a compressible and incompressible flow analysis of the air

Lehigh UniversityLehigh Preserve

Theses and Dissertations

1969

A compressible and incompressible flow analysis ofthe air-lift pumpKenneth A. SchoeneckLehigh University

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. . . . -

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A COivlPRES!~IBL.6 AND INCO~lPRESSIBLB .

PLOW ANALYSIS OF

TlIB AIR-LIPT PU)lP

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by

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A Thesis

Presented to the Graduate Faculty

of Lehigh University .. ,-:s,

in Candidacy for the Degree of

Master of Science

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Lehigh University

1969

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f~11ment of the requirements for the degree of Master of Science.

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date·

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Chairma:n of Department"

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The author would lilte to thank Dr. Alan H. S.tenning, ~-- ..... ..., . ,, . -

Professor of Mechanical Engineering, for supervising this . .

study .anct providing helpful suggestions during the course

of the program. ':d1e help of members of the Mechanical

Engineering Department is gratefully· aclcnowledged. ·· · #&~ ·

The writoer is indebted to the Ingersoll-Rand Company

and his supervisors for their aid and assist.ance. Tl1anks

· are d11e to ~lessrs. Arthur F. Stahl, Analyst, and Chester

L. Koppenhat1ver, Analyst, Systems Development, for for

mula ting the computer program •

Special gratitude is due to the author's wife, S~San,

who shouldered all domestic pr·oblems to permit time to be .

spent on graduate study.

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• - · TABLE OF CONTENTS

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ABS-TR>\CT SY~lBOLS

1. · INTRODlJCTION I ..

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The i1odel TI-IEORETICAL Ai~ALYSIS ..

. 3.1. Derivation of the Differential Pressure Gradient Equation Assuming Compressible Gas Flow in the Eduction Pipe ·

· 3 .2 ~'letl1od of Using Differential Pressure Gradient Equation in a Numerical

·Analysis _ 3.3 _Application of the Incompress.ible Flow

Equation for Calculating l"laximum

RESULTS 4.1 4.2

Theoretical· Isotl1ern1al Ef_f iciency

Tl1e Experin1ent Comparison of Incom-pressible Theory and Experiment

4.3 Compnrison of Compressible Theory and Experimental Data

.,4. 4 Su f!}m a r_y., DISCUSSION ~ "

5.1 Conclusions --·-····----·--·--·-------------------- _ _ ___ __ _ _ 5. 2 Sug:ge st io_ns for Pu ture ____ ,~_Qr_~ _____ -- .. ---~---- ... -------~, ______ _

FIGlJRES

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APPENDIX I. Computer Program II. Sample Print-Out

REFERENCES VITA

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. 1. The .~lode! Air-Lift· Pump

2. p·erf ormance of itodel .Air-Lift Pump·

-Pages

28

29 ·. .. - -·-··· ··- ·-~~~--_Jc.___~

'

3. Comparison·of Theoretical and Actual Outputs

-4. · Comparison between Compressible and Incom- · · pressible Theories

s.

6.

·7.

s.

9.

10-12.

Comparison of ~lode! Pumt> Performance and Theoretical Output · ·

·..,.

The Efficiency Curves

Comparison of Theoretical and ~laximum Eff ici·encies for 1 and 2 Inch Eduction Pipes ·

Comparison of ·T11eoretical and Actual Friction . Factors

Comparison of Theoretical and Actual Slip Ratios

Comparison of Theoretical and Actual Outputs, \\farcl and Kessler Test Data

I

30

31

32

33

34

35

36

37-39

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---------··· ·-------------1-3-16-. ·- Comparison of Theoretical and Actual. Outputs·, --- --- --- -- 40-~43·· ···----------·-······-· ·· P. Pickert Test Data

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Th_e operation of an· air-lift- pump, 14 ft. high- and -

2 inches in diameter, has, been studied. The resul t-s · show

tt1a t a present theory based on in-compressible flow and iri-_• , ·

eluding the effec,ts of friction and slip between tl1e gas .

and liquid acc.urately predicts· the performance character-

istics of these devices in shallow· submergence applications.

· ~leasurements of two-phase friction and sl~p haye been - ~

. "' ' "'

made. an(l correlations are presented._

A theoretical treatment of the pump operation in

deep submergence applications, takinginto account the

change i_n volume of the expanding air -is also presented •. . '

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As a first ·approximation, constant values of. slip and friction

factor have been assumed to exist in the eduction pipe.

Good correlation \\Fas f 011nd bet\'leen the compressible theory

-·-·---· -- ------------;-;--_-" _____ - and experimental data upto the point of maximum liquid flow.

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The assumptions of constant slip an<l friction factor prove

to be invalid beyond this p9int. No significant improve

ment is gained over the incompressible analysis unless point

values of slip and friction factor are used.

Suggestions for modifying the present comp~ter program

are included •

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D , Pripe diameter (ft)

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g Acceleration of gravity (32.2 .fps2)

gc Newton's _Law constant (32.2.lbm ft/lbf sec2)

H Submergence ~epth (ft)

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K Friction coefficient (dimensionless) (K= 4fL/D) ~

L· Pump length (ft) 1''

-----• ~lass f 10\-/ (lbm/sec) . m rate

P Pressure (psf)

Q Volume flow rate (cfs).

S Slip ratio (dimensionless)

T. Titne (sec)

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·7.., 1iall shear stress (psf)

\/ Volume (ft3)

V Velocity (fps)

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~ Density (lbm/ft3)

Subscripts

0 Pump inlet

1 Leaving injector

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g Gas at average pump pr~'¥ure

ATM Atmospheric condition

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ln 1966 work was· started at Lehigh ·University to exam,irie h ' ...

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- · :··. ; - · thoroughly the fundamental principles of the air-lift pump.

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Th:rougb the applicatio~ of basic fluid mechanics, ~n equation-

was derived which accurately predicted the performance

characteristics of these devices in shallow submergence·

applications [1] [2] • -The only limitation was that suit-·

•ble values for the slip· ratio and wail frictio_n factor had - - . ~

to be found before the ·eqtlation could-·be solved. Experiment~·

al. tests performed on a model pump, 14 ft. high and 1 inch·

in diameter, provided enough data to substantiate an existing

theory for calculating the slip ratio but did not provide

a· suitable means for predicting the friction factor.

This work is intended to extend that originally done

by ~1artin in two different ,-rays. Th~ first part of the

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program is concerned with rebuilding the existing model

air-lift pump using a 2 inch diameter eduction pipe. By

rebuilding the pump, additional experiments could be carried

------·····----------------out using submergence ratios from 0.692 to 0.478 to determine~--

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if the present theories for calculating pump performance,

slip ratio, and friction factor were still valid or invalid

for a pump t,.,ice the size of the original model. The ·

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numerical procedur~ for calcul~tfng pump performance -in

deep submergence.applications •. Under these circumstances,

it becomes nec~_ssary to account for, the change· in volume .

of the expanding air inside the eduction pipe. A comput~t'

program has been generated which performs the required '\ . . ,

·numerical integration while maintaining fixed values of the

slip ratio and friction factor as a first approximation •

Output from.the computer program is compared against avail

·able test data from previous authors.

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. -

• . .!,_.

. The ~1odel . ~

• .1'

· -- J·A sketch of the model· air-lift pump t·ested appears .

in Figure (1). The inj.ect~r allows air ·to pass into the '

dis·char$?e pipe thro1-tgh .56-7 /32'' diameter holes. The L/D

ratio of the apparatus was constant at 83.593 whereas the

H/L ratio could be varied from Oto 0.692. A rounded

entrance cone and large cross section area ·exit· elbo,1 were

used to minimize the losses iri these areas. The various

two-phase flow regimes of bubble, slug, and froth were

observed throu~h a 32'' length of plexiglass pipe located

near the discharge end of the model. A Vol-0-Flo meter

with~ scale range of Oto 10 CPM was used to read air

flow downstream of the pressure regulator. Air flow rates

up to 33 CPM were recorded through the use ·of a pressure

gauge and control valve downstream of t~he. f lO\\T meter.

i;.water flow was measured with the use of flow nozzles placed

in the bottom of ·the orifice tank. Both the air and water

- ------- --···-·---· ------ ··- - flow measurements were estimated to have .! 2% accuracy. .

1,

. .· 1-·_.;

'

f .,

.....

•,

. . .

' . i - --··· . -·- ---- . . ~

. I i f

.. I • f

l

. r ' (

. ' I l I

I I' t I 1·

. ,1 i f.

[ )"

i/ ·1· j.• :~:

. ,,

!I l1 .. r.::~··,

'/ti1 ,i1.1,,i l,l.11. wt 'I': t.::.-.

..

I

. , •·· ..

... ··. - - .-~~-· ... -·._._ .·.·.-~···-_ ·.,~.·c.:_.~. _ . .,,...-_· ___ .. ;-_,.,.;-._~----·----,......-;·-.

: . :·· --

.3. -.-,.--.CC_.C ,-C·.-_. .. ~.-·-~ ,0----·--- .... C-.--,-• .---• .-- ---,•-- •.. -- .... -..-.--".

\fllen. attempting to pre;_dict the_ pecf ormance of an -· . air-i'if t pump for suhmer~enc~epths exceeding 30 ft.,

-. .

..

--- --'·- -- . ' ..

.... '-, i- .

..J.

'I

- . ', .

-it.-be,comes pecessary to account for the variation in gas volume throughout the pumr,. Since isothermal flow is assumed to exist, the volume of,gas-in any given length of eduction pipehcan.he calculat~d providing the average pressure in that length is known. This section presents

.~ ... , .. _ .. -•~.-~ ... -.~-the derivation of ·a differential .. pressure equation ,1hich -can be used for analyzing the ope.ration of_ an air-lift pump through the.use of a step-by-step numerical computation •

)

Average pressures and pressure drops are calculated for any given finite length provided certain boundary conditions are satisfied.

•

_ 3.1 Derivation of the Differential Pre~sure Egu~.tion As.suming ComRressible Gas Flow in th~ Eduction Pipe The mathematical formu·lation of the conservation of -

· linear momentum for a material volume can be written as . ···------~----- - -- ~

- -----~

·· ·c1> where e_ is the fluid density, y the fluid velocity, and 'f the material volume being analyzed. The forces acting on ,

. '

,, •

1:1 1' •. ,I I

. ./': .

- .. %

-· ' .

,

·"

1'. .... . . ~--:-·· -\; ..

.,. ··- ., -·. ~ - . "

r

. , ........ _{:

. a'·,··,

. '·: :,, .•.. , .. --_,.J;- . ."., .• _.:. ,, .. p;.;,,_.,,: •.. .: •.•• _ ..... .. ":';,.

. . .- .

. . .

. ~the~-m-aterial vol'ttine can be cfividecl into t~o .gr:oups' ·the . -• ____ :~~--·-- . ~--·-'--:_.'.:.·'.·_,. --~.·: ........ ---· ~·- _: -·- .. .

... • . ,. • •. ,... . .. .. .. ...... c .. :-: .... ~,•-·->-··-·•· I r ., .... ,·>-•·•·-,

-surf ace forces .arid-- the body forces.· After making. -this . . .

6 . .

distinction_ between forces ~nd applying the Reynolds· .. · • • r-'- ·••••• ; -··•·•

Transport theorem to the left hand side, of (1) ,. the.

'principle of conservation of li-near momentum, when applied

to a control volume,·can be written [3] . _,,

(2) s s ~

,1here, f ·~denotes· the extraneous force per unit n_i3.Ss o't the -fluid, 1.., the wall shear stress, arid 1' the stress at the ·

fluid surface. A and A represent the pipe cross section s ,) p

area an<l wall shear stress area respectively.

In order to continue the· analysis, the following

assumptions are made: (1) steady flo\", (2)· one dimension-·

al flo\\1', and (3) changes in the linear momentu,n of thP

fluid are neglected. l\.pplying these assumptions to (2)

and substituting P the fluid static pressure for'\, the

momentum equation for a differential volume becomes

p~ + ~tegrating (3) yields

. . ~ ·, -

.•-

-s-• \ l , ·,

(4)

.

··-

/\

! .

. . • • l !

I !

'• j

•••. ('"' ·r., .......

• I ~ ·---------·--------· . -~

• • -• < r. ;·- ..• ,,---. ····,-•.·-••--;-· --:.--· ,~,_-.... ---·

0~ .' _,.,

..... _,_ ______ ,------- --

. \ ..

;.. .... .I . ,_

. . a

... •, . \

··":" '.

. . ' '••'-•·' -.•-,•-~•-•-'•·+ • ... ,.: .,__.;L,>•'•o''<.~-••,•-• •, .... ,.<oe,- ••--'~.---'+r •.. _, .. ,., .. _,_,,: •· •• ,_~,·---r-•,"C.,,;.-·, ~--· •,•,,;,,•'~.--.'.•-•-•'-

. . .

~ . .

·-~----·-·····--······. ··---···•>-"'•"'" --,·---···--·. . . ...... , wner·e· b i-s- tlie~ \,ret-tecr~,,-e-rimeterof ___ .the~--pipe~--and·-aw~·tne·-- -- ---_~--'. -" -- __ ·-_-- -- -·-

·-·' - . '--~---~-----~------....,.,...·-------·--··-·· ---·--- ---··-·-······ .

weight of the fluid in the differential volume~

The weight of the mixture in the pipe is the sum of

the weight of the gas a~~ the liquid~ . . ~

•I,(' dw :: dL °i ~~ Ai + {~ A.~ ·<s>·

~here A is the effective flow area of the gas, Af is the g

effective flow area of the liquid, and e~ the density of •

the gas. ~ince the gas and liquid do not rise with the

same velocity, the ratio of the average gas velocity Vg,

l

to the average liquid velocity Vf, is called the sl,lp ratio.

Also

• .i

(6)

.I

(7)

(8)

' \

•,

where Qf is the liquid volume flow rate and Qg the gas

volume flow rate. Suhstituting (6), (7), and (8) into ... . . ' -- ---·-- -- __________ ,.. ·-

(5) and neglecting the gas density in comparison with the

liquid density

_,

-9-

_J

_, ,,., . . . -- . ·-· -·---. ---· ·--··-· :···-·--- ·- - - . ·: . ... - -----. ·--"'----. . ... ---·-·

l --·-·-·------··---------·---------- ----·-·. -- ··---------- .. ·-- - - -··-··-·-------------~------ --··-··-·· . .

' .

.. . . : .. _ t, ..

. ... :·· - ,. . ~ ,.,

·-.;-. •'

. :_ :.,,:i:I · ... : i '~.

. '··. ~- •... • ·•. -- .•• ·'1- • .. ..

' .... ,' ·"';"

·@ .. ··

. .. . ...... -~,. . . . ~ \.·, .. ,.

"'~. ,.,._ .. l ·, . . ' -\~f .i;. . . . . . -~~ .

' . . ~ - . . ..... •, ':->. --~ ,; ' -. : ... ~

. . .. ··.< I

,.-_ . \'.-:~.,,' ·. ~ . .

·. · .... : ,-

, . . .. • . r,,:. ,·· '

- -~---~ ---r~ --~ ------- - ---- ------ -- -----:-i- - • --- - - - --- - -- -- -·- -

' . - --- --- ·· · --- --- -~ -- --Jw ~, dl e~ ~ . . ' .

1'.. -

~,.

. cs_·_·_·· __ ·· ~\

. .

•--~---- --· .

I ..

11- ~:.().\. .. ( 9)

··,_ . v~ where S is the slip ra t10 V+ •

· Following the analysis of [2] and [ 4] aqd assuming ·

the wall shear stress to be a function of the shear stress

for a single phase flow of liquid, tQe recommended expression

f o_r ~ becomes

+ e - .. ~ 'rw- -. 2~c. (10)

where f is a friction factor which can be evaluated using

the 1',loody Chart or equi val en t information •

With the aid of equations (9) and (10), equation

(4) can be rearranged and rewritten to yield an expression

for the pressure gradient at any point in the upper portion

of the pump

= ~ ~"'" + 4+ '4 v/· \ + o~ ~c. \+ ~ 1) 2!h. Q~

~Q~

dP --d\_ (11)

Examination of (11) shows that the pressure gradient can

be evaluated providing the slip ratio S, friction factor f, - ·

and gas volume flow rate are known at any point for a given

liquid flow rate.

,,

-10-

\

- . . '"" .'' .

', ..

-,

..

.' )

... , ...

---_.,.~----- ----- _ _. __ :._ ___ :;~:;;;:;:-~-=-= \.-.-. ~-:-:. -----· --.. ·->-:---· --·-- ----------·-------"~ - _"··- -·:_ ---~· -- . - ·:-- .----------·--- . ~-- . --· --- ---- _--- ---···-------·-·--·--·----·--·---·- ---~----- - - '-·- - --- --- --·--· -- ... __ : . ---- ·-- -----· --- -- --- --- -----··· -···--., ··- ----. -·---- - . ----- .. ---- --·----·--- .. -·--·----------- ----------·-------·------ ·--------------------------- ·------- ----- ------ -- -- -----· -----·--------

. \ t '. ' ' • . ~ • I

t·· l

l

_; _,__ .

fD .... _ ...

•

: .·. '

... -... , .

. \ ,- ..

--1.

_.,.;,_;. .

. ·• ~--

• _P

' . . /

---L-·•·•.3e2 · · Methoa of us~ing Oiff erential Pres.sure, Gradient c81fuat;1-on ~~- ~:::-.<-in ,a Nttpi~rical= Ahalysis

The analysis of Ste~~ing lind Martin [2] ~ showed that

. : the :pressure downstream of the _injector has. tl1e f_9rm

I _ 4~ VoQ~o

~ C. 'ti CI (12).

Although eqt1ation (12) was derived assuming incompressible

flow, it is sti-11 sui tahle for use _in a compressible gas ' aµalysis since the pressure dt6p across a n6rmal injectcit -

. ' is small compared to the pump inlet pressure. Thus, for

assumed gas and liquid flow rates at the pump inlet, the~ ./

_pressure and pressure gradient can be calculated at the

injector discharge provided the slip ratio Sand friction ..

factor fare known at this point.

As a first approximation, constant values of Sand f

can be assumed to exist at all points along the eduction

pipe. Appropriate S and. f values can be obtained for any

given suhmergence ratio and eduction pipe diameter by

extrapolating the combined results of .Martin [1] and the

experimental data contained in tl1is thesis. -

I '

Once the values of S an<l f hav·e been determined, the

-pressure gradient at the injector dischar~e and the absolute

pressure at some finite distance above the injector can be

-11- ·.;

·, .. ~ .

------=-.-·~, ... -~-~~- --- -- ,--~ ·-- ----·, ·--

I '•

-

-----------------------------~~-··. ~1

----:-------------------1!111_ !l!!!l[l! __ ~_ !!!l!IJ!I_ .... _._··--~-~~---"'---·-·--·--, ·-·--··---...• .,. ·-·-·-··--·-~---.. --•---~~~~~~~~~===!!= = .------- - ···---,-,.--:"----~---·---..:;_:_ ·-- .C-.- •• .- •.

- • - -· -- --- ··-. ..:.._· ___ -- ••• -. • ....c --- -- - -·- ·- ---- ------~- ------,----·--~- ' -----~ ------·-------·----·- ... ---- --------·-·------------·-----· - --- ---- ----·-··-·.!_ _________ ~----- ___ .:_ _____ _.,;_~----~--. -·------------- ---· . · _____ . ---·-·--· -·--------------~---------·-·- -·-------·.---·------·----------·--------------- -- -- ------· ·---------~- . -- -·- ---···--··· ·--- ----- --------------- -----·------- ------· -- - -······-----·-··· - -

. · .. ' ~

. '

. ·.· .

. . ~ . -· - - ..

. ,i -.

. . - . -' . . .

. '. . ' • - (j. ___ · '. - .. _ ----··------------- ______________ ;......_ _____ .:_.__ ____ .:..... _____ :.:..,...:__~-.'..----. ...:.....---------.-

- --'- ----- •••- ·-==· •• --~=----.~-.-~:-e-alcltl-a-ted ti-sing• -an·· ·.Bu-i.-er.~·c:au chy- me tli-od· of•- riume·:r-i-c-al • .. , .. C •••• ·'-· ::~.:,~··-c·•-c--,•'c •- .· , ,, --:• • -• • . - . .

··integration [s] • The gas volume flow r-ate at the first

. . : - . . . . . -point Q~ is simply the . flow rate injeccted in to the bottom -~ a .

of the pump multiplied by the pressure ratio. between the -~.-·--~---·.--,- ., , .- ,-·-····~--~---·--. - ~----'-- -~ - .. -· --- ----····-·-------'e-·~------__:____,.: --------------·, .

- # \

·~ - --- -------· - ... _--·--- .. -~- .

,-~,.; ~., ..__.,

•

·inject6r.discharge and the first point. Og~ then becomes

the proper_.value _to suhstitute into (11) when calculating

the pressure gradient at the first point. r~

The

for any

step-by-step ~

finite. numbe.r ' ' "l.

integration procedure can be repeated

of steps n along th·e complete 1·ength . .

of the eduction pipe.. At the p11mp exit, the calcul,a ted

discharge pressure shbuld equal atmospheric pressure

plus a pressure drop correction for the overall change in

momentum of the fluid. In mathematical terms

wher~

. - - -- -

and

:..

I ,--

V. - 4{ ~~"";- Q+) ~- ~ oi

•

Vo = 4( Q~o+ ~)

-tr O l

•

-12-

'·

(13)

(14)

., -·,t.

.. 1 ·1 --~ . \

•.

-------- a.--'~

.. -~---·-·~-~. ···----- '":- -:·-----~~------------ _ :_._., ,·--:~----- -- - ": .

, . . ' . . -- ·-·----- -·- -·----,- ------------··-·:---..:.c...:: .. ....:_· __ :-····----·----- -~--;-'·_---···· ·- ·--·-····--···---------------~----:----:------:. _______ :: __ . _ _:_~-----·-·_---- .......!..----~--'------- -- - .---- ------------ - -------·----·. ------~---· --·------------·--------··-----··-·---=-··------·-···-·--·--·--· -- ·•·---· ·----·-··----------,-----·--·--·--·-··-·····•··-•·•···--• ----•--· - -·-··-- ........ ···--- -----· ----·--a··--·--··-·------- - --- -- .. • ..

. -~--.

;,,;,

- · · · · · .. If· the ca\culated · f .:illai-discharge Pressure does not ;~tj_;fy .· ·• ' · ··-···.·· l ..

. '. ·.- - . ,, ' .

~' \....L;-

. . . (13), the complete step-by-step calculation is repeated.'.

using a different -value of Qf• ·: The Ilew value Of Qf is a -·. :·· . - ·. .

function _·of .. the magni tud~ and algebrai·c value·--of ... PERROR . -- ,. "-~-----_-. -· ---- ··---···-·--·-· - ----,,-- -~.- - .-- .~ ,.· .,.._-.., .---·-- ~--~~-'"--'-'

· · Where PBRROR is defined· as ;,

P •• ,o~ = P"" - PA'f"" - t:. P"' ~ (15)

A continuous curve of liquid pumped Qf vs. free air

~~ed_Q~~can be generated by. satisfying equation (13) .for.·. . .

the pump inlet. J

. . ·' ·_I

- ··~"'·v--= •.•;_ .. . ~ .. ,;

., .. ·,-'t ... '.:,~ ~-1:·.'.·

·_ .. {,· . ' ...

.. .. ,.

any number or' Q{ and Qg0

combinations at.

··Appropriate initial values of Qf and Qg0

can be-obtained --- - r "·" ••' -· ·---~~,

by .solving.the Stennin~ and Martin flow equation for

selected value.s of Q /Qf ratios •. The submergence ratio, g •

slip ratio, .friction factor, and pipe diameter to be used

in the compressible analysis should also be· used when solving

for the initial Of and Qg0

values.

Pigu~e (4) shows the comparison between a compressible

and inco~pressible analysis for a hypothetical pump operating I .

with a submergence of 150 ft. Figures (10-16) show results

f·rom the compressible and incompressible theories along

with Warc1 and Kessler [s] and P. Pickert [6] test data.

Tl,e effect of accot1nting for changes in the gas density ; '.

while maintaining fixed values of slip and friction factor

-13-..

. --:r .. -.

I '•

!--~~-. ' .. ·,______;____ .·----------~- . t --. ·-·----- -. . . -.---~' ·, ·,---.-·-~--. ··-~--~· . . . '-..• .__c...· _. ----

1... -- ---- · .. -- . __ -- . - . -- -•---. • - ---- ---- .. - - ·- - -··-· --- . --· ·- ---- •

. .

. . . . ·: -- . ·.

- ...• t!: ·,

-·--·-~·---····-·,-----~--·······---·--- -~---··~----- -·~~~--~-< -.. __ . ~- - ·-~ . . . .

. . ' . . - - . . . . .

- ' . . . ~ -- . -- . . . ... -~.~ . . ~ -.. - . -- -~. -~. ---·--··.----·--·- ---------- ·------------·-.-------·-----;:------

. -- . -- . ·- ---- ~...:. ,.-~,. ·. ,j:·-· :

· ..... . . . .

. ' ,:·· . '. ;, ·-:,·.

- --- OOes nC>t appear to yield results \1hich differ. ·signifiC8,11!~Y - .·. ·.-· . . - .. ""·· p~· -- -;'-- ' ... . . . . .

······-··---- --- - '--·-···-----··-· -·-·

-----/·.

..

' .

-- -- ···-·- - - -·----- ---- --

from tl,ose of the 1·ncompressible a.pproach. I . .,

All of the calcula"tions for·· the compressible gas

anafysis \'1ere. performed on an IB1vl 360/30~_digi tal computer. ,._ . . I --- J

'The, program was ,irri t ten in FoXra~ IV l~~uage and run /----~ .

.:using o.o.s. The computer program and a sample calculation· . ~

· using F. Pickert field data [6] are included in the Appendix •

3.3 Application, of th.e .Incom2ressible Flo1vv1. Equation for • C '

Calcula:ti ng .. ~l~ximum T}l.e.or.eti cal .Is.otl1e,rr.1al ~f f i ci~ncy

The efficiency of an air-lift pump is defined as the

ratio of the useful '"ork done on t11e \tater, VIL, to the energy .

available as. a result of the isothermal expansion of the .

gas from the injection pressure down to atmospheric pressure,

B • g

(16)

Bxa.mina tion of (16) sho,-1s that for a given set of opera ting

conditions, the highest efficiency is obtained \'lhen the

-· . . ·- ---- ---

, -~--------·---- .

... ~. ~-------- -· ____ ratio of Qf /Q ,.,"'\ is a maximum. -- - ---· ---·-·-------- - __ ,

J .

j . L...

(I\ • I .,

I "

. (

..

The flow equation for incompressible flow (17) can be

used to predict the perf orman,cc of an air-lift pump [2] •

•

(17)

·- ,

-14-

Q'· '

- . .'. _. ___ ... · ---·. - ·---' . . ~

.• .

. . --· ~---- ---,- ~ . - --:.,- -_- . -- -- - : ------ ,--· ________ :"" ---·--:;.---·-------=:;----~---------·----·-:----:- -----·-:----------~ -· ----- ------- -·---;----

·····-···--.~ --~---- : __ · ,,,: __

... : . . ;· ----- ·- : ... _ ·-· ..

r .,

~ -' . . '~ .. : -

. ; . . .

. . .

. !!'. . .... ·-~ .... -· .. ,: -. - .. --- --··.:

,

.. .' ·, ..

r. ·•: • . .

, : ... .-"where K.~.is,. called .. tt1·e friction coefficient .. · .. ., ··---- ------ ----- ... _ - ... -~------ ,. ___ .. ------· - ----·-· - ... - _, -- ---- · .. -·

K = 4.t\.. A 0

• , • : I

.. I ':

.,,, · ·An analysis of equ,ation (1.7)

,,

shows that the theory .predicts .:.,

-maximum-Qg/9r and hence maximum efficiencies at zero .

. liquid flO\\f rate. This is not in a·greement with experiment~l.

results, which show a ma.ximum Qg/Qf at .;_ finite flow rate

Figure (3). The analysis does not predict a meaningful

m·aximum efficiency point and. is .def ic~ent iri this respect. . . .

It- cottld nott'· be .expected to give good agreement wi tla

experimental ·data at low air and water flow rates.

:-; . /I'

:~

/

.. , .

•. .,

.•-_:,

-r---i

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

•

,,\

' I

-15-*'

...

·1.:

:;• :, .f.

; _ C • ,;, ; ~\<,.f ·.!!~

I ~ .;,.-· ·: . ...:. . . .·. . . ' --- ___ ,.,,..,.,_,,_..,.. _ _,, ___ ,-........, .. ~--·-·--. -.~..,_...--~--· · ........ _, ___ _._. --·· . ·.· ' ·,.-·' ',·: - ·. -. . : .. .. ·.··········-···-·-··-· -- .. , ,·.~.-, .... ·. -;. ----~·- ~.-......- - .... ~ ........ · -----·---.·-· ... .

-·.:._-;-... ---- - - ----~ -- ----- --------------~ - --....-- ----------··------·--····-·-·--·---- ·:::__ ______ _ --------------_ - ---------- . - ~----- - ----·--·--·----------------·-,_ ------------------~--- ----- - _.· ........ _. . . ' - ; ...... .

. . : C ·--·'-··· • • -- - - - - • - --- • . ···--------·-- . . ... - c:.·c..;cc---- - ·-;;-

' .- ·-, .. ·.

'<- -· ·, ... '., -, ' ·,··.,:--:. - f

. . •'"- . . . - >:_r·:;-· -_ -.: _-- - . '.', ., .. •. '

' . • __ ...• ,.,••··,,._:••j••·:;•••'-'•AO,>·,•,.c.,.

_,. ----- ---------····-·---·-----· -·--- --- .. ,. -."--.- :.... ---- .. .:.. ... --____ : ... ·'.--·-·· -·---·---· ··-~,·.-.·- : --- ' -,--··· - - :..... ____ --: - ----··· .. ____ - -·- . -- - ---·--·--- ·------------ -- -

- .

· ..

I -

. 4.1 T],e -·Experiments _ · ~ '

The results of_ the experimen-ts are presented graphically

in Figures. (2-9). Pumping experiments \iere carried out at

air injection rates ranging from 0-32 ft. 3/min. of free

·air. For each value of tl,e submergence ·ratio H/L, as the

rate of air injection was increased, the flow of water also

increased until a poi~t of maximum water flO\\f \'las attained.

The point of maximum water flow \\fas more clearly defined

at higher values of the submergence ratio.

As in the case with a 1 '' diameter eduction pipe, the

two-pl1ase flow regimes of bubble, slug, and froth flo\1

developed in the pump discharge line [1]. During the

experiment it was noted that the emergence of the various

flo\'I regimes occurred in a sequence consisting of the

following parts:

(1) A regime in which a small amount of air

could be bubbled through the pump with

- ' ------ . --- - _ .... , .. - ·-·. ~---- -- - - --· -· - -- --- -

no resulting liquid flow.

(2)

•·

A regime in which the air injection rate

was 10\.\1 and there \'las a small flo,-1 of

liquid. The air rose through the liquid in

the form of large slugs.

. .. :.;,, .....

•16-·

--·· ------~ -- - - -·-· . . . . . -- . --.) ..... ,. ........... .

/

~--f .

i

~ - - ---~------~-==------·-----'-'-== -----:---:-~---~-~-=--=---~----=---=-·-=-----=-=~--=··=·-=--=-=="=· ====

• <:::, -

I ; -

~ . ~----- -- ---~ ----------- ------ ---- -. ' -~-.·_··->--~-~-··-- -~:· __ :· ·: - - -- ~ - - - - - - - -- - - - _________ '.'. __ .· - .. ·:. ·:· --.-··_·_' '·, _________ ----------·----- ---- -~-~-~-------

_,. ' • ~ ... __ _,,_::___', --- ·1 - J •• - - •• ......... __ :· •• -.:·•,"-·-----~··-'··:._~:_._,. __ , ·-···,--:~·,.---, ••• •.:-:.,.'--·

. . . · ··- ... : .. ~r.:-.::c.·' ... ?~:-~~=~--:-:_;---·~~---~--- __ (3) 'A regime .in \\Thic'h the water···-- f lO\i" in .. -

- I - - ·-·' '• .--·-, e' <. .: • ••• ;_- ,.'._,_,l,_'_, -•·· ."

-- '---,- . ,---· -

'.

. c-

CD

.... ;.... -

.,o-·_1,1.·,,._-· _·. -. ,\ ..

............

•.~·,, - .,. -•• I· •',L"" '.,•_:,._•,,.,> 1 ••••- -,• .":\. ,- _:_

(4)

..

creased rapidly for small changes in ait .·. .

f 10,i rate •. -The slugs of air ,,,~;e greatly

redttced in -size and the flo,1 became a _)

·~ J .,

combination of slug _and annular with

·increased leakage along the inside diameter.

of tl1e pipe. \

The pump experienced smooth

operatidn during· this regime \'ll1ich lasted ..

almost to the -point of - • liquid gow. max1mum

A high air f lov, regime containing the

point of maximum water fl0\'1. The flow

pattern in this • froth and regime was

difficult to observe.

·, .

' .. . ':...

···-

, Figure (2) is a graph of tl1e test curves for submergence

ratio values of .692, .619, .543, and .478. Submergence

ratios·greater than .692 or less than .478 could not be

obtained due .) tl1e physical limitations of tl1e pump. The

general shape of the test curves or1 both sides of tl1e

maximum point is in ap,reeme11 t '"i th results produced by

previous experiments [1] [6] .

1.

. '

Figt1re (3) displays tl1e test data in tl1e dimensionless

form suggested hy the Stenn:ing and Martin flow equation (17). ·

When presented in this form, the initial portion of each

-11-.. ;

.,

·:- ·, ~ < -, · .. .-.,· ~--_. .. ·-···--·--·- ·- -. ·--·····-·····-·-·-

. . - =· ===----------- -· --- ------··---·---·----------·-- .. -=---------· -------·-----···---·------------~----------- - ---- li . - --- --- - ----:··-,-=-~- . . -"· _: - --: -- ; . -·.- ~ -. -.-· ---- -- ... -- ·----- -. -- -------------- --·- . - ------ ------ ----------- ----------------- ----_,--· - . -· ---- ----- ·-------------- -- -- -- ···--· - ----·-·'·----•-- --- -·----··-· ----'-·- ----- . -c-\ ---···--·· ·-·--.-·--·---·---·-----·--------·--··-·-·-·--- ·-------·-· --- --·------·--·-:··---- ···-·-··----·--- --··--··--·--·-··---------·-··-- ·---'··· --·-···- -·· -- ... ,.. . . ,-{

·} . ~

. li - f

111

. .r

. -· .. ···~< . . I . I . • .

. .. ,.·

·,.

....... ----·-·---·----·-- ~ --------- . · __ · __ · .. . _: .. :.._. ' --:--:-----.---

. ' .' ") . ---~--~. --· .. - ,·-

'. ' ,• .·:

~ r

~~- ·.- . .-.: .. ·.··,.c.-,:.. · .. -~ . . ' . - ., .. ·--"------.--,~----,~------:-'··--·-- '1·'

. . . .. ." .- .·· ' . . . . .- ~:/: - .-. ;,.,.,. ... ..,.· -· .. ·- : ... -• .. • • •••" .... • • ._ ••• i •. <-•

0

• Ao-,·_,._ .. • • • • /. ii• f \'

t I' i

. ' ll

. .. . · · cut".v_e __ t_ends to have a .negat;i.ve.sloi,e •. _T.hiS indicates tha.t-.. __ :· ---~----c· . I --- ----· --- -:,-----·-_-/-,·-·- ---; -. / h - --.: -: -·t

i"

. ~- ;--· -·

a :. ,

.Y

· the efficiency of_ the pump increases sharply to a _maximum .. f I

. . . and·th~n decreases at a more· gradual tate.

Figure (3) also indicates that for a give~ set of __../

.operating conditions, a theoretical analysis taking irito

acco11nt the com_p.ressi bili ty of the gas/Should al,..,ays pro~uce

a lO\'t'er peak liquid flow rate than th.e incompressible theory. "---- . . ~ . .

.---·~ressibiU, ty causes the bQttom of the pump to operate ~ '

at. ],.ow Qg/Qf r_atios and the upper portion to operate at

high Q /Qf ratios. The resulting mean value of Vo· will g "e~\..

always be lower than that predicted by the incompressible

flow ~quation (17).

4.2

'

Comparison of Incompressible Theory and Experin1ent

Figure (5) sho,is a direct comparison bet\\1een the test

data taken using a two inch eduction pipe and theoretical

pe··rformance curves drawn using the flow equation (17).

The values of S- and K used in drawing the theoretical

curves were calculated using the actual test data and the

Stenning and Martin max~mum flow equations. This method

yields Sand K val.ues which theoretically should only be

used for predicting the maximum point of the test curve.

However, as Figure ·(5) clearly illustrates, the specific I

-18·

!'

.I ,.. - . ~

. ., .,

·\.o. ..••

-~---------------· -·--- --- - ·- . . .

---- -.: -- ___ ___.:___; __ _ . .

-----'-- ----~---· __ . -~.------~~------~--~, --~-~-~-~----~r--· --------~--~~---·---~ ----~-~--~----••• ----~--~---- _·. - ••. , -.-_- _------- .------ .. --~----. - -------------------- _____ '_- ___ ---- --- - -----·--------··· --·--- .-.- ••• • --·----~ .• --0· ----·· ••.• - ---.--- - ----- - .... --,- -=- - ,. - -- --- . ---,--- - ------------

; . '

. _"I. _/, ·- ..

. . ..... --- .. ----._.

- . ., ' . .

. . . - . ' ~- ~-(' ···---~---· '• : . . --- a,-·--'..,.. ___ · - _- . -- . . -- . ---~---_ -.-----------------------~~- -- -- --- - -~-~-- - - -- -- :_' -. - - - -" -~- -

- . ~ - .- -.- __ _. .... ,•', - •• • -· ~~~~----- -=?--~-'----- - =----,----·-- -~--~.,.-~-~=..,.,,..,._,.=_..,,,,,.._...,.._~--=~--~---- .• .. ...,.,,~~~---··. . -- . -~--=,,., .. =-.#:- - .-·- ·- ···--·-- -----·---=--·--·-- '=-1==-rn=. ~--- :=-:"?7~~==~-=---.-- ··-- -- ----- -..

- ,__-__.__.._. .. ______________ _

•

accurat~ly predicts the complete shape and range of the

pump performance cur·ve in shallow submergence applications.

Deviation betweert,.:theory ·and· exper·i·nient is less than 1% . ' '

• ' - J_ --- -·· - - - - ........ -,- • -·· -··· . - -·- .. - - ... __ ,.

~ .. . -/--· ·-'--··- ---- ~-----··--·· ··- ~ ·------- -····--······ ···- - -·--·-·-·- ··-·----·'-···-- .. ·----¥--. -J·-·-·-·----~---·---~~"---J-.--.-~-'"---~~ ------~----·---'~~--~.-------~-----~------'-------,.......:...----~------

---------~~~-

0-..... r.__r;

~,,.o_ ·_- ·-.~

<..!., . I

._.,/

for all value~ of tl1e submergence,-ratio H/L. · . .

' .

As the air flow is reduced belo,i that req~~red for

maximum liquid flow, the efficiency of the ,pump increases

until a point with infinite slope is reach~d. Figu.re (6) • I

;. ' _; __ -.-, .. ,.'._ ...... - - . -·--...... ··---·---·--·--· ~-~----

"I;·

shows ~he maximum ~ff iciency ohtaii:ied qt1ring th~ experimen_t

for each of the foltr H/L settings~ For comparison purposes,

a second ·c11rve ·shov,ing the efficiency at the maximum· f 10,-1

point is als·o included. The '"ide variation in efficiency

between the two points is clearly noted.

Curves showing the maximum theoretical and experiment

al efficiencies for one and two inch diameter eduction

pipes are illustrated in Figure (7). The theoretical

efficiency points for each curve were calculated using the

flow equation (17) and Sand K test values for each sub

mergence ratio. ft.taximum efficiencies on the order of 53-

57% are predicted by tl1e· present' theory. Agreement between

the theoretical and experimental optimum conditions was

good for low val11es of H/L and poor for high values of H/L. 1

---Th is indicates equation (17) usin~ constant values of S

·.•

..

'

..

... -~ .

___ I ~ -- L ~ -~- ---~r-'---~ -~----- -~=•=--'~-~ .. . . ... ..... -~=-~---~-=···-····-.-~- -········--··--·-·-"·--r ~-~- -~~,.----~~-~:--:-····-···--···· ·'..:.:···· ... ·•

.· - . · .• ;.,.,,-,.y..·

' :· ,-- .. · .' -~./ . - .

.· . -: . l ,,,.,--•. , . : • . . : . . . '. . ..

,\,j·' J.,,, ·J... . ] . I :y;,£:' :';-c . C ••• _d _c..c~ ~ -~ .. :, . -· :~ ... :: - -~- -.--;....:..· __ ._ --··--·· .. ·---·· .. ~-:-... ~·-, .. ,_:.:.,..~ - .. ,: .. :.:.,.~,.,_ .. ·.~ -----~~---~-:,.. ..... ·-- .. .

' - -··· --- - ' - :.,'_ -~- ----·- ----- . --- .

---·-··--·-·---------- ___ ~~-· __ and K--Wi-1-1-----nGt----pr-edi-ct·--th-e·---~i:n ft·i-a.1···- po·r·t-i'on ·--of·--th·e-~ ac t·u-al _· . ·-- -<-· . -~~=~-------~- ~--------~-·./'

...... ··- ··--·--2·:--··c·· ~--·-· '··---···-···. .. . .. • .

perf orrnance curve or its maximum efficiency in this area,

. ----------~--~- -...... ____ u,.

. .

------ ···-----------~

particularily for submergence ratios above 50%. Since the ; \ '

flow is very unsteady along the initial portion of the • . . • .ff'J

curves~- it is possible ~hat agreement between theoretical

and experimen t_al op'timum conditions could be achieved if

refinements are made to the air and .liquid flow measuring ·

systems. ----- . _ __: ·--~--- - ----- --

The present theory and the test d~ t~,- ~<?th show

-· - -·---- ----r----··-·

a decrease in optimum efficiency ·as the ~duction.pipe

diameter increases.

The value off yielded hy the maximized equations

does not agree '"i th tl,at derived from '"all shear stress ~

calculations assuming a Single phase flow of water having~

a volume flow rate (Qf+Qg)• Figure (8) indicates errors

of 40% in t·his assumption. Since the resttl ts for the two

inch diameter eduction pipe show no.substantial improvement

over those reported by Martin [l] , it appears that a

• ~ -'.J

..

more basic approach than that of.a single phase approximation

must be used when calculating the friction factor.

Fip,t1re (9) illustrates the excellent agreement ob

tained ben1een the tDeoretical value of slip as calculated

by the Gri(fith equation [7] and that calculated from the

test data using the maximized equations. Deviations of less

. -

..

--20-

,

,: .. :,.:··

,.-

------- ---------t-~-------· -- ------------:--~·-_ -- ----- ------ -- ------------ - -------- ·- -- -----c..·.a.c:.-•-'------'----.- • -- - - " ---- -- -- - ~--------~-=-~~=:--_:_~~---:-:::-:-_-;-:-:::::_~ --=---==------ -----=---_:--~ ·- - -------- ----. ----- -----· ~ .:.'-·.

. .. '-._j . - _. __ -.. _ ·- _ ·; ~ _

- ·,. - - ~' - 1,;<- -~ ••

•a.,1 · .. :,· .• ·,_ . :

- ----,---, -----:-·- - --~-- - - --~----t - _.__'___-l-~-- . ... ,.

-~~----- '. : ;--~,. , - ,__ . ·--- - --- - .. _ ·-~-'.-: ... -,--·--........ -·--; - -' .-----'. - " ,. - . . . . - ·- ---- . -!-' -en·, I

• ~ •. .J . ,

, -

. ,-- :__ ·-- ::..;~ - - . ,...; . --- ,; ..

(~L:.-·j'_··. -· · .• __ -_._ ------ -- .. ·--· ... ~,.. . --- . -,~-~-- ~- ·: "" ~ ,.·. - -.. --_· ;-- .. ·-· ----- ·-' . ---- --·- ··:·-·-· . - ' • . • i . -~-- • ---- ' ••-" -·" ·-~·--·-••"--• ~~ .~._..__. ..... r~=-~•• - . . -· -~- ··"'= -r."-<=,- -- ·= ---- - -~=--·'-- =-.==··=·-s,=·~~-c·-::-·-''""'~~-""''

. .

...... '

. •'

. - - ..

. - than 5% betw(?~Il !h1

eQ;y a.rtCl e~per,iment were rec0-rded--f'~:r--al-l-· --~~-'---_ -. -~~--.. _··--~----~~a~e·_________ ... ····-·-···---- ---- -- -,---,

_· test points.

•.

I '

•

.. These results substantiate those reported by

Martin and:indicate the Griffith equation tO be a'Valuable ··, ·\.

tool for calculating the average or lotal value of the slip

4.3 Com arison of Com ressih1e Theor and

Data erimental

................ , .'. , , .. , .. ~ '"' , , , , ....... r

., ___ --- ----~-------~

.. The results of the ancl.lysis' t.i.1(-illg ,into account the.

· ... · . change in volume of the expanding air are shm'V'n in Figures j ' .

-.~,

(10-16). Output from the computer program described in

the Appendix.was compared against available data fr~

previous experimenters.

as follows: Input for the program w~.s obtained

(1) Slip ratios were derived by extrapolating

the combined data of ~Iartin for l'' ID I

eduction pipes and t~e experimental data

presented in this thesis for 2'' ID eduction

pipes. A straight line was drawn between

the two points and extended to include larger

diameters.

(2) Theoretical friction factors were evaluated

by substituting Qg and Qf f~~m the exper

imental maximum liquid flow points into

...

- -• - _. • ,__,:,,r.,.

';I.,,,. ,,

-. .. .. ---~- •• .. ····.,;:•••'! ,I

•. . ·-- . -. ,:~-.. ···-·-·

. . . . ., . . . .... - ··-·T- •-• -•a•-· ·------ -·· '' ,_. --- -- -- '··-·: ... ,,. ---- --:-··-·';.··

' ,' - . " . .. . ·;, .· '• • • . • -, •: . •- .. . ,':· .::~~-.-.--. • .• :.,,.-_~- '.=••,'.: ...... ~~. - ~. • ""-.· ... ,.m,"&-~n:e_,~-~-,,;...;,.::;·,•~~~-~~.,_;,.~_-..;,.'<=~~--~-,:.,~.T.-c=-• __ : __ . ,." =='=· - _f ·-. -==-~-----------::....~~~:- __ . --:-·=-~~=-==~~-. '~~~~ -:=.:=~ -~ .· ', ---0=:L.' . ·--=--~--_:

- .. ..:-.._

. . 'JJ ·- . . . ' ; .

" · -<· --~ - -~---.·_ · ~-::-~~--~_.___ --.. · ,--.-~--equation~~-(--1~8?·--.· ___ : .... . , ... ·: - . '.··'-· ~' ... :.;~ --- .:: .... - - . --. . ·----;- -· ___ ; -·--

. . r: -~--'-·-'-~-.. -~ ... -:. _··:. ----·~- .. _-· . -· _ ...... _. ~------···--·--- ·___ .. · ~-:_,__:__._;--~~-.:.-c·:_· -~-. .;.,._..· . -~~~~----...,--

' ·• . .' • '. .,:. l

--.,- ' ... ~- -- .. -~ .... ______ . -~,·---------·--··-·---,---.--

. '. ' . " . __ ,... ___ .__· _____ .____. ---· --- --- _· . , , ---- -- .... ·. --- .---;·_ :.- ' ·---· - -.. . • c, ·:, .. -· '-: - .- - ........... ,. ·.' .... ~ .: ----~~--- .: .... ~~- .. -~ .. ·. ',-.!.'.': -·c·~. ·· ___ -~,,·_ -·

I - . ' . . . . , .. •_ ...

- . .

. 0: '· t' .1-,, .

• I ---- -~- - ·---·- - - -·-····

. !' ' . ;_,;,

• t •

(18)

· The :theoretical values ~ere t~en ·multiplied' · · . . '. ,- . , ; .

_,.· ,. . .

' .

·by a·correc.tie>n f~ctor C?f ·1;.6. The correction :· -. - • - • .. .., •••• - - _, -.:c- -· ·• ··~

. ~ .. . .. .. _ ... _J .

factor ,,,as· derived from.· an examin.ation of the ' . ..

test data for ·1'' D and 2'' D eduction pipes. . ··-~ .'--"

(3) _Values_ of H/L, D, Og0 , and Of were obt_ain_e~---~--::--,---'---~-- . . . - ··- ----··-----·--- ··- .\.. .. ------ ···-··· -----.----·---------------------.............. -----r-~- . '> (;..,.L.__!. --'-'-' . ~·. ,..___~--- ,, ____ -- •

0~ -

· . from the rep<?r1;ed -r·e~11l ts of other experi~en t~ • . . -· ...

· ·.sttooi·er~-:e~ce depths gre_at~r than 160ft. were not considered

as the expansion ratios for the air became too large and

a slug flow regime could not be maintained over the ~ntire

eduction pipe lengt~.

Figures (10), (11?, and ( 12) compare output from the com-

puter program against experimental results reported by Ward

and Kessler [a] . The results sho\m are for a 2.5 inch

diameter eduction pipe \'Ii th s-t1bmergence depths ranging from

32 ft. to 46 ft. Excellent agreement was obtained over the

limited range of data presented.

.-- .. - ... ··-··-·----~-···. -· ------.. -.---- .. Figttres (13), (14), (15), and (16) compare the com-. .

OJ

pressil)le tl1eory rcst1l ts a~ai11st expcri1nental data rer,orted

by P. Pickert· [6] • The Pickert data was particularly

interesting since it v1as the only data found ,1hich presented

.. a complete ptunp performance ct1rve, including the maximt1m

-22-•

:i:

•

. r:

I}

(. . •.

;; . ~

' 0 .

-.--.- - ----:- .-.c.~-- .--, .-- ... • .. -_. - .. - ' --- -- -_ - .· - ... '.:""'' .. - .

_ I. ./ .

',: . : . . . . ' . - ; ,, ' . - .... . ....... · --,-~---,._,_ -.--· ---· .- ---~--- .'··_·:···:'·_--.-·~.:-"., ·,::= ·:. _·- .. :,..~~·~~--· ... ..,..: . .,., .. _ .,., . ~ ......... __ -· -___ ·· ... ----. . ·_· ,. :,;:·· :· . :. ' -. '

· .... J_ . :· .· ' ..

- ------- - liquid flo,\1 point. - The t'esul ts, shO\\'Il are for· ·a 3.937 inch _ --C.·--··--- --- ·- .. , ··: . .

from 62 ·ft. to 91 ft. Good ap,reement betw"een theory and .

experiment \'7as. found to ~,xist· up to the point .--of' maximum~ ~_, __ c ____ ·_ :

- ... ---~--------...,----1 i quid flo~. _ As the air v~lume· flow rate increased beyond

this-_ point, ho,'1'ever, large _deviations between .. tneory and

experiment were recorded. The errors occurred because the

·ori~inal assumptions· of constant slip and friction f actox:s, ~----·------ ------------ . -~· LL----............-':~-----~----------·

' . I . ' _;

. ' ....

~

particular_ly cons t-~n t s·li p factors , 1_ .. PrQ_Ye_g .. to_~ __ be i_nva.lid _ ........ ---------- ----,=:·~=~-~~~--~----

at·the larger gas flo,\1 r~ates~ Iri a11· cases, __ the assumption-----------------~-

of a· negligible mom en tt1m pres-s-u-r-e- -d-rop---,\J'a-s----a-cceptable-...... - - ---·- ---------- ·------·.

a

'

Calculated valu~s of A~ ,-rere al,'lays less than 1.5%~of the

overall pressure drop occurring in .·the pump.

4.4 Sum~ary \ .

The results of the experimental and theoretical work, "' as illustrated in P'igures (2-16), can be summarized as fallows:

(1) The ·\\TOrlc of Stenninp, and ~lartin has been

extended to incl tide 2'' diameter eduction

pipes in shallow submergence applications.

:a"'Ccellent agreement \·1as obtained bet\,,een

the performance predicted by the flow .

equation (17) and tl1at obtained from the ~ •

model air-lift plunp. sho\,rn in Figure (1).

',

I. '

• t,

-- . -- --"--~ ,._,. - . -.. -- ._ --" - --~-- -· -- - ... -- .

" . . . ' ' ·, -- •· ••• --·--- ··-- ·-- ---~---,-,-,.~-~---·~.,..._,,,,.. __ fC < ~....... .:cc: ... ·-----·-:-- c..-~---·· '' --.-. .;;-'..; .. :~~ ,.-::.:L2...:.~..:.C:;~ ·'

' . I·'.. •<•. ~ ,' . • • . - ,.._ . ~-. . ~. , • r ' ~ " •.. ;·· ·. , :." .: ·•·

·· .. ,· ..... -,

' ,, .. '

__ ,. '.

r ~:,- • ' ' • .· .• • ./ ~ ' ' ; .

..... ,,· ... ··.,.

. 41:i~f._ . / ,;,,l., ., .· ,. ~1.. ' ' .,

\~; . d1. . .· :;,-_ .

. ..... - -··,;. .. ~' .

.. ;, ..

.. -, .,_,. ' .. . . .. · -----.- .. --------._·_-----. . . . --·. ---.1.'"~'.-~-~~·--:~~,--.---:--,_-:-:·"·7·:...:...---·-:~- ----· .... --·.-·-'-'·----,- -.--- ~ """:'.;---:--· .. ----· ------- --· --· I"''' '.-,r;i--,-----:~--- c--,.-_-·-_, .

{)' . . . .

. - ,. . . '' _:..____ ______ ....;.__ '--~ ----- .~:1 .. .- .• '" .- .

. ·- .. I .

·' . . '

• • ! - --- ·- .•

- - -- - - -. . - .- - - -- . -- ,- - ·. - - - --- -- - - ---- --

· (2) . Accurate values of . sli_Q__J:~l'LJ?e_"O btaine.d~-• -. -~-__ -,--_.·, ____ . _ ~"· ·- ________ . ------~---- - - - --- ·- - ------ -- . - - - - - --·--.. ·········--·- .---· -----·--- . ----·------~·-·---...-:---------~-..-- . - ... , .....

··· -- .. --- -·

. . from theoretical calculations-. . I

(3) Use of single phase data to calculate a· . -

- .. --:- --- ~'--:. - ____ ....'l.. ·--~ -- - - •• -- ·-- -

- . -·-friction tactor;·y:i.eids ·results \\1h1ch do not . .. t..·

agree with experimental data~

,·: ... (4) · A theoreticai-arialysis taking into account 1

'

tl1e cha11ge in volume of the exp~nding air

~_:___:___~~_..;.;.;..:_~--~::_;·, -----'-----:-----· -:---.. M.-tt-uUroS~t~--a-1:s-t'.r pr OVid-e f Or 1 o-c-a1:--·· Ch art g es1n--t fie

........... - -. '·- -~

•'

slip ratio and friction fac:tor·~· . '

(5) · A compressible gas approach yi~lds no signi-·-··· --- - , .. ·- .

' J .

f'icari t improvement over the incompressible

analysis of Stenning and Martin unless point • . ...

values of slip and friction factor are ,1sed •

- .... , -~~- -·~- ·- ............ .

l''· .

. . - . •. -. . .. - ..

_ .. -- ''"""<---; . ··-- --·--a~·'"'.,-•----· .... -·•~ ..... ·-•,. .- •

·I

- ---··- -· ---.- -~ . ·-·.-·:-- -~---= .. ,_._ ... --

..

By combining the experimental results of Martin with those I

presented • this thesis, these additional observations be 1n can

made: ~J -,v -.. . /

-· (6) The efficiency o_f an air-lift pump de-'

. __,---. ·creases as the diameter of the eduction

• • p1pe increases. ·-- - --- - -- -- --. -------- -- . --~ --- ---- ---- --( 7 ) '

,~i th constant suhmergence ratio, tl1e slip

ratio increases as the eduction pipe ' ,·

. \,,

diameter increases.

(8) With constant submer8ence ratio, the friction

Q. .. ••

-24-J ~· ·.

'"-\~•-------. It. I ·.r » . ., .. .c- ·-----~

·-

(

--~~·-. - ·----·· -~--~-= . .=_ ---· ---··=· ·=·· ·=· =· ~· ----- -- ----

. (9)

·. - ··-- .. :.·

,. --~ --- -,;• ~

..

•

·.;.,

y. ,·; ..j ........... ...,.

•

L_. -=-=-=-=--_;=es

__ -_____ -~· - . ·- ------· .... -- ·---- ----·,-···--· ...

... ____ .~, ------ ... .~ . . .

--- . ·-~ . -·: :.:..

- . · ...... ' ; --.... •

factor deer-ease$ as 'the -eduction pipe. ---'---'-,----'-~-----··--------" .·-·-.· --·----~------···· - ---·------ --·- ----·-------·-····--''-'····~·-·--·-~----·-- ·-·--··---···----·--,·-,.·--~------~-- . --·------ -- . __ _____:__ -- - ----·

diameter increases •

For submergence. depths below 30. ft., the . --,

of constant s and K values • the use 1µ. . '<-'---"'-., I

fl0\\1 equation ~c 11) will result • '-.... j

1n an

accurate predictioh of the pump performance

characteristics.

~ .... ' .. - . :.,_'. ..• :.-,_ . . :.. .... -

·,',.

.,

..

~ .....

,· .;.

. . ·,

-2s ..

--------·.' - ,._ ___ . ___ -· - -------;,--·-,,----

,····

_ .

.. __ , ... --- --· ------.. -----· - . --· - -·---- -------·---·· - -·----- -------- ----. -- ------------ ---- -- -- ---- -.; _______ _li_ ____ -------------h~- y -,----~------------~---·------.-···------ --------~-----· - -------······----------------·- -·-···--------·------·------····- -·--·-·------ -- ·-·-------------------

. -.--~- -.;-· ... . ·-- -·--··· ----·--·- - ·-------- . - --- ---- - - -- - - - - - -- - -- .. ·----- ... - ---- . - ·-· ·----· - -

- . '. ,· . 1t . • . . J ._ ~. ·- .,'<.i'!'-·_-,t : . . .

·-~--' > ;-.· t . ·, -. '

. ~ . . ~-.

~----· .:. ... ----~---------.--···- ----:·-·-····.::·

....

.· ~ . . ;. :_·,

- •-.-- '": ___ ..,. __ ' .... - -- . - . -- - -----. ------- - --- -------~~--- --.-.-------- -,----- ----------·--··· ... ' ... ~ - .

· · S. -DISCUSSION - .. ,. .

. ~.t~ .. J. C

~'-----. . ' . - .. ___:__.......;._ ____ -,:........-_..,..:-__ ~-,----,..----:----,-~:----·-·----------

'. ' J - - ... -· - ---- - - ·-_:_ . .,_ --·-·--·-·- __ ·_._: ·- ·-·- -- -s.1 Conclttsions

. f - -- - - - -- - .. - -------- -- - - - -- - -~ . '

. -.··- --·-,·---·--- -~~--- - ___ ..,.._ ---- - -- -----:.~-- ---------- - - - - -

. '

The results- of tl,e air-lif-t pump experiments sub-

- ·. ·-··---·-·--- ~---- --·-:·-,----- ~-•,--- ----stantiate the fact that the ·flo~ equation (17), used in

- .,.~... . ......

conjunction with constant slip an~ frict_ion factors, can ·be-,"',.·. "+

' - - -, - r

,.

used to theoretically predict the performance characteristics 0

of these devices in shallow submergence applications. Local -

· or average _ y~ t!J.~_§ ____ 9f.. ~.l..ip ___ J~_~ __ n_ __ be ac -~u_r_a t_e_l y_,~ .. c.alcu_l __ at e_d __ _u_s __ ing__,__. _______ ~------..

t~e ~q¥ation suggested by Griffith. Values of the. two-· · · · -·~-- ..... -----

phase friction factor, l1owever, cannot be calcµlated using _____ - --

- - ....... -- ... - --- -- ---------- - - single phase approximations.

...... :

r\", u_,·

J . ,

--

' . , lthen attempting to theoretically calculate the pump

characteristics in deep submergence applications, it is

insufficient to acco11nt only for the changes in volume of

the expanding air~' Local cl1anges in the slip ratio and

friction factor must also be considered. Figures (13),

(14), (15), and (16) show the resulting deviation between

theory and experiment when constant Sand f values are used

in deep submergence calculations.

s.2 Sttggest.ions for Ft1ture \\Tork

The present computer program as outlined in the Appendix

should be modified to include a local calculation of Sand£

-26-

.. .:.,-

' ••·t',

l ' ....

"\ r

'i

---·-------······--·-·---··- -. :. ---· --·-~--·-·-·-_.._., ----·-----·--.--

. -- --····-·---- -··---- ----- --·----- ---~----'-- ---·-·---·--·-- ._

Ir •.• ,.

' . .. . . . . .

. . ' ·.. -·-:;---:- ,'

. , ..... . # ...

. . ~-.

. . . -.· :. ,•: .. : ,, . ~

.. : .. ·-- --.·-_<:'··::·.c.,;,.-- :;" --; .-. .· .. ·. . >., .:- ' .• , . '. . .. ' . ':--s. : .. :_ ,·. ;,: : .. :.:_:·,., . ~ ·, --- -. - --.··. - . ,.,.. . . j ·-· • -- ~-· - - •.• . : . , ,' ~::

. . . ' ... .. _:__ -~: . .'' .. . ' ,,./- ... ~·.· ' ..

. ~ ... -:-:.,. . . -

. --~ . " - . ' ...

-----····· -·"·.---.C...:...·. -- ·~-1-0-c·al:--·vai-u-e-s--;of-~. s--and----f·--s1iotI1d-~-:--then-1:nt-·-u-s--e<1--wh-e-n'."--me ........ : ... , '.... ·- .... · · .. ·· ..... -·-----·--··-· - -=------···--· - ___ :_- _____ __:_. __ : __ ·-·- ----- - --------- ··-·· . -.-------- ... : ________ ---

presst1re gradient dP/dL is calculated at· each of t·he n ~-

points· along the length of the eduction p~pe. \~lhile the . . . . . .. . ~ l . .

. _:__ , .-, . equaitii:m sugr,'ested .by .. ,Griffith could be used to calculate '

the individual slip ratio values, more:work ~ust be done

before accurate friction factors .can be predicted •

.....

" -~·-·•>a-···· -- --~ --·· --- -~--- - - .... -· . - -.· ·: ' --., 1· ..• . ·r·· _J ·- _ _. •• -,_ • •• -~--, -·'-: _ .... .., ' . '···"--.--- - ··-···-- ·--···-·-·-·

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.. . ..

:~.

•. ·.,.

'.•

.,

'.• .. '

~- -=-

. : o=, .•

-21-..

j.

- r. - ,.._ ;. • • r . .,__ ··--·-· •. ·-q -- ._ .......... .,....._._ •r-•·-"""'-C..• ,. -

: . -...... ·. [•.- 'f'

. •. . i • ·, ·. ·· ... · .. · £ ' c •· ; C ... ,. ;;-:: ,. '-s' •. ••'

I '.' ~ •'

0.~ .•... . ----·- _ .... ·· --~==~~~--~-·, ·····--·- -.... ---~~---. - . ·.·. -, ··--·:;· ..

(.· - .

i. ,.-.·._· ..

i i·:,:.-- ·. I' ·.·

._····o_-···_· __ ······ . . fil< ,.,

I ':' • 'j

\._.:)

.... -.'.~-------~----·------- .. -

. .--•- ··-·····--.------- ------. .:._ _____ ,_ ·---- ·.-------- -

. : . . - '~-=--- . : .. _ - ···- . ------------··---------~ ·----. . . . --· -- . · .. -- ·~~:- ·.· .. ··.···· ···.· .·

. . ~.

. . . "-~---~

--3 '' . ID . PIPE-----... ) I •

ORIFICE TM1K WITH SIGIIT GLASS --..... . . - ·--············. ----~-----· --~----·-· : ______________ .... ·•··· -~ ~---------- " ---· -- ---- .:. . -- ,--- - ----~------- -· --~·-······-· ... ---- ~- -·- •····• -- ......••.•. .., ..... ---- ------· .·. ··-· .. ··- .. ---·· - ·____ ----- ·_ ... ··-":

·-----------·-_·-····-· _···.···----3 ... 2" LENGTH OP CLEAR PLEXIGLASS PIPE ... J . . .... . - .

. '•,-- ·- ~·----~ ·-· ·····~--- -~ -~,~~- -' ... -,--~--- .. -..--~·-· -- _...._ __ ,, ... ____ ,_ -,- ·---~ .. _, ._.,.,. ...... ·-····,~- -~----·--···· -

---~--- I ""•

( )._ ..• ;

' ,f .._.

. -.

___________ WAT.BR SUPPLY----!.. --...

LM-1INAR 1YPB AIR FLOWMBTER-----...

' .

-{JI-'' .

AIR SUPPLY--__,

. PRESSURE REGULATOR _ __,

)

¥,-,.~·~+-·-··-- --'~~-- -~--'-'--·J-__ ,, ...... _,•,•.• -"~-•-.L.'>•••-•..--.,,-._.,. __ ,,, -~'•"-••.••• •.. •·-·- ·'••~---"•••-••-• • -•~ •-••-, .~. •• • • ~ .• ·-••.•• '•-••• > ••~"c• '"·••-· •.•

PRE_SSURE GAGE -----.1 ·- .

TH.ER~lO~lETEo--· __ _,

CONTROL VALV~----__,

OVERFLOW-------

1!1' ID AIR PIP~---_,

2'' ID DISCHARGE PIPE (D)

J\IR INJECTOR 56, 7 /32'' · D HOLE

'

4''

, . . ... ··- --··- -·-·-----

I :.. --·

167''(L)

... • •

l15''(H)

.. :-,---_.,__.... __ .._

•

-w---+---._t-+,.+.· .

ROUNDED ENTRANCE CONE----_:__+--__..,..

DRAIN---"

FIG. 1 THE t,lODEL AII{-LIFT PUJ\lP

-28-

•

• - f ..... -·.. -~- :,0 O

I ~

'° I

LJ __ .. ·· (~, ,.... ,,·

J-1 ,-... Cl)

rx. 0 '-"

Pit E-f

~ ~

6 H ~

~ ....1 0 :> ~ ~ 8

~

0.10

0.08

0.06

0.04

0.02

FIG. 2

I

0.1

I •

I "-..

0 ~- - o:_.

~-e

0.3

AIR VOLID1E FLO\v RATE (.CFS/) Q g

PERFOR!vIANCE OF IvIODEL AIR--LIFil' PUMP

. ' . ' i

0--0. . . 4:"---

o.5 o.6

'

\ ;)17 ,·

I

' .

I i

. i .

..

. · ·o-----._·.· __ -:_·· __ .. CC•;•, . ! ,•

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':·,,• .·, . .' -- . - . . .- ~- ._ ..... "

·".'··

. i:1

. f,l j( 1 hi, . /(,, (,f . p

. , . . . '! ~: . ' ;;J-,· '•\. . ... ··--· ·- -·-·· .- @ - ..• . . . . ii

... ·J. ;

' ( • . 1~:

. ····-···- -··--· -··-- ---~ ·' . --=,,· . : ... .:.~~·· . ·~-~-····---~,-'...-.... :...... .. ~--~-··---·-·. ·---. ----. ·-·-·. ····---~-------··------··--·~· --·-~--,·-----------·--··--·-·-'---···------····-··~-'-~---··-~--------__:_,, _____ ,.····----..,--·------·-------··--.!'::'.:_. -··- ~ --·---· -. ........._. . ·- -- --- ... l . . . --·-- --· ··-··-- -... -.~ ____ ~ ..... ~ __ --- . 0 .15 . . . . -.. ------ ~ . .

1'

0 .12 . .-· -----1----1f----------+--~iro--_.., _____ ,,1.-______ _.

. ' .! .

V2~t -· .---··-----·----- . -- -- -

,---., o.o.6 ()_·• ()

6 . '

t) 0

6 0

o.o. H/L DAT/\. THE RY

,_ ..

0.692 0 S= 1.956 T}"_ 2.115 ....... -0.619 6 S= 2.244 K= 2.127 0.543 0 S= 2. 72

.,. ,. 1.902 ~=

0.478 0 S= 3.05 K= 1.826

6.0 9.0 12.0 15.0 . ,,.

0 3.0

Og Qf

•. I

·• PIG. 3

,- '

C0~1PARISON OP THEOllETICAL AND ACID,\L OUTPUTS

.. 30.

I

I

H '·- I

~

f2 0 ~

M E-t

~ ~

I 0 w H ..... µ:., I E:J

f3 ..:I -0 > tr: µ:t 8 ~ ~

·., ·.

. J

- ' - I - - r

, I -l \ I

i ; I

0. 8 ----------------,-------;--~-.-----~---~-------, i

0.4

0.2

FIG. 4

,r-

5.o

:

. }

-------, -~

~-

CO}IPRESS BLE THEORY --INCOI·ll'RE SIBLE THEOR,

S= -2.45 C

f= 0.005 2 D= 0.333 FT.

10.0

•

15.0

0

FREE AIR VOLlTI1E FLO\v RA-TE (CFS) 1

l I C

i

20.0

COMPARISON BETWEEN COMPRESS!BiLE AND I

INCOJ\1PRESSIBLE THEORIES/

' .

i ) I J i ,

i ( ~ .

l

. '

' i i ' i

i 'I I I : ;

i '! ! I I

! i

'I • . i

25.Q .. I!

'. '' .; ;

I: I I I: ii

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L ! .

-,

(.

I •

't. .

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en ti. 0 ..._...

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~ I

e5 w w I a-l

Ji.

~ H 0 > Cl:!

~ ~ .,_;-

0.10

0.06

0.02

0 0

FIG. 5

!UL DATA

(i) 0 .692 0 S= 1.956 0.619 6 S= 2.241+ o.(i~3 0 S= 2.72 o. i 8 D S= 3.0., .

0.1 0.2 0.3

AIR VOLUME FLOW RATE {CFS) Qg

cor~IPARISOli OF MODEL PIDfP PERFORI',IltNCE AND THE01-illTICAL OUTPUT

j.

y

K= 2.115 K= 2.127 K= 1.902 K= 1·.a26

I

o.6

(

i

' j '(7 . • ·.

, I .-

·, .

T I • i •

. ;. . . :,

--·o· --·:· ,·' ~:···. ·:· :·. ' . . . . . '' :

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40

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·.30 . '· $

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t ·~\ E-t -

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fa 20 i,.;

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FIG. 6

'·-- ··-,· ... ·,·--··-'-·---·---,.-..-.~·- ------ ---·--.

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. / .. ...

J\t}\X I ~1U~1 •. ~·

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"-.. . ...

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:FICIE1~CY n.T ~ l:\ XI i1 U~1 ------ .

,,

·-,Olv POINT ti

-0 ,.,

~ I

-- - -~--- -- ---- -

\. ..

I

.

•

-•

20 40 60 80 100

PERCENT SU l3~lERr.:Ei~CE

-- .. ---------~-------------------· _, ___ ,_ ______ ._ . .,._.

THE EFFICIENCY ClJH.VES

~ -· I.

-33-,,

,· /

•

I·'

I I ~:i

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,i

I

f! Ii!

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, ____ ,,' . . ·.· .. '. ~ ...

. ·c-

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: ,, • . -· .• ,· ·1· __ •· ··-, .· ,. ~ .. -· ... -··--··· .• ·. ·.- ..• -···-·-·····.·--··. ·-.·-•• -"" ---·· -- .·.------- -,------ --------· ·--------··-----~---·· • ---------------·-. • I . . • .' . • :· .- ... .' • .. ·

. ··.. . . . . . . . . . - - - .. . - ·- - -· - - ·--------------.--·· ·~--'-- - - ----- . -- - .... - . ;-.' .. ·- .... · ... . ;_ :, . . . . . . . ' . . . . : '

. . .

60..---------,.......--~--------,.----------..---...;...-----.--------~

,.'

--'-~ •• - .......... ~·-,_____.;.._-~~ ........... ~ ~ - .~---,-,.···-.-··• .a• ...• •-•••--•-• --•• .,~--<'," - ··~-·· -, + ______ -~- - '""", ca· - + ···•- ·'• -.w- -~·-•••:._·_-• -•-•~ •---•.----~·····••• •.•· I i-·•-' .... - .,, __ -·.-- -- .---·--- -.. .· ...

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- --·- " -·- .. - -_ _,._., .. -~-· ,·. '---- -

,· - -· - -------- ·-·· --·--- ····-- -- - -- --·-- .. ----- - ··-

HEORETICA MAXIMUM FFICIENCY ERI}IENT ·L MAXI!1UM EFFICIENC -0-

10 t---~-+------h-----'"------'--------'

···FIG 7 • ,ii

{·:

20 • 40 60 ( i

PERCENT SUBMERGENCE

00?-IPARISON OF THEORETICAL AND Y.u\.XII'1u11 EFFICIENCIES FOR

l AND 2 INCH EDUCTION PIPES

.. 34_

. - ·----- -·- --------- ----.

0 100

---------·-- -·-·--.. --- .. _.

. !

.11

ir.

-r.J . ....... ' '· ' ' ·._,I

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f---

0.004 "

0

0.002

FIG. 8

·.'

.,

l.-' ..

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-•

-

EXT BRI~,1ENT.t\.T~ V . .\LUES

THI,ORETI C.-\I~ \ l\IJJES FlJT r.Y nEV.EI..OI ED Fl.OW

0.2 0.4 H I:"

.---

-.-o-

0.6

CO~'ll)_,\1lI SON OP 'TflEO!tETIC.'\L .t\CTUAL FRIC1,ION F,\CTOllS

..;35_

I

0.8

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1.0

. ...

,1,

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4.0 , .u •. x .• ··------=---~, --~- ,.,,.,_~•-· ..... ,.· --~'--T~---,c'-------·--....,,_,.~ -·-····:-:-·'-- --

,

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------ -- ------·-- -- -- --

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.

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FIG •. 9

-~

~ i/rlt ~ , ..

'

'U

- . ' . ·- ·- . .. -

' .EX1 >ERI~lENTAL VAI.lJES

Tl-l ~:ORETI C.t\L , r ,'\Ll JES ~

FU1 .LY DEVELO~ >Ei) FLO\·: . ..

'\

0.2 0.4 0.6 0.8

....

H r

C0~1PARIS0N OF TJlEOH..ETIC1\L AND ACTUAL SLIP RATIOS

..

-36-

~

----. ----- .. -

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--~--------· --

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1.0

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. __ ,_ .. _____ _

:( . ' ~ - .· .- ' ' -

. ' ,_ ~--~--~----=-~------""=-""--='=".......,---~---=-=""'--~--~--"=--=--= === - ---==--::::---=------=-:-~, -_----::: __ -.==----:::=-r-_-_--:_·__ _- '-:="":::·:::--.=.~:::':·~=--.,_::·-=-:---:-.=--=:-=:-~~~- . -~·--~~- ~.~.~---·· ~--- - .· _____ ,:_~:~~__:._ ... --~·~:-- ··-·--·- ·------- . --· ----------------~--. ' '

• ·-;··1 .~

.

; ·o .12 ------..------.,........-----,.-----.,....------,

\iARD ·& KESSLER D TA :0 -/ 0.10 ~~H-=--3-2-.-.J-r--FT---+;,~---+---------+---------1----~-------,

. -

L= 60. 8 FT. D= 0.2 833 FT.

0

0

--··-·-- +.- - --------·-·

"-. . . •.· .. .~· - .-~., .

- "'"' . - - -- - . . -·

- . ··- --~-- ------·-·

- . ~ . -------------------- ----------. --- ------------- ------0-- - ---

- ~ __,o .06 "------4-----+--H-----+-----,..---~-, . rxt

~

B r~

0 .... ...: ~

0 ~->

0.04

0.02:

0 0

FIG. 10

ro

0

0.1

I!fC QivIP. ~ SIB LE T !1PP~SSIBTJE THE

S= 2. 70 f= 0.004 5

0.3

FREE AIR VOLUME FLOW RATE (CFS)

COMPARISON OF THEORETICAL AND ACTUAL OUTPUTS

-37-

-··-

o.s ). .

•--·•-•c, ....... - ., .• --·~--~...,._.~ ,-.••.'··-·····-··•-~ ~-

·'·

- - ,.~---

OJ· . I '· ~ ' .· ' ·, '_,/

·'

I

•

. ' .. ------------ --··-···---~---------'"·--··-·--....-.--.-..-- ....... __.._......_~·- fi~._.,._--,~--.-.:c,.,·v_;.;~-~--- .-.;·;;;.·'.'-·;_ __ -:_; -_ .- __ ._, ... -_ · --~~;i.--, · ,-·. ~-----·;,z ... --~-~7_. ·

. -·----:--.. ·· .. ---"'·.

- .. . ----~-' ___ :.

-- ·-- -. - -

o.16

~

-~~~--~- ----··

e 0.12 fil

.8 - ------ - ~-- .- -----~ ----- ....

,_.,.-1,

"

. ' • •

' . . . ==~•=,,·•·-=• ~·=••,'-'••~•• =~~===f=~'==~='==~=~=~=~~-'-'-'~~: ,.~~·-> ___ ~-~ ··-.-~·---~-,.-·-·-:•····.··c-O-:~

--~-~--------- ------ - --·--- ,. ,. - - ----- ~------~--- . -. ------------~-- - - -

·-"··---~· .. ·· .. ··-· .. ....:.·;....;......-:.

WARD&.

H= 3_7.88 L='· 60 e88

SSLER

FT. FT.

0

-------~--------- -

0 0 .,,_..---

~---. -·-·:·---------· -------·- ---- ----------- -----------

0 ---------------·~·-----

~ 0.08

0 .04- .

0 0 0.1

• cv· Q . •

S= 2.2 t= o.o

SSIBLE T

0.2 0.3

ORY-··-

FREE AIR VOLUME FLOW RATE. :(CFS)

FIG. ll

.I

·O

COMPARISON OF THEORETICAL AND ACTUAL OUTPUTS

-38- •

-.

.. ·

...

. . . .:. . -- .. ___ . ------- -·--'--

---p-~-· ....... ~-·- .· . -. ' .. ___________ .,'-.,:..

----J----~ ... ~-~ ---~------ ·-- -- --

.-...,

(i· • , .

. . ' • :. 'q •.

' . . ---·- - -·· --- ,.··-· - . - --·---'-- ·- . . -· ........ · .. -- ·-·' ·-. '. . ,,-,---- ···-·-----· .._.

~:}J '·

. I

. . , .\J - ·. . .·.. . ,' I I ---· _-....._ ___ -c....:.....,...:..;;. --- --- ·. ·-~-----··,---------:--·---· • • , • ~~ -----~-------' ------------- -- ------------ -....: ----~--------··-----------~..:.:.. .. .___--'-'--' ___ ..:.:...~-_-'-:.:.. __ ·--?--':-~ .... -~.:...it..~- . .L~---·~_ .. ___ ~~· ·. -.... ~=-=:...~· ~ ------~---~-------------------~---- ---- - - ··---.-·-··.--------· ·------------ /--------._------·· ------ ------· -..------------ -__ · -~-.--- --0~ i6- -~--:_~-- --- -- -- -- ------- -/ ·---·· ..... -.-- .-·--·--· ,-.-:

O·~ • -\

J

- ---···-------

CT·

I'

l

· 1 ··-

0.88 FT' .20833 T.

0

O·

. - --------····--··--··· I

. __ ...,. __

o.12t--------..-------+--------+-------~'------~--------4

. ~ 0.10. E-t

~

0.06

0.04.

0.02 0

---r---~---- - -· .

,

-FIG. l2

'

•

·"

- -·- ------ - -·-- - --- -- ---- --- ---- ---- - -- -----~- --------·-- - ----------

0

•

,. I

I! cor1PRES IBLE T ORY-··c 11PRESSI LE THEO Y

-------- .. --

0.08 0.12 0.16 0.20 0.24 FREE AIR VOLUME FLO\v RATE (CFS)

C0?1PARIS0I~ OF TlIEORETICAL Al~D ACTUAL 01.JTPUTS

-39-

•

~

C/l r:r. t) ...._,

µ:1 8

~ I ~ ~ H 0 I

rx. E=1 E? ...:t 0 :>

ffi E-1

~

0.3

0.1

-

F PICKER DATA -O

H= 62.24 · • L= 137.8 T • D= 0.3280 FT.

I / \

(,..-....._' . I .·~· \ )

1.0 2.0

9-1 __

INCOMP SSIB:tE THEO COI·iPRES~ IBLE: THEORY

S= 4;30 _ · r= o.ooi 74

4.0

FRRE AIR VOLUI·lE FLOW RATE ( CFS)

o __ -----

., -··-

FIG. 13. cor-IPA11IS0!{ OF TIIEOIIBTICAL AND ACTUAL OUTPUTS

• •

•. j

-o . I • I

,. i

s.o -~. - ....

. I;<

·1 ' •.. ' '.

; ' :1-, ·' .:.

• . i . ·.

"

.y ·; . t

f

. /

\

' ·. !

I

I .

. ' . .

•

\

'" ' . \

I ~ ~ I

\ \ \

\

•

,...., Cl) ~ 0 """' M 8

cZ ~ 0

~ ~ p H 0 > ~ µl E-• ~ ~

F. PICiillRT DA A-O-

H= FT.

0.3

0.2

0 ', )

---o--+-~ o-- .0~1 --0 -t-- __ _

--0

INCOI\iIPRESSIBLE I-IEORY -··cor-fPRESSIBLE T ~ ORY

o.1 t--~~~..,:...-~~~-4--~~~~~~~--+-s~,=~2-=--.~~~~~-+-~------------~i f= 0.00494 I

I i I-

i i I .

. . . i

Ol ..

- ..

- ' f

0 o.___ _______ 1_ • ..__0 _______ 2_ ...... o-------3---'o.._0 _______ 4Jo. ,l

FRc;E AIR VOLffi,IE FLO\v RATE !(CFS) ,· ,

t

'

FIG. J.4 C0!:1PAnIS01I OF TIIEORETICAL AlID ACTUAL OUTPUTS

,

• I . i I

; I

:· J

I I I

I , I

I I I

i . . ' . .

i ' . -

' . . .

: " \

I . '

·i

l .

• ! I' •

• ! .

•

........_

E2 0 ~

~ 8

~

I 6

-~ ~ ~ N I ~ s

...:I 0 > P:: µq

~ ~

l ! ! ' . . }

4

o.5 F PICKE~ D TA 0- . . _,,,,,,,,-

I

0.4 IF= 137.8 Fr. D= 0~32808 F •

0.3 -o 0- 0 -0

. .{

. 0.2

INCO}!PRESSIBLE -··-0.1 .-~~7?-~~~~--t--~~~~~~~-1~========:::::;:========.=i=======:.....:...~-----.l..J,

i

FIG. 15

1.0

S= 2~70 i

:f= 0.00504

2~0 " \

i

RATE/ ··(CFS) I.

' ! .. ~

3.0

cor-1PARIS0N OF THEORETICAL AND ACTUAL OUTPUTS

i I

I .

. ..,

I

i I

. I

I

1 .. ·1. .

~o l - I I j I

I I 1 '

I I J I .·

·I ... 1· I. !

I I I . I

, I

.. !

· . . -.·-··-··o·.-···· !:,. --.·. ~; )! .. ·. . . . r .· ..

·. t· ! < . :,. . . I . r ....

. -•.:. ~_.-.-.-·· , t .

i

...

i . I I .

. ! ;._<

' ,,;i /1 .. · . . J

.. .·. l 11' .. '

· .. ·. I

. .

' ! • i

,, .. .r I .J . I . I .. ·.It: .

. : l . ' .lj .. I.

j ! .j .

. l

l i I •

! .i i .

,. 1

.- I. I i ..

i ' i

'"j

.· i j

. i i' .

. . ~l .

I •.

1· . i.

i .

! .

. ! ·. i . i

. I, . . 'i

. '.

- 1 ·

•

o.5 . F,

(

,-... H= Cl) rx.. 0.4 0 .__., D=

µ::t E-i

~ 0.3 ~....: b

I ....:l

.b. ~ w t) I ... ,_.

~ :::t

0.2

0 ::> er:: fit

~ ~

FIG. 16

PICKEl?I' DArA

91.38':F[r. • •

0.32808 FT.

o-

n. ·~)

~ 0

~

o----4)

l .

I I

. I .

INCOI'1PRE SIBLE THEORY-.- .. _ 7

C OI'·IP RESS LE THEO RY

1.0 2.0 3.0

FREE AIR VOLID·IB FLOW RATE (CFS)

C0}1PARISOli OF THEORETICA·L AND ACTrrAL OUTPUTS

. i

I I I

I

!

•· t· ..... 1-

· 1 · · .· 0 ·· I. :. ·t )•·;·.•····

";;, . • . I

'"i, .... .;;·· ·~ .

. ' .. , I

. i; . 'II .' ... :.

"' I· ..

. ·f .. t -·

·[_. .

- I ~- :

.. ·, [ .··

. t . I . . l .

. f

i . . ~ .

. .

i r . ·,

i: ; t

l . l . ~ l r ' i '. i

' ! r

I. I ..

i I I .

I I [ .

,/: •i ' : I J

. I

I I

. I, ·t. i . )

i j

:~ '.i

. :1 ·:[ -:I

'i -I 1 .

J, I . 'I ..

:1. ,I . ii ! r ... ;i ;1 ·

·, ' . , I . . ~ '

. ! ' . . ' i:

1: '

. ~: . . ;' . r . 1,~ .

t .

l .. · '{. r .

, : C . •I: ·.·· l l

. ~ ~ ; .

! ,: I -f-. ., .ii '. i.

I; . .(, '.

r'

Ii . 1; ·• .

. !' ·r

. ji' . l I:. ::

' /,':

t I

' l: . '

- . -- --· ~· - - .. . - .····-·~ _. .... ~ ~ ' ... ·~~

. - -~ '' . .·

-·: . . - ·' -~ :'• .·.·.;;: + • '

. '.. - _... - ·. .

' . .. ' •, ,. .

;--·,. ·-.~~:-. _--;..:_-:._-~ ... _ -·---·---"-•-:--·~,.-· · ...... :- .i·· /.-: .. . :...:._ ... :_(·-:--~- :./'-: __ ;-_- -~·~~~ /"-' ' v .. · .. · ~ ... _: . ~-.--·-.--· ----- -··.

L

- • --- - - • , - - - .~ C • ) ~ - - - • - ~ -. - <- ·· -~ -_ ; -r- •- . · -·· · -. . -1

.. ·~· .. -- -- ·~·~·~ .. -.. ~·-·---·~·--·-~~---:-~· ··~-·~APP:BNBI-X--- · --~-· · ~~ ~·. -..... · .... · .. ·· · ·:·' . . · ... ····-· ·---·~ ..... . . . ~~ ·--·-·-·- -------.-·------·-.-··--·····--·,--.4-----.. ~- . ··... - - ----1

I. Computer Program ' .

---· ····--·-- A· For t ran . c ompute-r----pr-~r ~-m-:-~~as---·wr-i t,-ten · t~pe-r-fo-rn·-~- ____ :.,_ ... ~_,._,,·~~-~--- .·. __ . ____ ~ _ ~ _

required numerical integration for the-·6ompressible gas

solution. The .. fallowing data was used as input:

a. Run identification

b. Submergence depth, H

. ~

,•,

~---------·- ------, -- ---- --- ---~. --

c. Pump length, L •.

·-··--

. .

d. Pump diameter, D .,· .. -· ·-;-- ····-;-· ~ ·- -

----------------------~-----e ... _____ Ac·celer.ation_ o.f __ gravity __ (32._~ fp~ 2)

... ..

.,·

- .. - ·- - - -·- - --- .. . . :· ---- ---- ' - ------ --- -

f. _Pluid density,~ f

g. . \'o i

Gas constant, R \ ~ ! b"' ·( ',',

-h. Fl11id temr,erat,1re, 0 R

i. Atmospheric pressttre

j. Number of integration stops, n

k.' Number of Q , Qf combinations to be cal c11J a ted g 0

1. Damping factor (normally 0.30)

m. Tolerance .(normally 12 psf)

n. Nt1mher of different diameter pipes making up edt1ction pipe

...

o •

.•

-44- ,

0

•.

·, , . , ...

. .

'·>iii

a::1.11 eer::c12··MW• • ;=m*'~·&tm u·,. ·,J - n, -- ;;. --~---~~~;:;::::::::=::::::::::::_=-=4 ::::::::~~~~-~-----~--~-·--~--... ~-'-... ___ ----......... ~~~~~-P£~Ei,..,¥~-·~~~~~~~ - . -----· ------ ----- -··-- -------··----·---------------·---~---------_ ------~- --- --- -----·-·----- ·- - - -- -- -- - - - - - - ' ' - - - --- _- - - - -·- - -- . --- - - - - - . . ~

i - -- - . • - . - -· - - -- ' ' -- - - - - -,~ - . • j: i~~-~-~·~~-~ ~·":~ ---••• -~- -- .,.... . ... ., .... ---~~~,.~---_-~"'r·"•·-~=-,--=c -, =•~~ - . ·-c:-- .-- ·":'-'-v~ .. ,--._ .... ,,---.,":.-~- - -- -•••···-·-·•·--•'---C.•--·-·····''··--·---~••--•-~•--------·-'-•••···-:·"··:••••··.-···•••••: __ ,,_.,, .. _.,'.'.,···t

- - - - D I SK O P E RA T I I\ G S 't. S T E tJ. 13 6·0 F GR T RA f\ -3 6 C l\ - F C -4 51 3 l - - - ; o • - - , , - - - - - • , I

. .

-_ _:__~-.--e -------'~--------~---~-K-E-N---S-€r.-(lEN:E~K -PRG GRA!'1- --- - - - ------ --- --- ~

DI~ ENS ION XL. E i\ { -10-l, -- 0 I-A(-lC-)--,--·---1-S l E-P--l--10) , PRESS ( l O) --_ .,_ -- --·-:' f . ·_ D I M E f\ S I ON X S < l O O > , X F < l C O ) , . Q G Z E Ra ( l O CJ , . C F I NC < l CC J , - l I y· LE ( 2 O > - f

- - 9999 FCRfv.AT(8.FlOo0) ~- __ - ____ '. ______________ .·- \ -_ 9998 FOR~AT(2IS, 2Fl0.0,I5) - - - t - · 9 9 9 7 - F C R r' AT ( 4 F 1 0 a O ) ,. -~-·

. .. . ' . -~~~~- ,, . ,. ·-· -·-· . -- -- -- . ···-,·- .. __,,' ---·-· --- ·-~- ·------·-·---:\-

·-·· --=~~-~==~-. --===~'J9-'l6- FO R M A T ( 2 0 ~ A -) - --- ------ ----------------- -- -------- --~ --- ----- ··· -- -- --- ,-- --~~---------~----------- --- -------"'":----------~------ -----------'--------,'--:·---------~ --'·:--: · 9995 FCR~AT ('l',T8,20A4) ~

9 9 9 4 F G R ~ AT ( T 2 , • NC • 0 F • , T 16 , • Na-. C F ' , T 3 0 , ' NO • CF ' , T 4 5 , • BO T TC M • ,~r 5 7 , 1 D I . 1SC~Al{GE',T71,'LIQU10') . .. _ .

:.:. - 9993FCRMAT (//TlO,'S = •. -,El_C.__~5_,_1_3c~_!£_.· = _ _t__,_no_.__s__,_rn_,___!_l-~CO~. CF= •_,ElO_~ --"'-·-·-···--------- · 1 • s / -,- ---- --- --- -~- ----------- - . -9992 FCRMAT (/T5,'CGZERO = ',ElC.5,T35,'QF = ',El0.5,T59,'P~ = ',El0.5,

-.. l / T 9 , ' Q N -= ' , E l O • 5 , T 3 2 , ' C E X I T = • , E l O • 5 ,· T 5 3 , ' D E L-T A P ~ = • ·, E l O • 5 / / 16 -2 , • e CT TO tJ. ,p RESS URE = ., , E 1 C. 5 , T 4 5, ' NO. CF I TERA TIC f\ S = • , I 3 )

q 9. ~ l -· F C R fJ AT ( / ' L E N G TH • , I 2 , • - : , E l 2 • 5 , ' D = ' , E 12 • 5 , ' ~ C • S T E P- S = 1 , I 3 ) g990 .FC~r-'AT( 2Fl0.0, I5 )" 9 9 8 9 F C R ~ A T t '.O I NT E R M E D I A T E P R E ·s SL R E S = • , E 1 2 • 5 , / ( 2 6 X ,. E 1 2 • 5 ) ) CJ q 8 8 F C R f\/, A T ( / / / T l C , ' L · = 1 , E 1 0 • 5 , T 3 t , ' H = ' , E l C • 5 , T 6 C , 1 D = 1 , E l O • .. 5 ) g997 FCR~AT (T8,'RCF = ',ElC.5,T36,'R = •,ElC.5,T60,'T = •,El0.5)

-~-~-·----------·- g9_a6 FCR~AT (T2, 'N~ •. STEPS = ',1·3 ,T3C, 'DAM'P ING -- = ·, ,Fc.4 ,T5E, 1 TCL = 1 ,,F4. 1·0) - ..

9985 FCH~AT (TlO,'G = ',ElC.5,T33,'PATl-4 = ',.ElC.5) ) g 9 8 4 F C R ~ AT ( T 2 , ' C Y CL ES ' , T l i.e , ' C I A M E T E R S '· , T 2 7 , ' I ~ TE G RA T I C ~ S 1 , T 4 4 , 1 P R-E S Su

lRE', T56, •PRESSURE•, T72, 'FLO\.\ 1 )

-------- ------------ 9 9 8 0 -F C R fl A T ( I 5 , T l 5 , I 5 , T 2 g , I 5 , T 4 3 , F 8 • l , T 5 7 1 F 8 • l , T 7 0 • F 8-. 5 ) _ -- 10 REAC(l,9g<;6)TITLE

C.

C

REAC(l,ggsg)XLEN(ll,H,CIA(l),G,ROF,R,l,PAlM -REAC(1,ggse)ISTEP(l),~Q,OA~P,TOL,~DIAS REAC(1,ggg7)(C~GZEROCI), QFit\C(I), XS(I), XF(·I), I=l,N,> WRITE(3,9SS5)TITLE . WRITE (3,g988) XLEN(l),~,CIA(l) WRITE (3,Sg87) ROF,R,T WRITE (3,S~86) ISTEP(l),DAMP,TOL WRITE (3,gg95) G,PATM

IF(~DIAS-1)45,50,40 45 NCIAS = l

GC TO 50 1

40 REAC(l,gqgc)(XLEN( I),ClA( Il,ISTEP( Ir,I=2·,f\0IAS)

~ ... -

49 wRITE. (3,~ggl)(I,XLEN(I),CIA(l),ISTEP(I) ,1=2,NOI~S)

50 ABCT = .?e~ilDIA{l)iiOIA(l) AT C P = • 1 E 6 * 0 I A ( ND I AS ) ~ C I A ( l\ C- I A S ) AAVG - ( ATCP + A~OT) * C.5 Xf\STEP - C

C C 4 7 I = l , r,~ C I A S ·47 X~STEP = X~STEP +ISTEP(l)

.....

DC 1000 I = ltN~ 51 S = XS(I) 52 F = XF ( I) 5 3 0 f = t~ F I N C ( I ) 54 WRITi(3,9S03)S,F,QF

WRITE (3,c;C,<;4)

•

';I: .. -.....

~45-

' •

4

-,,.

.. ··.

I I.

. .

.12./16/68 FCRTMAIN· . . .

·· WRITE (3,9984)

··: ''

N·CCYC = 0- . . .

ER R C L = 0 • . -- - . · -~ · ··"·r·-··-··--· ,._; .... _._.~.-- ··.·-·--~. ----~--- . --- - -- ---- -- ~-:--. -.. .n, '.

.'.- -._:_., . ·_ .. .

,· ·;;.:·;··:.--":, ",".p,"• ,, '.-' .. · .

,. - . ., . -- - . - _____ ,.. __ - -- -~ -- ~- - ~ . --- .. - -

DC . 7 8 3 IR= 1, l C : · ·

-~-i-=:-~~.-: __ 7 __ 6_3~. ____ PHESS (_IR l ;::C .• Q~~--.. _~··-__c_·~·.·_ =~:__•·~-··_··-·~~~.'.._. ~=====~~-:::-·:........:....····~·---:-· ._._. _· --:-:..C--_-~~~~···-··_····-~,=-··=--··=_--~ --·~·-=,,.·=-~·-=_;·=· ·=='~·-;-· _:.~,-:--·:'·=· --=·:::•--:::·~---·~;'·-,-'. ·.~>---·~· ··· .. _··. _-:_:~:~----~

~...:..___.:.__~-----~f\E w - = C .-· .. NOIAN =l

J=O : ·-· •. ·' .• r

100 QG =· CGZERO( I)

:,-... -·- ''-,---·-·- .-.. -·.·-- .. -:--w-R--I-T--E----"(--3:-,-y<,-e-e). N_Qg_v~~--'---~_C_JAN,, .. ~t.P_R~~_§J ! l, PNE.~_,Q_F ___ ----~--------~-=~----~.--- >'· · ~ . _ . .~- ... NDIAN =l

:NOCVC ~ NCCYC + l 110 V = CF/ABCT

. ' '

111 P = PATM + ROF•G/32.174•H - RCF•V•V/64.348 -lABGT

RCF .. V•CG/32.174/ . I

PRESS(l)·=· P 114 X~CCtG = (RCF•H + PATM)•QG/R/T 115 X~DCTL = ROF• QF I'

TGTSTP = C ----~-·--1·1·2----Ns·r-E·J)S = IS TEP ( ND I AN>

113 D E L T A L =. ·· X L E N < ND I AN ) /' I S T E P ( f\ D I A .f$) ... D = DIA(NCIAN)

C .......

116 OC 150 J = .l,~STfPS --------- ------11·1-· ---o PC L l = - : -G *RO F / 3 2. 1 7 4 / ( l • +C GI s1-cF r ---- ------··---------~------~--~------. -. ,--.. ,·.·., -- '' .· .. - ..... ~.

DPCL2= 4.•f*ROF*V*V/C/f4.348•(1. +QG/QF) .

J

IQ'._ ... ,: ...

·-,; - '

•

C

DPOL = -CPCLl -OPDL2 118 Pl\Eh .= P + CPCL*DEL TAL 119 QG = P/PNEW*CG 805 IF(PNEh)l25,15C,150 125 XJ = TCTSTP -1. + J

QF=CF•XJ/Xr\STEP GC TC lCO

150 P = Pt\EW

155

160

TGTSTP = TCTSTP + ISTEP(NCIAN)

IF(NCIAS-~CIA~)l60,l6C,155 NCI/lN = NCIAN +l PRESS(NCIAf$ ; P GC TC 112 V~ = (CF+CG)/AAVG

",i.

- 161 162 163 164 200

Vlt:RC = (C:GZERO(I) + CF)/AAvG

DtLP~ = (XMCOTJ -~ XMDCTL)*(VN - VZER0)/32.174/AAVG PERROR = P - ~ATM -DELP~ IF(ARS(PERR[R)-TOL )2CC,2CC,25C QEXIT = QGZERC(I)*(ROF*h+PAT~)/PAT~

C

210

WRIT~ (3,SSS2) CGZERU( I),CF,P,QG,CEXIT,DELPM,PRESS(l),~CC~C IF(~CIAS -l)lCOC,lC00,210 WRIT£ (3,S~R~) (PRESS(K),K=2,~CIAS) GC TO lCOC

250 IF(ERRCL)26C,3CC,260 260 QF~E~ = .9•(QFLAST - QF)•PERRCR/(PERRCR -ERROL) +,F

GC TC 4CO

--46-

.. .

.,

•

.. ,.

-·--.~-,·--·-- -.····---- - .. ::·· ,-•· ..... ... , .. :.·. -~---,., .... ~·--···~-·-··:&,;.:>·····,d,c.-.-·.,_, __ . , ... _· ____ .:.·....,_ ·.3~::~;__;-~--

·-~..---=-'--··"' •··· .-..=-=-:., •.• .-_,.•,.,.,...-c,,_ .,=.,·,-,--•. ·.~ ....•..• ;.. .. ,--:-,-, -'.C ..• , .••.•• -.,-.•• c,_.,

--~·12 /18/ 66 FORTMAIN

300 CF~ ROF*H/P~T~/CAMP QF···t\ EW -- } / C PA l M + D ,E l. P M ) l CF >: . . .

400 QFLAST = CF QF - QFNEW •

ERR C L = P E R·R OR. '·: ·; ....... ~ . ';..

GC TO 100 •·· - ··.-· ... ___ ...;:.:__. -· -------·-------·-----'-'--·-. -1-o-e·o--=c~e:1~1~~fN{JE··--~--:_.: ____________________________ ._. ______ . __ -__:_~---~·· •·__.._. ---- .. · ...... -·-·. --·--- ·- ·~-· ... · .. --· - ·.·. · ·-:~. ·, · ·-.---. ·. · · ·~- · · .. ·· · ·.. · .- ---------.---•"'"

1500 GC TO 10 CCt\TI~UE Et\ C

'"------~---,--,...--,-----~------------------..

\·

i, 1

,:

'

--· -···-- ·-· ~- . . --····,:-....... -- ·--··--~~ . - ··-·

·)

. . ..,,.._--~---·-'----c'--'---'---~-~:.......~--'~------- . - -- - -- . -----·---- -

. ' . j ·<

--·---------· ------ -----· -----

J , J I

. I

- i

-- . --- -- ... . -~----~-, ..

~:

0 ,.

,.

!• ·,

. ' ...... . . .- -. -~ " - . . ·--- - . -

.. --· - -~- .. ~- ,..:. -~-

-----··---------~~ .a-.·-·---~

.•

...

-47-

·-····--· ..

·- -·,·--·- --':¥'----·-~--·~-.-. --·----- ------- ---~- -.-.--·

:r

-r ... . '

.· ·--

. ......

·'

...

::·""~

i•

· ..

J.

·: ~ .. - - - ,·_ .. :.: ··, -· :---~ . _,_-'_ .. -... .' .... - - : . t trer:'11:"ll'Si:' 1:A'n;. !~ I£ ~ - •

. -·c····-···-':~-:~-----~-'---:.· ___ .·., .· . - ·_ ~--.. ·.··-~-~---· __ ··.' ___ :..·-~~--~------------, . _·:..·: .:~--1~ ---

- : '

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. ' ... - - .. '.. -··:"'-. ·-··--· .~ ... , : ... . ·-·:· _· -- ; - ,. . . . - . -·--·· ... - ~~--~-:~· ------- t ... . . . . .) . '-..

·. ;. · .. . , .

\ _; = -~,- ___ •- •···'--·-·· . ._ -~.......::l..'t" -•-•-• S:amgl~. · Pr in:t:--&t : . _ :.... __ ---.. -·· · ~ __ • -·~-: _;_ ·• . _ _ _,-:.. :.... ~ _,.__;___....:..:..:._.~.:-.:::.. _..'.:....:...,. _______ -- _: ----:...C---. ---:--._:_ ____ ·:----·:-~ -~::--: .. -"· . ' ·. . . . . . . - . . . . . . .

i :_ ' ;•

' . ,-a ', .· I

;:·- .. . .

't . . '

' ,·

... · 'I . The following is a sample print~Out comparing F. Pickert· • I

l · .. _ test data: agains;t the present theory. Thif data included in r., :_._- - - :__ ~'...;_-~~·,--.. -·-:--------_,:;~~~-,;.........._ _______ .. __ ···-·-····· .----···- ...... --- ....... .

1~,:, this print-011t was used._ to dra.\-t ~igure (14). Additional . . -1:;: ..

r·:·· ··: ,.. ' : :i: _,; . t- ... ·: r , l . . t, .

• i I ! . } i' f

i

' . ' ) ' .

' L r .

/

inf·6rmation in th~ print-out.includ~d: §f

a. ~Pressure at injector discharge (bottom piessure).

b.

- ----~-- -- -- ·- C •

i'lomentum pressure drop (AP~).

Number of times- the prcrgram ·'t-ecalcul ated the same

point hefore an answer was obtained which was within

the allo,'-'able tolerance. ' - -- - . ---- ---- --- - -- -------·· - -- --· -------·-----~-------·- --------------· -

!:,..;._.

-~ ....

- - - - ' .. ._r, ·-·-·-·- _ .......... ,. .••

• . :· . ~ .

-~

(J .... ,._ ., .

' ''_/,j

-48-

' ·. ~ .

., \:~.::-. ·'.-'--,:---;~C;-:.;_~=-:-··:~;~:-:=;:~:x_Z:: {~:;-=~r~=--_:,;~=~~-:-7.,_~-:--.~-<r~"."'"".'.:'.:/;;~-::-~--~~2~----'-,.:;-~7i..--__ _.:~::r.:.:--_.::a:2:.-·-:.1:----;~-~-~~~--~~:;:-_-,----·-z-: --,,..,·---~-.·--":'°::!-___ .~-.·---1~-----.~---.:-~_ ,-_ -i..~---..,."',.,.,....-..:....,.., -~~--..:..:::~-_=-=_. ___ -~=-:--""·-~-,.-·'-'-'c'---.:.---=-------~-,.-__ '. ;--.,-_, ---·-----!!!!!!· -· -~~--~. _!'l!!!!!! __ !II!!!! ___ ,.- -~!!!!!!:!!!!~~!!!!!!!!!!!!:==:;;;;;;;;;;~;;;;;1115_ . , -~ - -- ~ ----"'-~ --~-- ---.-.--.-- ...... ~-- __ __.,_,..,_ -- ..... ~.~-~-------··-··>·· --_,._ - ------·-·-·------- ___ _,;;..._~--~--v-~v:-:···.·;;::::;;:·'::-,.;:.:.\..;:.~~-.,,-.:r;·.~:..;::_·:0;~.-..:_--,-.. ,:

I - • • • ••• --· --- -· - ••.. - -

! __ ~-~-~---~ -_ ,,_,,,--c;- - ,__ -,---~-__ : __ ~ -~~·- ---~-~~"~-'~~~~~~,,~~--- --,--- ------·----.-~----,------ ----,-,~,-~=--~~--------------- --- -- - -----~-=--=

F PICKERT DATA ·s-uBfi'ERGENCE RATIO c.5736 SET 7 - - .

. . :.. ..... -.:.... .. ·---- -··- -- -- -· '-·--·--·- ~- -----'~ --~--- :_____ -----~--- ~.., ·---- . .. ---- ,, __ , _____ -··---·-. . .. . -

L = ·• 13800E 03 - H = • 7903<;E C2 - .. - ' '.

RG,F--~---,.6240CE -02 .' R = '.5363CE C2 . - D = .32808E CC ' T -= _ e 53CCCE C3 -

NO. '.:TQL=l8. STEPS = 138 '_', -- DAMPING,: 0.30CC , _I_ - - - , , ,

G = .32200E 02 '_ PATM =---~ __ Z__O_J_~~--E __ C't ,_,-__ -__ ·-""------· -,.,_ _____ -- _- - - - ~----~--------- ----- ------------=---~~~---------------~-----,

-.. _ ·. . . --~----.-.---~-•-·--·------- ------- .... -- ' - . -. -·-·· . ' . - . . ..

- -- -- - ---·---· ··-·-· -... ,. -- - .

. :: ... -..

- ' -~--. - ',, ' ' s = .2asooe or. F = .. • lt94CCE-02 ·1NCO~P QF· = . • 263COE 00 '~

·Ne. OF , , NC •. o F -NO-•. OF - ea 1 TOM o I SCHARGE ~: ____ , [.,. ~··: ... :e·v-et:E s·- .. ··_------~- .,..'., _____ O'l'A MET e·R:s· -~-~~--~'-l~N-T:E,GRlfTIO,NS -- . ---- ---·~PRE s Sl:R E ,-- ------ ------- -.--P'r~e-ss lfR E

L-ICUlO' -, ----~ ---------

- F LC k ,-·,'

0 - l O · o.o o.o C.26300 C.26'189 C.31166 C.31779

L .... _ , , _ , -l --~---_ -~-- - · :/" l 1 3 8 t: 9 c 1 • 3 2 5' a 4 • a -: -' 0 2,, ', - i 138 ~q5c;.1 2531.3 ·() '-,-' - 3 ' ' l - 138 c95C.3 2226.6

r··-·-----·-·-,· . ~.-.... ·- - , , , ' ' · CtZERC = .54700E- 00 1-.

~; QN = .l7400E 01 ~ - - . .

QF = .3177<;E CC QEXIT ~ .1848C~ 01

_ 'PN =· ,.2l843E 04 DELTA PM= .10343E 03

8GTT0f' PRESSURE =· .69488E 04 NO. OF ITERATICNS = 4 .'Q

~"Q_·· l . -~~-~--1\J O. 0,-F-

S· = .28500E 01

0 I"\'\

''J/

C Y CL c S 0 l 2, 3

--N-G • -~- 0-f--D I AME TE RS

l ' l l

·l

CGZERG- = .62660E 00 Q.~, = .19677 E 01

F · , = _ • 4 9 4 C C E- C 2 I NC C tJ P Q F = , • 2 6 5 lt OE C O - . " ..

-------NC-. ---0 F- -- --- -,------------·--B-O-f-T-O-M -. ----- -·- -- -{Jf-S CH A-R-G 1:--- ,- -- -------<--·t-I--Ctrtc--.------------INTEGRATIONS

0 138 138 138

PRESSLRE c.o

6'155.0 fg52.1 6'138.4

QF = .34027E CC. QEXIT = .2ll7CE Cl CELTA

PRESS~RE G.O

2757.5 2681.9 2268.3

F LO~ C.26540 C.275CO C.33156

. C.34C27

.22C8cE 04

.12455E 03

BCTTC~ PRESSURE~= .69361E 04 NO. CF ITERATICNS = 4

1,,, •

S = .28500E 01

NC. GF NC. C,F CYCLES DIAMETERS

- . ('' -- . - - --. -

l

,-, ( ~ ,.,_,,

. ···----- -------.. ---~---·- ---- .. - '. -· - ~ --- -

C l l l 2 l 3 l

,GZERC = .6607CE 00 CN = .2063SE 01

F = .~94CCE-C2 INCC~P QF = .266~0E CO

NC. OF BOTTOM DISCHARGE LICUIC INTEGRATIONS PRESSLRE .. --- -·- -·--· - PRESSLRE c-- FLC~ . ~ . -- - --

a c.o o.o C.266go 138 6<;52.2 2819.4 C.27150 138 c c; 4 <; • 5 2735.6 C.33S22 138 t933.3 2283.6 C.34872

QF = .34d72E CC PN = .2218~E C4 QEXIT = .22322E Gl DELTA PP= .1335~E 03

BCTTC~ PRESSURE= .69307E 0~ NO. a~ ITERATICNS = 4 - ...

S = .28500E 01 F • .~940CE-C2 INCO~P QF = .26070E CC I

NO. OF I

NC. OF 801 TOM DISCHARGE LICUIC \

' -49-•

··-~

- -·--- . .. .. --· .. - - ' .. - -~-·~· · · • .;....:...c: •..• c,'--=..-::..::;:":::.:::s;:c:•.:;...~,c:;_~.;.;, ... ~ -· ~""=--~~ ttc-=-_,._,.;_,=~-~=c~i,=; 'a<i5ri:;,;;;-ow·.,~;·:---~ .-..::;:;-;,:;:_~ · . _ ~~~~------ ----·--- -~---· ··-- ____ ..... , .· ... c __ . --- ---~ -·-····-'" '-· - ··- ..... . :. __ .. . . . --..,

- ------~-~--~-~ _ -~,..,.~,~-~,~~~=••-=="-•- --~~ .. ~a~~-a ... ·~a• .. ~.••-•~~""'·'··-·- ~-.~--. . -··· . = -.. - .. • . .. . . ·•· ~-=-':=--~--·····-~·-···- . • - • i

, ---~-"'C~-c~~S - . DIAMETERS' ~TEGR/.ITIONS PRESSLRE ·.. PRESSLRE FLCI-,

) ( · · ~ ·. -,. i 13 ~ C <; 4 ~: g. · 3 C 5 ~ : ~ . g: ~ ~ ~; ~ !l --· - -- ~--2 --~_ . - - ---1 . · · ....... •~ · rsa i- ·c , ........ -:§CJ 4 3 • l 2 9 4 0 • 9 C • 3 5 5 6 9 lil !{). __ .· 3 .· l. . , . 138 -- 6?19.,~ .Ji'; 2337 .5 C.36873

j.~.L ···.: .•. · CG)ERC : .756?0E ... 00 .... ·. QF = · .3.6873E oc ·. PN = .22472E 04 --i . · . · . f '1 N - • 2 3 2,o ~E_.O_L.~~---~- -·· .- .. -.. · --C-~-1-~T~ ·=-. ·•· i-5-5-1tl-E-~-1---·---=~e,_"T4----p~--=--.·~--1-~·e:,1)-~~=-~C--3~--~~, _ .. '.c ........ _

\:;-. =~=== ~-ac;'T~-" ~PRe;Su~ e- ~~ :~~-i-;1 E~4 -·- --- --·- No. o F- I TE RAT I c N s = 4 . _,, .. _ . .......... , '

,t ~: . . ' . ,._., .

- :' ~ ..... ~ .

---~-· .N·o.- OF NO. OF .

_ ....... , .. CYCLES ... DIAMETERS 1

NC. OF · · l-N-ft GR Af-1-e-N-S-- ------ · -

. BOT TOM . DI SC.HAR GE · l-I QUI C PRESSLRE ----: PREs·s-c-R·E- -- -- ----~-------~--- -

FLC\.t . ' i \.,f.,...,' .

.,-~. -

':"--'

)'1 0 .

l .... ~

·3 '

l l l

C 138 138 1·38

c.o. a.a 693g.6 3147.2 c934.q 3019.4 69C7.2 2361.4

C.26610 ·, C.28173 C.36960 C.38300

CGZERC = .83740E 00 _Q_E __ ~~-----183.CCE CC • •, ~ ----~--..-+----....-- -~ ---·-- . _,_ ---- ----- --:-·-- --- - Q N = • 2 5 4 9 0 E O l Q E X I T = • 2 8 2 <; 1 E -C l · - .. -PN = .22675E. C4 ' ~ .-. ' DELTA PM= .17Sl7E 03 { ··• ·._. ..

(' • I

\... .... --

- -t-· .

\..

.,,-..,

L

BCTTCf' PRESSURE = .6902SE 04 ,, . '

t\O. CF ITERATICNS· =

---~-Sv-......... =----..~8-5-e-e-E---e-t

NC.- GF CYCLES

0 l 2 3

NO. 0 F ' DIAMETERS

l 1 l l

CGZERC = .10750E 01 QN = .31?42E 01

-- ---------· ----------· ---- ---·-·------· .--··------ --- - -- --·-·--·-~-,--,--~-----· ---~---==--:··-~-4 9 4 CC E- 'C 2 I NC at' P Q F = - • 2 5 l 2 OE ·c 0 - .. ---- - --·--------

NC. OF I-NT E_G RA.TI ONS

0 138 136 138

BOTTOM PRESSLRE'

c.o 6'127.9 c92C.l

· t874.0

QF = .41'115E · CC QEXIT = .36319E Cl

0 I SCHA•RGE PRESSLRE

o.o 3589.3 34C4. 8 2455.5

LIC:UIC FLC~

C.25120 C.27259 C.39358 C.41115

PN = .23254E 04 DELTA PM= .23c28E 03

BCTTC~ PRESSURE= .6866qE 04 NO. CF ITERATICNS = 4

•

NO. OF CYCLES

0 l 2 3

S = .28500E Ol

- · NC. CF 0IA~ETERS

l 1 1 l

'CGZERO = .1034CE 00 Cf\= .34743E 00

•

NC. OF I ~ T E G r< A T I ON S

0 138 138 138

BOTTOM PRESSLRE

c.o 7CC7.o 7CC6.9 70C6.5

~ QF = .e1g24E-Cl CEXIT = .3493~E CC

DISCHARGE PRESSLRE

o.o 2 316. l 2293.4 2114.5

lICUIC FLC~

C.C7C66 C.C7168 C.C8C38 C.Cdl92

PN = .2CE5CE C4 CELTA F~ - .54~41E Cl

BCTTC~ PRESSURE= .70064E 04 ~a. CF ITERATICNS = 4 . ~-·

S = .28500E 01

"

F = .49~CCE-C2 I~CC~P QF = .98l~OE-Cl

-so-

·. . . 1' _['1

}- 1 - < -- -~- - ----·<-·· =- -·:-~: -=-· - ~--·-~-=.:..~_ --~~ · ·--.,··=-·-=--~-_=-- - -=~-~--. ----- -- .. ---- ·· -----~--,--.~-~ ~ ·-- -~---·--·--·- -----~---,,--~- -~"' -- -~- -· ·-- -----~- --,.-.-----.-. .,.-~ .. ----·-~---~~- -·------~--.,---·~---" -· -----~----,--. -~~ t;,I i ...... .

I - -

NG. GF "DI At' ET ER S ·

·l l . l

NC. OF · -- I t~ T E G R A T I ON S .

0 138 138 .

~OTTCM. PRESSLRE _

. . : ______ .. c·C e O • ---. 70C5.l

7CC5.2 .. ·

0 I SCt1ARGE PRESSLRE

.. , .. · .a 0 o .. 2021.1·

. 2026.8 _

LICUIC FLC"'

. C.09S.l4 .... C.09778

·_ C.09468. __

. ti

. Vi . • . . i'. ~ . '.!~·:··•-.'·····---~·., .. ,•.-, .. ---~---·_-.,,.-:"'!'o<·t;,J····-.· \ . - ~ '·(:

--, ..... : _______ ,_.:....._ _____ J;.GZ.E.8.Q __ _:_;; __ ... lJ..9.4-0-E----00 . ~C.....,-:;;~.S.4 6-S--lE-~-O-l-~-- P-N---~----.-2--G 7-&5 E---04--------~-- ----- -ti .,.., . . . 1,,

-- l~:./- . ----- -- ----- --.-·- ---------ON· -:::. · .40 2 71-E- --GG-- ---------~------- -- -~- QE-X I l-·:--~ --.-4-0 3 3 c; E -0-0- - ------C E-l TA-- PM-- =- --.1-3-G-6-l-E---01----- -·--· -- · -- 1 .. H

. > • . ,.. . . ··'·'' .. ' ••'· . .··· .;i ~-' .--~'' ' · ... '. . .· < . . ,. ., . ' .

BCTTC~ PRESSURE= .70053E 04· NO. OF ITERATICNS = 3 \·

. r·_ ~--. ,,,. ·:.: ... ;·s ~-: ..• --2as0Cie oi-- ._ ....... -.. - .---·----···,=···;c~,.~;~_4:94CC:E-C2 INC-Ot'P QF = . ,-:.\ C ,

• 208COE CO . fo.14'1" ' .

_;_~;:--:;:·:--N.O~.-- '.o f~~~ .: NO. CF DIAMETERS .....

l 0 CYCLES

0 l

. ·r---·"'- .•. ' "

2 J

:~· 1 . l

~-~~-' ~GZERG = • 24270E 00 QN = .81097E 00

0

NC. OF INTEGRATIONS

0 138 138

BOTTOM PR es·sLRE .

c.o t;gc;c.1 e:c;c;c.s

. DISCHARGE PRESStRE

a.a 1855.7 1881.5

LICUIC FLCh

C~20SCO C.20482 C.18C6l

Q-F ·= ;··1aC6lE cc PN .= .2C'i2eE C4 QEXIT = .el~~6E CC DELTA PM·= .27~5EE C2

BCTTC, PRESSURE= .69933E 04 NO. CF ITE~ATICNS = 3

C

0 -NG. OF NG. CF NC. OF BOTTOM DISCHARGE LICUIC ()'1 CYCLES DIAMETERS ;:~,;;, 0 l

C ~ ----~i 1-) , ~

...

INTEGRATIONS PRESSlRE PRESSLRE FLC~ 0 c.o o.o C.22770 .

138 cCJS4.8 lC,44.0 C.22532 13-8 ~~65.l 1962.4 C.20810 ..

0 QGZERO = .28850E 00 . CJF - .2081CE CC PN = .2lC16E 04 -QN = .959l5E 00 QEXIT - .g74c<iE CC OELTA p~ - .38C24E C2 - -r --· .. -

' ..

BCTTC~ PRESSURE= .• 69875E 04 ~a. CF ITERATICNS = 3

.,-, \_)

0

C

0

. ··") I ,........ ·i , i - ~

: ' ·'· i ~~,,, ,, .,....

NO . ._ OF CVCLES

C l 2

S :s .28500E Ol

--r··-- NO. CF OIA~ETERS

l 1 l

~GZERC·= .36410E 00 Cf\= .11916E 01

F = .49~CCE-C2 INCO~P QF = .24330E CO

NC. OF BOTTCM DISCHARGE LI,UIC INTEGRATIONS PRESSLRE PRESSLRE FLCY\

C c.o C.G C.2433C 138 6 CJ 1.7. 6 2154.'1 C.2436g

• 138 t:.<;77.5 2151.9 C.24634

QF = .2463LtE cc PN - .213l1E G4 -CEXIT - .123ClE Cl CELTA Pt-' - .5556CE C2 - -

BCTTC~ PRESSURE= .69771E 04 ~a. CF lTERATICNS = 3

NC. OF

S • .28500E Ol

NO. OF NC. OF

Fm .494CCE-C2 INCO~P Q~ = .25120E CO

BOllCM DISCHARGE LIC:LIC

-s1-

. -& . ~-. !l

. w . ~i,-

. . I • . .

i.:: f

. CY C .L E S · 0 I A M ET E R S - 1 N T E G RA T I ON S 0 · . l _. . 0

l. l · - 138 C • · 2 . · 1 · : . 'l 38 ·

i . :. 3 . 0 • • ••

0

• • • • • • : ' • • l - . . 1 ·3 8

PRESSLRE Co O ,.·· ·

6<l11o5· •- 6CJ7lo0 _ ·._-

- ~: -; -c)<)c7~ l -· · ··

PRESSLRE C. 0 -

. : 234405 2320.l 2168. 9.

fLCL\ C.25120 C.25435_

· ·C.-27468 c.21132 , ~ct· .. , . . . .

· C:GZERG = .4-3360E. 00 . QF = .27732E CC PN = .2150CE 04 .· . . . .· .' . ' .. ·· .. · .. QN = • l4049E ot .•. · .. · .. QEXIT = • l4l:49E Cl ··. DELTA PM = • 73A38E C2 ..•. ·· ·---r~~-.-. ;__-----'-~~".'."~:------'-----· ------------c.:, .. :.._ _______ . ________ ----· ''-··----;-~:---:---~------·---------------:-------··--·--:---- - - ----.=..-------- - - :..:--. -- -~.,;.'._:_~-~\--~_....._-____ . _. . --~------- --------,::-.-,.--~-----~----------'--.. _____ --'--·--~---------~~~--:-..:.C.....------'-------_ ---~-------- --,-----:-··- ... · ________ .. ______ ------------------'-~----- ------'-:·-:· .· ~~--.. _ -·~--~-~~----:

..... -1,_,_-···.• . . ·-- - .. --· - - -- ----- ·---- -- - - ------·. -------- -··-----·----- - ··- - -- - - - _____ _:_ ____________ ------ ---------~---- ------- --- - ---- --·----· .. ._ - --- --·- --------4-~-- -~---- - . ..

.. BCTTC~ PRESSURE= .69666E 04 ~

NO. Cf llERATICNS = 4

S = .2S.5-00E Ol f. = .• 494CCE-C2 INCQr,,p 9-f. = .25 1HOE GO , •• ·•·,- - !' -•••.' H ·-·~· • .,· . _ _,., . • •· .- • • ·0s•·-•-··. • - •: ' •. • - . ·-· · • .-.:.,-·-·.-;;:-· •·-• ...... •··-- • -···· ...... - ... ---··· • ._ ,. ...... _, __ ,__.:.. • ..:._s,_.._:_______ . .. ...... ··-···•·••·· -- ... . ............. ·····.·,· • .... ,,.. .. -· •· ............... • ...... ·-·----,..-· ...... _., •·-·•·· .. ,_ - ""°" ... -•• •. , ..

' . . ,.,·

-~~-

NO. OF .· CYCLES . ·

. 0 1 2

tr~:"·• .... ~.-- •. - ... --- l. 3 - ,-\

\. . --

·. NO. OF DIAMETERS

l l l l

NC. OF ,,

. <}. INTEGRATIONS 0

138 138 138

"BOTTOM - DISCHARGE LICUIC PR-ES SLRE , PRESSLRE FLC~

C;. 0 o.o C.25'il0 c'1t 7. 2 ·2411.6 C.26330 t9t6.4 237CJ.4 c.2ac;5g c9t1.o 2166.7 c.2q314

~"·'·'.

CGZERC = .47420E 00 . QF = .29314-E CO--_ -. ON---= .-152076·--0-h-· --~ ---- -·-QEX l T- -= .1602 lE--C 1

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~-lartin, C. 'B. ''A 'J.wo-Phase Flo~ -Analysi~ · of the AirLift Pump.'' ~1.s. ··Thesis~ 1'iechanical, Engineering, Lehigh

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. . 2. Stenning 9 Ao Ho and i1artin, C. B. ''An Analytical and Experi1nental Study of Air-Lift Pump Performance,'' J. of Pol'V'er, ·Trans$ ASlv1E, April 1968, PP• 106all0.

. 3. Owczarek, J. A.· Fundamentals .Qf Gas Dynamics. Scranton, ·Pennsylvania: Intern a tiona1· Textbook_ co·., 1964.

-------~----·-4-~ ____ Griffith, P. and \'lalli&, G. B. ''Two-Phase Slug Plow,'' . J. of Jieat Transfer, Trans. AS]w1E,- Vol. 83, August 1961,. pp. 307--320-.

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s. Golomb, M. and Shanks, M. Elements of Ordinary Differential. _ Egt1ations. Ne\\' Yorl<: i'lcGra\·i~Hill, Inc., 196.S. _________________ --. ____________ .: .. : _____ ··_

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6. Piclcert, F. ''The Theory of the Air-Lift Pump," Engineering, Vol. 134, 1932, pp. 19-20.

7. Griffith, P. ''The Prediction of Low .Qt1ality Boiling Voids,'' J. of Heat Transfer, Trans. ASf.IB, Vol. ~6, Augu~t 1964, pp. 327-333.

8. \\Fard, C. N. and Kessler, L. H. ''Experimental St1.1dy of Air Lift Pumps and Application of Results to Design," Bulletin of the_jlniversi ty of Vlisconsin, Engineering Series, Vol. IX, no. 4, Madison, 1924.

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Alleri·· Schoeneck is the son of ~-Ir.· and Mrs. Kenneth Otto.---~---· .. _::.-______ ---···· .. --··-····· - . -- . .--~-~:.:·.,-,-.•.. '- ' . ·--: -·.,--:---:--·--- .-----··-----

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'• C "\ •..;, • Scl,oeneck. / He attended grammar school in Illinois. and was graduated· from Dubuque Senior 1-Iigh Sct1ool in 1959.

~lr. Schoeneck· attended Iowa State University as an . , ·-•r,; . ·•. · undergraduate from September, 1959 to July, 1964, \-then he

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.Engineering- .and a Bachelor of· Scie11ce Degree in Industrial Engineering. I-Ie is a member of the Gamma Epsilon Sigma

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'· honor society. 'I

In August, 1964, Mr. Schoeneck went to work for the Ingersoll-Rand Company, Phillipsburg, New Jersey Plant. He is employed as a Project EnP,ineer, Turbo Products Engineering Depart~nt, where his functions include the

· design and development· of s~ngle and multistage centrifugal compressors and hot gas expanders.

He is married to the former Susan Burrowes and they - ' have one child, Kenneth Allen, Jr.

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a compressible and incompressible flow analysis of the air

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