2 2 .13 = YIYB+YZ -yL. (3.31) d6 33.1.: = 3’5 (3.32) dC dyS a?“ = 2 Y2Y4 + Y1YS (3'33) dy6 52— = 2YIY2 - 2YB (3'3“) The original boundary condions (3.24 — 3.25) became a = 0 : y1= O, y2 = 0, y3 = 0.510, yq = 1.0, y5 = —.6l6, Y6 = 0 (3.35) The values of the functions of velocity and pressure were calculated by the Michigan State University digital computer (CDC 3600), using library program D2 UTEX RKAMPDP, and are shown in Table l. The values are shown with four decimal places but the fourth figure is insignificant since some of the initial conditions were accurate to only three signi— ficant figures. Many of the very small values shown in Table 1 might be assumed to be zero. 18 TABLE 1. Functions for the velocity and pressure distri— bution as calculated by the digital computer using equations 3.29 — 3.34. C = ng’ -H F dF/d; G -dG/d; —P1 0 .0000 .0000 .5100 1.0000 .6160 .0000 0.1 .0048 .0462 .4160 .9386 .6113 .0924 0.2 .0179 .0836 .3336 .8780 .5988 .1673 0.3 .0377 .1133 .2618 .8190 .5804 .2273 0.4 .0628 .1363 .1997 .7621 .5578 .2745 0.5 .0918 .1535 .1465 .7075 .5323 .3112 0.6 .1238 .1658 .1013 .6557 .5049 .3393 0.7 .1579 .1740 .0633 .6066 .4765 .3605 0.8 .1932 .1787 .0315 .5604 .4478 .3761 0.9 .2292 .1805 .0054 .5170 .4193 .3873 1.0 .2653 .1799 -.0232 .4610 .3804 .3974 1.1 .3010 .1775 -.0329 .4387 .3643 .4003 1.2 .3361 .1735 -.0463 .4036 .3384 .4035 1.3 .3703 .1683 -.0565 .3710 .3136 .4052 1.4 .4034 .1623 -.0641 .3408 .2902 .4059 1.5 .4352 .1556 —.0694 .3129 .2681 .4058 1.6 .4656 .1484 —.0726 .2871 .2473 .4053 1.7 .4946 .1410 —.0748 .2634 .2280 .4044 1.8 .5220 .1335 —.0755 .2415 .2099 .4033 1.9 .5480 .1260 —.0751 .2214 .1932 .4021 2.0 .5724 .1185 —.0739 .2028 .1776 .4009 2.1 .5954 .1112 —.0721 .1858 .1632 .3997 2.2 .6169 .1041 —.0698 .1702 .1499 .3985 2.3 .6370 .0973 -.0672 .1558 .1376 .3974 2.4 .6558 .0907 -.0643 .1426 .1263 .3964 2.5 .6733 .0844 —.0612 .1305 .1158 .3955 2.6 .6896 .0784 —.0580 .1194 .1062 .3947 2.7 .7047 .0728 —.0548 .1092 .0974 .3939 2.8 .7188 .0675 -.0516 .0999 .0893 .3933 2.9 .7318 .0625 -.0485 .0913 .0819 .3927 3.0 .7438 .0578 —.0454 .0835 .0750 .3921 3.2 .7651 .0493 —.0396 .0697 .0630 .3913 3.4 .7833 .0419 —.0343 .0582 .0529 .3906 3.6 .7988 .0355 —.0295 .0485 .0444 .3901 3.8 .8119 .0300 —.0253 .0403 .0373 .3896 4.0 .8229 .0255 -.0216 .0335 .0313 .3893 4.2 .8323 .0214 —.0184 .0278 .0262 .3891 4.4 .8401 .0180 —.0156 .0230 .0220 .3888 4.6 .8467 .0151 -.0132 .0189 .0185 .3887 4.8 .8522 .0127 —.0112 .0155 .0155 .3885 5.0 .8569 .0106 -.0094 .0127 .0130 .3884 19 TABLE 1 (Continued) dF/dc G -dG/dc —P1 5.2 .8608 .0089 —.0079 .0103 .0109 .3883 5.4 .8640 .0074 -.0067 .0083 .0091 .3882 5.6 .8668 .0062 —.0056 .0066 .0077 .3881 5.8 .8690 .0052 —.0047 .0052 .0064 .3880 6.0 .8709 .0043 —.0040 .0040 .0054 .3879 6.2 .8725 .0036 —.0033 .0030 .0045 .3878 6.4 .8738. .0030 -.0028 .0022 .0038 .3877 6.6 .8749 .0025 -.0024 .0015 .0032 .3876 6.8 .8758 .0020 -.0020 .0009 .0027 .3875 7.0 .8765 .0017 - 0017 .0004 .0022 .3875 7.2 .8772 .0014 -.0014 .0000 .0019 .3874 7.6 .8780 .0009 —.0010 «.0006 .0013 .3872 7.8 .8783 .0007 —.0008 —.0009 .0011 .3871 8.0 .8786 .0006 -.0007 —.0011 .0009 .3871 8.5 .8790 .0003 —.0004 -.0014 .0006 .3869 9.0 .8792 .0001 -.0003 —.0017 .0004 .3867 9.5 .8792 .0000 —.0002 —.0018 .0003 .3865 10.0 .8791 —.0001 —.0001 -.0019 .0002 .3863 10.5 .8790 -.0001 -.0001 -.0020 .0001 .3861 11.0 .8789 —.0002 -.0001 —.0021 .0001 .3859 11.5 .8787 —.0002 -.0000 —.0021 .0000 .3856 12.0 .8785 —.0002 - 0000 —.0021 .0000 .3854 .866* 0* 0* 0* 0* .393* *These values were calculated by W. G. Cochran (1934). 3.2 Description of the Flow Pattern In the previous section the general flow equations were deduced for an infinite disk rotating in a semi— infinite fluid. In our special case we want to consider the motion Of.a limited quantity of liquid flowing on a rotating disk Of finite radius. This was done by studying the flow patterns in the layer Of the semi-infinite fluid nearest to the disk surface. 20 The flow on the disk can be described by means of streamlines, that is, the paths which the liquid particles follow as they move radially outwards due to rotation of the disk. In order to characterize the flow as mentioned it is necessary to know the kinematic viscosity of the liquid, the quantity of liquid flowing on to the disk, and the speed of rotation of the disk. The angular velocity is related to the speed of rotation by the formula w = Egg (3-36) where w = angular velocity N = Speed Of rotation Use of formula t = 27673—(eq. 3.18) enables the determination of the dimensionless distance (g) corres— ponding to an actual distance (z) above the disk (Appendix A). Note that z = 0 corresponds to the surface of the disk. Once the non-dimensionless distance is known, the corres— ponding velocity functions are obtained from Table l and the radial component (u) and tangential component (v) of velocity can be calculated by the relationships u = er(§) and v = rwG(;) (eq. 3.20). An example of the resulting velocity profiles obtained is shown in Figure 2. The area under the velocity distribution curve is prOportional to the quantity of liquid flowing through this section. If the thickness Of the liquid is known at a given radius then the average radial velocity of the liquid in this layer can be Obtained by using the expression Velocity (ips) 750 700 650 600 550 500 450 400 350 300 250 200 150 100 50 21 or N = 2400 rpm b v = .194 inZ/sec r = 3.0 in Tangential Velocity ’ Radial Velocity I l I 1 I l - - __.. .02 .04 .06 .08 .10 .12 .14 .16 .18 .20 Distance Normal to Disk Surface, z (in) Figure 2. Example of radial and tangential velocity profiles above a rotating disk. 22 uA = % udz (3.37) 2: :3‘ (D "S (D C. II A average radial velocity {3‘ ll thickness of liquid and also the quantity of liquid (Q) flowing on the disk can be calculated by using the expression Q = 2nrhuA (3.38) The surface of the disk where h = 0 represents one streamline. Another streamline can be found by the condition that Q = constant. The flow between two streamlines in the r-z plane is equal to Q. In our case we were therefore interested in finding this streamline. The values of radial velocity (u) were found numerically and therefore the average radial velocity (uA) could also only be determined numerically from the graphs of u versus z. The integral (equation 3.37) representing the area under a velocity distribution curve, was evaluated by using a planimeter and applying apprOpriate scale factors. A trial and error procedure was followed, that is, various h values were selected until the area required under a curve was obtained. After the depth of flow was Obtained, the average radial velocity component was determined by using equation 3.37. By determining the h—value at various radii, a streamline (thickness profile) in the r—z plane can be plotted for a given flow rate by assuming constant values of fluid viscosity and rotational speed. 23 We were also interested in the streamline pattern as seen in the r-e plane, that is, the direction of motion of particles in the surface 2 = h. The average tangential velocity component was determined from the tangential velocity distributions by direct use of a planimeter since the depth of flow had already been determined in the determination of the average radial velocity. By considering the geometry of a particle moving from point A to point B as shown in Figure 3 it can be seen that lim 1 9.3:- 19: ._u_ lid-*0 r A6 roe v (3.39) Or this may be written as de _ l 1 dr - r u (3.40) The velocity components (u, v) are functions Of the radius (r). By plotting the ratio of the velocities V/u, versus radius r (Figure 4) it was observed that there was approximately a linear relationship between this velocity ratio and the radial position along the disk for various flow conditions indicating that a first degree polynomial ought to provide a good approximation for this relationship. Thus assuming % = Ar + B (3.41) where A and B are constants which were calculated for each flow case by the Michigan State University digital computer using library program E2 UTEX LSCFWOP and are shown in Appendix B. Substituting equation 3.41 into equation 3.40. 24 Figure 3. Motion of a particle on a rotating disk. Tangential Velocity (v)/Radial Velocity (u) l—‘O 25 . Test NO. 2 (surface velocity components) Q = 10.2 gpm v a .194 inZ/sec »3 N = 2400 rpm 1 O . 1 . l . l . J .00 1.50 2.00 2.50 3.00 Radial Position, r (in) Figure 4. Example of how the ratio of tangential velocity to radial velocity varies with radial position on the disk. 26 yields the expression 99 = A + 3 (3.42) By integrating equation 3.42, one Obtains the equation a = Ar + B ln r + c (3.43) where C a constant of integration, which is evaluated by applying an initial condition. 8 angular displacement Thus by the use of equation 3.43 the curve along which a particle moves can be calculated and the resulting stream— line pattern in the r—e plane may be determined (Figure 3). Various streamline patterns will develop at different heights above the disk due to the variations in the magni- tudes of the velocity components (u, v). Two streamline patterns in the r—e plane were plotted for each test con— sidered. One plot was made using the average values of the radial and tangential velocity components at the various radii considered. The other streamline plotted was the one which existed on the surface of this layer. It was found that the term in equation 3.43 which contained the logarithmic function was usually relatively small. Neglecting this term will yield the equation for an Archimedes spiral. Thus a streamline in the r-e plane approximates an Archimedes spiral. IV. EXPERIMENTAL CONSIDERATIONS 4.1 Description Of the Apparatus An overall view of the testing apparatus is shown in Figure 5. A positive displacement hydraulic pump, powered by an electric motor, supplied liquid to a small reservoir mounted above the rotating disk. The small reservoir was vented near the tOp to permit oil to flow back to the main reservoir once the liquid reached the level of this Opening. This regulated the formation of hydraulic pressure in the system. The liquid from the small reservoir flowed by gravity through a flow—control valve which was used to vary the liquid flow to the disk. After the liquid passed through the valve it was directed down to the center of the disk through a section of 2—inch (2.0 inch outside diameter and 1.75-inch inside diameter) plexi—glass tubing with its lower edge taped toward the inside. Since it was important to minimize the axial velocity of the liquid as it flowed on to the disk and to keep the tube filled with liquid, metal screens were mounted inside the plexi-glass tube (Figure 6). The number of screens used in the tube depended upon the visco— sity of the liquid used. A 3—inch head of liquid was maintained in the tube directly above the disk. Other elements employed in 27 28 Heat lamp Main reservoir weight scale Dial indicator Thermometer Plexi—glass tube above disk Small reservoir Tachometer Variable—speed electric motor Hydraulic pump and filter OkOCfiNONU‘I-D-‘UUNH H Figure 5. The testing apparatus. 29 fit . I" 1..) 4/- Figure 6. The point gage and plexi—glass tube mounted above the disk. 30 the hydraulic circuit consisted of a filter mounted directly after the outlet of the pump and a thermometer mounted in the elbow above the plexi-glass tubing for Oil temperature measurement. The quantity of liquid flowing on to the disk was .determined by diverting the liquid flowing from the disk into a l-gallon can placed in the main reservoir (Figure 7). A stOp watch was used to measure the time of flow and the amount of liquid was determined by weighing the can with a scale. By knowing the specific gravity of the liquid, the liquid flow (Q) was determined using the relationship Q = —————7°§§3 w spm where s = specific gravity of the liquid w = weight Of the liquid, lb. t = time of flow, see. A 6—inch diameter steel disk was machined and mounted inside a cylindrical tank (Figure 5). The disk was driven by a variable Speed (~3200 rpm to +3200 rpm) electric motor and a 0—2500 rpm tachometer was used to indicate the disk speed. All four legs of the tank and the electric motor were bolted to the concrete floor to reduce vibration. The thickness of the liquid layer on the disk was measured with an Ames dial indicator, having a range Of 0 to 1 inch, graduated in .001 inch increments and accurate to within 1.001 inch, as shown in Figure 6. Initially the Figure 7. Equipment used for flow measurement. 32 dial indicator was set so that a zero reading was obtained when the tip of the point gage touched the tOp surfaCe of the disk. Thus when the pointer was set to just touch the surface of the liquid, the reading on the dial indicator was a direct measurement of the thickness of the liquid flowing. The accuracy Of this method of measurement was estimated to be 2.002 inch. The dial indicator was mounted in a support (Figure 6) so that it could be rotated around in a plane parallel to the surface of the disk. Thus the zero reading, representing the surface Of the disk, remained the same for every position at which the thickness of the liquid was measured. 4.2 Selection Of a Viscous Liqgid The basic requirement of the viscous liquid used in the experimental work is that it be a Newtonian fluid, that is,its viscosity remains constant at any given temperature regardless of the rate of shear. Georgi (1955), Klaus and Fenske (1955), and Appeldoorn §t_al. (1962) have shown that straight mineral Oils behave like Newtonian fluids at temperatures above 32°F. Motor oils which contain appreciable amounts of viscosity index—improving additives were indicated to be non—Newtonian fluids throughout the temperature range to which Oils are normally exposed in engines. Because of their suitable prOperties, straight mineral oils were used in the experimental apparatus to verify the theoretical results. 33 Two oils were used for the tests: one was an S.A.E. 20 mineral oil (Everest—Cutler Oil Co.) with a typical viscosity of about 300 S.U.S. (Saybolt Universal Seconds) at 100°F and a viscosity index of 45—50; the other Oil was an industrial hydraulic oil (Industrial No. 75— American Oil Co.) with a typical viscosity of about 750 S.U.S. at 100°F and a viscosity index Of 95—100. Small quantities (20 to 30 parts per million) of silicone foam suppressor (Dow Corning 200 Fluid) were added to both oils to prevent foaming. 4.3 Experimental Procedure The surface of the disk was sprayed with black paint to provide a dark background for photographing the flow pattern of the liquid. Concentric circles spaced one—half inch apart were drawn on the disk surface to enable a visual determination of radial position on the disk. This facilitated both the method for measuring the thickness of the liquid and the method for studying the streamline pattern on the photographs since the radial positions were clearly marked on the disk and could be observed because the oils used were transparent. The streamline pattern on the disk was photographed with a single—lense reflex camera (Pentax H—2), with a focal plane shutter, mounted about one foot above the disk. A short extension tube was employed between the lense and the camera body to permit focusing of the lense at shorter 34 distances. Various shutter speeds were used and it was found that slower shutter speeds of 1/30 sec. or 1/60 sec. were more suitable for obtaining a continuous streamline pattern and the high shutter speeds of l/250 sec or 1/500 sec. were more suitable for studying the relative velocities of particles as they moved radially outwards. Since the viscosity of oil is affected by temperature it was necessary to control the operating temperature of the Oil during the tests. By Operating the experimental apparatus for a period of 2-3 hours it was observed that the temperature of the oil became relatively stable when it reached 95°F. Four 250—watt heat lamps were placed around the oil reservoir to heat the oil to 100°F. The temperature of the oil was then maintained at 100°F (i.2°F) by varying the position and the number of heat lamps as required. All the tests were made employing 6 gallons of Oil in the system. After the equipment had Operated for 2 hours with the Oil at the Operating temperature of 100°F, two samples of the Oil were taken for analysis. The specific gravity defined as the weight of a product in relation to the weight of an equal volume of water was determined by the following relationship weight of the oil weight of an equal volume of water specific gravity (s) = Georgi (1950) stated that capillary tube viscosimeters are considered as the most accurate means available for 35 viscosity measurement of lubricating oils. A Series 200 Modified Ostwald Viscosimeter was immersed in a constant temperature bath maintained at 100°F and used for the viscosity determinations. The Viscosimeter tube was first calibrated using distilled water as a reference fluid, then the tube was charged with an oil sample. To equalize the liquid temperature with the bath temperature, the liquid was allowed to remain in the tube for 10 minutes before the actual tests were made. The tests consisted of drawing up the liquid sample by suction until the lower bulb was filled and the upper bulb was partially filled. After removing the suction, the liquid was allowed to flow down the tube by gravity and the time is seconds was recorded for the oil level to pass between two etched marks. The kinematic viscosity (v) of the oil sample was determined by using the relatiopship ”H— W v = kinematic viscosity Of the Oil @lOOOF vw = kinematic viscosity of the distilled water @100°F t = time Of flow Of the Oil @100°F tw = time Of flow of the distilled water @100°F The prOperties of the Oils were determined after the Oil was subjected to considerable usage since a small amount Of air was entrained in the Oil causing the viscosity to 36 increase slightly and the specific gravity to decrease slightly when compared with the values obtained by employing unused samples. The required prOperties for the Oils are shown in Table 2. TABLE 2. Values of specific gravity and kinematic viscosity obtained from samples of the oils used in the laboratory tests. Kinematic Viscosity Type Of Oil Specific Gravity (s) (v)in2/sec @100°F S.A.E. 20 Mineral Oil .90 .087 Industrial Hydraulic Oil .84 .194 The final consideration in the experimental pro— cedure was the determination Of what distance should be maintained between the plexi—glass tube and the disk to allow the liquid to flow on to the disk. Preliminary tests indicated that varying this distance from .20 inches to .40 inches produced some variation in the streamline pattern and in the thickness of the liquid only in a region within one-inch from the outside of the tube (radius = 1.0 in) with no noticeable effects produced beyond this region of :flow. It was thus decided to maintain a distance of 0.300 ichhes between the lower taped inside edge of the plexi— gglass tube and the surface of the disk for all the laboratory tests. 37 It should be pointed out that it was impossible to duplicate the theoretical flow conditions in the experi— mental work. The theoretical solution was obtained for the flow of a semi—infinite mass of fluid which could only be expected to be approximately correct for the experimental work where the fluid was a layer of limited thickness. V. PRESENTATION AND DISCUSSION OF RESULTS A series Of eight tests were conducted to study the effect of flow rate, viscosity, and rotational speed on the flow of a viscous liquid on a rotating disk. The tests are out-lined in Table 3. TABLE 3. Tests conducted to study the flow of a viscous liquid on a rotating disk. Test Kinematic Viscosity Flow Rate Rotational Speed Number (0) inZ/sec. (Q) gpm (N) rpm 1 .194 10.2 1000 2 .194 10.2 2400 3 .194 5.0 1000 4 .194 5.0 2400 5 .087 10.2 1000 6 .087 10.2 2400 7 .087 .0 1000 8 .087 5.0 2400 5.1 Streamlines in the r—z Plane The experimental and theoretical streamlines in the r—z plane (depth of flow versus radial position relationships) are compared in Figures 8 to 11. As was expected, there 38 Depth of Flow, h (in.) 39 ‘ Tests 1 and 3 Theoretical Plexi-Glass —__—Streamline Tube —---Experimental / Streamline -.300 """"" I l \ Q=5.0 gpm Q=10.2 gpm H250 “\ ‘\ ‘\ ‘ \ ‘ \ 77200 \ \\ —.150 p.100 #050 I l l I ll J 0 0.50 1.00 1.50 2.00 2.50 3.00 Radial Distance, r (in.) Figure 8. Comparison of experimental and theoretical stream- lines in the r-z plane for hydraulic oil (v = .194 inZ/sec. flowing on a disk rotating at 1000 rpm. Depth of Flow, h (in.) 40 Plexi—glass Tests 2 and 4 Tube Theoretical Streamline —30 ------_ ‘ ——--Experimental “ Streamline H L. ll '25 I) Q=5.0 =10.2 I‘fspm spm |\ ”.20 5.15 —410 —u05 l l l l l l 0 0.50 1.00 1.50 2.00 2.50 3.00 Radial Distance, r (in.) Figure 9. Comparison of experimental and theoretical stream- lines in the r—z plane for hydraulic oil (v = .194 inZ/sec) flowing on a disk rotating at 2400 rpm. Depth Of Flow, h (in.) 41 Plexi—glass TEStS 5 and 7 Tube Theoretical Streamline _ ...... J 300 I vv——Experimental | Streamline I A Q=10.2 0:5.0 gpm o=10.2 gpm \ gpm ‘\ \\ “-.20 _ \\ 0 6.2-5.0.9” gpm \\ \ \ \ \ \ \ \ \ F'.150 \ \ \ \ \ \ \ \ ".100 _ .050 I I I I I I 0 0.50 1.00 1.50 2.00 2.50 3.00 Radial Distance, r (in.) Figure 10. Comparison of experimental and theoretical stream— lines in the r—z plane for mineral Oil (0 = .087 inZ/sec) flowing on a disk rotating at 1000 rpm. Depth of Liquid, h (in.) 42 Tests 6 and 8 Plexi—glass Theoretical Tube Streamline e——-Experimental J Streamline —.300 """" \ \I I ‘.250 o\ ,5.0 gpm /10.2 gpm \\ \\ \\ \\ ".200 \\ \\ \ \ \ . \ \ ._.150 \ _.100 —.050 I I I I I I 0 0.50 1.00 1.50 2.00 2.50 3.00 Radial Distance, r (in) Figure 11. Comparison of experimental and theoretical stream- lines in the r—z plane for mineral Oil (v = .087,in2/sec) flowing on a disk rotating at 2400 rpm. 43 was considerable deviation between the eXperimental and theoretical curves in the area where the oil was initially introduced onto the disk. This was due to the face that in the theoretical solution it was assumed the disk rotated in a body of liquid where there is essentially no point of introduction Of the liquid onto the disk. Theoretically however, it is possible as earlier shown to determine a streamline representing a certain flew quantity, assuming constant rotational speed and liquid viscosity. It is inter- esting to note that theoretlcally for the velocities and viscosities used in this investigation, a rotating disk has little or no effect on the movement of liquid which is located 0.20 inches or more above the surface of the disk. This is due to the fact that the radial and tangential components Of velocity become negligible in this region as shown in Appendix A. The experimental curves in the area where the 011 was introduced onto the disk could have been made to correspond more closely to the theoretical curves by in— creasing the diameter Of the plexi—glass tube, used above the disk, until the inside surface of the tube coincided with the theoretical curve. Increasing the diameter of the tube has the other advantage of reducing the axial velocity of the liquld so as to agree more closely with the theoretical values. In the experimental tests conducted, the axial velocity of the Oil at the outlet was usually consider— ably higher than the theoretical values as shown in Table 4. .A: cesaoo .m oanmev msfipms who on o>fipmaos coapoosfip soap oomOsSm on» no unmoom on» an Am casaoo .m mapmev CH m u a pm mpfiooao> Amanda Hmpcosflsodxo owmso>m who wcHszapass an pentagoamo mmz mpfiooam> mficea 44 H.: o.m mwa m oozm ©.m o.w Hma w oooH o.m ewe. H.: :.oa oam w oozm m.oa m.m :.ma omH m oooa H.m o.m sea : oosm o.m 0.: o.w sea m oooa :ma. H.m :.wH mam m oozm m.oa 0.: z.ma mom H oooa osam> osam> mdfi xmeQ oz eds sz saw AGV .oom\~cH Apv awoaposoone HmpcoEHponm mo owpm um *mpfio pmoB poodm opmm zoam mpfimoomfi> Ioao> unmpHSmom HmCOHpmpom oaumsocfim mdfi sz uoapso Hapcoefipodxm 6939 he spaooao> Hedge .xmfip who mo omen on» on azpaooao> uQmOHSmon Hmpcoeflpodxo on» spa: .Xmfip on» mo oommLSm on» o>onm monocfl oom. popmooa .uoauso was» man um moHuHooao> Hmaxm Hmofipopoonu new amazosfisodxm mo comapmasoo .: mqm¢e 45 An increase in the axial velocity of the liquid would tend to increase the resultant velocity Of the Liquid on the disk. However, since the magnitude of the axial velocity is relatively small compared to the resultant velocity, as shown in Table 4, the difference in the initial kinetic energy may be assumed to have relatively small influence on the final result. There was good agreement between the experimental and theoretical streamlines at radial distances Of two inches or more when the rotational speed of the disk was 1000 rpm (Figures 8 and 10). At the higher rotational speed Of 2400 rpm (Figures 9 and 11) the experimental values for the thickness of the liquid were higher than the theoretical values at radial distances of two inches or more. In this flow region it is interesting to note that at the high rotational speed (2400 rpm) the experimental values of depth Of flow are affected more by changes in oil viscosity than.by changes in the flow rate. In all cases the real iJlfluence of flow rate is less than predicted by the theore— tical solution. The experimental results also indicated tfllat the boundary layer thickness increased slightly at the (edge of the disk at the higher rotational speed rather than cheerease as shown by the theoretical values. The variation txetween the experimental and theoretical curves Obtained at trma higher rotational speed (2400 rpm) can be at least par— tiially explained by considering boundary—layer stability. 46 The criteria used to determine the stability of flow is obtained by considering the ratio of inertia forces to friction forces as expressed by the Reynolds number Re. For the case of flow on a rotating disk the value of Re is determined by using the relationship 2 , Re - ”v“ (5.1) where Re Reynolds number r = radial distance w = angular velocity of the disk v = kinematic viscosity of the fluid Schlichting (1960) when considering the torque on a disk rotating in a body of fluid, stated there was good agreement between the theoretical values for laminar flow and the experimental values of torque for Reynolds numbers up to about Re - 3x105. At higher Reynolds numbers the flow was definitely turbulent. The curves plotted by Schlichting show that the onset of instability was present at measured Reynolds numbers (rzw/v) as low as 10“. The values of Reynolds numbers present in the experimental tests are summarized in Table 5. Gregory, Stuart and Walker (1955) discussed boundary- layer instability from both the theoretical and experi- mental points of view for a disk rotating in a compressible ' ‘. I] 47 TABLE 5. Values Of Reynolds number present in the labora— tory tests. Kinematic Rotational - . V \ r D All .\ 3 R Viscosity Speed (N) Test ILVJOld” ”dmbtr ( e) (V) inz/sec. rpm No. r=l.0 in r=2.0 in r=3.0 in 1000 l or .055x10‘+ .22x101+ .49xlO“ 3 .194 2400 2 or .13 x10“ .52x10“ 1.17x10“ 4 1000 5 or .12 x10“ .48x10“ 1.08xlo‘+ 7 .087 2400 6 or .29 x10“ 1.16x10“ 2.60x10” 48 fluid (air). They conducted experiments with a l2vinch diameter disk having one side coated with china clay for indicating the transition from laminar to turbulent flow. The boundary-layer velocity profiles were measured and compared with the theoretical laminar profiles. The shapes Of some of the curves obtained by Gregory and Walker are shown in Figures 12 and 13. The curves showed that for laminar flow the tangential velocity component was in good agreement with theory, while the radial component had a peak value which is somewhat lower than the theoretical. The experimental profiles (Figures 12 and 13) showed that for turbulent flow both the radial and tangential velocity components had smaller peak values which did not decrease as rapidly as the theoretical values with increasing distance from the disk surface. Gregory gt_gl. stated that on a one-foot diameter disk rotating in air the onset of boundary layer instability occured at Reynolds numbers of 1.8—2.1 x 105 and the highest value of Reynolds number at transition, 2.99 x 105, was obtained when the air in the room was at its stillest. The results of this work carried out in an aerodynamic laboratory can be applied to the jpresent investigation on the flow of a viscous liquid (oil) since similar theoretical considerations were made for .investigating the boundary layer on the rotating disk. .Also it is known that two systems are dynamically similar :if their Reynolds numbers are similar. WWW-m m .fi O i I l l I I l I J .01 .02 .03 .04 .05 .06 .07 .08 .09 .10 Distance Normal to Disk Surface, 2 (in.) Figure 12. Comparison of experimental and theoretical radial velocity profiles on a disk rotating at 2100 rpm in air (obtained by Gregory et a1. 1955). Curve 1, theoretical laminar profile; 2, 3, 4 experimental profiles; 2, laminar region; 3, instability region; 4, turbulent region. 1'1” ‘fi: . iii“... .08 .09 .10 Distance Normal to Disk Surface, 2 (in) Figure 13. Comparison of experimental and theoretical tangential velocity profiles 2100 rpm in air (obtained by on a disk rotating at Gregory et al. 1955). Curve 1, theoretical laminar profile; 2, 3, 4, experi- mental profiles: 2, laminar region; 4, turbulent region. region; 3, instability 51 Thus the variations between the experimental and theoretical curves (Figures 9 and 11) at the higher rotational speed were due to the differences in the shapes of the velocity profiles. At radial distances of two inches or more the experimental values for the thickness of the boundary layer were higher than the theoretical values indicating the actual radial velocity was lower than the theoretical value. This statement is in agreement with the Velocity distribution curves (Figure 12) Obtained by Gregory gt_§l. The values of Reynolds numbers (Table 5) present in the laboratory tests indicate there was limited turbulent flow present at the higher rotational speed which may have reduced the magnitude of the radial velocity. It is difficult, however, to explain why the thickness of the liquid, measured at the edge of the disk rotating at 2400 rpm was affected by kinematic viscosity but not by flow rate. In the theoretical consideration it was assumed the disk was an infinite rotating plane. The results were extended to include a disk of finite diameter by neglecting the edge effect. The significance of neglecting the edge effect is not known but it is normally assumed that if the boundary layer thickness is small compared to the radius of the disk then the edge effect can be neglected in a semi-infinite fluid. Since there is a finite quantity of liquid present in our case we must also consider the layer thickness. ‘4' huM'L‘EWH l ‘I 4.0.0 . "ofl'r-u'n ' 52 5.2 Streamlines in the r—e Plane The streamlines in the r-e plane, that is, the path followed by a particle moving on the disk as seen in the direction of the axis, was calculated for two positions in the liquid (Appendix B): the surface streamline was calcu« lated first by using the magnitudes of the velocity compon— ents present at the free surface of the liquid and secondly, for comparison, the streamline pattern was calculated using the average values of the radial and tangential velocity components at each radial position considered. A study of the velocity profiles indicated the "average streamline" corresponds approximately to a streamline at two—thirds of the layer height above the disk. The theoretical and experimental streamlines in the r-e plane are compared in Figures 14 to 21. All the curves are drawn through a common point located two inches from the center of the disk. The experimental curves were approximated by taking measurements from the photographs 5 taken in the laboratory. These experimental curves were i assumed to represent the flow on the free surface of the ‘Il-venrozmrv ‘3 liquid. Comparison of the experimental and theoretical stream— line patterns (Figures 14—21) existing on the surface of the liquid, indicates good agreement especially for cases of high flow (10.2 gpm) and low rotational speeds (1000 rpm). In every case the experimental curve falls between the two 53 STREAMUNES(TEST NOJ) \ ----— Experimental (surface) Theoretical (surface) —-— Theoretical (average) Figure 14. Comparison of experimental and theoretical stream- lines in the r-e lane for 10.2 gpn flow of hydraulic oil (0 = .194 inZ/sec§ onto a disk rotating at 1000 rpm (clockwise) (MSU Photo NO. 65772—2). 54 STREAMUNES (TEST NO. 2) ------ Experimental (surface) Theoreticd (surface) —-— Theoretical (average) \x Figure 15. Comparison of experimental and theoretical streamlines in the r—e plane for 10.2 gpm flow of hydraulic Oil (0 = .194 inZ/sec.) onto a disk rotating at 2400 rpm (clockwise). 55 /I STREAMLINES (TEST N0. 3) ————— Experimental (surface) 'Theorefical (surface) ----—— Theorefical (average) Comparison streamlines in the r—O .194 inZ/sec) ‘\ / I \ w/ I , / "xii ' 7’ ! I -J,/ (clockwise). Of experimental and theoretical plane for 5.0 gpm flow Of hydraulic onto a disk rotating at 1000 rpm 56 STREAMLINES (TEST NO. 4) ----- Expernnenfal(surface) Theorefbal (surface) ——-—Thecrelical (average) Figure 17. Comparison of experimental and theoretical streamlines in the r—e plane for 5.0 gpm flow of hydraulic oil (u = .194 inz/sec) onto a disk rotating at 2400 rpm (clockwise) (MSU Photo NO. 65772—1). 57 STREAMLINES (TEST NO. 5) ----- Expernnenfal(eurface) \‘\ -——Theoretical (surface) — - —Theorefical (average) Figure 18. Comparison of experimental and theoretical stream— lines in the r-e plane for 10.2 gpm flow of mineral 011 v = .0 7 inZ/sec) onto a disk rotating at 1000 rpm (clockwise) (MSU Photo NO. 65772-5). 58 I \ STREAMLINES (TEST no, 5) \‘f‘ .__ ------ Expernnental (surface) 5\g~ . Theorefical (surface) '\ — — — Theoretical (average) \ \ \ \- / I \p , f I \7 \, I,I ,7 I \ a y ‘1 ‘\ I ,(I I \'\ ’ / ,7" _ / [I I, ’* I I I I Figure 19. Comparison of experimental and theoretical stream— lines in the r—e plane for 10.2 flow Of mineral Oil (0 = .087 inZ/sec) onto a disk rotating at 2400 rpm (clockwise) (MSU Photo NO. 65772—6). 59 STREAMLINES (TEST NO. 7) ----- Experimental (surface) Theoretical (surface) —-—Theorefica| (average) Figure 20. Comparison of experimental and theoretical streamlines in the r—e plane for 5.0 gpm flow of mineral oil (0 = .087 inZ/sec) onto a disk rotating at 1000 rpm (clockwise) (MSU Photo NO. 65772-7). 6O ‘\~ / / /’ ‘ \ \ \ \ STREAMLWES (TEST No.8) u ----- Exper'menfal (surface ) ——Theorefical (surface) —-—Theorefical (average) \ I / s / I ‘ _ a? ’ ’/ J - r ’ Figure 21. Comparison of experimental and theoretical stream— lane for 5.0 gpm flow of mineral oil lines in the r—e (v = .087 inZ/sec§ onto a disk rotating at 2&00 rpm (clockwise) (MSU Photo No. 65772—8). 61 theoretical streamline curves: one for surface velocity components and the other for average velocity components at each radial position. The deviations between the experimental and theoretical streamline curves can be explained by a discussion similar to the one given earlier in eXplaining the variations in the thickness profiles on the disk. Since the radial velocity profile on the disk has a peak value which was smaller than the theoretical value (see Figure 12), the experimental surface streamline fell inside the theoretical curve on the disk. The. photographs taken at tie higher rotational speeds (2UOO rpm) did illustrate some transition in the boundary layer near the edge of the disk. Gregory and Walker (1955) took a photograph showing the process of transition on a disk rotating in air. This photograph showed that in an annular region near the edge of the disk there were stationary vortices which assumed the shape of logarithmic spirals. The inner radius of this region marked the onset of instability (Re = 1.9x105) and the transition to turbulent flow occurred at an outer radius corresponding to a Reynolds number of 2.8xlOS. The values of the Reynolds numbers present in our laboratory tests are shown in Table 5. _The maximum value of Reynolds number was 2.6x105 which seemed to indicate that in the present investigation the limit for “stability occurredat smaller Reynolds numbers than those indicated by Gregory et al. (1955). It is likely that there 62 was considerable turbulence in the oil as it flowed to the disk through the screens. The experimental and theoretical surface streamlines are compared in Table 6 by using two descriptive para— meters: one was the angular displacement of the particle when moving from a radius of two inches to a radius of three inches (the edge of the disk); the other parameter was the angle relative to the radius with which a particle leaves the disk. This comparison of the streamlines indicated, in every test, that the experimental values of both the angular displacements and the angles relative to the radius were larger than indicated by the theory. A final evaluation of the theory was made by comparing the experimental and theoretical average radial and average tangential velocities at the edge of the disk as shown in Table 7. This comparison of the experimental and theoretical velocities indicates relatively good agreement at low rotational speeds (lOOO rpm—Tests l, 3, 5, and 7) but at the high rotational Speeds (1 3“ rpm—Tests, 2, u, 6, and 8) the theoretical values of velocity were considerably larger than the experimental values. 5.3 Flow on Selected Areas of the Rotating Disk ‘ It was mentioned earlier that in certain applications it is desirable to limit the area of liquid flow from a 63 we as mm mm mm me me mm om me as mm mm om as on me mm m: mm mm 3 mm 5: am mm mm H: woa no me mm HNMZ’LOKONCI) osam> HmOHuopoona xmao mo>moH oaofiphmq coax: spa: msaomp map on o>HumHom oawc< Amoopwopv adam> Hmpcoeflsoqu msHm> Hecapopoone .GH mus on CH mus Eosw mo>oE osam> HmeoEHLoQNm .oz Amoopwoov oeoe macapumd can: pcmEmomHQmHQ mmazwc< .xmao on» mo owed on» pm..msfiomp on» on o>HpmHom .moawcm cam mpCoEoomHamHU smaswcm wcamd mmcHHEmopum oommndm Hmoauopoozp 6cm Heucmefipmdxo one we COmHLMQEoo .m mqm mzam> msam> msam> Amdavan.Cfi .oz Hoofipopooge Hmpcmefipodxm HmOHposoone Hmpcoefiuodxm mun um xmfio pmoe mo oommpsm Amafiv xmfic mo mwoo pm pm mpfiooam> mufiooam> Hmfiomg owmpo>< Amafiv smeo so owoo,oo mpfiooao> Hmaucowcmp owmno>¢ .meU on» we owoo one am mofipaooao> Hmfipcowcmp mwmpo>m use Hmfipms owmpo>m Hmoapoaomsp use Hmpcoefipooxo map mo comHanEoo .5 mqm<9 65 rotating disk. An attempt was made to produce a spray pattern similar to the one produced by a fan-type spray nozzle by controlling the introduction of the liquid on to the disk. To produce this form of spray pattern, nozzles with special-shaped Openings were constructed to introduce only a sector of the flow normally present with axially— symmetric liquid introduction on to the disk. The nozzle openings were designed on the basis of the shape of the streamlines existing in the r-e plane such that the liquid particles at various heights along an edge Of the opening would travel out and meet at a common point on the periphery. The horizontal Opening between the edges Of the nozzle controlled the width Of liquid flow from the disk. Two types of nozzles (Figures 22) were constructed using 3—inch (3.00-inch outside diameter and 2.U5-inch inside diameter) plexi-glass tubing: one nozzle had a F closed bottom which permitted liquid to flow out through an opening in the side, the other nozzle had an Open bottom which permitted liquid to flow against the disk within the nozzle. Both nozzles had parallelogram—shaped g Openings (Figure 23) designed on the basis of the theoreti- cal streamlines in the r—e plane from r=l.5 in (outside of the nozzle) to r=3 in (edge Of the disk) as shown in Figure 23. The lepeCKOf the edges Of the Openings was determined by measuring the angular distance between the surface 66 Figure 22. ClOsed—bottom nozzle (left) and Open—bottom nozzle (right) with parallelogram shaped openings. ( i L————-—-—-——- A if —_I Figure 23. Diagram of a parallelogram—shaped nozzle Opening. 67 streamline (AB) and the average streamline (AC) at r=l.5 in. when drawn from a common point A (Figure 23) at r=3.0 in. and using the relationship Slope (4) = tan"1 r191 = tan"1 1'561 h h where 61 = angular displacement between the surface stream- line and average streamline at r1 r1 = radial position Of the outside of the nozzle h = depth Of the liquid Calculations based on the methods presented earlier (section 3.2) showed that the "average streamline" corresponded to the particle movement at from approximately onevthird to two-thirds of the layer height above the disk. Thus in the present design considerations the streamlines corresponding to particle movement at less than one—third of the layer depth were not considered. The amount Of radial flow in this bottom layer is, however, small. A liquid depth of .20 inches was selected for the nozzle design. The horizontal distance between the sloping edges of the nozzle Opening corresponded to an angular displace— ment Of approximately 75 degrees, representing a typical Spray width for a hydraulic spray nozzle. By varying the flow rate, a 7—inch head of hydraulic Oil (v = l94 inZ/sec.) was maintained above the nozzle Opening in all the tests. 68 The flow patterns from the special nozzles are shown in Figures 2A to 27. A rotational speed Of 1000 rpm was employed for these tests since it was Observed that increasing the speed to 2000 rpm had little noticeable effect on the flow pattern. The resulting flow patterns indicate the presence Of high tangential velocity and low radial velocity of the liquid closest to the disk. In Figure 27 a large air bubble is clearly visible along the outside of the nozzle which shows that the calculated angular displacement of the layers at the inlet to the disk was as desired. After measuring both the depth Of the Oil and the resultant flow direction and by assuming a uniform velocity profile in the liquid at the edge of the disk the prOportion of Oil leaving the disk at various areas was determined. On the basis Of these results it was Observed that in the area Of highest concentration approximately two-thirds Of the liquid was leaving the disk in oneefifth of the circum- ference. It is felt that with further work and the addition Of shrouds to prevent the undesirable flow, the disk may be used to produce an acceptable fan—type spray pattern. 69 Figure 24. Flow pattern from the closed—bottom nozzle (N = 1000 rpm, Q = A.A gpm,v = .19“ inZ/sec.). Figure 25. Streamlines in the r—e lane for the closed— bottom nozzle (N = 1000 rpm, Q = .U gpm v = .19“ inz/ sec.). 7O Figure 26. Flow pattern from the Open-bottom nozzle (N = 1000 rpm, Q = 5.4 gpm,v = .19A in2 /sec) (MSU Photo No. 65772—15). Figure 27. Streamlines in the r—Bplane for the Open- bottom nozzle (N - 1000 rpm, Q = 5.4 gpm,\)= .194 in2 /sec) (MSU Photo No. 65772-16). VI. SUMMARY AND CONCLUSIONS 6.1 Summary The principle Of centrifugal distribution or atomization Of liquids has many inherent advantages. An investigation was conducted to obtain a more thorough understanding Of the factors affecting the flow Of a liquid on a rotating disk and to determine the feasibility Of using theoretical considerations to describe the fluid motion. In this investigation a solution to the Navier- Stokes equations, for the case of flow around a disk rotating in a fluid at rest, was applied to the present flow situation. The flow on the disk was characterized by means of streamlines in both the r-e plane and the r-z plane. In order to Obtain the thickness profile for a specified flow condition, the areas under the velocity profiles were found with a planimeter. A first order polynomial approximation for the ratio of tangential velocity to radial velocity as a function Of radius was integrated and used to obtain an equation for the path traced by a particle moving on the disk. An experimental apparatus consisting of a 6-inch (15.2 cm) diameter steel disk powered by a variable-speed 71 72 electric motor was used to verify the theoretical results. Two mineral Oils were selected for the tests because Of their Newtonian behavior: one had a kinematic viscosity Of .087 inZ/sec (56.1 centistokes) @100°F and the other Oil had a kinematic viscosity of .19A inZ/sec (125.0 centistokes) @100°F. The Oil temperature was maintained .".'7\l" at 100°F throughout all the tests by controlling the Operation of heat lamps placed around the main Oil reservoir. The Oil was introduced axially symmetric on to the disk through a 2-inch (5.1 cm) outside diameter plexi—glass tube designed to reduce the axial velocity component Of the Oil. A distance of .300 inches (.75 cm) was maintained between the disk surface and the lower edge of the plexi-glass tube for all the tests. Rotational speeds Of 1000 rpm and 2400 rpm and flow rates Of 5.0 gpm (.32 liter/sec.) and 10.2 gpm (.6A liters/sec.) were used in the tests. A series of eight tests were conducted to study the effect of rotational speed, flow rate and fluid viscosity on the flow of the liquid on the disk. The thickness of the Oil layer on the disk was measured with a point gage and the surface streamline patterns in the r—e plane were Obtained from a photograph. The theoretical and experimental curves were compared and the deviations between some Of the curves were explained. Preliminary experiments were made injecting liquid through special nozzles on a section of the disk. 73 6.2 Conclusions As a result of this study, the following conclusions are presented: 1. The Navier-Stokes flow equation can be success~ fully used to predict the laminar flow Of a Newtonian fluid on a rotating disk. The actual deviations from the theoretical solution are discussed below in point 6. Increasing the kinematic viscosity of the liquid increased the boundary layer thickness of the liquid on the disk and also increased the angular displacement Of the liquid on the disk which resulted in a higher resultant average velocity Of the liquid leaving the edge Of the disk, assuming constant flow rate and rotational speed. _ Increasing the flow rate of the liquid onto the ‘ disk increased the boundary layer thickness Of the liquid on the disk and also decreased the angular displacement Of the liquid on the disk which resulted in a lower resultant average velocity Of the liquid leaving the edge of the disk, assuming constant liquid viscosity and rotational speed. 74 Increasing the rotational speed of the disk decreased the boundary layer thickness of the fluid on the disk and also increased the angular displacement Of the liquid on the disk which resulted in a higher resultant average velocity Of the liquid leaving the edge of the disk, assuming constant liquid viscosity and flow rate. The angle relative to the radius with which a liquid particle leaves the disk increases with increased angular displacement of the liquid, increased disk radius and decreased height above the disk surface. This angle has a maximum value Of 90 degrees on the surface of the disk implying that a particle at this height has only tangential velocity. As the distance above the disk increases the flow becomes more in a radial direction. When compared with the theoretical results, the § experimental results indicated that the flow rate had less effect than theoretically expected, ‘I‘Vn.’ It- I while the viscosity and the rotational speed had the expected effect on the streamlines used to characterize the liquid flow on'a rotating disk. Comparison of the surface streamlines in the r—e plane indicated that in every test the 75 experimental values of both the angular displace— ments and the angles relative to the radius were slightly larger than indicated by the theory. When the liquid was introduced axially symmetrical onto the rotating disk, the liquid was uniformly distributed about the entire periphery of the disk. By introducing the liquid through a special- shaped nozzle on a section of the disk, approxi- mately two—thirds of the liquid left the disk on one-fifth of the circumference of the disk. \‘fwxmrnw L‘s. SUGGESTION FOR FURTHER STUDY Further investigations should be conducted to Obtain a better understanding in the following areas: 1. Edge Effect.——Tests should be conducted with F- disks having various diameters so that the edge 2 effect Of the disk can be determined. 5 2. Velocity Profiles.——Instrumentation should be i employed tO enable measurement of the velocity ' components at different positions in the liquid. 3. Liquid Introduction.-—The diameter Of the tube used tO introduce the liquid onto the disk should be varied to satisfy the conditions dictated by the theoretical solutions. A. Partial Flow.--More extensive laboratory tests should be conducted with special shaped nozzles and auxiliary devices should be added to provide better control of the liquid flow. 76 REFERENCES Appeldoorn, J. K., Okrent, E. H., and Philippoff, W. 1962 Viscosity and elasticity at high pressures and high shear rates. American Petroleum Institute Proceedings, Section III Vol. 42:163—172. Bainer R., Kepner, R. A., and Barger, E. L. 1955 Principles of Farm Machinery. John Wiley and Sons Inc., New York 571 pp. BOdewadt, U. T. 19A0 Die DrehstrOmung uber festem Grunde. Zeitschrift Fur Angewandte Mathematik Und Meckanik. Vol. 20: 2N1-253. Cochran, W. G. 193A The flow due to a rotating disc. Cambridge PhilOSOphical Society Proceedings. 30:365-375. Crowther, A. J. 1958 The distribution Of particles by a spinning disc. Journal Of Agricultural Engineering Research 3 (A):288-29l. Fettis, H. E. 1955 On the integration Of a class Of differential equations occuring in boundary layer and other hydrodynamic problems. Proceeding Of 4th Mid- western Conference On Fluid Mechanics : 93-llu Georgi, C. W. 1950 Motor Oils and Engine Lubrication, Reinhold Publishing Corporation, New York: 514 pp. Georgi, C. W. 1955 Viscosity characteristics of motor 0115 at higher shear rates. Proceedings Fourth World Petroleum Congress, Rome. Section VI/C: 211—221. Gregory, N., Stuart, J. T., and Walker, W. S. 1955 On the stability Of three-dimensional boundary layers with application to the flow due to a rotating disk. Royal Society Of London PhiIOSOphical Transactions A248zlSS-199. 77 78 Gunkel, W. W. 1957 Deposit of mist concentrate spray as influenced by drOplet size, air velocity, temperature and humidity. Thesis for the degree Ph.D, Michigan State University, East Lansing. (unpublished) Hannah, D. M. 1947 Forced flow against a rotating disc. Great Britain Aeronautical Research Council Reports and Memoranda NO 2772:1—17. Homann, Von F. H 1936 Der Einfluss grosser zahigkeit bei der Stromung um den Zylinder und um die Kugel. Zeitschrift Fur Angewandte Mathematik Und Mechanik. Vol. 16, NO. 3 : 153—164. Houghton, H. G. 1950 Spray Nozzle. pp 1170-1175 Chemical Engineers Handbook, third edition. John H. Perry, editor. McGraw-Hill Book CO. New York, pp. 1170—1175. Inns, F. M., and Reece, A. R. 1962 The theory of the centrifugal distributer (II), Journal Of Agricultural Engineering Research 7(4) = 345-353- Karman, Th."von 1921 Uber laminare und tubulente Reibung. Zeitschrift Fur Angewandte Mathematik Und Mechanik. Vol. 1, NO. 4 : 233—252. Karman, Th. and Lin, c. c. l96l On the existence Of an exact solution of the equations Of Navier-Stokes. Communications on Pure and Applied Mathematics 14 : 645-655. Klaus, E. E. and Fenske, M. R. 1955 Some viscosity-shear characteristics of lubricants. Lubrication Engineering, March-April : 101—108. Lance, G. N., and Rogers, M. H. 1962 The axially symmetric flow of a viscous fluid between two infinite rotating disks. Royal Society Proceedings 266 : lO9~l2l. Leigh, D. C. 1955 The laminar boundary—layer equation: a method of solution by means Of an automatic computer. Cambridge PhilOSOphical Society Proceedings 51; 320-332. 79 Patterson, D. E. and Reece, A. R. 1962 The theory of the centrifugal distributor (I). Journal Of Agricultural Engineering Research 7(3) : 232—240. Schlichting, Von H. and Truckenbrodt, E. 1952 Die StrOmung an einer angeblasenen rotierenden Schiebe.Zeitschrift Fur Angewandte Mathematik Und Mechanik Vol. 32 : 97-111. Schlichting, Von H. 1960 Boundary Layer Theory. McGraw-Hill, Inc., New York. 647 pp. Stewartson, K. 1952 The flow between two rotating coaxial disks. Cambridge PhilOSOphical Society Proceedings. 49 333—341. Stuart, J. T. 1954 On the effects Of uniform suction on the steady flow due to a rotating disk. Quarterly Journal Mechanics and Applied Mathematics 7:446-457 Truckenbrodt, E. n 1954 Die turbulente Stromung an einer angeblasenen rotierender Scheibe. Zeitschrift Fur Angewandte Mathematik Und Mechanik Vol. 34 : 150-162. Rogers, M. H. and Lance, G. N. 1960 The rotationally symmetric flow Of a viscous fluid in the presence of an infinite rotating disk. Journal Of Fluid Mechanics 7 : 617-631. T Tifford, A. N. and Chu, Sheng TO 1952 On the flow around a rotating disc in a uniform stream. Journal Aeron. Sciences Readers Forum 19 : 284. Wu, J. C. , 1961 On the finite difference solution of laminar ( E boundary layer problems. Proceedings Of Heat Transfer and Fluid Mechanics Institute, Stanford, California : 55-69. Yates, W. E. 1951 An analysis Of atomization by the rotating disk for controlled drOplet size. Unpublished thesis, University of California. APPENDIX A ‘ '1‘-- 80 81 H fi‘h‘m . wih‘fl . ,.. u; ”3v. m0000.0 3H00.0 030m.5 HOH0.0 03H0.0 0030.3 0m. 0000.0 0H00.0 0030.0 0mm0.0 55H0.0 00H3.3 0H. 0H00.0 5m00.0 0m03.0 m0m0.0 5Hm0.0 mm0H.3 0H. 3000.0 0m00.0 3mma.0 H0mo.0 30m0.0 0000.0 5H. 0000.0 3000.0 mm05.0 0m30.0 mmm0.0 30H5.m 0H. 0000.0 3500.0 0003.0 0000.0 0000.0 0003.m 0H. HNH0.0 m0H0.0 0m30.0 m000.0 H530.0 0m0m.m 3H. 35H0.0 03H0.0 0m00.3 0H00.0 0000.0 mHm0.m ma. 53m0.0 m0H0.0 3mmm.3 000H.0 0500.0 0005.m ma. 5300.0 m0mo.0 mm00.m H3ma.0 0H00.0 3000.m Ha. 3030.0 0000.0 0m00.m 000H.0 0000.0 03mm.m 0H. H500.0 0530.0 0H3m.m H50H.0 0HHH.0 0H00.m 00. 0m00.0 mm00.0 0H00.m 30mm.0 00mH.0 N000.H 00. H0ma.0 Hm00.0 3Hm0.m 3H0m.0 503H.0 00m0.a 50. 005H.0 000H.0 mHOH.m 3m3m.0 0m0H.0 330m.H 00. 0H3m.0 3mma.0 0H00.H m0H3.0 m05a.0 000H.H 00. m0mm.0 000H.0 0033.H 0000.0 000H.0 00m0.0 30. 0033.0 005H.0 0000.H 0000.0 0m5H.0 m500.0 mo. 0500.0 m05H.0 30m5.0 0005.0 053H.0 0303.0 m0. 3005.0 m0mH.0 m00m.0 0000.0 0300.0 3mmm.0 H0. 0000.0 H550.0 HO0H.0 5000.0 5000.0 0HH.0 000. 000.H 0.0 0.0 000.H 00.0 0.0 00. AHMHpComcmpV AHMHUMQV nm0.0mn Aamfipcmwcmpv Aamfivmhv N3m.mmn AQHV 0 m u 0 m u N eds ooem u z and OOOH u z .EQL 003m 0cm Eda 000a 3000 pm wcflpmpoa xmac a co wcfizoam A.oom\NCa 30H. u >0 HHO ofiasmaomn LOQ mucoCOQEoo mpfiooao> A00 Hmfiucowcmp 0cm Amv Heaven mmoHcoamcoEHQ .H< mam¢9 82 TABLE A2. Radial velocity components for hydraulic Oil (v = .194 inz/sec.) flowing on a disk rotating at 1000 rpm. Radial Velocity (ips) u=er in r=l.5 in r=2.0 in r=2.5 in r=3.0 in A H N :5 V '1 II H O 0.0 0.0 0.0 0.0 0.0 0.0 0.005 5.52 8.28 11.04 13.79 16.56 0.01 9.84 14.76 19.68 24.60 29.53 0.02 15.49 23.23 30.97 38.71 46.46 0.03 18.20 27.30 36.39 45.59 54.60 0.04 18.90 28.36 37.80 47.25 56.71 0.05 18.34 27.52 36.69 45.86 55.04 0.06 17.02 25.54 34.04 42.56 51.08 0.07 15.36 23.04 30.72 38.40 46.09 0.08 13.51 20.26 27.01 33.77 40.53 0.09 11.71 17.56 23.41 29.26 35.12 0.10 10.04 15.06 20.08 25.10 30.13 0.11 8.48 12.72 16.96 21.20 25.45 0.13 5.95 8.92 11.89 14.87 17.84 0.14 4.93 7.40 9.86 12.33 14.80 0.15 4.08 6.12 8.17 10.21 12.25 0.16 3.37 5.06 6-74 8.43 10.12 0.17 2.76 4.15 5.53 6.91 8.29 0.18 2.27 3.41 4.54 5.68 6.82 0.19 1.85 2.78 3.71 4.63 5.56 0.20 1.52 2.28 3.04 3.80 4.56 83 TABLE A3. Tangential velocity components for hydraulic Oil (V = .194 inZ/sec.) flowing on a disk rotating at 1000 rpm. z Tangential Velocity (ips) v=rmG (in) r=l.0 in r=l.5 in r=2.0 in r=2.5 in r=3.0 in 0.0 104.72 157.08 209.44 261.80 314.16 0.005 97.25 145.88 194.51 243.13 291.76 0.01 89.91 134.87 179.83 224.78 269.74 0.02 76.08 114.12 152.16 190.20 228.24 0.03 63.67 95.50 127.34 159.17 191.01 0.04 52.88 79.33 105.77 132.21 158.65 0.05 43.59 65.39 87.19 108.99 130.78 0.06 35.86 53.78 71.71 89.64 107.57 0.07 29.47 44.20 58.94 73.67 88.40 0.08 24.02 36.03 48.05 60.06 72.07 0.09 19.59 29.39 39.19 48.98 58.78 0.10 16.02 24.03 32.04 40.06 48.07 0.11 13.00 19.49 25.99 32.49 38.99 0.12 10.57 15.85 21.13 26.42 31.70 0.13 8.58 12.86 17.15 21.44 25.73 0.14 6.93 10.40 13.86 17.33 20.80 0.15 5.63 8.45 11.27 14.08 16.90 0.16 4.56 6.83 9.11 11.39 13.67 0.17 3.68 5.51 7.35 9.19 11.03 0.18 2.95 4.43 5.91 7.38 8.86 0.19 2.37 3.55 4.73 5.92 7.10 0.20 1.90 2.84 3.79 4.74 5.69 TABLE A4. .194 inZ/sec.) flowing on a disk rotating at 2400 (v Radial velocity components for hydraulic oil 84 rpm. Radial Velocity (ips) u=er (in) r=l.0 in r=l.5 in r=2.0 in r=2.5 in r=3.0 in 0.0 0.0 0.0 0.0 0.0 0.00 0.005 19.38 29.07 38.76 48.44 57.90 0.01 32.22 48.33 64.44 80.55 96.28 0.02 44.03 66.05 88.07 110.08 131.57 0.03 44.73 67.10 89.47 111.84 133.68 0.04 40.11 60.17 80.22 100.28 119.86 0.05 33.52 50.29 67.05 83.83 100.18 0.06 26.86 40.30 53.73 67.17 80.28 0.07 20.88 31.33 41.77 52.21 62.41 0.08 15.91 23.86 31.82 39.77 47.54 0.09 11.96 17.94 23.93 29.91 35.75 0.10 8.92 13.38 17.84 22.31 26.66 0.11 6.58 9.88 13.17 16.46 19.68 0.12 4.82 7.24 9.65 12.06 14.42 0.13 3.52 5.28 7.04 8.80 10.51 0.14 2.56 3.85 5.13 6.41 7.66 0.15 1.86 2.79 3.72 4.65 5.56 0.16 1.36 2.04 2.71 3.39 4.06 0.17 .95 1.43 1.91 2.39 2.85 0.18 .68 1.02 1.36 1.70 2.03 0.19 .48 .72 .96 1.19 1.43 0.20 .34 .51 .68 0.85 1.01 85 TABLE A5. Tangential velocity components for hydraulic Oil (0 = .194 inZ/sec.) flowing on a disk rotating at 2400 rpm. z Tangential Velocity (ips) v=rwG (in) r=1.0 in r=l.5 in r=2.0 in r=2.5 in r=3.0 in 0.0 251.33 376.99 502.66 628.32 750.99 0.005 223.56 335.33 447.12 558.89 668.01 0.01 196.89 295.33 393.78 492.23 588.33 .02 150.04 225.06 300.09 375.11 448.34 .03 112.09 168.14 224.19 280.23 334.94 .04 82.74 124.11 165.48 206.84 247.23 .05 60.65 90.97 121.29 151.61 181.21 .06 44.23 66.35 88.47 110.58 132.17 .07 32.20 48.29 64.39 80.49 96.21 .08 23.32 34.98 46.65 58.31 69.69 .09 16.86 25.30 33.73 42.16 50.39 .10 12.16 18.25 24.33 30.41 36.35 .11 8.72 13.08 17.44 21.80 26.06 .12 6.21 9.31 12.42 15.52 18.55 .13 4.37 6.56 8.75 10.93 13.07 .14 3.04 4.56 6.08 7.60 9.09 .15 2.09 3.13 4.17 5.22 6.23 .16 1.38 2.07 2.76 3.46 4.13 .17 0.85 1.28 1.71 2.14 2.55 .18 0.48 0.72 0.96 1.19 1.43 .19 0.21 0.31 0.41 0.52 0.62 .20 0.05 0.08 0.10 0.13 0.15 86 0000.0: 0000.0: ; 035.00 0000.0 0000.0 000.0 00.0 0000.0: 0000.0: 000.00 0000.0 0000.0, 000.0 ,, 00.0 0000.0: 000.0: 050.0 0000.0 3000.0. 330.0 00.0 . 5000.0: 0000.0 000.0 0300.0 5300.0 500.0,: »50.0. 0000.0: 0000.0_ 000.0, 0500.0 0000.0 000.0V .x00.0 0000.0: 0000.0 000.0 0000.0 0000.0 300.0 00.0 0000.0: 0000.0 300.5 5300.0 0000.0 500.3 30.0. 0000.0 5000.0 000.0 0000.0 0000.0 000.3 00.0 0000.0 0000.0 033.0 5000.0 0000.0 000.3 00.0 0300.0 5300.0 000.0 5000.0 0000.0 000.0 00.0 0000.0 0500.0 350.0 0300.0 0000.0 003.0 00.0 0000.0 0000.0 500.3 0350.0 3000.0 000.0 00.0 0000.0 0000.0 000.3 0000.0 0000.0 055.0 00.0 5030.0 0000.0 005.0 0000.0 0000.0 003.0 50.0 0000.0 0030.0 300.0 0000.0 0000.0 000.0 00.0 3000.0 0050.0 500.0 0000.0 3000.0 005.0 00.0 0550.0 0500.0 000.0 0330.0 0000.0 000.0 30.0 0300.0 0530.0 000.0 0003.0 0050.0 030.0. 00.0 0003.0 0050.0 050.0 3000.0 0050.0 300.0 00.0 0000.0 0000.0 500.0 0005.0 0300.0 530.0 00.0 0500.0 0300.0 5000.0 0000.0 0050.0 3050.0 000.0 000.0 00.0 00.0 000.0 00.0 00.0 00.0 A00002002000 00000000 N35.00u. A0m0pcm0cmpv A0m0wmsv N00.30u 0:00 0 m 0 0 m 0 N eds 003m ".2 . eds 0000 u z .EQL 0030 . 0cm Eds 0000 npoo pm wc00m90030m00 e no 0c03o0m Asoen\NC0 500. u >0 00o 00amC0E sow mpCmCOQEoo 0u00000>.A00 0000cmmcmun0cm A00.0m0umafmmm0CO0mcmE00 .0< mqm