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6..— UNITED STATES ATOMIC ENERGY COMMISSION

BMI-795

THE INERTIA PUMP

By

R. W. Dayton

E. M. Simons

W. H. Goldthwaite

December 18, 1952

Battelle Memorial Institute

'— Technical Information Service, Oak Ridge, Tennessee

UNIVERSITY OF'

ARIZONA LIBRARY

UNIVERSITY OF MICHIGAPOCumentS COHGCfiQn

WNW 4 AP“ “’55

3 9015 08646 6912

For sale by the Superintendent of Documents, U. S. Government Printing Office, Washington 25, D_ C. -

rice 25 can s

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Subject Category, ENGINEERING

Work performed under Contract No. W-7405—Eng-92

This report has been reproduced with minimum altera-

tion directly from manuscript provided the Technical Infor-

mation Service in an effort to expedite availability of the

information contained herein.

Reproduction of this information is encouraged by the

United States Atomic Energy Commission. Arrangements

for your republication of this document in whole or in part

should be made with the author and the organization he

represents.

Issuance of this document does not constitute authority

for declassification of classified material of the same or

similar content and title by the same authors.

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TABLE OF CONTENTS

Page

I I I I I I I I I I I I I I I I I I I 5

, I I I I I I I I I, I I I I I I I I 6

THEORETICAL ANALYSIS . . . . . . . . . . . . . . 6

Ideal-Performance Evaluation . . . . . . . . . . . 6

Pump Losses . . . . . . . . . . . . . . . , 12.

e e e e e e e e e e e e

Friction Loss . . . . . . . . . . . . . . l9

Angular -Momentum Loss. . . . . . . . . . 19

Cavitation . . . . . . . . . . . . . . . . . . ZO

I I I I I I I I I I I I I I I I I

e e e‘ e e e e e e e e e e e e

Description of Apparatus . . . . . . . . . . . . . 22

I I I I I I I I I I I I I I I I I

EVALUATION OF POTENTIALITIES OF PUMP . . . . . . . 29

APPENDIX I

INFLUENCE OF MASS OF FLUID EXTERNAL TO PUMP BODY . 31

APPENDIX 11

DEVIATIONS FROM SINUSOIDALMOTION IN MODEL TESTS . . 33

APPENDIX III

e e e e e e e e e e e e e e

APPENDIX IV

e l e e e e e e e e e e e e e e e

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_5_

ABSTRACT

A sealless, bearingless pump has been devised in which a variable

fluid pressure is developed by the inertia of the fluid in an oscillating

helix of tubing. The ends of the helix are connected by radial tubing to

flexible members at the center of oscillation, The theoretical analysis

reveals that Wide ranges of pressures and deliveries are possible.

Head -capacity measurements on an experimental model have shown

pressure values consistently higher than those predicted by theory. These

deviations can probably be attributed to deviations of the motion of the model

from the sinusoidal motion assumed in the theory.

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-6-

INTRODUCTION

In manyapplications of circulating pumps, it is important to prevent

leakage or contamination of the fluid. Frequently, the fluid is corrosive

and a poor lubricant. To eliminate the usual shaft—seal and bearing

difficulties under such conditions, a new type of pump has been conceived.

In this pump, the fluid is contained within a cylinder which is reciprocated

in a direction parallel to its axis. Owing to the inertia of the liquid, fluid

pressure will be generated at each end of the cylinder, alternately. It is

apparent that if check valves for intake and discharge are connected

properly into the system, pumping can be achieved.

Figures 1 and 2. show the reciprocating cylinder and valve installation

in typical single -acting and double -acting inertia pumps. The reciprocating

cylinder has been curved to form an arc of a circle and is reciprocated

angularly about the center of curvature. At its. center are two bellows which

serve as flexible connections between the moving legs leading to the recip-

rocating cylinder and the stationary piping which contains the check valves.

Since the reciprocating mechanism can be entirely external to the

pump, bearing and lubrication problems are avoided. This, together with

the fact that the pump requires no moving seals of any kind, suggests its

use for applications where freedom from contamination, leakage, and

mechanical failures are of paramount importance. '

THEORETICAL ANALYSIS

Ideal -Pe rformance Evaluation

To determine whether the characteristics of such a pump are

interesting for a practical set of design conditions, the performance can be

estimatedby a simplified analysis.

Assume that the container of Figure l is filled with a fluid whose

mass density is p *. Assume further that the container and fluid are at

rest. Now, accelerate the container angularly in a clockwise direction.

The fluid also will be accelerated in the same direction, but to a lesser

extent, if flow is permitted. This acceleration will cause p1-p2. The flow

which occurs will be in the direction opposite to the acceleration, and the

acceleration of the fluid within the legs will similarly be in the opposite

direction. Such accelerations will cause p1>po and p3-p2. If delivery is

occurring, it is necessary that po-p3, where theoretical delivery pressure

(-p)’ =po- p3 . The sign has been changed for later convenience.

'See Nomenclature at end of report.

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Check valves

Stationary piping

Point (3)

Inlet pressure p3 %

Flexible connectors

Outlet pressure p0

Axis of rotation

\

r (ii—1'3.) = Acceleration of fluid in legs

Legs r, Point, (2)

I ' ~ Pressure p 2

\\Angular acceleration of fluid in space

Maximum reciprocation

Point (I) /

Pressure p,

Reciprocating cylinder/

FIGURE I.SINGLE- ACTING INERTIA PUMP

Angular acceleration of container

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diam (EEMZ- 02-hoqlemDOn-d UMDQE

c3530 oezooocamoom .

35:36

@320 $2? .6on

code 328 .326

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-9-

If the container angular acceleration is a, and the angular

acceleration in space of the fluid in the circumferential arm is , the

linear acceleration radially of the fluid in the radial legs is r ( ii ), acting

in the opposite direction to along the length of the tube. From these

accelerations, the various pressure differentials can be calculated as

follows:

P 1-Pz = PI'2 933*

P3-P2 =pri&-B)-

Pl‘PQ

Theoretical delivery pressure, p, is

-p=po-p3=(6+2) profi -Zprzii. (1)

Now, assume that the container is being moved with simple harmonic

motion . Then

H

displacement (a)

velocity (a) "Ow cos wt

acceleration (a) = -aom sin mt

Let us concentrate our attention on the first quadrant of wt, in

which 6 and a are positive and ii is negative. In this case, delivery occurs

at (O), and intake at Introducing the value for a in (l), we have

P -__Z__ao.,2 sin. cut, (2)

B _ -i6 +Z-prz 6 +2

Note that if no flow is permitted in Equation (1), ii = , and introducing

2

ii =-aow sin wt,

2

-p =—6przaom sin (at.

This is a maximum for out = 2 , at which point Pm = p rzao 6oz, or

prz = Pm

a 6m:

Introducing this value for pr2 in (Z) and letting p' a: P , we have

m

" 6 z . z 2 -

3:- amp- arosincot.

0+2 0 6+2 0

This equation can then be integrated,

3=_ 9 aow2p'h 2 a0 wcosmt+a

0+2 6 +2

_ _ 6 a mzp' ta + Z aosinwt+at+b(3)

0+2 ° 0 +2

'All units must be consistent within some absolute system.

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..10...

To discuss the operation of the pump and to determine boundary

conditions from which Constants a and b May be evaluated, first consider

the system represented in Figure l. The fluid within the reciprocating-

pump body-will move with the container until its acceleration develops a

pressure equal to the pressure differential across the pump. At this time,

which we denote as t = to, the check valves will open, It is evident that ii =

S and d = at t = to. In addition, since the point of origin for measuring

B has not been prescribed, we may arbitrarily choose it so that B: a at t =

to. Because the motion of the pump body is sinusoidal, it must slow down

to‘reverse its direction, and it is clear that there will be some later time,

t = 1:1, for which it again will equal ,3 . At this time, the valves will close.

Using these boundary conditions, i.e.,

a=b, & =3, a = B' at t=to

a' = B at t=t1,

we arrive at the following evaluation of Constants a, b, to and t1:

a a0 (0((305 + P.

6+2

6

b (sin cuto- cuto cos alto - 1/z p‘ (02:5)

- a

0+2 °

sin wto = p'

cos wt1+p'ot1 = coscoto + p' (0110

Note that these values are all functions of the relative pressure p' .

Two types of operation of the pump are possible, depending on the

magnitude of p' . If mtl- mto< ir , the valves are open for less than a half

cycle, and flow will occur between times to and t1 during the first half

cycle of the motion of the pump body. It is clear that there will be another

time interval from (t°+ n/m ) to (t1 + 11/0, ) when the pressures developed by

the pump will be equal in magnitude but opposite in sign to those developed in

the time interval (to, t1). If a second set of pressure-relief valves is

provided as indicated in Figure 2., flow can also occur during this second

time interval. The pump will then be called a double -acting pump. The type

of operation just described, for which 0912— mto< 7;, Will be called Type II

operation.

If, however, mtl— wto> n , Type II operation is not possible in the

double -acting pump. (It would be possible in a single -acting pump, but then

only half the flow obtained in the double -acting pump can be achieved.) The

reason is that the two time intervals described in Type II operation would

overlap, and this would imply flow simultaneously in opposite directions.

Consequently, in the double -acting pump with mtl - 0110) 1! , a new set of

boundary conditions must be used. It is clear in this case, assuming balanced

operation, that t1 is determined by the condition cut1= wto+ rr , rather than

the manner in which it was determined for Type II operation. The set of

boundary conditions which can be used now is

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..11_

a=£i,a=/§,att=to

a=b att-t1

0t1= (0120+ 1].

With these conditions, Constants a and b have the same values as they did

formerly, but to and t1 are determined ‘by the equations

cos mto = " p',

2

ti = cello + '7.

Operation in this manner will be called Type I operation,

The transition from Type I to Type II occurs when to and t1,

respectively, have the same values for both sets of boundary conditions.

Thus mt1= (0110 + rr in the equations

sin @to = p' ,

cos will + p"0)t1= COS wto + pmto

2

"+4

If we set mtl = wto+ 11 in these equations, we find p' = = 0. 54.

For p'>0. 54, we have Type II operation; for p' < 0. 54, the operation is

Type I. '

The pump delivery can now be calculated. For Type II operation,

the angular delivery per cycle of a double -acting pump is clearly given by

D=|z(a - l att=t1.

Evaluation of this quantity results in

D = _2'_6___._ ao [(sin wtl-Sin mto)-( will -- alto) COS ratio

6 +2

+ _lzz'_ (mtl-mto)z]

When reference is made to the equation defining to and t1, this expression

can be reduced to

D= Ji—ao(— 1 sincewt1+5in®tl “2-1) ‘

6 +2 2p' 2

- z

a a 0 (sm th-P') _ (4)

a +2 p'

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..12_

The angular delivery per cycle of a single ~acting pump is one -half this

value. For Type I operation, the delivery is again given by Dal 2( a - B) l

at t = t1. When the defining equations for t1 in Type I operation are used,

this expression reduces to

D=36 a0 \/1-

0+2

"2

4

(P' )2. (5)

Relative delivery rates are obtained for both types of operation by

dividing the delivery rates given in these formulas by the quantity 40 a o

0 + 2

(i.e., the delivery rate for Type I operation at p' a: 0).

These relative delivery rates are tabulated in Table l for a double-

acting pump and in Table 2 for a single -acting pump. They are graphed

in Figure 3. In this figure, constant power curves are also drawn to give

an indication of the power output under various operating conditions.

The power output is proportional to the product of pressure and

delivery. It is easy to show that this product has a maximum at p' =

Therefore, the greatest amount of useful work is obtained from a double-

acting pump in Type I operation at p' = 7..

Since it appears desirable to operate a pump at nearly its maximum

power point, further calculations will be made only for a double -acting

pump in Type I operation, and especially for the case p' a:

l

;rut°= _.”_

For this case, cos (ate: 4

D = “0 ——L- .

0 +2.

Figures 4, 5, and 6 show the time variation of fluid and container dis-

placement, velocity, and acceleration.

Pump Losses

Just as in a positive -displacement pump, the theoretical efficiency of

the inertia pump is 100 per cent. The actual efficiency is less by an amount

which depends upon the design and configuration of each particular in-

stallation. In the discussion thus far, no losses have been assumed to occur

in the pump. Several sources of such loss are evident.

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-13-

TABLE 1. IDEAL CHARACTERISTICS OF DOUBLE-

ACTING INERTIA PUMP

Relative

Peak Relative

Relative Relative Fluid Valve Angles Output

Pressure Delivery Velocity @120 rot 1 Power

0.0 1.000 1.000 90.0° 270.0° 0.000

o. 1 0.988 0.989 81.0 261.0 0.099

$0.2 0.951 0.957 71.7 251.7 0. 190

80.3 0.881 0.898 61.9 241.9 0.264

EDA 0.778 0.808 51.1 231.1 0.311

E$0.5 0.618 0.676 38.2 218.2 0.309

0.536 0.536 0.608 32.5 212.5 0.288

<t§0.6 0.395 -- 36.9 201.9 0.237

g 0.7 0.213 -- 44.3 184.2 0.149

508 0.092 -- 53.1 165.2 ’0.074

$0.9 0.020 -- 64.1 141.0 0.018

E2:1.0 0.000_ 0.000 90.0 90.0 0.000

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-14..

TABLE 2.

IDEAL CHARACTERISTICS OF SINGLE-

ACTING INERTIA PUMP

Relative

Relative Relative Valve Angles Output

Pressure Delivery wto wtl Power

0.0 1.57 0° 360° 0.000

0.1 1.18 5.7 299 0.118

0.2 0.89 11.5 273.9 0.178

0.3 0.668 17.5 255.4 0.200

g 0.4 0.471 23.6 235.7 0.188

'

2 0.5 0.316 30.0 218.6 0.158

6)

8' 0.536 0.268 32.5 212.5 0.144

'53 0.6 0.196 36.9 201.9 0.118

a.

E?“ 0.7 0.106 44.3 184.2 0.074

0.8 0.046 53.1 165.2 0.037

0.9 0.010 64.1 141.0 0.009

1.0 0.000 90.0 90.0 0,000

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Relative pressure (p')

Relative power out

' 0.2 0.3

l.0

\

0.09 \ \

0.9 t

\ \

0.8 .00 1‘ -

e \

\ a. & \

or \\ \ ope» \ .

' \yordon \ /—*—Double—act|ng pump

‘ \ /

0.6 a

\ ,\ \-

\ O\\ \\ O L

o A 1 off\ \ O, '

j \ 0'7 0‘ I ’00

0.5 \ \ \\ '

\\ \> \\\

\-

s \

0.2 \\‘\ V/ \\ “O.2

Single—acting—P \\

6. 9198 ~ ~ ~- _ 199;; \

. - \

0 .

0 0.! 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I.0 l.l |.2 l.3 l.4 l.5 LS LT

Relative Delivery

FIGURE 3. IDEAL CHARACTERISTICS OF DOUBLE- AND SINGLE-ACTING INERTIA PUMPS

OF SAME DIMENSIONS AND OPERATING UNDER SAME CONDITIONS A_4“9

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.46-

4

1 —-—B Valve

reversal

// Displacement

/

// /

a //

Acceleration

FIGURE 4. TYPE 1 OPERATION

A-4l20

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Valve reversal

\

\

\\ \\

638m

7

\‘r

I

s

' “s

9

‘~”* ///

0.:

vl/ ////

Delivery ///4 lé/l”

a \-/e

00.

“I”! I" .

$sz 0

.. I-lll> \ 0”

*¢ B (A) ’0’! [I

//>. \ ' A

\‘r . Q / Acceleration

‘s‘ R

‘1 ,8 (s)

"

. f

’0 I 6 I (I

w we I, I

flmfl,’

FIGURE TYPE I OPERATION AT ZERO (A) AND MAXIMUM

(B) DELIVERY PRESSURE

A-4l2l

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/Volves closed

1/ Delivery

Valves open

\r...

Displacement

Velocity

l Acceleration

FIGURE 6. TYPE II OPERATION

A-4122

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-19-

Velocity -Head Lo 5 s

At bends and valves, losses of velocity head will occur. For that

reason, the velocity within the tubes is of interest. It is clear that this

velocity reaches a maximum during each cycle. This occurs when sin mt:

p' with wt in the second quadrant. From this, it can be shown that

Vm: 932 “6 @r (J 140')“2 -"2LP'+P' sin-lp' +P' °°s'1%"')'

JT

1!

Here, sinlp' and cos-1__"_ p' are in the first quadrant. For p':

v = a (Dr.

m 6+2, 0

It is desirable that the ratio of maximum velocity head to the pressure

head be low, so that each velocity-head loss will not contribute too largely

to inefficiency. If N1 velocity heads are lost, we find that for operation at

p'=

,

it

AP1 =0.62l2_§_ N1,

p (0+2)2

since the maximum single velocity-head loss is 1/3 p vm2.

Friction Loss

Friction losses will occur within the passages. This loss is equal to

the product of a friction factor, L/d , and the loss due to one velocity head.

For simplicity, assume a friction factor of 0. 02 and let L* = L/de, where,

for a rectangular section in which a is the small dimension and a/s the large

dimension, d6 = 123' . We can then find N2, an additional number of

+ 8

velocity-head losses resulting from friction, which is

N2 = 0.02 L*.

- and Ap2 =0.62 “00 N2.

p (0+2)Z

Angular -Momentum L05 5

Angular momentum must be conserved in the system. Thus, even

if no flow out of the pump is permitted, acceleration will cause circulation

of fluid within the container, and this circulation will lead to losses. No

detailed analyses of this source of loss have been made, although such might

I

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-20-

be desirable. Such an analysis would be difficult even for the case of zero

flow and probably impossible when flow is occurring. Nevertheless, it is

evident that by making the radial dimension of the container small compared

with the radius about which the container is centered, such effects would be

minimized.

Cavitation

Figure 7 shows the time variation of pressure at various sections of

a typical pump, operating at p' = 12 , from inlet to outlet, at various

parts of a cycle. The results showthat for wt .1 .5." the pressure at the end

of the inlet arm is considerably below the inlet pressure. If this low

pressure is below the vapor pressure of the fluid, cavitation will occur.

Therefore, it is necessary that this pressure be considered in any pump

design, and the inlet pressurized if the pressure is below the vapor pressure

of the fluid. This pressure can readily be calculated for any operating

condition; for operation at p' = \f2- , the maximum difference between

inlet pressure and minimum pressure is

.. l. 22

AF'C ~1—+2 P-

Pump Design

Using the relations which have been developed, calculations can be

made of the operating characteristics of different pumps. The relations

which are used are summarized below Egr convenience. These relations

pertain to Type I operation, at p' = a: 0.45, the maximum power point.

17

p = 0.45 p a o aw2r2 Ideal operating pressure

V = Q I 0 A a o or Volumetric delivery rate

r 0 +2 p

vm = 0. 748 __0___ a0 (01' Peak fluid velocity during a cycle

0 +2

A a 6 Relative pressure loss

__P_ = 0,62 __Q____ (N1 + N2) (N1: assumed number of velocity

p (0 +Z)z heads lost at bends and valves)

N2 = 0.02 L* (N2=friction.'loss in velocity heads)

L* = “G '1' z) (1 + s) For a rectangular cross section,

2a , whose short side is a and long

side a/s

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-2 |..

pressure

ll

~|-+~

ll

=1

wt -Valve opens

wt Maximum

cavitation

wt

wt =

/ \ _ 51 Valve

wt -— —

4 closes

Pump End inlet End outlet Pump

[ inlet arm arm outlet—N

Distance Along Pump

FIGURE 7. PRESSURE VARIATIONS THROUGH INERTIA PUMP ,

A-4123

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P = p - Ap Delivery pressure

A __ 1. 22 p - - -

pc .. o Maximum difference between inlet

0 + 3 pressure and vapor pressure of

fluid

Using these relations, the pump characteristics shown in Table 3

have been determined for large and small moderate -pressure pumps and for

a high-pressure pump. It appears that the designs as given are not un-

reasonable. It is likely that careful consideration will show how to improve

these designs.

EXPERIMENTAL WORK

An experimental program was undertaken to check the foregoing

theory and to extend the theoretical results, Accordingly, a small model

was built and tested.

Description of Apparatus

Figures 8 and 9 show the essential components of the model, The

pump body consisted of 1-1/4 turns of 3/4- inch copper tubing, 8 inches in

radius. Two radial legs of 3/4 -inch copper tubing connected the pump body

to two Monel bellows, 1-7/8 inches long and 1 inch in inside diameter,

located at the center of oscillation of the pump.

The reciprocating pump body was supported and driven by a mechanism

which was borrowed from another machine. It delivered an approximately

sinusoidal motion to a shaft, one end of which was modified to take the bracket

which supported and drove the pump body.

Four one -ha1f-inch rubber check valves, manufactured by the Grove

Regulator Company, were used in the model. Four gate valves were in-

stalled, adjacent to the check-valves, so that single -acting operation of the

pump could be studied by opening diagonally opposite gate valves and closing

the other two. This also permitted an empirical determination of the

average pressure drop through each side of the piping system, corresponding

to that occurring during each half of the pumping cycle, This was accom-

plished by noting the pressure drop across the system as flow at various

rates was forced through the stationary pump.

Surge tanks were provided at the inlet and discharge ends of the pump

to provide nearly constant pressure heads at these points, as assumed in the

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-22-

Check valves

To manometer To manometer

' (ZR—'- Discharge

( .

Bellows

Legs

I g\ A

Reciprocating cylinder

J\Reciprocating drive shaft

,1,

'FIGURE 8. INERTIA PUMP MODEL

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-24-

FIGURE 9 .

'eer

V

PHOTOGRAPH OF RECIPROCATING

CYLINDER OF MODEL

m,”

.L-xL’S ‘

~__

>1 _,‘</~_..,

I-

A.“

_.

L~

.1

o

1

I

_1

,3

f

n

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TABLE 3. CALCULATED PERFORMANCE OF A

VARIETY OF PUMPS

Assumed Values

r, feet 1.0 0,5 0,5 1.35 1,0

a o, radians 0, 175 0, 175 0,175 0, 175 0, 175

6, radians 2,66 6,0 6,0 2,5 120

a, feet 0,1 0,1 0,1 0,3 0,02

s 1,0 1,0 0, 1 0, l 1,0

speed, rpm 1800 2400 2400 1200 3600

Calculated Values

p, psi 100 100 100 76.3 18, 100‘

L* 46. 6 40 22 11. l 6100

N2 = 0,02 L* 0,93 0,8 0,44 0,2 122

N1, estimated 4,0 4,0 4,0 4,0 4,0

N1 + N2 4,93 4,8 4,44 4,2 126

Ap, psi 6,5 4.9 4.5 4.3 2000

P, psi 93.5 95. 1 95.5 72.0 16, 100

V, gallons per minute 38, 1 32,3 323 3000 5,2

Horsepower 2,22 1,94 19.4 133 55,4

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-26-

theoretical analysis, The pressure differential across the pump was

measured at the surge tanks with a mercury manometer, The discharge

head could be changed by throttling with a gate valve in the outlet piping,

An electric counter recorded the total number of cycles of the pump and

also determined its frequency, The time required for the pump to fill a

given volume was used as a measure of the volume rate of flow,

The dimensions that were used to obtain the theoretical head-

capacity curve for this model are summarized here:

0 = g- " = 7,86 radians

a = 0,26 radians

- 2,31 (10)"3ft2 = cross section of pump body

>,

I

0,677 foot

'1

II

V = 7 (See Appendix I)

G)

211

= f = 260 cycles-min

Test Results

The objective of the experimental program was to determine the

validity of the theory by a comparison of experimentally obtained head-

capacity curves with the theoretical values, About 25‘ head -capacity curves

were obtained under various conditions in an attempt to approximate the

conditions set up in the theoretical analysis.

Figure 10 shows the head -capacity curve predicted by theory, Curve

A, and the results of the experimental data, Curve B, taken with the apparatus

described here, running at a frequency Of 260 cycles/minute, A third

curve C, shows the pressure drops measured across the pump corresponding

to varying rates of flow through the pump produced by outside pressure,

These values are added 'to Curve B to give a corrected experimental curve

D for comparison with the theoretical, The reason for the correction is

that, in the theoretical analysis leading to the head -capacity curve, the

delivery pressure is the only pressure that the pump is working against,

Experimentally, the pump also works against an internal pressure drop,

which does not appear in the delivery-pres sure measurement but must be

added to it, This must be considered a first-order correction only, since

the actual internal pressure drop varies throughout the pumping cycle.

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~27-

Delivery pressure, psi

|3 \\ \Of

\ \

12 ‘ \ x01

0

l l ‘2 \\

\\ \Q Curve 0 — corrected experiment

lo \4-Curve 8 - observed <

(H‘\ \\_

\\\ K

9 Curve A-theory

\ i) \R

7 \\ \

n .\\ CL

9 ‘\

t \ l/

/’

4 ’J/><g\\ _

/ s

// \\ \

3 ~ ‘ a r

/r \o\ \\

2 Curve C—average internal pressure drop \\

b

o ‘ \

| \L‘ \

\\\ \

Q CL

0.0 DJ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 l.o l.l l.2 l.3 IA L5 LG I?

Volumetric Delivery, gal lmin

FIGURE IO. THEORETICAL AND EXPERIMENTAL HEAD-CAPACITY

CURVES FOR INERTIA PUMP MODEL c-4|24

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A comparison of Curves A and D reveals that the experimentally

observed delivery pressure is roughly twice the theoretical value for any ~

given delivery rate, This condition was present in previous runs and most

of the modifications of the apparatus were efforts to correct it.

If we look-at the equation (1) for the delivery pressure of the pump,

and let ii = B under the condition of no flow, -we have -p a: 0;; r2 a for the

cut-off pressure, Because of the surge tanks and check valves, only the

maximum pressure, developed during the cycle Pm = Aprza m, will be

measured, As sumin sinusoidal action of the pump, as in the theoretical

analysis, 6m = co m and the maximum pressure will be

pm as aprz aow2.

All of these quantities were easily and accurately measureable, and none of

them was responsible for the high pressures observed, The discrepancy

between observed and predicted pressures is very probably due to lack of

agreement between the assumed theoretical conditions and actual operating

conditions ,

When the high pressures were first noticed, it was suspected that

the rubber hoses that were being used for the flexible connectors, or the

spring-loaded check valves might set up a condition of resonance with the

reciprocating pump and produce the high pressures, Metallic bellows and a

different type of check valve were installed and, although the operation

improved, the high pressures were still present.

In the development of the expression for the cut-off pressure,

Pm == 6 p aor's (02"

the motion was assumed to be sinusoidal. If the motion of the pump body is

not sinusoidal, this equation does not define the cut-off pressure, Some

deviations from sinusoidal motion could be attributed to the kinematics of the

reciprocating drive mechanism, but the effect was not large enough to account

for the results, To obtain a better idea of the pump motion, a pointer was

attached to the pump body and high-speed motion pictures were taken of the

pointer moving-across a graduated scale, A displacement time plot was

obtained from the film, (See Appendix 11,) Certain portions of the cycle

differed noticeably from sinusoidal motion, Graphical differentiation of the

displacement curve in these regions showed that the pump body experienced

a maximum acceleration which was much greater than that to be expected

from sinusoidal motion. This could account for the excessively high cut-off

pressure, Moreover, the acceleration was greater than sinusoidal

acceleration for a considerable part of the' cycle, which would account for the

high pressures under partial delivery conditions,

Although pressures developed by the pump are higher than predicted,

these results show the inertia pump to be an apparatus which can pump

successfully, The head-capacity characteristics are certainly in the range

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-29-

of the predicted values, and use of an improved drive mechanism might

produce pumping more closely approaching theoretical,

The drive mechanism for this investigation was adapted from

apparatus which had been used for another purpose, Although it was suit-

able for a preliminary analysis of‘the pump performance, it was not

capable of providing harmonic motion, precisely, and was not adaptable to

precise measurements, Therefore, in addition to the difficulty in checking

the theory which has been discussed, it was not possible to make careful

measurements of the efficiency of the pump, For that reason, no good

estimates of pump efficiency can be made at this time, In view of the

necessity for devising an entirely new drive mechanism for more accurate

study of the performance of the pump, and also because the flexible-

connection problem has no certain solution at present (see Appendix III),

further work was considered unprofitable, The investigation has been

abandoned, pending better solutions to these auxiliary problems,

EVALUATION OF POTENTIALITIES OF PUMP

As a result of this study, some qualitative observations can be made

concerning the potentialities of the pump, Its advantages and disadvantages

can be tabulated as follows:

A, Advantage s

1, No shaft seals

a, No leakage

b. No contamination by packings

2, Isolation of pump from drive mechanism

a. No special-bearing or lubrication problems

b, Easily broken down for repair or sterilization of pump

3, Nonpositive' displacement pump with wide range of maximum

pressures and capacities

B, Disadvantages: flexible -connector problems

1, Bellows or torsionally twisted tube

2, Limitations

a. Temperature

b. Pressure

c. Size

d. Flexibility

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-30-

In general, the inertia pump seems ideally suited to pumping

applications where leakage and contamination are important factors, An

interesting possible application is its use as a blood pump, which was

investigated in some detail. In addition to features already mentioned,

its simplicity and lack of constrictions which might damage the blood give

it advantages over pumps now being used for this purpose,

It may also be interesting as a pump for developing very high

pressures,

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-31-

APPENDIX I

INFLUENCE OF MASS OF FLUID EXTERNAL TO

PUMP BODY

In the theoretical analysis leading to the delivery-rate equations, it

was assumed that the fluid in the lines beyond the check valves had no effect

on the delivery of the pump, In the experimental model, all of the fluid

between the surge tanks went through the same acceleration cycle as did the

fluid in the pump body, This must be considered in calculating the delivery

pressure, as measured at the surge tanks (see Figures 1 and 8),

We have, as before, the pressure differential at the ends of the

circumferential section p1 - p2 = prdrfl‘. The acceleration of the fluid

relative to the circumferential section of the pump body is again flat-ii) and

will be equal to the acceleration through any piping which has the same

cross-sectional area, To obtain the acceleration in a length of piping of a

different cross-sectional area, We can say:

For continuous flow of an incompressible fluid,

viii = vpap,

where

vi =: velocity in a section of pipe of length 11,

A1 a cross-sectional area of section of pipe of length 11,

vp :: velocity of fluid relative to the circumferential section,

Ap = cross-sectional area of the circumferential section,

dvi dv. dv -

__ A-=_P. A . But 2 =1-(a-p)

dt 1 dt p t ’

and therefore

dvi .-

A ..

-— n—E- " - 0

dt Ai r la Bi

Now the pressure drop Api in the section of pipe of length 11 is

Summing up the pressure drops from the surge tanks to the ends of

the circumferential section, we have

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-32_

- A

EApi=pr(&-,B)2 11.2.,

i 1 A1

and the delivery pressure appearing at the surge tanks will be

'0

ll

(P1 'PZ) " 2AIM

A

Pr9r1§ -Pr(5-5) 21i —-P

A1

p = przfilwii AP) -prz& Eli. _P_A

r A1 1' A1

or

._ Z" Z

P-(9+Y)Pr B—YPI' a (la)

where

y = AP

1' Ai

This is Equation 1 with y substituted for the 2 in the coefficients of and ti ,

When (la) is carried through to the delivery-rate equations, the factora 9

+Y

appears in place of a in both types of operation.

0 + 2

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_33_

APPENDIX II

DEVIATIONS FROM SINUSOIDAL MOTION IN

MODEL TESTS

Figure 11 is a plot of the angular displacement of the pump body as

shown in successive frames of a high-speed motion-picture film. The film

ran for 13 cycles of the pump action, and the sixth and seventh cycles, where

the camera speed was steadiest, were chosen for analysis, Every fifth

frame was inspected and the angular displacement recorded for those two

cycles, The sixth cycle appears in Figure 11. In a larger plot of these

data, the deviation from simple harmonic motion was quite apparent in two

regions: one from 100 to 130 degrees of the cycle, starting at zero dis-

placement, and one from 280 to 310 degrees of the cycle. These deviations

were present in both cycles which were measured completely and in other

cycles where this particular area was examined.

The region in the box of Figure 11 was investigated in detail, each

frame was inspected, and the displacement determined. This is plotted in

Figure 12. It is apparent from Figure 12 that the motion of the pump does

deviate from sinusoidal motion. To investigate. this quantitatively, a smooth

curve was drawn through the displacement points and new values of the

displacement were taken from‘this curve. These data and their first dif-

ferences are tabulated in Table 4. The first differences are plotted as

velocities-in Figure 13, and a curve drawn through the points.

For purposes of comparison, we may compute the maximum accelera-

tion of sinusoidal motion having the same frequency and displacement.

Maximum recorded displacement reading = 32, 2 degrees

Minimum recorded displacement reading r. - 0,4

2- 132.6

16.3-degrees

Amplitude of oscillation (No) ’

a O, 285 radian

Number of frames per cycle = 398

Frequency (f) = _1_ = 0. 00251 cycle/

398 frame

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nu-¢|<

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3 ~52... 2 l&

@0325 coco 25. mm

L\ mom

saaabep ‘ iueweomdsgo

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-I-

32.2

32 i

j-

3|

30.6

Angular Displacement, degrees

8

0

'

0

4b

30. 2

30

. ‘X

29.8P _ W‘

29.6 \

29.4

0

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Successive Picture Frames From Films

FIGURE l2. PORTION OF ANGULAR DISPLACEMENT OF PUMP

(a from 94° IO 130°) g-4me

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-36-

TABLE 4. ANGULAR DISPLACEMENT OF PUMP AND FIRST

DIFFERENCES, AS TAKEN FROM FIGURE 12

Displacement, Change in Displacement

Frame degrees per Frame, degree

0 32.17 0.02

1 32,15 0.01

2 32.14 0.03

2 22's;

5 32.05 8'3;

6 32.00 '

0.04

7 31.96 0.04

8 31.92 0 05

9 31.87 0:06

10 31.81 0.06

11 31.75 0.07

12 31.68 0 06

13 31.62 0:06

14 31.56 0.05

15 31.51 0.04

16 31.47 0.04

17 31.43 0.03

:2 21:2

20 31. 30 3'8:

21 31.25 0:07

:2 :1-1:

24 31.04 2' 83

25 30.95 0. 09

26 30.86 0.11

27 30.75 9.11

28 30.64 0. 13

29 30- 51 0: 13

30 30,38 0.13

31 30,25 0.14

32 30.11 0.16

33 29.95 0.16

34 29.79 0 19

35 29.60 0'20

36 29.40 '

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-37-

Z36 Apofi 2. ohm Eot e v

m2:& .mo >._._004m> 14.5624 .m_ UMDQI

8...... 69E 38:E 2:2.". 033305

0* on wm tn an on mm mm Qu mu ON _. Q S u_ o. o a e m“ O .

\\ .QNSQ

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v<\ \\ OAVIQO pod

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\l 2.8880328 EsEonE uco Fees-3.... oEom Ilu_o

O O 058... e022: 3282..» .3 2:3 2.3.2,

o 89.; 820030 on 2 coo-m 6.35.82 16

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swan-seem» 'AigoopA jdlfDUV

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-38-

2nf = 0.0158 radian/

frame

‘

Angular velocity ((0)

~

(1 = _a (1)2

m o

0.285 (1.58)2 (10—2)2

0, 712 (10) '4 radian/frame2

0, 00407 degree/frames

“m

This acceleration is indicated graphically by the dashed line in Figure 15,

Wherever the slope of the velocity curve exceeds the slope of this line, the

acceleration of the pump exceeds the maximum acceleration to be expected

from sinusoidal motion, For example, the slope of the velocity curve at

Frame 30 is indicated by the dotted line in Figure 15 and is equal to 0. 010

degree-frame2, about 2-1/2 times the theoretical maximum acceleration,

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-39-

APPENDIX III

FLEXIBLE CONNECTORS

The inertia pump Was thoughtof originally as apump suitable for

pumping high-temperature, corrosive liquids, As the work progressed, it

became apparent that all of its advantages would be 105}: unless a flexible

connector could be found which would withstand the conditions of tempera.-

ture, pressure, and corrosion imposed-by the contemplated application.

Thefollowing tentative specifications for a bellows-type flexible connector

were submitted to three bellows-manufacturing concerns.

Bending angle Plus to minus 5 degrees

Inside diameter 20 inches

Operating temperature 1500 F

Internal pressure Varying during cycle, 5.7-100 psi

Motion Sinusoidal, 200 cycles-minute

Life 100, 000, 000 cycles

Materials (in order of Silver -c1ad InconelX

preference) Monel

Inconel X

One company reported that Monel would not have the necessary

strength at the temperature of 1500 F and that Inconel X had proved un-

satisfactory as a bellows material.

Another company said that they had had good success with bellows

made of Inconel X and felt that our specifications could be met. However,

they have had no experience in the manufacture of multi-ply or internally

clad bellows.

A third company has manufactured multi ply bellows and bellows

made of Inconel X, but their hydraulic forming methods are not applicable

to bellows over four or five inches 'in diameter,

From these comments, it would seem that, though a suitable bellows

is not available at the present time, there is a fair possibility that one

could be developed which would meet requirements of thistype.

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-40_

P2

P3

Pm

AP

APPENDIX IV

N OMENCLAT URE

Cross-sectional area of the i'th section of pump passage

Cross-sectional area of reciprocating cylinder

Small dimension of rectangular section of pump passage

Angular delivery of pump per cycle

Hydraulic diameter of pump passage

L/de

Length of passage through pump

Length of i' th section of pump passage

Number of velocity heads lost at bends and valves

Number of velocity heads equivalent to friction 1055

Actual delivery pressure,

Fluid -pressure differential across pump (theoretical)

P = P3-Po

Ratio of pressure differential to cut-off pressure

Fluid pressure .at pump outlet.

Fluid pressure at reciprocating-cylinder outlet

Fluid pressure at reciprocating-cylinder inlet

Fluid pressure at pump inlet

Cut-off pressure differential across pump

Velocity-head and friction losses, Ap = Ap1+ Apz

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~41

am

APZ

Apc

Velocity-head loss

Friction loss

Maximum difference between inlet pressure and vapor

pressure of fluid

Inertia loss in i' th section of pump passage

Radius of reciprocating cylinder

Ratio of small dimension, a, to large dimension of

rectangular section of pump passage

Time of valve opening

Time of valve closing

Volumetric delivery rate, V = EL RAPD

rr

Velocity of fluid in i' th section of piping

Maximum linear velocity of fluid relative to pump

Linear velocity of fluid in reciprocating cylinder

Angular displacement, velocity, acceleration, and maximum

displacement of pump body

Angular displacement, velocity, and acceleration of fluid in

reciprocating cylinder, relative to space

Function of length and diameter of pump piping

y =Eli_Ap

ir A1

2w x frequency of oscillation

Angular length of reciprocating cylinder

Fluid mas s density

71' u. s. sovmmsm PRINTING 0FFICE-1955 o- 333405

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