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KfK 2863Februar 1981
Thermophysical Properties 01Sodium in the Liquid and
Gaseous States
K. Thurnay
Institut für Neutronenphysik und ReaktortechnikProjekt Schneller Brüter
Kernforschungszentrum Karlsruhe
KERNFORSCHUNGSZENTRUM KARLSRUHE
Institut für Neutronenphysik und Reaktortechnik
Projekt Schneller Brüter
KfK 2863
Thermophysical Properties of Sodium
in the Liquid and Gaseous States
K. Thurnay
-------------~-, ,.
1
Kernforschungszentrum Karisruhe GmbH, Karlsruhe
Als Manuskript vervielfältigtFür diesen Bericht behalten wir uns alle Rechte vor
Kernforschungszentrum Karlsruhe GmbH
ISSN 0303-4003
Abstract
A system of temperature and density dependent thermal-property-functionshas been developed and checked for the sodiumt using all of the accessibleexperimental data and the mutual relationships of the properties. Inextending these properties beyond the range of the measurements substanceindependent physical relations have been used.
The property-descriptions are valid for all temperatures above the meltingpoint of the sodium and for all densities below the melting density of theliquid.
The system consists of the following thermal properties: pressure t heatcapacity at constant volumet thermal conductivity.
Die thermophysikalischen Eigenschaften des flüssigen und gasförmigenNatriums
Zusammenfassung
Die thermophysikalischen Eigenschaften des Natriums werden mit Hilfe einestemperatur- und dichteabhängigen Funktionssystems beschrieben. DiesesFunktionssystem wurde aus den zur Verfügung stehenden Natriumdatenentwickelt durch Extrapolation der Eigenschaften mit stoffunabhängigenthermischen Relationen.
Die Zustandsdarstellungen sind gültig für alle Temperaturen oberhalb desSchmelzpunktes und für alle Dichtewerte unterhalb der Flüssigkeitsdichte amSchmelzpunkt.
Die dargestellten Eigenschaften sind der Druckt die spezifische Wärme beikonstantem Volumen und die Wärmeleitfähigkeit.
Contents
page
List of Figures
Glossary
1
3
1. Introduction 5
2. The Vapor Pressure of the Saturated Sodium 7
3. The Saturation Line and the Critical Point of the Sodium 9
4. Heat Capacities of the Sodium in the Saturated Area.
Enthalpy, Internal Energy, Entropy 12
5. The Pressure Derivatives of the Sodium on the Saturation Line 18
6. The Thermal Conductivity of the Sodium on the Saturation Line 22
7. The Static Thermal Conductivity in the Two Phase Region 25
8. The Thermal Properties of the Sodium as a Compressed Liquid 29
9. The Thermal Properties of the Sodium as an Overheated Vapor 31
10. The Thermal Properties of the Gaseous (Supercritical) Sodium 37
11. Conclusions 42
References
Appendix G. Fitting the gas equation at the critical point
Appendix I. The integral ~I(w,y)
Figures
Appendix P. The sodium property subroutines KANAST and KANAPT
44
47
50
53
94
- 1 -
List of Figures
Thermal Conductivity of the Sodium Vapor
The Conductivities at Low Pressures
Deviations of the Conductivity-Data from the Reduced Description
Thermal Conductivity of the Saturated Sodium Vapor
Thermal Conductivity on the Saturation Line
Conductivity versus Enthalpy Differences
Static Thermal Conductivity in the Two-Phase Area
Range of Validity of the Thermal Properties
Deviations from the Vapor Pressure Equation
P V T - Properties of the Sodium Vapor
Reduced P V T - Properties
Factor of Reality
The Saturation Line of the Sodium
The Derivatives of the Density on the Saturation Line
Marginal Heat Capacities in the Two-Phase Area
Cv on the Saturation Line
Cp/Cv on the Saturation Line
The Enthalpy on the Saturation Line
Density-Derivatives of the Pressure (Saturated Liquid)
Temperature-Derivatives of the Pressure (Vapor)
P(T*) on the Vapor Side of the Critical Isotherm
Temperature-Derivative of the Pressure (Liquid)
Density-Derivatives of the Pressure (Saturated Vapor)
The Reduced Density-Derivatives
P and ap/aT on the Critical Isotherm
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
The Correcting Factor in the Rarified Vapor
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33
Fig. 34
Hg. 35
Fig. 36
Fig. 37
Fig. 38
Fig. 39
Fig. 40
Fig. 41
- 2 -
The Pressure-Volume-Temperature Surface
The Near-Critical Part of the P(V,T)
The Z(V,T)-Surface
The Cv(V,T)-Surface (Liquid Side)
The Vapor Side of the Cv-Surface
The Gas Side of the Gy-Surface
The Near-Critical Part of the Cv-Surface
The Vapor Side of the Near-Critical Cv-Surface
The V-T-Surface of the Internal Energy
The V-T-Surface of the Entropy
The V-T-Surface of the Thermal Conductivity
The Pressure-Volume Chart
The Pressure-1/T Chart
The Gy -Volume Chart
The Gy-Temperature Chart
Side
Glossary
T
P
v = I/p
s
u
H
P Px,
PT 3P/3T
P 3P /3pP
Ps = 3P/3p!S
- 3 -
temperature
density
specific volume
entropy
internal energy
enthalpy
pressure, vapor pressure
temperature - resp. density
derivatives of the pressure
heat capacities at const. volume,
resp. pressure
Tr = - •
P~IClT P
P/RopoT
thermal conductivity
factor of reality
PT = Pr!Rep
PP/RoTp p
Ps PS/RoT
PL (T), Pv (T)
F(L,T) = F(PL[T],T)
T dP Lr = - 0
L PL
dT
CV(V,T) = CV(PV+o,T)
x x x xT = TL(P), T = TV(p)
pressure derivatives
in reduced form
densities of the saturation line
property in the state of the saturated liquid
sonlc velocity in the liquid
marginal heat capacities
of the two-phase area
the inverted saturation line
T 2508 Kc
0,23 g/cm3Pc =
P = 256,46 barc
a, ß, y
x = - T/Tc
w p/p c
y = T /Tc
R
- 4 -
the critical temperature
the critical density
the critical pressure
the critical exponents
distance to the critical temperature
reduced density
reduced temperature
the gas law constant
- 5 -
1. Introduetion
The development of the sodium-eooled fast breeder reaetors inereased
naturally the interest for the thermophysical properties of this metal. It
is espeeially the safety analysis of the breeders, whieh requires asolid
knowledge of a variety of the thermal properties of the sodium in a broad
range of temperatures and densities - mainly for modelling the Fuel-Coolant
Interaetion (FCI).
This phenomenon is a proeess of violent to explosive vaporisation,
resulting from inadvertent mixing of hot, molten fuel into the liquid
sodium. To describe the behaviour of this exotie mixture the pressure, the
heat capaeity and the thermal conduetivity must be available at tempera
tures as high as 3100 K and at densities near to the melting density of the
sodium.
Moreover, the spaee-dependent hydrodynamieal FCI-eodes'- reeently in use in
the safety analysis - assume these properties to be a thermodynamieally
consistent set of smooth funetions of both variables temperature and
density.
None of the thermal properties of the sodium had been measured at sueh high
temperatures so far, most of the experiments reaehed only just ~he vicinity
of 1650 K, so the construetion of the required Sodium-Thermal-Property
System (STPS) means a eonsiderable extension of the measured data known at
present.
Many attempts have been made in the past to obtain a eomplete deseription
for the thermal properties of sodium. Stone et ale at the NRL developed a
STPS for the region T < 1650 K, whieh also ineludes a pressure-deseription
for the overheated vapor /8, 24/. Miller et ale /25/ estimated the eritieal
properties of the sodium and extrapolated the saturation line and some
other properties of the saturated liquid. A. Padilla developed a STPS for
the suberitieal states using the Rowlinson-Approximation (eq. (72)) beyond
the two-phase region. In the earlier version /26/ of his STPS he used the
eritieal data of D. Miller, later on /27/ he ineorporated the data of Bhise
and Bonilla /4/.
- 6 -
All these STPS bear common insufficiencies in calculating the FeI: the
thermal properties given do not include the conductivity, a description of
the heat capacity in the two-phase area is also lacking and none of these
STPS cover either the supercritical or the high-temperature overheated
vapor area.
The following pages describe the STPS developed and in use in Karlsruhe.
The thermal properties included here are
the pressure P with both its derivatives PT and P~
the heat capacity at constant volume Cv and
the thermal conductivity Qr•.
The STPS (Karlsruhe) describe these properties for all temperatures, ex
ceeding the melting point of the sodium (TM = 371 K) and for all densities,
remaining below the density of the molten sodium at TM (see Fig. 1).
The presented STPS is actually the second edition of the original one from
1975. It differs from its predecessor mainly by taking into account the
vapor pressure measurements of Bhise and Bonilla /4/ and by using more
recent thermal conductivity dates for the sodium vapor /13/.
- 7 -
2. The Vapor Pressure of the Saturated Sodium
The saturation vapor pressure pX(T) is something like the backbone for the
whole STPS. On the saturation line (SL) and in the two-phase area It 18 the
sodium pressure
T Tc
for
(1)
In addition it allows to extend the heat capacity Cv along an isotherm in
the two-phase region via the equation (see e.g. /20/):
=T
-? (2)
It can be also used to substitute the saturated vapor density qv(T) with
the numerically more convenient "factor of reality"
(3)
It i5 also needed to calculate one of the pressure derivatives PT or P, on
the saturation line using the mathematical relation
dT::: +
d ~K
dTK L , V . (4)
Finally. the shape of the pressure-surface P(q,T) beyond the saturated area
depends directly on the values of P snd PT on the SL.
The vapor pressure description of the STPS (Karlsruhe) 15 based on the
2
of sI.
of Bhise and
@ 1255 - 2500 K@ From the
these data a
K
12153,:;: 11919-
To 1 5·ln
P in bars» r in K.
the deviations of the measured pressures from this
eq.
In a second
the
ion I 1s for
TB :: 1154 K
two pX-equations would mean either to use more formulas'
than eq. (5) (adding a rn-term to eq. (5), for in
and in) too at the (see eq.
». To avoid both of these alternatives and since the deviations are
- the maximal of eq. (5) fram the Makanski-
1s 2.75 i. - the vapor pressure i5 described
the whale range fram the to the critical with
eq. ).
- 9 -
3. The Saturation Line and the Critical Point of the Sodium
The shape of the saturation line in the "cold"-region
T < 11 0 0 K
of the STPS (Karlsruhe) is also based on the measurements of the NRL-Group
already quoted /5,6/.
The density of the saturated liquid qL(T) is described with the formula
given in /5/.
The density of the saturated vapor ~v(T) has been calculated via Zv(T) (eq.
(3». To do this the derivative of Z was set to
1
Zv
dZ V
dT=
3Cn= 0
nan . T (6 )
with Zv ( 334, 5 K ) = 1
Fig. 5 shows the shape of this derivative ( --- line) and the corresponding
Zv-function (_._._). The polynomial in the eq. (6) was determined by
fitting Zv to the corresponding values, developed from the NRL-measurements
(6-signs in Fig. 5). The pressures, measured by this group for various
isochores in the overheated vapor displays Fig. 3 (signs 0 - Z). The dashed
curve represents the pX(T)-equation (5). The remaining dashed lines are
linear approximations for the pressure
P(T) = +x X
(T - T )·PT(T ) (7)
using best fit PT-values for each isochor (as the reduced pressure
presentation - Fig. 4 - shows this equation is in fact not a good
approximation in this area, PT is in reality decreasing for increasing
- 10 -
pressures). From the eq. (7) and (5) the saturation of the
isochor, TX ean be caleulated. The saturation volume of the isochor
was eorrected to T • TX by assuming that the measured volumina in eaeh
experiment inerease slightly but llnearly with T. From pX, and V(TX) eq.
(3) gives Zv(TX). The eorrespondenee between these "measured" Zv""'Values and
the ealeulation via eq. (6) is better than 0.2 %.
N.B. Caleulating Zv from the virial-equation given in /8/ results in a non
monotonous dZv/dT with a loeal minimum at ~ 1000 K, leadlng to a Cv(V)
maximum of wS R at this point.
The high temperature part of the SL was eonstructed using the Hypothesis of
the Universality of the Critieal Exponents (HUCE). This theory states (see
e.g. /15/), that the behaviour of a material with phase-transition 1s - in
the vieinity of the corresponding eritieal point - determined by a set of
universal numbers, the eritieal exponents. The values of these exponents
are
cx = 0,11 ß = 0,325 1 = 1,24. (8)
aeeording to ealeulations using the 3D-Ising-model /16/ or to reeent
measurements /15/. For the near-eritical (T ~ Te) saturation line of the
liquids HUCE gives the following shape:
~ L ( T ) = ~ C . ( 1 "I- b· xß
b·x ß(9)
~v ( T ) = <Sc • { 1
with x = 1 - T I Tc (10)
A eomparison of the near-eritieal densities of the saturated cesium /7/
with these equations gives b • 2. Taking this b-value for the sodium too
results in the following high temperature SL:
- 11 -
(11)
~v (T) = ~ c . [ 1 - 2· xß + x· (o.V + x· gv + x 3 .hV) ]
for T 170 0 K
The x-polynomlals, added in this equation serve for a smooth connection
with smooth first T-derivatives at the sWitching point.
As a sodium critical temperature
Tc = 2508 K (12)
is used. This value corresponds via eq. (5) to the critical pressure,
measured in /4/:
Pe = 256 , 46 bar (13)
To achieve a smooth and monotonous dens1ty-connect1on at 1700 K a critical
density value of
~c =
was needed in the eq. (11).
0,23 g/cm 3 (14)
The complete saturation line 1s presented in Fig. 6.
- 12 -
4. Reat Capacities of the Sodium in the Saturated Area.
Enthalpy, Internal Energy, Entropy
The caloric properties of the STPS (Karlsruhe) are based - in the sub
critical region - on the heat capacity of the liquid with vanishing vapor
content:
-Cy(L,T) = CV{~L-o,T}
~L - 0 = ~L - E,E - 0
For other densities the Cv-s can be calculated either with eq. (2) or with
=T--.~2
(2A)
or, at the crossing of the saturation line, from the Cy-jump at this place.
The fact, that Cy jumps crossing the SL (see Figs. 30, 31, 33, 34) 1s not
broadly c1rculated in the thermal physics. The authors in 119/ - the single
quotation I found - declare the eq. (19) to be lIa well-known relation ll
without giving any further references.
The or1gin of this jump lies in the jump of the pressure derivative at this
same place:
dT
In the immediate neighborhood of 9L it is
dSCy = T· - =
dT
dST'
dTT'
dS d ~l
dT(5)
Setting
- J3 -
dS((LIT) = T'-(~L(T),T)
dT(16)
and using the thermastatic relation
s ~ = - PT I ~ 2
(see e.g. /20/) ane has
resp.
T d"L dpX
+ --'--~L2 d T d T
T dqL-. . PT ( S l )~L2 dT
(17)
(18)
These equations give a Cv-rise of
eV(Ll - [V(L)T d"L
= - -.[~L2' dT dT
] (19)
at the crassing fram the liquid into the saturated area.
Using the eq. (4) and wtth the reduced derivative
T d~Lrl = (20)
~L dT
eq. (19) can be transformed ta
...(V ( L ) r L
2 . P~( L) I T[V{L) - = (21)
- 14 -
The constant pressure head capacity Cp. can be expressed in a similar way
/20/:
(p(L) - (y{Ll = (22)
with (23)
From the eq. (21) and (22) one has
( p ( L) = Ey ( L) + (r 2 - r L2 ). p ~ ( L ) I T (24)
Since for the "cold" liquid the SL is practically "vertica1", i.e.
d~L O~L:::
dT oT
(see Fig. 7), so it 1s
-(v( L,T ::: (p( L,T ) for T « Tc (25)
On the other hand at near-critica1 temperatures the HUCE describes Cv as
-C(= C{O)·\X! for x « 1 (26)
The Cv(L,T)-shape, adopted in Karlsruhe (Fig. 8) was suggested by these two
equations as follows:
....(y{L,T) = (p{L,T) + c(-}·[ X-CA - &(T) ] (27)
For Cp(L) the formu1a of Ginnings et a1. /11/ was taken (t1-signs in Fig.
8). ~(T) is a polynomial subtracted in eq. (27) to equalize the term x-~at
low temperatures:
for T < 1100 K
The factor
- 15 -
c{-) = 5,S3·R
was chosen to achieve a smooth and monotonous Cv(L)-shape at near-critical
temperatures.
On the other side of the two-phase area the heat capacity of the vapor with
vanishlng humidlty
f. y (V I T) = CV( ~V + 0 I T )
(--- line in Fig. 8) can be calculated from Cv(L) with the eq. (2) to
Rere 1s
...Cy(V) =
...CV( L ) + ~ V· T . (28)
fjV = (29)
the volume-difference of the saturated states.
Actually» according to the eq. (2), for any volume
v = 1 I ~
inside the two-phase region the heat capac.ity is
[V(~,T ) = -[V(L,T) +1 1
~L
) . T . (30)
~V < <
The heat capacity of the saturated vapor CV(V) is calculated from the CV
step at this place:
- 16 -
(y(y) - (v(y) =
=T d~V. -.[
Q, y Z d T dT] (19A)
Cp(V) can be determined as on the liquid side. Fig. 9 displays the
saturated heat capacities Cv(L) and Cv(V), Fig. 10 gives the Cp/Cv
relations.
The enthalpy of the saturated liquid can be integrated using the SL
derivative
:: T· - + _. - ::dH( L )
dT
::
dS
dT
1
,..(V{L) +
dP
dT
(31 )
(see /20/ and eq. (16), (17». For the internal energy, U, one has a
corresponding derivative
d U( L )
dT::
,..
(V( L )Px
rL+ _.(
<SL T dT(32)
For the derivative of the entropy S a slightly transformed eq. (31) can
serve. At low temperatures (T < 1600 K) as H(L,T) and S(L,T) the respective
functions of the NRL-Group - cited in /12/ - can be used.
The properties H, U, and S in the saturated vapor are easier to determine
from the liquid properties than by using the derivatives. The equation of
Clausius-Clapeyron gives here
!1S = S(V) - S(L) :: !1V·--dT
:: !1 HIT (33)
resp.
- 17 -
LlU = T-LlV,(dT
(34)
Fig. 11 shows the enthalpy of the saturated liquid and vapor.
- 18 -
5. The Pressure Derivatives of the Sodium on the Saturation Line
In the following for the pressure derivatives reduced forms will be used:
,..
Pr :: Pr IR· ~
d p X
t :: I R·qdT
-P~ :: P~ J R· T
,.. oPPs :: -I I R·T
o~ S
(35)
(36)
(37)
(38)
These are more convenient in having no dimensions t giving a simpler form to
the physical relations and varying in a more restricted version - mostly
near to the unity - as the not reduced properties (compare the Figs. 12 and
16 with Fig. 17). Moreover, by using the properties (34) - (37) all the
heat capacities are given in R-units.
In the STPS (Karlsruhe) the pressure derivatives of the saturated liquid
are calculated - at low temperatures
T < 11 0 0 K
- from the measured values of the velocity of sound, Cs /9 t 10/. This can
be done by using the relations /20/
oP
Is2
:: (S (39)
O~
and
- (p -Ps = _.P~
(40)
(V
- 19 -
Wlth the equatlons (21)
(v = - 2 -(V - rL . p~ (21A)
and (24) (p = - 2 2-(V + (r - rL ). P~ (24A)
One can ellmlnate Cp and Cv from the eq. (40):
With the relation /20/
Cp - Cv
Cv=
2 r . p~
... 2 -Cv -rL .p~
(41)
PT I P~ = - r
and with the reduced form of the eq. (4)
...t .:: PT ... r L . P~
eq. (41) can be transformed to
(42)
(4A)
Cp - Cv
Cy:: (43)
!his equation and eq. (40) gives
...Ps :: -
p~ ... (44)
The inverted form of this relation
...P~ ::
...Ps -
) 2
-Cy - 2· t . r L +
(45)
- 20 -
together with the eq. (39). (4A) and (ZlA) allows to calculate the thermal
properties P~. Pr. ev. ep of the liquid from the sonie velocity. Cv(L), qL
and pX(r).
In the "eold"-statre of the saturated vapor
T 2320 K
-the basic property is Pr; Pq and Cv are determined by the eq. (4A) resp.
(21A).
-Pr in this region is deseribed with r-polynomials (--- line in Fig. 13)
fitted to satisfy the following conditions:
-1. for low temperatures Pr must eonverge to the ideal gas value
-PT ( V , T ) -_......=- 1 for T
2. Pr must suffice the values gained from the PVT-measurements /6/ (~-signs
in Fig. 13),
3. both PT and the ev(V) calcualted from it (0-0 line in Fig. 9) must have
the simplest possible form compatible with 1. and 2.
In the fitting of the polynomials to the PT-values only the first 8
best-fit PT(TX)-values (eq. (7» were used. The latest one, corresponding
to the exp. 7 (Z-signs in Figs. 3 and 4) has been omitted for having a
value much too high:
PT ( Tx = 16 14 K) =
At near-critical temperatures, i.e. at
1 ,86
T 1 '1 0 0 K
- 21 -
for the liquid end at
T 2320 K
for the vapor P~ was chosen as basic property. since the HUCE prescribes
for this function a shape:
for
P~(K,T) =
x - 0
K = L,V(46)
To have smooth connections with the respective low-temperature Pq-s form
polynomials ~(K.x) had to be included in the eq. (46). So the high
temperature functions are:
P~( L , T )
and
= (47)
with
(48 )
t.p(K,x 1 =4
1 + L: fn{Kl·x nn=1
K = L, V . (49)
As Po the fitting at the switching points gave
Po = 1150,89 j/g
- 22 -
6. The Thermal Conductivity of the Sodium on ehe Saturation Line
Thermal conductivity measurements of the sodium are scarce and the
temperature range of the data is very limited. T • 1100 K being the higheat
temperature aa weIl for the liquid as for the vapor. Correspondingly the
switching point between measured QT-functions and extrapolated ones lies
also low at
T = 1280 K
for both saturated states.
For the thermal conductivity of the cold liquid the T-polynomial
recommended by Golden and Tokar /12/ - is used in Karlsruhe.
The most recent data concerning the sodium vapor thermal conductivity has
been published by Timrot et al. /13/. This group measured the QT of the
sodium in the overheated vapor along five different isotherms as functions
of the pressure (see ~ - x signs on the Figs. 20 and 21). They calculated
the thermal conductivity for the saturated vapor. Qf(V.T) (--- line on Fig.
23) by extending the isothermal data up to the saturation points ( ! -linesI
on the Figs. 20. 21) via the equation
(SO)
...QT is here the conductivity of the monatomic vapor. The factors b and e bad
been determined for each isotherm individually.
Since Qf(V.T) has a vanishing derivative at T = 1200 K. it is not practical
to use it as saturated conductivity. giving no help in the QT-extrapolation
to higher temperatures. To avoid this hindrance. a temperature-independent
description was developed. instead of the eq. (SO) for the thermal
conductivity (- lines on Figs. 20. 21):
...= 0T ( T ) . [ 1 + b· TI + (51)
- 23 -
the reduced pressures
)
and the factors
e ::: -0,3353 (53)
pressure • Since Timrot et ale estimate theiras a
the
of the
the measurements from the eq. (51)
maximum error as about 5 % , the T-independent (50) is weIl
within the
(51) - (53)
of the
has the
uncertainties. From the equations
on(see
'"T(V T) ::: o.r(T)-1,5604
) .
)
There
thermal
a lack of ideas about the
the obvious notion
of the near-critical
T( C ) "'""=3 T(V ,T )
T ----e- TC
the near-critical conductivities
7/. He that for many substances
the and vapor conductivities vanishes as the
has been
the
for
. ~H if o < x )
for
0.05 for substances N and 0 but 0.14
for the near-critical thermal
(see eq. (33».
LlH =
- 24 -
F . ( ~ L ~v )
Hence the extrapolated thermal conductivities were
C1. T(L,T) -
= o.T ( c ) . [ 1 + 2· X ß + x· ( aL + x· 9 L + X 3. L)
resp.
=
C1. T(V,T) =
0. T( C ) . [ 1 - 2· Xß + x· (aV + X,9V + X 3. V) J .)
The conditions
1. the linearity-limit, XG had to be as large as possible (see
2. the connections at 1280 K had to be smooth and monotonous
• 25) and
determined the parameters ä L, "ä"t •••critical conductivity:
as weIl as the
o.T (c) = 0 , 05 WI cm I K )
Fig. 24 shows the reduced conductivities QT(T)/Qr( in the I/T·ede,pend4~nc:e
The dashed lines 1nd1cate the extrapolated properties. The vapor
function, given in /12/ resp. 114/ and used in the first version of the
STPS in Karlsruhe 1s also indicated here (x-signs).
- 25 -
7. The Statie Thermal Conduetivity in the Two-Phase Region
In the two-phase ares the substanee eonsists of a mass of saturated liquid
(ML ) dispersed in a mass of saturated vapor (Mv)' both having beside the
common temperature, T, a common pressure, pX(T). The quality of this
mixture depends at a given T on the mass relation of the vapor:
(59 )
Since the specific volume of the mixture is
(60)
with (61)
the mixture quality can be calculated directly from the densities:
= )
The transfer of heat can proceed in this mixture in four different ways
beside the radiation, convection and conduction heat can be also moved here
in latent form, i.e. by transporting and subsequently condensating
saturated vapor. AB the statie thermal conduetivity only the conductive
heat transfer 1s considered.
For the calculation of the eonduetivity in the
following, very simple liquid-vapor-mixture concept
sketch):
the
was devised (see
- 26 -
-' ",," / "" /./ /".I' "" ,,"
J
+
A- + +A'- , /'''
/" / '/ /
B'- // ,,/,-B- + +
"..,
" "" /
,," //
"
v"/
/ "/
1/ /.. "/
+ + + + +"""'"-,,"',-/-/'"' ,..1/-"..../-/""'''/-/"''1 D//
" v/ "" /' / /', "".I' .. /"" "" /'
+ +
'"/ ""/ //' /'
"" "...
The liquid is uniformly distributed in the vapor, as a number of little
cubes, each mixture volume 0 3 contain1ng a liquid volume d3 • In a 03-volume
there 1s ~ two-phase substance and qv' (l - 11,3) saturated vapor, i'\ being
rt = d I 0
the length-rat10 of the liquid. The mixture-quality is therefore
(63)
s =3
~V·(1-rt )/~
giv1ng with eq. (62) the follow1ng relation between length ratio and
mixture density:
ll.(<S,T) =3
(64 )
To facil1tate the calculation of QT(q) it 1s assumed, that the heat current
J crosses perpendicularly one O-layer of the mixture-lattice, on a surface
F. D is supposedly small enough to render QT(L) and QT(V) space-independent
inside of this layer. Further, since evaporation and condensation had been
excluded, the heat currents in all three sub-layers must be the same and
equal to J:
- 27 -
Denoting the temperatures on the surfaces A. A' •••• aB TA. TA' •••• the
currents in the first and in the third layer are:
2·TA - TA'
JA'A = ·F·o.T(V}D·(1-rt}
J S'B 2·TB - TB'
resp. = . ·F·o.T(V)D·(1-rt}
In the second layer there are two different heat currents, one in the
liquid part of the layer
(65)
(66 )
(67)
and one in the vapor part:
J V =
rendering the total layer-current to
J B'A' = J L + J V =
TA' - TB"·F·(Q.r(V) ll. 2 . 11 o.r ) (68)= +
D·rt
Here 19 (69 )
- 28 -
The equivalent heat current for the total layer AB i8
J = (70)
The identi ty
TA - T B = TA - TA' + TA' - TB' + TB' - TB
and the equation (65) in connection with the eq. (66) - (68) and (70) gives
or
= (71)
Mixture conductivities, calculated with this equation for various isotherms
are displayed on Fig. 26 (--- lines).
- 29 -
8. The Thermal Properties of the Sodium as a Compressed Liquid
In the state of the compressed liquid
T 0( Tc >
the sodium pressure is derived using a PT-approximation recommended by
Rowlinson /18/. It i8 assumed that in the compressed sodium - as in many
other compressed liquids - PT depends practically only on the density:
:: (72)
IX 18 here a "density-temperature", the inverted form of the saturated
liquid density, i.e.
::: resp. (73)
Wlth thie assumption the pressure of the liquid is
::
+
Tf dt·PT(q,t} ::
TX
x x+ (T - T ). Pr (L I T ) (74)
Ag a density derivative eq. (14) gives:
dp X
( T x )x
P~(~,T } :: [ Pr ( L I T ) +dT
( T - TX ) .
dP T (L,Tx
) ]1d qL
Tx ) .(15)
+ {dT dT
Since, due to the assumption (12) the second thermal derivative of P
vanlshes in thie state, the Cy-derlvative (eq. (2A» vanishes too and the
heat capaclty remains Cv(L,I) on the whole isotherm I in the compressed
- 30 -
In describing the thermal conductivity in this state - for the sake of
simplicity - also a temperature-independence-assumption was used:
-- = (76)
- 31 -
9. The Thermal Properties of the Sodlum as an Overheated Vapor
It is natural to try in this state
T < Tc < ~v ( T )
too a pressure-description with T-independent PT:
with
PT(~,T) ;:
X::: TV(q) resp.
:::
X::: <Sv (T )
(77)
(78)
The assumption (77) would give here - as in the compressed liquid -
P('3,T) ::: pX{T x ) + (T _ TX) . PT ( V I T x ) (79 )
dpx
( T X )x
with PQ, ( ~. / T ) ::: [ PT{V I T ;-
dT
( T - T x ) .dPT
(V/TX
) ] !d qv
Tx );- ( (80)dT dT
and CV(q,T ) ::: CV{V, T ) (8I)
The approximation (77) is unfortunately not fully adequate 1n the whole
vapor area; at low densities (rarified sub-region on Fig. 1)
the equations (77), (79) calculate too high values for PT and P. At
densities as low as
( to TX .... 1050 K) the sodium could be expected to behave aB an
ideal gas, especially far from the saturation line, on the critical
isotherm. So
Z(~,T) ;: 1 :: 1
- 32 -
were reasonable values here. As the Figs. 13 and 14 prove, the respective
calculated properties (---- line on Fig. 13, line on Fig. 14) are too
high for these low density states.
There is also an experimental evidence for the insufficiency of the eq.
(77) as PT-description in this sub-range. As the reduced representation of
the PVT-data of the NRL /6/ indicates, PT on an isochor is not constant but
decreases markedly with increasing T from a maximum at T • TX (Fig. 4).
To achieve a better pressure-description in the rarified region, a ('\,T)-
dependent correction-term was added here to the PT-equation:
,... ,... xPT ( ~ ,T ) = PT (V ,T ) + G(s)·H(u) (82 )
The T-dependence 1s included in the variable
u(~,T) = 0 , 2 . (83 )
S i8 only a function of the density:
s(~) = ln~c/ln~
(s(,\) has a similar shape to TX(~), but it 18 easier to calculate).
The selection of the functions
jJ(u) = U / ( e U - 1 )
and
H(U) = j.J (U ). [U + jJ ( lJ) ]
(84)
(85 )
(86 )
was dicta ted by the fact, that the necessary PT- and P-corrections on the
critical isotherm are of the same size, i.e.
xllPT(T ,Tc) -- x x
llP(T ,Tc)/(Tc-T ) (87)
- 33 -
A polynomial term for the correction:
=
would give
LlP= f . ---------
1 + m
Using here a smaII m-value (m< 1) tosuffice to the equation (87) would
render
o llPT / 0 T =x m-1
f·m·(T-T)
infinite at TX = T, giving an unphysical jump in the heat capacity (eq.
(2A» at this point, instead of a ev-shape, gradually decreasing with TX~O.
Integrating eq. (82) - the necessary derivatives are
and
dU
oT
djJ
d u=
=U
}J_.( 1 - U - jJ )
U
(88 )
(89)
the term G·H gives the following additional pressure:
TII P (~ I T ) = Jdt· R . q. G(s ). H (U) =
TX
x= R. ~. ( T - T ). G(s ) . jJ ( U ) ( 90 )
- 34 -
A comparison of H(u) with ~(u) for T • Tc results in a (87)-relation:
1 "" 0.9.
The pressure correction (90) gives, using
=u
---.( 1T _ TX
0,2--
u(91)
the following P~ -correction:
o~p _ TX dG= R· [ (T ) . ( G + ~'-)'jJ +
a~ d<3
TXH - jJ
. ( H - 0, 2 . ) . G ] (92)x
rV(T ) u
To the heat capacity the PT-correction supplies (eq. (2A»:
~(V(~/T ) = (V(~/T ) - (V(V, T ) =
~ G(s ) oH= - R· T· J d~· .- (93 )
~V~ oT
dH HWith
du= ---[2'(jJ-1) + U]
u
this can be converted to ~(V{~/T) =
=dt
t
T--·G·H·[2·(jJ-1) + U ]·rv(t) .T - t
(94)
- 35 -
The heat eapaeity, ealeulated by this equation for extremely low densities,
Cv(ü,T) is shown on Fig. 9 as a dashed line.
The boundary of the rarified sub-region of the overheated vapor in the STPS
(Karlsruhe) lies at
= 0,0 1 9 Icm3 (95)
(see -line on Fig. 18). This value eorresponds to a density-temperature
of
(on Figs. 13 and 14) or to
=
:::
1850 K
0,32
In the dense-area (~> ~G) P and Cv are deseribed with the equations (77),
(79 ), ( 80) and ( 81) •
For the density-fitting of the eorreetion-term in the rarified vapor a
(96)
form was used. The funetion ~(s) in this equation was shaped to give the
"right" P and PT-funetions on the eritieal isotherm and physieally meaning
ful Cv-values in the "vaeuum". Ag a requirement for eorreeting P(~,Te) and
PT(~,Te) a smooth erossing of the eritieal isotherm on the pressure-surfaee
is demanded:
PT ( ~ , Tc - 0 )
P{~ ,Tc o} = P(~ ,Tc + 0 }
I
(97)
(98)
for
(for the supereritieal funetions see the following ehapter).
- 36 -
As for the Cv(O,T) one expects this properey not to descend below the
Cv-value for the monatomic vapor, and yet to lie - at low temperatures
not much higher than this value, i.e.
CV(O,T)
Cv(O, T }
3 I 2 . R
3 I 2 . R
f. T ---ee=_ 0 (99 )
Fig. 19 shows the compromise-'1(s) chosen for the rarified vapor (- Une).
The dashed lines on this Fig. correspond to ~-s demanded by the eq. (97)
and (98). The maximal deviations in these equations - resulting from the
use of the actual ~(s) instead of the needed functions - are 0.65 % for the
pressure and 3.5 % for PT' Thell-departure from the connnon prescribed shape
for low s-values (s < 0.12) is due to the heat capacity values in the
vacuum. Without this departure Cv(O, T) would be less than 3/2 R at T" 900 K
(see Fig. 9) and even higher atT < 600 K than with the corrected ~.
Actually, it is not Cv(O,T), which 1s to~ large at low temperatures, but
Cv(V,T) - since the density of the saturated state at low temperatures is
very low - aud therefore
CVIO,T) -- CVIV,T} at T 400 K
is not unreasonable. This overestimated Cv(V,T) results probably from the
incorrect description of the vapor pressure with the eq. (5) in the low
temperature area (see eq. (28) and (l9A)). The dent in Cv(O,T) at T::1750 K
(Fig. 9) has no physical foundations and could have been eliminated with a
more complicated description of PT(V,T).
The corrected functions P(~,Tc) and PT(~,Tc) are shown on the Figs. 13 and
14 as solid lines.
The thermal conductivity of the overheated vapor has been set - as those of
the compressed liquid - temperature-independent:
(100)
- 37 -
10. The Thermal Properties of the Gaseous (Supercritical) Sodium
In the supercritieal or gaseous area
T Tc
of a non-ideal substance the easiest way to describe the pressure 1s to use
the equation of van der Waals /20/:
(P ... q I V 2 ). ( V - Vo ) = R· T (l0I)
q and Vo are the van der Waals constants representing the attraetive forees
between the gas-partieles resp. the self-volume of these. Unfortunately the
sodium is in the high-density eritieal states
T = Tc >
mueh softer than the "van der Waals"-gas. At the erit1eal point, for
example the sodium has a rea1ity of
0, 1 2 3
whereas the equation (101) gives here
3 I 8J
i.e. at the same pressure the sodium requires on1y a third of the volume of
the van-der-Waals substanee.
To derlve from the eq. (101) a more eompresslble P(V)-relation the self
volume had been rendered density-dependent:
Val (1 + ",2· B )
with w = ~ I ~C (102)
As a dens1ty-exponent in the denominator the smallest possible value, 2 was
- 38 -
chosen, which is able to give a monotonous P(~,Tc)-function in the whole
range of the liquid densities
<
A further requirement for the pressure-shape in the gas state is, to become
with increasing temperature aud with decreasing density more and more
ideal, i.e.
Here 1s
z (W ,y ) ---- 1
y =
f.
Tel T
w·y ---c:e:-_ 0 (103)
(104)
A
~ c 1
To comply with this behaviour the self-volume acquired a T-dependence:
Y-- . ------:~-+ W2. 8
(105 )
As for the coefficient of internal attraction, this property turned also
(q,T)-dependent, to allow a smooth crossing of the eritical isotherm on the
P-surface in the region of moderate to high densities. The following form
was seleeted:
with
and
q ---Ibt:- q(W,Y) = Q.(W,y )·R·T c I ~C
5Q(w , y) = L W n-1 . Fn ( y )
n=1
(106)
(107)
(l08)
These modifications of the self-volume and of the internal attractions
transforms eq. (101) into
Z lw,y) = 1 + W.y·[ AI S(w,y J - Q.(w,y) ] , (l 09)
with S(W,y) ::
- 39 -
1 - w·y·A + IN '2. B (110)
The respeetive pressure derivatives are here
-P~ :: Z + W·ZW
and
-PT = Z - Y. Z Y
with
Oll )
(112)
and
::
= w·y·A·( 11- w 2 ·B 1 J S2 +
5- y L w n . ( Fn + y.H n
n =1
(113)
(114)
As a base-line for the Cv-caleulation in the gas-area the eritieal isochor
was chosen
(see eq. (26». The eonstant C(+) was gained from Cv(L.T):
(115)
lim xcx.CV(L,T)x-=-+ 0
016 )
..Using here Cv(L.T) would result in much too large supercritical Cv-values
(see Figs. 30 or 32).
The density-derivative in this region 15 (see eq. (2A»:
- 40 -
oCv R 2 02
Z= -'y =
0') ~ dy2
R 2 (w·A)2.1 + w2. B 5
L: wn. H ] (117)= -2·_·y·[S 3 n=1
nq
Hence the heat capacity beyond the critical isochor can be given as
with
CV(~,T) = (118)
ßCV(q,Tl ::
')
f dCV ::Cl, C
=2 5
R·y .[ 2·r::n =1
----·H nn
(119)
The integral
ßI(w,y) ::
w
2.J1
dw·w-~.{ 1
S2
w+ A·y·
S(120)
is evaluated in the Appendix I.
The coeff1cients of the gas-equation (109) A, B, G1" •• ' HS had been
determined by adjust1ng the pressure to· the subcrit1cal values (see Fig.
18) and by demanding a "correct" pressure-shape at the crltlcal point:
- 41 -
02 P 0 3 PP = Pe P~ = 0 = 0 > 0 (121),
Oq 2 d~ 3
These four equations determine A, B, GI and G2. The rest of the set G3 - HS
were calculated to fulfill the requirements (97) resp. (98) in the domain
0,85 "<3c < < 1,1"~c
The remaining parts of the critical isotherm had been cut in four liquid
and four dense-vapor areas, and in each area the Gi-s and Hi-s had been
calculated (eq. (97), (98» separately. The maximal deviations in P, and PT
in these dense parts (q> 0.01) have the magnitude 0.1 %.
For the description of the thermal conductivity of the gas the same
approximations were set as in the compressed liquid resp. in the overheated
vapor:
f.
f. <
(122)
(123)
- 42 -
11. Conc1usions
The shape of the thermal and ea10rie properties of the sodium - eva1uated
in the foregoing ehapters - are shown in the Figs. 27 - 41. The Figs. 27
37 are property-surfaees. giving a eomp1ete aceount of the temperature
specifie volume dependenee. On the Figs. 38 - 41 one of the re1ationships
is given on1y in a parametrie way.
The pressure surfaee of the STPS (Kar1sruhe) is disp1ayed on the Figs. 27.
28. 29. 38 and 39. Fig. 27 shows the pressure for temperatures above the
boi1ing point in 10garithmie seale. Fig. 28 is a sma11. near-eritiea1 part
of the P-surfaee in linear sea1e. Fig. 29 presents the pressure in redueed
form. i.e. the faetor of reality. Z. Fig. 39 is. for the benefit of a
eustomary PX(T)-pieture. in the log P(l/T)-dependenee.
From all the sodium-properties given in this STPS it is the pressure. one
can most safely depend upon. Not on1y 1s PX(T) based in the who1e range on
measurements. but so are also - in a smal1er T-domain - the P-derivatives
on the two-phase-border. Moreover the surfaee shows the eorreet. idea1-gas
behaviour for V·T -> oo(see Fig. 29). Unappropriate P-va1ues are expeeted
only in two areas:
- in the eompressed. near1y-eritieal liquid far from the saturated state.and
- in the state of very hot gas.
The ealorie equation of state is presented on the Figs. 30 - 36. 40 and 41.
The Figs. 30 - 32 show three different views of the Cv-surfaee from the
liquid. vapor and from the gas side in 10garithmie sea1e. Figs. 40 and 41
are the eorresponding (V.Cv)- resp. (T.Cv)-projeetions. Figs. 33 and 34
show a b10wn-up part of this surfaee near to the eritieal point in linear
seale (the vertiea1 walls on the Figs. 31 - 34 should indieate the
infiniteness of the Cv on the eritieal isotherm). The interna1 energy and
the entropy - eorresponding to this Cv (eq. (22). (30» are presented on
the Figs. 35 and 36.
The Cv-surfaee of this STPS is seeured by on1y one direet1y-measured data-
- 43 -
set /11/. It depends also heavl1y on the exaet deserlptlon of d2pX/dT2,
espeelally at low temperatures. So this surface is eorreet probably only in
a qualitative way. In any ease the eurrently available measured Cy-surfaces
(/21/ deseribes the Cy of the C02 in the twe-phase and in the gaseous area,
/22/ glves the two-phase Cy-surface ef the n-Heptane) shows great
similarities with the Figs. 30 - 34 (ehe Cy-ridge of the supereritieal C02
lies mueh nearer to the eritical isochore as it i8 on the Fig. 32).
As for the surface of the thermal conductivity (Fig. 37, for the (V,QT)
projection see Fig. 26), the greatest part of it 18 conjectural. Not only
the measured data are scarce for thisproperty but also there are not any
cross-relations available with other thermal properties to check the
calculated values. Still, the electrical conductivity measurements of the
cesium /23/ disclose, that this property 1s also practieally temperature
independent at densities in the liquid range.
So an assumption cf a proportionality between the two conduetivities at
high temperatures too would give some support for the temperature
independent description of the QT at high densities.
- 44 -
References
11/ R. W. Ditchburn and J. C. Gilmour,
The Vapor Pressures of Monatomie Vapors, &eva. Mod. Phys. 11, 310
(1941).
/2/ M. M. Makansi, C. H. Muendel, and W. A. Selke.
Determination of the Vapor Pressure of Sodium. J. Phys. Chem. ~, 40
(1955).
/3/ J. P. Stone. C. T. Ewing. J. R. Spann. E. ,Wo Steinkuller, D. D. Williams,
R. R. Miller,
High-Temperature Vapor Pressures of Sodium, Potassium, ar,d Cesium.
J. Chem. and Eng. Data, 1l, 315 (1966).
/4/ V. S. Bhise and C. F. Bonilla.
The Experimental Vapor Pressure and Critieal Point of Sodium,
Proc. lot. Conf. on Liquid Metal Technology in Energy Production,
CONF-760S03-P2 Champion, PA, May 1976.
/5/ J. P. Stone et al.,
High Temperature Specific Volumes of Liquid Sodium, Potassium. and
Cesium, J. Chem. and Eng. Data, ll, 321 (1966).
/6/ J. P. Stone et al.,
High Temperature PVT Properties of Sodium. Potassium. and Cesium,
lbid •• p. 309.
/7/ C. T. Ewing. J. P. Stoue. J. R. Spann. R. R. Miller.
High-Temperature Properties of Cesium, J. Chem. and Eng. Data, ll, 474
(1966).
/8/ C. T. Ewing et al.,
High-Temperature Properties of Sodium, Ibid., p. 468.
/9/ L. Leibowitz. M. G. Chasanov. R. B1ornquist.
Speed of Sound in Liquid Sodium to 10aO·C. J. of Appl. Phys. 42,
p. 2135 (1971).
- 45 -
/10/ M. G. Chasanov, L. Leibowitz, D. F. Fischer, R. Bl~mquist,
Speed of Sound in Liquid Sodium from 1000 to 1 J. of Appl.
Phys. ~, p. 748 (1 ).
/111 D. C. Ginnings, T. B. Douglas, A. F. Ball,
J. Res. Natl. Bur. Stds., ~, 23 (1950).
112/ G. H. Golden, J. V. Tokar,
Thermophysical Properties of Sodium, ANL-7323, Argonne (1967).
/13/ D. L. Timrot, V. V. Makhrov, V. I. Sviridenko.
Method of the Heated Filament for Use in Corrosive Media ••••
Temperature. li. 58 (1976).
/1 B. I. Stefanov, D. L. Timrot, E. E. Totskii. and Chu Wen-hao,
Viseosity and Thermal Conductivity of the of Sodium and
Potassium, High Temperature 4, 131 (1966).
A. L. Sengers, R. Hocken, J. V. Sengers,
Critical-Point Universality and Fluids, Physlcs Today, Deeember 1977,
p. 42.
/16/ Yu. E. Sheludryak, V. A. Rabinovich.
On the Values of the Critiea! Indices of the Three-Dimensional Ising
Model, High Temperature, ll. 40 (1979).
/1 P. E. Liley,
Correlations for the Thermal Conductivity and Latent Heat of
Saturated Fluids near the Critieal Point, Proc. Flfth • on Therm.
Prop., Boston, 1970.
8/ J. S. Rowlinson.
Liquids and
1969.
Mixtures, 2nd Edition Plenum Press, New York.
- 46 -
/19/ Kh. I. Amirkhanov, B. G. Alibekov, B. A. Mursalov, G. V. Stepanov,
Caleulation of the Derivatives of Thermal and Calorie Quantities on a
Saturation Line, High Temperature, 10, 415 (1972).
/20/ o. A. Hougen, K. M. Watson and R. A. Ragatz,
Chemieal Proeess Prineiples - Vol. 11, Thermodynamies, 2nd Ed.,
John Wiley (1959).
/21/ Kh. I. Amirkhanov, N. G. Polikhronidi, B. G. Alibekov, R. G. Batyrova,
Isoehorie Specifie ~eat-Capaeity of Carbon Dioxide, Thermal
Engineering, ~, 87 (1971).
/22/ Kh. I. Amirkhanov, B. G. Alibekov, D. I. Vikhrov, V. A. Mirskaya,
L. N. Levina, V. A. Antonov,
Investigation of the Heat Capacity ev of n-Heptane in the Two-Phase
Region••• High-Temperature, ll, 59 (1973).
/23/ H. Renkert, F. Hensel, E. U. Franck,
Elektrische Leitfähigkeit flüssigen und gasförmigen Cäsiums bis
2000·C und 1000 bar, Ber. Bunsenges. physik. ehem. 12, 507 (1971).
/24/ J. P. Stone et al.,
High-Temperature Properties of Sodium, NRL-6241, Naval Research
Laboratory (1965).
/25/ D. Miller, A. B. Cohen, C. E. Dickerman,
Estimation of Vapor and Liquid Density and Heat of Vaporisation of
the Alkali Metals to the Critieal Point, Proc. Int. Conf. on the
Safety of Fast Breeder Reactors, Aix-en-Provence (1967).
/26/ A. Padilla Jr.,
High-Temperature Thermodynamic Properties of Sodium, ANL-8095 (1974).
/27/ A. Padilla Jr.,
Extrapolation of Thermodynamic Properties of Sodium to the Critieal
Point, Proe. 7th Symp. on Thermophysical Properties, Gaithersburg
(1977).
- 47 -
Appendix G
Fitting the gas-equation at the eritieal point.
For the redueed pressure the deseription of the eritieal point (eq. (121»
takes the following shape:
dZ 02Z 03 ZZ :::; Zc = - Zc :::; 2'Zc :::; 6 . ( A - Zc )
ÖW ow2 dW3
Zc3
03
P<icwith A =
.__ .
6 Pe o~ 3
Z and az/ow are defined with the eq. (109) and (113), the two other
derivatives are
and
2·y·A 2-_.[ y'A - w·B·(3-w 'B}] +
53
5y.L., n·(n-1 )·F .w n- 2
n =1 n
=6·y·A-_.[ (y·A - 2·w·B)2
S4+
5y.L., n·{n-1) ·(n - 2 )·F '\oIn-3
n =1 n
To simplify the eritieal-description new variables are introdueed for A. B,
GI and G2:
- 48 -
0' = 1 - B
:x::: 1 +B-A
+ Zc - 1\
5- t·u :: C (n-1 )'G n - 2'Z c + 1
n::2
along wtth the abbreviatlons
and
95 n - 2
= L (n-1 )· __ ·G nn =3 2
;- 3 'Ze - 1
5 n - 2h = L. (n-1)'--
n=4 2
n-3--'G n - 4·Z c ;- 1 + i\
3
These equations turn the eritieal point descrlption into the following
form:
A :: X· t
'J
=
= :x:·(1-u
x. 2 .{gJ t - u)
3CI - 2-0'-x + ::
- 49 -
The first two of these equations give
:x: = 2/N with N = 2 + t-u
so A and B can be caleulated from t and u:
A = 2·t I N
B = (t+ u)/N
The second two equations - from which t and u ean be ealculated - transform
by elimina ting l. and (J' to
t 2 - t·u·(2·u -1) + 2·g = 0 resp.
t 2 .( 2·u -1 ) - t·u·( 2·u 2 - 4·u + 1 ) - 2·( g + h) = 0
or in a simplified form
t 2 u 2·g - 4·( g' u + h )+ t . + = 0
1 + 2·u 1 + 2·u
and
U 2 + u·( t - 2·g I t ) - 2· h / t = 0
To fit the gas equation at the eritieal point this system of coupled
quadratie equations was solved (by iterations) using the best fit
coeffieients G3 - GS in g and h. As A 0.0001 was used.
The solutions t and u give then beside A and B the eoefficients GZ and GI
via the equation for-t·u resp. for t.
Appendix I
- 50 -
The Integral ß I =
'vi
2 ·f1
dw·w--.( 1
52
'vi+ a·-
S
The abbreviations sand aare here
s = 1 - a'w . + B·w 2
resp. a = y'A
To evaluate Öl the following primitive functions are needed:
J 1 (w ) =W dw
J-S
=2
--'areIrr
so'tg (
Irr
1 so'_. ( +
CI S
W dw 1 S'J 3 (w) = JS3 = - .( + 3'8 0 J 2 )
2· s 2rr
with S' = 2· B·w - a
and
- 51 -
With these functions the first term in ~I turns to
Jw dw·w
52::
1 Q'W - 2_.(
a S
and for the second one get
w dw-w'2
J S 3=
2'0 - Q2 )·w + 2·Q
S 2) +
1 S" - w·a+ J 3 ] = -_. ( f·J 2 +
2-B a-S 2 =
1 S" - w· a-_.( f·S" + ) ]
2·B·S S
with
so the <"vWjJ"''''
f ::: + 2 B ) I 0
to 1::.1 i5
I ( w) :: 2-W dw.w--.( 1
S2
w+ Q'-
S= + tp(w) ]
with ~ = 6·Q·ß I {a
- 52 -
1 S'" - a-wand I.P ( w) = _. ( ~. S'" - er + a - ----
8'S S
The difference corresponding to a w-difference
I1w = W - 1
1s in the first function
11 are tg (S' I Irr) = ar e tg (~w·/rJ I u )
Here is
u(w) = 2-x + (2-8 - a)'ßw
with
x = 1 +8-a
The corresponding ~-d1fference 1s
/).wßlP = _.[g
S·x
with
+ 2·(a+rJ)-~·u
a+ _. ( rJ'W - U ) ]
S
g(w) = [a·a + B·( 4 - 2-a + a}'~w] I:x:
so the integral ~I turns out to be
fiI1 4·~ ~w·la
= _.{ -_. are tg (cr la u
+
t~
::EI-
CCJMPRESSEDLIQUID
GAS
TVCJ PHASE REGICJN
OENSE RAR I F I E 0
CJVERHEATED VAPOR
VV(T)
v ::: I/RH
LnLV
========================================================================================~*============================
FIG. 1 SCJDIUM • RANGE CJF VALIDITY CJF THE THERMAL PRCJPERTIES
3.00E-02I
2.00E-02 + I;'>o..Ju..u......0
1.00E-02
LNC P/BAR ) = 11.919
- 12153./T - 0.195-LNC T )
(!)
V1-!O-
2600.
(!) (!) (!)
(!)
(!)
O. 2@0.~(!) (!) (!)
e>
(!) (!) (!) (!) (!)(J~ i>(!) (!)
(!)
(!)(!)
~ ~OO. (!) (!)2(!) ~ (!)
(!) (!)
(!)
(!)
d600. (!) 1~OO .(!)(!)
e>(!) (!)
(!)
(!)(!)
(!)+
(!)
1000.80v.0.0
-2.00E-02
-1.00E-02
-3.00E-02
-4.00E-02
+ + EVING,STONE ET AL.
e> (!) BHISE AND BONILLA~__~ MAKANSKI ET AL.
-5.00E-02(!) (!)
==================================================================================:*===========================
fIG. 2 SCDIUM. DEVIATIONS fROM THE VAPCR PRESSURE EQUATlCN
LnLn
TS==SAT. TEMP.
PS =P SRT.( T)
______ P == PS+( T-TS)wOP/OT
[!J [!J EXP. 4
C) C) EXP. 19
li li EXP. 18
+ + EXP. 3X X EXP. 20~ ~ EXP. 23
+ + EXP. 25~ ~ EXP. 17
Z z EXP. 7
0..
[,~.
,'....~,I,,,,,,,
" fI1!I',p""'--- ..., ~...-,~~',,,,," ",.. ... --
,,' -+++, .......'--'-,~- ......+,,,,,,,,,
" .---,., ~ ~_-~<r---
,,' ~,-~-~~--' -'
.. , ..,,,"!~ --
""..," _v---*--x-------, V. _-.4- --* - '", X v __-~- ~
..'v. ~---- ---,..... _,..:;--7"1;
",","" A-----------..' -L ..1-_-+-----+---'.... +-----~---r-----"
.......:-t+H-+- - - - - -+-.. '.... ~ _-L--------_...... ~ -6------------~---------~-- ~~:----A-------------~----------
,"....~- ..-e-..- .. e ~ Jb AL 6L----~-----Jb------JD----er----~----e--------·... _ _ ~-----~---~ er ~ ~ ~
p"-----",_,---~~~~i9------i9-------~-----er---~----er---i9----------4!r------4!r-----i9----f;---------f;-------
T ( K )
.
2.
4.
8.
6.
14.
12.
10.
16.
18.
26.
22.
20.
O.1200. 1250. 1300. 1350. 1400. 1450. 1500. 1550. 1600. 1650. 1700.
======================================================================================*=========================
FIG. 3 P V T -PROPERTIES OF THE SOOIUM VAPOR ( STONE ~ AL )
1.0I
0.9 +LI
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
lJ1-....J
2600.800. 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400.600.
~---~ SAT.lIQUIO_____ SAT. VAP~R
~ ~ AFTER ST~NE ET Al.
-.001-0Z,V/CZ,V-OTJ
-----------.------~~-""-'" -
~~
"',"-.'",-
-"-",
-""--"'-
-"'-'\,
\,
\\\
\jJ
J-'(eK~~i---~ ;)100
E!J-------~---------.,,---_ ... --400.200.
0.0
================================================================*=========================
FIG. 5 FACT~R ~F REAlITY ( Z=P/CRGAS-RH-TJ ) ~F THE S~DIUM
1.0I
0.9 + li"-t!)
0.8 1\-J
::ca::
0.7
0.6
0.5
[9 eJ SRT. LIQUID(9 E!) SRT. VRPelR______ (RH, L + RH, V )/ 2
0.4
0.3
0.2
------------------------------------------------------------"------------------------------------V100
0.1
0.0200. 400. 600. 800. 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600.
======================================================================================:;*============================
FIG. 6 THE SRTURRTIelN LINE elF THE SelDIUM
=================-==============================================================~*=========================
FIG. 7 S~OIUM . T-OCRHJ!CRH-OTJ ~N THE SAT.LINE
40.
35.
30.
25.
20.
15.
10.
5.
t~a:.....>u
VAP~RISING LIQUID______ C~NDENSATING VAP~R
A A CP AFTER GINNINGS &AL.
\\ ..
\ ..\,
\
\\\\\\\
\\
\\ ,
\
\ ,\ ..\ ..........
"...... ...... ..'\, ..,
""
" ...' ... ... ... ......
............._-,
T C K )
0\oI
o.200. 400. 600. 800. 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600.
=======================================================================================*==========================
FIG. 8 CVCRH,LCTJ-O,TJ & CVeRH,VCTJ+O,TJ ~F THE S~DIUM
800. 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600.
[!J I!l SAT. LIQUI0G 0 SAT.VAP~R
______ VACUUM
0\
T ( I< )
-------_ ...-
600.
-,"', "' ........ , ...... -------- ---'. _// --', ~/....... ..-,... '.....,... ,./'"
........ - ..._---_.......... """'......,.,.#'
400.
6.0I
5.5 +t~5.0 r '-'
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0200.
=====================================================================================*=========================
FIG. 9 S~OIUM • HEAT CAPACI C~NST. V~LUME
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
t~a..u
~ ~ SRT.LIQUIDC9 E!) SRT. VRPClR
(J'\
N
0.0200. 400. 600.
T ( K )
800. 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600.
=================================================================================*============================
FIG. 10 S~DIUM • CP/CV ~N THE SATURATIClN LINE
11- 63 -
·0
,.e--s..,
I "-/ '\,
~l:' \/ \
/ ' '0
, \ NN
.I \/ ' ·0r ~
0 I.I.J0 ZN ........ \ -1
I ' .· I-. \ 0 CI:
I \0 Cf)co..... I.I.J. :c::
I,
\ I-
· z
\0 0
\ 0c.o >-, , ..... CL
\ \ -1CI::c::
\ · I-
\ 0 z0 I.I.J'<t'".
\ ..... w
\ :c::• I-
~ \ ·00 EN ::>
\ ..... .......0.0I
\\Cf)·0
\ 0
\0 0....... a::: .....::> 0 •CI CL \\ ....... CI: .....-1 > .....• ·• . .
\0
\I- I- 0CI: CI: coCf) Cf)
\.. t!)
\ ~ EP .......,IJ...• \ ·•
~0
\I
Eh 0,c.o. \\ ,
• \ ·~
02J 0
'<t'"
( JH-<
·00N
0 Ln 0 Ln 0 Ln 0 Ln 0 Ln 0 Ln 0. . . . . . . . . . . .c.o Ln Ln '<t'" '<t'" n n N N ..... ..... 0 0
0\.j::--
T (
800. 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600.
DP/DRH______ DP/DRHJS
A A FR~M THE VEL~CITY ~F S~UND
600.400.
1:I: A..........A.0:: ......Q ...~ ..." ......a..~ ~ ...!! --~-,--~co ' __ <co ~,co __,... '-
...~ ..... ,
...~ ......"''A......
......A........
...~......
....~......
......ä........ ......
"''-" ,
........."',
"""...... ..,
'\., .." .. ..
" ..,...)",
--_::)0." .........
200.
1.0
1.5
5.5
4.5
5.0
6.5
4.0
2.5
2.0
6.0
3.5
0.5
3.0
0.0
=====================================================================================*=======================
FIG. 12 DP/DRH ~ DP/ORHJS IN THE SATURATED LIQUID S~DIUM
~~~-
~~~~~---~--~-~~
______ PT == PT( T.) (SAT. VAPOR)
(!) (!) AFTER STONE ET AL.PT == PT(T.) +PT.CORR.(T.,TC)
SOOIUM
(j\
V1
400. 600.
TII ( K )
800. 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600.
==========================================================================~*======:======:======:====
FI 13 PTCT.,TJ ON THE VAPOR SI THE ITICAL ISOTHERM
0\0\
T. ( K )
I t---~ I2000. 2200. . 2600.
I I+-:1800.liDO. 1600.
T = TC
1200.1000.
P = PlIC TlI)P =PlICTlI) + (T-TlI)lIPTCTlI)
VITH PT.CORR.CTlI,T)
800.600.
---------------------- ~~~- ------ '.,
~---~--- "--- .,--- . '",------ ''\-.................. :
........ ,: '\~ '\:,. "
"" \\" \,
',\,'~,~\\
\400.
1.10I
t~1.00 + :!ceII(f)(.!)
0.90 + ce\J........C-
I0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.0200.
W==============================================================================================:, /1n.:==.=========================
FIG. li SODIUM .PCT-) ON THE VRPOR SIDE GF THE CRITICAL ISOTHERM
4.0I
3.5 + I~......Cl"-Cl..Cl
3.0
2.5
2.0
1.5
1.0
0.5
T ( K )
0\-...J
0.0200. 400. 600. 800. 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600.
==========================================================~*========================::
FIG. 15 DP/CRGAS-RH-DTJ IN THE SATURATED LIQUID S~DIUM
o 4
0.3
[!] !!l
(!) (!)
lIQUIDlIIO.
SAT. VAPClR
0\\.D
0.2
0.1
0.0200. 400. 600. 800. 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600.
=================================================================================*==========================
FIG. 17 OP/CRGASlIITlIIORHJ ClF SRTURRTED SClOIUM
.(!)....
....I-....
Cl.Cl
,,II,II
Q.N
LO.N
Q.M
N,....,I-,...., a
I- IIIIII :I::I: a::a:: III• (J1(J1 Cl:Cl: t!)t!) a::a:: '-''-' "-"- Cl..Cl.. a
....
LO.M
·031:1 ld "3 d-<
,", ..., ... ...,
'" ...
"" ,'"" ,,
... ,," ,,,,,,..,............,,
'\,
.-rCl
5.0
-...J-
lNC VC J/LNC V)
--_ ..... --
ETA USED IN PT. (T-,T)NEEDEO F~R P-C~RRECTI
",,,,oOC"d"'TItJ,----
5
5
4.5 + I4 0
1.5
1.0
0.00.0 0.020.040.060.080.100.120.140.160.180.200.220.24 0.26 0.28 0.30
======================================================================================*============================
FI 19 S~DIUM VRP~R . C~RRECTING FRCT~R ETA
5.20E-04I
5.00E-04+ 1~ , I
II
II
~I
, I
4.80E-04 + ..... II
.... II
0 ~I
IIII
4.60E-04 + I ~ / • / ~
I. IIIII
4.40E-04 + /!I
/ I / X vYI
II II I
II
4.20E-04 + / : / /f /I
:_/ II
, III
4.00E-04 + I I. / :/ v?Y" !QTeX) = QTEeT).el+X.B+X.X~)
IIIIII
3.80E-04 + I / : / ,.! / .. • , T 900 KI
I [!] I!J = II I '-J
I N
+11 t/;/'/ i(!) (!) 950 K
3.60E-04 I ;I I 1000 K/). /).
3.40E-04 +_ff~: I + + 1050 KI
X x 1100 K
3.20E-04 ~ I X =P/PSeT)
3.00E-04 4f QT-VAlUES FR~M TIMR~T ET Al PC BAR)
~......
2.80E-04 I I I I I I I I I I I
0.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65
===================================================================================~*==========================
FIG. 20 THERMAL C~NDUCTIVITY ~F THE S~DIUM VAP~R
-...IW
3.83.4
LOGC P ) + 4
3.02.62.21.81.41.0
QTeX) :: QTECTJ-Cl+X-S+X-X-C)
~
QT-VALUES FR~M TIMR~T ET AL
l!l l!l T == 900 K
(!) (!) 950 K
~ ~ 1000 K
+ + 1050 K
x: x: 1100 K
X :: P/PSeT)
20E-04I
5.00E-04 +r4.80E-04 r4.60E-04
4.40E-04
4.20E-04
4.00E-04
3.80E-04
60E-04
3.40E-04
3.20E-04
3.00E-04
2.80E-040.2 0.6
========================================================================================::;*========================
FIG. 21 THERMAL CONDUCTIVITY OF THE SODIUM VAPOR
3.50E-02
3.00E-02
2.50E-02
2.00E-02
I!'-'I.L.I.L.-Cl
x
(!)
C!J
"'-J.l:'-
0.60
X
X
0.55
P/PSCT)::-- X
(!)
(!)(!]
X (!) X6- (!)
(!) 6- X(!)
+ X6>5<
+++ X
6 X
I I
0.15 X O.~• '1" . "-
~.25 0.30 0.35 0.40 0.45 +0.50X
X(!)
(!)i- C!J
X
+
XX XX
+
+
(!]
(!] 6 6-
X(!]
:t:
X '*X
+" XC!J 0.05 X 0.10., +
6 6X
66
X
1.00E-02
0.0
5.00E-03
1.50E-02
-1. 50E-02
-1.00E-02
-5.00E-03
-2.00E-02 XX X
-2.50E-02
======================================================================================~*==========================
fIG. 22 DEVIATI~NS ~f THE DATA fR~M THE REDUCED DESCRIPTI~N
"Ln
1100.
TC K )
1080.1060.1040.
x = P/PSCT)
TIMR~T,MAKHR~V ~ SVIRIDENK~
~ ~ QTCX) =QTECTJuCl+XuS+XuXuC)
1020.1000.9S0.960.940.
5.05E-04I
5. +r4. 95E-04
ri.90E-04
4.S5E-04
4.S0E-04
4. 75E-04
4.70E-Oi
4. 65E-04
4.60E-04
4. 55E-04
4.50E-04
i.i5E-Oi900. 920.
=================================================================================~*======================
FIG. 23 THERMAL C~NDUCTIVITY ~F THE SATURATED S~DIUM VAP~R
-1. 2 -0.8 -9A""--~ o 0 '---tJ:-+-_ 0.8 1.2 1.6 2.0". -....". ........
'" ........ .,..'" ........
/1.2
........ -LClGCQT/QTCJ/ ........
/I ........
/........ ...
I ...I ...............
I 1.4I ....I
E!l SAT. LIQUID ....",I [9I "I (9 E!) SAT. VAPClR "I "/ 1.6 "X STEFANClV f.. AL. "I X \
I \I \
I \
1.8 \
X
2.0 '-lC1'
2.2
2.4
2.6
t~
CT IN IJ/CM/K X
X
X
====================================================================================rvJfurz~ --
FIG. 24 SCiDIUM • CT ClN THE SATURATICIN LINE CREDUCED PRClPERTIESJ
-..J-..J
2.01.8
H,V-H,L>"
1.6
QT IN V/CM/K
H IN KJCJULE/G
T/TC = .99, .98, - - - .90
1.4
+
1.2
+
1.00.8
)
0.6
____ L
O.i0.2
O.
1
0.0
o.
========================================================================================:::*======================
G. SCJOIUM • AS A LINEAR fUNCTICJN fCJR T-- TC
......(Xl
4.54.0
LeJGC V )
3.5
QT IN IJ/CM/KV IN CM--3/G
3.02.52.01.5
SAT. lINECRIT.Pl'INT
T = 1000 K
T = 1800 K
T = 2200 K~--~
A
+---+~---+<
1.00.5
II
\\
\\\\\\
\,,~ '~
" ------"" --------------------------~------1;
-0.2
-0."1
-0.6
-0.8
-1.0
-1.2
-1."1
-1.6
-1.8
-2.0
-2.2
-2."1
-2.6
-2.8
-3.0
-3.2
-3."1
rvA1vz========================================================================================:, .<'1 n,;::=,=======================
FIG. 26 STATIC THERMAL Cl'NOUCTIVITY l'F THE TIJl'-PHASE Sl'OIUM
- 79 -
P (Ba r )
LOOOO
T ( K )
4. 0 0 0
1.
1000
3000ITc
20001 1541 0 0 0
1 0 0
v( m3 I g )
FIG. - V~LUME - TEMPERATURE SURFAC~
v , T )
51 0
- 80 -
24 00
P (Bar)
2500
T ( K.)
400
3 0 0
200-2600
==========================================================*==================
FIG. 28 S~DIUM . THE NEAR-CRITICAL PART ~F THE PRESSURE - SURFRCE
PC v , T )
- 81 -
Z:: P·V/(Rgas·T)
1 000T(
1 1 5 4
I . v - T - I
v , T )
- 82 -
( V (R gas )
100
~ 5 0
..- 1\
f\1\~
t\ 20- 1\
1\1\
- 10
5_
Vc10100
1000
V
l cm 3 I g )
1154 2000
Tc
3000
T ( K )
============================================================*==================
FIG. 31 S~OIUM . THE CV - v - T - SURFACE , VAP~R SIOE
- 83 -
(v ( Rgas )
100--50
10-5-----
1,5-40 001
T (J< }v (
c m3 I 9
------
--
FIG. 30S~OIUM . THE CV - V - T - SURFACE , LIQUID SIOE
- 84 -
(v { R ga s }
10
\ Vc
10\\100 0 100
V t cm 3 1 9 )
\
1" t K )
\ \
1154 1c \'LO OO
3000
4000
=:===============================================================$===============::::::::::::=
FIG. 32S~OIUM . THE CV - V - T - SURFACE , SUPERCRITICAL SIOE
- 85 -
-(v (Rgas r
15-..;;;;.
10
5-2
I jTc I I 12,5I Vc7,5 102 5 00
5)K )
24 00 5 m3 I 92,T (
V { c
-
FI 33S~OIUM • THE NEAR-CRITICAL PART ~F THE CV - SURfACE
LIQUID SIDE
- 86 -
(v (Rgas)
20
1~
10
5
2500 \ Tc
2,5 \Vc
1,5 I12,5 2400
T ( K )
==============================================================*================
FIG. 34 S~DIUM • THE NERR-CRITICRL PART ~F THE CV - SURFRCE
VRP~R SIDE
U ( kJ Ig )
- 87 -
4 Vc 1 0001 00T )
1 0
11 5 4 V ( c m3 I 9 )
*FIG. SCjOIUM • THE V - T - SURfACE (jf THE INTERNAL ENERGY
U( V , T )
4-
2---
- 88 -
S ( J/g/K )
4000 Tc
T l K )
1 1 54
Vc1 0
V l
1 0 0
cm3 I 9
1 000
==================================================$==================
FIG. 36 SOOIUM • THE V - T - SURFACE OF THE ENTROPY v, )
- 89 -
0,5
0,0 5
Ur (W/crn/K)
0,0 0 5
Vc 4000
1 0 3 0 0 01 0 0 2000
1 00 0 1 1 54
V r K )3 I 9 )
(ern
==============================================================~=================
FIG. 37 SODIUM . THE V - T - SURFACE OF THE THERMAL CONDUCTOVITY
eH v , T )
P IN PASCALS
6 6 T = 1000 K
I I T = 1500 K
x >< T = 2000 K
~ ~ T = 2508 K
1- 1- T = 3000 K
~ ~ T = 3500 K
z Z T = 4000 K
LOG V
\0o
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
==============================================================================================:*===========================
FIG. 38 SODIUM. THE PRESSURE - VOLUME CHART
10.
1.4
4. I
I.D
1.
2.
o.
3.
5.
7.
8.
9.
6.
0.4
T = TC
P IN PASCALS
0.60.81.01.2
[9 E!l RH = •88C9 E!) = 1. E-7
6. ö. = .81
I I = 6.E-5
)( >( = .61
~ ~ = .05
t t = .23 G/CM--3
1000/T
2.0
~=======================================================================================:, /1n~ ~l======================
FIG. 39 S~DIUM. THE PRESSURE - TEMPERATURE CHART
2.2I
+ i~[!] E!J T = 1000 K
2.0 (9 E!) T = 1250 K
6, 6 T = 1500 Ku'-I T = 1750 K1.8 + (,.!) I I0..J T =2000 K)( )(
1.6 t ~ T =2250 K
• .- T =2500 K
1.4 + ~ ~ T = 2505 K
1.2
1 ~ #/1/ / / I
\0
1.0 N
0.8
0.6
0.4I
V IN CM1IUlI3/GI
0.20.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
========================================================================================*============================-.
FIG. 40 SOOIUM. THE SPECIFIC HERT - VOLUME CHRRT
1.0VJ
I!I E!J RH :::: •88(9 E!):::: 1. E-7
A A::::. 81I I:::: 6.E-5
X >( = .61
~ ~ = .05t t = .23 G/CM--3
2.6r
2.4 +112.2 + u
......c.o0...J
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2O. 500. 1000. 1500. 2000. 2500. 3000.
T ( K )
3500. 4000.
==============================================================================~*==============================
FIG. 41 S~DIUM. THE SPECIFIC HEAT - TEMPERATURE CHART
- 94 -
Appendix P
The sodium property subroutines KANAST and KANAPT.
For the benefit of potential users of the STPS-Karlsruhe the following
property-codes have been developed and tested:
KANAST calculates for given T and ~ the properties p. p~. PT' Cv '
and So •
KANAPT calculates for given T and P the properties ~. Pq • PT' Cp • QT'
px. ~L' ~v.
To accelerate the calculations KANAST uses not all the descriptions given
in Appendix C directlYj the saturated properties Cv ' PT' Qr and the
saturated densities are stored in the code point for point. corresponding
to the temperatures 370 - 402 - 434 - ••• - 2386 - 2418 K. For a tempera
ture between these values the properties are calculated by cubic interpola
tion. The use of the time consuming high-temperature property-formulas 15
restricted to the remaining part of the 8L.
Also for the reason of time-saving the integral ~Cv (eq. (94)) - needed for
the heat capacity in the overheated vapor - is pre-calculated and stored in
the code as a function of both variables T and TX!T (a typical running time
for this code 1s a half msec on the IBM 3033).
KANAPT calculates the sodium properties by iteration: the first dens1ty.
corresponding to a given T and P is estimated, then KANAPT calls KANAST to
check this value. If the corresponding KANA8T-pressure does not agree with
P, the dens1ty 1s eorrected using ~p and P~. On average one KANAPT run
gives rise to three KANAST ealls.
All the property-surfaces and charts of the Figs. 26 - 41 have been
calculated with KANA8T.
