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Interim Report I-A2049-32

SOME REFINEMENTS OF THE THEORY OF THE VISCOUS SCREW PUMP

by

H. G. Elrod

March 1969

Prepared under

Contract Nonr-2342(00) Task NR 062-316

Jointly Supported By

distributton v

ovod

im

t^ G Department of Defense

Atomic Energy Commission National Aeronautics and Space Administration

Administered by

Office of Naval Research Department of the Navy

Reproduction in Whole or in Part is Permitted for any Purpose of the U. S. Government

liife THE FRANKLIN INSTITUTE RESEARCH LABORATORIES ■ ENJAMIN FRANKLIN PARKWAY • PHILADELPHIA. PENNA. 1910S

I I I

Technica

Interim Report I-A2049.32

SOME REFINEMENTS OF THE THEORY OF THE VISCOUS SCREW PUMP

by

H. G. Elrod

March 1969

Prepared under

Contract Nonr-2342(00) Task NR 062-316

Jointly Supported By

Department of Defense Atomic Energy Commission

National Aeronautics and Space Administration

Administered by

Office of Naval Research Department of the Navy

Reproduction in Whole or in Part is Permitted for any Purpose of the U. S. Government

i THE FRANKLIN INSTITUTE RESEARCH LABORATORIES BENJAMIN FRANKLIN PARKWAY • P H I L A D E L f H I A . PINNA. 10103

■-t^f«.«*?

ABSTRACT

Recently-performed analysis for herringbone thrust

bearings has been Incorporated Into the theory of the vis-

cous screw pump for Newtonian fluids. In addition, certain

earlier corrections for sldewall and channel curvature effects

have been simplified. The result Is a single, refined formu-

la for the prediction of the pressure-flow relation for

these pumps,

?•> provide a self-contained presentation, some of the

results of FIL Interim Report I-A2049-31 are also given here.

TABLE OF CONTENTS

Paai

ABSTRACT 1

TABLE OF SYMBOLS fv

I. INTRODUCTION 1

II. THE INTERIOR PRESSURE SOLUTION 3

III. THE EDGE CORRECTION 5

IV. FLOWRATES ACCORDING TO REYNOLDS EQUATION 6

V. 5IDEWALL EFFECTS 8

VI. EFFECTS OF CHANNEL CURVATURE 13

VII. EDGE SOLUTIONS FOR RECTANGULAR GROOVING 15

VIII. COMPARISONS WITH OTHER INVESTIGATIONS 20

IX. PUMP PERFORMANCE FORMULA 23

X. A NUMERICAL EXAMPLE 26

REFERENCES 31

APPENDIX A - SIDEWALL EFFECTS . 33

APPENDIX B - CHANNEL CURVATURE EFFECTS 36

APPENDIX C - SINGLE-GROOVE EDGE CORRECTION 39

11

»yy'WW 'WWW

I I I

LIST OF FIGURES

Fig. No. Page

1 Schematic Diagram of Pump (Adapted from Ref. 2) 1

2 "Unwrapped" Groove-Ridge Combination, Showing Coordinates and Dimensions Employed 3

3 Sketch of the "Sawtooth Function 5

4 Control Region ABC Used for Secondary Method of Calculating Mass Flow 8

5 Groove-Ridge Cross-section 9

6 Sidewall Correction Factor - Drag Flow 11

7 Sidewall Correction Factor - Pressure Flow 12

8 Conformal Transformation 17

9 Sketch to Assist Understanding of Inter-relation of Inlet and Exit q's 18

10 Pressure Correction Factors for Edge Effect with Rectangular Grooving (Versus Groove Angle) 19

11 Dimensions of Experimental Screw Pump 26

lii

»■^MWKjJMfc«

TABLE OF SYMBOLS (Equation Number Indicates Where Symbol First Used)

c Nominal film thickness9 (2,03)

tit) Pressure ripple function, (2.05)

fit) Hippie function, (2.12)

F(«0 Radius ratio function (Booy), (B.03)

F Hyperbolic function, (A.04)

Fp.Fp Correction factors for pump formula, (9.03)

FDC,FPC Booy's correction factors, (8.01)

FpE Booy correction factor, (8,02)

g Constant In ripple solution, (2,0?)

G(|) Ripple function, (2,12)

G Radius ratio function (Booy), (B.04)

h Film thickness, (2.01)

H Dlmenslonless film thickness, h/c, (2.03)

K Radius ratio function (Booy), (B.05)

L Axial distance (z^) through pump. Fig. 2

m fm Components of mass flux, (4,01)

M Total mass flowrate thnough pumping section, (4,05)

p Pressure, (2.01)

P* Dlmenslonless pressure, (9.12)

Ap Muijdermann's pressure correction factor. Fig. 10 corr q* Homogeneous solution of eq. (2,04), (3.01)

Q Volumetric flowrate, (5.02), QD due to drag, Qp due to pressure gradient.

Q» Dlmenslonless flow, (9.12)

r Assignable constant, (A.12)

R0,R1 Outer and Inner radii of channel, resp,, (6,01)

iv

I i s

~ St tu

s

'■ w s(*)

■' T(a£)

•• ül

V

• - VTh • •

V2

■m *

Xk

mm. X

.. yk

t » c^

.* oc

« A V r ' *

A

»«. e

* r

•(» ^

..-, 7

1 7

.'>- A r *

i T

i

width of channel or groove, Pig, 5

Sawtooth function, (2.14)

Filmthlckness constants, (9.1^)

Radius ratio function (Booy), (B.02)

Radius ratio function (Booy), (B,02)

Velocity of surface In x^-dlrectlon, (2,01)

Velocity of fluid In y-coordlnate system, (5,01)

Particular and homogeneous solutions of (A,01), (A.02) and (A.03)

Velocity of surface In y2-direction, (A,02)

Cartesian coordinates. Fig, 2 and (2,01).

Dummy variable for cosine function, (A.14)

Coordinates for Inclined Cartesian coordinate system, (2.02)

Pressure-gradient constant in interior solution, (2,05)

Radius ratio, RJ/RQ» (B,01)

Groove angle, Fig, 2

Fraction of A devoted to groove or channel, (2,13)

Wavelength of groove-rldge cycle, Fig, 2

hG/R0, (6.03)

Dlmenslonless distance In y-i-direction, scaled by (2,03)

Coordinate In complex plane, (C,02)

73/^}. (A-01)

Coordinate In complex plane, (C.02)

Viscosity, (2.01)

Dlmenslonless distance In X2-dlrectlontscaled by L, (2,03)

Dlmenslonless pressure, (2,03)

^ 3.14159

f Density, (4.02)

-r y^/hc, (A.01)

$ Groove angle at radius R0. (Booy's notation)

^ Dlgamma function, (7.06)

vi

s^v*; i ■ *"?.•*?**t, aajMUHWI

I. INTRODUCTION

Figure 1 shows a schematic diagrau of a viscous screw

pump. This device is used to pump relatively small quantities

of viscous liquid against relatively high pressure differences•

It is used, for example, in the melt extrusion of plastics.

In operation, as the screw rotates, the barrel surface drags

by viscous shear a ribbon of liquid between the "flights*1 of

the screw from a feed port at low pressure to a discharge port

at high pressure.

-BARREL JL FEED PORT j—

SCREW-' FLIGHT ^-DISCHARGE PORT

Figure 1. Schematic Diagram of Pump (adapted from ref. 2)

A substantial body of literature pertaining to the opera-

tion of the screw pump already exists, as the reader who pur-

sue? tne subject will discover. The references cited herein

are believed, however, to give a state-of-the-art picture of

the theory of viscous screw pump operation with Newtonian

fluids. The purpose of the present work has been both to gen-

eralize and to simplify some of the prior theoretical results.

In particular, it has been possible to Incorporate almost

directly some recent analysis performed to treat herringbone

thruct bearings.

The simplest theoretical treatment of the viscous screw

^•"*^sua*

»—

pump involves an "unwrapping" of the pumping channel to create

o passage between parallel planes. Plow over the flight tips

Is neglected. The channel depth Is taken as very small compared

with Its width, so that Reynolds* equation can be employed for

computation of mass fluxes. A simple formula relating flow (5)

and operating pressure difference can thereby be obtained

Successive Improvements of the simplified theory are pos-

sible. For the Isolated, rectangular, parallel-plate channel

the effects of finite width ("sidewall effects") can be taken (^.6.2)

into account . Also, for the same geometry compatibility

between the interior transverse pressure ripple and uniform

extejrior pressures (at Inlet and discharge) can be achieved (2,3)

. Without unwrapping, the isolated rectangular channel

of large width-to-depth aspect ratio can be analyzed for the (1)

effect of channel curvature , If sidewall effects, channel

ourvature and entrance-exit effects are neglected, Reynolds*

equation can be solved Jointly for groove and ridge film thick- (3.7)

nesses . Moreover, variation of the channel depth in the (7)

direction of flow can be accommodated

In the present work, a Reynolds* equation treatment for

arbitrary transverse ridge and groove shape is given for the

pump Interior, and the general theory for the corresponding

edge effects is set up. The edge-effect analysis is numerical-

ly implemented for the case of rectangular grooves and ridges,

an exact solution being obtained in the case of an Isolated

groove. Simplifications of the treatment of sidewall and

curvature influences are offered.

2

MfSflP^V r**&¥* ■

II THE INTERIOR PRESSURE SOLUTION

The starting-point for the present analysis Is Reynolds'

equation for a constant-property Newtonian fluid, applied to

the unwrapped groove-rldge combination shown In Figure 2.

*~ xt

Figure 2 "Unwrapped* Groove-Ridge Combination, Showing Coordinates and Dimensions Employed

Thus:

«0.'£&'''&+&'&} (2,01)

Conversion of this equation to the skewed coordinate system

(y.,*i) gives:

Novr take: (2.02)

Then!

/»^ fyiAtefi. y rr /= ^- /= X- Ush (2.03)

^=^ "'iT+^ ^- ^^ «;4 -^^ f2'wl Asymptotically (away from Inlet and exit) It Is antici-

pated that the pressure will rise linearly with distance

along the groove. Substitute the following "Interior" solution

3

-fimniiimi

Into eq. (2.0^).

«r= -H^L- + f(t> (2.05)

An ordinary differential equation for the transverse pressure

ripple, fCf), Is thereby obtained. Thus:

Ää- AH3*** cc dH3 <2 06) Tf- M* Tf T( (2-06)

One Integration gives:

H=Hdf~e(H+-2i a a C0*dt**t (2.0?)

On

*i = A/"V o^ - gH* (2.0?)

Since the net change In "f •• over one cycle In " f " must be

zero, "g" must satisfy the following condition.

0~ fF + * ~ fJF (2.08) Therefore:

W3 df = (H'H3- tFti') + * (ff1- W) <2.09)

fP/CO* -iKV-xGtO {2-10)

Where:

F(t)= 2 S(FH'- iPN2) dc ; Cft)= fClJ'- ipjdt (2.11) o o

The "Interior" pressure solution Is therefore given by:

For rectangular geometry such thati

H= Hl} o* fir . U= HZiy* t<l (2.13)

the pressure solution Isi

ir^ t£cH4)

~{fa3Hz-»3tf) + oc(H'~y)} £(J^ (2.1^)

Here ^fCfyr) is the "sawtooth function" depicted In Fig. 3.

Ir

Figure 3

0 Y 1

Sketch of the "Sawtooth Function"

III THE EDGE CORRECTION

Although solution (2.11) Is an exact solution of the

differential equation (2.0^), and Is adequate for the pump

Interior, It does not satisfy conditions of pressure uniformity

at ^ = 0 and ^ = 1. Therefore, in accordance with standard

procedure in the solution of linear differential equations,

a homogeneous solution is sought which can be combined with

the inhomogeneous solution to satisfy boundary conditions.

Let q'(^,f) be a solution of the following homogeneous

differential equation:

Boundary conditions on "q" appropriate to the inlet region

are:

fat) - ctMl^iW H

5

(3.02)

■"■-■«»*wS»i*|

Likewise, boundary conditions appropriate to the exit region

aret

n (3.03)

Corresponding functions (2*- f* ) and (|t- ^ ) vanish In the

pump Interior, but possess the required "ripple" at the exit

and/or entrance. Evaluation of specific " f^S " will be deferred.

As the general solution satisfying both differential

equation and boundary conditions, we have, then:

The constant "<*," In eq. (3.0^) can be found by taking the

difference between results at / = 0 and ^=1. Thus:

In view of the definition of "^" and the fact that the "fjs "

are Independent of (A/L), It can be seen that, as (A/L)

approaches zero with finite pressure difference between en-

trance and exlt,"o6" Is ascertalnable with Increasingly high

precision. In other words, the edge phenomena have vanlshlng-

ly small effect on the general linear pressure rise within

the pump as the groove-to-length ratio of the channels becomes

small.

IV FLOWIUTES ACCORDING TO REYNOLDS EQUATION

The mass flux per unit transverse distance In the

x^-dlrectlon is given, according to Reynolds' relations, by:

I I 1 ^--^kl i

X

i

(4.01)

Conversion to the dlmenslonless variables "ft, f, and f n

gives:

Since eq, (2.05) gives an accurate evaluation of M1tH In the

pump Interior, and since the total throughput must be Invariable

at all axial locations, this equation Is now used In (^,02),

Therefore:

Use of eq. (2.09) then yields:

The total mass flow per groove-ridge combination Is found by

integrating over a distance "a." Thus:

j. The pressure-discharge performance of the screw pump can now

-#. be found — according to Reynolds' equation — by the insertion i

in eq, (4.05) of the pressure-gradient value of "OL" from

eq. (3.05).

An alternate way of arriving at the expression for "M"

^ consists of calculating the flow components separately across

I lines AB and BC shown in Fig. 4. These flows combine to give

"M", of course, but it will be helpful for later purposes to

T possess this flow breakdown.

7

I ■ ■'■■ --'-v^/»^,

Figure ^ Control Region ABC Used for Secondary Method ' 2£ Calculating Mass Flow

The mass flux In the xJ-direction Is» according to

Reynolds* approximations:

Components of the mass flux vector are now readily found

In the y, and y^-dlrectlons by use of eqs, (4.01) and (4.06).

In particular, It Is found that: o

^ fJ^-r^fft, (..06)

fMy d/t = PC U(c~/3) C*±HVA (4.07)

andi c _ _^ ,

/M/a^= fcL!fc*ß)lH~*JPiiutfi)}A (4,o8)

V SIDEWALL EFFECTS

Viewed In the y^-direction, the groove-rldge cross-section

within the pump appears like that In the sketch of Fig. 5

Reynolds' equation cannot take account of the fact that fluid

velocities must vanish on the sides Pg and RS of this configura-

tion. To accommodate such boundary conditions, the full Stokes'

equation must be used.

There can be little doubt that even within the more pre-

cise theory of Stokes equation, an interior region of the

.-»-v, r-.^- ■.*««al^n tmMitH

i

I I 1 I I I

/

I

I I I I

I I I I

s. Y~*

K

I t a

A—* Figure 5 Groove^Ridge Cross-section

pump exists where velocity patterns nre established and where

vyA is constant. For the coordinate system (y^ ,y# .y^ ) Stokes'

equation for M#^,, becomesi

(5.01)

Although this Is a simple form of Polsson's equation, the boun-

dary conditions on the groove-rldge corrugations are awkward.

Since usually the ridge or flight film thickness Is much

less than the channel or groove film thickness, examination

of the solution for an Isolated groove of dimensions (h^ x s).

Exact solutions have been given by a number of authors in

several different forms. Here we write:

(? =r fl^ "f 0/

With: &~i(l%)l

fHVS

(5. 02)

' «— *******

The series terms In the foregoing equations represent devia-

tions from the results of Reynolds* equation. For h /s ~ 3A,

the following approximations are accurate within 2,5%, Improv-

ing very rapidly as the depth-to-width ratio diminishes.

^.-^^[i-^OZ^jJ (5.051

A derivation for the flow deficiency along the sldewalls

from the groove bottom to some arbitrary position " y3" Is given

In Appendix A, For this more general case, expressions (5.0^)

and (5.05) become:

O.-iVth^ll-k tifyj (5.06)

Q'=- & & {i - ¥ fum (5.o?i The functions Mf|M and "f^' are shown In Figs. 6 and 7 respect-

ively.

The foregoing sldewall analysis Is strictly applicable

only to the case of an Isolated (or closed) rectangular groove.

However, with PQ/PT taken as the argument (y,/h^) In Figs. 6

and 7. the flow deficiencies for open rectangular grooves are

approximated. This particular mode of approximation will lead

to somewhat greater-than-actual absolute values of both flow

deficiencies, since some of the flow constraint along gT Is

missing In the case of the open groove.

10

r i i i

i i

f 1

CfAtt

I u

S

I k

O

I

4 ^1

8

nAft

12

„■ .'...--v*,,^^.

I I I

il

VI EFFECTS OF CHANNEL CURVATURE

It Is usually justifiable to treat the flows In the

grooves and over the ridges as though they took place be-

tween parallel flat plates. However, It Is possible for the

radium of the screw root, R. , to he sufficiently different

from the radius of the pumt) barrel, RQ, SO that the unwrap-

ping process can introduce some error. This error is, of

course, principally associated with the groove. (1)

Booy, in an analysis which will not be repeated here,

considered the effect of channel curvature. He treated

the case of infinitesmal film thickness over the flights,

and presumed an "Infinite" groove (or channel) width.

Using Stokes* equation, Booy showed that it is possible to

obtain a flow-field solution in the following form.

Radial velocity and pressure gradient negligible.

Axial velocity and pressure gradient inter-related as

follows:

- *£[£)%{,-*')$§>-,]>*.% ,6..., Tangential velocity and pressure gradient inter-related

as follows:

The tvro undetermined pressure derivatives appearing in

the foregoing equations were determined by requiring that:

(a) the net flow normal to the ridges (or flights) be zero.

(b) the volumetric throughput be appropriately related to

13

'''■'■•*'-'f*«*-.**

the Integrated pressure gradient from entrance to exit at

ny fixed transverse position In the groove. (Note: all such

Integrated gradients, or pressure differences, must be the

same, although transverse pressure ripples remain to be can-

celled by edge corrections.)

In the case of helical grooves, the assumption of "Infin-

ite" groove width Is Justified only when the groove depth Is

small compared v.Tlth the screw root radius. Accordingly, It

Is reasonable to restrict consideration to those cases where

the ratio (R^/KQ) IS greater than 0,5 In Appendix B it is

proved that a virtually "flat-plate" formula results If a

"mean" radius Is selected two-thirds of the way from the ^jt

to the barrel radius. The helix angle at this mean radius

Is taken as "fi ." The simplified formula Is: t

Tills formula shows that within the limits designated, the ef-

fective groove width varies by about plus or mlnu? 8^, Fur-

shows that the curvature correction virtually vanishes

for a groove angle of ^5 degrees.

Booy's solution of Stokes* equation does not yield zero

local velocity normal to the groove edges - only zero net

flow normal thereto. This same type of average conformity

to the velocity boundary condition would be provided by a

solution of Reynolds' equation In the case of a more shallow

groove. Hence sldewall corrections are In order. Rigorously,

such corrections depend on the radius ratio,ot. However,

U

I I

If correctlonr; on corrections are dismissed by reason of

1 their pecond-order smallne^s, the analysis of Section V

is applicable,

VII hDGiL SOLUTIONS FOR RECTANGULAR GROOVING i * At thlr; point we revert to further consideration of the I

entrance-exit edge problem. It suffices to treat this aspect

} of the pump problem vTlth Reynolds* equation. Moreover, atten-

tion will be directed to the rpecial, but extremely important,

• cape of rectangular grooving with equal groove and ridge

j width.

The analysis of Section III demonstrates that the princl-

j pal result required from solution of an edge problem is the

value of tf . Some information concerning this value is pro-

vided by the mass-content rule. Thus, Integrate eq, (3.01) i i i i

f I I

1 ^fö^-^ltir^t (7-03)

r i

I

over one groove-rldge cycle:

0 o * o > (7.01) Integration of the second term by parts results in some cancel-

lation, and the following simpler equation.

This equation can be integrated once with respect to "% ,"

The integration constant must be zero if uniformity is to be

achieved at infinity. Therefore:

15

In the care of rectangular grooving, the fllm-thlckness

derivative becomes a delta-function at /" = 0 and f = K .

Thus:

o (7.04)

It would appear that further progress cannot be made along

these lines for the case of arbitrary groove angle, since

qQ and 'q depend on both "£" and the fllra-thlckners ratio,

H-^/Hp. In the special case of £ = fr/2, It Is seen from

eq. (7.03) that in all cases:

JH9f^$ = constant (7.05) o

With rectangular grooving (and any/B) *q must become the saw-

tooth function along ^--r 0. (See eq. 2,14 and Fig. 3) For

convenience, assign unit amplitude (rather than K*) to this

function. Then It Is easy to shov; from eq. (7.05) that:

For arbitrary "g" a solution for "q " probably cannot be f * CO *

obtained in closed form, even for rectangular grooves and

ridges. However, for an isolated rectangular groove (zero

ridge or flight film thickness), the author has been able to

obtain such a solution. Details of the analysis are given in

Appendix C. The procedure Involves transforming by conformal

mapping problem A of Fig. 8 into problem B. .Vlth constant film

thickness, "q"' satisfies Laplace's equation in the (/, ,)^)

plane. By virtue of the properties of conformal mapping, it

will also, then, satisfy Lanlace's equation in the («,&")

16

I

f )

I

B

0

Figure Q Conformal Transformation

plane. Zero normal derivative along the groove sides is also

prererved.

Mov the average value of "jp1 at any level v = const, is

alro "q*" In the (u,v) plane, riovever, both planes possess

the same Mq V Therefore, the determination of "q^" is reduced

to an Integration along v = 0. The final result is:

£= ^ {^(i) - f(i)} (7.06)

vjhere "Y" ^^ the ^igamma function, tabulated (e.g.) in ref. 7,

In the ca.^e of rectangular grooving, the asymptotic value

of "KD" at the exit can be found from that at the inlet.

The inlet and exit configurations, as shown in Fig. 9^are help-

ful in understanding the relation. The pressure ripples which

Ferve as boundary conditions for the respective "tf 's" are

shown. Now the "responseM to the pressure ripple of the inlet

configuration Is "q'4 f,, The response to a uniform pressure of

unity would be, of course, Just 1.0. Because of the linearity

and homogeneity of the differential equation for "5"*, the res-

ponse to (1.0 minus the Inlet pressure ripple) would be 1 - ^T*

17

r

prettore

Inlet Configuration bxit ConfiRuration

Figure 9 Sketch to Assist understanding: of Inter-relation of Inlet and Exit TJ's

However, a? can be seen by examining; the ripple for the exit

configuration, this newly-created compound ripple at the inlet

is precisely the same. Therefore:

?> i-f- (7.07)

In particular, for the Isolated rectangular groove:

ft** ^lf0-^)-^(i)j (7.08)

Reference to eq. (3.05) shoves that the difference ^^T feo

is the quantity needed for over-all pump-performance calcula-

tions. Accordingly, this is the quantity presented in Fig. 10

as a function of groove angle and film-thiclcness ratio. The

groove and ridge widths are taken as equal. The curve for

H2/H1 = 0 corresponds to eq. (7.06). The curves for other

film-thickness ratios were obtained by use of the finite-element (8)

numerical approach developed at The Franklin Institute . For

this data the author is much indebted to Dr. T. Y, Chu.

Appendix D gives the numerical values on which the cürv.:r uf

rlC. lr ''*?< K'iccd, Cross-plotting war employed with the

18

10 20 JO 40 BO «O TO SO 90 IOO

GROOVE. ANGLE, ß

Figure 10 Pressure Correction Factors for Edge Effect with Rectangular Grooving

(Versus Groove Angle)

i 19

-^--.t, isv

assumption of linearity In (H /H )^.

The asymptotic levels of "ff" can be found from Fig, 10,

but no Indication Is provided concerning the distance along

the groove or channel required to attain them. A reasonable

estimate can be formed by further consideration of the case of

the Isolated groove vith 4 = T/2. In this particular case, a

standard Fourier treatment yields: op -filtV*

3-= y"/4„ c*,(*ri<)e (7.09)

Here A- = ^ = ^ , Deviations from uniformity approach zero

exponentially, with the least rapid attenuation associated with

the first mode, containing the factor € , When v = 1,

this factor has already been reduced to only ^.3$ of its value

on the boundary v = 0, Translated into a more general con-

clusion, this result means that end-effects persist for only

about one channel-width into the channel. In the case of

oblique channels, measurement towards the interior should be

made from the apex of the obtuse angle. To achieve asymptotic

conformity within the pumD interior, there is a minimum axial

length required. It is given by: (approx.)

^^ ZM/S + CH.fi (7.10)

VIII COMPARISONS WITH OTHER INVESTIGATIONS

In order to develop a suitable pump-performance formula,

it has been necessary to repeat some material presented by

earlier investigators. Here an effort will made to indicate

the aspects of novelty of the present work, to inter-compare

20

I I I I

v:lth earlier results, and to report various checks for accuracy

which vrere made.

Accurate account for cross-flight leakage (within Reynolds' (9) (10)

lubrication theory) vTas made by Chippie, Vohr and Pan and by

others treating herrln^bone-Dattern bearings, as well as by (3)

Wuijdermann in work cited earlier. However, the derivation

piven in Section II is believed to be new, and is simpler than

the matching of solutions at groove-ridge Interfacial discontin-

uities. For the special case of rectangular grooving, eq, (2.14)

fCives Muijdermann's "approximate pressure distribution,"

i^xact solutions for established laminar flows have a ven-

erable part. "Pressure-flow" expression (5.03) for a rectangu-

lar flov; crops-section was given by Carley and Strub as a

form preferable to that given by ßoussinesq — although mathema-

tically equivalent. In the same way, expression (5.02) for

"drag flew" is somewhat to be preferred to equivalent series

given in refr. (1), (4), and elsewhere. In each case, rapid

series convergence with attendant computational simplicity is

obtained by treating the sidewall flow deficiency as a residual,

boundary-layer effect, brought out further by the approximate

expressions (5.0^) and (5.05). The partial deficiency functions

In cqs. (5.06) and (5.0?) capitalize on this simplicity,

Booy*s analytical results for the influence of channel

curvature are reproduced by eq (6,03) in the (hG/R ) range of

most interest, u'ith virtually the same degree of approximation,

his flow factors are given by: (his notation)

r ^ = "r £c= -J±i— (8.01)

i

I

r i

I

I 1 *A>-V. 3 *ih*-tf. / 5 21 a

Values from eqs. (8.01) agree adequately with those from Booy's

Figs, 3 and k of ref. (1), The very wor.^t agreement, corres-

ponding to a radium ratio of 2.0 and extremes of groove angle,

is shovn belov.

F (8.01) DC

F (Booy; DC

F (8.01) PC

F (Booy) PC

4 = 5 1.08 1.0? 1.2^2 1.308

A- 60 0.722 0.693 0.833 0.847

Agreement improves rapidly with diminishing "B ", (3) (2)

Both Muijdermann and Booy have dealt with the edge

effect at the inlet and exit of an isolated groove. Muij-

dermann's electrical-analog results are shown on Fig. 10.

His results ap;ree with the present exact results within 20^,

and the agreement is considerably inside this limit for most

measurements. It should be recalled that any deviations

here are in a correction term. A less demanding comparison

is possible with Booy'r finite-difference computations of

his pressure-flow correction factor, Fp^. This factor satis-

fies the following relation: (Booy's notation)

r,, 1 (Wi)**i ( * Values from this equation are imperceptibly different from

those presented in Booy's Fig. 12, where curves for /^-values

of arctan(1/3,1/2,1) were presented.

For the case of rectangular grooving with ^/H.. ^ 0,

there have hitherto been no accurate corrections available.

Muijdermann tentatively offers a formula based on heuristic

reasoning. He expresses all pressure corrections in terms of

22

I I I

I 1

the pressure correction for an Isolated groove. Thus:

A?.„= *$.* irip, (8.03) * Although thir formula fflves correct

1

Although thir formula gives correct limiting values at

H./FL = 0 and H^/ri =1, It would appear to be considerably

in error for rome intermediate values. For example, with

Jh = 15* and (H^/^)3 = 0.or>f Fig, 10 gives Afa^ 0.43.

T"Therear eq . (8. 03) gives :

1 - 0.0^ ^\b s 0.?9 = 0,70

rc#,r 1 + 0.06

Actually, the curves in Fig, 10 for ^g/^l ^ 0 must be

regarded as somewhat tentative, inasmuch as the finlte-element

calculations were not "converged" by running with diminishing

grid size. Checks with the exact con formal-mapping solution

warrant an error estimate of less than 3/.

IX FUWF PERFORMANCE FORMULA

The individual results of the preceding sections will now

be combined to yield the overall relation between pressure dif-

ference and quantity pumped. The development will be for one

groove-ridge (channel-flight) pair, and it will be assumed

that the channel width has been adjusted for curvature prior

to "unwrapping" by use of the factor in eq. (6,03),

With reference to Fig. 4, the flow across AC is equal to

the flow across AB plus the flow across BC. Kow Reynolds'

equation is deemed satisfactory for the computation of all

flows exceDt that alon^ the groove. (The transverse flow over

AB \r small, In any event, and the flow along the ridge 1n-

■n

■

volve? a very small film thlckneps.) The sldewall flow defic-

iency In the groove Is given by the second (subtractIve) terms

In eqs. (5.0M and (5.07). Thus the total flow, Including

sldewall effects, Is given by:

Q = Q - Q (9.01) Rey def

The flows will, of course, be evaluated In the pump Interior.

Again:

A» a, -/'!£A*£- ^sj^)^l (9.02)

The factors F and F will ordinarily be given well enough by:

The required pressure derivative can be put In dlmenslonless

form. Thus:

When this derivative Is used on the Interior pressure solution,

given by eq. (2.14)f and Inserted Into eq. (9.02), the result

Is:

Q« 0^- U;f^)(rA)h^F^ taJf HiFpJ (9>o5)

Combination with the Reynolds1 flow now gives:

2A

I I

Further straightforward algebraic rearrangement gives:

' \9.07:

T Z ' m9 Let us turn now to the pressure-difference expression

tflvon by eq, (3.05). The Mff*sH of this equation must corres-

pond to the ripple term in eq. (2.1^), since rectangular

irroovlnff" is no^ assumed. For unit savrtooth ripple, Fig, 10

" Kl'/o.: tfi-J SAß.,, Hence, for rectangular grooving, cq. (3.05) becomes:

rOJ-tb) =* —* ^ (am/>^udt) A^,. (9.08)

where the arrplitude is:

^///*</e « ritfH*-Jr3l£)**(H?-H-')] (9 09)

T Apain, some algebraic rearrangement gives:

1 cM/> .= cc - ^V^.. r^^//;j;^JA"^-^/| (910)

Klirnination of the pressure-gradient term "OL" between eqs.

(9.10) and (9.0?) yields the followin.c: final pump performance

formula. Thuss (with c = hG, so that IIQ = H1 =1)

I ^PVä «/I- xle*MA*,äb.tSX{Sn.S>--4L U9.il

lie re:

T AP~ cAt> . Q"^ a (9.12)

(9.13)

9.1^)

25

A NUMERICAL EXAMPLi,

1-0

1". O.oSo

vz. &rrr[t rTTi^rrA* rrrx ran

uff/y

/ / / ' *,//'/.

flischorye Jlffip/j

Figure 11 Dimensions of Experimental Screw Pump

The dimensions of an experimental screw pump tested at

Columbia University are shown above In Fig. 11. Data for

this test were kindly supplied the author by Prof. D. D. Fuller.

The pump possesses two single-thread pumping sections working

In opposition. The channel-grooving Is In the barrel. Per-

tinent characteristics are listed below.

Pitch of Threads; four per Inch

Shaft Diameter; 2.1192 Ins,

Barrel Diameter; 2.123B Ins.

Width of Channel; 0.125 Ins.

Width of Flight?: 0.125 Ins.

Depth of Channel Grooving: 0.050 Ins.

The corresponding film thicknesses are:

Ridge Film Thickness: hR = radial clearance = 0,0023 Ins,

Groove Film Thickness: h = 0.0023 + 0.050 = 0.05231ns.

Also: HG = H1 = 1 and H2 -.: 0.0023/°. 0523 = 0.0^4

Unwrapping of Pump Channel

Although, In this instance, the effect of channel curvature

26

r i i

i i i i i

1." very pmall, for the sake of Illustration Its magnitude will

be comnuted.

Rm = 0.5 * 2.1192 + (2/3)» 0.0523 = 1.09^5 Ins.

Therefore:

1 1 1 1 T rind = O.25O ^ 0.03^ = tan^ 1 / 2tr* 1.09^5

-r in add it I on: i * € = hr/no = 0.0523/(0.3*2.1192 + 0.0523) = 0.04?

The effective groove vidth Is no-.-J, according to eq. (6.03),

seff = 0.125 (1 + 0.04?/6) = 0.126 r

1 SMenall Factors ;

^ For determination of the factors F and Fp, we have:

* hG/s = 0.0523/0.126 = 0.415

T and: yi/hr, = 0.05/0.0523 = 0.956 JL

With the definitions in eqs. (9.03) and the use of Figs. 6

and ?, vc obtain:

FD = 0.415 « 0.516 = 0.214

Fp = 0.415 * 0.624 = 0.259

Film-Thickness Constants:

The follov-'inp; different averages of film thickness are

I required,

H" = 0.5 (1 + 0.044) = 0.522

?= 0.5 + 0.5*(0.044)3 ^ 0.500

H2= 0.5 + 0.5/(0.044)2 = 260

tf*= 0.5 + 0.5/(0.044)3 = 5R65

evaluation of the Sk:

The four G-constants can now be evaluated. Thus:

27

s1 = 1 - I/5B65 = 1.000

S2 = 1 - 260/5865 = 0.955

S = .522 - 260/5865 - .5*.21^ = 0.370

F/f = 1/5^65 + .5#(.036^)2 - .5*.259*(.0364)2

= 6.62*10"^

Final Funp Formula;

Direct sul—Mt-Li! l^n *• - t:» .q. (9.11) nov: ^iver the final

formula relating operating pressure difference and quantity

of liquid belnp- pumped. (^P is read from Fig. 10.) Thus:

Ap* + *251 = fi - t?*t2^2*t97^in]r 370 4 ^1 - , 9* I ^ 1 I 1 J \6.62»10"^ 1 6.62no^ *l/

Or: ^P* = 488 - 1320 Q* (10.01)

It will be observed that the numerical constants In the above

formula depend on the geometry only. The speed and the fluid

viscosity appear solely In the dlraenslonless variables /^P* and

Q*# and are assignable over wide ranges.

Comparison Kith Experiment:

In an experiment conducted with the pump of Fig. 11, the

following measurements were made:

Shaft Speed: 3000 rpm

Shut-off Pressure: 42.5 psl

Fluid Viscosity: 4.57»10^ lb-sec/ln2

The viscosity value was inferred from a viscosity-temperature

curve for the oil with measured temperature used as argument.

A prediction of the shut-off pressure is simply made with

the use of eq. (10.01) and tho definition of ^P* in eq. (9.12).

28

T

\ 4-

T I

4i

The velocity at the outer radlu? of the groove (with the shaft

treated ar rtationary) lr:

Uj = (3000/^0)^3.142*2*1.112 = 3^p' In/sec

Then, plnco:

Af> rr 6/tLtj ia*f £p* (10.02)

the prediction of rhut-off prerrure Is:

& -. fi^ijnöV^j)^. 488 , 56J3 psl

In a recond experiment with the same pump, the data were

as follov^s:

Thaft Speed: 3000 rpm

Flow: 0.525 ln3/sec

Pressure Difference: 31.5 psl

Fluid Viscosity: 4,5? lb-sec/ln2

With the piven values of orcssure difference and viscosity:

ZiP* = 2^8 and Q* = (488-248)/1320 = 0.182

Reversion of the definition of Q* in eq. (9.12) gives:

(JU gelt** ct~ LUcA- ct (10-03)

Therefore:

Q = (3^8/2)*.0523*(.250/1)*.182 = 0.415 ln3/sec

= (for two Dumping sections) 0,830 in^/sec

Compared with the measured values of 42.5 P^l and

O.525 ln-Vsec for the shut-off pressure and flow, the predic-

tions of 5^.3 psl and O.830 ln3/sec are optimistic. However,

29

' •~--:m*:l«M

If only cro.«s-flight leakage had been allowed for, with no

consideration of channel curvature, sidewall effects, and inlet

and exit correction.«?, the values would have been 62.8 psi and

1.12 in-vsec. Clearly, consideration of these additional ef-

fects greatly improves accord between theory and experiment.

An examination of possible sources of the residual dis-

crepancy leads to the conclusion that the viscosity values

employed in the calculations are too high. Such values would

aripe through measurement of temperatures lower than actually

existed in the oil films — an eventuality which Prof, Fuller

suggests as highly likely in the foregoing experiments. In

fact, it is notevjorthy that if the viscosity valuer used in

the theoretical predictions are arbitrarily lowered by 25!-,

the predicted shut-off pressure becomes the same a? measured,

and the predicted flow is a mere 0.025 ln3/sec higher than

measured.

ACKWwLrrC-MbJMT

The author is happy to acknowlege the benefit of several

discussions of Prof. D. D. Fuller, who also made available

unpublished data.

As previously noted, the author is also much indebted to

Dr. T. Y. Chu for the finite-element computer runs which pro-

vided data for Ap when H0/HT is different from zero, corr ^ J-

This work was performed under Contract Nonr-23^2, jointly

sponsored by the Department of Defense, The Atomic Energy Com-

mission and The National Aeronautics and Space Administration,

and monitored by Mr, S. Dcroff of the Fluid Dynamics Branch of

the Office of Naval Research.

30

References:

1. Booy, M. L.t "Influence of Channel Curvature on Flow,

Pressure Distribution, and Power Requirements of Screw

Pumps and Melt Extruders," Trans. A. S. M. E., Jnl. of

Eigrg. for Industry, Vol. 86, Series B, pp. 22-30,

(196^).

2. Booy, M. L., "Influence of Oblique Channel Ends on

Screw-Pump Performance," Trans. A. S. M. E,, Jnl. of

Basic Engrg., Vol. 83, Series D, pp. 121-131, (1966).

3. Muljdermann, E. A., "Spiral-Groove Bearings," Thesis,

Technological Univ., Delft, The Netherlands, (1964).

Published as Philips Research Report Supplement No, 2.

Available from Philips Technical Library, Springer-

Verlag, N, Y., (1966), 199 pages. See also: "Analysis

and Design of Spiral-Groove Bearings," Trans, A. S, M.

E., Jnl. of Lubrication Technology, Vol. 89, Series P,

pp. 291-306, (1967).

4. Carley, J. F, and Strub, R. A,, "Basic Concepts of Ex-

trusion," Indus, and Engrg. Chemistry, Vol. 45, No, 1,

pp. 970-3. (1953).

5. Carley, J. F., Mallouk. R. S. and McKelvey, J. M,,

"Simplified Flow Theory for Screw Extruders," Indus,

and Engrg. Chemistry, Vol. 45, No. 1, pp. 974-8, (1953).

6. Plgott, W. T., "Pressures Developed by Viscous Materials

In the Screw Extrusion Machine," Trans. A. S. M. E.f

Vol. 73. PP. 947-955. (1951).

31

/

7. "Handbook of Mathematical Functions," National Bureau

of Standards. Applied Mathematics Series 55, June 1964.

For sale by the Supt. of Documents, U. S, Gov't. Print-

inp; Office, Wash.. D. C., 20^02.

8. Reddi, M., "Finite Element Solution of the Incompressible

Lubrication Problem," to be published in Journal of Lub-

rication Technology. Trans. A. S. M, £., (1969).

9. Whipple, R. T. P.. "Herringbone-Pattern Thrust Bearing."

Atomic Energy Research Establishment. Harwell. Berkshire,

England. T/M 29. (1951).

10. Vohr, J. H.. and Pan. C. H. T.. "On the Spiral-Grooved.

Self-Acting, Ga.^ Bearing." Mechanical Technology, Inc.,

Latham. N. Y, MTI Technical Report MTI 6352, prepared

under 0. N. R. Contract Nonr-3730(00), Task NR 061-131.

(1963).

11. "Smithsonian Mathematical Formulae and Tables of Elliptic

Functions," by E, F, Adams and R. L. Hipplpley, publica-

tion 2672. Wash., D. C., 1939. Published by the Smithson-

ian Institution.

32

1 I I I I I I I

APPENDIX A

SIDEWALL EFFECTS

It 1«^ convenient to rolve eq. (5.01) In terms of dimen-

sion loss variables:

Then: k' ?-*;

£ (A.01)

A particular solution 1P provided by heynoldp* equation. Thuss

(A,02) ^= ^t-jikd-^ A homogeneous solution for "^2 must now be found which vanishes

on ^ = 0 and 1. This solution is formed from the series:

(A.03)

where:

^rW= C*1{**(T~3;)}

c*ijl) (A.O^)

On T- = 0 and 1:

Then:

2^^^)^ Vay-A^ cf-j) (A.05)

a» h-tl

•*s

-A2 (A.06)

oinf? Vrj must be subtracted from vp to give the true v«, It

33

•■- »■-'*"'.'w»i<-*,a^ij};j

represents the velocity deficiency occasioned by the sideKallc,

The flow deficiency per unit distance along one sidewall

is given by: JL

- £ % iS1^-'^^ M'%t> ,Ä•07,

The total flovr deficiency for one sidewall is:

O O k*lt3,S-

2^ 2/* 7t^^— —g *** (A.08)

**f,*,s

When this last result is multiplied by two, and subtracted

from the usual Reynolds flow, eqs. (5.02) and (5.03) are the

consequences.

As stated in the main text, when Iv/r is less than 3A.

sidewall effects are essentially confined to the channel ends.

For this case, all the hyperbolic tangents in the foregoinp;

series can be taken as unity. Accordingly, let us make this

simplification and compute next the partial flow deficiency:

PA HYA^T- WA'-gttAV 'i■09,

Then:

t(i>~£2%;{''c~("-pj ,A■10, A'/

34

I I I

I I

I

and :

I

I Ar*/ ^C* v. j

Tho rerler for ^2 converter exceedingly rapidly, and can

be urcd for computation directly, with the results shown In

Ki«?;. 7. The rerler for f-. IF best converted, ar follows.

From ref. 11, n, 139, >Tp have:

(**) (A. 12)

•p .'ettinp r - -1, and performing two integrations, one finds: i xx

Again, from ref. 11, p. 123:

U(c<**) = -T Z 6?" /j g xtlC (A. 14)

where the B. are the Bernoulli numbers, a fev: of which are

\ ^iven bcloT/-.

B] = 1/6; B2 = 1/30; E3 = 1A2, B4 = 1/30; B5 = 5/66

X The integrations of eq. (A.13) are readily performed with the

IneK rerie^, and the result is:

Finally:

i (A.16)

»■

This function is rraphed in Fig, 6. i

35

APFüNDIX B

CHANNEL CURVATURE EFFECT:

(1) ßooy glvoF the following formula for the flow, Q, In

term? of the overall pressure difference, P0, the radius

ratio, o(= R^/H , and other variabler. Thus:

a - Tsf A/ Ms) - 7f * r^.^;JL (B. 01)

Here:

!<(*)=: 1 + Ziit« O-uV (B.05)

In succeeding- manipulations, use will be made of the

fact that:

n =■ ^-/^ ; IRilu.if - ^ Ä^^ for any R (B 06)

Booy's notation used in this Appendix. A factor of 4 has been

introduced into the denominator of the second term of (B.Ol)

to correct for misprint In (1),

36

Further, y? ir choren as dlrtance parallel to the flights at

radius, R, and V2 ar the correspondinr component of barrel

velocity,

Algebraic rearrangementr and rubrtltutions give succes-

rively;

(B.o?)

0~*)iF(*)~t6C*j&i}

0-<){F(*)~2W)t**%tJ

(B.08)

(B.09)

liquation (li.OO) 1- valid for any "IV Let us choose

H to effect, as much ar possible, correspondence with standard ^m

flat-plate formulas. Thus, take:

Or:

(ß.10)

(B.ll)

Because, ir. most practical circumstances, V" can never

deviate much from unity, it is helpful to write: ^« f~U

and to expand the various functions of MoC ' in terms of M6 ",

Thu.~:

Kt^-^O-Z), W-ftCH*)) KC*)~ tO+f) (B.12)

37

J~ „. /-. .1 (B,13) ^ 3

yormuldis (B.12) are very ^atlFfaci ory approximations for

the practical range of V". For example, even when Hj/Hg = 1/2

(««. = 1/2), tf, exact and approximate valuer of these functions

differ little. (Tee the table belov;. )

bxact Approx.

F{.5) -.126 -.125

C(.5) 0.108 0.104

K(.5) 0.53B 0.5^3

Ure of eq. (B.ll) in eq. (B.09) rives:

When the t-approximations are uped, the following simple

equivalent flat-plate formula rerult'-:

^ . fvtk JL dt 2 a= ^^/^f- j-T- '?**/J(j+€) (B.i5)

38

APPENDIX C

DINGLL-GROOVE EDGE CORRECTION

The mapping function

W

converts the real axis of the a-plane into the form shown

b^low in the w-piüne

D »

f

0 0 1 x

;?-b/one

^ x+ iy

'TT' When & = -7- we take t **> f V

Along y = o

t = A^J**~'(i-*~) "dxj R = I T^-'V^

Now in the (u> v) plane

39

I

%~ [f^oU = f('>- i*fäi*

I

o =1 -M^

0

[Note: %' t^

- *- $ f^tä;'$'**"0-*)***

Next let *=• j' . f =.r*~

then:

Write tan'^t) in the following integral form. Thus:

t^'W - i M^

J!=-o l*o

40

Resolution into pörtiai fractions gives:

His

Use is made of the infinite integral

Oo

/

*-'

and the refLection formula for the Gamm^ function;

^)^-n)=Ä Then:

%*>- "IF4 J/,-5*) *

The above definite integral converges, but not with

m the terras 1, 5 treated separately. Accordingly, the behav-

ior of an integral with the denominator "shaded" to id?

0- slJ T will first be examined.

rm 41

■■' -■•■ .-

Let ^ sr h+J . , and examine Mi" as e - 0. 2^

I- -L ePCe) *■ P&iWlp^l

pqjpfr* "■£')-rf^We+i) e

i rft)rC^')

TOrir^-rc^p^)

Hence:

r-= !- T r'0-1) _ r'ft) rfi-A) ret)

i - V^1-*)- ^l 'More conveniently for computation:

!„= i- 1-^(^-1) .^i)-^]

A2

I

J

UNCLASSIFIED S»cmity Claiiificaüon

DOCUMENT COKTROL DATA - R&D (Smeurtiy elMtlMrtMon ol tltf. bo4y ol »totmtt mnä Inäimtn* mmtmUmn mutt bm tnfnä mhm m§ OTWjfl fßfl I» *I—Ul04)

I OmGINATiNC ACTIVITY rCeiyoM«« MiUMrJ

The Franklin Institute Research Laboratories Philadelphia, Penna. H. G, Elrod

I«. nil»ONT »ICUMITV CkAMiriCATlCM

lINn.A.S.STFTF.T) t* 9*0U*

mm 1 RIECHT TIT^t

An Analysis of the Side-Leakage Effect in High-Speed Gas Lubricated Slider and Partial ARC Bearings

4 OCSCRirnvi NOril (Typt ot fßort mnd lAthtttv 4tf)

S *\irHOn(S) (Lmtt mm; Hnt namt. Mllmt) Interim Report I-C2049-32, March 1969

Elrod, H. G.

ff HI PORT OATf

March ±J69 • a. CONTRACT on «HANT NO.

Nonr-2342 6. »AOJiCt NO.

Task NR 062-316

56 7»- NO. OP NIFS

11 If »»OAT NUM.ENfj;

I-A2049-32

f*. 9TMIN mmßonr HO(S) (Any othtrnumbtn *•!may *• •••Iff*

10 AVAILAIILITY/LIMITATION NOTICIt

Reproduction in whole or in part is permitted for any purpose of the U.S. Government.

11. tURPLIMIMTANY NOTll It. tPONSORINO MtUTARY ACTIVITY

Department of Defense Atomic Energy Commission National Aeronautics & Space Administrctio

IS ABSTRACT

Recently-performed analysis for herringbone thrust bearings has been incorporated into the theory of the viscous screw pump for Newtonian fluids. In addition, certain earlier corrections for sidewall and channel curvature effects have been simplified. The result is a single, refined formula for the prediction of the pressure-flow relation for these pumps.

To provide a self-contained presentation, some of the results of FIRL Interim Report I-A2049-32 are also given here.

DD^-1473 ecur^ 'Uitltic Sec cation

jaamum m Stcurity Clfaificition

14- KtY «09IOS C < A

POUK

LIMK •

NOLI »T UNICC

ROCI

Theory, Analysis Viscous Pump Pump Screw Pump Herring Bone Thrust Bearing Thrust Bearing Hydrodynamlc Bearing Newtonian Fluid Lubricant Sldewell Corrections Channel Curvature Corrections Pressure Distribution Flow Rates Edge Correction Pump Performance

INSTRUCTIONS

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(4) "U. S. military agenciea may obtain copies of this report directly from DDC Other qualified uaera ahall requeat through

(5) "All distribution of thia report ia controlled Qual- ified DDC uaera ahall request through

If the report hss been furnished to the Office of Technical Services, Department of Commerce, for aale to the public, indi- cate this fact and enter the price, if known.

1U SUPPLEMENTARY NOTES: tory notes«

11 SPONSORING MILITARY ACTIVITY: Enter the name of th« departmental project office or laboratory sponsoring (pay ing tor) the reaearch and development. Include address.

13. AdSTRACT: Enter sn sbstract giving • brief and factual aumma-y of the document indicative of the report, even though it may alao appear elaewhere in the body of the technical re* port. If additional apace ia required, a continuation sheet «hell be attached.

It is highly desirable that the abatrcct of classified reports be unclaaaified. Each paragraph of the abstract ahall end with an indication of the military aacurity claaaification of the in- formation in the paragraph, r«pr«s«Rted as rrs,), (S). (C). or (V)

How-

Use for additional explana

There ia no limitation on the length of the abstrsct. dvar, the suggested length is from 150 to 225 words.

14. KEY WORDS: Key words are technically meaningful terms or short phraaea that characterise a report and may be used as index entriea for cataloging the report. Key words must be aelected ao that no security classification is required. Identi- 'fiers, such aa equipment model designation, trad« name, military project code name, geographic location, may be used ss key words but will be followed by sn Indication of technical con- test- The assignment of links, rttles, and weighta is optional.

UNCLASSIFIED Security Clssaification