UNCLASSIFIED

AD 27 5626

ARMED SERVICES TECHNICAL INFORMATION AOTCYARUNGTON HALL STATIONARLUNGIN 12, VIRGINIA

UNCLASSIFIED

BestAvai~lable

Copy

NDTICE: When goverment or other drawings, speci-fications or other data are used for any purposeother than in connection vith a definitely rentedgoverment procurement operation, the U. S.Government thereby incurs no responsibility, nor anyobligation whatsoever; and the fact that th, Govern-ment Wey have formilated, furnished, or I , way

ed the said drawings, specificatiov otherdata is not to be regarded by implication jr other-vise a.s in any manner licensing the holder or anyothsr person or corporation, or conveying any rightsor per,.ission to -nnufacture, use or sell anypatented invention that may in any way be relatedthereto.

ARL 62-318 2 5-

LAMINAR FLOW BETWEEN TWO PARALLELtI ROTATING DISKS

MARK G. BREITERKAR. POHLHAUSEN

APPLIED MATHEMATICS RESEARCH BRANCH

AERONAUTICAL RESEARCH LABORATORY

OFFICE OF AEROSPACE RESEARCHUNITED STATES AIR FORCE

rrI

ii

FOR ERRATA

A275562AD

THE FOLLOWING PAGES ARE CHANGES

TO BASIC DOCUMENT

ARL 62- 318 March 1962

ERRATA - July 1962

Correct the last three lines on page 11 to read:

1 1

r i wr

AERONAUTICAL RESEARCH LABORATORIESOFFICE OF AEROSPACE RESEARCH

UNITED STATES P IR FORCEWRIGHT-PATTERSON AIR FORCE BASE, OHIO

ADI 275562

END CHANGE PAGES

NOTICES

N'e'-to G )-Jerlment drawings, specifications, or othecr tin .irc u-sed for any purpose otherthan in euic1ion with a definitely related Government procurement operation, the Unite Statesit-iirtiment thereby incure no responsibility nor any obligation whatsoever; and the fact thatthk 1L.11;.rnn ent may have formulated, furnished, or in any way supplied the said drawingg,spe.lfiicatiow, cr other data, is not to be regarded by imprlication or otherwise as in any mannerlicernsing th( h~ider or any other person or corporation, or conveying an~y rights or permissionto nmanufactirE, use, or sell any patented invention that may in any way be related thereto.

Qualifil'-d 7rquesters may obtain copies of this r o p o r t from the \,rmed Services TechnicalTnformatiClL Agency, (ASTIA), Arlington Hall Station, Arlingoui 12, Virginia.

Thi :report has been released to the Office of Technical Services, U. S. Department of Cm-nierce, Waishington 25, D. C. for sale to the general public.

Copies of AR. 'Technical Reports and Techn!ca! Notes should not be returned to Aero-autlcalR-!x.arrc, Laboratory unless return is required by security con siderations, contractual oblgations,-,r notic s on a specific document.

ARL 62-318

LAMINAR FLOW BETWEEN TWO PARALLELROTATING DISKS

MA.9K C. BREITERKARL POHLHAUSEN

APPLIED MATHEMATICS RESEARCH BRANCH

"MARCH 1962

Project No. 7071Task 70437

AERONAUTICAL RESEARCH LABORATOBYOFFICE OF AEIROSPACE RESEARCH

UNITED STATES AIR FORCEWVRIGHT-PATUERSON AIR FORCE BASE', OHIO

300 - May 1962 - 31.366 9 1367

FURE WORD

This report was prepared in the Applied Mathematics Research 13)

Aeronautical Research Laboratory, Office of Aeroispace ResearC-., Wr

Patterson Air Force Base, Ohio~.

The authors extend grateful acknowledgement to Dr. K. G. Guderlt

the Applied Mathematics Research Branch, for encouragement and mu-

advice.

The numerical calculations were carried out partly with the Recon

graciously made available by Professor R. T. Harling of the Air Forc,

of Technoulogy, and partly with the 1103A computer of the Digital C-Friip

Branch, Systems Dynamic Analysis Division. Captain R. V. Hendon of

Computation Branch rendered valuable assistance in programming.

The work was carried out un. ?roject 7071, "Methods of Mathev

Physics", Task 70437.

BESTAVAILABLE COPY

FOREWORD

i

This repo t was prepared in the Applied Mathematics Research Branch of the

Aeronautical Research Laboratory, Offlce of Aerospace Researc'., Wright-

Patterson Air Force Base, Ohio.

The authors extend grateful acknowledgement to Dr. K. G. Guderley, Chief of

the Applied Mathematics Research Branch, for encouragement and much helpful

advice.

The numerical calculdtions were carried out partly with the Recomp computer,

graciously made available by Professor R. T. Harling of the Air Force Institute

of Technology, and partly with the 1103A computer of the Digital Computation

Branch, Systems Dynamic Analysis Division. Captain R. V. ilendon of the Digital

Computation Branch rendered valuable assistance in programming.

The work % L , .a, ried out under Project 7071, "Met. of Mathematical

Physics", Ta k 7043

ii

ABSTRACT

The report inrestigates the viscous flow between two parallel disks rotating in

the same direction with the same velocity. The fluid enters the space between the

two disks at a certain radius In the radial direction. fcause of the shear forces,

it assumes a rotating motion with about the velocity of the disks. The centrifugal

forces then build up a pressure increase in the radial direction. The arrangement

corresponds to a centrifugal fluid pump, which may be advantageous if cavitation

is ai problem.

The general equations of viscous flow are simplified by the assumption that

the pressure difference normal to t..le disks is negligible (boundary layer assump-

tions). One obtains a system of parabolic partial differential eq,,ations. For large

radii the deviation from rigid body rotation (with the angular velocity of the disks)

is smail. The lsi, _:- i equations which then result are s. , analytically. The

velocity profile. dePenc! ipon a parameter containing the kinematic viscosity, the

angular velocity and the distance of the disks, but not the radius.

The non-linearized parabolic differential equations are approximm.Led by a

difference scheme and solved numerically. The results are given in non-dimen-

sional form with the entrance velocity and the distance of the disks as parameters.

Furthermore, the efficiency of the pump is computed from the gain of the total

pressure and the torque at the shaft of the rotating disks,

iii

TABLE OF CONTENTS

Section Page

1. Introduction I

11. Statement of the Problem 3

111. Linearized Treatment and A symnptotic Solution 5

IV. Numerical Solution of the System of Equations

(The Inlet Problem) 10

V. Results and Conclusions 14

VI. Appendix 17

iv

LIST OF ILLUSTRATIONS

Page

Figure

1. Schematic diagram of the flow between two rotating disks 21

2. k I and k2 versus ad 22

3. g(ad) versus ad 23

4. Radial velocity u and relative tangential velocity V for

various values of ad 24

5. Radial velocity 5 and relative tangential velocity V for

ii0= 1.0 d=0.5 25

6. Radial velocity 5 and relative rangential velocity Vfor

0 =o. 5 d=0.5 26

7. Radial ve jct,.. a ard relative tangential velocity N or

d 0 = 0 , .5 C ..5 27

8. Radial velocity d anid relative tangential velocity V for

di0= 0. 1 a = 0.5 28

9. Radial velocity il and relative tangential velocity V for

ao = 1.0 a=l.O 29

10. Radial velocity ai and relative tangential velocity V for

i0 = 0. 5 d=l. 0 30

11. Radial velocity i and relative tangential velocity V for

5 0' 0. 25 = l.0 31

12. Radial velocity 5 and relative tangential v,-locity V for

0o= 0. 1 =l.0 32

v

LIST OF ILLUSTRATIONS (Continued)

PageFigure

13. Radial velocity i and relative tangnetial veloc*.Ly V for

a 0= 1.0 d- 1.5 33

14. Radial velocity i and relative tangential velocity V for

f0 =0.5 d=l.5 34

'A 5. Radial velocity d and relative tangnetial velocity V for

10- 0.25 d =1.5 35

16. Radial velocity 5i and relative tangential velocity V for

a 0=0.1 j= 1.5 36

17. Radial velocity 5 and relative tangnetial velocity V for

r,0 = 0= n j - i7

18. Radin. velocity i and relative tangential velocity V for

u0 =0.5 d=2.a 38

19. Radial velocity 5 and relative tangnetial velocity V for

i0=0.5 d=2.0 39

20. Radial velocity 5 and relative tangential velocity V for

5 o= 0. d 2=2.0 40

21. Radial velocity 5i versus r/r. for d=0. 5 -&1ni 1

22. Radial velocitv 5 versus r/r. for d=l.0 41m 1

23 Radial velocity 5i versus r/r. for d=.5 42

24. Radial velocity 5G versus r/r. for d= 2 42

25. Fi() for various values of 6 0 versus r/. for d=0.5 430v1

vi

LIST OF ILLUSTRATIONS (Continued)

Figures Page

26. T"(Q) for various values of 50 versus r/r. for d= 1.0 44

27. F(0 for various values of -0 versus for d:.5 44

28. P(r) for various values of a0 versus r/ri for d=2.0 45

29. Moment, pressure, and efficiency versus r/r i for

d =0.5 46

10. Moment, pressure, and efficiency versus r/ri for

d= 1.0 47

31. Moment, pressure, and efficiency versus r/ri for

S=1.5 48

32. Moment, pressure, and efficiency versus r/r i fo-

d=2. 0 49

I. INTRODUCTION

Investigated in this report is the viscous flow between two parallel disks,

which rotate with the same constant angular velocity in the same direction. The

fluid enters the space between the disks in radial direction through a cylindrical

surface (Figure 1). It is set into a spiral motion whose tangential and radial

velocity components are due to the friction forces and to the centrifugal forces

respectively. The pressure will increase with the radius because of the centrif-

ugsl forces, thus the arrangement can serve as a centrifugal pump. One might

suspect that the absence of blades leads to low values of the efficiency, but if the

blades are spaced very closely and the mass flow is low, the efficiency may be

quite acceptable. On the other hand, such a pump would not encounter the problem

of cavitation and also its characteristics over a wide operatin ° -ange might be

more favorable than in i conventional turbo pump. These advd,,tages were recog-

nized by Mr. S.: . Hasinrer, Mr. L. G. Kehrt and Dr. J. P. von Ohain of the

Thermomechanics Research Branch of this laboratory. Technical details and

experimental results will be published by them in a future report.

The present mathematical analysis will determine the velocity distribution

between the disks, the pressures, the torque applied at the shaft and the resulting

efficiency.

There exists an extensive literature on the single disk rotating in an infinite

medium*. Assuming that the disk extends from radius zero to infinity, it is

*Advances in Applied Matheratics, Academic Press, Inc.. New York, Vo) IV,

1956, p. 166. F. K. Moore: Three-Dimensional Boundary Layer Theory.

1

possible to reduce the partial differential equations governing this problem to a

sy3tem of ordin:'y differential equations by means of a similarity bypothesis. But

an approach of this kind is not feasible in the present case because of the presence

of the seccid disk and the fact that the flow enters the space between the disks at a

radius different from zero. However, another simplification can be made here.

For any practical application the distance between the disks must be very small,

for otherwise the friction would not be sufficiently effective to produce a tangential

f1jv,. But then the approximations made in boundary layer theory apply reducing

the original system of Navier-Stokes equations, which is elliotic, to a system of

parabolic differential equations. They are derived in Section II.

A first insight into the basic characteristics of the proble n is obtained if one

assumes that the velocities of the fuid relative to the disks are small so that

quadratic ternr. , -.- gligible. This is a condition whici 'ell satisfied for

large radii; F a idea ii carried out in section Ill. It gives valuable insight into

the general character o" the velocity profiles; these profiles are governed by one

dimensionless pararmeter which contains the angular velocity, the vi-cosity and

the distance, but not the mass flow and the radius. Naturally this parameter is

important for the practical design. The solutions thus obtained can be considered

as an asymptotic expression for the solution of the actual non-linear system.

In the • icinity of the entrance radius the linearization is not justified, in par-

ticular the tangential velocity relative to the disks is certainly not small. This

region is most critical from the point of view of cavitation, and also the principal

losses are encountered there. Therefore, the system of differential equations has

been integrated numerically. The methods and the results are shown in section

IV and V and in an appendix.

2

fl. STATEMENT OF THE PROBLEM

We use a systrn of cylindrical coordinates r and z (Figure 1), where the

z axis coincides with the axis of symmetry and r is the distance from it. We

denote by

ai the component of the velocity in the radial direction

v the component of the velocity in the tangential direction

measured in a system of coordinates fixed in space

w the component of the velocity in the axial direction

2d the distance between the two disks

ri the radius of the entrance hole

p the density

i the viscosity, v = the kinematic viscos'"

p th. pru sure

w he commant anguldr veloci 'y.

An incompressible axi-symmetric steady viscous flow is described by the Navier-

Stokes equations

Ov +wav + u. v 4v- a + 8Vur wT: r 2z z -Or r 2 28z 8r

w 8w .! + V __ + . --. + - (2.1)

ur 8z p Oz 8z r Or 8r2

+ &u A.. u 8 uu au v z I Ip + V 8 L !

+r a-: r p Or 8z 2 Or r r

and the equation of continuity,

Ou + a__2w =0 (2.2)

Or r 8z3

By applying the boundary layer approximations, this system of equations is reduced

toOU 2 au 2 2u

_U M- V+ w 2-=.-iu -r O z p dr Oz2

(2. 3)

u V u. .v + w w K Xv

Pr r z 2--

uu 8w+r 2 + L X 0 (2.4)Or r' r 8z3

It is convenient to introduce instead of v, which is the tangential velocity in a

fixed system of coordinates, the tangential velocity V relative to the disks

V1, v - rw (2.5)

One then obtains

u2 2 u 1 dp 2uOu V Vrr w * =- 3

U r iz Pdr O(2.6)

8V u[V+,rw] + V av (2.7)r z = az 2

Along the disks, given by plane z - d, one has the boundary cnditions

u (r, *d) =0

(2.8)

V(r, *d)= 0

4

To determine the boundary value problem com.pletely, one must also prescribe

the velocity disti ioution in the entrance cross section. One has

u(ri, z)a u0

(2.9)

V(r i ,z)-rw

IMI. LINEARIZED TREATMENT AND ASYMPTOTIC SOLUTION

Omitting quadratic terms in u, V and w one obtains from Eqs (2. 6) and (2. 7)

V u : d (3.1)0 8z2 p dr

u a z (3.2)

In these equations does not depend upon z becauL che boundary layer

p&w dr

assumption thai rhe pre sure does not depend upon z. The unknown w does not

appear, and so it is possible to solve these equations separately; w is determined

from Eq (Z. 4) afterwards. It is ecmarkAble that in these equations .!,ere appear

no parlai derivatives with respect to r. This means that, except for the expres-

sion

rw = F (r) (3. 3)p'aw dr

which depends upon r a nd is undetermined so far, the equations can be solved

for each value of r independently. The expression (3. 3) is the only inhomogeneous

term that occurs in the equations and in the boundary conditions; therefore, it will

enter the solution for u and V a a a factor which depends only on r. This factor

5

is determined by the condition that the mass flow between the disks is constant for

every cross section r =const. Thus u and V as well as - can be found.p dr

After elimination of u one obtaiv, from Eqs (3. 1) and (3. 2)

2 4VP d V + 2V z F(r) (3.4)2rZ dz 4

Setting V = one has finally

dV + 4a 4V 24F(r) (3.5)dz

4...

An the most general solution of Eq (3. 5) that satisfies the symmetry conditions at

the plane z v 0, a ac .,: 1

V=A sin!, (az) sin (oz)+A cosh(az)cos(az)+ F(r) (3.6)2 2

where A I and A2 are two constants which will be determined presently. The

boundary conditiona (2.8) together with Eq (3.2) give fcbr z =d

V=0

(3.7)

1 d2 V

2 2 dz 2

6

Hence, by inserting Eq (3. 6)

A, F(r) k A -F(r) k2 381 2 1 2 2 (38

with

k 2 asi a). sin (ad)I cosh'.r.)+cos42ad)

(3.9)

k2 2 cosh(ad).cos (ad)2 cosh (2ad)+cos(2ad)

Figure 2 shows kI and k2 as a function of ad

The shear force at the wall is proportional to dV/dz. One finds

dV F(!) ! (k I , ) cosh(az) sin(az)+(kl +k 2 )einh(az)coo(az) (3.10)dz *a2 (320

Let t)he flow of liquid (measured by volume) be given by 0. The flow through

a surface r = const extending between the disks is given by

+d 2 +dQ= 27r f udz= I df dz = dV (3.11)

-d a -d dz a -d

Hence ith Eq (3. 10)

Q . .. F(r). g (ad) (3.121a

where:

g(ad) a sinh(Zad)- sin (2ad)cosh(Zad)+coo (2ad) (3.13

Figure 3 shows g(ad) as a function of ad

irrom Eq (3. 13) we find the function F(r)

IF W 4 eQw- r.g(ad) r (3.14)

where

q -w. g(ad) (3.15)

One thus obtain, finally tor the velocity components, expressed in terms of the

mass flow

v U3 [ k , ,inh(az) sin (, z)+k. cosh(z) cos(az)- 1 (3. 16)(3. 16)

U a -[kl cosh(az)cos(aL.-k 2 sinh(a-) sin (az)]

Inserting these expressions into the continuity equation Eq (2.4) one finds w = 0.

The velocities in the middle plane between the two disks are found by setting z = 0.

Z, =0 r(3. 17)

5U klzZO = r

8

Figure 4 shows the radial velocity component for different values of ad as a func-

tion of z/d for q/r = I. For small values of ad, the profiles have a near para-

bolic shape with maximum velocity in the middle (z = 0). For ad = w/Z the profile

is flat in the rviddle. for higher valie of ad the flow is moro and more concentrated

in the neighborhood of the disks.

The tangential velocity V is represented in Figure 4 for q/r = 1. For small

values of ad it also has a parabolic shape, but with increasing ad the slope in the

vici;:ity of the disks becomes steeper and the relative velocity in the middle plane

ajroaches -1.

Actually the profiles for high values of ad must be viewed with considerable

caution. One sees that in this case the velocity u may become n 1gative in the

middle; i. e. for such cases one has a radial outflow close to the disks with a

radial inflow in t'k roddle. Obviously it would take a rathe -ial arrangement

to produce such • o, pattern physically, Theoe results ought to be disregarded

for technical aFplications. Important in any case is the role played by the dimen-

sionless parameter ad which determines the character of the profiles. This

character is neither inflaenced by the mass flow Q nor by the radius r. For a

given w and a given v, I. e. . a given fluid, the only quantity that can be influencid

in thi3 parameter is the distance of the disks. If one wants to use high angular

velocities, one must try to lower the value of d by skillful design.

The pressure distribution is obtained from the function F(r) in Eq (3. 5).

Inserting Eq (3. 14) into (3. 3).

one finds d pw [r -2q (3.18)

dr r

9

and by integration

2

p = [ r. . 2qlnr +const] (3.19)2

The momentum required at the ahaft of the rotor can be determined in two

different ways, either from the moment of the shearing forces on the inner side

of two disks, or by applying the law of conseration for the moment of momentum.

One has in the first case

r rz I dVdM =-2- z. ' f r '-dr (3.20)

ri iz zd

in the second

2 dl4 u2,rr p f u.vdz (3.21)

-d

For the numerical ealuation the second one is preferable.

IV. NUMERICAL SOLUTION OF THE SYSTEM

OF EQUATIONS (THE INLET PROBLEM)

The approximation shown in the previous section is unsatisfactory in the

vicinity of the entrance, where the velocity components are certainly not small.

Moreover, the lineariration causes all r-derivatives to vanish, i. e. it introduces

such i strong change of the character of the flow equations, that it becomes

impossible to satisfj the boundary conditions at the inlet. But the inlet is im-

portant for it accounts for most of the losses.

10

As was mentioned above, the introduction of the boundary layer assumption

gives us a parabolic system of differential equations where lines r = const are the

characteristics, therefore, it is possible to proceed in the numerical solutions

from one line r = const Lu the next one without any need to go haclk. Thus, we may

adopt a numerical scheme for the integration of the parabolic differential equations.

For our computations we assumed that the rotating disks lie in horizontal

planeR, and introduce a new system of coordinates x, y, whose origin lies in the

entrance cross section at the lower disk. Thus the x-axis coincides with the

loder disk. We have

x~rr

y = z+d

0

uO

ri r

We introduce dimensionless variables in the system of ,'Iferential equations

(2.6), (2.7), and (2.4)uo +

r. r.1 1

r. Y=Y"; r~ d1

-m W

11

One thus obtains

a " -[ + + +2 !: +

u + -+ ] = 8 (4.2)

where for abbreviation

F'(r) a r. 1 d (4. 3)142 r Z dr

The functions 5 ind 7 and v are given at the entrance (i01. The channel

extends from j = 0 to -=d. The boundary condition at y = Zd can bd replaced

by a symmetry coiidition at i = d. Thus, one has as boundary conditions

0) 0 a,! (q(,a) = 0

0)=0 aV(x,d) = 0 (4.4)

V,(; 0) =0 V(;, a) = 0

The natural method of solving such a system numerically is the introduction

of a difference scheme. Here caution is needed from the point of view of stability.

12

The coefficients of the derivatives with respect to ;., in Eq (4. 2) may become very

sma)l for large values of i and it also is small in the vicinity of y d, but then it

is necessary to use a very fine mesh size in the ; direction, if one applies a

'"iirect" difference scheme where the derivatives with respvct to are computed

along the line I = const which is already known. Otherwise the procedure would

become unstable.

For this reason it was decided to use an ";nverse" difference procedure where

the Ierivatives are formed along the line ; a const for which the state is to be

-Imputed. This brings about a rather severe complication, for the stat. along

the new line ; = const is determined by a system of simutaneous equations.

Since the original differential equations are non-linear, this system of equations

is also non-linear. At first the non-linear system was solved directly by an itera-

tion method. Sirw this method proved to be inconvenient, ?proach vras modi-

fied in the follo, ng manner: In going from one line = const to the next one we

computed the chanies of i, V and Z rather than the values themselves. If a

difference procedure is admissible at all, these changes ought to be small and

second order terms in these changes are negligible. In other words we use a

linearization which considers as a basic approximation the values at the line

= const which has Leen computed previously.

The state along the new line can then be obtained by a suitable super-position

of particular solutions of the linear equatioa which arises in this manner. The

linear system of equation which arises in this process has a rather simple struc-

ture insofar as the matrix of systems has elements that are different from zero

on only a few lines that are paraUel to the main diagonal. This brings about

certain simplifications in the inversion of the matrix. For further details of the

13

numerical analysis see the appendix.

V. RESULTS AND CONCLUSIONS

The numerical procedure described above gives a nearly complete description

of the flow pattern. Assuming constant velocity u0 at the entrance, we give the

results for a set of four initial radial velocities 1i-- 1. 0, 0. 5, 0. 25 and 0. 1. The

initial tangential velocity V 0 is always V0=-I because of Eq (2. 5). For the

dimensionless distance between the disks we choose d = 0. 5, 1. 0, 1. 5, and 2.

Figures 5 thru 20 show the dimensionless velocity profiles versus z/d with

r/r i as parameter. These p:ofiles have a common characteristic: at the wall the

inlet velocities 0 and ;00 wb ch are constant over the cross section, are immedi-

ately reduced to zero because oz the boundary conditions. The reduction of the

radial velocities at the wall causes the velocities in the middl. *.o increase. There-

fore the radial ve)c -.iE , & the profiles close to the entrance c. us section over-

shoots the entra, "e prnf;,-, in the middle. The effect decreases with increasing r

because of the increase of the available cross section with r, finally the profile

approaches the form given by the linearized theory. Figures 21 thru Z4 show this

behavior again; here the radial velocities V in the median plane of the i41.m

(z = 0) are plotted versus r /r.. The average radial velocities are obviously de-I

termined by the condition of continuous flow.

The average tangential velocity is directly connected with the moment M of

the shear forces and thus with the work that must be performed to drive the pump.

Further.more it gives the main contribution to the dynamic pressure of the fluid

particles as they leave the pump, thus it influences rtrungly the efficiency. As

mentioned before shear forces can be obtained from the slope of the velocity

14

profiles at the wall Eq (3. 18); but it appears to be preferable (from the numerical

point of view) to compute it from the radial and tangential velocities Eq (3. 19).

The pressure distribution is computed from the function F() which is found

in the numerical procedure as a function of r/r i . Here an integration in the

direction of r is required. In our figures the contribution to p due to the term

in Eq (4. 3) is shown separately, (as a straight line). The pressures are given by

the difference of the curve ;(M) from this straight line.

Properly speaking the dynamic pressure varies from streamline to streazn-

Ii-.. For technical purposes only the average velocity rark be utilized. The

dynamic pressure for the average velocity is slightly lower than the a-erage of

the dynamic pressures; in other words in defining the dynamic pr :ssure with

average velocities one takes into account the mmxing losses. Thus, we have accord-

ing to Bernoulli'p e-,tion

PLI 2 U v av

The efficiency ?I of the process in I.he pump is best defined by

Ps + P d

11= - . O

where Q = Zirri . 2du0 denotes the amount of fluid entering at r.. Figures 29 thru

31 show, for - p = L, the moment M, the total pressure p8+ Pd and the efficiency

versus for the chosen values of 40 and d.r1 0

According to the linearized approach the profiles at large values of r do not

depend upon the flow field at smaller radii, thus dynamic pressure and torque are

determined only by the exit cross section. However the stz.tic pressure arises by

15

an integration over the pressure gradient in the radial direction. The losses

encountered appear in this analysis as Icsse in static pressure.

The exit cross section does not play a special role in the computation; in

other words, each value of .r can be considered as the other radius of the rotor.

In the vicinity of the erncrance, one has always a slight pressure drop, for

large volume flow the pressure drop is more pronounced, and this fact is of

interest from the point of view of computations. High mass flows are also detri-

nr ental from the point of view of efficiency.

16

APPENDIX

For the numarical solution of the system of equations (4. 2) we set up a net-

work of lines x-constant and y constant with a spacing of h and I respectively

along the x and ; axes. The length I is chosen as uzi twentieth of the distance

from the wall to the median plane of the disks. The derivatives in Eqs (4.2) are

replaced by difference quotients, e. g.

-u - w - -

u i+l,kmUi, k rv - i+l,k+l 1 i+l, k-I

(A 1)

a2 j i+l, k+l l2 I+l, k + 5I+1, k-I

8j2 t 2

Here i and i+ I correspond to successive lines i = constav, and k - 1, k, and

k+ 1 to success4 e linei = conqtant with k- 0 at the wall. Starting with given

values of fi, V, 7 at che line x = const (i = 0) corresponding to the inlet, we

determine the unknown values of a. V, 7 on the (+ 1 )st line from the known

values on the i-th line.

It is convenient to introduce as new ,.wikowna Lho- differences Aa, k'

Arvk , defined by

Ak 'ui+ lk "ui,.k

and similarly for AVk , and &Ck' and neglect all terms in the equations of higher

order in the A's. (Naturally these A's will also depend on i, but only one value

of i is needed at a time).

17

Equations (4.2) then lead to the following linear difference equations in the

unknown differences ,ii, AV, and A and the unknown pressure function f(i).

- - 1,k ) +I 1k+l I2 Wi,k) + Ui, I i, k-1(1+ -" Wi,k)

2 - i,k -. k +a)+ (2 + 4L rl,) A-k

-( L k k-+ (ik+li, k ;wk

-

Lz k

2(l-w. )A wik+-i-+ (1+Z)

2 ik VkI zx 2 k h ik(

v itkl (+ IZVk- I ) x, k-

2~ ~ itk~ Itk1

itk x k2i,kk-i

A2t I h z-I a v + VkI &d k- I -h X Auk_ 1 ik i,k+l iLk-l

In addition to Eqs (A2) the complete specification of the problem requires

Eqs (A3), (A4), and (A5) resulting from boundary conditions.

18

Since W = = = 0 along the wall, one has

A 50 = A *0 = A 0 = 0. (A 3)

Since and V are symmetric and v antisymmetric with respect to the

median plane of the disks,

AGz - AG1 =0,A ,21 Ai19 0

AV -21 = 0, (A4)

a Iv20 0 .

In the continuity equation there appear only first derivatives with respect to

', whiie the otl . r .':' "ns contain second derivatives. 0, .11 notice that in

Eq (Al) the fir. derivtive of 'v with respect to j has -iot been formed from

values wv at adjacent points. This was done in order to obtain a central difference

a a2- ;-formula for ; - jsa for andB2fom , just as a-- As a consequence or. equation

is lost. Therefore, the following approximation is made in the vicinity of the wall.

In the last of equations (4.2) the term a(I is zero at the wall; a may be approxi-

mated by its linear term. Thus

5= f(4 ~

= f',( ). ,

(A5)8w,- f,(i) K -I ) =A

-27v =A(;) y--2

19

From the last equation one finds

-~1

wi,l 4 - 1.w,2

and so

- 1 -

J. 4 2

An alternate form i.s

I i , 2 -

A*,' i "A' + ( "1 wt, ) )

Here the second term ought to be zero, but by proceeding in this fashion we are

more sure th.L C '-: equation of (AS) is not violated bec of error accumu-

lation.

By numerically integrating the radial velocity a with respect to j for any

fixed value of ;, we obtain the total fluid volume passing through thf- cylindrical

surface characterized by ;, and comparison with input volume provides some

check of the numerical results.

20

r

-c-d 0 +d

r.r

Fig 1. Schematic diagram of the flow between two rotating disks

21

1.0-'- -

-I I i f I

kk

0.8i ~ I' -H

K,~~~~4 -:u1-2 1~

1 41

+

0 if

0. T

-ti+g4.4+ k, an versus ad

1,4

I.0 --- --

10

I 5

0.223

AVAILABLE COPY

41, -,-1

-T- I-..,,-T,

FT 4It 1 L4 ------ H

140

Z-t-

- -

444

414

C; -444-

Il lo I

or

I I

doT fl

v 2.7 1

c;"

---- ---ME

IU M I I I

a~ I-

44

I I f

44J

4-3

-I-!0H t1

-4 - 4 N 0T N

goC;

0; 0; >--.. 83

U.'1T1v 1-- 0-1

t I

A I

AR*

4-H

A

I ~~ I 1 I I:mIT-T I -I - -T

V. .. .. .

414+

I *-7T c4

Afl

ca N

-tC 4-Z C

--- --- ILa t- L A I

+4-

,~in I

I i lu

9 1 1 1Ao

0I I

[i - 4 I

-I J-Iof I

I IA

I f I

oPol

I I

N0 go

0 S S S I

NI I

N .1

0 a4

41,,

a I

4?P1

Lil I II I L0

d d

* 41

d d

0*1

xt I

N 4' '44~ ~ N 0_ _ a

o o -U

26I ~ 01

0 f I I

. 1 1 f i l I I

I I Lu 444-44 1

* M ffisdI

I I I

IllI

Ill In I4'

0r 0 4

it I

IfI

44

I I

Ifni

at I

IF .. .. ..L0 -

H R4

0 d A

26

0.',

0. 1

r1. 109 .0.0 . 0.6 05 -. 4 .. .2 ..

0.241

27.

0.4~~~~ a/dM. II

o0. 4 ~ -

T/ M0

IC___4

F1. - g9.08 -0.ia veo -0. .0. and4 reai.tneta euiy Vf 0. I 0. -01

'&/d

I I I [ f i l

0TT fil I I CI

4+1i0-i

N I

-1 1 f f l

.4:pl

-IN11 IfJill .. .

L I

LL I -f 1

j449

Vti

p41

T- 141 - 1 1 Il I

1-u I I 1

I V

LX i

I 0L -I

I I II

~O

1-0 0 -_ _ f

t4 TR

IIIIi

I I f I

If I' I Iit 0 0

A IA

I Yv

C; .*-

I I I

00

J I A

0 L SI I I

It.9

0.8 I IsIIh

0. i4fw

0.

~iiif

-1.0a .0.9al -0.d .0.7iv .0.6nia veo0t.V5o -0. d0. -0.0

0.40

0. 6 . ....... 1 17 1 .. -1 __

1010

04

.1.0 -0.9 .0.5 .0.'7 -0.6 -0.5 -0.4 -0.3 -044 .0.1

ai

-0. L

~1.01

Fig. 11 Radial velocity Z and relative tangential velocity Vfor 0. 25 ;,ilit

31

0. T

.10s .0. 0 . .0..6 .. .. 03 .0Z ..

.1.0 .0.9 .0.8 .0.7 .0.6 -.. -0.4 -0. 3 -0. .0.1 0

-0 1 4_ _ __ _ 0_

Fig. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ -i ! 1i 44t.vlct adrltv a~e~a eoi' Vfr ~ 1d .

33-

I t -

ki- 4-1-

0 It I I I

H44 -,1T

T§-T

t I

IT I

TRIT

- T I IN LI I

T7 T

Jo II

+H0

c;I

* aT1

4.4

VIIa

A' t

UIA ITT

1 4.

14 7

ILI

01 _4 33

0.4 7.'i2 i -

. 1 .. . q o0 . o. . 0 . 0 3. ..0.0

r/r. . 1..

080

'344

I IaI. - ."fIT

0.',

.1.0 .0.9 -0.8 .0. ? -0.6 .0.s .0.4 .0.3 .0.~ . 0.1 0

.- a/d

.1.0 .0.9 .0.9 .0.7 ? .0.6 .0.5 .0.4 .0.3 -. 4 .0.1 0

.04-

.0. .

T0 I

Fag 1 Rdil elciy i ndreatvetanenia vloit Vfo .. S *t.

+1 11 f

0. 4

.1.0 .0.9 .0.1 .0.. . .0. 5 .0. .0. .0...0

01

00

.1.0 -0.9 -0. .0. .06 O. -0. .0 3 .0 a

4 N-

* -

IV

1 U.

414,

4--

1:4- 4q

-- 44>

37r

-0 0711

I I tt4

41

4-44

4 T

I~fl I 4j2

4 ~~4 IT U44

CI 0

37

0.6 .0

00

33:. 1.0

t.4.

-l.0

17.1 ~da eoiy~adrltv tneta eoiyVfri *5 d ,

00

-1.0 -0.9 0.8 .0.7 .0.6 -0. .0. .0.3 .. 2 -.38

0.4 6 i

0.

15.0 .0.9 .0.8 .0.1 .0.6 .0. ~ .0.4 .0.3 .0.3 *0. 1 0

,qp- 9/d

-3.0 .0.9 .0.8 .0.? .0.4 .0,9 -0.4 .0.3 .0. .. 0.1 0

0

Fig. 19 Radial velocityT and ritlative tangential velocity V for 700.25 I.2.0

)90

.1.0 .0.9 .0.8 .0. 7 -0.6 .0.5 .0.4 .0.3 .0.2 .0.1 0

"1 4~tF~ 11 'UHHiMi

Pa,~0 ad..I ~d,',y ani .'at~.. v~o~,y fo 0.I a* . ol

4. .4 W 111 1 Z.;

-0.2 61144

44 11-

r t I

I I f [if .'

-4- -4- --- 1 4- - 4 - C -

4t-

-4- -,- , 4 -

*~~~~~ Ar~ '-i I2T~~

-- 4-4i

tK :> 1 F 3 & _I -H- -pr .. l.. -.. -

4-;;

:1;.7 rJrn-'

K~ ~ ..... ....>:- :.&-:. jj;~ :~

T.~j~ LC2

___ _____

4- ,--~.1-E

.4 44

LA

to

-fill

0 E

44'4

41,

LA~

4. 4

14* *

I R I _

F +14

Im-

.7 + .21

I loll

- in1 -

jqpfT '4- 1 _H__14

U4 I+I I

+4-4-I I, F!-

C; 0

r-42

InI

AA

4' T- of

'4l

o #4#II I v I 1I Io I I I I

'4II

ItI

Iv I

f1 go -

10 1 1 1 11 r Tr

r 4441+4+ ifI Ir

if I

lv++H

1111 -FT_8

-T,

Q.; 44 4 rT7 I . T I

Tt -H

41 Ti I +4+ 4- _7I -,-

T -t t-*t-

-t-0444 4-

Tl

I I fill

ttT

4

J4-

m3UO 1. 0

_tit

0.5

I-i 14- 7-t-44- -

T

.25

-4

0.1 ;-i-.6

3 4

r/r III,

Fig. 2 5 F(7) for Ydrious val,,es of

TO versus r/r fo r d 0. 5

43

IF I

TILLT+4-4

T-T4-

45

T -' T

........... KTTh

in --- r -t T T-t -i--t- r -N

N:7-Tr -5-TEH

*t- -"11

44 I v

-44

*I 11

$44-H+ -. i$4

I -- 4

744

_1444

o -H 110 11-14 1- 1 1 AS IA0

. . . . . . . . . .

EIU

I : I

RMi

4--4

IfI

If 0

NI I If

If I I I

T Tk.. .. , I - I

I I I TT-N

>4

'.0 14 0HEH HI

FT-FT45

tfJillL

zr, -4 ' if I LIit

147-

7 1r

-4 t1

L

f 1* 0LI I I

0t

4 i-446 111<

-tt

4--4wi-

~44-L~iU4I

I . ! I- - -,-;

1 '1 1 111UT-r "I -- - l l 1 1 1 1 1

4 1 +I -[Ill[ I4 I I I i Il

t44-

II I4k

i-I-m

4 -4 A - W 1 L-4

-H [i -- l - I

LA LIN

UM

1-t

0rrT7

UII

I I TVI 1 -

44 40 0 0 'V

LN -LL U'I'LiT

InI0 n

H I*T "0 11 1 V-l_ I II I I Iaio fa l 1 tII

4+0 In-

fillfillA II ..

A _.4-L

r &

If.11 L

47

r - +% I

4! 4 000

0 0 0fil0

*1ji

IL

lf A 11

11

II

II f i l l -

IH

07 i ,IIIIit ±IM0

UIN

C;

- 0H

1~ 11 it at-~-

yrT , ~

IIt.

N

p IN

JITT- -l

f 1:3 ]if

,0 N 0 00

48

I

C;C

in 0f-T ll II I I I I II I

miiIn*

I I L Nor

I I. ft . 4-_ -r

14

-I-F

ri I I V

-ne

C;

P-V VT

li f*l, I I

4-i

4 ~ i f

IL N .

T- J 0U0

-I 0-0--0 -N-

FU I

N LfN

H II

in .. fa

N Ili

T441, t

4-4-

IiA'I

-T I

t,1 Itt4h L ti I~4

-1 49 _

1.54n

k~c I'D C a.06 - - c .,49!

> ->b.--.c 'o v 'm I.. I.

C Sj a ~ X ".6 0 a a;~.

X CL'U 4 0. '"

Aj - " Ia so 2a, 0 N w

-4 AtS -

N W~ a r.' 0 0-;,

£ .4a~ m w c v I"

o , t .0~. v 0 c 4o

10 .4 - * a a v

1. a: 17Z : - ; oo

v u 9

0'.'

vs S.

0 9 a aU'V~~ 0I aleb .u a j d 4 cUV V a -

-0 2. .S4N

0 -6a * W -

00.c a 4'. v54 &1a Ia

-oJ -I v -6

-6j 0 0.

.5 . 4o

&~- r. 4'4j a O 4)~.

0~ a C C

v 4, I Is. 1.-£31,'-- ~ *2 - 0 ~. '- a a 0 4'

-6 1 ..4~' 2. loS v 0

I0 :2-, 'U

cc Zo Is:i 0 r'~ . . f

c

4,O0*CuI., "Cc 'o- 3 4. U~a *~a 0'-. :

I.~~6 m 4. tti. c 440 Cc a 1rb I 1 0.3.. .0 0~UC b. I '~ - 0 hi c

9L. - A 0CV M. at U - a 09 = --4 J - Q,.0 r A . r-41 U

=- - 0

0. 0 b. L.l -

go 0 -A-w a -- ' s X. - 0c

:F CIO cr4.

0

V aa

b.- vs

I" c

fe n* V b..- $. :1. =

w U l-0

0~ 0U -ClXJo 0l-t b 40 1- ~ . U

0~~~.-~0 b. 414 j C~. i £ 4-u uq- *U. r 0

U7a -0 1~ *V0. U. lw04

-Aw 3 4.. 4 ~ .C -4 .71 ,. > I

a j96 C . r'i a 0 ' 4 64.t~ U0 cc r3-~~ 6 aJ 4.4 w94.C. * -cl

LCI-CtI 0 10 C i .

dc IV, 4 a a,~ul - 3- *

Ji ~ M 0 C, 4iC al.", . c 4.

10Cj .2f a