Drag Coefficients for Flat Plates, Spheres, and Cylinders Moving At

Home , Drag coefficient

DRAG COEFFICIENTS FOR FLAT PLATES , SPHERES, AND CYLINDERS MOVING AT LOW REYNOLDS NUMBERS IN A VISCOUS FLUID

ALVA MERLE JONES

A THESIS submitte

OREGON STATE COLLEGE

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE June 1958 APFROVED T Redacted for Privacy

In 0hrrg* of laJar Redacted for Privacy

Redacted for Privacy

Drtc tbrclr la prrrontr a h'4u.*l*^r-J-;t,r,,.lqql lfypcd by ftrcdeetr Or Joncr i

ACKNOWLEDGEMLNT

The author wishes to express his appreciation to Dr . J . G. Knudsen for helping with this investigation and to the Do Chemical Company for aiding this work through a Research Fellowship. ii

TABLE OF CONTEN TS

Pa ge

Introduction•••••• • • • • • • • • • • 1 Review of Literature . • • • • • • • • • • • 3 Theoretical Po ssibilities . • • • • • • 3 Experimenta l Data•••••• • • • • • 11 Analysis of Theoretical Solutions and Experimental Data . • • • • • • • • • • 12 Literature Containing Ge neral Theory • 14

Theoretical Considerations . • • • • • • • • 16 Definition of the Drag Coefficient • • 16 Obtaining Drag Coefficient by Dimensional Analysis • • • • • • • • • 19 Exact Solutions for Drag Coefficient • 21

Moving Bodies and Moving Fluid • • • • 25 Description of Apparatus ••••• • • • • • 26 Force Measuring Equipment . • • • • • • 26 Spheres, Cylinders, and Plates • • • • 30 Experimental Procedure ••••••• • • • • 35 Viscosity and Density Ca l ibration. • • 35

Velocity Measurements . • • • • • • • • 35 Foree Measurements • • • • • • • • • • 36 Experimental Results . • • • • • • • • • • • 37 iii

TA BLE OF CONTf!.N 'l'S (CONT .)

Page

Discussion of Results •••• • • • • • • • • 48 Correction and Accuracy of Measurements •••••••• • • • • • • 48 Analysis of Results • • • • • • • • • • 50 Comparison of Results with Other Data and Theoretical Solutions • • • • • • • 53 Summary and Conclusions . • • • • • • • • • • 57

Nomenclature • • • • • • • • • • • • • • • • 60 Bibliography • • • • • • • • • • • • • • • • 62 Appendix • • • • • • • • • • • • • • • • • •

Experimental Data • • • • • • • • • • • 64 Density and Viscosity Calibration • • • 89

Sample Calculations • • • • • • • • • • 92 iv

LIST OF I LLUSTRAT I O{S

Fi gure Page 1 Drag Coefficients for Spheres •••• 5

2 Drag Coefficients for Cylinders ••• 6 Dra g Coefficients for Flat Plates Parallel Flow •••••••••••• 8

4 Drag Coefficients for Fl a t Plate s Perpendicular Flow • • • • • • • • •

5 Block Diagram of Apparatus • • • • • 27

6 Apparatus - Le ft View. • • • • • • • 28

7 Apparatus - Ri gh t View • • • • • • • 29 8 Photograph of Spheres , Cylinders, and Plates • • • • • • • • • • • • • 33

9 Drag Force on the Wires - Ligh t Oil. 38

10 Drag Force on the Wires - Heavy Oil. 39

11 Da ta for Spheres • • • • • • • • • • 40 12 Data for Cylinders - L/D : 16, 24, 32 • • • • • • • • • • • • • 41 13 Data for Cylinders L/D c 2 and 4. • • • • • • • • • • • 42 14 Data for Cylinders L/D : 6 , 8 and 12 • • • • • • • • • • 43 15 Data for Fl a t Plates - Parallel Flow 45 16 Data for Flat Plates - Perpendicular Flow - W/L : 2 • • • • • • • • • • • 46 v

LIST OF II.LUSTRI''. TIONS ( CONT . )

Figure Page 17 Data for Flat Plates - Perpendicular Flow - W/L : 1, 4 • • • • • • • • • 47

18 Dependence of Viscosity O'!l 'l'emperature - Light Oil • • • • • • 90 19 Dependence of Viscosity on 'l1emperature - He.avy Oil • • • • • • 91 vi

LIST OF TA BLES

Table Pa ge

I Description of the Sphere s , Cylinders , and Plates •••• • • • 31 II Data for Spheres • • • • • • • • • 64 III Data for Cylinders • • • • • • • • 67 IV Data for Flat Plates - Para l lel Flow • • • • • • • • • • • • • • • 77 v Data f or Fl a t Plates Perpendicular Flow • • • • • • • • 82

VI Dependence of Density on Te mperature•••••••••••• 89 DRAG COEFFICIENTS FOR FLAT PLATES , SPHERES, AND CYLINDERS MOVING AT LOW REYNOLDS ~UMBERS I N A VISCOUS F LUID

LJTRODUCTI ON

The study of laminar flow of very viscous fluids over immersed bodies is important in many engineering problems . In the field of aerodynamics the study is becoming in creasingly important because as the speed of aircraft in creases, there is a tendency for the occurrence of a re gion of laminar flow on their surfaces, due to the low density of the air at the high speeds . Furthermore, the mainte nance of extensive laminar flow is desirable in order to minimize the friction drag . Other problems include the theory of lubrication and the flow over banks of tubes in heat exchangers . Many of the polymers formed in the field of plastics are highly viscous materials and problems, such as the power requirement for mixers, are encountered in flow over immersed bodies at low Reynolds numbers . At present there are only a few theoretical solutions and approximations and almost no experimental data on flo over spheres, cylinders, and flat plates in the range of Reynolds numbers from 0 . 01 to 10. The force of resistance is related to the r.eometry of the immersed body and the properties of the fluid by 2 a non-dimensional drag coefficient which is defined by the following equation:

{1)

The drag coefficient is also a function of the Reynolds number for geometrically similar bodies. Thus, if the drag coefficient and the Reynolds number are known, the force of resistance for flow over immersed bodies or bodies moving in a fluid can be predicated . The present investigation involved a determinati n of the drag coefficient as a function of the Reynolds number and geometric ratio for spheres , cylinders and flat plates at Reynolds numbers ranging from 0 . 01 to 10. The drag coefficients were determined by measuring the force of re sistanco and calculating the drag coefficient by the use of 'Equation (1) . For each drag coefficient a Reynolds number las calculated . From a plot of the data it was possible to determine an e xpression relating drag coefficients , Reynolds numbers , and L/D and W/L. The data and empirical equations have been compared to other available data and theoretical solutions . 3

REVIEW OF LITERATURE

Theoretical Solutions

A large number of investigators have analyzed laminar flow of a viscous fluid past various immersed bodies . Their analyses have resulted in expressions for drag coef... ficients and boundary layer velocity profiles . In their work they have made various assumptions which account for fairly wide discrepancies bet een the results of individual investigators . In addition, little experimental data are available to compare with theoretical work. Stokes (14, p . 55) was one of the first investigators to study the motion of a very.vfscous fluid over an immersed body . In 1850 he published the well-known solution for the motion of a sphere, whereby the force of resistance is given by the following equation: F : 6ffA vr (2) • By substituting the definition given in Equation (1), the drag coefficient for fluid flowing past a sphere at low Reyno l ds numbers is fd - 24 (3) -re • Equation (3) holds for Reynolds numbers up to nearly 1 . 0 . Oseen (11, p . 122) improved Stokes' analysis 4

by linearizing the Navier· Stokes equations . The drag coef ficient of the sphere by Oseen 's analysis is

f - 24 1 d - Re (1 r 3Re) . (4) I6 Equation (4) is good for Reynolds numbers u p to 5 . Vfuile Oseen's work wa s published in 1910, his method of linearizing the equations of flow has been used by recent investigators in studying the flow- of fluids over elliptic cylinders and flat plates . Horace Lamb (8 , p . 112-121) as another early con tributor td the study of the flow of viscous fluids over immersed bodies. He presented a simpler demonstration of Oseen 's results and further developed their scope and significance . Also, he a pplied the same method to flow past a circular cylinder. Lamb's solution for the drag coefficient of circular cylinders is

f - 8 ff (5) d - Re (2.002 - ln Re) Equation (5) is good only for Reynolds numbers up to 0 . 5 . Bairstow, Cave , and Lang (2, p . 383- 432) extended

Lamb ' s solution to eover lar ~. er values of Reynolds numbers .

Their solution is plotted in Fi5~re 2 . Goldstein (3 , p . 225•235) has solved Oseen's equations completely for fluid flow at small Reynolds numbers past spheres. His solution take s into account the higher 5

It I I ·i.

. - -· . . ·- ~. . . : L! : ~. · _: <· :--·-~ ::i. -- -- I' STOKES OSEEN I LIEBSTER 0 0 I 00 I GOLDSTEIN-·-·- - i "'"' -.:. ". -" 50 ~-+~~-+--+i'."~~ H-',. ~~--~-4~. ...+. ~- ~· ·~· ~· ~-_~HH I "\ .:·.1:-_·

• •··-tf'-

! 1"1\. . . . ~ ~. . - I . • .J!.

I' : · ~ . t--i 2 ~--~+-+-~4-4-~ -~H---~~~~~~~~~

f .L • l

0.1 .2 .5 1.0 2 5 10 Re DRAG COEFFICIENTS FOR SPHERES

Fl GURE I 6

• • WIESELSBERGER o o INAI --LAMB ... • • ALLEN a SOUTHWELL r.r - · - TONOTIKA a AOI - · BAIRSTOW,CAVI a "" . LAN I 1 ' .. ,, - ~· ... - -· .. . 'a ' - ' ... e .. -= J ·..: • • • J'Io.

I l---,_:-::::_ :+-~.:;__-+--_~..-+-+-+-l-+-+-+--+-+--'H---t:::::::---i-7--+-+---t---t--tl•·· t-t--t-t--r-t--r.::t ,. • , 1 I,. ~-':-. -...... ::::::-- . .r. '1 "t:t1j iffilfl .' if rt:C: =~; · h:: ' .:tn ~ ~ .... :r~ wrw~ ;~ ~ u · ~~ 1~ ·~-t ·•·• 'tl=t f.§ ;s ~ 1 o.L-•~~~~~~~~~~~~~~.~.o~--~~~~~~~~~,~~~~.o .2 .e 1 Rt DRAG COEFFICIENTS FOR CYLINDERS

FIGURE 2 7

powered terms in the series solution that were omitted by Oseen . The solution is plotted in Figure 1 . It covers values of Reynolds numbers up to 10 . In recent years several people have developed approxi mate solutions of drag coefficients for flow at a low Reyno l ds number over ell iptic cylinders for various ratios of major and minor axes and angles of incidence . For the major axis equal to the minor axis, the result is a circu lar cylinder . For a ratio of major axis to minor axis of infinity, the resul t is a flat plate with parallel flow for a zero anglo of incidence and a f l at plate ith perpen dicular flow for an angle of incidence of ninety degrees . Tomotika and Aoi (15 , p . 290-312) have obtained e xact ntJm'3rical solutions of Oseen ' s equations for steady flo past an elliptic cylinder in terms of elliptic coordinates . When the calculations are based upon Oseen's equations , they found that the total drag can be analyzed into pressure and friction drag proportional to the axes of the cylinder for any Reynolds number . Their solutions are plotted in Figures 2, 3 and 4 and cover Reynolds numbers from 0 . 4 to 4 . 0 . Imai (4 , p . 141- 160) has presented a numerical solution to flow past an inclined elliptic cylinder for Reynolds numbers of 0 , 1 and 1 . 0 . His method is essentially one of f I t '

501----+--+-+--+-JU..L

o o INAI - JANSSEN •• ••• TONOTIKA a AOI

~ ...... • .. t ... •

. ~ .

. ~ .. ~ ...... - ..... ' ·.

:-=:.. .. :-...!'. . · ~

.... ' ...... •t. ! .. . •.:J • . • I ' ! 1 0 .1 I I '!" ·;· T p 0 .1 .2 .5 1.0 2 10 Re

DRAG COEFFICIENTS FOR FLAT PLATES PARALLEL FLOW FIGURE 3 9

100 I I I - I I ! I I I --- -r f-- --;-- - r- :l -- - ' --+- I - ---- ~0 . - ·-- -· + --f.- -- ·-t- ~-f--l I MAl -f--·t-- I I 0 0 . - -- .--- - r--'- --r - t- -- TOMOTIKA a AOI - - 1-t-- ' ! ··- f------~ I I I . ' I I 20 -- H~ I l I . I i I 1 )., ...... I i ~ I I I t-... I I I ! ~ 10 ---~-- I " ...... I -· -- 1'-. I ., I ' ' r-..... - I f'... I ...... - -- - l - r---1--t- ...... --r-' f---- c·------r-- ' --·~ f-1---f- ' 1 -~-~ - i·-- --l=l-- -- f-.- : - - --- f-.- _,__ - - -- 1---J. ~- --r- I I I I I I I I I I I J --r-f--1- 1-- -- r-f- -- -1- I : I H- I I I I I 2 I I I I I ~- I ' I I I I I I i 0.1 .2 1.0 2 10 Rt DRAG COEFFICIENTS FOR FLAT PLATES

PERPENDICULAR FLOW

FIGURE 4 10 successive approximations in poter series of Reynolds numbers . The solution is shown in Figures 2, 3 and 4 . Allen and Southwell (1, p . 129-145 ) have used the relaxation methods to determine the motion of a viscous fluid past a fixe d circular cylinder. Their solution covers Reyno l ds numbers from 0.1 to 10 and is plotted in Figure 2. Blasius (7, p . 66) investigated the laminar flow in the boundary layer of a thin flat plate immersed in a stream flowing parallel to the surface of the plate. By making several assumptions he obtained an exact solution of the simplified flow equations . One of the most recent developments in the study of flow over immersed bodies at low Reyno l ds numbers is that t y Janssen (6, P • 173-183), who used an analog computer to determine drag coefficients for flat plates in parallel flow . By defining vorticity ( ..<.. ) as

o1... : d v_ J u (6) d X d Y and the stream function ( t.f } as u = d~ ' v = .. Jtf' (7) d y d X where u is the velocity in the direction of the x - co• ordinate and v is the velocity in the direction of the y coordinate, and making the proper substitution in the Navier-Stokes equation ; he obtained the following two 11 equations : Vlo(_ • _1 [- d

Experimental Data

Very little experimental data has been obtained for drag coefficients of flat plates, cylinders, and spheres in the range of Reynolds numbers from .01 to 10. There is no data for flat plates in perpendicular flow. Janour (5, p . 1-40) obtained drag coefficients for parallel flow over flat plates. However , his data only covers Reynolds numbers down to twelve, which is above the range being considered in the present work. One significant result of Janour's work is establishing a lo~er limit for the well-known Blasius formula,

fd : 1.328 1/2 (10) (Re ) ' 12

4 of about 2 , 0 X 10 • The equation proposed by Janour for Reynolds numbers of 12 to 2335 is : 2 . 90 fd (""He) . 601 {11) Drag coefficients for flow over cylinders have been experimentally determined by Wieselsberger (16, p. 22), His data covers Reyno lds numbers from 4 to 100. The data for very long cylinders is plotted in Fi£Ure 2 . VJ iesels berger also studied the effect of the len gth ~to- diameter ratio on drag coefficients . He found that the drag coef ficient decreases with a decreasing L/D r a tio at a constant Reynolds number. However, his data for L/D other than infinity was obtained at Reynolds numbers above 40 . Relf (13, p . 47-51) measured the resistance of flow over cylinders but only for Reynolds numbers above ten . Liebster ( 9 , p . 541-562) measured the resistance of flow over spheres . His data cover s the range of Reynolds numbers from 0 .13 to 10.1. His data is plotted in Fi£ure 1.

Analysis of Theoretical Solutions and Experimental Data

The data of Liebster (9 , p . 548) provides a good check for the solutions of Stokes (14, p . 55), Oaeen {11, p , 122), and Goldstein (3, p. 234) for flow over spheres at Reynolds numbers less than 0.5. As Figure 1 shows, the results are 13 in good agreement in that range . As the Reynolds number becomes greater than 1 . 0 it is known that Stokes' formula does not hold true . The results of the other workers are very close up to a Reyno l ds number of 2, so that all of their data is probably very good in that range . Above a Reynolds number of 3, Oseenta solution is proba bl y not very go od since it was only an approximation. At a Reynolds number of 10 Liebster's data is about 25~ lower than Goldstein's solution, so the true solution is probably somelhere between the two values . Since Lamb's solution (8, p . 112-121) for flow over a cylinder was based upon the method of Oseen, his solution is probably very go od for Reyno l ds numbers of less than 1 . The solutions of Tomotika and Aoi (15, p . 302), Imai (4, p . 157 ), and Bairstow, Cave, and Lang (2, p . 404) seem to substantiate this fact since they all agree with each other as shown in Figure 2 . The only solution which does not agree is that _of Allen and Southwell (1, p . 141). For the range of Reynolds numbers from 1 to 10 the different results vary considerably. Lamb's solution is not correct. The results of 'l'omotika and Aoi, and Bairstow, Cave, and Lang, as shown in Figure 2 are very close. How ever, the data of Wieselsberger (16, p. 22), the only experimental work for cylinders, is 30t below the results 14 of t he other workers . It is interesting to note that the s olution of Allen and Southwell coincides with Wiesels bergers data. in this range .

Very little ~ork has been done for flow at low velocities over f l at plates , both parallel and perpen• dicular to the flowing stream. For parallel f low at very low Reyno l ds numbers , the solutions of Imai (4 , p . 157) , Tomotika and Aoi (15 • P • 302 ), and Janssen (6, p . 183 ) are very close as shown in Figure 3 . For Reynolds numbers near 10, Janssen's solution is below that of Tomotika and Aoi . For flat plates perpendicular to flow, there is only the theoretical data of Tomotika and Aoi (15, p . 302 ), and I mai (4 , p . 157) . Their solutions , as before, nearl y coincide.

Litera ture Containing General Theorx

Several excellent books and monographs containing the general theory of flow over immersed bodies , particul arly at low Reynolds numbers , are available . Knudsen and Katz (7, P • 64 ...105 ) give a good discussion of flow turbulent and laminar, past thin flat plates , circular and elliptical cylinders , and spheres . Boundary l ayer theory and boundary-layer equations are included. 15

The Blasius solution is described in detail. There is a section on drag coefficients with many graphs of different data . However , most of these do not cover low Reynolds numbers . Several chapters of the book by Pai (11, P • 100- 260) pertain to drag at low Reyno l ds numbers . In addition to the fundamenta l equations of f luid dynamics , there is excellent material covering the Navier-Stokes differential equations, theory of very slow motion, and the boundary layer equations . His description of the Oseen method of linearization (11, p . 122) is particularly good . Prandtl (12, p . 98-196) has several good sections on flow past immersed bodies . Among these are the sections on the motion of bodies in viscous fluids (12, p . 105-110) and the resistance of bodies immersed in fluid (12 , p . 174 178 ) . There is also a section containing the experimental results of fluid resistance . Included is drag coefficient data for spheres, cylinders, and plates at all Reynolds numbers , Though short, Janour {5, p . 1-40) has a good discussion of the general theory of the resistance of bodies in l aminar flow . 16

THEORETICf\ L CONSIT'ERATI 01\TS

Definition of the Drag Coefficient

The resistance or drag of a body movin g in a liquid or gas or exposed to a medium flowing past it is a compli cated function of the geometric properties of the body and physical properties of the medium . The resistance depends upon the size of the body, geometric shape and position, quality of surface a nd the velocity, viscosity, and de nsity of the medium . Newton postulated that the resistance with which a fluid opposes the motion of a body immersed in it through the force of its inertia must be proportional to the area of the section of the body at right angles to the direction of flow, and also proportional to the density of the fluid and to the square of its velocity. This result may be explained by the following simple ar~nnent (12, p . 174). In a unit of time the body must move a mass of fluid

m : f av (12) out of its way and in doing so imparts a velocity to each element of the fluid . This velocity is proportional to the velocity of the body . The resistance is equal to the momentum imparted to the fluid and is therefore proportional to 17

2 mv : p av , (13 ) where a is the projected area of the body on a plane normal to the direction of flow . In Newton 's theory the laws of collision of elastic bodies are applied to the resistance of a fluid. Jewton regarded the medium as consisting of particles free to move but at rest, which are regularly reflected by the moving body . The detailed results, however, have proved unsound . The Newt onian concept of fluid resistance has been replaced by the hydrodynamical theory; hereby, the re sistance consists of the pressure differences and friction stresses arising from the fluid flo ing around the body.

These resistances are sometime~ referred to as form drag and surface drag. A fundamental difference between the old and new theories is that in the former only the shape of ~ front portion is considered; whereas , it is known that the phenomena giving rise to resistances are largely due to the shape of the rear portion. In general the pressure differences predominate and may be taken as proportional to the dynamic pressure corresponding to the velocity; that is, as proportional to 1/2 f v2 • The resistance, being the product of pressure differences and the area exposed to it, is proportional to 1/2 f av2 • 18

There are several methods of defining the drag coef ficient . In Germany , the United Statea, and most countries the drag coefficient is defined as

where F ...- force of resistance, density of the fluid, ~ = ap - projected area, v : velocity, and fd - drag coefficient . This is the definition used in the present work . In soma countries, particul arly England, the drag coefficient is defined as

{14 ) where the symbols are the same as defined in Equation (1) . The data of Tomotika and Aoi (15, p. 302), Goldstein (3, P• 234), and Bairstow, Cave , and Lang (2, p . 404), based upon Equation (14), has been changed so that it is defined as in Equation (1) and can be compared easily with that of other investigators . For the flat plates in parallel flow the drag coef ficient is defined as 19

2 F : 1/2 f f aw v go , (15) where F, , and v are the same as in Equation (1) and aw : wetted area.

Some investigators define the drag coefficient as follows 2 F : 1/2 fd f b v (16) go where F : force of resistance par unit width , and b : a characteristic dimension such as diameter for cylinder and length for a flat plate . It is easily seen that when Equa tion (16) is multiplied by the width, it reduces to Equation (1) for cylinder s and flat plates in perpendicular flow . Also Equation (16), when mul tiplied by the width , reduces to Equation (15) for the case of flat pl ates in parallel flo 1f only one side of the plate is being considered.

Obtaining Drag Coefficient by Dimensional Anal ysis

The drag coefficient may also be obtained by di mensional analysis. There are several methods for getting dimensionless groups butthe meth od used here is the ;r 20

Theorem described by McAdams (10, p. 30).

The factors involved are b, v, F , ~ and g • It is f , 0 necessary to include gc since both mass and force terms are involved. If the dimensions are solved in terms of the dimensionally incompatible factors, the following is obtained:

L : b , (17)

g - L b .. -v --v , (18) 3 M : f L3 =f b , (19) F e: F . (20 ) Each of the remaining factors, { g0 , ~ ), must produce a dimensionless group when its dimensions are eliminated by one or more of tho above four equations. Thus ,

gc : ! !!2 -- f b2 v2 (21) F e F and

A : _!_ :: {22 ) Le fbv Equations {21) and (22 ) yield the following dimensionless groups : 1T 1 = F g c - (23) and 21

1T 2 : P bv : Re • {24) .A If a is substituted for b2 and 1/2 f v2 for f v2 then Equation {23) is the same as Equation (1). Also one dimensionless group may be expressed as a function of another so that

f : ¢ (Re) • (25) d

Thus , drag coefficients for constant Reynolds numbers and ge ome tric similarity have the same value . Dimensional analysis lacks the pictoral quality of dynamic similarity considerations, but it has the ad vantages of not using the knowledge of the equations governing the problem .

Exact Solutions for Drag Coefficient

The possibilities of an exact theoretical solution of the laminar, steady flow about bodies and the calculation of the resistance are examined . The laminar motion of a viscous fluid is governec by the Na vier-Stol{e s equations , which for two - dimensional, incompressible flow in the absence of external forces are

- .=.c.g (26 ) f and 22

{27)

where x and y : distances in the coordinate direct1oqs, u and v : velocities in the x and y directions, respect!vely, t • time, p : static pressure, and

2 '\1 : Laplacian operator. For the case of steady flow the terms Ju and dv are Jt Jt zero . The Na vier-Stokes equations are supplemented by the, equation of continuity which for an incompressible fluid is J u f J v : 0 . (28 ) Jx n Pal (11, p . 37) gives a good derivation of Equations (26) and (27) . The following boundary conditions may be applied: (1) As x approaches I- "'and y approaches I- cP the velocity equals a constant, and (2) At the wall, the · normal and tangential components of the velocity v nish. A solution to the Navier-Stokea equations would give u, v, and the pressure distribution. The drag force could be calculated from these unknown quantities . The equations are non-linear and their general solution is unknovm 23 because a superposition of particular sol utions is impossible . Howeve r, solut ions can be obtained if the equations are simplified . If viscosity is assumed zero, the Euler equat ions of motion for an ideal f luid du j. du I v d u: -~ ~ (29) d t U d X c) Y ( J x and

(30) are obtained. The integral of these equations a long a streamline gi ves t he Bernoulli equation, which expresses the law of the conservation of energy . A streamline is tangent to the velocity vector at every poin t . For the case of steady flow Blasius assumed that the 2 thickness of the boundary layer is s ma ll, J u is less than I JY!Z d 2u and that v is less than u . With thes e assumptions the ;r-y;; following equation is obtained: d u f ) u urx VTY (31)

Equation (3l)t along with the continuity equation, completely describes the flow in the laminar layer. Blasius obtained an exact solution of these equations. The non-linearity of the Navier-Stoke s equations lies in the terms on the left side of the equations . If these 24 terms are neglected the equations simplify to

2 = g ~ (32) .AAV u c(JX and

2 = g ~ (33) 'V v c J y • The solutions of these equations for flow about a sphere was derived by Stokes (14, P - 55) . Equations (32) and (33) are good only at very low Reynolds numbers when the viscous forces are large compared to the omitted inertia forces .

Oseen improved upon the Stokes solution by replacing the inertia terms u d u v du u d v and v dv by the rx, JY, rx 7Y approximate terms u d u v Ju u J v and v dv o rx, o e} y , o ;;rx , o d Y where u and v are the constant value of the velocity 0 0 components, u, and vat an infinite distance from the body . Near the body where the values of u deviate from u the 0 inertia terms are small compared with the viscosity terms so that the Oseen equation becomes the Stokes equation. Thus , for very low Reynolds numbers, high viscosity or small dimensions, neglecting the inertia forces will give a good solution to the Navier-stokes equations of flow . In all cases, this t ype of flow has the property that the resistance to motion is proportional to the velocity, which 25 means that the drag coefficient must be inversely pro• portional to the Reynolds number .

Moving Sodies and Moving Fluid

The question arises as to how the resistance of a body moving in fluid at rest is related to the force exerted by a moving fluid on a body at rest . Prandtl (12, p . 179) explains that as long as the fluid is moving perfectly uniformly there is no difference between the two cases, The superposition of a common uniform motion (equal and opposite to the velocity of the body , so that the latter is brought to rest) makes no difference to mechanical phenomena . If flo is not perfectly uniform with respect to the body or if the flow is turbulent, the resistances are usually greater for a moving fluid on a body than for a body moving through a fluid . 26

DESCRIPTI ON OF APPARATUS

Force Measuring Equipment

The force measuring equipment was connected as shown in the diagram in Figure 5 . Figures 6 and 7 are photo• graphs of the apparatus . The apparatus is constructed to move various bodies vertically through a viscous fluid . It consisted of a 1/6 horsepower motor coupled to a Revco speed reducer . A four-step V-pulley with diameters of 3/4, 1-1/4, l-3/4 and 2-l/4 inches was installed on the speed reducer. The drag force as measured by means of a 2-pound spring scale with 1/2 ounce divisions purchased from Scientific Supply Company . This scale was calibrated on a platform scale measuring to the nearest 0 . 001 pound . It was connected to the four step pulley by means of a nylon cord. A capstan arrangement with a single turn around the pulley as used to connect the scale to t he pulley. A wei ght was placed, as shown in Figure 5 , at the end of the cord . Several different wei ghts were used in order to counterbalance the varying weights of the cylinders and spheres . With this arrangement a wider range of velocities was obtained. A fine wire 0 . 003 inch diameter was used to connect 27

SPEED REDUCER

MOTOR

WEIGHT

-SPRING SCALE

SPACER -F====t

.-FINE WIRE

I I COOLING WATER I

I I EXIT L ___ J 1PLA1'E 1 L_-- J

I I OIL DRUM

I I I I L------' WATER "'ACKET COOL lNG W•TER INLET

BLOCK DIAGRAM OF APPARATUS

FIGURE 5 28

APPARATUS .. LEFT VIEW FIGU'RE 6 29

APPARATUS- RIGHT VIEW FIGURE 7 30 the plates, cylinders, and spheres to the scale. Fifteen gallon oil drums set inside of a 31 gallon barrel we~e used for performing the experiment . The oil drum was set upon a bracket inside the barrel so that cool ing water could be circulated all around the oil except for the top. Two types of heavy duty gear oil were used; Shell SAE 140 and Richfield SAE 250 . Viscosities of the two oils are shown in Figures 18 and 19 and densities in Table VI .

Spheres, Cylinders, and Plates

The objects for which drag measurements were obtained are described in Table I . Figure 8, wi th two exceptions, is a photograph of the spheres, cylinders, and plates studied in th~ experiment . A 1-1/2 and a 2 inch sphere were substituted for the 1/4 and 1/2 inch spheres since the small spheres were too small to register a force on the scale . Also, the 1 x 2" plate for perpendicular flow is not shown . Holes were drilled in the spheres and the ends of the cylinders . Ordinary household cemen t was used to connect the 0 . 003 inch diameter wire to the objects . Small holes were drilled in the corner of the plates and the wires were tied to the plates . For the plates in parallel flow, three 31 TA BLE I Description of t he Spheres, Cylinders, and Plates

s;ehe res No . D-in. Material

1 3/4 steel 2 1 steel 3 1 1/2 steel 4 2 steel

Cylinders -No . L-in. D-in. Material 1 2 1/4 steel 2 2 1/2 steel 3 2 1 steel 4 2 1 1/2 aluminum 5 4 1/4 steel 6 4 1/2 steel 7 4 1 steel 8 4 1 1/2 aluminum 9 6 1/4 steel 10 6 1/2 steel 11 6 1 steel 12 6 1 1/2 aluminum 13 8 1/4 steel 14 8 1/2 steel 15 8 1 steel 16 8 1 1/2 aluminum

Flat Plates - Parallel Flow -No . W•in . L-in. Th-in. Material la 4 1 3/64 steel lb 1 4 3/64 steel 2a 4 2 3/64 steel 2b 2 4 3/64 steel 3 4 4 3/64 steel 4a 4 8 3/64 steel 4b 8 4 3/64 steel 32 Flat Plates - Per12endicular Flow !.2..:. W-in. L-in. Th -in . Material 1 8 2 7/64 aluminum 2 5 1 1/2 7/64 aluminum 3 4 1 3/64 steel 4 2 1/2 3/64 steel 5 8 4 7/64 aluminum 6 6 3 3/64 steel 7 4 2 3 / 64 steel 8 2 1 3/64 steel 9 4 4 3 / 64 steel 10 3 3 3/64 steel 11 2 2 3/64 stee l 12 1 1 3/64 steel .... ~

• J~ ' -- __4t

1 I l ,,,, ,11'''" i~.. ------

---·-1""~ II ~ ------' ~ FIGURE e- PHOTOGRAPH OF SPHERES, CYLINDERS, AND PLATES 34 holes were drilled so that each plate could be used for two geometric ratios by changing the wires . (See , for example, plates la and lb in Table I} 35

EXPERI MENTA L PROCEDURE

Viscosity and Density Calibration

A calibrated hydrometer measuring to the nearest . 0002 was used to measure the density. Table VI shows that the effect of temperature on density is practically negli gible in the small temperature range used . A Brookfield Synchro-lectric viscometer was used to measure the viscosity of both the light and heavy oil. Figures 18 and 19 show the effect of temperature on vis cosity. In addition, the viscosity of the light oil was checke d using the falling ball method and the equation 2 --

Velocity Measurements

The velocity of movement through the oil was measured by determining the rate of rotation of the pulleys with a stop watch . Usually the time for 10 revolutions was measured at the highe r ve locities and for 5 revolutions at the low velocities . From this information and the di... amaters of the pulleys, the velocities ere calculated. 36

The time was measured to the nearest tenth of a second . Since the measured time was usually between 20 and 40 aeconds 1 the error in ~easuring velocity was considered to be less tha~ 0 . 5~ .

force Me asurements

The object connected to the scale 1 w.as dropped to the bottom of the oil drum . The motor was started and the scale was read as the object vms being pulled towards the top of the drum . Two or three readings were taken for each object at each velocity. In nearly all cases these readings were the same . 37

ti: XPER I MENTAL RE STJLTS

The drag coefficient and the Reynolds number were calculated by the use of Equations (l} or (15) for each of the spheres, cylinders, and plates from the measured

quantities of force and velocity a~d the values of the vis cosity and density corresponding to the temperature of the

oil. It was necessary to ~ubtract from the measured force the force on the wire . The corrected force measurement was then used to determine the drag coefficient . The force on the wire has been determined as being proportional to the velocity. A correction curve relating force on the wire and ve l ocity is plotted in Figure 9 for the light oil and Figure 10 for the heavy oil. The calculated drag coefficients, Reynolds numbers and velocities along with the measured force for the spheres, cylinders, flat plates - parallel flow, and flat plates perpendicular flow have been tabulated in Tables II, III, I V and v, respectively . The calculated drag coefficients have been plotted as a function of the Reynolds numbe r on logarithic graph paper with geometric ratios as a parameter . Drag coefficients for the spheres are plotted in Figure 11 . The data for the cylinders are plotted in _,CD .03 ..•. 0 G: T ..0 . .02

.01

.10 .20 .30 .410 .50 .60 .70 .80 VELOCITY- FTJSEC.

DRAG FORCE ON THE WIRE-.LIGHT OIL

FIGURE 9 I '-, ! : I ' : -· ! .-- !- -1 _i -i . I --~ I I' _, .. ..-·- , I : ' i

_I_ .. - _; . .... ·- !.. LL I l . l t· - ·1.· . . ~- :,- .. -- -+i' · ! ,... : ! I -. -.;.-+-.cl . - ! . l ,...... ! ! 1 ! 1 I I : I!V j c,---;- --r--··t·' r-··--t··--:- ______L __ --~- --1 ..., ... ··r-r-·t- 1 -f-f-T- _::..~. +-L--1---~- 1-- l ' ' I ~I - - I I-+-l --R·-- ' . 'i , ". ~~ ..i. -~~ ~- -T f .. i r ~-- -- i- ----~-- , - ·1 .. . . I i ! ' ! I i I /,' . I 1--- '-· . !- f· i----1---+- - i-· -~+--: .. --~- . --~-- --;-- -t+ ! I . v-~~ -· . j

1- - · --t-: 1---t- i t----t· ' --~--I -· ,· i-' '1!_,.., _ I ', t·--'-+-+1-+--liI /.~/. '+· -+--+-+-1-+-'+'-+-+--tc--1-+-t-1.1i . - ~ i : I ! ' , i ! v i ··· [:!~ ,:v": +L~ + ~. -! I ~~j- + :.r V " I ~ t.--- -~-- I +---~-- I f-· --,-1-- ~ -- --!-) Li--+--+--+-+-+-:+-:-1--+--+---t---4 -1--1--+-+--+-l-i tl~ I . ' ' I Q Y +l'~~!-+-++++-·HH-++-+-+-+-'-H--+ii + -i. t I· i i : 1 . ! j _...V I . :: f 1 r-t~-· l--r-t -~ 7 ·. 1 - ; · ·· .... I

DRAG FORCE ON THE WIRE- HEAVY OIL FIGURE 10 40

+ ' l i~ <> !~ '• r-r ' I ' ' . ; i t , ,_l 1 lf-1-1 l+r+ ,..., fJ-Ct I+ I I t li 1~t rtH r+l rf-l It llil : l l~£: 11 1..; ~· ~ ~ ·,t ~ ! It,: lqf. :L ':: :~ I _; ~ _;:£~- ' §' I:· t I:!+-:-= r 17: ' -E;'r\ I+ r t::"" - ~ 111=:::: h= iU=ff=t! 1 +.~ :t_ I "· . .: 1:;:. t= I:E:' I'" 1:::: ±~ , .. k±::i!-;;; -STOKES EQ. l h+· (~ ru: :. H·Hti+H1 1'1 l t~ 4 ';:, c l'ffii l;:ri ~ ~· ~ ff l ~' :.: ~h i lt 1 ' ¥~ · ., I"'' ~ I'!"' I T ~ >"' ...... , . .;.,.., l+t H+h l+ ' i ,.,.j l tfl-l I'·· ft+++ lf+' I·· I+ I+

·t •i I 1·1:, I'" ftt-1 · I · r 11 I IH Ij ~,.; ,, ~ ' *~ ·.i .J F 1::=: 6'= =f l ~:iit rtti l lit~ I FS l:f~ l=i-+ tt. lr 1 ~ 1 -t.., !=l=;:R' ttl l- 11ffi 1ft :.!i-' 1 ~ I+ I { ~~ l .f!.lJ ::t I ;lfl: m ~!~W FB L:t IF I Hi'r tt: 1-+++· 41±811. ft;. i.t:tt± i :" I ~ I (~' ffi:trHf1 Itt· ~ l r i ; H- t-r r . I • I 11 H++ HHt m''' I ' I ~ 1._... ,...,_. F •· I·· t-"- 1- 'T , ... h iT f-t+ ftt I+ I l:t .... 1 + "T I·! 1 t , _± . ·~~ ~- 1:.1'!: '!: ?.=a~ 1~ '" - ::;:: =!itf lttti H I ...... , = DATA FOR SPHERES

FIGURE II 41

I :-1---1:-1-+-.;..-+--Ti-+------+--:--r-- --r--- - + t----+! ----4-~---+-f----f--+-f--l--1 I t-- --t-- ---+- 'J-+-~f--'~~ -___l_ ~- - - 5000 4 i 1 L~L~-~tr-l----H~ -----~-f----.--+------+-----+----+---+·-t-·-H ~--~--~----;:---+------+-+--+--+- +-~-~---'-----'----'--'-- ! -1 : r - ~ -~- i - -- -+---!---- f--- f- LID =16,24, 32 2 0 0 0 1---i------+---,-+---+--,---1---t--+-+ LID =12 I I I L/ D= 8 t---"'~1 - -- j _j - -+--+-if-++ ,, I L/ D = 6 ~ _0 '!- '\, : --- LID=4 r-- 1000 I I LID= 2 r- I-

~00 p

-""0

--+-l-+-1--+------+--+---+---4-1-

10~--~~~~~~~~~~~~~~--~~~~ .01 .02 .05 .10 .20 .50 1.0 Re DATA FOR CYLINDERS- LID= 16,24, 32 FIGURE 12 42

:1:-: • F - t~::.: S:R rtf!:; f$ .,, ...... -~ .• .._ · !±'-! 11. ~ 't:' ;± jit 1 ~1 ftl ·. . l '"~r I I '• ~- -J

t-+ t ttt. l+i ti ~ Ill 1111 --1)-0-- L/ 0 • 2 -- o-o L/0•4

,..,

.I.; I

~ . 1:ill ie ~ :± - *' - lq .t- I I ·r- I II

I I I ..... "' h4 l ~j:j IR:: 1:!_ t:;:!t - ~t== 't It-' n ~ tt,~ I"'it ' 1 -h~ I T ~ ...~t· £ -'- r-+-' .F. 7 1ffi I T"'1 r - - 11 - · ~ 1='::

It ti: H --+++ +t, ~ * Ll·· T jT' 11111 lt lttn .02 .05 .ro .20' .50 1.0 2.0 Re DATA FOR CYLINDERS- L/D= 2 AND 4 FIGURE 13 L_ 43

I I

' I ~000 I I I i i .. . . -~ f- --' ~ . . ··•1 • 2000 ; -- I •• LID • 6 ~ -· - --o--o-- L/ D • 8 ' ~,_- ' 1000 --o-0-· L/Dc 12 _"\..' ,'-..·, 0 ,, ... \.. ;.,.' . :'\. ~00 p -- '-' ~ ( ~~' ~ · li '!.'~ ~cp, ~ ~ Qi"'!y_ 200 "' (~ . ~~0' ( r:l-~~ ~~~ ~~ 13 y I f'-~ .. c ~ );j ~.. .. ·- 100 ::1' !1<• . - ::> r-' "" -~ " • I> • ~ ...... - c. ~- ·"- t> 4 ,,.,'"' 11 ...l-~I) 50 • 'c~"'~ • I " ~ • ~ -. l:i p~ .., -:: ' 1,;.~ ~ ~ -~. • ~ ...'

Figures 12, 13, and 14. The data for L/D values of 16, 24, and 32 were nearly the same and have been plotted together i n Figure 12. In addition, the curves for the other L/D ratios determined from Fi b~res 13 and 14 have been drawn in Figure 12 so that the effect of the length-to-diameter is clearly shown . Figure 13 shows the data for L/D values of 2 and 4 and the curves determined from this data. Firure 14 shows the data for L/D values of 6, 8, and 12, and the curves determined from this data . The data for flat plates in parallel flow are plotted in Figure 15. A correction factor for the edge effect has beon used so that the width-to-length ratio is not a parameter in this plot. A portion of the data of Janour (5, p . 31) is also shown in the diagram . The data for fla t plates in perpendicular flow is plotted in Figures 16 a nd 17 . Figure 16 shows the data for W/L values of 2 . Also the curves for the three W/L ratios, 1, 2 and 4, have been drawn in the figure . Figure 17 shows the data for W/L values of 1 and 4 . The curves determined from the data have also been dravm in the figure . 45

10~ ~ ~--- - t=;=Ff1TR=+ iJ+-,-_-_-'-r_-_---+-+-..:.--+--+-+--_---"_;-~r-=r~'=.~+:!--=---=---=---=--~'=--=;_~1=:_--:=._~_::;-·~~--+-+-,t~

1'\. Ll~+--+--' ---jt I I l ~t L,--+ I Ir--'--;----+-f------+--+1----+-+-+---J-++------r-1-+-----'-1-t-+----t---+----:-+1--+--1 P------_l -- --1---L i

20 ~ -- I ~g. ; I --- -: ---+-- : r t L_ [-rl-' I_--t--+---+-t---i-- ~- t-r-t-t--1

' .}' ~ ~B 1,.) I : --o-o- JONES -' ..., '() - . - ~~ p f---j- -~-- e e JANOU R c :> ~c ? ~ ------JANSSEN I ,.0 0 ~ I IO ~2=i~~~~~~a=~~f=:j:= ---- TOM OTIKA • AOI ---t-+--+---t---t-H l= ' ' I ~~n ~~--~~~~~~o~~~~~--4- ' " NDCI>tl o ; -

~--~~~~~+--+~+--4,-r-~1+-~-·+1~-H--~~-~~-~~~i-+---t-~-+---r+~ o.s : I i ! i i'-4. ---~T I I ! : ~~. I

f-- t --- li-:----,--~--+-_--+--t-----~~-~::_;+---_-_-._-'-+------+-+;-__+-[- __+_- __+---___+-r-+--H----_ +--r------+_____-+-i~-t_----"'~_!l'i;-:-:::111~1.t-,--t"'+t--l

0.2 1---+!, ----+!------'-1--+---t-----t--+--++t-+---+-+--+----r----t-----t-t--++i-t------t------t-----t1-t---t--+----r--t"'NN;; --, --~- ~ +-~~-~~~4---t----+-++!~~~11~+-f-~~

0.1L----1.---l___..-J...-.J..-...I....J._I-.L...J....J....-.!-.....L..-.J...... l-.l-..L...... L.....I...-I.-'-L-~--L..-'--'---'-...... _...... ~ 0. 1 0.2 0.5 2 10 20 50 100 Re DATA FOR FLAT PLATES- PARALLEL FLOW

FIGURE 15 46

..... + r -'"~··· ......

''- • +"""""""1;. + ' ...... -t-t·. ·~ - .... '.' •

• • 0 • • ...... I · ""-

-- W/L =2 W/L: 4 I"' ... ~ .. 'V --- W/ L• I

h, <6 • ' I ' - ~ .., • .., - ~- • o ..,_ ·•• • .., • • • + I'"' . . • I j- • • •• • •••• ...... J. .... ' .. ' I' . ~ .: : ~ : : ~- :::-:· :. : ...

~ ' l'n ...... , ,.C • ·r· ......

,... . r'\.. :'-. . : :._ ~.: :--~ :-;.: . ..::~ .. ··: -· . .,... . . , . ~,. y' ::: ~ ..:.. -:::..;: :::::::.:·. . .. - -

- 1' ' I:X ~ .. 11: ...... - . 1,.._ IC . 0\,. ~_;j ·:: ~rf ::· ·: ... :::: ·::: !--:. ,;.. .v : II" ... : ":""DS· ~~ -.. .. l... .,.. ... ' - - ...... :.. . :. . :.. :....: - • • ..... -- +- ... • • • • • • • • • +

IV ...... ,...... ·-" ...... , .. .

1.: ...... ~ ...... ~. . .. • ,.,1., n I an. : - ,~ ::: :;:. ·n. .:. :-::: ·n : . \- , ...... , ' ...... --...... - .... I DATA FOR FLAT PLATES PERPENDICULAR FLOW- W/L=2 FIGURE 16 47

1 ...... _ •• I T +1t LL ..J'-t+fi:Ft= ;.;:I' I H~ -·H- f-J ±!i -1t~'"-

'f r.;. f+ ::r=::r:_ I -ttt+ ~- •tt i=f- 3:: ~ . e e W/ L = I +·

- -<>-o-- W/L = 4 ..l. I .., + ~ iH-H ·i' T • it! I+ ~ •. t :u:: 1 nH ~ ri j t++t+t++tft •m H--~+H-t+ t-++H-'f+t+~Httt H t •~*H-I.rtt!I-H :m I

I I

t" H+t-~ 1-r f'-!tj I i it 'iT" ' -t ·Ht I I I I Ill II I ~· ...... ,_._ - 'I -

t r · 1 r • - r 1 ~- l:tt++l=t:U, tt~S-"t+t+++~-++U +HJ:Jm;.;..;,~\-fl~HH}tt t ttn ll +t-Tt-'~ - ~ i I r : fH r- T --r -1 t -'--t-' -'t ...w..,.... ,_.,+ ,_, I- ·I --r- + . . . H• H ' t- f H- j.Lj f I --r++ -t iHr -1 H-e-.- -t 1 r Hrr t,~ t f f-l -t+tt I 1IT 1 'H-rf-I IJftJ J;f.+i+ ~ L . ~ =;=:;,_ = ~· irE:::;. =+ - t:::.;j , :rt.:.· ~ - H 1-J.t I tt' .o:: =;:::t::t: ~- + ..,... ~'1. l +f: i!ll ' l\ ±: ~ f± -' + I t-

DATA FOR FLAT PLATES PERPENDICULAR FLOW- W/L= I, 4 FIGURE 17 48

DI SCUSS ION OF RESULTS

Correction and Accuracy of Measurements

After a few preliminary force measurements with the spheres and a check with Stokes ' law (Equation 2) it was apparent that the drag force on the wire was appreciable and needed to be considered. It was decided to take a series of measurements with the spheres and calculate the difference between the measured force and the force calcu lated from Stokes ' law. The difference in force could then be attributed to the drag on the wire . If Stokes ' law is followed, the force on the wire should be proportional to the velocity. A series of twenty measurements of the force on the spheres was taken for each oil and the difference between

the measured force and that c.alcula ted by Stokes 1 law was determined. For each oil this difference as plotted vs . the velocity . The points grouped fairly ell around a straight line nearly passing through the origin. The method of least squares was used to determine the equation of the line best fitting the da t a . The equa tion of the line for the libht oil tas found to be Fe ••05605v - .oooa , (35) which was determined at about 62 . 7°F . Since the intercept 49 of the line is very close to zero, it is believed that the line is a good indication of the drag on the wire . The equation of the line for the heavy oil was found to be F . 19llv . oo2o1, (36 ) c - I which was determined at about 64 . 2° . The intercept of this line is also quite close to zero . These lines plotted in Fi£ures 9 and 10 were used throughout the investigation for the correction factor of the drag on the wires . For the cylinders and flat plates in parallel flow, which were pulled by two wires , the values determined from Equations {35) and (36) were doubled . For the plates in perpendicular flow pulled by four wires, the correction force was multi plied by four . The spring scale had 1/2 ounce divisions but could be read to the nearest sixth of an ounce . Some of the measure ments of force were under an ounce; hence, a considerable spread of the measurements was noticed in the preliminary data and throughout the experiment . However , sufficient points were obtained so that it was possible to draw a reliable curve through the data in all casas . An analysis was made to determine the average deviation from Stokes' equation for the spheres . It raa found that the average deviation wa s 15 . 1% for the light oil, 16 . 6% for the heavy oil and 15 . 9% overall . The maximum deviation was 89% . 50

Inspection of the other data shows that these deviations are also representative of the cylinders and flat plates . The force measurement is the least accurate part of the experiment . Other insignificant errors are introduced by a small variation in the temperature . This variation was held to about 1 0 from the temperature of the calibrated correction curve . The velocity measurements and the dimensions of the cylinders , spheres , and pl ~ tes are con sidered go od enough so that no appreciable errors occur. In order to e l iminate the W/L parameter for flat plates in parallel f l ow , an additional factor for the effect of the edges was subtracted from the measured force . Janour (5, p . 27) presented the foll owing equation for the edge correction for one edge of a flat plate in parallel flow:

F : ~ lv~ • (37 ) edge gc

In present work this equation as doubled because both edges of the plates were submerged in fluid . It is assumed in appl ying this correction that the lowe r limit of a Reynolds number of 10 proposed by Janour can be extended close to 0 .1.

Analysis of Results

Forty of the points for the spheres were used to get 51

the correction factor for the wires . The remaining thirty points are well erouped about Stokes' law . The data for cylinders for L/D ratios of 16, 24, and 32 did not seem to be segregated; therefore, these data were plotted together . It would seem that in the low range of Reyno l ds numbers an L/D of 16 and greater can be con

sidered an ~.nfini tely long cylinder. The other L/D ratios of 2, 4, 6, a, 12 provided fairly distinct and separate lines. The best straight lines were drawn through the data for each of the L/D ratios . It was evident that in eaeh case a slope of -1 on a log-log graph gave the best straight line , which would indicate that the force varies directly as the velocity. It was possible to develop an empirical expression relating drag coefficient, Reynolds number, and L/D . The following equation was obtained from the straight line plots of Re vs fd for the various L/D ratios: ( 38 )

Equation (38) applies for Reyno l ds numbers from . 01 to 1.0 and for L/D ratios of 2 to 16 . For L/D ratios greater than 16 re10 . (39 ) The data for flat plates in parallel flow is plotted in Figure 15 after the correction factor for tho edge 52 effect was subtracted . When the edge correction is made no effect of W/L ratio is indicated. This result would be expected. The data followed a straight line with a slope of -1 up to a Reynolds number of 2 . After that a curve was dravm connecting the line to that obtained by Janour . The equation for the straight section of the curve is

f - 6 (40) - Re , which applies for Reynolds numbers of 0 .1 to 2 . 0 . Here , a gain, the force is proportional to the velocity. Vfuen determining drag force for flat plates in parallel flow, the force is first calculated from Equations (40) and (15 ); then the edge correction is added . The effect of the geometric ratios is clearly shown in the data for flat plates in perpendicul ar flow which are plotted in Figures 16 and 17. As with the other data, the best straight line was drawn through the various points for eaoh of the W/L ratios . Again, the line had a slope of -1. The equation relating fd' Re , and w/L was found t o be rd : 37 (w) -o. 3o (41) Ire"'l:, which applies for Reynolds numbers of about . 05 to 2 . 0 and W/L ratios of 1 to 4 . It is possible, but it has not been proved that Equation (41) is suitable for higher W/L ratios . The exponent on W/L in Equation {41) is very close to that 53 on L/ D i n Equation ( 38 )~ It i s possible t ha t these exponents are t he same but this cannot be sho ~~ de£1nitely until more accura te da ta are available. It would be ex pected that a s the Reynolds number approaches zero t he effect of geometric ratios would be the same for cylinders and fla t pla tes in perpendicula r flow.

It is seen in the t a bles of data that occasionally a ne gative force was obtained because the correction applie d due to t he wire drag was greater than the mea sured force . These points obviously are incorrect . This occurred only for the smallest plates in the heavy oil at t he highest velocities . However , these kno\m bad points occur in less tha n 5~ of the data . It is clearl y shown that for cylinders and plates the fd increases as L/ D or W/ L decreases . This is in direct contrast to Wiesel aberger ' s investigation. However , his work is for higher Reynolds numbers at which a turbulent wake forms •

Comparison of Results with Other Data and Theoretical So l utions

The data for spher e~ a grees , of course , with Stokes ' l aw since that law was used to determine the correction factor for the wire . Liebster (9 , P• 548) has 54 substantiated Stokes' equation. There are no experimental data with which to compare the results of the cylinders . Wieselsberger's minimum Reynolds number of 4 is above the range covered in the pre sent investigation. The da ta for the highest L/D ratios (16, 24, and 32) does agree almost exactly wi t h the solution of Allen and Southwell (1, P • 141) (L/D =00) in the range of Reynolds numbers from 0 . 1 to 1 . 0 . Allen and Southwell's solution a greed with the data of Wieselsberger (16, p . 22) . However, the present data is above the theoretical solutions of Lamb (8 , p . 112-121) throughout the range of Reynolds numbers from 0 . 01 to 1 . 0 and above the solutions of Bairstow, Cave, and Lang (2, p . 404); I mai (4, p . 157); and Tomotika and Aoi (15, p . 302) for Reynolds numbers of 0 . 1 to 1 . 0 . Allen and Southwell's solution a grees 'dth both Wieselsberger 1 s a nd the present data . Their solution and the present data represent the best means for predicting drag coefficients for flow over long cylinders for Reynolds numbers of 0 . 01 to 10 . It should be remembered that the o t her solutions should a gree with each other since they were all essentially derived by linearizing the Na vier Stokes equation. The data for flat plates in parallel flow is 55 considerably above the theoretical solutions of Janssen (6, p . 183 ) and Tomotika and Aoi (15, P• 302) . However ,

Fi f~re 15 shows that a smooth transition occurs bet een the present work and the data of Janour (5 , P • 31) . The present data considerably extend the experimental infor mation previously available for laminar flow paral lel to flat plates . In the re gion of Reynol ds numbers less than 2 the drag coefficient is shown to be inversely proportional to the Reynolds number . Janour's data covers a range of Reynolds numbers from 11 to 1000 . The results of the present investigation line up with Janour's results which in turn on extrapolation to higher Reyno l ds numbers (greater than 1000) make a smooth transition into Blasius ' curve represented by Equation (10) . At Reyno l ds numbers greater than 20 ,000 the drag coefficient is inversely pro portional to the square root of the Reynolds number . The data for flat plates in perpendicular flow is con siderably above the solutions of Tomotika and Aoi (15, p . 302) and Imai (4, p . 157} . However , their solutions f or cylinders and plates in parallel flow are also below the present data . Also , it should be remembered that their solutions are for infinitely wide plates. If a value of W/L of above 100 is used in Equation (41) then the present data and the solutions of Tomotika and Aoi are fairly close . 56

The present results indicate that Equation (41~ can be used with an accuracy of 15 to 20% within the limitations of the equation (W/L: 1 to 4, Re = 0 . 05 to 2) . 57 ...

SUM RY AND CONCLUSIONS

Only a small amount of work has been done in the past on the study of laminar flow over immersed bodies . There are many areas in the chemical process industries and the field of aeronautics where this information would be very helpful . The purpose of the present investigation wa s to study the almost totally unexplored range of Reynol ds numbers from 0 . 01 to 10 . Drag coefficients have been determined for spheres, cylinders, and flat plates in parallel and perpendicular flow . The drag coefficients have been plotted as a function of the Reynolds number, with dimension ratios as a parameter, on log-log graphs . The best straight lines have been drawn through the data . In all cases these lines had a slope of -1, hich shows that the drag coefficient is inversely proportional to the Reynolds number at very low Reynolds numbers for all shapes and dimension ratios . The following equations have been determined from the data. For cylinders

fd - 27 L -0 . 36 (38 ) - Re (!)) , which applies for Reynolds numbers of 0 . 01 to 1 and L/D of 2 to 16 . For L/D greater than 16, the equation is 58 (39) .

For flat plates in parallel flow a correction factor has been applied to account for the edge effect. The equation which applies for Reyno l ds numbers of 0 .1 to 2 is

f :: 6 (40) Re. For flat plates in perpendicular flow 0 30 f - 37 (w) - • (41) d - Re t wbieh applies for W/ L of 1 to 4 and Reynolds numbers of 0 .05 to 2 . It is concluded tha t Equations (38-41) give the best values of drag coefficients within an accuracy of 20~ for the range of Reynolds numbers that were considered. Also, it is evident that the dimension ratios are a n important factor in determining the drag coefficient for a given Reynolds number. Furthermore, the drag coefficient in creases with decreasing values of L/ D or W/ L for a constant Reynolds number . The da ta obtained in this investigation compare favorably with the other experimental data and with some of the theoretical sol utions . It should be remembered that when comparing the experimental data with theoretical solutions that practically all of the solutions are for an infinitely long cylinder or an infinitely wide plate . It is recommended tha t the present apparatus be 59 modified so that a force of . 001 pound can be measured . Also , it would improve tho accuracy to set up a constant temperature bath so that the temperature of the oil can not vary over 0.2°F . A few check points on the present data is all that is necessary to confirm the validity of Equations (38- 41) . It is also r ecommended that only SAE 140 oil be used and that 2 inches should be the minimum plate width and cylinder length to be studi3d. These conditions would help to maintain the accuracy of the correction force for the wire . 60

~WMENCI.A TURE

Symbol Dimensions

A area sq. ft . D diameter ft . F force lb. f L length ft .

M mas s lb. m Re Reynolds number Dv.f = -:::<:r w width ft . a area sq. ft . b characteristic length ft . d diameter ft .

f, fd drag coefficient gc gravitation constant l b . mft . 2 = :32 .17 l b _. rsec . 1 length ft . m mass l b •m p pressure lb.r/sq.ft . r radius ft . t time see . u velocity ft ./sec . v velocity ft ./sec . w width ft . 61

Symbol Dimensions

X x•coordinate ft . y y- coordinate ft . o(, vorticity time sec . viscosity lb. m/ ft .-sec .. kinematic viscosity ft .2/sec . circumference/ diameter = 3 . 1416 density lb. m/ ft . 3 function stream function Laplacian operator infinity

Subscripts

c corrected

f force 1 l iquid m mass p projected s solid w wetted

} 62

BI BLIOGRAPHY

1. Allan, D. N. de G. and R. v. Southwell . Re laxation methods applied to determine the motion, in two di mensions , of a viscous fluid past a fixed cylinder . Quarterly Journal of Mechanics and Applied Mathe matics 8 :129-145 . 1955 . 2 . Bairstow, L., B. M. Cave and E. D. Lang . The re sistance of a cylinder moving in a viscous fluid . Philosophical Transactions of the Royal Society of London, ser. A, 223:383- 432 . 1923 . 3 . Goldstein, Sidney . The steady flow of viscous fluid past a fixed spherical obstacle at small Reyno l ds numbers . Proceedings of the Royal Society of London, ser. A, 123:225-235. 1929. 4 . Imai, I. A new method of solving Oseen's equations and its application to the flow past an inclined elliptic cylinder . Proceedings of the Royal Society of London, ser . A, 224 :141-160 . 1954. 5 . Janour, Zbynek . Resistance of a plate in parallel flow at low Reyno lds numbe rs. Washington, Nov. 1951 . 40 p . {National Advisory Committee for Aeronautics. Te chnical Memorandum 1316) 6 . Janssen, E . An analog solution of the Navier-Stokes equation for the case of flow past a f l at plate at low Reynolds numbers . In: 1956 Heat Transfer and Fluid Mechanics Institute (Preprints of Papers) . p . 173-183. 7 . Knudsen , James G. and Donal d L. Katz. Fluid Dynamics a nd Heat Transfer . Ann Arbor, University of Michigan, 1953. 243 p . (Michigan . University. Engineering Research Bulletin no . 37)

8 . La~b , Horace. On the uniform motion of a sphere through a viscous fluid . Philosophical Magazine and Journal of Science , s~r . 6 , 21:112-121 . 1911. 9 . Liebster, H. Uben den widerstrand von kugeln . Annalen Der Physik , ser. 4, 82 :541- 562 . 1 927 . 63 10 . McAdams , William H. Heat transmission . 3d ed . New York , McGraw- Hill, 1954 . 532 p . 11 . Pai , Shih- I . Viscous f l ow theory . I . Laminar flow . Princeton, D. Van Nostrand, 1956 . 384 p . 12 . Prandtl• Ludwi g . Es sentials of fluid dynamics . London, . Blackie & Son, 1954 . 452 p .

13 . Relf, i . F . Discussion of the results of measure ments of the resistance of wires, with some addition al tests of ' the resistance of wires of small diame ters . In; Technical report of the Advisory Committee for Aeronautics {London) March 1914 , p . 47 - 51 . (Report and memoranda no . 102 ) 14 . Stokes , George Gabriel . Mathematical and physical papers . Vol . 3 , Cambridge , University Press , 1922 . 413 p . 15 . Tomotika, s . and T. Aoi . The steady flow of a viscous fluid past an elliptic cylinder and a flat plate at smal l Reynolds numbers . Quarterly Journal of Me chanics and Applie d Ma thematics 6 : 290- 312 . 1953 . 16 . Wieselsbergo r , c. Versuche Ube r der luftwiderstand gerundeter und kantiger korper . Er gebnisse der Aero dynamischen Versucbsanstal t . Vol . 2 . G~ tti nge n , 1923 . 80 p . APPENDIX 64 EXPERI~ffiNTAL DATA

TABLE II Data For S;Eheres

(1) {2) (3) {4) {5) (6)

Velociti Force Temp . Re fd Measured Corrected

S;Ehere No . 1 Lisht oil

. 2250 . 0230 . 0112 62 . 2 . 384 87 . 3 . 2539 . 0178 . 0044 62 . 2 . 432 25 . 4 . 2892 . 0283 . 0 129 62 . 2 . 493 57. 0 . 4228 . 0387 . 0158 62 . 2 . 720 33 . 0 . 5919 .0543 . 0219 62 . 2 1.008 23 . 3 . 7610 . 0700 . 0246 62 . 2 1 . 296 15. 8 Sphere No . 1 - Heavy oil . 05496 . 01562 . 00311 64 . 3 . 0381 378 . 5 . 0916 . 02604 . 00653 64 . 3 .0635 286 . 2 .1282 . 03646 . 00995 64 . 3 . 0890 222 . 6 .1649 . 04887 . 01535 64 . 3 .114 207 . 6 . 09843 . 03125 . 01043 63 . 6 . 0633 395 . 8 . 1641 . 05208 . 01871 63 . 6 .106 255 . 5 . 2297 .07292 . 02701 63 . 6 .148 188. 3 . 2953 . 08854 . 03010 63 . 6 .190 127.0 Sphere No . 2 - Light oil . 09639 . 01050 . 00570 62 . 2 . 219 125 . 8 .1606 . 01600 . 00780 62 . 2 . 365 63 . 4 . 2250 . 01900 . 00720 62 . 2 . 512 30 . 0 . 2892 . 02600 . 0106 62 . 2 . 658 26 . 2 . 2539 .02600 . 0126 62 . 2 . 576 41 . 2 . 4228 .04500 . 02210 62 . 2 . 960 26 . 0 . 5919 . 08900 . 05660 62 . 2 1.344 33 . 9 . 7610 .10400 . 05860 62 . 2 1.730 21 . 3 Sphere No . 2 - Heavy oil . 05496 . 02083 . 00832 64 . 3 . 0508 570 . 0 . 09160 . 03125 . 01174 64 . 3 . 0848 289 . 7 .1282 . 04687 . 02036 64 . 3 . 119 256 . 3 .1649 . 05208 . 01856 64 . 3 .153 141.3 65 (1) (2) (3) (4) (5) (6) Sphere 'No . 3 - Li ght oil . 09~29 . 01042 . 00599 62 . 3 . 310 65 . 95 . 1555 . 01562 .00770 62 . 3 .519 30 . 51 . 2177 . 03125 . 02005 62 . 3 . 727 40.54 . 2799 . 04167 . 02678 62.3 .935 32 . 76 . 1343 . 01562 .00889 63 . 1 . 463 47 . 22 . 2238 . 03125 . 01951 63 . 1 . 772 37 . 32 . 3134 . 04687 . 03010 63 . 1 1 . 082 29 . 37 . 4029 . 04687 . 02509 63 . 1 1 . 390 14 . 81 Sphere No . 3 - Heavy oil . 05496 . 03125 . 01874 64 . 3 . 0754 585 . 5 . 09160 . 0~646 . 01695 64 . 3 . 126 190. 7 . 1282 . 05729 . 03078 64 . 3 . 176 176 . 8 . 1649 . 06250 . 02898 64 . 3 . 226 100. 6 . 03974 . 01562 . 00602 65 . 8 . 0598 359.9 . 06624 . 02604 . 01139 65 . 8 . 0997 245 . 1 . 09273 . 03125 . 01152 65 . 8 . 140 126 , 5 . 1192 . 03646 . 02479 65 . 8 . 180 77.53 . 09843 . 04687 . 02605 63.6 . 125 253 . 8 . 1641 . 07812 . 04475 63 . 6 . 209 156. 9 . 2297 . 09896 . 05305 63 . 6 . 292 94 . 90 . 2953 . 10940 . 05096 63 . 6 . 375 55 . 18 Sphere No . 4 - Lit,:ht oil . 09329 . 01562 . 01119 62 . 3 . 416 68 . 86 . 1555 . 02604 . 01812 62 . 3 . 694 40 . 13 .. 2177 . 03125 . 02005 62.3 . 973 22.65 . 2799 . 03646 .02157 62.3 1 . 249 14 . 75 . 1343 . 02604 . 01931 63 . 3 . 623 57 . 34 .2238 . 03125 . 01951 63 . 3 1 . 040 20 . 86 . 3134 . 04167 . 02490 63 . 3 1 . 454 13.58 . 4029 . 05208 . 03030 63 . 3 1 . 8 70 10 . 00 Sphere No . 4 - HeavY oil . 05496 . 02083 . 00832 64 . 3 . 101 145 . 3 . 09160 . 03125 . 01174 64 . 3 . 168 73 . 83 . 1282 . 04687 . 02136 64 . 3 . 235 68 . 55 . 1649 . 05208 . 01856 64 . 3 . 302 36 . 01 . 03974 . 02604 . 01644 65 . 8 .oao 549 . 1 . 09273 . 03646 . 01673 65 . 8 . 187 102. 7 . 06624 . 03125 . 01660 65 . 8 .133 199 . 6 66 (l) ( 2 ) (3 ) ( 4 ) ( 5 ) ( 6 )

. 1192 . 03646 . 01167 65 . 8 . 241 34 . 06 . 09843 . 05729 . 03647 63 . 6 .167 198 . 6 . 1641 . 08333 . 04996 63 . 6 . 279 97 . 85 . 2297 . 09375 . 04784 63 . 6 . 391 47 . 85 . 2953 . 11460 . 05616 63 . 6 . 502 33 . 98 67

TABLE III Data For C;y:linders {1) (2) (3) ( 4 ) (5) { 6 )

Ve locit:t:: Force Temp . He fd Measured Corrected

Cylinder No. 1 L/D = 8 - Light oil . 09329 . 02083 . 01197 62 . 7 . 0537 454 . 2 . 1555 . 03125 . 01541 62 .7 . 0895 210 . 3 . 2177 . 04167 . 01 927 62 .7 . 125 134.2 . 2799 . 04167 . 01189 62 . 7 .161 50 . 10 .1343 . 02604 . 01258 62 . 5 . 0765 230 . 2 . 2238 . 04167 . 01819 62 . 5 . 128 119 . 9 . 3134 . 05208 . 01854 62 . 5 . 1'79 62 . 33 . 4029 . 06250 . 01894 62 . 5 . 230 38 . 51 Cylinder No . 1 - Hea~ oil . 05496 . 03125 . 00623 64 . 8 . 0129 670 . 5 . 09160 . 05208 . 01306 64 . 8 . 0216 506 . 0 .1282 . 06250 . 00948 64 . 8 . 0302 187 . 5 .•1649 . 08333 . 01629 64 . 8 . 0388 194 . 8 . 03974 •02604 . 00684 65 . 8 . 0101 1409 • . 06624 . 03125 . 00195 65 . 8 . 0168 144 . 5 . 09273 . 04167 . 00221 65 . 8 . 0235 83 . 58 . 1192 . 05729 . 00771 65 . 8 . 0302 176.4 . 09843 . 05208 . 01094 63 . 6 . 0211 350 . 2 .1641 . 08333 . 01659 63 . 6 . 0352 200 . 3 . 2297 . 11460 . 02278 63 . 6 . 0493 140.4 . 2953 .14060 . 02372 63 . 6 . 0633 88 . 48 Cylinder No. 2 - L/ D = 4 Light oil . 09329 . 01562 . 00676 62 . 7 .108 128 .2 . 1555 . 03125 . 01541 62 .7 .180 105. 2 . 2177 . 03125 . 00885 62 . 7 . 250 30 . 82 .1343 . 02083 . 00737 62 . 5 .153 67 . 43 . 2238 . 04167 . 01819 62 . 5 . 255 59 . 93 . 3134 . 06250 . 02896 62 . 5 . 357 48 . 68 . 4029 . 07292 . 02936 62 . 5 . 460 29 . 85 68 (1) (2) (3) (4) (5) (6) Cylinder No . 2 - Heavy oil . 05496 . 04167 . 01665 64 . 8 . 0258 896 . 0 . 09160 . 05729 . 01827 64 . 8 . 0432 354 . 0 . 1282 . 08333 . 03031 64 . 8 . 0604 299 . 7 ,1649 . 09375 . 02671 64 . 8 . 0776 159 . 7 . 03974 . 02083 . 00163 65 . 8 . 0202 1 67 . 8 . 06624 . 04167 . 01237 65 . 8 . 0336 384 . 2 . 09273 . 04687 . 00741 65 . 8 . 0470 140. 1 . 1192 . 05208 . 00250 65 . 8 . 0604 28 . 60 . 09843 . 05208 . 01044 63 . 6 . 0422 175 . 1 . 1641 . 09375 . 02701 63 . 6 . 0704 163 . 0 . 2297 . 11460 . 02278 63 . 6 . 0986 70 . 2 . 2953 . 14580 . 02892 63 . 6 . 127 53 . 93 Cylinder No . 3 - L/D = 2 - Light oil . 09329 . 02083 . 01197 62 . 7 . 215 113 . 5 .1555 . 03646 . 02062 62 . 7 . 360 70 . 35 . 217'7 . 04167 . 01927 62 . 7 . 502 33 . 55 . 2799 . 05208 . 02230 62 . 7 . 644 23 . 49 . 1343 . 03646 . 02300 62 . 5 . 306 105 . 2 . 2238 . 06250 . 03902 62 . 5 . 510 64 , 28 . 3134 . 07292 . 03938 62 . 5 . 714 33 . 09 . 4029 . 07292 . 02936 62 . 5 . 920 14 . 92 Cylinder No . 3 - Heayy oil . 05496 . 03646 . 01144 64 . 8 . 0517 307 . 8 . 09160 . 06250 . 02348 64 . 8 . 0864 227 . 4 . 1282 . 07812 . 0 2510 64 . 8 . 121 124 . 1 . 1649 . 08854 . 02150 64 . 8 . 155 64 . 27 . 03974 . 0 3 125 . 01205 65 . 8 . 0404 620 . 3 . 06624 . 03646 . 0071 6 65 . 8 . 0672 132 . 6 . 09273 . 05729 . 01783 65 . 8 . 0940 168. 5 . 1192 . 0625 . 01292 65 . 8 . 121 73 . 87 . 09843 . 06771 . 02607 63 . 6 . 0844 218 . 6 . 1641 . 10940 . 04266 63 . 6 . 141 128 . 7 . 2297 . 1 5100 . 05918 63 . 6 . 197 91 . 14 . 2953 . 16150 . 04462 63 . 6 . 253 41.60 Cylinder No , 4 - L/D :: 2 - Light oil . 09329 . 02604 . 01738 62 . 7 . 322 109. 9 . 1555 . 04167 . 02583 62 . 7 . 538 58 . 75 . 21 77 . 05729 . 03487 62 . 7 . 755 40 . 50 69 (1) (2) (3) {4) (5) (6)

. 2799 . 05729 . 02751 62 . 7 . 967 19 . 32 . 1343 . 04167 . 02821 62 . 5 . 459 86 . 03 . 2238 . 05729 ,. 03381 62 . 5 . 765 37 . 14 . 3134 . 07292 . 03938 62 . 5 1 . 071 22 . 06 . 4029 . 08854 . 04498 62 . 5 1 . 380 15 . 25 C;y:1inder No . 4 - Hea~ oil . 05496 . 04687 . 02185 64 . 8 . 0775 392 , 0 . 09160 . 06771 . 02869 64 . 8 . 130 185. 3 . 1282 . 08854 . 03552 64 . 8 . 183 116 . 1 . 1649 . 0 9896 . 03192 64 . 8 . 233 63 . 61 . 03974 . 03125 . 01205 65 . 8 . 0606 413 . 6 . 06624 . 05729 . 02799 65 . 8 . 101 345 . 8 . 09273 . 06771 . 028 25 65 . 8 . 141 178 . 1 . 1192 . 08854 . 038 96 65 . 8 . 181 148. 6 . 0 9843 . 07812 . 03648 63 . 6 . 127 204 . 0 . 1641 . 12500 . 05826 63 . 6 . 211 117 . 2 . 2297 . 17190 , 08008 63.6 . 296 82 ,. 29 . 2953 . 20310 •.08622 63 . 6 . 3 80 55 . 95 Cylinder No ~ 5 - L/ D = 16 - Li ght oil . 09329 . 02083 . 01197 62 . 3 . 0525 227 .1 . 1555 . 03646 . 02062 62 . 3 . 0875 140. 7 . 2177 . 05208 . 02960 62 . 3 . 123 103 . 3 . 2799 . 6250 ,. 03272 62 . 3 . 158 68 . 94 .1343 . 03125 . 01779 62 . 5 , 0765 162. 7 . 2238 . 04687 . 02339 62 . 5 . 128 143 . 0 . 3134 . 06771 , 03417 62 . 5 . 179 57 . 43 . 4029 . 08854 . 04498 62 . 5 . 230 45 . 74 Cylinder No . 5 - Heavy oil . 05496 . 03125 . 00623 66 . 7 . 0148 335 . 2 . 09160 . 06250 . 02348 66 . 7 . 0247 454 . 8 . 1282 . 07812 . 02510 66 . 7 . 0346 248 ,. 2 .1649 . 09375 . 02671 66 . 7 . 0445 159. 7 •03974 . 03125 . 01205 65 . 8 . 0101 1240 • . 06624 •04687 . 01757 65 . 8 . 0168 651 . 1 . 09273 . 06250 . 02304 65 . 8 . 0235 435 . 6 . 1192 . 06771 . 01813 65 . 8 . 0302 207 . 4 . 09843 . 06671 . 02607 63 . 6 . 0211 437 . 2 . 1641 . 11980 . 05306 63 . 6 . 0352 320 . 3 . 2297 . 16150 . 06968 63 . 6 . 0493 214 . 7 , 2953 . 18750 . 07062 63 . 6 . 0633 131 . 7 70 (1) (2) (3) (4) (5) (6) Cylinder No . 6 - L/D : 8 - Light oil . 09329 , 02083 , 011 97 62 . 3 . 105 113 . 5 . 1555 , 04167 , 02583 62 .• 3 . 175 88 . 12 . 2177 . 05208 . 02968 62 . 3 , 245 51 , 67 . 2799 . 06250 , 03272 62 . 3 . 315 34 . 47 . 1343 . 04167 . 02821 62 , 5 . 153 129 . 0 . 2238 . 06250 . 03902 62 . 5 , 255 64 . 28 . 3134 . 08333 . 04979 62 . 5 . 357 41 . 83 . 4029 . 06250 . 01894 62 . 5 . 460 9 . 63 Cylinder No . 6 - Rea!:! oil . 05496 . 03646 , 01144 66 . 7 . 0297 307.8 . 09160 , 0625 . 02348 66 . 7 . 0494 227 , 4 ,1282 . 06771 . 01467 66 . 7 , 0692 72 . 64 . 1649 , 08333 . 01629 66 , 7 . 0890 48 . 7 . 03974 , 03125 . 01205 65 , 8 . 0202 620.3 . 06624 ., 04167 , 01237 65 . 8 . 0336 192 . 1 . 09273 . 05208 . 01262 65 . 8 . 0470 119 . 3 . 1192 . 06250 . 01292 65 . 8 . 0604 73 . 87 . 09843 . 07292 , 03128 63 . 6 . 0422 262 . 3 . 1 641 . 11460 , 04786 63 . 6 . 0704 144 . 4 . 2297 . 16150 . 06968 63 . 6 , 0986 107 . 3 . 2953 . 18750 . 07062 63 . 6 . 127 65 . 8 Cylinder No . 7 - L/p : 4 - tieht oil . 09329 , 03125 . 02239 62 . 8 . 215 131 . 7 . 1555 . 0468 7 . 03103 62 . 8 . 358 52 . 93 . 2177 . 06250 , 04010 62 . 8 , 502 34 . 90 . 2799 . 07292 . 04314 62 . 8 . 646 22 . 72 . 1343 , 04167 . 02821 62 . 5 . 306 64 . 50 . 2238 , 06771 . , 04423 62 . 5 .510 36 . 43 . 3134 , 09375 . 06021 62 . 5 . 714 25 . 29 . 4029 , 09896 . 0554 62 , 5 . 920 14.08 Cylinder No . 7 - Heavy oil . 05496 . 03646 . 01144 66 . 7 . 0594 153 . 9 . 09160 . 06250 , 02348 66 . 7 . 0988 113 . 7 . 1282 . 07812 . 02510 66 . 7 . 138 62 . 05 . 1649 . 09375 , 02671 66 , 7 . 178 39 . 92 71

(1) (2) (3} (4) (5) ( 6 )

. 03974 . 03125 . 01205 65 . 8 . 0404 310 . 1 . 06624 . 05208 . 02278 65 . 8 . 0672 211 . 0 . 09273 . 06771 . 02825 65 . 8 . 0940 133.5 . 1192 . 07292 .02334 65 . 8 . 121 66 . 74 . 09843 , 09375 . 05211 63 . 6 . 0844 218 . 5 . 1641 . 14580 . 07906 63 . 6 . 141 119 . 3 . 2297 . 17710 . 08528 63 . 6 .197 65 . 89 . 2953 . 19270 . 07582 63 . 6 . 253 35 . 35 Cylinder No . 8 - L/p = 2 - Light oil . 09329 . 03646 . 02760 62 . 3 . 315 8 7 . 24 .1555 . 06250 . 04666 62 . 3 . 524 53 . 06 . 2177 . 08333 . 06093 62 . 3 . 735 35 . 35 . 1343 . 05208 . 03862 62 . 5 . 459 58 . 8 7 . 2238 . 08333 . 05985 62 . 5 . 765 32 . 85 . 3134 . 10420 . 07066 62 . 5 1 . 071 19 . 79 . 4029 . 11460 . 07104 62 . 5 1 . 380 12 . 04 C~linder No . 8 - Hea V:f. oil . 05496 . 04687 . 02185 66 . 7 . 0891 196 . 0 . 09160 . 0 78 12 . 03910 66 . 7 . 148 1 26 . 3 . 1282 . 09896 . 04594 66 .7 . 208 75 . 71 . 1649 . 11980 . 05276 66 .7 . 267 52 . 58 . 03974 . 03646 . 01726 65 . 8 . 0606 296 .1 . 06624 . 05729 . 02799 65 . 8 . 10 1 172 . 9 . 09273 . 07812 , 03866 65 . 8 .141 1 21 . 8 . 1192 . 09896 . 04938 65 . 8 . 18 1 94 . 14 . 09843 . 10420 . 06256 63 . 6 ,127 174 . 9 . 164 1 . 16670 . 09996 63 . 6 . 211 100 . 6 . 2297 . 218 10 .12688 63 . 6 . 296 65 . 15 Cylinder No. 9 - L/ D = 24 - Light oil . 09329 . 03125 . 02239 62 . 7 . 0537 283 . 0 . 1555 . 0468 7 . 03103 62 . 7 . 0895 141. 1 . 2177 . 06250 . 04010 62 .7 . 125 93 . 05 . 2799 . 07292 . 04314 62 . 7 . 161 60 . 57 . 05441 . 01 042 . 00592 63 .1 . 0315 220 . 0 . 09068 . 02083 . 01218 63 . 1 . 0528 163 . 0 . 1270 . 03125 . 01 861 63 .1 . 0738 126 . 9 . 1632 . 03646 . 01976 63 . 1 . 0948 81 . 60 .1343 . 03646 . 02300 62 . 6 . 077 1 40 . 21 72 (1} (2) ( 3 ) ( 4) ( 5 ) ( 6 )

. 2238 . 06250 . 03902 62 . 6 . 1 28 85 , 68 . 3134 . 08854 . 05500 62 . 6 . 179 61 . 60 . 4029 . 09896 . 05540 62 . 6 , 230 37 . 54 Cylinder No . 9 - Heavy oil . 05496 . 03125 . 00623 66 .7 . 0148 223 . 5 . 09160 . 05208 .. 01306 66 . 7 . 0247 168 . 6 .1282 . 07292 .01990 66 .7 .0346 13 1 . 2 . 1649 . 08333 . 01629 66 . 7 . 0445 64 . 93 . 03974 . 02604 . 00684 65 , 3 . 0097 469 . 7 . 06624 . 05208 . 02278 65 . 3 . 0162 466 . 5 . 0 9273 . 06250 . 02304 65 . 3 . 0227 290 . 3 . 1192 . 07292 . 02334 65 . 3 . 0292 177 . 9 . 09843 . 08854 . 046HO 63 . 6 . 0211 524 . 4 . 1641 .13020 . 06346 63.5 . 0352 255 . 3 . 2297 . 17190 . 08008 63.6 . 0493• 164 . 6 . 2953 . 21350 . 09662 63 . 6 . 0633 1 20 .1 Cylinder No , 10 - L/D :: 12 - Lirht oil , 09329 . 03646 . 02760 62 . 7 .108 174 . 5 . 1555 . 05208 , 03624 62.7 .180 82 . 43 . 2177 . 06250 .04010 62 . 7 . 250 46 .. 53 . 2799 . 07292 .04314 62 .7 .. 322 30 . 27 . 05441 . 02083 . 01633 63 .1 . 0630 303 . 5 . 09068 .03125 .02260 63 .1 . 106 151 . 2 . 1270 . 03646 . 02382 63 .1 .148 81 . 25 .1632 . 04167 . 02497 63 .1 . 1 90 51.58 , 1343 . 04687 . 03341 62 . 6 . 154 101 . 9 . 2238 , 07812 .05464 62 . 6 . 256 60 , 01 . 3134 .10940 .07586 62 . 6 . 358 42 . 50 . 4029 . 13020 . 08664 62 . 6 . 461 29 . 37 C;y:11nder No . 10 - Heavy oil . 05496 . 04687 . 02185 66 . 7 . 0282 392 . 0 . 09160 . 06771 . 2869 65 . 7 . 0469 185 . 3 . 1 282 . 09375 . 04073 66 . 7 . 0658 134 . 3 . 1649 . 11980 . 05276 66 . 7 . 0846 105 , 2 . 03974 . 03646 . 01726 65 . 3 . 0 195 592 . 2 . 06624 . 05729 . 02799 65 . 3 . 0329 345 . 8 . 09273 . 07812 . 03866 65 . 3 . 0454 243 . 6 . 1192 . 09375 . 04417 65 . 3 . 0584 168 . 5 . 09843 . 09375 . 04164 63 . 6 . 0422 291 . 5 73

(1) (2) (3) (4) . ( 5) (6)

. 1641 . 15100 . 08426 63 . 6 . 0704 169 . 6 . 2297 . 20310 . 11128 63 . 6 . 0986 114 . 3 . 2953 . 23440 . 11752 63 . 6 . 127 73 . 07 Cylinder No . 11 - L/ - 6 Li ght oil . 09329 . 03125 . 02239 62 . 7 . 215 70 . 75 . 1555 . 05729 . 04145 62 . 7 . 360 47 . 13 . 2177 . 06250 . 04010 62 . 7 . 502 23 . 27 . 2799 . 06771 . 03793 62 . 7 . 644 13 . 32 . 05441 . 01562 . 01112 62 . 8 . 124 103 . 3 . 09068 . 03125 . 02260 62 . 8 . 208 75 . 62 . 1270 . 03646 . 0238 0 62 . 8 . 291 40 . 62 . 1632 . 03646 . 01976 62 . 8 . 374 20 . 40 . 1343 . 05729 . 04383 62 . 7 . 308 66 . 81 . 2238 . 07812 . 05464 62 . 7 . 512 30 . 00 . 3134 . 09896 . 06552 62 . 7 . 716 18 . 35 .4029 . 10940 . 06584 62 . 7 . 922 11 . 16 Cylinder No . 1 1 - He a :Y:il oil . 05497 . 05208 . 02706 66 . 7 . 0594 242 . 6 . 09160 . 08333 . 04431 66 . 7 . 0988 143 . 1 . 1282 . 09896 . 04594 66 . 7 . 138 75 . 71 . 1649 . 11460 . 04756 66 . 7 . 178 47 . 40 Cylinder No . 12 - L/D : 4 Light oil . 0 9329 . 05729 . 04843 62 . 7 . 322 102 . 0 . 1555 . 07812 . 06228 62 . 7 . 538 47 . 21 . 2177 . 08854 . 0661 4 62 . 7 . 755 25 . 58 . 2799 . 09896 . 06918 62 . 7 . 967 1 6 .19 .1343 . 07292 . 05946 62 . 7 . 462 60 . 42 . 2238 . 11460 . 09112 62 . 7 . 768 33 . 35 . 3134 . 13540 . 10 186 62 . 7 1 . 074 19 . 02 . 4029 . 14580 . 10224 62 . 7 1 . 383 11 . 55 Cylinder No . 12 - Heavy oil . 05497 . 06250 . 03748 66 . 7 . 0 891 224 . 2 . 09160 . 09375 . 05473 66 . 7 . 148 117 . 8 . 1~ 82 . 10940 . 05638 66 . 7 . 208 61.95 .1649 . 13540 . 06836 66 . 7 . 267 45 . 41 . 03974 . 05729 . 03809 65 . 3 . 0585 435 . 7 74

(1) (2) (3) (4) ( 5) (6)

,.06624 . 07812 . 04882 65 . 3 . 0972 201 . 0 . 09273 . 09896 . 05950 65 . 3 . 136 125.0 .1192 ,.13540 ,. 08582 65 . 3 .. 175 109.1 Cylinder No . 13 L/D - 32 - Light oil . 09329 . 03646 ,. 02760 62 . 7 . 0537 261 . 7 . 1555 . 05729 . 04145 62 .7 ,. 0 .'3 95 141.4 . 2177 , 07812 . 05572 62 . 7 .125 96 ,. 98 . 2799 . 08854 . 05876 62 .7 .161 61 . 89 . 05441 . 01042 . 00572 63 . 0 . 0310 165.0 ,. 09068 ,. 02083 . 01218 63 . 0 . 0520 122.2 .1270 ,. 04167 . 02903 63 . 0 . 0728 148.5 .1632 . 04687 . 03017 63 . 0 . 0935 93 . 45 .1343 . 05208 . 03862 62 .7 . 0770 176 . 6 . 2238 ,08333 , 05985 62 .7 .128 98 . 55 . 3134 .11460 .08106 62 . 7 .178 68 .11 . 4029 .13540 ,. 09184 62 .7 . 230 46 . 69 Cylinder No . 1 3 - Heavy oil . 05497 . 04687 . 02185 66 .7 . 0148 588 . 0 .091 60 . 072£2 ,. 03390 66 .7 . 0247 328 . 4 .1282 . 09375 . 04073 66 .7 , 0346 201.4 .1649 .10420 . 03716 66,.7 .0445 111.1 . 03974 . 04167 . 02247 65 . 3 . 0097 1157 . . 06624 . 05208 . 02278 65 . 3 ,. 0162 422 .0 . 09273 . 07292 . 03346 65 . 3 . 0227 316 .3 .1192 . 08333 . 03375 65. 3 . 0292 193.0 . 09843 . 09396 ,. 05732 63 . 6 . 0211 480 . 8 ,.1641 ,.16 1 50 , 09476 63 . 6 . 0352 286 .0 . 2297 . 22400 .13218 63 . 6 . 0493 203 .7 . 2953 . 26560 .11688 63 . 6 . 0633 138.7

Cylinder No. 14 L/ - 16 Light oil .09329 . 05208 . 04322 62 .7 .108 204 . 9 .1555 . 07292 ,. 05708 62 .7 .180 97 . 37 . 2177 ,08854 . 06614 62 .7 . 250 57 . 56 . 2799 .10420 ,. 07442 62.7 ,. 322 39 . 20 ,. 05441 . 02083 ,. 01633 63 . 0 . 062 227 . 6 . 0~068 . 04167 ,. 03302 63 . 0 .104 165 . 7 .1270 .05208 . 03944 63 .0 .146 100 ,. 9 .1632 . 06250 . 04580 63 .0 .187 70 . 95 75

( 1 ) ( 2 ) (3) ( 4 } ( 5 ) ( 6)

. 1343 . 06250 , 04904 62 . 7 . 154 112 . 1 . 2238 . 09375 . 07027 62 . 7 . 256 57 . 88 . 3134 , 12500 . 09146 62 . 7 . 358 38 . 41 . 4029 . 1354 . 09184 62 . 7 , 461 23 . 34 Cylinder No . 14 - Heavy oil . 05497 , 05208 . 02706 66 . 7 . 0297 363 . 9 . 09160 , 08333 . 04431 66 . 7 . 0494 214 . 6 . 1282 . 09896 . 04594 66 . 7 . 0692 113 . 6 .1649 . 12500 . 05796 66 . 7 . 0 8 90 86 . 63 , 03974 , 03646 . 01726 65 . 3 . 0195 444 .. 2 . 06624 , 06250 . 033 20 65 . 3 . 0324 307 . 5 . 09273 . 08333 . 04307 65 . 3 . 0454 207 . 3 . 1192 . 1146 . 06502 65 . 3 . 0584 186 . 0 . 09843 . 10420 . 06256 63 . 6 . 0422 262 . 3 . 1641 . 16670 . 09996 6~$ . 6 . 0704 150 .. 8 . 2297 .. 22920 . 13738 63 . 6 . 0986 105. 8

Cylinder No • 15 L/D : 8 Lieht o i l . 09329 •05208 . 04322 62 . 7 . 21 5 102 . 4 . 1555 . 062f0 . 04666 62 . 7 . 360 39 . 79 . 2177 . 08333 . 06093 62 . 7 . 502 26 . 51 . 2799 . 10420 .. 07442 62 . 7 . 644 19 . 60 . 05441 . 03125 . 02675 63 . 0 .124 186 . 4 . 09068 . 04167 , 03302 63 . 0 , 208 82 . 84 . 1270 .. 05208 . 03944 63 . 0 . 292 50 . 43 . 1632 . 05729 . 04059 63 . 0 . 374 31 . 4 4 . 1343 . 06250 . 04904 62 . 7 . 308 56 . 06 ~ 2238 . 09375 . 07027 62 . 7 . 51 2 28 . 94 . 3134 , 12500 , 09146 62 . 7 ,. 716 19 . 20 ,4029 . 13020 , 08664 62 . 7 ,922 11 . 01 Cylinder No . 1 5 - HeayY oil . 05497 , 06771 . 04269 66 . 2 . 0 576 287.1 . 09160 . 09896 . 05994 66 . 2 . 0960 145. 2 . 1282 .13020 . 07718 66 . 2 . 134 95 . 39 . 1649 . 14580 . 07876 66 . 2 . 173 58 . 86 Cylinder No. 16 - L/ D : 6 Light oil . 09329 . 06250 . 05364 62 . 7 . 322 84 .. 77 .1555 . 09375 . 07791 62 . 7 . 538 44 . 3 1 76

(1) ( 2 ) ( 3 ) (4) ( 5 ) (6 )

.2177 . 10420 . . 08180 62 . 7 . 7 55 23.74 . 2799 . 1 2500 . 09522 62 . 7 . .967 16.72 . 1 343 . 08330 . 06984 62 . 7 . 462 53 . 25 . 2238 . 13540 . 11192 62 . 7 . 768 30 . 73 . 3134 . 17710 . 14356 62 . 7 1 . 074 20. 10

C~11nder No . 16 - Hea~ o~!_ . 05497 . 07812 . 05310 66 . 2 . 0864 238 .1 . 09160 . 11460 . 07558 66 . 2 . 144 122 . 0 . 1282 . 14580 . 09278 66 . 2 . 202 76 . 46 . 1 649 . 17190 . 104 66 . 2 . 259 52 .25 . 03974 . 04687 . 02767 65 . 3 . 0585 237 . 4 . 06624 . 08333 . 05403 65 . 3 . 0972 166 . 8 . 09273 . 10940 . 06994 65 . 3 . 136 110 . 2 . 1192 . 14580 . 09622 65 . 3 . 175 91 . 74 77

TA BLE IV Data For Flat Plates - Parallel Flow

( 1) (2) (3) (4) {5) {6)

Velocit:,y: Force Temp . Re f .1e asured Corrected -

Plate No . l a - W/ L = 4 - Light oil . 09329 . 02083 . 01038 62 . 4 .. 212 24 . 60 .1555 .03125 .01276 62 . 4 . 353 10.88 . 2177 . 0468 7 . 02075 62 . 4 . 494 9 . 03 . 2799 . 06250 . 02794 62 . 4 . 634 7.36 . 05441 . 01562 . 01021 63 . 1 .126 71.14 . 09068 . 02083 . 01067 63 .1 . 211 26 . 77 . 1270 . 03125 . 01650 63 .1 . 295 21 . 09 .1632 . 04167 . 02225 63 .1 . 379 1 7 . 25 .1343 . 03125 . 01552 62 .7 . 308 17.74 . 2238 .. 05208 . 02482 62 .7 . 512 10 . 22 . 3134 . 07292 . 03408 62 . 7 . 716 7 .16 . 402~ . 08333 . 03296 62 .7 . 922 4 .19 Plate No . la - Heav:,y: oil . 05496 . 03125 . 0041 2 65 . 8 . 0563 27 . 71 . 09160 . 04687 . 00433 65 . 8 . 0936... 10.49 . 1282 . 06250 . 00455 65 . 8 .... - ~2- •1649 . 0781 2 . 00474 65 . 8 .168 ..__ 3•54 . 0 9843 .06771 . 02176 64 . 2 . 0885 45 . 63 .1641 .10420 . 03027 64 . 2 .147 22 . 84 . 2297 .13540 . 03352 64 . 2 . 207 12.92 . 2953 .177 1 . 04729 64 . 2 . 265 11.02 Plate No . lb W/L - 1(4 - Lisht oil .09329 . 02083 . 00559 62 . 4 . 848 13.25 .1555 .03125 . 00429 62 . 4 1.412 4 . 08 . 2177 .04167 . 00441 62 . 4 1.976 1.92 . 2799 . 05208 . 00318 62 . 4 2 . 536 0 . 84 . 05441 . 01042 . 00238 64 . 0 . 516 16.59 . 09068 . 01562 . 00108 64 . 0 . 865 2 . 72 .1270 . 02083 64 . 0 1.212 .1632 . 03 125 . 00394 64 . 0 1 . 560 3 . 06 . 1343 . 02083 62 .7 1.232 73 (1) (2 ) ( 3 ) (4) (5 ) (6)

. 2238 . 04167 . 00:306 62 . 7 2 . 048 1 . 26 . 3134 . 06250 . 00776 62 . 7 2 . 864 1 . 63 . 4029 . 07292 . 00211 62 . 7 3 . 688 . 27 Plate No . lb - HeaY,I oil . 05496 . 03125 65 . 8 . 255 . 09160 . 04167 65 . 8 . 374 . 1282 . 06250 65 . 8 . 524 . 1649 . 07292 65 . 8 . 672 . 09843 . 06250 . 00362 64 . 2 . 354 7 . 59 . 1641 . 09375 64 . 2 . 568 . 2297 . 13540 . 00334 64 . 2 . 828 1 . 29 . 2953 . 15620 64 . 2 1 . 060 Plate No . 2a - WLL : 2 ... Light Oil . 09329 . 03125 •01920 62 . 4 . 424 2 2 . 75 . 1555 . 04687 . 02572 62 . 4 . 706 10 . 97 . 2177 . 06250 . 03267 62 . 4 . 98 8 7 . 11 . 2799 . 07292 . 03358 62 . 4 1 . 268 4 . 42 . 05441 . 02083 . 01452 63.1 . 252 50 . 59 . 09068 . 0 3125 . 01958 63 .1 . 422 24 . 57 . 1270 . 04167 . 02480 63 . 1 . 590 1 5 . 86 . 1632 . 04687 . 02474 63 .1 . 758 9 . 58 . 1343 . 04167 . 02367 62 . 7 . 616 13 . 53 . 2238 . 0625 . 03146 62 . 7 1 . 024 6 . 48 . 3 1 34 . 08333 . 03919 62 . 7 1 . 432 4 . 11 .4029 . 10420 . 04701 62 . 7 1 . 844 2 . 98 Plate No . 2a - Hea!I oil . 05496 . 03125 . 00211 65 . 8 . 113 7 .10 . 09160 . 05729 . 01122 65 . 8 .187 13 . 59 .1282 . 07812 . 01524 65 . 8 .. 262 9 . 42 . 1649 . 09375 . 01402 65 . 8 . 336 5 . 24 . 09843 . 07292 . 02266 64 . 2 . 177 23 . 77 . 1641 . 12500 . 033B9 64 . 2 . 284 12 . 79 . 2297 . 17710 . 06516 64 . 2 . 414 12 . 56 . 2953 . 20830 . 06556 64.2 . 530 7 . 64 Plate No . 2b - W/L : 1/2 - Light oil . 09329 . 03125 . 01601 62 . 4 .848 18 . 97 . 1555 . 04167 . 01521 62 . 4 1 . 412 6 . 49 . 2177 . 05208 . 01482 62 . 4 1 . 976 3 . 25 79

(1) (2) (3) (4) (5) ( 6)

. 2799 . 06250 . 01460 62 . 4 2 . 536 1 . 92 . 05441 . 01042 . 00238 64 . 0 . 516 8 . 29 . 09068 . 01562 ,. 00108 64 . 0 ...s&5- 1 . 36· . 1270 . 02083 64 . 0 1 ,. 212 .1632 . 03125 . 00394 64 . 0 1 . 560 1 . 53 .1343 . 03125 . 00871 62 . 7 1 . 232 4 . 98 . 2238 . 05208 ,. 01347 62 . 7 2 . 048 2 . 77 . 0134 ,. 00333 . 02859 62 . 7 2 . 864 3 . 00 . 4029 . 09375 ,. 02294 62.7 3 . 688 1 . 46 Plate no . 2b - Heavy oil ,. 05496 . 03646 . 00298 65 . 8 . 225 10 . 02 . 09160 . 05208 65 . 8 . 374 .1282 . 07292 ... - 65 . 8 . 524 .1649 . 08333 65 . 8 . 672 . 09843 . 0 6 771 . 00883 64 . 2 . 354 9 ,. 25 . 1641 . 10 420 64 . 2 . 568 . 2297 . 15620 . 02414 64 . 2 . 828 4- . 65 Plate No . 3 ... W/ L = 1 - Light oil . 09329 . 03646 • 02122 62 . 4 . 8 48 12 . 58 . 1555 . 05208 . 02562 62 . 4 1 . 412 5 . 46 . 2177 . 07292 . 03566 62 . 4 1 . 976 3 . 88 . 2799 . 08333 . 03443 62 . 4 2 . 536 2 . 27 ,. 05441 . 02083 . 01279 64 . 0 . 51 6 22 . 28 . 09068 . 03125 . 01671 64 . 0 . 865 10 . 48 .1270 .03646 . 01557 64 . 0 1.212 4 . 98 .1632 . 04167 . 0 1 436 64 . 0 1.560 2 ,.78 . 1343 ,. 05208 . 02954 62 . 7 1 . 232 8 . 45 . 2238 . 08333 . 04472 62 . 7 2 . 048 4 . 60 . 3134 . 1146 . 05986 62 . 7 2 . 864 3 . 15 . 4029 . 1354 . 06459 62 . 7 3 . 588 2 . 05 Plate No . 3 - Heavy oil . 05496 . 05729 ,. 02381 65 . 8 . 225 40 . 05 . 09160 . 07812 . 02500 65 . 8 . 374 1 5 .14 .1282 . 09896 . 02621 65 . 8 . 524 8 .10 .1649 .11980 . 02738 65 . 8 . 672 4 . 74 . 03974 . 03646 , 0108 7 65 ,. 3 . 156 34 . 98 ,. 06624 . 06771 . 02776 65 . 3 . 260 32 .15 . 09273 . 08333 . 028 96 65 . 3 . 363 17.10 . 1192 .12500 . 05625 65 . 3 . 468 20 .11 80

(1) ( 2 } (3 ) (4) ( 5 ) (6)

. 0 £:1843 . 09375 . 03487 64 . 2 . 354 18 . 28 .1641 . 1615 ., 06602 64 . 2 . 568 12 . 46 . 2297 . 2292 . 09714 64 . 2 . 828 9 . 35 Plate No . 4a - W/_L : 1/_2 - LiEht oil . 09329 . 05208 . 03056 62 . 4 1 .. 696 9 . 05 . 1555 . 07292 ., 03584 62 . 4 2 . 824 3 . 82 . 21 77 . 09375 . 04163 62 . 4 3 . 952 2 . 26 . 2799 ., 10420 "03618 62 . 4 5 . 072 1 •.19 . 05441 . 02604 . 01430 63 . 1 1 . 008 1 2 . 46 . 09068 . 04167 . 02094 63 . 1 1 . 688 6 . 57 .1270 . 05729 . 02773 63 . 1 2 . 360 4 . 43 .1632 . 06250 . 02407 63 . 1 3 , 032 2 . 33 . 1343 . 06250 . 03088 62 . 7 2 . 464 4 . 4 1 . 2238 .10420 . 05046 62 . 7 4 , 096 2 . 60 . 3134 .13540 . 05946 62 . 7 5 . 728 1 . 56 . 4029 .. 15620 . 05814 62 . 7 7 . 376 . 92 Plate No . 4a - Hea:YI oil . 05496 . 05208 . 01014 65 . 8 . 45 8 , 52 . 09160 . 08333 . 01611 65 . 8 . '149 4 . 88 . 1282 . 11460 . 02212 65 . 8 1 . 048 3 . 42 . 1649 . 1354 . 01760 65 . 8 1 ., 344 1 . 65 . 03974 . 05208 . 02010 65 . 3 . 312 32 . 34 . 06624 . 07292 . 02232 65 . 3 . 520 12 . 92 . 09273 . 08854 . 01926 65 . 3 . 726 5 . 69 . 1192 . 1250 . 03708 65 . 3 . 935 6 ., 63 . 09843 . 1250 . 04888 64 . 2 . 708 12. 81 . 1 641 . 20830 . 08408 64 . 2 1 . 136 6 . 46 . 2297 . 26040 . 08810 64 .. 2 1 . 656 4 . 24 . 2953 . 30210 . 08178 64 . 2 2 . 120 2 ., 38 Plate No . 4b - w/_L = 2 ... Light oil . 09329 . 05729 . 04205 62 . 4 . 848 12 . 46 . 1555 . 08854 . 06208 62. 4 1 . 412 6 . 62 . 2177 .10940 . 07214 62 . 4 1 . 976 3 . 92 . 2799 . 11980 . 07090 62 . 4 2 . 536 2 . 33 , 05441 . 02604 . 01800 64 . 0 . 516 15. 69 . 09068 . 04687 . 03233 64 . 0 . 865 10 .14 . 1270 . 05729 . 03640 64 . 0 1 . 212 5 . 82 .1632 . 0625 . 03519 64 . 0 1 . 560 3 . 41 . 1343 . 06771 . 04517 62 . 7 1 . 232 6 . 45 81

(1) {2) {3) ( 4) (5) (6)

. 2238 . 11980 . 08119 62 . 7 2 . 048 4 . 18 . 3134 . 1615 . 10676 62 . 7 2 . 064 2 . 80 . 4029 . 2031 . 13229 62 . 7 3 . 688 2 .10 Plate No . 4b - Heavy oil . 05496 . 07812 . 04464 65 . 8 . 225 37 . 53 . 09160 .11980 . 06668 65 . 8 . 374 20 .19 . 1282 . 14060 .. 06785 65 . 8 . 524 10 . 48 .1649 .15620 .06378 65 . 8 . 672 5 . 96 . 03974 . 05729 . 03072 63 . 5 . 135 49 . 40 . 06624 . 08854 . 04695 63 . 5 . 225 27 . 17 . 09273 . 11980 . 06314 63 . 5 . 315 18 . 66 .1192 . 15100 .08931 63 . 5 . 405 1 5 . 97 . 09843 . 1 5100 . 08712 64 .2 . 354 22 . 84 .1641 . 22400 . 12852 64 . 2 . 568 12 . 10 . 2297 . 28650 . 15444 64 . 2 . 828 7 . 43 82

TABLE 'l Iata For Flat Plates - Per£endicular Flow (1) (2) (3) (4) (5) (6)

Velocit:z: Force Temp . Re fd Measured Corrected

Plate No . 1 - W/L = 4 - Light oil . 09329 . 07812 . 06040 62 . 9 . 432 71 . 60 . 1555 . 13020 . 09852 62 . 9 . 720 42 . 03 . 21?7 . 16150 . 11630 62 . 9 1 . 010 25 . 31 . 2799 . 17180 . 11224 62 . 9 1 . 296 14 . 78 . 05441 . 04687 . 03787 63 . 6 . 255 131 . 9 . 09068 . 06771 . 05059 63 . 6 . 428 63 . 47 . 1270 . 08854 . 05326 63 . 6 . 599 40 . 46 . 1632 . 10940 . 07600 63.6 . 770 29 . 43 . 1343 . 11980 . 09288 62 . 7 . 616 53 . 11 . 2238 . 19270 . 14574 62 . 7 1 . 024 30 . 01 . 3 134 . 25520 . 18812 62 . 7 1.432 19 . 76 Plate No . 1 - Heavy oil . 05496 . 11980 . 06976 65 . 7 . 113 234 . 7 .09160 . 18230 . 10426 65 . 7 . 187 126 . 3 . 1282 . 25000 . 14396 65 . 7 . 262 88 . 98 . 1649 . 30730 . 17322 65 . 7 . 336 64 . 73 . 03974 . 10420 . 06580 63 . 5 . 0676 423 . 5 . 06624 . 15620 . 09760 63 . 5 . 112 226 . 0 . 09273 . 20830 . 12938 63.5 . 157 152 . 9 . 1192 . 25000 . 15084 63 . 5 . 202 10'7 . 8 . 09843 . 21870 . 13542 64 . 2 . 177 1 42 . 0 . 1641 . 35420 . 22072 64 . 2 . 294 83 . 28 . 2297 . 42710 . 24346 64.2 . 414 46 . 89 Plate Nv . 2 - W/L • 4 Light oil . 0 9329 . 06250 . 04478 62 . 6 . 319 94 . 37 . 1555 . 09896 . 06728 62 . 6 . 532 51. 01 . 217'7 . 13020 . 08540 62 . 6 . 745 33 . 04 . 2799 . 15620 . 09664 62 . 6 . 960 22.62 . 05441 . 03646 . 02746 63 . 1 . 188 170 . 1 . 09068 . 06250 . 04538 63.1 . 315 101 . 2 .1270 . 07812 . 05284 63 . 1 . 441 60 . 06 83

(1) (2) (3) (4 ) (5) (6)

. 1632 . 08854 . 05514 63 . 1 . 566 37 . 97 . 1343 . 07812 . 05120 62 . 7 . 462 52. 04 . 2238 . 14060 . 09364 62 . 7 . 768 34 . 28 . 3134 . 20310 . 13602 62 . 7 1 . 074 25 . 39

Plate No . 2 .. Heavy oil . 05496 . 09375 . 04371 65 . 6 . 0825 261 . 3 . 09160 . 1458 0 . 06776 65 . 6 . 1 38 145 . 9 . 1282 . 1 8230 . 07626 65 . 6 . 192 83 . 79 . 1649 . 23960 . 10552 65 . 6 . 248 70 . 10 . 03974 . 06771 . 02931 63 . 5 . 0507 335 . 4 . 06624 . 11980 . 06120 63 . 5 . 0843 252 . 0 . 09273 . 15100 . 07208 63 . 5 . 118 151 . 4 . 1192 . 20310 . 10394 63 . 5 . 152 132.1 . 09843 . 16670 . 08342 64 . 2 . 133 1 5 5 . 5 . 1641 . 27080 . 13732 64 . 2 . 221 85 . 39 . 2297 . 35420 . 17056 64. 2 . 310 58.40 Plate No . 3 - W/L = 4 - Light oil . 09329 . 04167 . 02395 62 . 6 . 213 113.5 . 1555 . 07292 . 04124 62 . 6 . 355 70 . 34 . 2177 . 09375 . 04895 62 . 6 . 497 42 . 62 . 2799 . 10420 . 04464 62 . 6 . 640 23 . 51 . 05441 . 02083 . 01183 63 . 1 . 125 164 . 9 . 09068 . 03125 . 01413 63 . 1 . 210 70 . 91 . 1270 . 04167 . 01639 63 . 1 . 294 41 . 92 . 1632 . 05208 . 01868 63 . 1 . 377 28 . 93 . 1343 . 05208 . 02516 62 . 7 . 308 57 . 52 . 2238 . 08333 . 03637 62 . 7 . 512 29 . 95 . 3134 . 11980 . 05272 62 . 7 . 716 22 . 15 . 4029 . 14580 . 05868 62 . 7 . 922 14. 91

Plate No . 3 ... Heavy oil

•05496 . 06250 . 01246 65 . 6 . 0550 167 . 6 . 09160 . 098 96 . 02092 65 . 6 . 0918 101. 3 . 1282 . 13020 . 02416 65 . 6 . 128 59.72 . 1649 . 16150 . 02742 6 5 . 6 . 165 40.96 . 03974 . 04687 . 00 8 47 63 . 5 . 0338 218 . 0 . 06624 . 07812 . 01952 63 . 5 . 0562 180 . 8 . 09273 . 10940 . 03048 63 . 5 . 0788 144 . 1 . 1192 . 1 3 020 . 03104 63 . 5 . 101 88 . 77 . 0 9843 . 1250 . 04172 64 . 2 . 0885 174 . 9 84

(1) (2) (3) (4} (5) ( 6)

.1641 . 20830 . 07482 64 . 2 .147 112 . 9 . 2297 . 27080 . 08716 64 . 2 . 207 67 .13 . 2953 . 33330 . 09954 64 . 2 . 265 46 . 4 Plate No. 4 - W/L = 4 - Light oil . 09329 . 02083 . 00311 62 . 6 .107 58 . 99 .1555 . 04167 . 00999 62 . 6 . 178 68 .17 . 2177 . 06250 . 01770 62 . 6 . 249 61 . 64 . 2799 . 07292 . 01336 62 . 6 . 320 28 .15 . 05441 . 01042 . 00142 63 .1 . 0628 79.18 . 09068 . 02083 . 00371 63 .1 .105 74 . 48 .1270 .. 03125 . 00597 63 .1 .147 61 . 09 .1632 . 04167 . 00827 63 .1 .188 51.25 .1343 . 03125 . 00433 62 .7 .154 39 . 62 . 2238 . 05208 . 00512 62 .7 . 256 16.86 . 3134 . 07812 . 01104 62 . 7 . 358 19.24 . 4029 . 09375 . 00663 62 .7 . 461 6 . 99 Plate No . 5 - W/L - 2 - Li ght oil .09329 .14580 .12808 62 . 6 . 852 76.01 .1555 . 20830 .17762 62 . 6 1.420 37 . 88 . 2177 . 23960 .19480 62 . 6 1.988 21 . 20 . 2799 . 28120 . 22164 62 . 6 2 . 560 14.59 . 05441 . 07292 . 06392 63 . 2 . 508 111.4 . 09068 .1198 .10268 63 . 2 . 852 64 . 40 .1270 .15620 .13092 63 . 2 1.192 41 . 86 ,1632 .18230 .14890 63 . 2 1.532 28 . 83 .1343 .1979 .17098 62 .7 1.232 48 . 87 . 2238 . 30210 . 25514 62 . 7 2 . 048 26 . 27 Plate No. 5 - He a~ oil . 05496 . 19790 . 14786 65 . 6 . 220 248 . 7 . 09160 . 31250 . 23446 65 . 6 . 367 142.0 .1282 . 41 670 . 31066 65 . 6 . 514 96 . 01 . 03974 .16150 .12310 63 . 5 .135 396 .1 . 06624 . 23440 .17580 63 . 5 . 225 203 . 6 . 09273 . 31770 . 23878 63 . 5 . 31 5 141.1 Plate No. 6 - W/L = 2 - Light oil . 09329 . 08333 . 06561 62 . 9 . 648 69 .13 .1555 .12500 . 09332 62 . 6 1.070 35 . 38 . 2177 .17710 .13230 62 . 6 1.498 25 . 59 8 5

(1) (2 ) ( 3) (4) (5 ) ( 6)

. 2799 . 18750 . 12794 62 . 6 1 . 944 14 . 98 . 05441 . 05208 . 04308 63 . 6 . 383 133 . 4 . 09068 . 07292 . 05580 63 . 6 . 642 62 . 23 . 1270 . 09375 . 06847 63 . 6 . 899 38 . 92 . 1632 . 10420 . 07080 63 . 6 1 . 155 24 . 37 . 1343 . 12500 . 09808 62 . 7 . 924 49 . 84 . 2238 . 18750 . 14054 62 . 7 1 . 536 25 . 72 . 3134 . 25000 . 18292 62 . 7 2 . 148 17. 08 Plate No . 6 - Heavy oil , 05496 , 12500 . 07504 65 . 6 . 165 224 , 3 . 09160 . 17710 . 09906 65 . 6 . 275 106 . 6 . 1282 . 23960 . 13356 65 . 6 . 385 73 . 38 . 1649 . 31250 . 17842 65 . 6 . 495 59 , 26 . 03974 . 10420 . 06580 63 . 5 . 101 376 . 4 . 06624 . 15620 . 09760 63 . 5 . 169 200 . 9 . 09273 . 21350 . 13458 63 . 5 . 236 141 . 4 . 1192 . 26040 . 16124 63 . 5 . 303 102. 5 . 09843 . 22920 . 14592 64 . 2 . 266 136 . 0 .1641 . 37510 . 24152 64 . 2 . 441 81.0 Plate No . 7 - W/ L : 2 - Light oil . 09329 . 04687 . 0291 5 62 . 9 . 432 69 . 10 . 1555 . 0781 2 . 04644 62 . 9 . 720 39 . 61 . 2177 . 09896 . 05416 62 . 9 1 . 010 23 . 57 .. 2799 .10940 . 04984 62 . 9 1.296 13 . 12 . 05441 . 02604 . 0 1704 63 . 6 . 255 118. 7 . 09068 . 03646 . 01934 63 . 6 . 428 48 . 52 . 1270 . 04687 . 02159 63 . 6 . 599 27 . 60 . 1632 . 05729 . 02389 63 . 6 . 770 18. 50 . 1343 . 06771 . 04079 62 . 7 . 616 46 . 63 . 2238 . 10940 . 06244 62 . 7 1 . 024 25 . 72 . 3134 . 16150 . 09442 62 . 7 1 . 432 19 . 83 . 4029 . 19270 . 10558 62 . 7 1 . 844 1 3 . 42 Plato No . 7 - Hea~ oil . 05496 . 08333 . 03:329 65 . 7 . 113 223 . 9 . 09160 . 11980 . 04176 65 . 7 . 1 87 101 . 1 . 1 282 . 15100 . 04496 65 . 7 . 262 55 . 56 . 1649 . 18230 . 04822 65 . 7 . 336 36 . 03 . 03974 . 05729 . 01889 63 . 5 . 0676 243 . 1 . 06624 . 10420 . 04560 63 . 5 . 112 211 . 1 86

(1) (2) (3) (4} (5) (6)

. 092'73 . 14580 . 06688 63 . 5 .157 158 . 0 .1192 ,.17710 . 07794 63 . 5 . 202 111.4 . 09843 .15620 . 07292 64 . 2 .177 1 52 . 9 .1641 . 25000 . 11652 64 . 2 . 294 87 . 91 . 2297 . 31250 .12886 64 . 2 . 414 49 . 64 Plate ·No . 8 - wi_L = 2 - Lifiht oil .09329 .03 125 . 0 1353 62 . 6 . 21 3 128.3 .1555 . 05208 . 02040 62 . 6 . 355 69 . 60 . 2177 .07292 . 0281 2 62 . 6 . 497 48 . 95 . 2799 ,. 0833:3 . 02377 62 . 6 . 640 25 . 04 . 05441 . 01042 . 00142 63 ,. 2 .127 39 . 54 .09068 . 02083 . 00371 63 . 2 .• 213 37 . 24 . 1270 . 03125 . 00597 6:3 . 2 . 298 30 . 54 .1632 . 04167 . 00827 63 .. 2 ,. 383 25 . 62 .1343 . 04467 . 01475 62 . 7 . 308 67 . 46 . 2238 . 06771 . 02075 62 . 7 . 512 34 .18 . 3134 . 09375 . 02667 62 . 7 . 716 22 . 40 ,. 4029 .11460 . 02748 62.7 . 922 13.97 Plate No . 8 - Haa!I oil . 05496 . 05208 . 00204 65 . 6 . 055 54 . 88 .09160 . 07292 65 . 6 . 0918 . 1282 .10420 65 . 6 .128 . 03974 ,. 03646 63 . 5 . 0338 . 06624 . 06250 . 00390- 63 . 5 . 0562 72- . 21 .09273 . 07292 63 . 5 . 0788 . 09843 . 09375 . 01843- 64 .. 2 . 0885 87- . 47 .1641 .16150 ,. 02802 64 . 2 .147 84 . 59 . 2297 . 21870 . 03506 64 . 2 .207 54. 02 . 2953 . 26040 . 02664 64 . 2 . 265 24 . 84 Plata No . 9 ... W/ L • l - Light oil . 09329 ,.0'7292 . 05520 62 . 6 ,. 852 65 . 44 .1555 .12500 . 09332 62 . 6 1.420 39 . 80 . 2177 .15620 .11140 62 . 6 1.988 24 . 25 . 2799 .16670 .10714 62 . 6 2 . 560 14.11 . 05441 . 04167 . 03267 63 . 2 . 508 113.9 . 09068 . 06771 . 05059 63 . 2 . 852 63 . 47 . 1270 . 08333 . 05805 63 . 2 1 .192 37 .12 .1632 . 09375 . 06035 63 . 2 1.532 23 . 38 .1343 . 10420 ,. 07728 62 . 7 1.232 44 .19 . 2238 .16670 . 11974 62 .7 2 . 048 24 . 66 . 3134 . 22920 .16212 62 .7 2 . 864 17.03 87

( l) (2} (3) (4) ( 5) (6) Plate No . 9 - !Ieavy oil • . 05496 . 10940 . 05936 65 . 6 . 220 199 ,6 . 09160 . 16150 . 08346 65 . 6 . 367 101.1 . 1282 . 21350 . 10746 65 . 6 . 514 66~41 . 1649 . 28650 . 15242 65 . 6 . 660 56 . 96 . 03974 . 08854 . 05014 63 . 5 . 135 322,7 . 06624 . 13020 . 07160 63 . 5 . 225 165 . 8 . 09273 .17190 .-09298 63 . 5 . 315 110.0 . 1192 . 21350 . 11434 63 . 5 . 404 81 . 76 .09843 . 21350 . 13022 64 . 2 . 354 136.5 .1641 . 3281 ;19462 64 . 2 . 588 73.43 . 2297 .40100 . 21736 64 . 2 . 828 41 . 8 7 Plate No . 10 - wLL • 1 - LiBht oil . 09329 . 05208 . 03436 62 . 9 . 648 72.40 . 1555 . 08333 . 05165 62 . 9 1 . 080 39 .17 - 2177 . 10420 . 0 5940 62 . 9 1 . 515 22 . 98 . 2799 .11460 . 05504 62 . 9 1.944 12 . 88 . 05441 . 03125 . 02225 63 . 6 . 383 137 . 8 . 09068 . 05208 . 03496 63 . 6 . 642 77 . 97 . 1270 . 06250 . 03722 63 . 6 . 899 42 . 31 . 1632 . 06771 . 03431 63 . 6 1.155 23 . 61 . 1343 . 07292 . 04600 62 .7 . 924 46 . 75 . 2238 . 12500 . 07804 62 . 7 1 . 536 28 . 57 . 3134 . 16670 . 09962 62 . 7 2 .148 18 . 61 Plato No . 10 - Heavy oil . 05496 . 08333 . 03329 65 . 6 . 165 224 . 3 . 09160 . 12500 . 04696 65 . 6 . 275 101 . 1 .1282 .16670 .06066 65 . 6 . 385 66 . 66 . 1649 . 19790 . 06382 65 . 6 . 495 42 . 40 . 03974 . 06771 . 02931 63 . 5 . 101 335 . 4 . 06624 . 09896 . 04036 63 . 5 . 169 166 . 1 . 09273 . 13540 . 05648 63 . 5 .236 118. 7 . 1192 .16670 . 06759 63 . 5 . 303 85 . 66 .09843 .15600 . 07272 64 . 2 . 266 135 . 6 .1641 . 25000 . 11652 64 . 2 . 441 78.15 . 2297 . 33330 .14966 64 . 2 . 621 51 . 25 Plate No . 11 - wLL : 1 - L1f3ht oil . 09329 . 04167 . 02395 62 . 6 . 426 113 . 5 .1555 . 06250 . 03082 62 . 6 . 710 52 . 59 88

(1) (2) ( 3 ) (4) ( 5) (6)

. 2177 . 09375 . 04895 62 . 6 . 994 42 . 62 . 2799 . 10420 . 04464 62 . 6 1 . 280 23 . 51 . 05441 . 02083 . 01183 63 . 2 . 254 164 . 9 . 09068 . 02604 . 00892 63 . 2 . 426 44 . 76 . 1270 . 04167 . 01639 63 . 2 . 596 41. 92 . 1632 . 05208 . 01868 63 . 2 . 766 28 . 93 . 1343 . 04687 . 01 995 62 . 7 . 616 45 . 61 . 2238 . 08854 . 04158 62 . 7 1 . 024 34 . 25 . 3134 . 11980 . 05272 62 . 7 1 . 432 22 . 15 . 4029 . 14060 . 05348 62 . 7 1 . 844 13.59 Plate No . 11 - Heavy oil . 05496 . 05729 . 00725 65 . 6 . 110 97 . 52 . 0 9160 . 09375 . 01571 65 . 6 . 184 76 .10 . 1282 . 11980 . 01376 65 . 6 . 257 34 . 00 . 1649 . 14580 . 01172 65 . 6 . 330 17 . 52 . 03974 . 05729 . 01889 63 . 5 . 0676 486 . 3 . 06624 . 0781 2 . 01952 63 . 5 . 112 180 . 8 . 09273 . 09896 . 02004 63 . 5 .157 94 . 5 . 1192 . 10940 . 01034 63 . 5 . 202 29 . 57 . 0 9843 . 11460 . 03132 64 . 2 . 177 131 . 4 . 1641 .17710 . 04362 64 . 2 . 294 65 . 82 . 2297 . 24480 . 06116 64 . 2 . 414 47 . 12 . 2953 . 30730 . 07354 64 . 2 . 530 34 . 28 Plate No. 12 - W/ L • 1 - Light oil . 09329 . 03125 .01353 62 . 6 . 213 256 . 6 .1555 . 04167 . 00999 62 . 6 . 355 68 . 17 . 2177 . 05208 . 00728 62 . 6 . 497 25 . 35 . 2799 . 06250 . 00294 62 . 6 . 640 6 . 19 . 05441 . 01042 . 00142 62 . 9 . 125 7 9 . 18 . 09068 . 02083 . 00371 62 . 9 . 210 74 . 48 . 1270 . 03125 . 00597 62 . 9 . 294 61 . 09 . 1632 . 04167 . 00827 62 . 9 . 377 51 . 25 . 1343 . 03125 . 00433 62 . 7 . 308 39 . 62 . 2238 . 05208 . 00512 62 . 7 . 512 16 . 86 . 3134 . 07292 . 00584 62 . 7 . 716 9 . 81 . 4029 . 08333 62.7 . 922 89

DENSITY AND VISCOSITY C LIBRATION

TABLE VI

rependence of Denaitx on Temperature

Temp . • °F Density-lb.m/cu.ft.

Light oil {SAE 140) 60.0 56 . 2 61. 4 56 . 2 63 . 6 56 .1 65 . 8 56 .0

Heavy oil (SAE 250) 63 . 4 57 .0 65 .8 57 .0 66 .7 56 . 9 90

, ' I ! . . J...... 1'\._ . : : :·· . ' .. . - __L-9---r--+--,:--- ·:-,·--.- --_1---1--+ . _· . . . :· :· r ' ·::~_.;...:·+·~-+-+-4---+-l ~~ ~ :....:. ;·.: ··: ::1 ..;:,.·..

I _c· "': --+···. i !·1 .. . . · ·: :" . ';.!, •, .• :.: ,·h -· .. :.... j t-.-+~-:' . ·,: ·: ~t ~:..,.... ~!-'-: ': · D I.:J··!--;-:-, · ' , ; .· , · ,,D iJ · ,. ·, ::,.; ,'· . - ~ : . ; . . . I·. -.' ,'· .:• ..... ·. . .-

DEPENDENCE OF VISCOSITY ON TEMPERATURE- LlGHT OIL

FIGURE 18 i' . t ~ . . ,,, .ir :••; ,.· ... H

.. - ~ - ii'' .. '!li , .

. v I , ..... '-+- -- H -+-c+--f=t-4-+-+~~---+--1- -+-+- --:- -~-- ~ -0~"-"" ~-.+-+-~:.,..·.+· --+.:.:J-_-:f',:..,.:,,:+-c,,-1]'.. -+-+-i~-'--1·-, -,L... -'-'-l-+-+++...;..1-1- ~ --t~--1- --~- ~+-· j----:- -~* ,-i- -1-~ -·,..,-:.·~--+-'-f'-+-+- ~.-- h 1:-+-+-t-+-+--1-+-t-+- +- L .. ~~ : --~ ----:---t- -1- f,~ r . i , J , -...... 1 I . T -, --'- r-I.·.+,--~- -,l·-;-- ' -~-~-, T .. .po.;..d---l>-1-'-'+-t-+-+-+-t------r·-i ·+-- :~ .e +-- --+- ~- ~~ 3 t- - . ~- "". -. 6!5 +- -f- - 1 .. e1-l --- -· -+-+~•a+;_,_~J·, -.- 1 I , I +-+rH-r~'1 ~ - i~-t-f- : ·' l I c.....Lt DEPENDENCE OF VISCOSITY ON TEMPERATURE HEAVY OIL FIGURE 19 92

SAMPLE CA LCULA TIONS

1. Calculation of Drag Force on the Wire

Example : l-inch sphere (62 . 4°) 1.29 rev./sec . 3/4 in. pulley Light oil Velocity - 0 . 196 ft . x 1 . 29 rev. : 0 . 254 ft ,/sec . rev. sec . Density - 56 .1 lb.m cu . ft . Viscosity • 2 , 06 lb,m ft .-seo . (Figure 18 ) Diameter - 0 . 0833 ft . Reynolds number

~a 0 , 0833 ft .(.254 ft ./sec ,)(56 .1 lb.m) : 0 , 576 .A 2 .06 l b . m ft3. ft.-sec. Total measured force including weight - 0 .156 lb.

Wt . of ball - !(485- 56 lb.m)(l ft .)3 : 0 .130 lb. 6 3 12 ft . Measured drag force on sphere - 0 .156 - 0 . 130 : 0 . 026 lb . fd (Stokes) - o . ~~S : 41 . 6

Force (Stokes ) 3 2 41 . 6 (56.1 lb.m/ft . )(0 . 254 ft ./sec . t(0. 00545 ft .) : 2(32 . 2 lb .m ft ./lb . f sec.2

o . ol29 lb.f 93

Drag force on wire - 0 . 026 lb . - 0 . 0129 lb . : 0 . 0131 lb.

for F ow)

Example : 4 11 l cylinder, 1/2" dis. ., 0 . 4751 rev./sec . 3/4" pulley Light oil Ve locity - (Same method as part 1) : 0 . 09329 ft ./sec. Density • 56 . 1 lb.m/cu . ft .

Viscosity - 2 . 05 l b . m/ ft . -sec . (Figure 18 ) Diameter - 0 . 0417 ft . Reyno l ds number - (Same method as part 1) : 0 .105 Measured drag force - 0 . 02083 lb.

Correction force f or wire (Figure 9 ) : 0 . 00886 lb . Drag force on cylinder - 0 . 02083 1b.-0 . 00886 lb. : 0 . 01197 lb. fd - o . 01197 l b . r ( 32 . 2 lb.mft ./l b . rsec~}(2) 2 ( 56 .1 lb. /cu.rt.) (0 . 09329 ft ./sec .) (0.01389 ft . 2 ): m 113 . 5

3 . Ca lcul ation of Dra' Coefficient for Flat Plate - arallel Flow

Example: 1 in. 1 ., 4 in . w, (62 . 4° ) 0 . 4751 rev./sec . 3/ 4 u pulley Li ght oil 94

Ve locity - (Same as part 2) : 0 . 09329 ft . /see . Density - 56 . 1 lb. / cu. ft . m Viscosity - 2 . 06 l b . m/ft . - sec . (Fi gure 18 ) Length - 0 . 0833 ft . Reynolds nutlber - (Same method as part l ) : 0 . 212 Measured drag force - 0 . 02083 lb. Correction force for wire - 0 . 00886 lb. (Fi gure 9 ) Correction force for edge effect 3 . 2(0. 09329 ft . /sec . ){0 , 0833 ft.)(2 . 06 l b . m) = 0 , 00159 lb. 2 (32 . 2 lb. ft . /lb. sec . ) ft . -sec . m f Drag force on plate 0 . 02083 lb. - 0 . 00886 lb . - 0 . 00159 lb . = 0 . 01038 l b . fd - 0.01038 (2} (32 . 2 lb.mft ./lb.fsec . 2 ) - 2 - (56 . 1 lb. / cu . ft .)(0 . 09329 ft . 2 ) m ft . /sec . ) ( 0 . 055~6

24 . 6