Calibration of flow nozzles using water with extrapolation to steam

FoLro

T47

~

CE'2-ci•E,q-37

41'·2.

CALIBRATION OF FLOW NOZZLES

.

USING WATER WITH EXTRAPOLATION TO STEAM

by

Albert G. Mercer

for

Westinghouse Elect

r

ic Corporation

Engineering Research Center Civil Engineering Department Colorado State University Fort Collins, Colorado 80521

CERB-69AGM37

f

1

l

I

ENGINEERING

RESEARCH CENTER

Civil Engineering Department

Colorado State University Fort Collins, Colorado 80521

CAL

IBRATION OF FLOW

NOZZLES USING WATER WITH EXTRAPOLATION TO STEN I

Pr

epared

by Albert G.

M

ercer

June 1969

for

We

stinghouse Electric

C~rporation Pl

ant

Apparatus Division

Bo

x

1047

Pittsburgh, Pa. 15230

CER68-69AG~l37

CALIBRATION

OF

FLOW NOZZLES

USI1

G WATER WITH

EXTR.A..POLATION TO STEAM

Introduction and

Summary

This

report presents

the

results

of

calibration tests performed

on two

steam flow nozzles at the

Engineering

Research Center of

Colorado

State University in

accordance with Westinghouse

Electric

Corporation

Purchase

Order

56-IC-46472-G dated April 22, 1969. The

aim of

t he calibration

was

to relate the stea m flow rate through the nozzles to

the

difference in pressure sensed at the nozzle taps. The

calibration

w as performe d ,vi th

water

in place of steam

and

the results

of the

calibration

were

corre ct ed for ste am analytically. The

primary

results for

water

are presented in Table II as a set of data

relating

discharg e coefficient to Reynold's number.

The

Reynold's numbers for the

tests with

water

are lower

than

those

that will be experienced

with

steam.

Figure

4 shows the

empirical

curve determined for the extrapolation of the data to the

higher

Reynold's numbers

.

The calibration data for the two nozzles were so close that one

curve

fits both sets.

The

relationship between pressure differeilce and steam flow

. given

in Table III and Figure 5

were obtained

from

the

empirical

curve

of Figure 4

with

analytically

determined

corrections for the

thermal properties

of steam.

The · calibration

of

Nozzle l

was performed May 8, 1969 and the

calibration

of

Nozzle

2 was done on

May

17,

1969.

The

calibration data

was collect ed by

Mike Wittington, and

the data reduction

was

done

by Ikram

Ul-1-iaque

, both graduate

stud ents in Enginee ring.

Theor etical Conside r ations

The basic differential equation for th e flo w of a compressib l e fluid i n a pipe of varying cr oss section

(as

for example a flow nozzl e) . with negligib l e h eat lo st to the pipe wa lls, is

2

Jdu + d(

pv)

+

d(~)

= 0 (1)

2g

where: u is th e intern a l energy

p

is th e absolute pressure

V

is th e sp ecific volume

V

is

th

e m ean velocity across the pip e

g

is th e accelerati on of gravity and

J

is th e conversion fro m mechani ca l en ergy to heat

.

The t erms of Equation

1

r epre s ent ch anges in int erna l energy, flo w

work

and kin etic energy in th 2.t order. The middle t erm of Equat ion 1

can

be divided into t w o parts to give

2

Jd U + p

d

V + V

d

p +

de~) 0

2g =

The first t w o t erms of Equa tion 2 appea r in th e defi nition of entropy s

JTds =

Jdu

+

pdv

(2)

(3)

The

t

wo t erms on th e 1eft in Equation 3 are conside r ed t herma l t erms

and th e two t erms on th e r ight are conside red m echanical t erms .

Friction al ways causes a

lo

ss of energy from th e me chanica l terms

to th e th

erma

l t erms

.

Th ere is no proc ess in a flow

-conduit

that

can cause energy t o flow in th e r everse dir ection.

3

For frictionl

ess

flow, Equation 2 becomes

2 d d(y_) = O

V p + 2g ⁽⁴⁾

Under

conditions of incompressibility vis constant and Equ

ation 4 can

be integrated to give

(S)

Equation S can be expressed in ter

ms

of the fl01v rate

F,

the cross sectiona l areas A

1 and A

2 , and a calibration coe fficient Kin the form

(6)

U

nder ideal conditions, the coefficient K would be unity but in practice it is somewhat less. The major cause for this difference is the bound

ary

layer w hich has the effect of making the noz

zle seem

smaller than it is (displacemen t thickness). Since the boundary

layer

thickness is less with higher flo

ws

, the coefficient character- istically approaches unity as

.

the flow is increased. The boundary layer thickness can be shown to be a function of the Reynold's number R

· defined

by

e

VD

\) (7)

where D is th

e

pipe diameter and

v

is the kinematic viscosity

.

It foll

•

::iws that

K

is basic

a

lly a function of

R •

e

The pri

mary

calibration usin

g wa

t

er

obt

ained

relat

ed

va lues of K and

R •

e

As

long as

t

he expansion of steam is not great the

boundary layer th

ickness is no

t

materially affe ct

ed

and the K - R

e rel

a tionship

obtained for water will

be app

licable

to st eam as we

ll. On

ce the

id

eal f l

ow

r ate pr essur e

di

fference

r

elati

onship i

s obtained for

steam

the

coeffi

ci ent K

can

be

appli

ed.

To obt

ain

t

he pr essure differenc e for ste

am,

Equat ion

4

mus

t be int

eg=ated with t he constraint that

t

he entropy of the steam

be

~onstant

. I

ntegration of t he

l

eft hand t erm. cannot be

performed unless

the p-v

r

elati

onship

is kno1m . However th e m ean value

t

he

orem

for int egra ls state s that there is a

con

stant k betwee n

zero

and

1

such th at

v2 v2

1 2

z:g- - 2g

⁽⁹⁾

w

here flv is

th

e change, in specific volume

. T

hen to get an expression

parall

el

to

Equation

6

we

can write

Equation

9

as

F = -

- - - - - - --- --

⁽¹⁰⁾

- /

_ (v+6v)2

Al A2

If 6v is

sma

ll

enough

th

e assumption can

b

e ma de that v

varies

linearly with p meaning that k

= 0.5.

If flv is

too l

arge

the no

zz

le

can be

treate

d in steps with each step hav ing a sma ll enough

change in area

that the assump tion of

l

ineari

t

y

i

s good.

To

solve Equat ion

10, t

he follo

l'l

in g st eps can be performed·:

Step 1. Set K = 1

St

ep 2.

C

ompute

v

corre

sponding

to p

1 and

th

e percent

dry

ness x

1 fro

m steam

tabl es

St

ep

3.

Set ~v

= 0

j

I

_I

·1

i

6

Step

4.

Compute

P2

St

ep 5. Comput

e

th

e

percent dryness

_x2

for zero entropy

change

f

rom

steam tables.

Step 6. Compute

tw

for P2 and

_x2

Step 7. Rep

eat

steps

4

through 6 to convergence

Step 8.

Correct

F

for co

e

fficient

K.

D

escription of Nozzles

The geo

me

try of the noz

zles

are shown in Figure 1. The nozzles when received had identical m

ark

ings:

1 s. o. #3395-1

i Mil -

S2319

4A

C/S

HT

#213060

To

distin

guish

between them th

ey

were stamped 1 and 2 on the outside face of th

I e

pressure

manifo

ld. ~licrometer r

eading

of ~1

e

thro

at

diam- eter averaged

I

4.516 inches at room t

emperature (about 20°C)

for both

nozzles. The upstr

eam

pipe and t

ap

was furnished by CSU and th

e

diame ter at the t

ap

was measured as 6

.

081 inches

.

Under 1000 psia and 544°F it is estimated that th

e

di

~1eter wo

uld expand by 0.34 percent which would

mean

an increase in flmv rat

e

of 0.68 percent.

This expansion was

a

llo\'Jed for in the computations .

D

escription of th

e

Test Facility

·

Th

e

tests were

performed

at the

m.:iin

flow calibration facilities

of th

e

Enginccr ine Rese

ar

ch Center whi

ch

arc shown diagram

ma

tically in

Figure 2. The

no

zzles were insta ll ed

i

n the

12-inch line

as shown in

Fi

gure 3.

To find th

e flo w rate, water

fro

m

the no

zz

l

e is

diverted

into

volumetric

tanks

for

a sp ecified period of

ti

me meas

ur

ed and

controll

ed by a puls e coun ter op er a ting

off th

e AC power line

. Th

e volume of w a ter collec t ed in

th

e tank is measured by a prec ision hook gauge . Th e w ater temp er a ture is m easur ed

on s

amp

l

es

coll

e cted from the flo

^·1

div ert er. The pres sur e difference was meas ured by two

separa t e sys

t

ems s e

t up

especia

ll

y for th ese test s (Mercury manometers usually

used

w ere not a

ll

owed und er the sp ecific ations). For high

flows, a

set

of

three differenti al manometers each 12

f

eet

lo

ng with acetyl ene t e trabromide

(S.G.

2.95) as th e indicating

flu

id were pro- vided. Th ese we r e con ne ct ed to th e no zz

l

e

t

aps in seri es with each

oth

er

. Th

e

tubes

had

l

arge bor es

(9/1

6-

inch) for accurat

e meniscus

form

ati

on. · For

low flows, an

invert

ed air-water manometer was

us

ed

.

for

, l

arger manometer de

fl

ections

.

I

The

propos al for this study st a t ed

t

hat open wel

l

hook

_

guages would b e

used.for low

flo

w

s but ci1e se proved

to

b e impract ic al. The air-water manometer is

no

t quit e as accurate as th e

op

en well system was predicted

t

o be and

,

as a

r

esu

lt, th

e data for

the lowes

t flo ws

show

the most scatte r .

D

es

cri

pti on

of t

he Test Proc edure

Each of

th

e flo w no zz l es were calibrat ed for 15 di s ch arges

, adh

ering as c

l

ose as possible to th e quantities outlined in the

pr

e-calibration data of Tab

l

e I. Th e only differen ce was

th

e

- - __

^..._..

___

"""'"_.,.._ ....

Table I. Pre-calibration Data for Calibration of Nestinghouse Steam Flow Nozzles

Run No. 1 2 3 4 5

.

6 7 8

9

10 11 12 13 14 15

Flow rate

(gpm) 450 675 900 1125 1350 1575 1800 2025 2250 2475 2700 2925 3150 3375 3600 Run time

(min) 10.0 7.0 5.0 4.0 3.5 3.0 2.5 2.5 2.0 2.0 1.75 1.75 1.5 1.5 1.5 Sample

(gals) 4500 4725 4500 4500 4725 4725 4500 5062 4500 4950 4725 5120 4725 5060 5400 Volume error

per rdg. (gals) 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2

Number of

readings 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Percent volume

\

error

.09 .08 .08 .09 .08 .08 .09 .08 .09 .08 .08 .08 .08 .08 .07 Percent time

error

. 0 0 .00 .oo ^.01 .01 .02 .03 .03 .04 .04 .05 .05 .06 .06 .06 (Ac curacy of 60 cps)

Manomet. er system I Hook gauges 1 manometer

.

2 manometers 3 man.

Total

Man

.

Def. (ft) 0.87 2.0 3.5 5.5 4.0 5.5 7. 2 · 9.1 11.2 14.1 16.9 19;0 22.1

.25. 2 28.8

Error per

rdg. (

ft

) .001 .001 .001 .001 .005 .005 .005 .005 .005 .005 .005 .005 .005 .005 .005 Number of

readings (ft) 2 2 2 2 2 2 2 2 2 4 4 4 4 6 6

Percent pres-

sure error

.22

.10 .06 .04 .25 .20 .14 .11 .09 .14

.12

.10 .09 .12 .10

substitution of an inverted air-\'/ater manometer for the hook guage

. .

arra_

ngement indicated for Runs

1 through

4. The

r

eading error for

this

man:)meter nominally would be . 005 feet, as for the oth

ers·, but the lon

g r n

ti

me for these runs allm1•ed rep

eated readings for high

er accuracy.

The step-by-step procedure adopted was as follows:

St

ep

1.

With

th

e pumps operating and with water recirculating

through

the nozzle but bypass ing the volume tric tank,

th

e valves were adjusted to obtain the desired flo

w

rate as indicated by the precalcul

ated manome

t

er deflections of Table I.

St

ep 2. The follow ing data was record

ed:

a. the tem

pera ture of flowing water

b. th

e tempe ratur

e

of aiD

c. th

e temperatur

e

of w

ater

bled from m

anome

t

er

lines

d. the l

evel of \'later in the volume tric tanks

e. the

manome t

er l

evels.

For

some runs the manomete r lines were bled to check for air accumulation and the manometers were short circuit

ed to check

the

null

balance.

Step

3. The timer was started which automatically div

erts t}:le

flow into the volume tric

tanks,

St

ep

4

. Manometer

l

evels were measured repeatedly durin

g th

e run

to get

as many readings as time allowed.

St

ep

5. Th

e tim

er was

stopped which automa tically div

erts th

e flow to the sump .

Step

6. St

ep

2 was

repea

t

ed.

10

St

ep 7. The volumetr ic tanks

were emptied

by pumpi_

ng

into the

s

ump and pre li minary cal cul ations

were made

to catch

gross errors. Sev

era l r uns were r epeat ed on the bas is t hat th er e were gross

errors but only tho

se for \vhich th e

source of th

e

error

could be id en tified

.

E

valuation of Syst emati c

Errors

The

calibrati on

equipm

ent

has

been designed to m inimiz e or

correct for

most causes of s ystem at ic

error.

Th e sourc es of

possible s

ystemati c error

are:

1.

calibr ati on of th e vol umetric tank

2.

vari ations i n th e 60 cycl e A C

curr

ent fo r th e timer

"3. non-uniformity

of th e flow

_

div erting proc es s

4.

variations

of

th e densi ty

and

vi scosi ty of th e flo wing

\-l

at er

5.

variation

of

t he s pecific we i gh t

of

th e fl uid

in

t he

manom

et

er

and t

he connect ing

l

in es

6.

ina ccuracy of the

manome

t er sca l es

7.

men iscus

effects

8.

local vari at ion of the

acceleration

of grav ity

9.

bouyancy of t he flu ids du e to loc a l barom etri c

pressure 10.

tank leakage

and

evaporation.

The evaluation

and

treatm

ent

of th ese error sourc es we r e

as

follo ws:

1.

Volu me tric t ank ca lib ration

The

t anks

have

b een ca l ibrated periodically by filling

t

h em with water in

240 pound

incr ements

weigh ed on

a

sp ecia l scal e ca libra t ed b y

weights checked

by th e

National Burehu

of

Standards

in

Boulder, Colorado.

The depth

volw

e

_

re

lationship

is

known

accurate to

at

le ast ±2

gallons over

th

e

6000 ga llon r ange

.

2.

Timer variati ons The accuracy of the timer depends on the accuracy of the 60 cycle A.C. power supply. The power supply has been checked periodically against th e gov ernmen t time signal from Station WWV in Fort Collins. The average error experienced has varied from

.09 percent for a period of 1 minute to .0003 percent for 10 minutes.

3.

Div ersion proc

ess

The

· tim

e r equired to complete th e divers ion from th e sump to the volumetric tank, or vice-versa, is less than t wo second s . The equipment is designed so that the amount of flo w diverted vari es lin ear ly with time during th at int erval .· If it is perfectly linear, th ere is no error introduced by th e diversion process. How- ever, there is no way to evaluate this possible source of error except

•to .

see if there is a systematic error associated with sampling ti me and none h as been detected.

4.

Density and viscosity of the flo

w

ing water The density and viscosi ty of water was de t ermined fro m th e water t emperature using stand ard t ab l es* . Temperatures \v ere r ead with a precision thermome ter to 0.5° F. This perm itted density to be computed to

.

02 percent and

I

viscosi ty to ~bout 1,0 percent. This accuracy for visco sity is acceptab le since th e discharge co e ffic ient was found to vary about .01 pe~c ent for a one perc ent change in viscosity. Incidently , th e mass density is the import an t prop erty of the flo wing fluid rather than the specific weight as f ar as th e flow no zz le is concerned .

5. Specific weight of manometer fluid The specific weight of the manometer fluid was dete rmined from t

emper:i ture

r eadings of th e

* Meyer, C.A.

,

~!cClintock, R. 13

.

, Siverstri, G .J. and Spencer, R.G. JR.,

Thermodyn

am

ic and Tr

ansport Properties

of Steam,

AS~IE

Research Committee

on th

e Pro perties of Steam.

^AS~lE,

Un ited Engineering Center, New York,

1967.

12

s

urrounding air

.

The r el ationship of

th

e sp

ecific weight

of the

ac

etylene tetrabromide to t empera ture was evaluated from pycnometer

determinations.

6.

Manomete rs scales The m anometer scales ,-,ere

st

andard st ee l

m

eas

uring

tap es accurate to at

l

east

.

002 feet over ten f eet

.

7.

Meniscus effects

The

capillar ity associat

ed with

the meniscus is idea

l

ly balanc ed out in a

di

fferentia

l

m

anometer.

Never-

t

he

less,

large

t

ub

e

di ameters (9/16-lnch) were used and the tubes were

t

horoughly cleaned and fill ed with fresh

man

ometer fluid.

8.

Acceleration of gravity

Th

e var iation of g

ov

er th e

U

nited St~tes is 0.2 percent

. In

Colorado the value differs about

.04 percent

fro m a value of 980.0 cm /se c

2

:

9.

Bouyancy of air

The

bouy ancy of th

e

a

tmo

sphere affects

the specific weight

of ivater by about 0

.

13 percen t. The difference

in thi

s

bouyancy at

Fort

Coll

ins (Elev 5000) and at sea level is about

.0

2 percent

.

10.

Tank le akage and evapora tion Th e amounts of water lost due

to tank l

eakag~ and evaporation are

comp

let e ly negligible.

Evalu

ation of R andom Errors

R

andom errors are

th

ose tha

t

cause scatter of the data. They arise mainly fro

m the

determination of the volume of

th

e sample and the

deflection

of th

e

manometer.

The

volume of w

a

t

er collec

t ed in t he vo

l

ume tric tanks is

determi ned by me

as

uring

th

e height of th e wat

er

surface with a hook

guagc befor

e

and aft

e

r th e run

.

Th e hook

·

guagc has a

vernier which

reads t o

0.001

feet, equiva l

ent

to about

0.8

ga llo ns of t ank

capacity. However , it is di fficul t to repeat r eadings to this accura cy and

±2

gallons is a reasonabl

e

estima t

e

of th e maximum r andom error for each meas urement. T ab le I gives f or each run the r esulting percen t error in volume expect

ed

fr om each of t he two mea

surements

r equi r

ed.

Th

e manomet

er defle

ction is determined by

measur

ing the eleva t ion of

th

e top of both menis cuses r e lative t o the t ape scale. Th e tap es are gr aduat ed in hundr

edths of a foot and

r eadings are estimated to thousandths of a foot. However

,

flu ctuation of th e

me

niscu s level and error s of para ll ax make

.005

f ee t a r easonab le maximum error for

manometer readings. The l ast r ow of Table I shows th

e

expected

.

percent error i n pressure f or this order of accuracy. Thes

e

figures are r easonab l e exc

ept

for th e first fou r run 6 which are based o"n ho ok guages readable to

.001

i nch

.

Without hook guages , the p

ercen

t

pressu re error can be expected to b e about five ti

mes

the v2

.

lue

given. It

shoul d be remembered t hat th

e

square root of pressure is

only

one ha lf as much as fo r pressure itself

.

Furthermore, the mano meter 4eflections r ecorded were th

e

average of a number of readings for th ese first runs , which should i

mprove

the accuracy

.

.

Results for Wat er

The va lue of K and R obt ain

ed for

th e

15

runs for each fl ow

e

nozzl e are giv en in Tab l e II. The va l ues are also shown plotted in Figure 4. The scatter of t he data is emphasized by th e larg

e

scale chosen for K. The data for th e t wo no

zz

l es were s o clo se tha t on

e

curve could be fi tt

ecl

t o both sets . Th e curve chosen was th

e

exponc1:tial

14

curve discussed earlier. The coefficients A and B were chosen to produce a least squares fit. With A = 0. 35 and B = 0 .45, the data of Nozzle 1 fitted with a root-mean-square error of 0.34 percent and Nozzle 2 with an error of 0.32 percent.

Table II

Discharge Coefficients and Reynold's Numbers for the Flow Nozzles

Nozzle 1 Nozzle 2

K R K R

e e

.9681 1.94(105

) .9778 2.51(105

)

.9686 3.33 .9656 3.64

.9761 4.38 .9746 4.86

. 9.712 5.41 .9736 6.15

.9670 6.37 .9742 7.28

.9740 7.37 .9770 8.47

.9696 8.11 .9770 9. 79

.9700 9.08 .9761 11. 23

.9736 9.91 .9773 12.42

.9770 11. 83 .9788 14.08

.9809 13.10 .9802 15.83

.9788 14.06 .9816 16.95

.9810 15.03 .9831 18 .10

.9778 15.99 .9824 19.15

.9801 16. 89 .9818 19.97

~

Results for Steam

Pressure differences and flow rates are related in

Table

III

for

dry saturated

st

eam.

Th

e

t

ab

le covers

flow

rates from

30,000

to

300,000 pounds of

st

eam

per hour

,

with

pipe line

pressures

varying from 500 to 1000

psia.

The

pressure

diffe

r

en

ce

values were

determinc.d

by comp

uter

using

th

e

step-by-st

ep

proc

edure

outlin

ed earlier.

Pressure differences assuming incompressibility arc

shown

in

brackets

in Table III for

comparison.

The

steam prop

er

ti

es including

viscosity were

obtained

from

tables

in

the

ASME

publicat1on

r

e

ferr

ed

to

earlier. Key values of Table III

arc plo_tt

ed in

Figure

5 to provide a more convenient form for use.

Of course,

th

e steam

in

the

pip

e line may

not be in

a

dry

saturation

condition.

Table IV

sh0\•1 s

₁ for

comp aris

on,

pres sur

e dif- ferences

for steam ,-1h ich

is 99 percent

dry

.

. .

C)LOAADn STATE UNIVfRS!TY

STEA14 QJAL1TY 100 PERCENT ORY - - - -- - - --'-T_A~RULATEO VALJES ARE PRESSURE OIFFI tN PSI FOR ~Tr.AM

B~ACKETEn VALJES ARE PRESSURE I..,--?ST ASSU14I~G t~r.o.,.PREss-i~tLlrv PIPE PAF:SSURE

PSIA

- -- - _ -2.0J) - -·. -

(

560

_ _ _ ...:,6(10

FLOw RATEi TN ll'OUNOS Pl:R ~nUR

30000 60000 9ooon 120000 l'iOOOO 180000 210000 2i.onno 300000

,i.o5 1,617 ;1~-5~ -.,57?. 10,419 15 294 21,327 2A,7n6 37,7n• 48,736 --- -· ·--

;i.0111 1,6011 3,604> 6,396T. -9,979T 14,355> 19,5211 1 25,477> 32,2n1 1 39,759;·

• 389 1,553 3,512 6, 30? 9,979 14,621; 20. 351 27, 1l 4• 35, 7111 45,956 ,388) 1,544) 3,463) 6,145) 9,58A) 13,792) ( 1 ~. 7<;5) ( 24,478) 30,9<;91 18,200) ,3111 1,49ft 3,376 6,05?. 9,57?. 11;.001 19,1tc;4 2-..050 33,9A1 43,521 ,373) t l,41'!51 I 3,332) ( S,912) I 9,2251 (13,270) ( \R,045) I 23,"i5ll ( 29,781'!) 36,754) .. • J.60_ _ _ l_._4..3_0 _ ~-3..Zll _ ___

s---~- °--- -

..9_...il L _ _ u .. 4.!t.O . _l 13 ._~J 6 ___ 24_190 0 _ 32, )Q J 41 •. 3 '._l9_

,360) ( 1,4311). 3,210) ( 5,b9t,) ( A,81Hl ( 12,7R4) ( \7,3A4) ( 22,,r,A9) ( 2~,&97) 35.408)

,347 l,3A7 3,131 5,605 A,849 12,917 t7,BR4 23,A'i2 ]0,9c:;6 39,384

,3471 l,3!10) 3,096) 5,4941 8,572> 12,3'301 16,767) I 21,RA3) 27,6781 I 34,151>

,335 1,339 3,021 5,405 R,526 12,433 17.190 _.?2,_f:!!38 __ 2..? .. ~4...3 ___ 3.L .6.17 t ,3351 I 1,3331 .( 2,9901 I 5,3051 I l'l,277) I 11,90!',·l ( \6,191) I 21,1311 26,72,r,) I 32,9771 - - - -~c.""'2"'"'0,___ _ _ __ _.32~ ____ l..291t __ . .2.21.a ___ s...z...Le. .... ___ 8L2.26 ______ l.L.9..B.3 ___ -1.b...S.SJ ___ 22.o.01t-_ 2~.1;41 36~o'J.5

I ,3241 t 1,2~8) I 2,890) ( 5,1281 ( l'l,000) I 11,501'!1 I JS,649) I 20,425) I 25,8]3) 31,R74l

,314 1,251 2,822 S,043 7,945 ll,564 15,955 21,lAS 27,344 3/t,553

,3131 t 1,246) I 2.796) 4,9611 7,7411 I 11,1141 I 15,1411 19,71,ll I 24,9Q4) 30,8391 ,304 1,211 2,731 4,879 7.682 11,174 15,402 2n,4?7 2&,3;>9 33,211 l't60

.3031 I 1,207) I 2,708) 4,804) ( 7,496) I 10,782) ( 14,662) I 19,136) I 24,204) I 29,864)

- - - - -"'~B-'-n _ _ __ _ ____,,,..,2,_9'--'--4 ___ _..!..!.ll..4 _ __ 2 646 .~ .! 7?_5 ___ ___ 7.~:3~-- __ 1_9.._8QJ_ ___ 4 8':J_~-- _l9..t22l ____ ]!.?-'38~ _ __3l ,9!7 ________ _ ,29,.> I l.170) I 2,624) 4,6St,l I 7,265) I 10.450) I 14,211> I 18,5471 I 23,4S8) I ?8,943)

_ _ _ _ 1=0=0 _ _ _ ___ ,285 1. JB 2,s&s ,~579_ 1,201 101461 j,1391 19,051 z4,So6 30,024

~ 851 1,135) 2,545) 4,516) 7,047) 10,116) 13,784) 17,990) 2'-2.753) 28,0741

- -_ !. 2 7 7 _ ·-- ! • 1 0 5 _ 2 • 4_ 8 9 4, 4 4 I 6, 9 !:12 l O , 1 3 6 l 3, 9 3 9 l 8, 4 15 2 3 • & f!J 2 9 , 7 4 8 I ,277> l,l'lll 2,471) 4.3841 6,840) 9.819! \3.JAO) 17,463) 22,0Abl ( 27,252) _ _ _ _ _ ,~4._c.ll _ _ _ __ _ ,_~6.'!_ . . . 1.on 2.417 4,311 6,774 9,830 1:t.So9 l7,A"i2 22.._90~ _28J45 __

.269> 1,070) 2,400) 4,259) 6,&451 9,5581 12.9971 ln,9631 21,4S51 I 2&,4721

,262 1,043 2,348 4,188 ,r,,57A 9,541 13,104 17,104• 22,l'l7 27,811

,2611 1,0401 2,333> 4,1401 6,459> 9,2QJ> 12.614> 16,4A9> 20,sss, ?5,732>

,254 1,014 2,283 4,071 6,392 9,267 12,721 ll't,7R8 21,509 2&.937

- _7]JO ___ _

,254) t 1,011 l 2,269) 4,026) 6,282) 9.036) 12,288) 16,018) . ( 20,284) 25,028) _ _ _ _ ...,_,A_,..n..,,o _ __ _____ .~48___ L91}7 2.221 3~9~0 !a216 9_.Qo~ 12,)59 l"',300 _ 2.0.~10 .. 26.,ll3

1 .2471 ( ,9A4) 2,208) 3,919) ,r,.114) R.7951 11,959) 15,,r,09) 19,74?) 24,3581 -!l..2.._l'i __ - ---

.. _ ri_4 f)

. 880,_

91)0

'140 .

,241 ,961 2,163 3,854 l't,048 8,761 12,016 .l5,A40 20,2(18 25,341

,2411 I ,9591 ?.,1511 3,81,;1 r;.954) 8,5Ml ll,t.46) 15,1991 19,2241 23,719) ,235 ,91& 2,106 3,753 'i,B!:IR 8.5?.7 ll ,,r,R9 15,402 19,695 24,608 ,235) ^t ,9341 2,095) 3,718) 5,BO!l B,3441 11,346) .( 14,AOBl 18,710) 23.1101 ___ _ ,Z2J ___ ,912 2,053 3,b57 5,73£, B.303 11.379 · 14,9~5 19,152 23,914 ( ,22'i) ( ,910) 2,042) 3.b24) c;.655) 11,113) 11.060) ( 14,435) }8.25A) 22,5271

,223 ,890 2,002 3,565 5,590 A,OQ(l \l,0112 14,51'19 }8,&35 23.25S ,223) ,BAl3l 1,99?) 3,534) S,515> 7.9121 l0,7A7l l4,n7Bl 17,801,) 21,970)

,218 ,868 1,953 3,477 5,451 7,887 \0,800 14,?ll !B,145 22,630

1 .2101 ,8661 1,9441 3,4491 1 s,301, 7,7401 10,525> 11.1111 17,31111 21.,H>

____ _._2_1_3 ,847 1,906 _3,393 __ 5,318 _ 7,693 10,531 l1,A52 17,&7B 22,036

I ,2131 ( ,846) ( l,A97) 3,367) ( 5,253) 7,555) \0.274) 11,41'191 lb,96nl 20,926)

.,.?QB __ .827 __ 1,861 3,313 ·- 'i.19! . 7,5n7 10,273 13,"iOB 17,2)2 21,471

, 2 O 8 l I -;a 26 1· I l • B 5 3 l 3, 2 8 B I I c; , l 3 O l 7 • 3 7Q l I O , 014 l l 1 , n 9 & l \ti, 5 6 1 l 2 0 • 4 3 7 l

96p _ ,203_ _ ,808 1,IH8 3,235 . 'i,069 7,32Q 10,!l27 13,\AO !6,Bn7 20.932

,203) I ,807> I 1,810) I 3,212) I 5,0121 I 7,2091 9,'l03l 1?,7941 I lb,l'l?l 19,9661 - - - ·- - '9'-'8"-'0'--- - - - - __ . .JJ~. ____ _..19 0 ____ l...1I.6. ____ 3 -!. 61 _ ____ 4 • 952 _ _7 • l 5A _ 9 1.791 . l? • A~S . 1bt400 20 • 41 7 . ( , l 9 8 l I , 7 A 9) ( I • 7 6 9) I 3, 13 9 l I 4 , 8 9 Bl I - 7 • 0 A f,) 9 , 5 Fl l l l?, c:; O 4·1 ( I 5 • B I ~ l \ 9, 514) ,l94 _ ___ L77~ .. _ l .. 736 _ _ 3,09p ___ 4,839 t,,994 9,564 12,564· 16,010 19,923 ,1941 I ,771) 1,730) I 3,069) I 4,789) 6,8AQ) 9.3671 l;>,;>;>61 15,111,3) 19,n79)

_t

ooo

Table III