Approximate Function for Heavy Water Properties Computation-Durmayaz

Nuclear Engineering and Design 178 (1997) 309–329

Approximate functions for the fast computation of thethermodynamic properties of heavy water

Ahmet Durmayaz *Institute for Nuclear Energy, Istanbul Technical Uni6ersity, Maslak, Istanbul TR-80626, Turkey

Abstract

A set of some approximate functions derived for the fast computation of the thermodynamic properties of heavywater at saturation, in subcooled liquid and superheated vapor states is presented. To derive these functions, the datagiven in the steam tables by Hill et al. AECL 7531 (1981) were accurately and successfully fitted with curves by usingthe least-squares method. Specific volume (or density), specific enthalpy, specific entropy, constant-pressure specificheat and temperature at saturation were approximated by a number of piecewise continuous functions of pressurewhereas pressure at saturation was approximated by a piecewise continuous function of temperature for heavy water.Density in subcooled liquid state, specific volume in superheated vapor state, specific enthalpy, specific entropy andconstant-pressure specific heat in both of these states were also approximated as piecewise continuous functions ofpressure and temperature for heavy water. The correlations presented in this study can be used in the two-phasethermalhydraulic system analysis of CANDU-PHW reactor with confidence. © 1997 Elsevier Science S.A.

1. Introduction

The fast, accurate and efficient generation ofthe thermodynamic properties of heavy water atsaturation, in subcooled liquid and superheatedvapor states is of importance for the design, engi-neering applications and LOCA analysis with thetwo-phase flow models of nuclear reactors thatare cooled and/or moderated by heavy water. Thethermodynamic properties used in these analysesare generally calculated by proper interpolationmethods (linear, hermitian, etc.) applied to thethermodynamic tables which are stored in com-puter memory. These interpolation methods re-

quire a search algorithm to select the properthermodynamic values from the tables in additionto inputting a large amount of data. Hence, theuse of the interpolation algorithms requires con-siderably large computer CPU time. As an exam-ple of using the interpolation polynomials for thispurpose, the thermodynamic properties of heavywater (D2O) were approximated by piecewise Her-mite interpolation polynomials in the CATHENAcode. A more complete description of this code(formerly called ATHENA) can be seen inRichards et al. (1985). Additionally, a descriptionof the piecewise Hermite interpolation polynomi-als used in the code cited above can be found inLiner et al. (1988).

The values of the thermodynamic propertieslisted in the tables for heavy water by Hill et al.

* Fax: +90 212 2853884; e-mail: [email protected]

0029-5493/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved.

PII S0029 -5493 (97 )00235 -5

A. Durmayaz / Nuclear Engineering and Design 178 (1997) 309–329310

(1981) were generated from the Helmholtz freeenergy function which is an analytic equation ofstate. The use of the Helmholtz free energy func-tion for this purpose was also given by Kestin etal. (1984), by Kestin and Sengers (1986), and byHill et al. (1982). Equations for all thermody-namic properties can be generated by using thecombinations of appropriate derivatives of thisfunction. The formulation set given by these refer-ences, was also given provisional acceptance byIAPS. However, it is very important that thethermodynamic properties and their derivativesbased on the Helmholtz free energy function forheavy water in subcooled liquid and superheatedvapor states are given as functions of two inde-pendent variables, namely density and tempera-ture, which are different from the independentvariables used in the thermalhydraulic systemanalysis in two-phase flows. Therefore, such kindof formulation set is inappropriate for the two-phase flow analysis.

An alternative choice is to use approximatefunctions for the fast computation of the thermo-dynamic properties. Such kind of functions devel-oped for light water at saturation were given byGarland and Hoskins (1988) and for light waterin subcooled liquid and superheated vapor stateswere given by Garland and Hand (1989).

Fig. 2. Density and its derivative with respect to saturationpressure for gas phase at saturation.

In this study, a set of approximate functions forthe fast computation of the thermodynamic prop-erties of heavy water at saturation, in subcooledliquid and superheated vapor states is presented.This set of functions was derived by Durmayaz(1995) to use in the LOCA analysis for the nearstagnation flow in the fuel channel of a CANDUreactor following a postulated small break in theinlet feeder. Detailed information is given belowabout the approximation method, which was alsoused by Garland and Hoskins (1988) and by

Fig. 1. Specific volume and its derivative with respect tosaturation pressure for liquid phase at saturation.

Fig. 3. Specific enthalpy and its derivative with respect tosaturation pressure for liquid phase at saturation.

A. Durmayaz / Nuclear Engineering and Design 178 (1997) 309–329 311

Fig. 4. Specific enthalpy and its derivative with respect tosaturation pressure for gas phase at saturation. Fig. 6. Specific entropy and its derivative with respect to

saturation pressure for gas phase at saturation.

Garland and Hand (1989), and about the set offormulation.

2. Approximation method

The data given in the steam tables by Hill et al.(1981) for the thermodynamic properties of heavywater at saturation, in subcooled liquid and su-perheated vapor states were accurately and suc-cessfully fitted with curves by using the

least-squares method in order to obtain someappropriate functions. The approach used in de-veloping these correlations is to minimize the sumof the squares of the deviations from the datagiven in the reference steam tables. During thecurve fitting operations, the pressure or tempera-ture range for each of the properties at saturationand the pressure-temperature ranges for each ofthe properties in single-phase liquid or vapor

Fig. 7. Constant-pressure specific heat and its derivative withrespect to saturation pressure for liquid phase at saturation.

Fig. 5. Specific entropy and its derivative with respect tosaturation pressure for liquid phase at saturation.


Tab

le1

Coe

ffici

ents

and

expo

nent

sus

edin

Eq.

(7)

for6 f

for

diff

eren

tra

nges

ofus

e

bc

a 0e

a 1R

ange

ofus

e(k

Pa)

a 2a 3

a 4a 5

0.0

0.0

0.0

0.66

015

Psa

tB2.

02.

8120

7×10

−6

−3.

2891

4×10

−7

−9.

7302

9×10

−6

1.73

177×

10−

5−

1.53

382×

10−

59.

0947

2×−

4

2.59

83×

10−

10

3.91

703×

10−

12

0.0

0.0

0.0

2.05

Psa

tB10

.08.

0453

5×10

−7

9.02

885×

10−

46.

8869

2×10

−8

−8.

9194

2×10

−9

0.0

0.0

0.0

10.05

Psa

t546

.09.

1823

9×10

−1

41.

2555

6×10

−6

9.02

33×

10−

4−

1.46

553×

10−

11

9.85

586×

10−

10

−3.

8373

2×10

−8

0.0

0.0

0.0

7.28

531×

10−

60.

4086

46.0B

Psa

t545

00.

00.

00.

08.

9331

1×10

−4

0.0

9.10

067×

10−

43.

3377

6×10

−6

0.0

0.50

2345

0BP

satB

1500

0.0

0.0

0.0

0.0

0.0

0.0

0.0

15005

Psa

t545

002.

2574

6×10

−2

39.

5447

1×10

−4

−4.

5584

2×10

−1

93.

8641

4×10

−1

5−

1.79

758×

10−

11

7.77

743×

10−

8

0.0

0.0

0.0

4500B

Psa

t515

000

9.82

43×

10−

44.

6682

7×10

−8

−3.

4232

4×10

−1

23.

2632

4×10

−1

6−

1.52

425×

10−

20

3.49

64×

10−

25

0.0

0.0

0.0

1500

0BP

sat5

1690

00.

0−

1.53

85×

10−

40.

04.

354×

10−

16

−1.

73×

10−

11

2.72

3×10

−7

0.0

−7.

6062

1×10

−6

0.0

0.0

0.0

1690

0BP

satB

1900

06.

6377

×10

−1

0−

2.57

563×

10−

14

3.78

083×

10−

19

0.03

4053

5918

990

0.0

0.0

1900

05P

sat5

2020

02.

6041

7×10

−2

11.

1393

7×10

−1

11.

7653

4×10

−3

9.60

015×

10−

8−

6.04

877×

10−

18

1.07

981×

10−

14

8.46

354×

10−

20

1.63

387×

10−

720

190

0.0

0.0

2020

0BP

sat5

2140

0−

1.69

935×

10−

11

1.80

208×

10−

13

−1.

7314

7×10

−1

61.

9094

195×

10−

3


Tab

le2

Coe

ffici

ents

and

expo

nent

sus

edin

Eq.

(8)

for

rg

for

diff

eren

tra

nges

ofus

e

a 4a 5

bc

eR

ange

ofus

e(k

Pa)

a 2a 0

a 3a 1

0.0

−3.

2994

×10

−5

8.55

78×

10−

30.

00.

9459

0.66

015

Psa

t52.

00.

00.

00.

00.

00.

08.

6254

6×10

−3

0.94

162.

0BP

sat5

7.5

0.0

0.0

−1.

1532

4×10

−4

0.0

0.0

0.0

0.0

0.0

0.0

8.76

874×

10−

30.

9365

7.5B

Psa

t550

.00.

00.

00.

0−

4.85

16×

10−

4

0.0

0.0

0.0

8.67

814×

10−

30.

9382

50.0B

Psa

tB25

08.

73×

10−

40.

00.

00.

00.

08.

0415

4×10

−3

0.94

9825

05P

sat5

500

0.0

0.0

0.0

0.01

9659

80.

00.

00.

00.

00.

06.

9825

2×10

−3

0.96

9150

0BP

sat5

1000

0.0

0.0

0.0

0.08

1991

40.

00.

00.

05.

2152

1×10

−3

1.00

5710

00B

Psa

tB19

500.

00.

00.

00.

2991

30.

03.

3858

1×10

−3

1.05

632

19505

Psa

t530

000.

00.

8026

150.

00.

00.

00.

00.

00.

00.

01.

6995

2×10

−3

1.13

3330

00B

Psa

tB48

001.

9264

20.

00.

00.

00.

06.

256×

10−

41.

2407

48005

Psa

tB65

000.

00.

04.

081

0.0

0.0

0.0

0.0

0.0

0.0

1.84

6×10

−4

1.36

8765

005

Psa

t583

000.

00.

00.

07.

1843

0.0

3.02

275×

10−

51.

5549

8300B

Psa

tB10

400

12.3

5935

0.0

0.0

0.0

0.0

0.0

0.0

2.55

289×

10−

61.

8059

1040

05P

satB

1240

00.

00.

019

.779

160.

00.

00.

00.

00.

00.

07.

4047

4×10

−8

2.16

2312

4005

Psa

tB14

400

0.0

0.0

0.0

30.2

155

0.0

0.0

0.0

5.84

747×

10−

10

2.64

714

4005

Psa

t515

900

43.4

059

0.0

0.0

0.0

0.0

0.0

0.0

1590

0BP

satB

1820

01.

9075

3×10

−1

96.

7764

4×10

−1

13.

9176

1×10

−7

6.22

×10

−3

7.76

793×

10−

9−

6.95

623×

10−

15

0.0

0.01

8561

418

190

0.0

0.0

1820

05P

sat5

2000

02.

0115

2×10

−6

1.03

421×

10−

10

1.95

229×

10−

13

155.

798

−4.

446×

10−

11

1.74

657×

10−

14

1999

00.

00.

020

000B

Psa

t521

200

198.

348

0.03

3670

7−

8.96

932×

10−

64.

4850

1×10

−8


Fig. 8. Constant-pressure specific heat and its derivative withrespect to saturation pressure for gas phase at saturation.

Error=100)Fapp−Fref

Fref

)(%) (1)

while Fapp is the value of any thermodynamicproperty calculated by using the approximationfunctions and Fref is the value of the same prop-erty seen in the reference steam tables.

2.1. Properties at saturation

The properties at saturation can be representedby a number of functions containing only oneindependent variable: pressure or temperature. Inthis study, specific volume (or density), specificenthalpy, specific entropy, constant-pressure spe-cific heat and temperature at saturation were ap-proximated by a number of piecewise continuousfunctions of pressure whereas pressure at satura-tion was approximated by a piecewise continuousfunction of temperature for heavy water.

Some of these properties obtained by piecewisecontinuous functions and their derivatives withrespect to pressure are plotted in Figs. 1–9. It caneasily be seen that each of these functions andtheir derivatives gives a smooth continuous curve.

2.2. Subcooled liquid state

In subcooled liquid state, density, specific en-thalpy, specific entropy and constant-pressure spe-cific heat were approximated by some piecewisecontinuous functions of pressure and temperature.If pressure and specific enthalpy are considered asthe independent variables, first, temperature iscomputed from the specific enthalpy correlation,Eqs. (18) and (18a), in subcooled liquid state byusing root finding algorithms such as NewtonRaphson or Regula Falsi methods. Then, theother properties are computed by using pressureand temperature as the independent variables.The method to derive the approximation func-tions in subcooled liquid state is given in detailbelow.

Consider an arbitrary thermodynamic propertyF=F(P, T) for heavy water in subcooled liquidstate and expand it into a Taylor series. The firsttwo terms of the Taylor expansion of F in pres-sure about the saturation point corresponding tothe temperature T are:

states were divided into several subregions toobtain higher accuracy since it was observed thatany given property could not be fitted accuratelyover the entire range with a single function. Spe-cial care was also taken to ensure that the slopesof the fitted curves are continuous across theboundaries since discontinuities in the slopes cancause undue distortions at the boundaries of thesubregions.

The error for each approximate function is alsodefined as:

Fig. 9. Saturation temperature and its derivative with respectto saturation pressure.


Table 3Coefficients and exponents used in Eq. (9) for hf for different ranges of use

a0 fa1 a2 c e Range of use (kPa)

0.6601BPsatB2.40.00.0−463.1372 0.00.1246487.73680.0 0.0 474.3497−449.6439 0.1277 0.0 2.45Psat56.5

−405.974 0.0 0.0 431.97675 0.1375 0.0 6.5BPsatB24245Psat5600.00.15091377.364−346.8479 0.00.0

0.0 0.0 318.12035 60BPsat51500.167315 0.0−277.930.0 0.0 246.077 150BPsat54000.1914 0.0−184.2785

−74.8603 0.0 0.0 175.094 0.2228 0.0 400BPsat59000.0 0.0 102.40472.2187 0.2717 0.0 900BPsat53000

353.2485 0.0 3000BPsatB85000.00.390827.19280.0721.6373 0.0 85005Psat513 7000.64641.63160.00.0

0.0 1.41484.834×10−4 0.01148.3684 0.0 13 700BPsatB18 6004.465691.492×10−170.0 0.00.0 18 6005Psat520 2001506.2541

−4886805.5 −57.0067 7.01378×10−4 0.0 0.0 580415.0 20 200BPsat521 400

F(P, T)=Ff(T)+(F(P

)T

[P−Psat(T)]. (2)

Since properties at saturation, Ff(T), presentedin Section 3 were approximated as functions ofpressure rather than temperature, the temperatureT in the first term of Eq. (2) will have to bechanged to its corresponding saturation pressure,Psat, by using Eq. (16). Therefore Eq. (2) becomes:

F(P, T)=Ff(Psat(T))+(F(P

)T

[P−Psat(T)]. (3)

In subcooled liquid state, the thermodynamicproperties are strong functions of temperaturewhereas they are weak functions of pressure.Therefore the derivative of the function F withrespect to pressure will be a slowly varying, al-most constant function. Hence, Eq. (3) can berearranged to give:

R(T)=(F(P

)T

=F−Ff

P−Psat

. (4)

For a given temperature, this derivative wascalculated over each of the subdivided ranges andthen averaged. The resulting averages were fittedwith functions of temperature.

2.3. Superheated 6apor state

In superheated vapor state, the thermodynamic

properties may vary strongly with both pressureand temperature. Consider again an arbitrarythermodynamic property F=F(P, T), then ex-pand it into a Taylor series. The first two terms ofthe Taylor expansion of F in temperature aboutthe saturated vapor point corresponding to thepressure P are:

F(P, T)=Fg(P)+(F(T

)P

[T−Tsat(P)]. (5)

Since the derivative is a sensitive function ofboth pressure and temperature, Eq. (5) can berearranged to give:

R(P, T)=(F(T

)P

=F−Fg

T−Tsat

. (6)

This derivative was evaluated and fitted withsome functions of pressure and temperature. It isdifficult to fit this derivative since R(P, T) is afunction of two independent variables.

The reason for using Eqs. (2) and (5) is that thefirst terms in each of these equations are thedominant terms, which were fitted very accuratelyat saturation, and errors in the fitting of R(T) inEq. (4) and R(P, T) in Eq. (6) will not be as largeas they would if these properties were fitted di-rectly, without reference to the value of the satu-ration properties.


Tab

le4

Coe

ffici

ents

and

expo

nent

sus

edin

Eq.

(10)

for

h gfo

rdi

ffer

ent

rang

esof

use

cd

a 0e

a 1R

ange

ofus

e(k

Pa)

a 2a 3

a 4b

168.

6551

0.05

0.13

740.

6601B

Psa

t510

0.0

0.0

0.0

0.0

0.0

2166

.013

1.96

7122

11.1

156

1.0

0.0

0.15

6210B

Psa

tB50

0.0

0.0

0.0

0.0

299.

5196

1.0

0.08

93505

Psa

t516

00.

00.

020

29.3

920.

00.

00.

013

890.

05.

00.

0029

5116

0BP

satB

500

1159

7.97

40.

00.

00.

00.

00.

080

.293

20.

00.

250

05P

satB

1000

0.0

2284

.995

0.0

0.0

0.0

−0.

0293

0.0

0.0

73.2

730.

00.

210

005

Psa

tB30

0023

07.7

9186

50.

0−

0.02

410.

063

.256

60.

00.

2130

005

Psa

tB50

000.

023

29.0

0.0

0.0

0.0

−0.

0233

204

0.0

0.0

0.0

50005

Psa

t510

500

2582

.395

650.

0143

079

−3.

3696

9×10

−6

1.96

082×

10−

10

−5.

5558

4×10

−1

50.

00.

00.

00.

010

500B

Psa

tB16

000

0.0

2689

.69

0.0

−4.

7783

1×10

−1

18.

7408

9×10

−7

−0.

0200

070.

0−

0.24

5479

0.0

0.0

0.0

1600

05P

sat5

1880

01.

4809

7×10

−5

−3.

3553

×10

−1

00.

039

08.3

009

0.0

0.0

0.0

1880

0BP

sat5

2040

00.

01.

4764

×10

−4

1988

1.20

41−

2.76

788

0.0

−2.

6681

2×10

−9

2039

0−

0.07

2497

60.

00.

00.

020

400B

Psa

t521

467

−4.

4964

5×10

−5

4.97

61×

10−

8−

4.38

706×

10−

11

2207

.595


Tab

le5

Coe

ffici

ents

and

expo

nent

sus

edin

Eq.

(11)

for

s ffo

rdi

ffer

ent

rang

esof

use

bc 1

c 2e 1

a 0e 2

a 1R

ange

ofus

e(k

Pa)

a 2a 3

a 4a 5

0.0

0.0

0.0

0.0

0.0

−0.

1229

850.

66015

Psa

tB1.

80.

0159

342

0.39

8931

−0.

7185

470.

8674

39−

0.35

2894

0.0

0.0

0.0

−0.

0198

780.

00.

0198

273

0.0

1.85

Psa

t53.

00.

1837

48−

0.10

5636

0.02

5212

5−

2.26

244×

10−

3

0.0

2.66

210.

00.

0815

0.0

0.0

3.0B

Psa

t515

1.54

8×10

−7

−2.

5720

39−

2.34

7×10

−3

1.61

1×10

−4

−7.

631×

10−

6

0.0

0.48

71−

13.8

068

0.16

66−

0.01

13.4

0375

415B

Psa

t585

0.0

0.0

0.0

0.0

0.0

0.0

0.12

75−

20.2

062

0.26

45−

0.01

0.0

85B

Psa

tB33

020

.133

290.

00.

00.

00.

00.

01.

2869

0.0

0.12

8−

1.04

667

0.0

0.0

3305

Psa

t593

00.

00.

00.

00.

00.

00.

8106

0.0

0.16

150.

00.

093

0BP

sat5

2500

−0.

4043

20.

00.

00.

00.

00.

00.

3763

0.0

0.21

990.

00.

3614

225

00B

Psa

tB57

000.

00.

00.

00.

00.

00.

00.

00.

00.

00.

00.

057

005

Psa

t512

000

2.03

4696

−1.

7958

6×10

−1

78.

6763

7×10

−1

3−

1.71

232×

10−

82.

2136

×10

−4

0.0

0.0

0.0

0.0

0.0

1200

0BP

sat5

1720

02.

4522

548.

1057

2×10

−5

8.85

135×

10−

10

−1.

7870

6×10

−1

35.

2401

1×10

−1

80.

00.

00.

00.

00.

00.

00.

017

200B

Psa

t520

000

2.24

902×

10−

16

24.6

3978

7−

4.88

947×

10−

34.

1885

1×10

−7

−1.

5816

2×10

−1

1

2090

0.0

0.0

0.0

0.0

0.0

2000

0BP

sat5

2100

03.

9334

91.

2444

3×10

−4

5.21

091×

10−

83.

7846

7×10

−1

11.

4322

6×10

−1

40.

020

900.

00.

00.

00.

00.

021

000B

Psa

t521

600

0.0

3.94

149

−1.

4205

4×10

−5

1.25

436×

10−

12

−1.

5632

3×10

−9

7.86

877×

10−

7


Tab

le6

Coe

ffici

ents

and

expo

nent

sus

edin

Eq.

(12)

for

s gfo

rdi

ffer

ent

rang

esof

use

c 1c 2

c 3e 1

e 2a 0

e 3a 1

Ran

geof

use

(kP

a)a 2

a 3a 4

b

10.9

702

−1.

9693

0.0

−0.

048

−0.

10.

00.

00.

00.

66015

Psa

t530

.00.

00.

00.

0−

0.74

8064

0.0

0.0

1403

.587

96.

291

−67

2.09

16−

0.00

46−

0.1

−0.

0130

.0B

Psa

t530

00.

0−

729.

4911

0.0

0.0

7514

8.6

−72

55.5

10.

0−

0.00

01−

0.00

10.

00.

030

0BP

sat5

700

0.0

−67

885.

030.

00.

00.

00.

00.

042

.252

8−

5.24

151

0.0

−0.

015

−0.

10.

070

0BP

satB

1200

0.0

0.0

0.0

−29

.353

619

56.3

724

−9.

1127

90.

0−

0.01

5−

40.1

4351

9−

0.1

0.0

0.0

12005

Psa

t520

000.

00.

00.

00.

00.

00.

00.

00.

00.

00.

00.

020

00B

Psa

tB50

006.

4021

85.

8249

6×10

−16

−1.

0772

7×10

−11

8.17

719×

10−

8−

3.79

694×

10−

4

0.0

0.0

0.0

0.0

0.0

0.0

6.15

393

50005

Psa

tB11

000

−1.

7296

6×10

−4

1.48

637×

10−

8−

8.60

568×

10−

131.

9537

8×10

−17

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

1100

05P

sat5

1740

05.

7353

8−

8.93

649×

10−

183.

5207

4×10

−13

−4.

9467

2×10

−9

−2.

5837

2×10

−5

0.0

6.48

023×

10−

30.

00.

00.

00.

00.

00.

017

400B

Psa

tB20

000

−5.

5629

8×10

−7

2.11

454×

10−

11−

3.03

427×

10−

16−

23.0

8881

0.0

0.0

0.0

0.0

0.0

0.0

0.0

2000

05P

sat5

2100

04.

8644

8×10

−7

66.1

4712

3−

9.42

612×

10−

30.

0−

8.44

907×

10−

12

2055

0.0

−1.

6072

7×10

−4

0.0

0.0

0.0

0.0

0.0

0.0

2100

0BP

sat5

2146

70.

00.

0−

5.84

047×

10−

144.

5496

1


Table 7Coefficients and exponents used in Eq. (13) for cp,f for different ranges of use

a1a0 a2 ea3 a4 a5 c Range of use (kPa)

0.0 10.05Psat530.00.00.0−245.38258 −1.121×10−4249.67040.00.0−1.186×10−3 8.025×10−6 −2.487×10−8 2.997×10−114.2201 0.0 0.0 0.0 30.0BPsatB300

3005Psat510000.00.04.1458 0.0−2.335×10−6 0.0−6.694×10−111.645×10−7

1000BPsat510 0000.02.08×10−124.1019 −8.873×10−9 0.06.218×10−21−1.552×10−161.46×10−4

3. Correlations for the thermodynamic properties ofheavy water at saturation

The correlations given below are the approxi-mations to the heavy water thermodynamic prop-erties for the liquid and gas phases at saturation.

Specific volume of heavy water for the liquidphase at saturation, 6f (m3 kg−1), is

6f= %5

i=0

ai(Psat−b)i+cP esat. (7)

Density of heavy water for the gas phase atsaturation, rg (kg m−3), is

rg= %5

i=0

ai(Psat−b)i+cP esat. (8)

Specific enthalpy of heavy water for the liquidphase at saturation, hf (kJ kg−1), is

hf= %2

i=0

aiPisat+cP e

sat+ f ln(Psat). (9)

Specific enthalpy of heavy water for the gasphase at saturation, hg (kJ kg−1), is

hg= %4

i=0

ai(Psat−b)i+c(Psat−d)e (10)

Specific entropy of heavy water for the liquidphase at saturation, sf (kJ kg−1 K−1), is

sf= %5

i=0

ai(Psat−b)i+ %2

j=1

cjPejsat. (11)

Specific entropy of heavy water for the gasphase at saturation, sg (kJ kg−1 K−1), is

sg= %4

i=0

ai(Psat−b)i+ %3

j=1

cjPejsat. (12)

Specific internal energies for the liquid and gasphases at saturation, uf and ug (kJ kg−1), can be

calculated by using the correlations for specificenthalpy, specific volume and the equation u=h−P6.

Constant-pressure specific heat of heavy waterfor the liquid phase at saturation, cp,f (kJ kg−1

K−1), is

cp,f= %5

i=0

aiPisat+cP e

sat. (13)

Constant-pressure specific heat of heavy waterfor the gas phase at saturation, cp,g (kJ kg−1

K−1), is

cp,g= %5

i=0

aiPisat. (14)

Saturation temperature of heavy water, Tsat

(°C), as a function of saturation pressure is

Tsat=a+cP esat. (15)

Saturation pressure of heavy water, Psat (kPa),as a function of saturation temperature is

Psat=�Tsat−a

cn1/e

. (16)

The coefficients a, b, c, d, f and the exponent eseen in Eqs. (7)–(16) for different thermodynamicproperties at saturation are given in Tables 1–10for different ranges of use. The coefficients a, cand the exponent e seen in Eq. (16) for thesaturation pressure of heavy water are given inTable 10 for different ranges of use. Starred co-efficients or exponent seen in Table 10 gives lesserror in calculation of saturation pressure thanthose of the counterparts seen in Table 9 that areused for calculation of the saturation temperature.

The range of use and the maximum error foreach of the piecewise continuous functions pre-sented above for the thermodynamic properties of


Table 8Coefficients and exponents used in Eq. (14) for cp,g for different ranges of use

a3 Range of use (kPa)a2a1a0 a5a4

0.01.7349 4.288×10−3 −6.287×10−5 10.05Psat560.04.403×10−7 0.01.777275 1.7×10−3 −6.128×10−6 1.832×10−8 −2.821×10−11 1.688×10−14 60.0BPsat5400

2.163×10−181.8534 6.987×10−4 −2.457×10−7 400BPsat510001.101×10−10 −2.477×10−14

1000BPsat510 0001.864×10−205.066×10−4 −4.975×10−161.9209 6.837×10−12−3.985×10−8

heavy water at saturation are given in Table 11. Asummary of the approximation functions for thefast computation of the thermodynamic proper-ties of light water at saturation derived by Gar-land and Hoskins (1988) is also given in Table 11in order to compare the ranges of use and theerror limits.

Comparison of the two correlation sets with theaid of Table 11 shows that the error limits of theapproximate functions presented in this study forthe thermodynamic properties of heavy water atsaturation are much less than those developed forlight water previously. Additionally, the ranges ofuse of the correlations for heavy water mostlybegin at 0.6601 kPa whereas the ranges of use ofthe functions for light water at saturation mostlybegin at 75 kPa.

4. Correlations for the thermodynamic propertiesof heavy water in subcooled liquid state

The correlations given below are the approxi-mations to the thermodynamic properties ofheavy water in subcooled liquid state when usedin conjunction with Eq. (16) for Psat. The coeffi-cients a, b and c seen in Eqs. (17a), (18a), (19a),(20) and (20a) for different thermodynamic prop-erties in subcooled liquid state are given in Tables12–15 for different ranges of use.

4.1. Density, liquid phase

The piecewise continuous function given belowis an approximation to the density, r (kg m−3), ofheavy water in subcooled liquid state in the tem-perature range of 3.8°C to Tsat and the pressure

Table 9Coefficients and exponents used in Eq. (15) for Tsat for differ-ent ranges of use

c Range of use (kPa)a e

122.399 0.1174−112.77175 0.66015PsatB1.10.1204 1.15PsatB2.4−109.6602 119.285

109.249 0.1303−99.56149 2.45Psat56.5−87.6554 97.6839 0.1424 6.5BPsatB23.0

84.9989 0.1567−73.9183 23.05PsatB60.073.64−60.649 0.1708 60.05Psat515061.5568 0.1877 150BPsat5400−45.0052

−30.1035 400BPsat590051.5347 0.2037900BPsatB14000.2114−22.8884 47.1923

40.0043−9.39078 0.2254 14005PsatB30000.2353 30005Psat585000.70657 35.4213

8500BPsat511 1000.2253−12.5021 40.49570.2105 11 100BPsatB13 700−35.812 49.76250.1812 13 7005Psat516 500−95.7685 76.4539

16 500BPsat521 6600.1051−420.0882 276.9554

Table 10Coefficients and exponents used in Eq. (16) for Psat for differ-ent ranges of use

ca Range of use (°C)e

3.85TsatB11.0−112.77175 122.399 0.117411.05TsatB22.890.1226*−107.3874* 117.01548*

0.1303 22.895Tsat539.86−99.5614* 109.24997.6839 0.1424−87.6554 39.86BTsatB65.01

−73.9183 84.9989 0.1567 65.015TsatB87.5487.545Tsat5112.650.170873.64−60.649

61.5568 0.1877−45.0052 112.65BTsat5144.530.2037 144.53BTsat5175.90−30.1035 51.5347

−22.8884 175.90BTsatB195.370.211447.19230.2254 195.375TsatB233.7540.0043−9.390780.2353 233.755Tsat5298.460.70657 35.4213

298.46BTsat5317.730.2253−12.5021 40.495749.7625 0.2105−35.812 317.73BTsatB333.7476.4539 0.1812−95.7685 333.745Tsat5348.46

348.46BTsat5370.74276.9554 0.1051−420.0875*


Table 11Comparison of the approximation functions for the fast computation of the thermodynamic properties of heavy water at saturationpresented by this paper to those of light water at saturation derived by Garland and Hoskins (1988)

Light water at saturation by Garland and HoskinsHeavy water at saturation by DurmayazProperty(1988)

Range of use Psat (kPa), Tsat (°C) Maximum error, Maximum error, (%)Range of use Psat (kPa)(%)

0.66015Psat520 400 0.0086f 75–21 500 0.14020 400BPsat521 400 0.013

rg 0.66015Psat520 000 0.014 85–21 500 0.22020 000BPsat521 200 0.056

75–21 700hf 0.95Psat521 400 0.1000.0300.6601BPsat520 600 0.004hg 75–21 550 0.066

0.01620 600BPsat521 46765–21 2500.066 0.1200.66015PsatB3.0sf

3.05Psat521 600 0.00925–21 500sg 0.66015Psat520 400 0.002 0.100

0.01220 400BPsat521 46775–21 5000.029 0.0801.05Psat521 400uf

0.005 85–21 500 0.110ug 0.66015Psat520 80030–13 300cp,f 10.05Psat510 000 0.009 0.080

13 300–20 300 0.60010.05Psat510 000cp,g 0.018 50–16 000 0.120

16 000–20400 0.6000.048 0.0200.66015PsatB23.0 70–21 850Tsat

23.05Psat521 660 0.006Psat 3.85Tsat5370.74 0.032

The maximum error in the ideal range of use is typed bold face for each property.

range of 0.8–18 000 kPa. The maximum error isnot worse than 0.097% up to 300°C and below0.323% between 300 and 355.21°C.

r(P, T)=rf(Psat(T))+R(T)[P−Psat(T)] (17)

where

R(T)=0.17

375−T− %

7

i=0

aiTi. (17a)

4.2. Specific enthalpy, liquid phase

The piecewise continuous function given belowis an approximation to the specific enthalpy, h (kJkg−1), of heavy water in subcooled liquid state inthe temperature range of 20°C to Tsat and thepressure range of 2–21 000 kPa. The maximumerror is not worse than 0.079% up to 12 000 kPa,below 0.135% between 12 000 and 19 000 kPa andbelow 0.28% between 19 000 and 21 000 kPa.

h(P, T)=hf(Psat(T))+R(T)[P−Psat(T)] (18)

where

R(T)=1.32696 · 10−3−0.1656

%4

i=1

ai(b−T)i

(18a)

In determination of subcooled liquid tempera-ture by using Eq. (18), Eq. (18a) and NewtonRaphson algorithm, the maximum error is below0.29% in the temperature range of 20°C to Tsat

and the pressure range of 2–21 000 kPa.

4.3. Specific entropy, liquid phase

The piecewise continuous function given belowis an approximation to the specific entropy, s (kJkg−1 K−1), of heavy water in subcooled liquidstate. The maximum error is not worse than0.087% for the temperature range of 25°C to Tsat


Tab

le12

Coe

ffici

ents

used

inE

q.(1

7a)

for

r(P

,T)

inliq

uid

phas

eat

diff

eren

tra

nges

ofus

e

a 4a 5

a 6a 7

Ran

geof

use

Ta 2

a 0a 3

a 1(°

C)

[for

0.85

P5

1800

0kP

a]

−5.

293×

10−

16

4.41

3×10

−6

0.0

0.0

3.85

T5

205

−2.

881×

10−

86.

827×

10−

11

1.00

8×10

−1

3−

1.18

×10

−4

3.05

672×

10−

17

0.0

205B

T5

320

−4.

7425

6×10

−1

4−

8.95

929×

10−

34.

2927

1×10

−8

1.00

123×

10−

42.

4431

8×10

−1

1−

4.81

305×

10−

9

−1.

1116

2×10

−1

2−

2.28

991×

10−

16

3.49

11×

10−

15

−3.

6572

3×10

−1

832

0BT5

355.

211.

0852

7×10

−1

41.

4308

9×10

−1

21.

1802

9×10

−1

00.

0


Table 13Coefficients used in Eq. (18a) for h(P, T) in liquid phase for different ranges of use

a4 b Range of use P (kPa) [for 205T5T sat° C]a1 a2 a3

0.815094 −3.61545×10−5 6.20387×10−85.69834×10−3 368.0 2.05P519 0000.0 0.0 369.0 19 000BP521 0001.0 0.0

and the pressure range of 3–19 200 kPa and be-low 0.147% for the temperature range of 20°C toTsat and the pressure range of 2–21 200 kPa.

s(P, T)=sf(Psat(T))+R(T)[P−Psat(T)] (19)

where

R(T)= %5

i= −1

ai(b−T)i. (19a)

4.4. Constant-pressure specific heat, liquid phase

The piecewise continuous function given belowis an approximation to the constant-pressure spe-cific heat, cp (kJ kg−1 K−1), of heavy water insubcooled liquid state in the temperature range of20°C to Tsat and the pressure range of 10–10 000kPa. The maximum error is not worse than 0.24%up to 6000 kPa and below 0.46% between 6000and 10000 kPa.

cp(P, T)=cp,f(Psat(T))+R(T)[P−Psat(T)]+c1

(20)

where

R(T)=c2−489.3951

(378.05−T)3.832 . (20a)

4.5. Specific internal energy, liquid phase

Specific internal energy of heavy water in sub-cooled liquid state, u (kJ kg−1), can be calculatedby using the correlations for specific enthalpy,specific volume (inverse of density) and the equa-tion u=h−P6 with the maximum error being notworse than 0.179% for the temperature range of20°C to Tsat and the pressure range of 2–18 000kPa.

5. Correlations for the thermodynamic propertiesof heavy water in superheated vapor state

The correlations given below are the approxi-mations to the thermodynamic properties ofheavy water in superheated vapor state when usedin conjunction with Eq. (15) for Tsat. The coeffi-cients a, b, c, d, f, g and m and the exponent eseen in Eqs. (21a), (22a), (23a) and (24a) fordifferent thermodynamic properties in super-heated vapor state are given in Tables 16–19 fordifferent ranges of use.

5.1. Specific 6olume, gas phase

The piecewise continuous function given belowis an approximation to the specific volume, 6 (m3

kg−1), of heavy water in superheated vapor statein the temperature range of Tsat to 450°C and thepressure range of 0.8–18 200 kPa. The maximumerror is not worse than 0.28% up to 4000 kPa andbelow 0.99% between 4000 and 18 200 kPa.

6(P, T)=6g(P)+R(P, T)[T−Tsat(P)] (21)

where

Table 15Coefficients used in Eqs. (20) and (20a) for cp(P, T) in liquid phase fordifferent ranges of use

Range of use P (kPa), T (°C)c1 c2

105P510 000 and 205T530−0.0189 −2.8514×10−6

−4.59×10−60.0 105P51000 and 30BT5Tsat

1000BP54000 and 30BT5Tsat−2.8514×10−60.04000BP56000 and 30BT5Tsat−3.385×10−60.0

0.0 6000BP58000 and 30BT5Tsat−4.028×10−6

8000BP510 000 and 30BT5Tsat−1.51×10−6−0.02244


Tab

le14

Coe

ffici

ents

used

inE

q.(1

9a)

for

s(P

,T)

inliq

uid

phas

efo

rdi

ffer

ent

rang

esof

use

a 3a 4

a 5b

Ran

geof

use

T(°

C)

a 1a −

1a 2

a 0[f

or0.

66015

P5

2120

0kP

a]

0.0

1.64

5×10

−17

1.70

2×10

−7

0.0

3.85

T5

150

1.77

3×10

−8

2.00

8×10

−1

01.

759×

10−

12

8.23

9×10

−1

5

0.0

369.

7515

0BTB

300

0.0

−2.

967×

10−

40.

03.

454×

10−

70.

00.

00.

05.

179×

10−

70.

037

0.08

630

05T5

368

0.0

0.0

0.0

−3.

113×

10−

4


Table 16Coefficients used in Eq. (21a) for 6(P, T) in gas phase for different ranges of use

a b c d Range of use P (kPa)f g[for Tsat5T5450°C]

0.416955 0.0311582 0.85P51204.00512×10−5 1.0 0.0 0.803876120BP57500.419864 0.01165 7.93992×10−6 1.836131.0 0.0

1.47732×10−5 0.0 1.32549 750BP540000.425415 0.0138294 1.00.408275 −64.1458 4000BP518 200−0.532726 0.0205 1.0 0.0

R(P, T)=aP

+

� bT+100

−c�

P0.1[f+ (1− f )[d(T+8)4−P2]0.5]+g

[d(T+8)4−P2]0.5 . (21a)

5.2. Specific enthalpy, gas phase

The piecewise continuous function given belowis an approximation to the specific enthalpy, h (kJkg−1), of heavy water in superheated vapor state

with the maximum error being not worse than0.095% in the temperature range of Tsat to 450°Cand the pressure range of 0.8–19 600 kPa.

h(P, T)=hg(P)+R(P, T)[T−Tsat(P)] (22)

where

R(P, T)=a0+a1

T+a2 exp[a3(T−a4)]+

bP� %3

i=0

ciPi+d1P2T+d2PT+d3PT2+ %3

j=1

fjT jn0.5. (22a)

In determination of superheated vapor temper-ature by using Eq. (22), Eq. (22a) and RegulaFalsi algorithm, the maximum error is below0.31% in the temperature range of Tsat to 450°Cand the pressure range of 1.6–19 600 kPa.

5.3. Specific entropy, gas phase

The piecewise continuous function given belowis an approximation to the specific entropy, s (kJkg−1 K−1), of heavy water in superheated vaporstate with the maximum error being not worsethan 0.078% in the temperature range of Tsat to450°C and the pressure range of 0.8–19 600 kPa.

s(P, T)=sg(P)+R(P, T)[T−Tsat(P)] (23)

where

5.4. Constant-pressure specific heat, gas phase

The piecewise continuous function given belowis an approximation to the constant-pressure spe-cific heat, cp (kJ kg−1 K−1), of heavy water insuperheated vapor state in the temperature rangeof Tsat to 440°C and the pressure range of 10–10 000 kPa. The maximum error is not worse than1.384% up to 4000 kPa and below 2.751% be-tween 4000 and 10 000 kPa.

cp(P, T)=cp,g(P)+R(P, T)[T−Tsat(P)] (24)

where

R(P, T)= %3

i=0

aiPi+b1P

2T+b2PT+b3PT2+ %3

j=1

cjTj+dP−0.5

+fP1.2� %

3

k=0

gkPk+h1P2T+h2PT+h3PT2+ %3

l=1

mlTl+m4(35.85+T)5n0.5. (23a)


Tab

le17

Coe

ffici

ents

used

inE

q.(2

2a)

for

h(P

,T)

inga

sph

ase

for

diff

eren

tra

nges

ofus

e

Ran

geof

use

P(k

Pa)

[for

Tsa

t5T5

450°

C]

300B

P5

550

550B

P5

1000

1000B

P5

2800

2800B

P5

4000

4000B

P5

6000

6000B

P5

1000

010

000B

P5

1500

015

000B

P5

1960

016B

P5

650.

85P5

1665B

P5

300

8.77

936

8.77

936

8.77

936

2.99

797

2.99

797

2.99

797

12.0

484

12.0

484

9.78

095

a 010

.067

610

.347

3−

14.5

654

−66

8.02

5−

668.

025

−66

8.02

5−

6388

.74

−14

.565

4−

6388

.74

a 1−

14.5

654

4.20

364

0.46

1977

0.59

222

−6.

9074

5a 2

4.06

839

−8.

0289

94.

0683

94.

0683

936

.695

836

.695

8−

8.30

33−

8.57

778

−6.

9074

5−

6.90

745

−0.

0000

5−

0.00

8−

0.00

8−

0.00

8−

0.00

8−

0.00

005

−0.

008

−0.

0000

5−

0.00

005

a 3−

0.00

005

−0.

0000

516

0.0

160.

016

0.0

162.

016

2.0

162.

016

2.0

162.

016

0.0

160.

016

0.0

a 41.

00.

0852

264

1.0

1.0

1.0

1.0

1.0

1.0

0.65

3461

0.62

1974

1.0

b4.

8694

9×10

646

01.6

323

25.5

210

33.2

1−

1.83

077×

108

2.02

606×

106

9.25

914×

107

c 0−

5.48

124×

106

0.0

0.0

0.0

4894

9.9

5455

.09

−15

3.16

6−

3876

6.8

−21

539.

8−

1060

3.3

−26

599.

8−

3619

5.5

c 10.

00.

00.

00.

7219

992.

7233

41.

5760

50.

7188

50.

5169

17−

4.58

711.

1931

3−

1.0

c 2−

95.7

573

−1.

0−

1.0

0.0

0.0

−2.

7124

3×10

−4

0.0

0.0

0.0

2.33

741×

10−

6−

1.36

951×

10−

60.

00.

00.

0c 3

0.01

5299

90.

01.

4385

1×10

−3

−7.

3022

1×−

3−

4.42

643×

10−

3−

2.04

974×

10−

3−

1.65

955×

10−

3−

2.81

054×

10−

30.

00.

00.

3955

42d 1

−14

.420

810

0.62

542

.748

111

.895

689

.434

6−

2.29

869

76.7

847

d 236

.865

40.

00.

00.

0d 3

0.03

6002

10.

0−

1.67

785×

10−

30.

0353

695

0.03

6947

4−

0.04

7782

90.

0180

701

0.0

0.0

−0.

9551

19−

0.04

7090

3−

5011

1.9

6387

97.0

3883

76.0

2218

22.0

2.35

87×

106

−48

355.

577

2825

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4236

0.6

f 110

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00.

023

6.83

7−

2329

.48

−12

51.2

8−

590.

03−

7500

.61

−34

14.0

2f 2

0.0

10.0

30.0

157.

418

213.

866

0.04

1193

82.

2880

41.

1584

80.

6003

497.

0823

92.

8733

60.

1674

9f 3

0.0

0.0

0.83

4372

0.0


Table 18Coefficients used in Eq. (23a) for s(P, T) in gas phase for different ranges of use

Range of use P (kPa) [for Tsat5T5450°C]

0.85PB22.0 225P5140 140BP51000 1000BPB11 000 11 0005P518 000 18 000BP519 600

6.16662×10−3 5.9217×10−3 6.03405×10−3 4.81826×10−3 4.81826×10−3 4.81826×10−3a0

a1 −5.03219×10−5 −1.48048×10−6 1.74264×10−6 0.0 0.0 0.0a2 0.00.00.06.89893×10−103.19711×10−83.05565×10−6

0.00.0−2.99361×10−13 0.0−9.57186×10−11−6.23545×10−8a3

−7.3344×10−14−1.13805×10−12−9.36868×10−10b1 0.00.00.00.0b2 1.38783×10−8 −1.31701×10−8 −9.67509×10−9 0.0 0.0

0.00.0b3 2.37152×10−11 0.01.82997×10−11 1.12257×10−11

−2.87624×10−6 −2.87624×10−6 −2.87624×10−6c1 −9.55276×10−6 −9.85418×10−6 −1.28919×10−5

1.79269×10−8 2.80426×10−8 0.0 0.0 0.0c2 1.58514×10−8

0.00.00.0−2.48958×10−11−1.47079×10−11−1.23685×10−11c3

0.0 0.0 0.0d 1.00104×10−6 1.00104×10−4 1.00104×10−4

8.52194×10−78.52194×10−78.52194×10−70.0f 0.00.01.0 1.0 1.0g0 0.0 −8.34545×10−7 −4.28135×10−6

g1 0.0 0.0 0.0 0.0 −3.60396×10−3 −0.02073510.0 0.0 −1.0×10−6 −1.07037×10−6−2.0114×10−6g2 0.0

g3 1.63299×10−101.95565×10−100.00.00.00.01.70265×10−80.00.0 −1.67564×10−80.00.0h1

0.0 0.0 0.0 3.03573×10−42.44047×10−4h2 0.0−6.49484×10−83.82669×10−8h3 0.00.0 0.00.0

m1 0.0 0.0 0.0 −1.34×10−4 −5.89338×10−40.00.00.0m2 −9.88442×10−3−9.71333×10−30.00.0

2.51487×10−50.00.00.0 2.59554×10−50.0m3

m4 0.00.00.0 0.00.03.0×10−11

R(P, T)=a+bp(c−T)2.1+%4

i=0

dip i+ f1p3T+ f2p2T2+ f3pT3+ f4p2T+ f5pT+ f6pT2+ %4

j=1

gjT j

[h(T+8)2−p ]1.5 (24a)

where p=P/1000.

5.5. Specific internal energy, gas phase

Specific internal energy of heavy water in super-heated vapor state, u (kJ kg−1), can be calcu-lated by using the correlations for specificenthalpy and specific volume and the equationu=h−Pv with the maximum error being notworse than 0.096% for the temperature range ofTsat to 450°C and the pressure range of 0.8–18 200 kPa.

6. Conclusions

A set of correlations has been presented in thispaper in order to make possible the fast computa-tion of the thermodynamic properties of heavywater at saturation, in subcooled liquid and su-perheated vapor states. These correlations are ac-curate enough and can be used with confidence inthe two-phase thermalhydraulic system analysis ofCANDU-PHW reactor.

A computer program, called D2O, was alsocoded by using these correlations in FORTRAN


Table 19Coefficients used in Eq. (24a) for cp(P, T) in gas phase fordifferent ranges of use

Range of use P (kPa) [for Tsat5T5440°C]

10.05P56000 6000BP510 000

a 4.72439×10−36.44886×10−4

b 2.52944×10−7−4.49835×10−9

550.0650.0cd0 −1.82782−1.50168×10−8

4.39832d1 −4.05154×10−6

−1.92734×10−6 −0.0115614d2

6.20089×10−5 0.0134129d3

d4 5.89434×10−4 0.0−4.21103×10−5−1.98528×10−5f1

2.84327×10−7f2 2.63761×10−8

f3 −3.7646×10−9 −2.32163×10−9

−3.77898×10−5f4 3.04821×10−4

−4.22794×10−4f5 −0.0298064f6 1.27748×10−6 3.81522×10−5

g1 2.15835×10−6 1.62507×10−3

−1.55305×10−7g2 1.71995×10−6

g3 2.34051×10−11 1.17518×10−8

g4 7.10664×10−13 0.01.4441×10−4h 1.474×10−4

u specific internal energyspecific volume6

Greek lettersr density

Subscriptsf liquid phase at saturationg gas phase at saturation

saturationsat

AcronymsAtomic Energy of CanadaAECLLimited

ATHENA Algorithm for THErmalhy-draulic Network Analysis

CANDU- CANada Deuterium Uranium-PHW Pressurized Heavy Water

CATHENA Canadian Algorithm for THEr-malhydraulic Network Analysis

CPU Central Processing UnitLOCA Loss-of-Coolant AccidentIAPS International Association for

the Properties of Steam

References

Durmayaz, A., 1995. A Loss-of-Coolant Accident Analysis forNear Stagnation Flow in the Horizontal Fuel Channel ofCANDU Reactor. Ph.D. Thesis, ITU Institute for NuclearEnergy, (May 4, 1995) Istanbul.

Garland, W.M.J., Hoskins, J.D., 1988. Approximate functionsfor the fast calculation of light-water properties at satura-tion. Int. J. of Multiphase Flow 14, 333–348.

Garland, W.M.J., Hand, B.J., 1989. Simple functions for thefast approximation of light water thermodynamic proper-ties. Nucl. Eng. Des. 113, 21–34.

Hill, P.G., MacMillan, R.D., Victor, L., 1981. Tables ofthermodynamic properties of heavy water in S.I. units,(December 1981) AECL 7531.

Hill, P.G., MacMillan, R.D. Chris, Victor, Lee, 1982. Afundamental equation of state for heavy water. J. Phys.Chem. Ref. Data 11, 1–15.

Kestin, J., Sengers, J.V., Kamgar-Parsi, B., Levelt Sengers,J.M.H., 1984. Thermophysical properties of fluid D2O. J.Phys. Chem. Ref. Data 13, 601–609.

Kestin, J., Sengers, J.V., 1986. New international formulationsfor the thermophysical properties of light and heavy water.J. Phys. Chem. Ref. Data 15, 305–320.

77 on a 486DX2-66 personal computer. D2O canbe obtained freely by corresponding with Dr Ah-met Durmayaz.

Appendix A. Nomenclature

a, b, c, d, f, g, coefficientsh, m

specific heat at constant pres-cp

suree exponentFapp value of any thermodynamic

property calculated by usingthe approximation functionsvalue of any thermodynamicFref

property seen in the referencesteam tables

h specific enthalpyP pressures specific entropyT temperature


Liner, Y., Hanna, B.N., Richards, D.J., 1988. Piecewise Her-mite polynomial approximation of liquid–vapour thermo-dynamic properties. Am. Soc. Mech. Engineers, FluidsEng. Div. FED 72 99–102.

Richards, D.J., Hanna, B.N., Hobson, N., Ardron, K.H.,1985. ATHENA: A two-fluid code for CANDU LOCAanalysis. Presented at the 3rd Int. Conf. Reactor Thermal-hydraulics, October 15–18, Newport, Rhode Island.

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Approximate Function for Heavy Water Properties Computation-Durmayaz