c 4 .3 r-v .—\ l r. n . . f .K“, ~\- - . “; 23' :g‘; b ‘I\.: st \ l 1 I . . ”a U. '_\ 4_,.,,g,,,:,,,, ,5, .6. .,,,, on, g .,,, 51.3 3.0.- ,“ E fig .5? 9%.... 5. . _.-_ LIBRARY Michigan 5‘3” University ‘7 .a-n. _.~_h_.. _ A STUFV OT: FLOW T?RO”CH VTQY STOET, 7? LL CYLIND?IC L TU??? k Submitted to the Collefe of Engineering Michiren State University of egriculture fine Applied Science in partiel fulfillment of the recnirements for the degree of IVLQS'T‘FQ OF SCIF‘NCT Derertment of Mechfiniefil Enéineering 19:0 _ 1 - L. Smith (‘1’- Flow through very shor cylindrical tubes oiffers from flow through long ripes in thct the entrence eno exit pienomena are the significant flow conditions. Eoth a theoretical end in experimental approach were employed in ‘ fie flow 055P¢0t”rietice throuTh (f Li (D an attempt to predict t tube. The theoretical Opproach requiree a anthem tic 1 description of each flow zone within the tube. Then, to trcdiet the flow cherecterintics of the entire tube, these methemeticel deecriptionfi UUQt be linked to§ether. Inc to momentum coneideretione, the etre mline boundinr the lJo m“ n etrerm, or core flow, Pepir”tes from the tube well at the efiuere edged entrcroe o? the cylindricel tube. This sepiretion continues to increase for a short dietfince downetrewn from the entrance. This is celled the contrection zone. Downstream of the cortrfiction zone, the strevnlinee oiverge “no “fibro”oh the twee will — the evp"nrion zone. Hormel rife flow velocity profiles rnd frietior coneifieretione ere conefioerrfi to rpply fron this point dowretre-m to the exit. ”0”P¢, ° ”“theneticel representation of Flow throufh n vrrf nhort, nrnll cylindricel tube recuiree the joining of mnthon“tictl Beficrintione of four flow vonefi: l. Contrfiction eone 9. prfinoion zone 3. Pipe flow zone A. Exit zone [‘0 - L. “31th KathemetiC"l solution, fire Presentlv “veil“ble for 7' he ripe flow and exit zones. A numerical ap*rorimetion to the streenline pattern in the contraction zone is also available. lowevor, mathemoticel solutions to the eoustions describing the contrrction end exrdnsion zones will recuire s better underst“ndinr of the bound ry lover find 2 mothe- mcticel representction of the flow boundsry stre“hlines in these zones. Further investijrticn is recuired in these eress. An exrerimentsl spjrosch to flow throujh very sh-rt, Stall cylindricel tubes approximdtes this flow condition by a long orifice. Fxperinentol determinrtion of the orifice coefficient of discherfe indicotes the coefficient is mrrhedlv effected by the velocity of flow and the jeometric chcrscteristics of the cvlindriC“l tube: lenfth, diPmeter, and surfece finish. Tests were conducted to deternine the relationship between the geometric choreo- teristics of thu tube, the fluid prorerties, the fluid velocity, end the coefficient of discharge. The dots ohteined was plotted nondimensionolly, and is considered to be gcncrdlly applicxble to onv fluid flowing through any dismeter tube subject only to the recuiroment of geometric similirity, i.e., eoudl L/D and e/D ratios for cylindrical tubes with a scusre ed:ed entrnnco end exit configuration. q) TUEY OF FLOW A ”RFCUGH vsev secs? s: LL CYLINETIcAL TUevs ’ Pv t LOWELL C. fM TV THESIS Submitted to the Collefe of Engineering Michigen Stete University of flericulture ind frnlied Science in pfirtisl fulfillment of the reeuirenents for the degree of I STE? OF SCIENCE Depertment of Mechanicel Eneineerinq 1959 Tflet of fimvres '3 vii? OI‘ err muknni T*TCT.*‘TT" 727*:0 Int'wq8‘104kfiqn . O O O O 0 O O O O C O . Foqnj‘wdj ,j'nc‘ n y \ I '5 I1 ’3 "Vr 1 (:1 7'“? O " r“1fisie mhfore‘ic'l Investic~tions fiscnseion Scone of anerimental Investir:tions Previous Lrannw T 0 Q o a a a O O O O O O ’7 1’:- 7'7‘ A fit? Peserrch . . . . . . o o o o o o o o o o o o o o o o o o o o o 9 o o o o o c Q‘Il'f‘an?-T TT __ FVL‘C‘:TW7L'1\"TV T O C O O O O I O O O O O 0 O I . \J ‘ Peview of Previous Pesesrch . . . . . . Ln'1v*is TTv'nowfl h an + n. 1 ‘ g . c . ‘,‘ Fvn”rinent'l Feninnent T3044 Front“ Discussion List of Wef Frnendix innendix ;, p n if .1“ n " 0t} 00 o o o o o o o o o o O O O Q C O O O Warn 0 O C O O O 0 O O O O O O O O O O O O O O O l O C O Ruwvuwrvw: a mleT.TJ F)renqeq O O O O O O O O O I - Flowneter Cilihr-tion . . - Psrts List 'nd Drawirfs of O O O O O O I O O O O I O O C O O I O O O O O 0 O I O I O O O O O 5'17“““77‘ O O 0 O o o 0 o O O 0 O O a o 0 O O 0 I O o " )\ f...’ \1 w tn ‘\ \J K Symbol LIST or SYMBOLS Description ’ ggigg cross-sectional area of cylindrical tube in2 cross-sectional area of the fluid in2 cross-sectional area at station 1 in2 discharge coefficient - contraction coefficient velocity coefficient diameter of flow'boundary ‘ _ in diameter of cylindrical tube . a in surface roughness - , microinches rms pressure force ‘-'- A ' 1b shear force lb acceleration of gravity in/sec2 a constant dimensionless ‘length of cylindrical tube in mass :' M} "’ lb-sacz/in momentum rate N '~ lb momentum rate in the boundary layer lb momentum rate in the core lb pressure downstream from the tube exit lb/in2 pressure upstream from the tuba'dntrance lb/in2 total velocity: ;‘= 3|+‘; ": in/scc flow rate in3/soc or gpm flow rate in boundary layer ' in3/sec flow rate in the core =‘ ins/soc coordinate-radial distance outward from ‘ in axis 5 C2 .4 ’o-<'cpn a 'O m ii w~Description radius of cylindrical tube time: temperature velocity component in x direction‘? core velocity component in x direction velocity component in r direction average velocity in x direction axial distance radial distance inward from tube wall relevation with respect to datum QBEEK SYMBOLS asymptotic radius of contraction jet temperature coefficient of viscosity weight density of fluid boundary layer thickness temperature coefficient of density' viscosity of fluid ' kinematic viscosity function ofwr and x mass density of fluid shear stress " average shear stress local shear stress . .stream function Units in” 806 in/sec' in/sec‘ in/sec in/sec in in in in 1/”? lb/in3 ‘ in l/°F lb-sec/in2 in2/sec dimensionless lb—secz/in4 lb/in2 lb/in2 lb/ing in2 . Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 10 ll 12 13 iii LIST or FIGURES C versus 29~Plot of Experimental Points “D Flow Zones Plot of Streamlines near Orifice Plot of Streamlines near Orifice Plot of Streamlines near Orifice Control Volume Cross Section of Uniform Pipe Streamline Pattern for a Circular Laminar Jet Volume Element Characteristic Curves for Square- Edged Jets with Different Values of L/D Coefficient of Entrance Loss of Square-Edged Inlet. Calibration Curve - Potter Meter Flow Pickup Calibration Curve - Waugh Flow Pickup 10' 11 12 16 17 23 24 4O 41 l FLGV THEGJGH VERY SHORT. SMALL CYLINDRICAL TUBES “I INTRODUCTLQQ Flow through very short cylindrical tubes differs from flow through long pipes in that the entrance and exit phenomena are the significant flow conditions. Actually, a’very short pipe could be termed a long orifice. The prediction of flow characteristic through very short, small‘ cylindrical tubes may be approached in two ways. One approach.- is to mathematically describe each flow zone within the tube. Then, to predict the flow characteristics of the entire tube, these mathematical descriptions must be linked together. Vere ification of theory, by experiment, would be desirable. ,‘1._ Another apprdach is to consider the entire tube to be a long orifice, described by the flat eguation; .. '55P}. Q'CA/T The value of the coefficient of discharge, C, may be determined experimentally. ' CONCLUS IONS Theoretical Analysis 1 An exact mathematical description of flow through a very short, small cylindrical tube is not now possible. A good mathemati- cal approximation of the flow phenomena will require a better understanding of the boundary layer downstream of the inlet and a mathematical representation of the flow boundary streamlines in this region. Further work in these areas must precede sol- ution of the equation describing this region. Experimental_Determination Experimental determination of the orifice coefficient of dis- charge indicates the coefficient is markedly affected by the velocity of flow and the geometric characteristics of the cyl- indrical tube: length, diameter and surface finish. (See figure 1). (Savitation occurred when the difference between the supply pressure and exhaust pressure, Ps - Pe,was very large. When cavitation was occurring, variation of P8 did not affect PS. P5 was approximately constant during cavitating flow until P8 was increased to a critical pressure at which the cavitation ceased. The pressure difference, P - Pe' was approximately 5 constant when the flow was without cavitation. Thus, the orifice coefficient of discharge is a function of PS and Pe at low back pressure, Pe, when Ps - P6 is large and cavitation occurs. The coefficient is not a function of P5 or Pe when non-cavitating flow occurs. Figure 1 also indicates the coefficient of discharge is a lin- ear function of 5% for values of 5% greater than 9,000. It is the contention of Zucrow (8)* that the data plotted in figure 1 will apply to any fluid flowing through any diameter tube subject only to the requirement of geometric similarity, i.e., equal L/D ratios for cylindrical tubes with a square edged entrance and exit configuration. Further experimenta- tion would be required to verify that contention. ~--.—..-._‘-._ -- 4-— .5”. ”Mc- .n- c..- * Numbers in parentheses refer to the List of References. -4- SECTION I THEORETICAL APPROACH Scope of Theoretical Investigation When fluid flows into a cylindrical tube through a square edged entrance, a contraction of the flow boundary occurs. The inflow- ing fluid particles have a radial velocity component as well as an axial velocity component. Particle inertia permits only a gradual turn at the square corner of the tube entrance, so the flow must separate from the tube wall. Between the wall of the tube and the streamline bounding the core flow area, a field of eddies is generated. Very little linear movement of these eddies occurs a Downstream from the contraction, the flow boundary expands. A tube of sufficient length will allow enough expansion of the flow boundary to completely fill the tube. A portion of the problem is to determine the effect of various tube lengths upon pressure .drop and quantity of flow through the tube. With long tubes, a normal boundary layer begins to develop at the point of reattach- ment. The problem is to relate tube dimensions, fluid properties, and pressure difference to the quantity of flow. (See figure 2). A mathematical representation of flow through a very short, small cylindrical tube requires the joining of mathematical descriptions of four separate flow zones: ' l. Contraction zone near the tube entrance . Expansion zone Pipe flow zone . Exit zone 2 3 4 -5- Preliminary analysis was conducted assuming-laminar flow. No attempt was made to represent theoretically, a turbulent or a cavitating flow condition. [ -—%£; *4Ef-—Contraction Zone JF‘ ' /’-——¥- "4%-+ Expansion Zone .i p. / . .~ 4-— Pipe Flow Zone ——3~/ ' é ‘ ' /[ Exit Zone .\ . _ 7 \‘I\ /// ‘ Z / Z A / / “1‘1; 1 / j / / ‘ ,.-/ ~\ ~,\ M“, r; ,- V) ”/7 ‘- \«v " -,-./ “:\. \ “a _ H 1‘)” "f M -..,,"" -““" -- -—...___._ \ -._————--~"'""/- _. ‘ - ".~‘\—-....-_ ‘MM' ”...—w ... ‘ -~. ~VWNk~~c-..--.-. ~~ "M." Q, _ _ ..; 1- ,/ " .— ......” 11".; ~"’""" ~--- w- ~ .. ”...... .......______,,,_.-....—..-.—~-.-~~- -. . Pe / I, “‘-~.—._.,__, - P 2”" . ; ’a--..~._ “Ks-....“ -—-———-" ‘Mam‘ 8 -. f / "‘1'. .1”..‘ ...;::~:‘\-_ "“Ma‘ ~ - ' ‘ 2—...“ ........ ..." “- K ,x/ /’ /(1 4) {j Z‘~..\ M". “W ‘_.' f” "---... ' ”"" \ x / / / /‘ / f/ / { 7 l / / 7 f 7 f // \. ‘ / v | / / / \ / \ _/ /’ . * ‘< L St} \\ fl / 2/ Flow Regions . Q = 1 (D, L/D, u.-p, PS — re, fe,e) Order of magnitude c.122 S D S c.133 ,1S L/DS 8.122 “15099 = 14.77 x 10’7 lfliigfi 2 8.37 x 10 5 lE—EEE- p in4 lawr= ‘Ps S 3000 psi Po 2 40 psi Figure 2 Review of Previous Research A survey of previous research produced”treatments for three of the four flow regions discussed previously. 1. Contraction Zone The contraction zone near the inlet may be likened to ori-L fice flow in three dimensions with axial symmetry. This is very similar ththe flow of fluid through an aperture of infinite length-~a two dimensional problem. An exact sol- tion to this problem was arrived at by Kirchkoff in 1896 (1). Many authors have since repeated Kirchkoff's conformal transformations in the complex plane including a recent book-; in English byStrieter (2). Unfortunately, no exact solution to the similar three dimensional orifice has yet proven feasible (3). In 1913 Trefftz (4) determined, by successive approximation, that the coefficient of contraction for the three dimensional case of a circular orifice in an infinite plane boundary was between 0.60 and 0.62 as compared to a coefficient of 0.611 for the two dimensional case. A description of the three dimensional jet profile with axial symmetry was accomplished by relaxation and published by Southwell and Vaisey (5). They consider an orifice plate in a circular tube. They compute a boundary error n defined %$—k (1) and reduce this error to zero (sensibly) along the stream- by . ' 1 11:; line by modifying the streamline shape and the constant k. -7- Far downstream, the asymptotic solution was assumed to hold. 1 J. m = g k r2 .H , (2) Let .¢B = the jet boundary streamline constant- a2 = coefficient of contraction' Then 2nt is the total flow through the orifice- a is the asymptotic radius when the orifice radius is unity. From equation 2: 2 ‘ ‘ “ By assigning a value to m8, k may be accurately estimated, since a2 is approximately known from hydraulic experiments. The solution, by relaxation to negligible error at all points is: u, . ; ' k = 118.34 a2 s (0.78)2 = 0.6084 See figures 3, 4 and 5 from (5). Expansion zone The expansion zone and reattachment of the flow boundary to the tube wall has not been previously described in .mathematical detail. Pipe Flow Zone Brenkert (6) presents a calculational procedure for laminar flow and development of the boundary layer in a circular tube.' Two assumptions are made: a. Bernoulli equation (neglecting friction) can be applied to central core. b. Velocity is constant across central core. n oasawm oowufiao see: meeuaaeoaam we wean q oasoflm com we use: mocwdseoaum no «can ll; /]f U a a// /]J // / / / // j. / / / / T/ / v m -11.- Consider a control volume, figure 6: Control Volume Figure 6 Apply Newton's Second Law: -11. ZFX - dt (mv) PnA - Pn+lA - 2flRxTa = Mn+l — Mn Solve for x: (P -P )A+N -M x = n n+1 n n+1 (4) ZWRTa Evaluation of each of the terms in equation 4 follows: m = “BL + MC (5) P 2 b 2 MBL = p u 2nrdr = 2flp u (L-y) dy (6) - R-b 0 Me = p17 (R-b)2 112 (7) Substitute equations 6 and 7 into 5: b 2 2 2 M = 2wo u (R-y)dy + D" (R-b) U (8) ,.o The velocity is assumed constant across any core cross- section, i.e., viscous effects are considered appreciable only in the boundary layer. The modified cubic profile used by Shapiro, Siegel and Kline (4) most accurately de- scribes the boundary layer velocity profile. E~£1+E<§>~Gfl£fl F- .— 3 “ = 3:31 >— «a, -12- As % increases from zero to unity, the cubic velocity pro- file changes to the parabolic form characteristic of fully developed flow through a circular pipe. Rearrange equation 9 and let k = % u = [IE-(3" _21_(%>3+%k<1.) 11G)?" +ék (1)1 (10) Substitute equation 10 into equation 8, integrate and sim- plify: = 2-_36 677 2 _1___o7 3, 1 4 “Av L1 35 k 1680 k 840 k 560 k] (11) Thus, for each assumed boundary layer thickness, the momentum rate may be calculated. Pressure is assumed constant over the cross-section. In the central core apply Bernoulli 5 equation: P +U2 = constant ' (12) Y 29 In order to calculate pressure, the velocity must be first calculated. To satisfy continuity, the flow rate Q must be a constant at every cross-section of the pipe. Q = QBL + QC _ (13) Consider the cross-section of a uniform pipe shown in figure 7. Cross-section of a Uniform Pipe Figure 7 -13... QBL =,j/ u 2flrdr = 2m2/b u (R-y)dy (14) R-b C h-‘ H.- 23 Qc =_v Tr(r+6x 2) the average shear force, is obtained by considering the Therefore: Similarly: Fs, definition of shear stress. _ d ‘ T~Mfi ' Therefore: FS = TA where A = 2wrdx Substituting: _ du s p dr rdx Assume the shear stress at the midpoint between stations n and (n+1) is equal to the average shear stress between stations n and (n+1). Momentum change, the right hand term in Newton's Second Law, is obtained from the following considerations. From the definition of a total derivative: 9 -1 951 dt (mu) - 6x (mu) dt + or -19.. From the assumptions of steady state and negligible mass inter- change between core flow and the boundary layer. d man dx ' au dr _ __ + __ __ dt (mu) max dt m 8r dt dx dv Since u >>> v, i. 6., dt >>> dt' and q is considered constant across any core cross section, then %%'~ 0 and the last term in the preceding equation may be neglected. d — 92 dt (mu) - m 6x u .EL = 2 __ dt (mu) pnr u 6x dx _ where mass = density x volume = pwrzdx Substitute the above quantities in Newton's Second Law: _ g3 g; _ 6r dx er dx g5 g; 2 (P ex 2)"(' 3x 2x)2 "P“ W)( 6x 2 - 9.9-29.2... 2wpr dr dx par u 6x dx 0 Expand squared terms and multiply through by % -9292 2.9.: +9.39st 9..de 2 9.2. 6r 2 (P 6x 2>if r ax dx (fix 2)] (P+_ 6x 2 —)E +1. 6x dX+(ax dx), _ 22 - 2 22 = 2pr dr dx pr u 6x dx 0 6r dx 2 . '.- 6r '° Neglect (6x 2) in compar1son with 5; dx, an infinitesimal of, higher order in comparison with 95-dx. 6x Expand equation: -292- 2.1:- 9.9-29.2: r 6x dx 2Pr 6x dx 2pr dr dx pr u 6x dx 0 Multiply through by - ——- : 3‘9'» -. .— M -20- '0? ~~6r . -ndu . bu = - r7+2P5§+2pdr+pruax 0 (32) Discussion of Physical Significance gg is negative and mainly a function of the velocity increase through the contracting flow regionwand, to a negligible extent, p a function of frictional losses. The second effect may be ig- 92 nored in comparison with the magnitude of the first. Thus, fix is a function of the flow streamline boundary. g; is negative and definitely a function of the flow streamline boundary. %% is negative and refers to the sizeable change in axial vel- ocity as r increases from just within the core section to just inside the eddy field bounding the core section. %% is positive. This is the change in velocity required by continuity with a change in flow cross section--thus a function of the flow streamline boundary. D Isc‘bss Ion?" The difficulties encountered in solving the partial differential equation can readily be seen. To date, the streamline bounding core flow has not been described mathematically. The numerical solution is approximately correct at the relaxation nodes, but it is not continuous nor does it have derivatives. Thus, %§, 3% and %% cannot be obtained, since all depend directly or in- directly on the partial derivatives of the equation of the flow boundary. No method of deriving this equation is presently available. -21- Evaluation of %% is also very difficult, due to our present lack of understanding of the eddy field bounding the core flow and the interaction between the two flow regions. An analysis of the expanding flow zone is very similar to the preceding development. The force equations are identical. How-.. ever, when evaluated, the gradients will be opposite in sign. Just as in the contracting_flow zone, the lack of a mathematical equation of the streamline boundary of core flow and an under- standing of the eddy field bounding the core flow, prevent sol- ution for the expanding flow Zone. -22.. SECTION II EXPERIMENTAL APPROACH Scope of Experimental Investigation An experimental determination of the flow characteristics through short, small cylindrical tubes delves into variations of arbit- rary coefficients which modify theoretical flow equations. The .yrigice flow equation is: i Q = CA 2%: gThe.coefficient of discharge, C, modifies a theoretical descrip— .tion of frictionless flow through an infinitely thin orifice. .This coefficient is affected by the properties of the fluid, LAfinite thickness of orifice wall, surface roughness,and the inlet and exit configurations in practical applications. This experimental investigation is intended to: l. Isolate unimportant factors. 2. Determine magnitude of each effect. Review of Previous Research Zucrow (8) investigated the flow of benzol through submerged tubes of small diameter (0.020 S D S 0.088). Among his impor- tant conclusions are: 1. If C, the coefficient of discharge, is plotted as a function of W¢r, the characteristic curve is valid for all jets which are geometrically similar. Where: W = actual rate of discharge ¢ = fluidity of the liquid = & r = reciprocal of diameter = % 2. Square-edged jets are geometrically similar when they have equal values of L/D. -23- o\q no mo:~e>.a=onouuwa saw: meow uoavmuoeasum new me>asu swamwceaoeaano om ensued I .amH . em «a «a '3 em a mm m om .oH m e.. a m .. v7 . ti . .. O . .1 .. . n+0 ., . \ x - Nfio \ ...m w 0 Dmb.d . .02 con llOl\1I\\\\l i 1 1T ill... ‘ xiii. mWI. _ - - mme.~ M M \‘J indwno. 1\\\\L -en ado .tltilnrllixllli em.o~ .Vne JTLHU \\e 0 9 l||l\ .62 90h. 4 . on poo H 61f .. N6 .. He o as , n. mam.g 2 sea .e sen .llldwllln -0. xlsre nlldwrlllllew a meme.e o\a no ezaa> on touch whenasz 5.2 eh need so: an??? a} .w meme madam-mm<:om do w. mmbmso oHemHmmao>> A1, and thus qs <<< ql. Therefore qs’VO. Also, since the pipe is very short, Z ~ Zl, regardless of the orientation of the pipe. P P Thus. p p + 2 q1 . (PS - P1) Rearranging: ql = p (35) To find total flow through the small cylindrical tube, consider the continuity equation: Q = q A (36) Substitute equation 35 into 36: P-P ' Q=A/2(spl) (37) Equation 37 applies only to frictionless flow. For actual flow conditions, the equation must be modified to account for en- trance losses, the reduction in effective tube cross sectional area due to flow contraction just downstream from the tube en- trance, fluid friction along the tube, and exit losses. Thus, Q = CA fig; P1) (38) for actual flow: where: C = C C Cc, the contraction coefficient, is attributed to the inability of the radially moving fluid particles to instantaneously change direction of travel to an axial direction at the sharp-edged -27- entry of the tube. Hence, the fluid flow area contracts near -.the tube entrance. For a very short tube, an orifice, C = 0.6, c meaning that the flow area is only 60% of total orifice area. In longer tubes, an expansion follows the contraction but size- able losses occur in the contraction-expansion process. Cc is a function of fluid momentum and the tube geometry. C the velocity coefficient, is attributed to frictional losses v! along the tube wall and fluid friction. Wall friction is a func- tion of fluid properties, surface finish, and geometry. Fluid friction is a function of fluid properties and flow quantity. Thus: C = f(p, V, p, D, L, e) Since: C = Cccv where: Cc = f(p, V, L, D) Cv = f(p, u, e, L, D, V) For actual experimental purposes, the quantity of flow, Q, may be measured directly. Velocity, V, is obtained by calculation from the equation Q = VA. It was decided to consider C = f(Q) for the purpose of simplifying the recording of experimental data. Therefore: C = f(p, Q, u, D, L, e)' Dimensional analysis _ 2 - Dimensions: 9 AQ—EEE- = FL 4 T2 in4 . 3 _ Q in : L3 T 1 sec in D, L, e (in) = L ~28- Dimensional Matrix: p Q n D L Force: F . - l 0 0 0' 0 Length: L -4 3 42 l ‘1 1 Time: T 2 -l_ . l 0 0 0 From F: 0 = p + u (39) From L: 0 = ~49 + 3Q - 2p + D + L + e (40) From T: 0 = 2p - Q + u (41) Solve equation 39 for u: p = -p (42) Solve equation 41 for Q: Q = 2p + u Substitute equation 42: Q = 29 ‘ P Collect like terms: Q = p I (43) Solve equation 40 for o: D = 49 so + 2n - L -'e' Substitue equations 42 and 43: 8 D = 4p 3p - 2P - L - e Collect like terms: D'= -p L -‘e (44) From equations 42, 43 and 44 construct the n matrix. p Q n D #2 (L) o o .. o -1 '1 0 n3 (e) o o o -l n‘ 1 Therefore: 2 HI = fig - L 1T2 - 5 .. 8 -29- The coefficient of diScharge can now be expressed in terms er~ nondimensional parameters thusly: _ L e CD ‘ f(&3 . ‘D’ . 5) (45) Experimental Method Recall the relationships: . H ' - . keg) = A ___§..___9__ Q C p and C = f 5% l % t 3") V ‘ (45). p and p are determined by the fluid and fluid temperature. A, L, D and e are dimensions which may be measured directly or computed from simple basic formulas. Experimental values of Q and P8 - Pe may be measured directly. The values of C may be computed from equation 38 and experimental data. By varying Q,. L, D and e, the relationship suggested in equation 45 may be investigated. Experimental Equipment. Small cylindrical tubes were made by drilling, reaming and, in some cases, polishing a small hole through a special test block. (see Appendix D). The L/D ratio was varied between 8 and l by_ reducing the length of the hole. The sharp edge of the tube inlet and outlet was compared by microscopic inspection and, protected from damage at all times. Steel and plastic testifix blocks were used. Holes in the steel test block were drilled and reamed yielding a microinch finish of between 50 and 55 rms. Holes in the plastic test block were drilled, step reamed and ~30? polished, resulting ina microinch finish between}. and 6 rms'--. an optically clean surface, permitting visual observation of the flow phenomena. Finished hole sizes ranged from 0.122 inches to 0.133 inches. The plans of the entire test fixture are included in Appendix D. A forty power microscope and microscope light were mounted in the microscope mount, which could be indexed along the length of the cylindrical tube. Careful, accurate observation of flow conditions was therefore possible. Pressure taps were located 48 diameters upstream from the tube entrance and downstream from the tube exit to measure Ps and Po respectively.v Thermocouples, located at the same position, ' measured Ts and Te. Two turbine type flow pickups located down- stream from the pressure-tap and thermocouple were employed. The capacity of the smaller one was from 0.3 to 2.2 gallons per minute; the capacity of the larger one from 2 to 24 gallons per minute. Valving permitted flow to be directed into either of‘ the flow pickups. f2hus, flows from“0.3 to 24 gallons per minute could be accurately measured. The flow output of the variable delivery pump was manually con- trolled from negligible flow to 14 gallons per minute. The pump required a supply of oil at a pressure of 40 to 50 psi. This was accomplished by pressurizing the reservoir with compressed air acting on the fluid surface. Thus, the reservoir pressure, pump supply pressure, and reservoir return pressure was 45 t psig. All reservoir connections were submerged. Pump output -31- pressure was determined byr resistance to flow in the system, pressure drop across the test block, and adjustable pressure drop across the needle valve. Line losses were considered negligible. The cross-sectional area of the lines was no less than 25 times the cross-sectional area of the cylindrical tube under investigation. The assumption of negligible line losses 'is therefore quite good. The pressure drop across a given test block is determined by the quantity of fluid flowing.‘ A needle valve, downstream of the test block,-was_employed to vary the back pressure on the test block m-hence, the pressure range at which the pressure drop across the test block occurred. Test Procedure Warm up of the system was accomplished by an electric heater in the reservoir. Temperature was maintained by heating or cooling the fluid asrequired. The heater was controlled by a thermo- stat device whose sensing element was in the reserVOir.fl Cooling Was controlled manually by adjusting the quantity of cooling water flowing through the heat exchanger. For each length and diameter of tube investigated, a series of runs was made. ‘Runs were conducted at flow quantities of 1, 2, 43 6, 8, 10, 12 and 14 gallons per minute. The test block exit pressure was varied from 50 psig to (3000 psig - AP) by adjust- ‘ing the needle valve. After each adjustment in back pressure, “the flow quantity and the temperature of the fluid were adjusted, 'if necessary, before the data were recorded. ”-32- Each test block was run and checked in the reversed direction of flow to detect any noticeable discrepancies due to edge sharpness or surface irregularities. Discussion A nondimensional plot of C versus Eg-indicates the coefficient increases as the quantity of flow increases--a rather sharp initial rise and less steep linear rise for further increases in fig..'rhe surface finish of the cylindrical tube appears to have a very important influence on the coefficient-~certainly as important as the L/D ratio.: See figure 1. It must also be noted that the cylindrical tube in the plastic test block was not truly cylindrical. When the tube walls were polished, the ‘5 'entrance and exit of the tube were enlarged slightly. " 1948, pp. 174-177. 10. -33- LIST OF REFERENCES Kirchkoff, 6., "Zur Theorie Freier Flfissigkeitsstrahlen", Crelles Journal, Vol. 70, 1869, p. 289. Strieter, V. L., Fluid Dynamics, McGraw-Hill Book Co., Inc., Rouse, H. and Abdel-Hedi Abul-Fetouh, ”Characteristics of Irrotational Flow through Axially Symmetric Orifices”, Journal 2; Applied Mechanics, Dec. 1950, pp. 421-426. Trefftz, V. E., "fiber die Kontraktion Kreisformiger Flfissig- keitsstrahlen", Zeitschrift fur Mathematik and Physik, Vol. 64, 1917, p. 34.“““"“‘ '“”‘ Southwell, R. V. and Vaisey, 6., ”Relaxation Methods Applied to Engineering Problems - Fluid Motions Characterized by Free Streamlines", Proc. Roy. Soc., Vol. 240 A, pp. 117-127 Brenkert, K., "A Study of Pressure Variation in the Region of Boundary Layer Transition in Cylindrical Tubes”, Thesis, Stanford University, 1955. Schlicting H. 'Boundar La er Theor McGraw-Hill Book Co. Inc., 1955: pp: I§§=TET‘%ha‘%BZ 3353531: Zucrow, M. J., "Discharge Characteristics of Submerged Jets”, Bulletin No. 31, Purdue Engineering Experiment Station, 1928. Chen, Chia-Tsun, "Unestablished Flow Patterns Downstream from Sgggre-Edged Pipe Entrances", Thesis, Stanford University, Reethof, G., Goth, C., and Kord, H., ”Thermal Effects in the Flow of Fluids between Two Parallel Flat Plates in Relative Motion", American Society of Lubrication Engineers, preprint number 58 AM 4A-l. -34.. APPENfigx A SAMPLE CALCULATIONS properties of petroleum base aircraft hydraulic oil Y = YTOOF (1 + KAT) x = 4.26 x 10‘4011; lb = 0.0312 —- Y70°F 1n3 _ -fiAT fl ‘ ”70°F 6 3 = 0.01036 03- F = -7 lb-sec “70°F 26 x 10 -_;;§- late: -4 Y150°F — 0.0312 1 + 4.26 x 10 (80) = 0.0323 $93 in = 26 x 10-7 e-0.0lO36(80) “15001? -7 lb-sec 11.35 x 10 --§- in :3 equation: = CA/gzl Q P lculate C, given Q, D, AP and y, the following form of the ion would be more convenient: C = __Q._. /_i__ I [)2 2(AP)9 4 yhout these tests: Y = 0.0323 %-3 1n -35.. . = ——2.:: For the steel test block: .’D = 0.122 inches Substituting them values: Eél . c = Q (I) 1. x670§23 MA? },I-’(0.122)2 2(336) : 2.13 .9.— ‘JZF From experimental results: for Q = 6 gpm, AP = 267 psi Substituting: 6 C = 2.13 = .782 V 2 ;67 The abscissa of the experimental plot is easily calculated: '*“ “231 - 6 '60 0.0323 29.: 9;! = - 2 #0 QFD (386)(11.35 x 10")(0.122) ——- -—__. ~—_—_ ____.~,____— —— —_— —— fl...“ _— -36.. APPENDIX B DATA . Series A - Plastic Test Block D = 0.126 inch e = 5 i 1 x 10-6 inch rms T8 = 150 :l: 5°F L/D = 7.86 @ L = 0.990 inch Q (gpm) Ps - Pe (psi) C 5% 1 10 .631 22.5 x_lOO' 2 28 .755 44,9 x 100 L-J" 4 90 .842 90,0 x 100 6 180 .893 135. x 100 8 304 .918 179.5 x 100"Ti~' 10 418 .980 225. x 100 12 551 1.020 250. x 100 Series B - Steel Test Block D = 0.122 inch e = 52 i 2 x 10'6 inch rms Ts = 150 3: 5°F L/D = 8.12 @ L = 0.991 inch Q (gpm) Ps - Pe (psi) C 5% 1 10 .672 23.25 x 100 2 35 .720 46.7 x 100 4 133 .738 93.5 x 100 6 267 .782 139.5 x 100 8 456 .798 186. x 100 10‘ 619 ..855 232.5 x 100 12 833 .887 279.0 x 100 14 1150 .880 326. x 100 10 12 14 L/D = 1.00 Q (99m) Hl-l MOODOwat-a -37- @ L = 0.490 inch P — Pe (psi) 10 35 130 263 475 626 867 1149 @ L = 0.1225 inch P S - Pe (psi) 10 33 117 245 422 594 816 .672 .782 .850 .868 .880 1 .672 .720 .790 .818 .830 875 :887_. 23.25 46.7 93.5 139.5 186. 232.5 279. 326. xxxxxxxx 80 23.25 46.7 93.5 ' 139.5 232:5 279. XXNXXX): -38- APPENDIX C . FLOWMETER CALgBRAngfl ibration curve for a turbine type flow pickup is a plot of low rate versus the flow pickup output reading in pulses econd. The pulse rate of the flow pickup is measured by ectronic counter. The flow rate through the flow pickup nputed from data for each pulse rate. The time required llect a given weight of fluid is measured. Given the t of fluid flowing per unit time and the weight density 3 fluid, the volume flow rate may be computed. g Flowmeter story number - 2011 ight oil rature: 150°F t 5°F fic gravity = 0.840 t density (water) = 8.345 lb/gal Yoil = Ywater x specific graV1tyoil 8.345 x 0.840 1b oil gal 7.0 ) cycles/sec: 1 gpm = 400 cycles x 60 sec x 50 lb oil sec min 7 1b oil gal = 171,500 Data 400 Cycle Count 88625 46045 31313 23503 18641 15602 13499 11800 10482 9388 8543 7808 7206 6685 -39... PulseslSec 25 50 75 100 '125 150 175 200 225 250 275 300 325 350 gpm 1.94 3.73 5.48 7.30 9.20 11.00 12.7' 14.5 16.4' 18.4 . . 20.1 21.9. _ 23.81 25.8~ 11-21-58 Date -40- Curve No. 8021-022-7 k _ 95:5 Vt....ean- 1gi ‘qil [lg Jill. 1. III. Figure 12 -41.. wve No. .1 -—o22- 7 Figure 13 -42 ... APPENDIX D PARTS EJST AND DflAflgNGS 0F APPARATUS Parts List 8021-022-0101 Drawing Drawing Drawing Drawing Drawing Drawing Drawing Drawing Drawing 67198€X 67199-X 67200-X 67201-X 67202-X 67203-X 672042X 672052X Microscope Mount Outlet End Plate Inlet End Plate Outlet Bracket Inlet Bracket Test Block Microscope Mount Slide Orifice Test Fixture Circuit Diagram 30 DATE POSTED k'llcxu: Inc. NAME DATE ML. "’ PA RTS LIST 021nm: 1‘sz Funny: PART NUMBER 8021-022-0101 SHEET NO. 1 OF E, o. 802 I -022 W. O. NO. I -l SH EETS APPLICABLE “3}? r5”, 7" ~ " , 1. 1 SPECS. IICHOFILMED MACH. PER V. SPEC. MT- UNLESS OTHERWISE SPECIFIED THE FOLLOWING NOTES APPLY: MACHINED SURFACE FINISH I \/ ' h - Ii i l‘ K E R ’ VICKERS INCORPORATED 0"- - ‘ ’ DIVISION OF SPEBBY RAIIO conPORATIOII . . DETROIT. MICHIGAN u. s. A. HEAT THEATIENT MACHINED SURFACES INTERSECTING AT RIGHT ANGLES TO BE 80. WITHIN ~- PER INCH CHK. ECCENTRICITY TOLERANCES TO BE WITHIN PROD. ENG. DEC. -- — FRAc. T. I. R. T. 0. REL. [I TOLERANCE on IIIACII. mus. . 050.: '~ ‘ FRAc.-I__- ..- Ami ---- T'°°"°' MATERIAL ' PROTOTYPE . TOLERANCE on CAST. mus. ——- REL- :230' SCALE REVISION I REVISION 73} . I , ' _ C Twig-1.30.12 worm-Id “92 0001 83.0: .vaa Ios H3 Mfio‘i . Q‘! ‘1 l ' . ' ‘-_-.-; ,V - . .- , .- - - _ - . _ ,.-.. r .-.—.4- » “1)»- ww ‘\ I .\ , I ...-«h..- 5'... ._.. ..-__._.._. - i l i ' I I I I I I I APPLICABLE SPECS. IICROFILNED MACH. PER V. SPEC. MT- UNLESS OTHERWISE SPECIFIED THE FOLLOWING NOTES APPLY: MACHINED SURFACE FINISH \/ . Ii i l‘ K E R ’ VICKERS INCORPORATED DR. " . 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WITHOUT WRITTEN AUTHORIZATION OF ma ,. , . , , REVISION VICKERS INC. ' V j. -~: , j _ , 0 DATE *1 ba-I 008 .A.a.u muammq 85:: M3 .1: 4: r‘ 'I" . } A. 5:: J I): $ IIIIIIIIIIIIIIIIIIIIIIIIIIIII