Normal shock wave properties in dissociated chlorine

Ju1y, 1970.

NORMAL SHOCK WAVE PROPERTIES IN DISSOCIATED CHLORINE

by

Y. Kondo and C. K. Law

UTIAS

Technica1 Note No. 14

9

,

NORMAL SHOCK WAVE PROPERTIES IN DISSOCIATEq CHLORINE

by

Y. Kondo and C. K. Law

Manuscfipt received June, 1970.

ACKNOWLEDGEMENT

The authors would like to express their gratitude to their supervisor

Professor I. I. Glass, for bis helpful direction in the preparation of this

work.

Stimulating discussions and advice from Dr.

M. Bristow are also

ack-nowledged with thanks.

The financial support for this work

was

provided

by

the National

Re-search Council of Canada and the Air Force Office of Scientific ReRe-search

under

grant AF-8FOSR 68-1368A.

SUMMARY

Accurate tab les are presented for the gasdynamic and thermodynamic

properties behind incident and reflected normal shock waves of gaseous chlorine

in dissociation equilibri urn.

The values of the temperature and pressure behind

the respective shocks have been computed to within 0.1% of the exact root.

Second order correction factors for the molecular partition·

function have

also been taken into account.

The intial temperature is taken to be 298.13

0

K for all cases, the

initial pressures cover the range from 0.01 to 2000 mmHg and the incident shock

Mach number is varied in increments of 0.1 from

6

to 18.

The computer program was written in such a way that only slight

modi-fictions are needed for its use with

ot~er

gases and other states of equilibrium.

Summary

Notation

Discussion of Results

Figures

Tables

Program Listipg

TABLE OF CONTENTS

1

A

B

P

c

_p

C

v

h

H

k

M

s

N

p

R

S

T

u

v

z

E

n

Pl

NOTATION

Speed of Sound

Characteristlc temperature of molecular rotation for

electronic level p in oK

Specific heat at constant pressure

Specific heat at constant volume

Statistical weight of n

th

electronic energy level

Planck's

const~nt

h/27T

Specific Enthalpy

Boltzmann's constant

Equilibrium constant for dis.

sociation

Mass of one ID9+e of atom X

Inc

~

dent

shock Mach number

Flow Mach number in state 2 in laboratory reference frame

Flow Mach number behind reflected shock

Avogadro' s number

Pressure

Gas constant defined as (Nk)/(2m)

Specific entropy

Temperature

Veloci ty in

shbok,

sta.j?iDnfu;>y

·~ne;fefenll.e

frame

Velocity in laboratory reference frame

Compressibility factor defined as

(1

+

a)

Degree of dissociation

Ratio of Specific Heat

th

Second order correct ion factor to the n

electronic state

of the molecular partition function

th

Characteristic temperature of electronic excitation of n

electronic state in oK

p

SUBSCRIPl'S

n

x

2

5

21

51

SUPERSCRIPl'S

I

R

Characteristic temperature of dissociation in oK

Characteristic temperature of molecular vibration oK

Density

Electronic energy level

Atomie Species

Molecular Species

State ahead of Incident Shock

State behind incident shock or ahead of

ref~ected shock

State behind the reflected shock

Denotes ratio of property of shock state 2 to that of state 1

Denotes ratio of property of shock state

5

to that of state 1

Incident Shock

Reflected Shock

v

.

Vl.

INTRODUCTION

The computer program, developed by Law and Bristow (Ref.l), was

modi-fied in order to obtain the present set of tab les for chlorine in dissociation

equilibrium.

In obtaining the normal shock properties by assuming a dissociation

equilibrium model, the only modification needed was to change all the gas

con-stants in the subI9utine

.

FTHMO to those for atomic and molecular chlorine.

The program is listed in Ref.l, but with the subroutine FTHMO

·

for chlorine

in~uded

in the present report.

To obtain the tables for the normal shock properties with the

assump-tion of vibraassump-tional equilibrium in all states, the statement (in FTHM

O

)

AL

=

SQRT ((XK/(KX

+

4.

*

p)))

was changed to read

AL

=

O.

which dictates that the degree of dissociation remain at zero for all states

,

viz., 1, 2 and

5.

DISCUSSION OF RESULTS

The results are presented in both tabular and graphical forma

Table 2 presents the no

r

mal shock properties with t

h

e assumption of

dissociation equilibrium.

The initial temperature Tl is fixed at 298.13°K

,

whereas Pl assumes the values 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10,

20, 50, 100, 200, 500, 760, 1000

and 2000 mmHg.

Table

3

is similar to Table 2 except that bot

h

the initial temperature

and the initial pressure are systematically varied.

Results in Table

4

assume a vibrational equilibrium model in all

states, where the shock

~~ch

nurnber M is limited to a maximum value of 10.

s

The thermodynamic properties behind the incident shock for translational

(T) and rotational (R) equilibrium (T 7 R

+

V(T

l

)) and as well as for vibrational

(V) equilibrium (T

+

R

+

V(T

2

)

are presented in Figs. 2.1, 2.2 and 2.3. It

should be noted that even at room temperature chlorine exhibits a marked deg

r

ee

of vibrational excitation i.e.

7

1

=

1.326

for Tl

=

298.13°K.

The shock

'

properties of chlorine differ from those of other diatomic

gases (e.g. O

_2,

N

2

) in that it becomes vibrationally excited and dissociated

at much lower temperatures due to its relatively small characteristic vibration

and dissociation temperatures. For an initial pressure of 1 mmHg the gas is

completely dissociated at M

~

15

behind the incident normal shock and at

M

~ 10 behind the reflecteä shock. Since the effects of ionization have not

b~en taken into account, care should be taken in utilizing the present data for

conditions of significant dissociation, where some ionization may be present

,

i.e.

for

a

>

0.98~

The dependence of the normal shock properties on the initial temperature

Tl are

il~ustrated

in Table 3.

The deviation of these groperties within the

range of the usual laboratory temperature (290

o

K to 300 K) is around 1% for

the most sensitive parameters, i.e., Tand Z.

The differences in T

2

, T

₅

, Z2 and

o

0

8

0

0

Z~

between Tl

=

298.13 K and 295.0 K and between Tl

=

29 .13 K and 290.0 K are

plotted in Flg'S. 4.1 to 4.4 for initial pressures of 1, 10, and 100 mmHg.

The behaviour of the gasydnamic properties behind the reflected shock

for conditions of high shock Mach number and high initial pressures, e.g.,

Ms

~ l~,

and Pl

~

760 mmHg was observed to be unusual in that the dissociation

fraction

a

was lower behind the reflected shock than ahead of it. In other words

some of the chlorine atoms recombine af ter being processed by the reflected shock

.

Secondly, it was also noticed that, under these extreme conditions, as the

streng

th

of the incident shock was increased, the dissociation fraction behind the

reflec

t

ed shock reached a maximum without attaining a value of unity.

To check for possible errors in the computer program, values from the

computer output were substituted back into the formulas defining thermodynamic

properties as well as the normal shock relations (see Chapter 2.3 in Ref. 1).

The agreement was exact, implying that the computer results obtained do represent

the behaviour of chlorine for conditions of dissociation equilibrium. We

~herefore

attempt to offer a qualitative explanation of

t

his phenomena as follows.

The degree of dissociation is defined as

a

=

₍

Kd

)1/2

Kd

+

4p

( 1)

where Kd, the equilibrium constant for dissociation, is given by

_ {M k

5/ 3 } 3/?- 5/

2

Kd -

N

~

47T

T

exp

(1-exp(-8 /T))

n

(2)

Kd is therefore most sensitive to variations in the Boltzmann factor exp(-8

D

/T),

so that at low temperature a small change in T produces a large change in

exp(-8D/T), whereas the reverse is true at high temperatures. For chlorine

8

D

=

2e,735

0

_{K, so that for temperatures of l4,300oK and l430 0K, an increment of}

10000K

produc~s an increase in the Boltzmann factor by amounts of the order of

unity and 103b respectively.

Thus

i t

is not inc:önceivable that under certain

circumstances, as the incident shock strength is increased, the rate of increase

of Kd is considerably slower than that of

a

in the region behind the reflected

shock, enabling us to obtain a smaller value of a.

The criterion that an ensemble of dissociated gas at give conditions

of Tand p will further dissociate or recombine upon a slight increase in tempe

r

a-ture and pressure, has been formulated as follows*.

From Eq. (1) we have:

2 (

Kd

k

+

d

4p )1/2

(Kd

+

1

4

p

)2 Po

{ ( d Kd )

\<it

'

.IJ

,

11

LlT _

Kd 6

P

}

U

0

Thus if the gas recombines upon an increase of temperature and pressure, 6a

<

0,

implying that

[

d Kd ]

P

~

6T

<

Kd 6p

P

(4)

because

[

dd~dJp

~

-

[6Kd

6T

J

p

Hence Eq. (4) becomes:

~>

6Kd

for 6 a

<

°

p

Kd

(6)

For fut her dissociation to occur,

~

<

6 Kd

for 00

>

°

P

Kd

The criteria stated in Eqs.

(6)

and

(7)

have been tested for various

cases in the present tables and were found to be valid suggesting that the

dissociation fraction

a

5

behind the reflected shock does attain a maximum value

as the incident shock strength is increased.

* The authors are indebted to Professor S. C. Lin of the Dept. of Aerospace

and Mechanical Engineering Sciences, University of California, San Diego,

for suggesting this explanation.

1.

Law, C. K.

Bristow, M.

2.

Herzberg, G.

3.

McBride, B.

J.

Heimel, S.

Ehlers,

J.

G.

Gordon, S.

4 •

Moore, C. E.

RE:FERENC ES

"Tables of Normal Shock Wave Properties for Oxygen

and Nitrogen in Dissociation Equilibrium", UTlAS

Tech. Note No. 148, 1969.

Spectra of Diatomic Molecules, Von Nostrand,

New York, 1950.

"TherIlX)dynamic Properties to 6000

0

K for 210

Substanees lnvolving the First 18 Elements", NASA

SP-3001, 1963.

"Atomie Energy Levels," VoLl, Circular 467, NES,

(1949).

CONTACT

SURFACE

INaDENT

SHOCK

(A) INCIDENT

SHOCK

IN

STATIONARY ( LABORATORY ) REFERENCE FRAME

CONTACT

SURFACE

-f-U~

INCIDENT

SHOCK

2 I

..

(B)

INCIDENT SHOCK

IN

MOVING

(SHOCKFRONT)

REFERENCE FRAME

CONTACT

REFLECTEO

SURFACE

SHOCK

(C) REFLECTEO

SHOCK

IN

STATIONARY (LABORATORY) REFERENCE FRAME

CONTACT

REFLECTED

SURFACE

SHOCK

(0) REFLECTED

SHOCK

IN

MOVING

(SHOCKFRONT) REFERENCE FRAME

FIG. I

SCHEMATIC

SHOCKS

IN

DIAGRAM

SHOWING

INCIDENT

a

REFLECTED

500

p

=

0.01

torr --...,.-2

I

'

.

:...HiI~-

1000

100

T

+

R

+

vn.,

10

T. •

298.13 •

K

I--~---~---~----~---~----~

I

2

4

6

8

10

12

14

MI

Fig.

2-1

Pressure Ratio

(P2/P1)

Across Normal Incident Shock Wave

Against Incident Shock Mach Number Ms in Chlorine

22

20

18

16

14

12

10

8

6

4

2

2

TI •

298.13 • K

0.01

torr

--

~l1J.1.---1 ..

--

R.!_

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

---/",,'"

_---" +

R

+

VIT.)

...

--~

",,-'

.

" , ,

-,,/

"

,,/'"

/ " /

4

6

8

MI

10

12

14

16

Fig. 2-2 Density Ratio

(f

2 /Yl> Across Normal Incident Shock Wave

J

TI

T,

24

I

I

I

I

I

I

I

I

,

I

22

·

I

I

I

_I

20

18

16

14

12

10

8

6

4

T. • 298.13 -K

I

_I

I

_I

I

I

_I

I

I

_I

I

_I

l

_I

I

_I

,

I

I

I

I

T

+

R

+

vn.,

• I

,

I

I

.

,

I

I

,

_I

/

I

I

I

t - - T +

R

+

V(T

_a,

I

I

I

I

1

1

I

1

1

1

, ,

I

1

1

1

1

1

1

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I I

I I

I I

I

/

I

I

I

I

I

I

I /

I

I

I

I

I I

I I

I I

I /

I I

I /

I /

/ /

/ /

/ /

/ I

/ I

I /

/ /

/1

/ 1

1 1

1 1

I/.

I

I

I

I

I.

P

2

0.01

torr

,::

I

MI

Fig. 2-3 Temperature Ratio (T2/Tl) Across Normal Incident Shock Wave

..

50r-~----~----~~----~----~----~----~---45

TI •

298.13 • K

40

35

30

PI

=

0.01 -

1000

torr

20

"

15

o

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

14

16

I

2

4

6

8

10

12

M.

Fig. 2-4 Enthalpy Ratio (h2/hl) Across Normal Incident Shock Wave

Against Incident Shock Mach Number Ms in Chlorine

Q2

a.-4.5

----...,..---..,.----,~--r---,__--~--ï

4.0

r. •

298.13 eK

3.5

3.0

PI

:

1000

torr

2.5

2.0

1.5

1.0

2

4

6

8

10

12

14

16

Fig. 2-5 Sound Speed Ratio (a2/al) Across Normal Incident Shock Wave

Against Incident Shock Mach Number Ms in Chlorine

2.1

r - - -...

- -...

--...,.---r---"T"'""""---,----2.0

1. •

298.13

0

K

1.9

1.8

PI

: 0.01

torr

1.7

0.1

1.6

1.0

Z2

10

1.5

100

1.4

~--IOOO

1.3

1.2

1.1

1.0

2

4

6

8

.

10

12

14

16

MI

Fig. 2

'

-6 Compressibility Factor (Z2)Againstlncident Shóck Mach

70

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

60

5

2

50

R

40

30

TI •

298.13· K

PI •

o.

Ol

torr

25

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

1 : 2

4

6

8

10

12

14

16

M.

Fig. 2-7 Entropy (S2/R) A

'

gainst Incident Shock Mach Number Ms

in Chlorine

104r-_r---~----_r---~----_r---~----_r---~

p.

&

0.01

torr - -...

O.l---~

...

1.0

--r-",4oy".

10

~'---

100

/ ' f o l - - - -

1000

la'

T.

Ir

298.13

oK

1.5 '---"'---...

---"'---'"""'""'---.1.---... ---.1.---'

I

2

4

6

.

8

la

12

14

16

Ms

Fig.

3-1

_{Pressure Ratio (P 5}

!P1)

for Reflected Normal Shock Wave Against

160

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

1'40

PI : 0.01

torr

---<~

120

100

80

60

40

20

T, • 298.13· K

o

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

I

2

4

6

8

10

12

14

16

MI

"

Fig. 3-2

"

Density Ratio (.J'5/)'1) for ReHected Normal Shock Wave Against

SO---~----~---r----~---~----,

, >,

TI • 298.13 • K

40

0.01 ---.."

/ ' 4 - -

1000

30

20

PI : 1000

torr

- - - ' 7 '

10

I

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

I

2

4

6

8

10

12

14

16

MI

Fig. 3-3 Temperature Ratio (T5/Tl) for Reflected Normal Shock Wave

Against Incident Shock Mach Number Ms in Chlorine

120

-_--...,..---,---~--..,._--"""T""--___r--___,

TI

=

298.13· K

90

PI : 0.01

torr

- - - I

f1~-

10

,,-..--- 1000

30

_{1 0 - - - u}

~---0.01

Ol

~~·2--~4~--6~---~8---10~--~12--~14~-~16

I

.

Ms

Fig. 3-4 Enthalpy Ratio (h5/hl) for Reflected Normal Shock Wave

Against Incident Shock Mach Ntimber Ms in Chlorine

as

a;-10

9

T.

= 298.13

0

K

8

0.01

7

1000

6

s

4

PI

=

1000

torr - - - , /

3

2

I

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

I

2

Fig. 3-5

4

6

8

10

12

'

14

MI

Sound Speed Ratio (a5/ a ) for

·

Reflected Normal

Shoc~

Wave

Against Incident Shock 1Mach Number Ms

in

~

Chlorine

Zs

-,1

2.1

r---~---~----~---~---~----~----~

2.0

0.0 I

----'.1-./

1.9

O.l--~

I.O--~~

1.8

1.7

PI

:

₁₀₀₀

torr

1.6

100

10

1.5

1.4

1.3

1.2

TI

=

298.13

0

K

1.1

1.0

L..-."""","~IIi;:;",I,. _ _ _ _ _ _ - ' - -_ _ _ _

---L ______

- ' -______

J.-. ____

-J..

_ _ _ _

- - . I

2

4

6

8

10

12

14

16

Ms

Fig. 3 -6 Compressibility Factor (Z5) for Reflected Normal Shock Wave

Against Incident Shock Mach Number Ms in Chlorine

Ss

R

70

~~---~---r----~---~----~---~----~

T,

• 298.13 • K

60

p.

a

0.01

torr

- - - . y '

50

40

30

25

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

I

2

4

6

8

10

12

14

Ms

Fig. 3-7 Entropy (S5/R) for Reflected Normal Shock Wave Against

Incident Shock Mach Number Ms in Chlorine

103---T----~~----~----._----~----~r_----~----ï

10'

I~L-~---L---~----~----~---~----~----~

I

2

4

6

8

10

12

14

16

Ms

Fig. 4-1 Influence of Initial Conditions on Incident Temperature T 2

in Chlorine

IOI~~---~----~---~---~----~---~---10'

100~

__

~

____

~

______

~

______

~

______

~

____

~

______

~

______

~

I

2

4

6

8

10

12

14

MI

Fig. 4 -2 Influence of Initial Conditions on Reflected Temperature T5

in Cliorine

IÖI~--~---~----~----~~---r---~----~

PI •

I

torr

---~

p

11

10

tor r

----:~

.

I

~---

PI • 100

torr

.."..".'--- P, •

10

torr

P, •

I

torr

IÖ4~~----

__

~

____

~

____

~

______

~

____

~

____

~

3

4

₆

₈

₁₀

₁₂

₁₄

₁₆

M.

Fig. 4-3 Influence of Initial Conditions on Incident Compressibility

Factor Z'2 in Chlorine

IÖIr-~~---r---~---~----~---~---~---,

~f---

PI

11

10

torr

PI •

I

torr

~-

PI •

100

torr

'~--

PI • 10

torr

J-H'----

PI.

I

torr

4

6

8

10

12

14

16

Fig. 4-4 Influence of Intiial Conditions on Reflected Compressibility Factor Z5

in Chlorine

PROGRAM LISTING OF SUBROurINE FTHMQ AB USED ON IBM 1130

*

Tl • 298.13 IC. MS T2 1.1 1.3

I.'

I.'

1.1 1.8 I •• 2.0 2.1 2.2 2.3

2.'

2.'

2.'

2.1 2.8 2 •• 3.0 3.1 3.2 3.3

,

..

,.,

'.0

301 308 lo.

'.0

T> 31. 331 330 '03 345 3" 301 '21 311 '01 393 '95 '10 531 '21 508 '45 606 .0. 6.0 483 088 503 131 52' 11> 5'0 820 568 80. "1 '00 615 ,945 0'0 .80

.0.

1011 0.2 1038 11' 1063 141 1085 115 1106 803 1125 832 1143 800 1161 081 1111 912 1193 .36 1209

.5.

1224 P21 P'I 1.24 1.53 1.51 2.22 1.79 3.08 2.10 4.14 2.44 5.40 2.19 6.89 3.11 8.61 3.58 10.57 4.00 12.79 4.45 15.27 4.92 18.02 5.42 21.04 5.94 24.33 6.48 21.89 7.05 31.11 7.64 35.18 8.25 40.07 8.88 44.58 9.54 49.31 10.23 54.26 10.93 59.44 11.66 64.86 12.42 10.55 13.20 16.53 14.01 82.87 14.84 89.63 15.11 96.89 16.61 104.75 11.54 113.Z8 18.51 122.53 Tl • 298.13 K MS T2

'.1

•• 2

'.3

...

•• 5 •• 0 •• 1 •• 8

•••

5.0 5.1 5.2

5.,

5 •• 5.5 5.0 5.1 5.8

,

..

0.0 0.1 0.2 0.3

0.'

0.'

•••

6.1 •• 8

•••

1.0 15

'"

123' •• 8 1253 1010 1267 1032 1281 1041 1295 1061 1309 1014 1322 1087 1335 1099 1348 1110 1360 1121 1373 1131 1385 1141 1398 1151 1410 1160 1422 1169 1434 1118 1446 1181 1457 1195 1469 1203 1481 1211 1492 1219 1504 1221 1515 1234 1521 1241 1539 1249 1550 1256 1562 1263 1574 1270 1585 1276 1597 P21 P51 19.50 132.53 20.53 143.31 21.60 154.87 22.69 167.24 23.81 180.0\3 24.96 191 . . 42 26.13 209.25 27.34 221t.92 28.57 241.44 29.83 2'8.81 31.12 217.06 32.1t4 296.20 33.78 316.23 35.15 337.17 36.54 359.03 37.96 381.82 39.41 405.55 40.88 430.24 42.38 455.89 43.90 482.52 45.45 510.14 47.03 538.75 48.63 568.36 50.26 598.98 51.92 630.63 53.60 663.30 55.30 697.01 57.03 731.75 58.79 767.55 60.57 804.39 PI • U.Ul TUHN HZI ROH21 A21 l2 GAMAl CP2/R CVZlR S2IR M2

H!)l R0l151 A~H l5 GAMA; CP~/R eV;/N SS/I< MN

1.06 1.177 1.u3 1.0001,) 1.323 4.1U 3.1U 3Ü.U4 Uel61 1.12 1.379 1.05 1.0000 1.320 4el3 3.IJ HI.U4 U.li36

1.12 1.360 1.05 1.0000 1.320 4.13 3.13 n.05 0.302

1.24 1.823 1.10 1.0000 1.315 4.t8 3.18 3ts.06 0.8tn

1.18 1.547 1.U7 1.000U 1.317 4.I5 3.1; 3Ü.U6 U.4~Ü

1.36 2.328 1.14 1.0000 1.311 4.;12 3.~2 lb.ut; U.Ü49 1.24 1.737 1.10 hOOOO 1.315 4.17 h17 3ihU9 0.542 1.49 2.888 1.19 1.0000 1.3u7 4.25 3.2; 3tj.lZ u.ü18 1.30 1.928 1.12 1.000U 1.313 4.2U 3.20 3H.12 0.645 1.62 3.497 1.23 1.0000 1.305 4.26 3.28 3~.ltl 0.193 1.36 2.119 1.14 1.0000 1.311 4.22 3.22 36.16 0.7 .. 0 1.75 4.149 1.28 1.0000 1.302 4.31 3.31 38.25 0.772

1.42 2.308 1.16 1.UUOO 1.309 4.23 3.23 3b.il u.tin

1.89 4.836 1.32 1.000U 1.300 4.33 3.33 3b.)2 U.755 1.49 2.496 1.19 1.0000 1.308 4.25 3.25 38.21 0.908 2.04 5.552 1.37 1.0UOU 1.299 4.35 3.35 311.41 0.741 1.56 2.680 1.21 1.0000 1.306 4.27 3.27 3a.33 U.99Z 2.19 6.290 1.41 1.0000 1.297 4.37 3.37 39.~1 0.730 1.63 2.862 1.24 1.0000 1.304 4.28 3.2t1 31h4U 1.U52 2.35 7.044 1.46 1.0000 1.295 4.39 3.39 3i;h61

u.no

1.71 3.038 1.26 1.0000 1.303 4.30 3.30 311.48 1.117 2.51 7.808 1.50 1.0000 1.294 4.41 3.41 39.12 0.711 1.78 3.211 1.29 1.0000 1.30l 4.31 3.31 39.56 1.111 2.69 8.578 1.54 1.0001 1.291 4.45 3.44 3af.94 0.7U4 1.87 3.378 1.31 1.0000 1.301 4.33 3.33 Jij.64 1.ZH 2.87 9.353 1.59 1.0001 1.2t16 4.52 3.52 3W.96 0.6C:i7 1.95 3.541 1.34 1.0000 1.300 4.34 3.343 ... 73 1.2a6 3.05 10.134 1.63 1.0004 1.276 4.68 3-61 39.08 0.691 2.04 3.698 1.31 1.0000 1.299 4.35 3.35 3ij.ti2 1 •. ;35 3.25 10.928 1.66 1.000Y 1.261 4.99 3.95 39.20 t.h&83 2.13 3.850 1.39 1.0000 1.298 4.36 3.36 38.92 1.382 3.44 11.745 1.69 1.0020 1.239 !5e50 4.43 39.33 0.614 2.23 3.997 1.42 1.000U 1.297 4.37 3.31 lW.Ul 1.425 3.64 12.595 1.70 1.0036 1.216 6.25 5.13 3 ... 46 U.662 2.32 4.138 1.45 1.0000 1.296 4.38 3.3t1 39.11 1.466 3.85 13.486 1.12 1.0060 1.194 7.24 6.05 39.59 0.64'1 2.43 4.275 1.48 1.0000 1.294 4.40 3.40 39.21 1.505 4.0614.417 1.13 1.0091 1.175 8.43 1.1439.72 U.634 2.53 4 .... 06 1.50 1.0000 1.293 4.42 3.42 39.32 1.542 4.27 15.396 1.15 1.0128 1.161 9.79 8.38 39.tl5 0.61W 2.64 4.533 1.53 1.0000 1.291 4.44 ).44 39.42 1.511 4.49 16.392 1.76 1.0171 1.149 11.26 9.72 3W.98 0.603 2.75 4.656 1.56 1.00Ul 1.289 4.49 3.49 39.52 1.611 4.11 17.433 1.71 1.0220 1.140 12.83 11.13 4u.12 0.581 2.87 1 . . 777 1.59 1.0002 1.283 4.57 ).56 39.63 1.645 4.94 18.510 1.18 1.0273 1.134 14.48 12.61 40.21 0.~72 2.99 4.896 1.61 1.0004 1.276 4.69 ).68 39.74 1.680 5.17 19.626 1.19 1.0330 1.128 16.19 14.12 4U.42 0.S~9 3.11 5.011 1.63 1.0007 1.265 4.89 3.86 39.84 1.711 5.41 20.790 1.80 1.0392 1.124 17.95 150.61 40.~1 0.546 3.14 5.141 1.65 h0012 1.252 5.18 4.14 39.95 1.757 5.66 22.013 1.81 1.0457 1.121 19.75 17.25 4U.13 U.535 3.37 5.271 1.66 1.0021 1.235 5.60 4.52 40.0b 1.801 5.92 23.309 1.62 1.0527 1.118 21.60 18.85 40.89 0.526 3.50 5.410 1.68 1.0032 1.218 6 . n 5.04 40.l1 h848 6.19 24.689 1.84 1.0601 1.116 23.48 20.47 41.05 0.518 3.64 5.558 1.69 1.0041 1.201 6.85 5.69 40.28 1.896 6.47 26.163 1.85 1.0679 1.114 25.40 22.10 41.23 0.512 3.78 5.717 1.10 1.0067 1.185 7.68 6.46 40.39 1.947 6.75 27.736 1.86 1.0761 1.113 21.)4 23.74 41.40

o.sç.

PI • 0.01 TURN

H21 ROH21 A21 Z2 GAMAZ ep2/R cv2/R S2/R M2

HSl ROH51 A51 Z5 GAMA5 CP5/R Cv5IR SS/I( MI(

3.93 5.a85 7.05 29.406 4.08 6.062 7.35 31.112 4.23 6.241 7.66 33.029 1t.39 6.439 1.99 34.973 4.55 6.635 8.32 36.991 1t.11 6.837 8.65 39.098 0\.88 1.042 9.00 41.210 5.05 7.251 9.35 43.509 5.23 7.462 9.72 45.811 5.41 7.676 10.0948.172 5.59 1.891 10.46 50.588 5.78 8.108 10.85 53.055 5.97 8.326 11.21t 55.511 6.16 8.545 11.64 58.130 6.36 8.165 12.05 60.730 6.56 8.9a5 12.47 63.368 6.16 9.205 12.89 66.039 6.97 9.425 13.32 68.140 7.18 9.61t5 13.76 11.468 7.39 9.865 14.21 74.220 7.61 10.084 14.66 16.992 7.83 10.303 15.12 79.181 8.06 10.521 15.59 82.582 8.28 10.738 16.07 85.394 8.52 10.955 16.55 88.213 8.75 11.170 11.04 91.034 8.99 11.384 17.54 93.855 9.23 11.597 18.05 96.673 9.48 11.80a 18.56 99.483 9.13 12.01a 19.08102.2H3 1.70 1.0090 1.111 8.64 7.3440.51 1.998 1.al 1.0847 1.11Z 29.30 25.37 41.58 0.500 1.71 1.0117 1.160 9.71 8.32 40.62 2.050 1.88 1.0936 1.111 31.28 27.00 Itl.76 0.495 1.72 1.011t8 1.15010.86 9.37 4U.71t 2.101 1.89 1.1030 1.110 33.25 28.63 41.9!:1 0.490 1.13 1.0182 1.14i 12.09 10.49 40.tl6 2.U3 1.90 1.1126 1.110 35.23 30.23 42.15 (.).486 1.73 1.0218 1.136 13.38 11.66 40.98 2.204 1.91 1.1226 1.110 37.21 31.82 42.34 0.482 1.71t 1.0258 1.130 11t.12 12.87 41.11 2.255 1.92 1.1329 1.110 39.17 33.38 42.55 0.418 1.15 1.0300 t.126 16.11 14.11 Itl.23 2.306 1.93 1.1436 1.110 41.12 34.91 42.75 0.474 1.76 1.0344 1.122 17.54 15.38 41.36 2.356 1.91t 1.1546 1.110 43.05 36.41 42.91 0.411 1.16 1.0391 1.119 19.00 16.67 4L.49 2.407 1.95 1.1658 1.110 44.95 37.88 43.ttj 0.468 1.71 1.0440 1.116 20.48 17.97 41.63 2.456 1.95 1.1774 1.110 46.83 39.32 43.40 0.466 1.78 1.0490 1.114 21.99 19.29 41.76 2.5U6 1.96 1.1893 1.110 48.61 40.11 43.63 0.463 1.18 1.0543 1.112 23.52 20.62 41.90 2.555 1.97 1.2014 1.1H 50.48 42.07 43.85 0.461 1.79 1.0597 1.111 25.07 21.95 4Z.04 2.605 1.98 1.2139 1.111 ~2.24 43.37 44.09 0.4S9 1.80 1.0654 1.109 26.63 23.30 42.19 2.653 1.99 1.2266 1.111 53.96 44.64 44.32 0.451 1.80 1.0712 1.108 28.21 24.64 42.33 2.702 2.00 1.2397 1.11l 55.63 45.65 44.;6 0.4~6 1.81 1.077l 1.101 29.79 25.99 42.48 2.750 2.01 1.2530 h112 S7.24 47.01 44.81 0.454 1.82 1.0833 1.106 31.39 27.33 42.63 2.799 2.02 1.2666 1.113 58.16 49ell 45.U6 0.453 1.82 1.0896 1.106 32.99 28.67 42.79 2.846 2.03 1.2b04 1.114 60.21 49.16 45.31 0.4~2 1.83 1.0961 1.10~ 34.61 30.01 42.94 2.994 2.03 1.2946 1.114 61.68 !:IO.14 4).!:I6 U.451 1.83 1.1027 1.104 36.22 31.35 43.1u 2.941 2.04 1.3090 1.11) 63.01 51.06 45.tl2 IJ.4H 1.84 1.109S 1.1u4 31.84 32.68 43.2b 2.988 2.05 1.3237 1.116 64.26 51.92 46.U9 0.450 1.84 1.1164 1.104 39.46 34.00 4l.42 3.035 2.06 1.33&7 1.116 65.42 52.7u 46.35 0.4S0 1.95 1.1235 1.103 41.0tl 35.32 43.~9 3.081 2.07 1.3539 1.111 66.48 53.41 46.62 0.450 1.86 1.1307 lelU3 42.10 36.62 43.16 3.128 2.08 1.3694 1.119 67.45 54.04 46.90 0.450 1.86 1.13U 1.10344.3237.92 4l.93 3.174 2.09 1.3852 1.119 68.30 54.58 41.ltf U.450 1.87 lel456 1.103 45.93 39.20 44.10 3.Z20 2.10 1.401) 1.12U '''.04 ;'5.0~ 41.46 IJ.4;'0 1.87 1.1S3) 1.103 47.54 40.48 44.l7 3.265 2.10 1.4176 1.121 69.66 )5.42 47.74 0.451 1.88 1.1611 1.103 49.14 41.13 44.4; 3.311 2.11 1.4342 1.122 7Uel' )S.10 4b.IJ3 0.451 1.a8 1.1691 1.10) 50.73 42.98 44.63 3.3~6 2.12 1.4510 1.123 70.49 ;5.t:l8 4d4~2 0.452 1.89 1.1772 1.103 52.31 44.21 44.81 3.401 2.13 1.4681 1.t24 70.10 55.95 4b.bl U.4~3 Tl • 298.13 K MS T2 P21 P51 1.2 1.3 1 •• 1.0 7.1 7.8 1 •• •• 0 8.1 8.2 8.3 8 ••

8.'

8.0 8.1 8.8

•••

9.0 9.1 9.2 9.1

...

..

,

'.0

9.7 9.8

•••

10.0

l'

1283 62.38 1609 842.30 1290 64.22 1621 881.21 1296 66.08 1633 921.30 1303 67.97 1645 962.41 1309 69.88 1658 1004.59 1316 71.82 1671 1047.86 1322 73.78 1683 1092.20 1328 75.77 1697 1137.63 1334 17.78 1110 1184.15 1341 79.82 1124 1231.77 1347 81.89 1738 1280.48 1353 83.98 1753 1330.29 1359 86.10 1769 1381.22 1365 88.24 1785 1433.25 1371 90.41 1802 1486.42 1317 92.60 1820 1540.11 1383 94.82 1839 1596.16 1389 97.06 1859 1652.79 1395 99.33 1882 1710.62 1401 101.63 1906 1169.70 1407 103.95 1934 1830.10 1412 106.29 1966 1891.92 1418 108.66 2004 1955.34 1424 111.06 2051 2020.62 1430 113.48 2112 2088.25 1436 llS.93 2194 2159.10 1442 118.40 2313 2234.52 1448 120.90 2480 2315.64 1454 123.42

~::~

(2

~~t;~

2915 2491.44 Tl a 298.13 IC. MS T2 P21 P'I 10.2 10.5 10.6 10.7 10.8 10.9 11.0 11.1 11.2 11.3 11.4 11.5 11.7 11.8 11.9 12.0 12.1 12.2 12.5 12.6 12.7 12.8 12.9 13.0 15 1466 3154 1472 3399 1478 3649 1485 3'01 1491 4157 1497 4416 lS04 4678 BID 4942 1'11 5210 1523 5479 1530 5752 1537 6027 1544 6305 1551 6585 1559 6868 1566 7153 1574 7441 1582 7731 1590 8024 1598 '8318 1607 8616 1616 8915 1626 9211 1636 9521 1647 9826 1658 10134 1670 10444 1683 10756 1697 11069 1713 11384 128.55 2583.26 131.14 2676.85 133.77 2772.02 136.42 2868.61 139.09 2966.13 141.19 3066.15 144.51 3166.89 141.26 3268.88 150.03 3312.07 152.83 3476.42 155.65 3581.85 158.50 3688.31 161.37 3195.73 164.27 3904.02 167.19 4013.10 170.13 4122.89 113.10 4233.27 176.09 4344.14 179.11 4455.36 182.15 4566.19 185.21 4618.25 188.30 4789.56 191.40 4900.49 194.53 5010.18 197.68 5120.11 200.86 5228ell 204.05 5334.33 207.26 5438.21 210.49 5539.04 213.74 5635.94 PI • 0.01 TUf 0< H21 ROH21 A21 Z2 GAMAl CPZ/R CVVR S2/R M2 H)l ROH51 A51 lSo GAMA5 CP5/R c.v!:l/Iot S!t/N MR 9.98 12.227 1.89 1.1854 1.103 53.87 45.42 44.99 3.446 19.61105.069 2.14 1.4854 1.125 70.75 55.92 48.91 0.454 10.24 12.434 1.90 1.1938 1.103 55.43 46.61 4; • .16 3.490 20el5107.837 2.15 1.5031 1.126 70.65 55.78 49.Z1 0.455 10.50 12.639 1.90 1.2023 1.103 56.96 47.78 45.31 3.535 20.70110.583 2.16 1.5209 1.121 70.38 55.51 4W.~1 0.456 10.76 12.843 1.91 1.2109 1.103 58.46 48.94 45.56 3.519 21.25113.304 2.17 1.5390 1.128 69.93 55.13 49.82 0.458 11.03 13.045 1.91 1.2191 1.1u3 59.96 50.07 45.7~ 3.623 21.81115.9972.18 1.5514 1.13069.31 54.61 50.12 0.459 11.30 13.244 1.92 1.2267 1.104 61.46 51.18 45.94 3.666 22.38118.655 2.19 1.5760 1.131 68.49 ~3.95 50.44 0.461 11.58 13.442 1.92 1.2377 1.104 62.91 52.26 46.14 3.710 22.96121.271 2.20 1.5948 1.133 67.4& 53.16 50.15 0.462 11.86 13.638 1.93 1.2469 1.104 64.3453.32 46.34 3.753 23.55123.856 2.21 1.6139 1.134 66.26 52.22 51.01 0.464 12el4 13.832 1.93 1.2563 1.104 65.74 54.35 46.54 3.196 24.14126.389 2.22 1.6332 1.136 64.83 51.12 51.39 0.461 12.42 14.024 1.94 1.2651 1.104 67.11 55.36 46.14 3.839 24.74128.869 2.Z3 1.6528 1.138 63elS 49.S6 ~1.11 0.469 12.71 14.214 1.941.2153 1.10568.4556.3446.95 3.682 2S.35131.292 2.24 1.6126 1.140 61.31 48.44 52.U3 0.411 13.00 14.401 1.94 1.2851 1.105 69.75 57.28 47.15 3.924 25.97133.650 2.25 1.6925 1.143 59.Z2 "t6.84 5i.36 0.474 13.30 14.587 1.95 1.2949 lel05 71.01 58.20 41.36 3.966 26.60135.936 2.26 1.1121 1.145 56.88 45.06 ;4:.69 0.417 13-60 14.169 1.95 1.3049 1.106 72.21t 59.08 47.57 1 . . 008 27.24138.142 2.28 1.7331 1.148 54.31 43.09 53.02 0.480 13.90 14.950 1.96 1.3151 1.10613.4259.9247.18 4.050 27.88140.251 2.29 1.1536 1.151 51.51 40.94 53.35 U,483 14.21 15.128 1.96 1.3253 1.106 74.56 60.73 48.00 4.092 28.53142.270 2.31 1.7743 1.155 48.46 38.58 53.69 0.4t11 14.52 15.304 1.97 1.3351 1.107 75.65 61.50 48.22 4.133 29.20144.166 2.32 1.1952 1.159 45.18 36.03 54.u2 0.491 14.83 15.477 1.97 1.3462 1.107 76.6962.2348.434.114 29.87145.921 2.34 1.8161 1.165 41.68 33.29 54.36 0.495 15.15 15.648 1.98 1.3569 1.101 77.67 62.92 48.65 4.215 30.56141.5282.361.8311 1.171 37.9630.35 54.7u 0.5uo 15.47 15.816 1.98 1.3617 1.108 78.61 63.57 48.88 4.256 31.25148.938 2.38 1.85&1 1.119 34.0427.22 55.04 0.506 15.8015.982 1.99 1.3786 lel08 79.4864.1749.10 4.296 31.96150.1112.41 1.8791 1.18929.96 23."3 5~.39 u.512 16.13 16.145 1.99 1.3896 1.109 80.29 64.73 49.33 4.337 32.69150.986 2.44 1.8998 1.202 25.75 20.50 55.73 0.519 16.46 16.306 1.99 1.400a 1.109 81.03 65.23 49.55 4.377 33.43151.46tj 2.49 1.92U1 1.22U 21.50 17.OU 56.U6 U.528 16.79 16.463 2.00 1.4121 1.109 81.71 65.69 49.78 4.416 34.19151.409 2.54 1.9398 1.247 17.31 13.50 5th43 0.539 17'}3 16.618 2.00 1.4235 1.110 82.32 66.10 50.01 4.456 34.98150.566 2.62 1.9582 1.289 13.)6 10.16 56.19 O.5~3 17.47 16.771 2.01 1.4350 1.110 82.8566.44 50.25 4.495 35.81148.569 2.74 1.9743 1.355 9.92 7.23 51.15 0.572 17.82 16.920 2.01 1.4""67 1.111 83.30 66.74 50.48 4.534 36.71144.938 2.91 1.9868 1.452 1.38 5.05 57.52 0.599 18.17 17.067 2.02 1.45&5 1.111 83.68 66.91 50.72 4.513 37.69139.599 3.12 1.9943 1.552 5.96 3.83 57.90 0.637 18.52 17.211 2.02 1.4704 1.112 83.96 67.14 50.96 4.611 38.14133.433 3.31 1.9917 1.613 5.3ü 3.33 5ts.2t1 0.613 18.88 17.351 2.03 1.4824 1.113 84.16 67.25 51.20 4.649 39.8.H27.465 3.48 1.9990 1.638 5.19 ).16 5H.b3 0.733 PI • 0.01 TORI( H21 ROH21 A21 Z2 GAMAl CP2/R CV2lH S2/i-( M2

HSL ROH51 A51 l5 GAMA5 CP;'/R Cv5/H S5/R Mk

19.24 17.489 2.03 1.4946 1.113 84.27 61.30 !>l.44 4.687 40.94122.106 3.63 1.9996 1.648 5.10 3.10 58.91 0.184 19.60 17.624 2.04 1.5068 h114 84.2961.27 Sl.6ö 4.124 42.08117.3933.771.9998 1.653 '.07 ).0759.26 0.835 19.9711.7562.041.5192 1.11484.2167.1851.934.162 43.22113.255 3.91 1.li999 1.655 5.06 3.05 !J9.~. 0.885 20.3417.8842.05 1.5317 1.11584.0267.01 52.11 ~.798 44.38109.608 4.U4 1.9999 1.657 5.0S 3.04 59.83 0.936 20.12 18.010 2.05 1.54~4 1.11683.73 66.77 52.42 4.835 45.55106.377 4.18 2.0000 1.658 5.04 ),04 60.1J9 0.985 21.09 18.132 2.06 1.5571 1.116 83.33 66.44 52.61 4.871 46.73103.497 4.31 2.0000 1.6S" 5.03 3.03 60.33 1.035 21.4818.2512.061.5699 1ell1 82.82 66.U4 52.'12 4.9U7

47.93100.917 4.43 2.0000 1.660 5.03 3.03 60.55 1.083 21.86 lB.367 2.07 1.5829 1.11t1 62.11i 65.55 5J • .!1 4.942 49.1398.5924.562.0000 1.661 5.03 3.0360.16 1.131 22.25 18.419 2.07 1.596U 1.119 81.44 64.98 53.42 4.971 50.35 96.488 4.68 2.0000 1.661 5.02 ].02 60.97 1.179 22.64 18.588 2.08 1.6092 1.120 80.58 64.31 53.68 5.011 51.58 94.514 4.80 2.0000 1.66l 5.U2 ),02 61.16 1.226 23.04 18.693 2.08 1.6225 1.121 79.5ij 6h55 53.94 5.045 52.83 92.825 4.92 2.0000 1.66l 5.0l h02 61.34 1.l12 23.44 lB.794 2.09 1.6359 1.121 78.46 62.10 ''-'.19 '.079 54.08 91.220 5.03 2.0000 1.663 S.02 3.02 61.52 1.318 23.84 18.891 2.09 1.6494 1.123 77.2061.74 54.45 5.111 55.34 89.742 5.15 2.0000 1.663 5.U2 3.02 61.69 1.363 24.25 18.985 2.10 1.6630 h124t 75.81 60.69 54.71 5.144 S6.62 88.314 5.26 2.0000 1.663 5.02 3.02 61.tl5 1.407 24.66 19.074 2.11 1.6161 1.t2~ 14.28 ~9.53 54.97 5.175 57.91 87.103 5.38 2.UOOu 1.664 !i.Ol 3.Ul 62.ul 1.451 25.07 19.159 2.11 1.6905 1.126 12.61 58.26 ~5.24 5.2U6 59.20 85.917 5.49 2.0000 1.664 5.01 3.01 62.16 1.4"4 25.49 19.240 2.12 1.7044 1.127 70.19 56.88 55.50 5.236 60.51 84.806 5.6U 2.00UO 1.664 5.01 3.Ul 6Z.31J 1.536 25.91 19.315 2.13 1.7164 1.12968.83 5).3S'5S;17 5.265 61.83 83.761 5.70 2.I.WOO 1.664 5.01 3.ul 62.44 1.578 26.34 19.386 2.13 1.7324 1.130 66.73 »).11 56.03 5.294 63.16 82.774 5.81 2.000u 1.664 5.01 3.01 62.58 1.619 26.76 19.452 2.14 1.7466 1.132 64.48 52.03 56.30 5.321 64.50 81.836 5.92 2.0000 1.665 S.Ol 3.01 62.71 1.659 27.20 19.511 2.15 1.7608 1.134 62.08 50.18 56.;7 5.347 650.85 80.942 6.02 2.0000 1.665 5.01 ).01 62.IH 1.698 27.63 19.565 2.16 1.7751 1.136 59.53 48.2U 56.83 5.312 61.21 80.0846.13 2.000U 1.665 5.Ul 3.ul 62."6 1.731 28.07 19.612 2.16 1.1894 1.138 5(uS3 46.10 57.10 ~.3"5

68.59 79.257 6.23 2.0000 1.665 5.01 3.01 63eUÜ 1.774

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