THE UNIVERSITY OF MICHIGAN College of Engineering Department of Mechanical Engineering Cavitation and Multiphase Flow Laboratory Report No. 02643-PR-3 Progress Report No. 3 ON MODES OF OPERATION OF AUTOMATED WATER-SLUG GUN by: Edward E. Timm Frederick G. Hammitt July 1969

ABSTRACT This is the second report on an apparatus developed for studying erosion by water jet impact. It describes modifications made to the device since the last report, new equipment developed for studying the performance of the device, and a mathematical model of the device. ii

ACKNOWLEDGMENTS The author gratefully acknowledges the assistance of Charles Kling and Gilbert Huang, both graduate students at the University of Michigan, for their assistance in computer programming. iii

TABLE OF CONTENTS Page ABSTRACT...................... ii ACKNOWLEDGMENTS................... iii LIST OF FIGURES.................... v I. INTRODUCTION.................... 1 II. DEVELOPMENT OF A MATHEMATICAL MODEL OF THE JET GUN'S MODE OF OPERATION............... 2 A. List of Symbols..................... 2 B. General Discussion of Modeling Problems....... 2 C. Jet Gun Mathematical Model (No. 1).......... 4 D. Quasi-Steady State Model for Gun (Mathematical M odel No. 2)...................... 7 III. MEASUREMENT OF JET VELOCITY.............10 IV. DETERMINATION OF BOLT VELOCITY........... 1 1 V. ALTERATIONS IN BOLT.................... 13 VI. SELECTION OF DIAPHRAGM MATERIAL.......... 14 VII. CONCLUSIONS......................... 15 APPENDIX.............................. 16 REFERENCES............................ 17 Figures............................... 18ff iv

LIST OF FIGURES Figure Page 1 Schematic of Jet Gun.................. 18 2 Schematic of Model....................19 3 Solution Curve for Limiting Case Model.......... 20 4 Solution Curve for QSS Model...............21 5 Schematic of Velocity Measuring Device......... 22 6 Typical Oscilloscope Trace From Velocity Measuring Device........................ 23 7 Schematic of Timing Circuit................ 24 8 Bolt Position vs. Time.................. 25 v

I. INTRODUCTION The impact of a high velocity liquid drop on a solid surface produces stresses of sufficient magnitude to damage the surface. This phenomena occurs when raindrops strike a high velocity aircraft and in the low pressure end of steam turbines. Since different materials have different resistances to liquid drop impact erosion, it is desired to develop a laboratory scale test for ranking erosion resistance. A device for conducting such a test has been developed in this laboratory. It does so by impacting material specimens with high velocity liquid jets. Fig. 1 is a schematic of this apparatus. The operation of the device can best be explained by momentum considerations. A large mass moving at a low velocity (the bolt) gives up a part of its momentum to a small mass (the working fluid) which exits from the device at a high velocity. A detailed description of this apparatus was given previously (1). This report deals with the work done on this apparatus since December 1968. 1

II. DEVELOPMENT OF A MATHEMATICAL MODEL OF THE JET GUN'S MODE OF OPERATION A. List of Symbols a proportionality constant A orifice area 0 A piston area P B liquid bulk modulus D orifice diameter o D piston diameter P g gravitational acceleration h height above a datum plane K liquid compressibility P pressure Jf density t time from bolt impact t stroke time m V velocity w frequency factor = 2rrf y piston displacement f frequency V chamber volume c B. General Discussion of Modeling Problems Due to a possible interest in further increasing the velocity of the jets produced by this "jet gun", it was decided to formulate a mathematical model of the mechanism of liquid jet production so that realistic requirements could be evaluated if higher velocities than presently attainable (about 550 m/s) were to be reached. This section of this report contains two mathematical 2

models of this process,of increasing realism although still of considerable simplicity, and the results obtained from them. The following typical parameters for the gun were chosen for the development of a model: Orifice Diameter = 0.0625" Chamber Included Angle = 120 Chamber Base Diameter = 1" Bolt Velocity at Impact = 950 cm/sec Bolt Mass = 0. 76 lb. Resultant Jet Tip Velocity = 500 m/s 3 Volume Expelled Per Shot = 0. 13 cm From this data the most simple estimate may be made of the chamber pressure necessary to achieve a jet velocity of 500 m/s by applying Bernoulli's equation, i. e., assuming the jet expulsion can be approximated as a quasi- steady-state phenomena. Assuming that the water in the chamber is incompressible: 2 2 P VV 2 1 + 1 + gh = 2 +2 + gh -— (1) 2 Pf 2 with subscripts 1 referring to the condition in the chamber and subscripts 2 referring to the condition in the jet outside the chambers in the atmosphere. Neglecting gravitational effects, atmospheric pressure, and liquid velocity in the chamber, eq. (1) becomes: 2 P = V 1 2 f 2 3

or, dropping the subscripts, p yV V2- - - (2) For H2 (P 1 gr/cm ) with a velocity of 500 m/s (1) (5 x 104)2 P 2 9 2 - P = 1. 25 x 10 dyn/cm 18, 000 psi Since this magnitude of chamber pressure is typical, it is obvious that some account of liquid compressibility must be taken. A more realistic model is discussed in the following section. C. Jet Gun Mathematical Model (No. 1) The diaphragm of the jet gun presents an obstacle to the formulation of a precise mathematical model. Thus a model in which the diaphragm was replaced by a piston driven by an appropriate forcing function was used. Fig. 2 is a schematic of the model. The forcing function of the piston was chosen to be of the form y = a sin wt, which is consistent with the known gun performance. The constants a and w were evaluated from the following conditions: 1. The volume of water expelled per shot is 0.13 cm 2. The bolt velocity as it strikes the diaphragm and hence the initial diaphragm (piston) velocity, since the mass of the diaphragm is much less than the mass of the bolt, is 950 cm/sec. 4

Applying condition 1 with y = a sin wt, the displaced volume = A y, P so that the maximum displaced volume occurs when wt=Tr/2 or sin wt = 1. 0. 13 =A a (1) P a.- 0.13 A P Setting the piston diameter equal to the base diameter of the chamber, A =5.07 cm. P 0.13 -2 a - = 2. 565 x10 cm 5.07 Applying condition 2 at t = 0, dy = 950 cm/sec dt y = a sin wt dy dy = aw cos wt dt 950 = aw (1) 9.5 x 102 4 -1 w = 565xlO = 3.71x10 sec 2. 565x10 Knowing w, the time required to discharge 0.13 cm may be computed. Wtmax = r/2 t = TT/2w = TT 2(3.71x10 ) -5 t = 4.24 x lO sec =42.4 fs m 5

If the water is assumed incompressible, direct liquid displacement may be applied. A mass balance around the chamber then gives: d (V ) = mass flux out dt c mass flux out = - A V ~~that iso that is, dt ( Vc) = -A V since is constant dV = -A V dt - - -(3) - - - (4) with y = a sin wt V =V c c 0 - A a sin wt, where V p c = initial chamber - - (4-a) o volume dV d-c = -A aw cos wt dt p so that eq. (4) becomes A aw cos wt = A V P o or A aw V = p A o cos wt - - - (5) Thus, V occurs at t = 0 with V = A aw/ A = 2.55 x 10 m/s. P o max Fig. 3 is a plot of this result. max 6

D. Quasi-Steady State Model for Gun (Mathematical Model No. 2) From eq. (2) it was found that the chamber pressure 4 must approximate 2 x 10 PSI. Thus the effect of compressibility of water must be considered. Starting with the mass balance, eq. (3) d d(p Vc) = - AV and differentiating 0d% + Vc. - i dt dt - A oV - - - (3) - - - (6) An equation of state is now needed. For liquids like water, a permissible approximation to the equation of state is: B'= dP " =Pdy where B is the bulk modulus of the liquid, and B = the liquid compressibility. Then, 1/K, K being dP - - - (7) Separating variables, df = KdP and integrating from a reference pressure (P = O) 0 appropriate density to P and the corresponding reasonably constant over this range: P Ah Ji Kdp = 0; and the assuming K - - (8) KP = ln- p J 7

so that ~f =ye and dt dt KP - - - (9) KP p e d(KP) 0 dt d dt PKe KP dt - - - (10) Substituting (9) and (10) into (6): KP KP P0 4 c + V Ke dP dt c o dt 7 ~~dt = - A V J o - - - (11) A rate relation is necessary in order to relate the chamber pressure P to the jet velocity V. Although this is actually an unsteady flow problem, the actual jet has a relatively large L/D so that a quasi-steady-state analysis may give a reasonable approximation. With this assumption, eq. (2) is applicable, and V in this form may be substituted into eq. (11), along with p from eq. (9): 1/2 Ke dV /~ dtc KP + V Ke dP = c j dt KP - p e A 0 o o - - - (12) Now, from eq. (4-a) dV = -A aw cos wt dt p and combining with eq. (12) KP - Dee A aw cos wt + (V o p 1/2 KP o 2o P 12 =- e A \ KP - A asinwt) D Ke dP p Jo~ dt - - - (13) 8

KIP dividing through by k f e -A aw dP ____c p dt - -A ___P \ 1/2 7P ) -- - (14) which, upon rearrangement yields, dP = A aw A / 2P dt P cos wt- l__ - K ( KP e K —-K —-p KP —-— )-v 0 -- - - (15) V - A a sin wt c p 0 Results were computed from this equation using the digital computer. The solution curve found is shown in Fig. 4. The maximum chamber pressure calculated from equation (15) was 20, 500 psi which compares quite closely with the 18, 000 psi chamber pressure computed for a steady-state 500 m/s jet when compressibility is neglected. That this agreement is obtained from these simplistic models tends to indicate that the models do reasonably represent the jet gun behavior. It thus seems reasonable to assume that the jet gun operates essentially by a straight-forward compressible displacement principle and that complicating factors such as shock wave reflections in the chamber can be neglected to obtain at least a first-order approximation of jet gun behavior. These conclusions could be further substantiated by the use of a pressure transducer in the chamber, if this is deemed desirable at a later date. 9

III. MEASUREMENT OF JET VELOCITY As previously reported (1), the velocity of the jet tip was determined from high speed motion picture sequences taken with a Beckman and Whitley 2 x 10 f/s model 330 camera. Once the jet shape was known from this method it was decided to find a more convenient and less expensive means of determining the jet tip velocity. The method adopted consisted of focusing two light beams across the path of the jet, and then focusing these beams on a high speed photodiode. The output of the photodiode is fed to an oscilloscope which gives a photographic record of the transit time of the jet tip between the two light beams and hence the mean jet velocity. Fig. 5 is a schematic of the velocity measuring apparatus, and Fig. 6 is a typical oscilloscope trace generated by this velocity measuring device. It has been found that velocities measured with this device agree quite well with velocities as measured by high speed photography. 10

IV. DETERMINATION OF BOLT VELOCITY In order to develop a theoretical model of the jet gun (see Appendix) it became necessary to know the bolt velocity as it impacts the diaphragm. It may be shown that for a spring-mass system of this type that the velocity of the mass is given by the expression V = h(k/m') /cos(k/m)/t - (1) where V = velocity t = time h = initial spring compression k = spring constant m= m +m b v m- mass of bolt b m- virtual mass of spring which is equal to half the mass of the spring These parameters for the jet gun are: h = 3.49 cm mb =. 73 lb. m = 1/2(.241) lb v k = 167 lb-f/in Since at impact cos(k/m) /t = 1, the velocity at impact becomes V = h(k/m)l/2 -.. -.(2) o or substituting numbers V = 961 cm/sec o 11

In order to verify this figure it was decided to measure the position of the bolt at various times after its release and thus determine V from the slope of this curve. This measurement was o made by taking a photographic time exposure of the bolt under stroboscopic illumination. However, it was found that the bolt rebounded after impacting the diaphragm and this produced confusing pictures. Thus a means was necessary to limit the stroboscope illumination to one stroke of the bolt. To accomplish this, the circuit shown in Fig. 7 was constructed. The curve of bolt position versus time obtained in this manner is shown in Fig. 8. From Fig. 8 the bolt velocity as it impacts the diaphragm was found to be 944 cm/sec. The agreement with the theoretical frictionless value of 951 cm/sec is thus excellent. 12

V. ALTERATIONS IN BOLT It was found that during repeated testing the bolt head tended to seize in the bore. Furthermore, it was found that the bolt assembly showed substantial wear after approximately 10 impacts. It was therefore decided to make the following alterations to the bolt assembly: 1. An oilite bronze bushing was fastened to the bolt head (type 4340 steel) to provide a better bearing surface. 2. The bolt guide bushing was changed from a ball type bushing to a simple oilite bronze bushing in order to eliminate the tendency of the former to groove the bolt shaft. These modifications are expected to give a useful bolt 5 6 life of the order of 10 -10 impacts. 13

VI. SELECTION OF DIAPHRAGM MATERIAL Since the diaphragm is exposed to the working fluid (water in all work done to date), corrosion is a problem. The diaphragm material initially selected for this device was type 4340 steel, heat treated to a Rockwell C hardness of 50 to 52. Rusting of this material proved to be a nuisance although not a serious problem, so that it was decided to construct a diaphragm not having this problem. A new diaphragm was, therefore, first constructed of type 440-C stainless steel, heat treated to a Rockwell C hardness of 48-50. This diaphragm did not corrode. However, after it had been subjected to approximately 200 impacts it developed a radial pattern of cracks and it use was discontinued. A second diaphragm was constructed of type 4340 steel, heat treated to a Rockwell C hardness of 50-52. To the face of this diaphragrna sheet of 0. 006" thick phosphor bronze was soft-soldered. This composite diaphragm did not corrode. It, however, failed after it had been subjected to approximately 30 impacts by delamination of the bronze sheet. Since both of these attempts at constructing a corrosion-f ree diaphragm have resulted in low diaphragm life, the use of the original type 4340 steel diaphragm has been continued at least for the present. The corrosion of this diaphragm can be minimized by emptying the chamber of water and coating the diaphragm with oil when the gun is idle. 14

VII. CONCLUSIONS An automated device for the projection of high speed liquid jets against a target has been further developed. The detailed performance of the perfected device has been further analyzed and some mechanical improvements made. It is therefore concluded that as soon as minor developmental modifications are completed on the apparatus to ensure its performance, the device will be ready for "production testing' of rain erosion or other materials at velocities up to about 550 m/s. 15

APPENDIX Calculation of Constants _ Eq. (16) dp dt A aw p P K A 0 2P 1/2 cos wt - - K ( PKP e A P 2 = TrD P 4 = Tr(2.54)2 4 = 5.07 cm A o 2 = TrD 0 4 = (TT)(1.586 x 10 ) 4 -2 2 = 1.975 x 10 cm V = TT 2 - rh o 3 = (1. 72 (1. 27 tan 30) = 1. - (1.27) (1.27 tan 30 ) =1. 573 cm 3 3 4 -1 w = 3. 71x 10 sec a = 2.565 x -2 a = 2. 565 x 10 cm R = 4. 58 x1011 Dyn/cm2 so that, A aw P K = (5. 07)(2. 565 x 102)(3. 71 x 104) 4.58 x 10 = 10.53 x1013 cm /Dyn sec A F 2 = AKri o (1.975 x 10 2)(1.414) /,,.8 11//2 = 6.11 x 10 cm/2 Dyn G1/2 (4.58 x 1011)(1) -2 -1 3 A a = (5.07)(2. 565 x 10 ) =1. 301 x 10 cm3 P 16

REFERE NCES 1. "An Apparatus for Studying Erosion by Water Jet Impacts", University of Michigan Office of Research Administration, Report No. 02643-PR-1, University of Michigan, Ann Arbor, Michigan, 1968. (Also see E.E. Timm, Term Paper for Chem Met 690, University of Michigan, same title, by E. E. Timm) 2. Pitek, Martin T., and Hammitt, Frederick G., "Hypervelocity and Fluid Impact Studies, A Literature Review,' ORA Technical Report No. 08153-1-T, Laboratory for Fluid Flow and Heat Transport Phenomena, Nuclear Engineering Department, The University of Michigan, Sept. 1966. 17

PLATE )IAPHRAGM FEED WATER 2765 Fig. 1. Schematic of Jet Gun 18

ORIFICE y Fig. 2. Schematic of Model 19

(A E I - I w I LLI > I w I 1 I 1 1 I I I I I 12903 1\ 4 8 12 16 20 24 28 32 36 40 TIME FROM BOLT IMPACT, (As) Fig. 3. Solution Curve for Limiting Case Model 20

21 REQ'D PRESSURE FOR 500 m/s TEST VELOCITY I I I I 315- a. 14 4 "n 13'0 x 12 C L 0n 50 a. 9 w8 GO 7- 5 6- 5 4 3 2 I,______________, 2904 0 5 10 15 20 25 30 35 40 45 50 55 TIME FROM BOLT IMPACT, (/ts) Fig. 4. Solution Curve for QSS Model 21

TO LAMP POWER MIRROR SGD 100 PHOTODIODE MIRROR I N) o Ln PATH OF JET TO OSCILLOSCOPE Fig. 5. Schematic of Velocity Measuring Device

VERITICAL A 20 us/cm HORIZON'AIL. 20 Tmv/cm I JIE1TI INTIi:RUrTI:JS JETi' I NTlIERiRUPTS I0l OWEA R BEA Mi UPPEIIR BEA M Delta t 10 105.0 ts, which is equivalent to o jet velocity of 259 m/s. P'ig. 6,''Typical Oa1scilloscope Tir1ac(c P-::romn Velocity Measuring IDevice 3

I-7 t 45 V N) NJ -_j 0 RELAY TO 2N441 TO GUN Fig. 7. Schematic of Timing Circuit

4' SLOPE AT X - 3.50, = 944 cm/sec E 0 2 / 5-_ // 0 0. H- / GJ 0 m / I / 0/ 0 I 2 3 4 5 TIME (ms) Fig. 8. Bolt Position vs. Time 6 7 25