KfK 2863 Februar 1981
Thermophysical Properties 01 Sodium in the Liquid and Gaseous States
K. Thurnay Institut für Neutronenphysik und Reaktortechnik Projekt 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 vervielfältigt Für diesen Bericht behalten wir uns alle Rechte vor
Kernforschungszentrum Karlsruhe GmbH ISSN 0303-4003 Abstract
A system of temperature and density dependent thermal-property-functions has been developed and checked for the sodiumt using all of the accessible experimental data and the mutual relationships of the properties. In extending these properties beyond the range of the measurements substance independent physical relations have been used.
The property-descriptions are valid for all temperatures above the melting point of the sodium and for all densities below the melting density of the liquid.
The system consists of the following thermal properties: pressure t heat capacity at constant volumet thermal conductivity.
Die thermophysikalischen Eigenschaften des flüssigen und gasförmigen Natriums
Zusammenfassung
Die thermophysikalischen Eigenschaften des Natriums werden mit Hilfe eines temperatur- und dichteabhängigen Funktionssystems beschrieben. Dieses Funktionssystem wurde aus den zur Verfügung stehenden Natriumdaten entwickelt durch Extrapolation der Eigenschaften mit stoffunabhängigen thermischen Relationen.
Die Zustandsdarstellungen sind gültig für alle Temperaturen oberhalb des Schmelzpunktes und für alle Dichtewerte unterhalb der Flüssigkeitsdichte am Schmelzpunkt.
Die dargestellten Eigenschaften sind der Druckt die spezifische Wärme bei konstantem Volumen und die Wärmeleitfähigkeit. Contents
page
List of Figures 1
Glossary 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 44
Appendix G. Fitting the gas equation at the critical point 47
Appendix I. The integral ~I(w,y) 50
Figures 53
Appendix P. The sodium property subroutines KANAST and KANAPT 94 - 1 -
List of Figures
Fig. 1 Range of Validity of the Thermal Properties
Fig. 2 Deviations from the Vapor Pressure Equation
Fig. 3 PVT- Properties of the Sodium Vapor
Fig. 4 Reduced PVT- Properties
Fig. 5 Factor of Reality
Fig. 6 The Saturation Line of the Sodium
Fig. 7 The Derivatives of the Density on the Saturation Line
Fig. 8 Marginal Heat Capacities in the Two-Phase Area
Fig. 9 Cv on the Saturation Line
Fig. 10 Cp/Cv on the Saturation Line Fig. 11 The Enthalpy on the Saturation Line
Fig. 12 Density-Derivatives of the Pressure (Saturated Liquid)
Fig. 13 Temperature-Derivatives of the Pressure (Vapor)
Fig. 14 P(T*) on the Vapor Side of the Critical Isotherm
Fig. 15 Temperature-Derivative of the Pressure (Liquid)
Fig. 16 Density-Derivatives of the Pressure (Saturated Vapor)
Fig. 17 The Reduced Density-Derivatives
Fig. 18 P and ap/aT on the Critical Isotherm
Fig. 19 The Correcting Factor in the Rarified Vapor
Fig. 20 Thermal Conductivity of the Sodium Vapor
Fig. 21 The Conductivities at Low Pressures
Fig. 22 Deviations of the Conductivity-Data from the Reduced Description
Fig. 23 Thermal Conductivity of the Saturated Sodium Vapor
Fig. 24 Thermal Conductivity on the Saturation Line
Fig. 25 Conductivity versus Enthalpy Differences
Fig. 26 Static Thermal Conductivity in the Two-Phase Area - 2 -
Fig. 27 The Pressure-Volume-Temperature Surface
Fig. 28 The Near-Critical Part of the P(V,T)
Fig. 29 The Z(V,T)-Surface
Fig. 30 The Cv(V,T)-Surface (Liquid Side)
Fig. 31 The Vapor Side of the Cv-Surface
Fig. 32 The Gas Side of the Gy-Surface
Fig. 33 The Near-Critical Part of the Cv-Surface Side
Fig. 34 The Vapor Side of the Near-Critical Cv-Surface
Hg. 35 The V-T-Surface of the Internal Energy
Fig. 36 The V-T-Surface of the Entropy
Fig. 37 The V-T-Surface of the Thermal Conductivity
Fig. 38 The Pressure-Volume Chart
Fig. 39 The Pressure-1/T Chart
Fig. 40 The Gy -Volume Chart
Fig. 41 The Gy-Temperature Chart - 3 -
Glossary
T temperature
P density v = I/p specific volume s entropy u internal energy
H enthalpy x PP, pressure, vapor pressure
PT 3P/3T temperature - resp. density P 3P /3p P derivatives of the pressure Ps = 3P/3p!S heat capacities at const. volume, resp. pressure
T r = - • ~I P ClT P
thermal conductivity
P/RopoT factor of reality
PT = Pr!Rep pressure derivatives PP/RoT p p in reduced form Ps PS/RoT
PL (T), Pv (T) densities of the saturation line F(L,T) = F(PL[T],T) property in the state of the saturated liquid
T dPL r = - 0 LP dT L
sonlc velocity in the liquid
marginal heat capacities of the two-phase area CV(V,T) = CV(PV+o,T) x x x x T = TL(P), T = TV(p) the inverted saturation line - 4 -
T 2508 K the critical temperature c 3 Pc = 0,23 g/cm the critical density P 256,46 bar the critical pressure c = a, ß, y the critical exponents x - T/T distance to the critical temperature = c w p/pc reduced density y T /T reduced temperature = c R 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
(1)
for T Tc
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
d ~K ::: + K L , V . (4) dT dT
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 of sI. of Bhise and
K @ 1255 - 2500 K@ From the 2 these data a
12153, :;: 11919- o 1 5·ln T
P in bars» r in K.
the deviations of the measured pressures from this eq.
In a second ion I 1s for the
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 dZ V 3 n = C an . T (6 ) Zv dT n= 0
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
x X P(T) = + (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:
ß ~ L( T) = ~ C.( 1 "I- b· x (9) ~v ( T) = 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 3 ~c = 0,23 g/cm (14) was needed in the eq. (11). 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 --. (2A) = ~2 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 ll quotation I found - declare the eq. (19) to be lIa well-known relation 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 dS dS dS d ~l Cy = T· - = T' T' (5) dT dT dT - J3 - dS Setting ((LIT) = T'-(~L(T),T) (16) dT and using the thermastatic relation s ~ = - PT I ~ 2 (see e.g. /20/) ane has X T d"L dp + --'-- (17) ~L2 d T d T T dqL resp. -. . PT ( S l ) (18) ~L2 dT These equations give a Cv-rise of T d"L eV(Ll - [V(L) = - -.[ ] (19) ~L2' dT dT at the crassing fram the liquid into the saturated area. Using the eq. (4) and wtth the reduced derivative T d~L rl = (20) ~L dT eq. (19) can be transformed ta ... 2 P~( [V{L) - (V ( L) = r L . L) I T (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 - 15 - The factor 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 ...... Cy(V) = CV( L ) + ~ V· T. (28) Rere 1s 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 1 1 [V(~,T ) = [V(L,T)- + ).T. (30) ~L ~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 = . -.[ ] (19A) Q, y Z d T dT 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 dH( L ) dS 1 dP :: T· - + _. - :: dT dT dT ,.. :: (V{L) + (31 ) (see /20/ and eq. (16), (17». For the internal energy, U, one has a corresponding derivative x d U ( L ) ,.. rL P :: (V( L) + _.( (32) dT 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·-- :: !1 HIT (33) dT - 17 - resp. LlU = T-LlV,( (34) dT 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· ~ (35) d p X (36) t :: I R·q dT -P~ :: P~ J R· T (37) ,.. oP Ps :: -I I R·T (38) o~ S 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 2 :: (S (39) O~ Is and (p - _. (40) Ps = -P~ (V - 19 - Wlth the equatlons (21) - 2 - (v = (V - rL . p~ (21A) - 2 2- and (24) (p = (V + (r - rL ). P~ (24A) One can ellmlnate Cp and Cv from the eq. (40): Cp - Cv r2 . p~ = ... 2 - (41) Cv Cv -rL .p~ With the relation /20/ PT I P~ = - r (42) and with the reduced form of the eq. (4) ... (4A) t .:: PT ... r L. P ~ eq. (41) can be transformed to Cp - Cv :: (43) Cy !his equation and eq. (40) gives ... Ps :: -p~ ... (44) The inverted form of this relation ) 2 ...... P~ :: Ps - - (45) Cy - 2· t . r L + - 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) = 1 ,86 At near-critical temperatures, i.e. at 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: P~(K,T) = K = L,V (46) for x - 0 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) = (47) and (48 ) with 4 n t.p(K,x 1 = 1 + L: fn{Kl·x K = L, V. (49) n=1 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 ( ! -lines I 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) the the measurements from the eq. (51) as a of the pressure • Since Timrot et ale estimate their maximum error as about 5 % , the T-independent (50) is weIl within the of the uncertainties. From the equations (51) - (53) has the T(V T) ::: o.r(T)-1,5604'" ) (see on ) . There a lack of ideas about the of the near-critical thermal the obvious notion T( C) "'""=3 T(V ,T ) for T ----e- TC the near-critical conductivities has been 7/. He that for many substances the the and vapor conductivities vanishes as the . ~H if o < x ) 0.05 for substances N and 0 but 0.14 for for the near-critical thermal - 24 - LlH = F.( ~ L ~v ) (see eq. (33». 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· X ß + x· (aV + X,9V + X 3. V) J. The conditions 1. the linearity-limit, XG had to be as large as possible (see • 25) and 2. the connections at 1280 K had to be smooth and monotonous determined the parameters ä L, "ä"t ••• as weIl as the critical conductivity: 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 the following, very simple liquid-vapor-mixture concept was devised (see sketch): - 26 - J + + + + + + .. , -' " """'"-,,"',-/-/'"' ,..1/-"..../-/""'''/-/"''1 " ,," / " D// " / "" / ./" / " v/ "" /' / /' / ".I' " ,," / , "".I' .. /"" "" /' " " ,," A- + + A'- , /''' / v" / /" / ' / " / / 1/ .. / // ,,/,- / B'- " B- + + '" / / "" / /' /' "" " + + ... 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 (63) the length-rat10 of the liquid. The mixture-quality is therefore 3 s = ~V·(1-rt )/~ giv1ng with eq. (62) the follow1ng relation between length ratio and mixture density: 3 ll.( 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 - (65) Denoting the temperatures on the surfaces A. A' •••• aB TA. TA' •••• the currents in the first and in the third layer are: TA - TA' JA'A = 2· ·F·o.T(V} (66 ) D·(1-rt} TB - TB' resp. J S'B = 2· . ·F·o.T(V) (67) D·(1-rt} In the second layer there are two different heat currents, one in the liquid part of the layer and one in the vapor part: J V = rendering the total layer-current to J B'A' = J L + J V = TA' - TB" 2 = ·F·(Q.r(V) + ll. . 11 o.r ) (68) D·rt Here 19 (69 ) - 28 - The equivalent heat current for the total layer AB i8 J = (70) The identity TA-TB = 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 T + f dt·PT(q,t} :: TX x x :: + (T -T ). Pr (L I T) (74) Ag a density derivative eq. (14) gives: X d p x (T x ) P~(~,T } :: [ Pr (L I T) + dT dP X T x d qL (15) + (T- T ) . (L,T ) ]1 { Tx ). 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: PT(~,T) ;: ::: (77) X X with ::: TV(q) resp. ::: The assumption (77) would give here - as in the compressed liquid - x X pX{T ) _T P('3,T) ::: + (T ). PT ( V I T x ) (79 ) x dp x with ~. ( T X ) ;- PQ, ( / T) ::: [ PT{V I T dT dPT X d qv ;- (T-T x ) . (V/T ) ]! ( T x ) (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: ,... ,... x PT ( ~ ,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~ (84) (s(,\) has a similar shape to TX(~), but it 18 easier to calculate). The selection of the functions jJ(u) = U/( e U - 1 ) (85 ) and H(U) = j.J (U ). [U + jJ ( lJ) ] (86 ) was dictated by the fact, that the necessary PT- and P-corrections on the critical isotherm are of the same size, i.e. x x x llPT(T ,Tc) - 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 x m-1 o llPT / 0 T = 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 dU U = (88 ) oT djJ }J and = _.( 1 - U- jJ ) (89) d u U the term G·H gives the following additional pressure: T (~ II P I T) = J dt· 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 0,2 ---.( 1 (91) = X -- T_T u the following P~ -correction: o ~p dG _TX = R· [ (T ).(G + ~'-)'jJ + a~ d<3 X T H- jJ ).G] (92) x .(H- 0, 2 . 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· d~· .- (93 ) J ~ oT ~V dH H With = ---[2'(jJ-1) + U] du u this can be converted to ~(V{~/T) = dt T = --·G·H·[2·(jJ-1) + U ]·rv(t) . (94) t T- t - 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 = 1850 K (on Figs. 13 and 14) or to ::: 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: P{~ ,Tc o} = P(~ ,Tc + 0 } (97) I PT ( ~ ,Tc - 0 ) (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} 3 I 2 . R CV(O,T) 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 8 J 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. ---c:e:-_ 0 (103) z (W ,y ) ---- 1 f. w·y Here 1s y = Tel T (104) To comply with this behaviour the self-volume acquired a T-dependence: A Y -- . ------:~- (105 ) ~ c 1 + W2. 8 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: ~C q ---Ibt:- q(W,Y) = Q.(W,y )·R·Tc I (106) 5 with Q(w , y) = L W n-1 . Fn ( y ) (107) n=1 and (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) ] , (l09) - 39 - with S(W,y) :: 1 - w·y·A + IN '2. B (110) The respeetive pressure derivatives are here Oll ) -P~ :: Z + W·ZW and PT- = Z - Y.ZY (112) with :: (113) and 2 = w·y·A·( 11- w ·B 1 J S2 + 5 (114) n - y L w . ( Fn + y.H n n =1 As a base-line for the Cv-caleulation in the gas-area the eritieal isochor was chosen (115) (see eq. (26». The eonstant C(+) was gained from Cv(L.T): lim xcx.CV(L,T) 016 ) .. x-=-+ 0 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 - 2 oCv R 2 0 Z = -'y = 0') ~ dy2 R 2 1 + w2. B 5 wn. H ] (117) = -2·_·y·[ (w·A)2. L: n q S 3 n =1 Hence the heat capacity beyond the critical isochor can be given as CV(~,T) = (118) with ') ßCV(q,Tl :: f dCV :: Cl, C 2 5 = R·y .[ 2·r:: ----·H n (119) n =1 n The integral w dw·w w ßI(w,y) :: 2.J -~.{ 1 + A·y· (120) S2 S 1 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 - 0 2 P 0 3 P P = 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. (122) f. < (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 Mixtures, 2nd Edition Plenum Press, New York. 1969. - 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 02 Z 03 Z Z :::; Zc = - Zc :::; 2'Zc :::; 6 . ( A - Zc ) ÖW ow 2 dW 3 3 3 Zc . Z and az/ow are defined with the eq. (109) and (113), the two other derivatives are 2·y·A 2 -_.[ y'A - w·B·(3-w 'B}] + 53 5 y.L., n·(n-1 )·F .w n- 2 n =1 n and 6·y·A = -_.[ (y·A - 2·w·B)2 + S4 5 y.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 5 n - 2 = (n-1 )· __ ·G 9 L n ;- 3 'Ze - 1 n =3 2 and 5 n - 2 n-3 h = L. (n-1)'-- --'G n - 4·Z c ;- 1 + i\ n=4 2 3 These equations turn the eritieal point descrlption into the following form: A :: X· t 'J = :x:·(1-u = x. 2 .{gJ t - u) 3 CI - 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 eliminating 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 u 2·g - 4·( g' u + h ) 2 t + 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. - 50 - Appendix I 'vi dw·w 'vi The Integral ß I = 2 --.( 1 + a·- ·f 52 1 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: W dw 2 so' J 1 (w ) = = --'are tg ( J-S Irr Irr 1 so' _. ( + CI S W dw 1 S' J 3 (w) = = - .( + 3'8 0 J ) JS3 rr 2· s 2 2 with S' = 2· B·w - a and - 51 - With these functions the first term in ~I turns to w dw·w 1 Q'W - 2 :: _.( J 52 a S and for the second one get w dw-w'2 2'0 - Q2 )·w + 2·Q = ) + J S 3 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 f ::: + 2 B ) I 0 so the <"vWjJ"'''' to 1::.1 i5 W dw.w w I( w) :: 2- --.( 1 + Q'- = + tp(w) ] S2 S with ~ = 6·Q·ß I {a - 52 - 1 S'" - a-w and 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 /).w a ßlP = _.[g + 2·(a+rJ)-~·u + _. ( rJ'W - U)] S·x S with g(w) = [a·a + B·( 4 - 2-a + a}'~w] I:x: so the integral ~I turns out to be 1 4·~ ~w·la fiI = _.{ -_. are tg ( + cr la u t~ GAS OENSE RAR I F I E 0 CCJMPRESSED LIQUID Ln CJVERHEATED VAPOR LV TVCJ PHASE REGICJN VV(T) ::E I- v ::: I/RH ======~*====== FIG. 1 SCJDIUM • RANGE CJF VALIDITY CJF THE THERMAL PRCJPERTIES 3.00E-02 (!) I (!) LNC P/BAR ) = 11.919 2.00E-02 + I; - 12153./T - 0.195-LNC T) '>o..J u...... u.. 0 (!) (!) (!) 1.00E-02 e> (!) (J~ (!) (!) (!) (!) (!) (!)(!) (!)i> 0.0 80v. 1000. d600. (!) 1~OO .(!) ~ ~OO. (!) (!)2 O. 2@0.~ 2600. (!) (!) ~ (!) (!) (!) (!) e> (!) (!) (!) (!) -1.00E-02 (!) (!) (!) (!) (!) (!) (!) V1 -2.00E-02 -!O- (!) + (!) -3.00E-02 + + EVING,STONE ET AL. e> (!) BHISE AND BONILLA -4.00E-02 ~ __~ MAKANSKI ET AL. (!) (!) -5.00E-02 ======:*====== fIG. 2 SCDIUM. DEVIATIONS fROM THE VAPCR PRESSURE EQUATlCN 26. ,~. ,,'.... ~ ,I . ______P == PS+( T-TS)wOP/OT TS==SAT. TEMP. ,, ,, [ ,, [!J [!J EXP. 4 22. " fI1!I',p""'--- ... 0.. , ~... - C) C) EXP. 19 ,~~' , 20. ,, li li EXP. 18 ,, " ",.. ... -- 18. + + EXP. 3 ,,' -+++, ...... '--'- ,~- ...... + X X , EXP. 20 ,, ,, 16. ~ ~ EXP. 23 ,, ,, " .--- + + EXP. 25 ,., ~ ~_-~ ======*====== FIG. 3 P V T -PROPERTIES OF THE SOOIUM VAPOR ( STONE ~ AL ) 1.0 I ------.-- 0.9 ----~~-""-'" +L - I 0.8 ~~ "',"-. 0.7 '",- 0.6 -"-", -""- 0.5 ~---~ SAT.lIQUIO -"'- _____ SAT. VAP~R -"'- 0.4 ~ ~ AFTER ST~NE ET Al. '\, lJ1 -.001-0Z,V/CZ,V-OTJ \, -....J 0.3 \\ 0.2 \ \ 0.1 jJ J-'(eK~~i ---~ E!J------~- ;)100 0.0 ------.,,---_ ... -- 200. 400. 600. 800. 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600. ======*====== FIG. 5 FACT~R ~F REAlITY ( Z=P/CRGAS-RH-TJ ) ~F THE S~DIUM 1.0 I 0.9 + [9 eJ SRT. LIQUID (9 E!) SRT. VRPelR lit!)"- ______(RH, L + RH, V )/ 2 0.8 \-J::c 1 a:: 0.7 0.6 0.5 0.4 V1 ------"------00 0.3 0.2 0.1 0.0 200. 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. t~ \ \ \ a: \ ..... \ \ \ > \ u \ \ 30. \ \ , \ \ , \ .. 25. \ ...... 20. " ...... '\, VAP~RISING LIQUID .., 15. " ______C~NDENSATING VAP~R " 0\ " ... o ' ... I A A CP AFTER GINNINGS &AL...... _-, 10. 5. T C K) 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 6.0 I 5.5 +t~ [!J I!l SAT. LIQU I0 5.0 '-' G 0 SAT.VAP~R r ______VACUUM 4.5 4.0 3.5 3.0 2.5 ------_ ...- -,"', ...... 0\ "' ...... , ------'. _// 2.0 , ~/ --' ...... -,...' .....,... ,./'" 1.5 ...... - ... _---_...... """'...... ,.,.#' 1.0 0.5 T( I< ) 0.0 200. 400. 600. 800. 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600. ======*====== FIG. 9 S~OIUM • HEAT CAPACI C~NST. V~LUME 5.0 4.5 t~ ~ ~ SRT.LIQUID a.. u 4.0 C9 E!) SRT. VRPClR 3.5 3.0 2.5 2.0 (J'\ N 1.5 1.0 0.5 T( K ) 0.0 200. 400. 600. 800. 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600. ======*====== FIG. 10 S~DIUM • CP/CV ~N THE SATURATIClN LINE - 63 - 11 0 · ,.e--s.., I "- / '\, ' \ / \ ~l: '0 / ' N , \ N .I \ / ' 0 · 0 I.I.J 0 Z ~ N ...... r -1 . \ . I ' I- \ 0 · CI: . 0 Cf) .....co .I \, :c::I.I.J I- I z \ 0 · 0 0 \ c.o >- , \, ..... CL -1 \ :c::CI: \ I- 0 · z 0 I.I.J \ \ ' ======*====== FIG. 12 DP/DRH ~ DP/ORHJS IN THE SATURATED LIQUID S~DIUM ~~~~~---~--~-~~ ~~- ~ ______PT == PT( T.) (SAT. VAPOR) (j\ (!) (!) AFTER STONE ET AL. V1 PT == PT(T.) +PT.CORR.(T.,TC) SOOIUM TII ( K ) 400. 600. 800. 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600. ======~*======:======:======:==== FI 13 PTCT.,TJ ON THE VAPOR SI THE ITICAL ISOTHERM 1.10 I t~ 1.00 + :! ------~~~ ce ~---~---- - II -- " (f) (.!) 0.90 ce --- '.,., + \J --- ...... '", C- --- I ------...... ''\ 0.80 - ...... : ...... ,: '\ 0.70 ~:, '\ . " 0.60 "" \\ 0.50 " \, P = PlIC TlI) ',\, 0\ 0.40 P = PlICTlI) + (T-TlI)lIPTCTlI) '~, 0\ VITH PT.CORR.CTlI,T) ~ 0.30 \\ T = TC 0.20 \ 0.10 T. (K) 0.0 I I+-: I t---~ I 200. 400. 600. 800. 1000. 1200. liDO. 1600. 1800. 2000. 2200. . 2600. W ======:, /1 n.:==.====== FIG. li SODIUM .PCT-) ON THE VRPOR SIDE GF THE CRITICAL ISOTHERM 4.0 I 3.5 + I~ ...... Cl "- Cl.. Cl 3.0 2.5 2.0 0\ 1.5 -...J 1.0 0.5 T( K ) 0.0 200. 400. 600. 800. 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600. ======~*======:: FIG. 15 DP/CRGAS-RH-DTJ IN THE SATURATED LIQUID S~DIUM 0\ o 4 [!] !!l lIQUIDlIIO. \.D (!) (!) SAT. VAPClR 0.3 0.2 0.1 0.0 200. 400. 600. 800. 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600. ======*====== FIG. 17 OP/CRGASlIITlIIORHJ ClF SRTURRTED SClOIUM , , " ... , ...... , '"... "" , '" " ,, ... ,, " ,, ,, ,, .., ...... I-...... ,, '\, ,...., N ,...., I-a I- III III :I: :I: a:: a:: III (J1 (J1• Cl: Cl: t!) t!) a:: a:: '-' '-' "- "- Cl.. Cl.. a .... , . , (!) I .... ,I I I ·031:1 ld "3 d -< Cl. Cl Cl. LO. Q . LO. Q. -r M M N N 5.0 4.5 + I 4 0 ETA USED IN PT. (T-,T) NEEDEO F~R P-C~RRECTI ",,,,oOC"d"'TI 5 tJ,---- -...J- ..... -- 1.5 --_ 1.0 5 lNC VC J/LNC V) 0.0 0.0 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-04 I I , I 5.00E-04 I I I I 1~ I + , I I I 4.80E-04 ..... I I + .... ~ ~ I 0 I I I I I I • 4.60E-04 ~ / I / ~ I + . I I I I I I I 4.40E-04 /! / I / X vY I I + I I I I ,I I 4.20E-04 + / : / /f / :_/ I I QTEeT).el+X.B+X.X~) QTeX) = I I I 4.00E-04 + I I. / :/ v?Y" ! I I I I , I • T 900 K '-J 3.80E-04 ,.! / .. I I [!] I!J = I + I / : / I N (!) (!) 950 K i I ; I 3.60E-04 +11t/;/'/ I /). /). 1000 K 1050 K +_ff~: I + + 3.40E-04 I 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)...... ~ I I I I I I I 2.80E-04 I I I I 0.50 0.55 0.60 0.65 0.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 ======~*====== FIG. 20 THERMAL C~NDUCTIVITY ~F THE S~DIUM VAP~R 20E-04 I 5.00E-04 +r QTeX) :: QTECTJ-Cl+X-S+X-X-C) 4.80E-04 l!l l!l T == 900 K 4.60E-04 r (!) (!) 950 K ~ ~ 1000 K 4.40E-04 + + 1050 K x: x: 1100 K 4.20E-04 4.00E-04 X :: P/PSeT) 3.80E-04 QT-VALUES FR~M TIMR~T ET AL -...I W 60E-04 3.40E-04 3.20E-04 3.00E-04 ~ LOGC P ) + 4 2.80E-04 0.2 0.6 1.0 1.4 1.8 2.2 2.6 3.0 3.4 3.8 ======::;*====== FIG. 21 THERMAL CONDUCTIVITY OF THE SODIUM VAPOR 3.50E-02 3.00E-02 (!) '-' 2.50E-02 I!I.L. I.L. -Cl C!J 2.00E-02 x 1.50E-02 (!] X (!] 6 6- X (!) 1.00E-02 6 6- (!) (!) 6- X 6 X(!] (!) + X X X + 6>5< ++ 5.00E-03 :t: X 6 P/PSCT) + ::-- X + I • '1" . "- I 0.0 +0.50 0.55 0.60 "'-J +" X X O.~ ~.25 0.30 0.35 0.40 0.45 .l:'- C!J 0.05 X 0.10 0.15 X ., + (!) -5.00E-03 X (!)i- C!J X 6 6X -1.00E-02 + X X X (!) '* (!] -1. 50E-02 X (!) X X XX X -2.00E-02 XX -2.50E-02 ======~*====== fIG. 22 DEVIATI~NS ~f THE DATA fR~M THE REDUCED DESCRIPTI~N 5.05E-04 I 5. +r 4. 95E-04 i.90E-04 r 4.S5E-04 4.S0E-04 4. 75E-04 4.70E-Oi TIMR~T,MAKHR~V ~ SVIRIDENK~ "Ln 4. 65E-04 ~ ~ QTCX) = QTECTJuCl+XuS+XuXuC) 4.60E-04 x = P/PSCT) 4. 55E-04 4.50E-04 TC K ) i.i5E-Oi 900. 920. 940. 960. 9S0. 1000. 1020. 1040. 1060. 1080. 1100. ======~*====== 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 ". .... ". - ...... '" ...... ,.. / ...... -LClGCQT/QTCJ / '" 1.2 ...... / ...... I ...... / ... I ... I ...... I I 1.4 .... I .... I [9 E!l SAT. LIQUID ", I I (9 E!) SAT. VAPClR " I " / 1.6 f.. "" I X X STEFANClV AL. \ I " \ I \ \ I 1.8 \ X 2.0 '-l C1' CT IN IJ/CM/K X 2.2 X 2.4 t~ X 2.6 ======rvJfurz~ -- FIG. 24 SCiDIUM • CT ClN THE SATURATICIN LINE CREDUCED PRClPERTIESJ 1 o. ) ____ L QT IN V/CM/K -..J H IN KJCJULE/G -..J + + T/TC = .99, .98, --- .90 O. H,V-H,L >" 0.0 0.2 O.i 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 ======:::*====== G. SCJOIUM • AS A LINEAR fUNCTICJN fCJR T-- TC 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 -0.2 LeJGC V) -0."1 -0.6 SAT. lINE -0.8 A CRIT.Pl'INT -1.0 +---+ T = 1000 K -1.2 ~---+< T = 1800 K I I ~--~ T = 2200 K -1."1 -1.6 \ -1.8 \ -2.0 QT IN IJ/CM/K ...... \ (Xl \ V IN CM--3/G -2.2 \ \ \ -2."1 \ \, -2.6 , ~ '~ -2.8 -3.0 1; ------~------3.2 " "" ------3."1 ======:,rvA1vz.<'1 n,;::=,====== FIG. 26 STATIC THERMAL Cl'NOUCTIVITY l'F THE TIJl'-PHASE Sl'OIUM - 79 - LOOOO 1000 P (Ba r ) 1. Tc 4. 0 0 0 I 3000 1 0 0 2000 1 0 0 0 1 154 v T(K ) ( m3 I g ) FIG. - V~LUME - TEMPERATURE SURFAC~ v , T) - 80 - 400 P (Bar) 3 0 0 200- 2600 5 2500 1 0 24 00 T( K.) ======*====== FIG. 28 S~DIUM . THE NEAR-CRITICAL PART ~F THE PRESSURE - SURFRCE PC v , T) - 81 - Z:: P·V/(Rgas·T) T( 1 000 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_ Vc 10 100 Tc 1000 3000 V 1154 2000 l cm 3 I g ) 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 v ( 3 T (J< } c m I 9 FIG. 30 S~OIUM . THE CV -V - T- SURFACE , LIQUID SIOE - 84 - (v { R ga s } 10 \ \ 1154 1c \ \ Vc 'LO OO \ 3000 \ 10 \ 100 0 100 4000 V t cm 3 1 9 ) 1" t K) =:======$======::::::::::::= S~OIUM . THE CV -V - T- SURFACE , SUPERCRITICAL SIOE FIG. 32 - 85 - (v (Rgas r - -..;;;;.15 - 10 -5 2 I Tc j Vc 12,5 2 5 00 I I I 7,5 10 5 K ) 00 ) T ( 24 2,5 3 { cm I 9 V FI 33 S~OIUM • THE NEAR-CRITICAL PART ~F THE CV - SURfACE LIQUID SIDE - 86 - (v (Rgas) 20 1~ 10 5 2,5 Vc \ 1,5 I 2500 \ Tc 12,5 2400 T(K) ======*====== S~DIUM ~F FIG. 34 • THE NERR-CRITICRL PART THE CV - SURFRCE VRP~R SIDE - 87 - U( kJ Ig ) -4 ---2 4 Vc 1 000 1 0 1 00 T ) ( c m3 I 9 ) 11 5 4 V * FIG. SCjOIUM • THE V - T- SURfACE (jf THE INTERNAL ENERGY U( V , T) - 88 - S( J/g/K ) 4000 Tc Vc 1 000 1 0 1 0 0 T l K ) 3 1 1 54 V l cm I 9 ======$====== 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 3 0 0 0 1 0 1 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) 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 \0 o P IN PASCALS LOG V 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 ======:*====== FIG. 38 SODIUM. THE PRESSURE - VOLUME CHART 10. 1000/T 9. 8. 7. [9 E!l RH = •88 C9 E!) = 1. E-7 6. 6. ö. = .81 II = 6.E-5 5. )( >( = .61 ~ ~ = .05 4. I t t = .23 G/CM--3 I.D T = TC 3. 2. P IN PASCALS 1. 2.0 1.4 1.0 0.8 1.2 0.6 0.4 o. ~ ======:, /1 n~ ~l====== FIG. 39 S~DIUM. THE PRESSURE - TEMPERATURE CHART 2.2 I [!] E!J T = 1000 K 2.0 i~ (9 E!) T = 1250 K + T 1500 K u 6, 6 = '-I (,.!) T 1750 K 1.8 0 II = + ..J )( )( T =2000 K 1.6 t ~ T =2250 K • .- T =2500 K 1.4 + ~ ~ T = 2505 K 1.2 I ~ \0 1.0 1 #/1/ // N 0.8 0.6 0.4 I V IN CM1IUlI3/G I 0.2 0.0 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 2.6 r I!I E!J RH :::: •88 2.4 +11 (9 E!):::: 1. E-7 A A::::. u 81 2.2 ...... + c.o I I:::: 6.E-5 0 ...J 2.0 X >( = .61 ~ ~ = .05 1.8 t t = .23 G/CM--3 1.6 1.4 1.0 1.2 VJ 1.0 0.8 0.6 0.4 T( K ) 0.2 O. 500. 1000. 1500. 2000. 2500. 3000. 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.